Theory Review for Cylindrical Shells
and
Parametric Study of Chimneys and Tanks
Theory Review for Cylindrical Shells
and
Parametric Study of Chimneys and Tanks
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties
in het openbaar te verdedigen
op maandag 22 maart 2010 om 15.00 uur
door
Jeroen Hendrik HOEFAKKER
civiel ingenieur
geboren te Amersfoort
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. J. Blaauwendraad
Samenstelling promotiecommissie:
Rector Magnificus,
voorzitter
Prof.dr.ir. J. Blaauwendraad,
Technische Universiteit Delft, promotor
Prof.dr.ir. L.J. Ernst
Technische Universiteit Delft
Prof.dr. A. Metrikine
Technische Universiteit Delft
Prof.dr.ir. L.J. Sluys
Technische Universiteit Delft
Dr.ir. W. van Horssen
Technische Universiteit Delft
Dr.ir. P. Liu
INTECSEA
Ing. H. van Koten
Gepensioneerd, eerder TNO Bouw
ISBN 978-90-5972-363-4
Eburon Academic Publishers
P.O. Box 2867
2601 CW Delft
The Netherlands
tel.: +31 (0) 15 - 2131484 / fax: +31 (0) 15 - 2146888
[email protected] / www.eburon.nl
Cover design: J.H. Hoefakker
© 2010 J.H. HOEFAKKER. All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording, or otherwise, without the prior permission in writing from the
proprietor.
Acknowledgement
The majority of the research reported in this thesis was performed at Delft University
of Technology, Faculty of Civil Engineering and Geosciences under the supervision of
my promotor Prof. Johan Blaauwendraad in the Section of Structural Mechanics.
I am deeply indebted to Prof. Blaauwendraad for the journey we have travelled so far
together. I am really proud that I have been able to work with such an excellent mentor,
who in turn has been a challenging sparring partner and the source of much valuable
inspiration over these last few years. I am especially thankful for the chance to teach
students together with him on the application of shell theory, which has been of crucial
importance in my understanding of shell behaviour and in the focus of my research.
I am very grateful to Carine van Bentum for her valuable contribution to the
development of the computer program as part of her graduation project.
I would also like to thank my family, friends and colleagues at INTECSEA and the
Delft University of Technology for their interest, encouragement and support. Special
thanks go out to my colleague Pedro Ramos for the numerical simulations to validate
the computer program and to Frank van Kuijk for his help during the creation of the
cover design.
I am sincerely grateful for the sacrifices my parents have made and the possibilities
they have offered me. Dear Mother, I am sure that Dad would be as proud of this result
as you are!
Mirjam, my gratitude to you is beyond words. Your continual sacrifice, endurance and
cardinal support throughout these years have been truly admirable. At last I hope to
devote more time to you and our wonderful daughters, whom I daily thank for
enriching my world.
Utrecht, February 2010
v
vi
Table of Contents
Acknowledgement
Summary
Samenvatting
List of symbols
1
Introduction
1.1
Motive and scope of the research
1.2
Research objective and strategy
1.3
Outline of the thesis
1.4
Short review of the existing work within the scope
2
General part on shell theory
2.1
Introduction to the structural analysis of a solid shell
2.2
Fundamental theory of thin elastic shells
2.3
Principle of virtual work
2.4
Boundary conditions
2.5
Synthesis
2.6
Analysis by former authors
2.7
Proposed theory
3
Computational method and analysis method
3.1
Introduction to the numerical techniques for a solid shell
3.2
The super element approach
3.3
Calculation scheme
3.4
Introduction to the program CShell
3.5
Overview of the analysed structures
4
Circular cylindrical shells
4.1
Introduction
4.2
Sets of equations
4.3
The resulting differential equations
4.4
Full circular cylindrical shell with curved boundaries
4.5
Approximation of the homogeneous solution
4.6
Characteristic and influence length
4.7
Concluding remarks
5
Chimney – Numerical results and parametric study
5.1
Wind load
5.2
Behaviour for a fixed base and free end
5.3
Influence of stiffening rings
5.4
Influence of elastic supports
6
Tank – Numerical study
6.1
Introduction
6.2
General description of large liquid storage tanks
6.3
Load-deformation conditions and analysed cases
6.4
Content load cases
6.5
Wind load cases
6.6
Settlement induced load and/or deformation cases
v
ix
xiii
xix
1
1
2
3
4
7
7
10
21
26
28
32
42
51
51
53
60
60
64
65
65
66
68
71
84
89
92
93
93
94
112
137
149
149
150
151
155
159
166
vii
7
Conclusions
Appendices
Literature
Curriculum Vitae
169
175
245
250
List of Appendices
Appendix A
Results from differential geometry of a surface
177
Appendix B
Kinematical relation in orthogonal curvilinear coordinates
183
Appendix C
Equilibrium equations in curvilinear coordinates
185
Appendix D
Strain energy and Laplace-Beltrami operator
187
Appendix E
Expressions and derivation of the stiffness matrix for the elastostatic
behaviour of a circular ring
191
Appendix F
Ring equations comparison
199
Appendix G
Semi-membrane concept
203
Appendix H
Solution to MK and SMC equations
215
Appendix I
Back substitution for MK and SMC solutions
223
Appendix J
Program solution for influence of stiffening rings
233
viii
Summary
Since the considerable effort in the development of rigorous shell theories – dating
back to the early twentieth century – many approximate shell theories have been
developed, mainly on the assumption that the shell is thin. With the development of the
numerical formulations and the continuously increasing computing power, a gradual
cessation of attempts to find closed-form solutions to rigorous formulations has taken
place. This has led to an increasing lack of understanding of the basic and generic
knowledge of the shell behaviour, the prevailing parameters and the underlying
theories, which is obviously required for the use of numerical programs and to
understand and validate the results.
Objective and scope of the research
This research project intended to combine the classic shell theories with the
contemporary numerical approach. The goal was to derive and employ a consistent and
reliable theory of shells of revolution and to present that theory in the context of
modern computational mechanics. The aim of the project was to derive an expeditious
PC-oriented computer program for that by reshaping the closed-form solutions to the
rigorous shell formulations into the well-known direct stiffness approach of the
displacement method. The objective was to conduct a generic study of the physically
and geometrically linear behaviour of the typical thin shells of revolution, i.e. circular
cylindrical, conical and spherical shells, under static loading by evaluating both the
closed-form solution to the thin shell equations and the output of the computer
program.
This research concentrated on the behaviour of circular cylindrical shells under
static loading while accounting for the axisymmetric, beam-type and non-axisymmetric
load-deformation conditions. Due to required effort identified during the development
of such a program for circular cylinders and upon inspection of the sets of equations for
conical and spherical shells, it has been decided to fully focus on circular cylindrical
shells as a first, but complete and successful step towards more applications.
Review of the first-order approximation theory for thin shells
Based on previous work, it was envisaged to employ the so-called Morley-Koiter
equation for thin circular cylindrical shells. The Morley-Koiter equation fits in the
category of the first-order approximation theory, viz. only first-order terms with respect
to the thinness of the shell are retained, resulting in an eighth order partial differential
equation. To understand the assumptions and simplifications, which are introduced to
obtain such a thin shell equation, the set of equations resulting from a fundamental
derivation for thin elastic shells is reproduced. The formulations for thin, shallow, nonlinear and cylindrical shells by some former authors are discussed and, as a result of
the comparison, a set of equations for thin elastic shells within the first-order
approximation theory is proposed. This set comprises kinematical and constitutive
relations that are complemented by the equilibrium relation and boundary conditions,
which are derived by making use of the principle of virtual work. To arrive at a
consistent and reliable theory of shells of revolution, the expansion of the strain
ix
description that adopts the changes of curvature has been considered and, while
simultaneously approximating the constitutive relation, the combined internal stress
resultants of the boundary conditions are congruently approximated.
Computational method and expeditious PC-oriented computer program
The concept of generating the stiffness matrix of shell elements on basis of closed-form
solutions was already proposed as early as 1964 by Loof. Since then little effort with a
similar approach has been reported and to date the method has been employed only to
study axisymmetric structures subject to loads that are also axisymmetric with respect
to the axis of symmetry of the structure.
For shells of revolution with circular boundaries under general loading, the
numerical procedure to be performed by a digital computer is described. This approach
avoids the shortcomings of most existing element stiffness matrices and attempts to
minimise the number of elements needed to model a given problem domain. Similar to
the conventional method, the first and crucial step is to compute the element stiffness
matrix but for the super element, this is synthesized on the basis of an analytical
solution to the governing equation. The precise formulation of the classic approach is
reshaped into the well-known direct stiffness approach of the displacement method
enabling the calculation of combinations of elements and type of elements while the
valuable knowledge of the classic approach is preserved. In addition to the
conventional transition and end conditions, the method enables implementation of
stiffening rings, elastic support, prescribed displacement and various load types. Based
on the proposed solution procedure and with the mentioned functionalities, an
expeditious PC-oriented computer program has been developed using the Fortranpackage in combination with graphical software. The formulations that are
implemented in this program are based on the approximated solution to the MorleyKoiter equation for circular cylindrical shells.
General solutions to the circular cylindrical shell equations
The proposed set of equations is formulated for circular cylindrical shells with circular
boundaries and the resulting single differential equation has been derived. An
approximation of this exact equation is introduced to arrive at mathematically the most
suitable equation for substitution with the same accuracy, i.e. the Morley-Koiter
equation.
The exact roots to the Morley-Koiter equation have been obtained and, albeit being
surplus to requirements, the presented solution is a unification of former results by
other authors. To progress towards generic knowledge of the shell behaviour based on
closed-form solutions, approximate roots have been derived for the axisymmetric,
beam-type, and non-axisymmetric load-deformation conditions. The associated
characteristic and influence lengths have been derived and discussed to facilitate
insight in the prevailing parameters of the shell response to the respective loaddeformation conditions.
x
Parametric study of long circular cylindrical shells (chimneys)
Design formulas, based on closed-form solutions to the Morley-Koiter equation and an
equation derived by the semi-membrane concept, and numerical solutions obtained by
the developed program are provided for long circular cylindrical shell structures, i.e.
long in comparison with their radius (for example industrial, steel chimneys).
The design formula that describes the stress distribution at the fixed base of long
circular cylindrical shells without stiffening rings subject to wind load has been
derived, which is a marked improvement of the existing formula that is based on the
Donnell equation. This formula relates total membrane stress σ xx ,total to the “beam”
stress σ xx ,beam .
For the specified wind pressure distribution around the cylinder, this formula reads
2

 a  a
σ xx ,total = σ xx ,beam 1 + 6.39 1 − υ2   
 l  t 

in which the radius, length and thickness of the shell are represented by a , l and t ,
respectively, and υ denotes Poisson’s ratio of the shell material. Alternatively, this
equation can be written as
σ xx ,total
4


2 a 
= σ xx ,beam 1 + 6.39 1 − υ   

 l2  
in which the characteristic length l2 is defined by l2 = 4 atl 2 .
New design formulas, which describe the effect of (centric and eccentric)
stiffening rings and elastic supports (in the axial and planar directions), are presented
such that the respective influence is represented by inclusion of an additional factor in
the formula for the fixed base case without stiffening rings.
The formula for the case with stiffening rings reads
4

a 
2
σ xx ,total = σ xx ,beam 1 + 6.39 1 − υ λ r   

 l2  
in which the stiffness ratio λ r represents the ratio of the bending stiffness of the
circular cylindrical shell only to the modified bending stiffness of the shell (with the
contribution of the ring stiffness per spacing).
It has been concluded that, in case of an elastic support to a long circular cylinder,
only the axial spring stiffness has to be taken into account. The formula for the case
with axial elastic supports reads
4

a 
2
σ xx ,total = σ xx ,beam 1 + 6.39 1 − υ λ xn   

 l2  
in which the normalised stress ratio λ xn is introduced, which depends on the respective
factors and mode numbers of the load and the parameter ηx , which in turn is mainly
described by the geometrical properties of the cylinder and the ratio of the axial elastic
support to the modulus of elasticity.
xi
From the comparison with the numerical results, the range of application of the
improved and new design formulas has been obtained within which a close agreement
is observed. These formulas have been shown to be applicable to cylinders for which
the characteristic length l2 is larger or equal to its radius. For ring-stiffened cylinders,
the formula has further been shown to be applicable to cylinders with ring spacing
shorter than half of the influence length of the long-wave solution for circumferential
mode number n = 2 .
Numerical study of short circular cylindrical shells (tanks)
For short circular cylindrical shells (lengths in the range of 0.5 to 3 times the radius),
numerical solutions have been presented with the intention to demonstrate the
capability of the developed program to model the shell of large vertical liquid storage
tanks. Additionally, tentative insight into the response of such tank shells to the
relevant load and/or deformation conditions is provided, which is obtained by several
calculations (for the response to content or wind load or due to full circumferential
settlement) and by comparison with the insight as obtained for the behaviour of the
long cylinder.
Conclusions
This study has focused on a thorough analysis of the behaviour of circular cylindrical
shells with the following main results:
o The first-order approximation theory for thin shells and the various approaches
discussed in the literature have been reviewed and a consistent set of thin shell
equations has been proposed. On basis of the proposed set, the Morley-Koiter
equation has been identified as being the most suitable single differential
equation for deriving closed-form solutions.
o On basis of these closed-form solutions, an expeditious PC-oriented computer
program has been developed for first-estimate design of long and short circular
cylindrical shells, e.g. chimneys and tanks.
o In the literature, a design formula for the stress at the base of a chimney
subject to wind load has been developed by combining a solution obtained on
basis of the Donnell equation with finite element analysis. On basis of the
closed-form solutions to the Morley-Koiter equation, this formula has been
confirmed. As an advantage of the new solution, the design formula is
generalized with respect to the wind pressure distribution around the chimney.
o The above mentioned design formula has been extended for the influence of
elastic supports at the base of the chimney.
o The above mentioned design formula has been extended for the influence of
stiffening ring properties and spacing along the chimney.
o The range of application of these formulas has been conclusively and
conveniently obtained by comparison with results obtained with the developed
computer program.
xii
Samenvatting
Sinds de aanzienlijke inspanningen in de ontwikkeling van strenge schaaltheorieën –
die teruggaan tot het begin van de twintigste eeuw – zijn er veel benaderende
schaaltheorieën ontwikkeld, voornamelijk op basis van de veronderstelling dat de
schaal dun is. Door de ontwikkeling van de numerieke formuleringen en de continu
toenemende rekenkracht is er geleidelijk mee gestopt om voor strenge formuleringen
oplossingen in gesloten vorm te vinden. Dit heeft geleid tot een toenemend gebrek aan
begrip van de fundamentele en algemene kennis van het schaalgedrag, de dominante
parameters en de onderliggende theorieën. Dat is een spijtige ontwikkeling omdat juist
dat inzicht vereist is voor het gebruik van numerieke programma’s en om de resultaten
te begrijpen en te valideren.
Doel en reikwijdte van het onderzoek
Dit onderzoeksproject beoogde om de klassieke schaaltheorieën te combineren met de
hedendaagse numerieke benadering. Het aanvankelijke doel was het afleiden van een
consistente en betrouwbare theorie van omwentelingsschalen en deze theorie te
presenteren in de context van de moderne numerieke mechanica. Het project beoogde
de ontwikkeling van een snel PC-georiënteerd computerprogramma door de
oplossingen in gesloten vorm voor de strenge schaalformuleringen te herstructureren
en onder te brengen in de bekende aanpak van de verplaatsingsmethode. De
doelstelling was de uitvoering van een generieke studie van het fysisch en geometrisch
lineaire gedrag van de meest voorkomende dunne omwentelingsschalen – dat wil
zeggen de cirkelcilindrische, conische en bolvormige schalen – onder statische
belasting door de beoordeling van zowel de oplossing in gesloten vorm van de dunne
schaalvergelijkingen en de uitvoer van het computerprogramma.
Het hier gerapporteerde onderzoek is afgebakend tot het gedrag van
cirkelcilindrische schalen onder statische belasting, waarbij drie specifieke
belastingstoestanden zijn betrokken: axiaalsymmetrie, liggerwerking en asymmetrie.
Gezien de inspanning die tijdens de ontwikkeling van een dergelijk
computerprogramma voor cirkelcilinders vereist bleek te zijn, en na beoordeling van de
sets van vergelijkingen voor de conische en bolvormige schalen is het besluit genomen
het onderzoek volledig te richten op cirkelcilindrische schalen als een eerste, maar
volledige en succesvolle stap naar andere toepassingen in de toekomst.
Terugblik op de eerste-orde benaderingstheorie voor dunne schalen
Op basis van eerder werk was de aanwending van de zogenaamde Morley-Koiter
vergelijking voor dunne cirkelcilindrische schalen beoogd. De Morley-Koiter
vergelijking past in de categorie van de eerste-orde benaderingstheorie waarin alleen
eerste-orde termen met betrekking tot de dunheid van de schaal worden meegenomen,
hetgeen resulteert in een achtste-orde partiële differentiaalvergelijking. Om de
aannames en vereenvoudigingen, die tijdens de afleiding van een dergelijke dunne
schaalvergelijking ingevoerd zijn, te kunnen begrijpen is de set van vergelijkingen
gereproduceerd die uit een fundamentele afleiding voor dunne elastische schalen volgt.
De formuleringen van enkele eerdere auteurs voor dunne, licht gekromde, niet-lineaire
xiii
en cilindrische schalen worden besproken en, als gevolg van de vergelijking, een set
van vergelijkingen binnen de eerste-orde benaderingstheorie voor dunne elastische
schalen is voorgesteld. Deze set bestaat uit kinematische en constitutieve betrekkingen
die gecomplementeerd worden door de evenwichtsrelatie en randvoorwaarden, welke
door gebruik te maken van het principe van virtuele arbeid zijn afgeleid. Om tot een
consistente en betrouwbare theorie van omwentelingsschalen te komen is de
reeksontwikkeling van de rekbeschrijving op basis van de krommingveranderingen
beschouwd en, onder gelijktijdige benadering van de constitutieve relatie, zijn de
gecombineerde interne spanningsresultanten van de randvoorwaarden overeenkomstig
benaderd.
Numerieke methode en snel PC-georiënteerd computerprogramma
Het genereren van de stijfheidsmatrix van schaalelementen op basis van oplossingen in
gesloten vorm werd in 1964 reeds voorgesteld door Loof. Sindsdien is er weinig
inspanning met betrekking tot een soortgelijke aanpak gemeld en tot op heden is de
methode slechts toegepast om axiaalsymmetrische structuren te bestuderen onder
belastingen die ook axiaalsymmetrisch zijn met betrekking tot de symmetrieas van de
structuur.
Voor omwentelingsschalen met cirkelvormige randen onder algemene belasting is
de numerieke procedure beschreven die door een digitale computer uitgevoerd moet
worden. Deze aanpak vermijdt de tekortkomingen van de meeste stijfheidmatrices van
bestaande elementen en beoogt om het aantal elementen dat nodig is om een bepaald
probleemdomein te modelleren tot het minimum te beperken. We noemen zulke
elementen super elementen. Net als in de standaard eindige-elementenmethode (EEM)
is de eerste en cruciale stap het berekenen van de stijfheidsmatrix per element, maar
voor het super element is deze synthese uitgevoerd op basis van een analytische
oplossing van de heersende differentiaalvergelijking. De precieze formulering van de
klassieke theorie is omgevormd tot de bekende aanpak van de verplaatsingsmethode
hetgeen het mogelijk maakt om combinaties van elementen en type elementen te
berekenen, terwijl de waardevolle kennis van de klassieke theorie bewaard is gebleven.
In aanvulling op de conventionele overgangsvoorwaarden en eindvoorwaarden maakt
de methode de implementatie van verstijvingsringen, elastische ondersteuningen,
voorgeschreven verplaatsingen en verschillende soorten belasting mogelijk. Op basis
van de voorgestelde oplossingsprocedure en met de genoemde functionaliteiten is, met
behulp van Fortran in combinatie met grafische software, een snel PC-georiënteerd
computerprogramma ontwikkeld. De formuleringen in dit programma zijn gebaseerd
op de benaderde oplossing van de Morley-Koiter vergelijking.
Algemene oplossingen voor de cirkelcilindrische schaalvergelijkingen
De voorgestelde set van vergelijkingen is voor cirkelcilindrische schalen met
cirkelvormige randen geformuleerd en de daaruit voortvloeiende enkele
differentiaalvergelijking is afgeleid. Een benadering van deze exacte vergelijking is
ingevoerd om te komen tot de mathematisch meest geschikte vergelijking voor
terugsubstitutie met dezelfde nauwkeurigheid, dwz de Morley-Koiter vergelijking.
xiv
De exacte wortels van de Morley-Koiter vergelijking zijn verkregen en hoewel
deze expressies de vereisten overtreffen is de gepresenteerde oplossing een unificatie
van eerdere resultaten van andere auteurs. Om te komen tot generieke kennis van het
schaalgedrag op basis van oplossingen in gesloten vorm zijn de benaderde wortels
afgeleid voor de drie eerder genoemde specifieke belastingstoestanden
(axiaalsymmetrie, liggerwerking, asymmetrie). Bijbehorende karakteristieke lengtes en
invloedslengtes vergemakkelijken het inzicht in de parameters die het schaalgedrag in
de betreffende belastingstoestanden bepalen.
Parametrische studie van lange cirkelcilindrische schalen (schoorstenen)
Ontwerpformules zijn verstrekt voor cilinders die lang zijn in vergelijking met hun
straal (bijvoorbeeld industriële, stalen schoorstenen). De formules zijn gebaseerd op de
oplossingen in gesloten vorm van de Morley-Koiter vergelijking en van een
vergelijking die is afgeleid met behulp van het semi-membraan concept. Ook
numerieke oplossingen hebben een bijdrage geleverd.
De ontwerpformule voor de spanningsverdeling aan de onderkant van lange
cirkelcilindrische schalen zonder verstijvingsringen onder windbelasting is afgeleid.
Deze is een duidelijke verbetering van de bestaande formule die op de Donnell
vergelijking gebaseerd is. De formule relateert de totale membraanspanning σ xx ,total aan
de spanning σ xx ,beam volgens de liggertheorie. Bij de gebruikte winddrukverdeling rond
de cilinder luidt de formule
2

 a  a
σ xx ,total = σ xx ,beam 1 + 6.39 1 − υ2   
 l  t 

waarbij de straal, lengte en dikte van de schaal door respectievelijk a , l en t worden
vertegenwoordigd en υ de dwarscontractiecoëfficiënt van het materiaal weergeeft
(Poisson verhouding). Deze vergelijking kan tevens geschreven worden als
4

a 
σ xx ,total = σ xx ,beam 1 + 6.39 1 − υ2   

 l2  
waarin de karakteristieke lengte l2 is gedefinieerd door l2 = 4 atl 2 .
Nieuwe ontwerpformules worden gegeven voor het effect van (centrische en
excentrische) verstijvingsringen en elastisch ondersteuningen (in axiale en
omtreksrichting). Het effect is beschreven met een extra factor in de formule voor de
spanning onderin de schaal bij afwezigheid van verstijvingsringen. De aangepaste
formule luidt:
4

a 
σ xx ,total = σ xx ,beam 1 + 6.39 1 − υ2 λ r   

 l2  
waarin λ r de verhouding is tussen de buigstijfheid van alleen de cirkelcilindrische
schaal en de gewijzigde buigstijfheid van de schaal (met de bijdrage van de
ringstijfheid per afstand).
xv
Voor het geval van een elastische ondersteuning van een lange cirkelvormige
cilinder hoeft alleen de axiale veerstijfheid in rekening gebracht te worden. De formule
luidt
4

a 
σ xx ,total = σ xx ,beam 1 + 6.39 1 − υ2 λ xn   

 l2  
waarin de genormaliseerde spanningsverhouding λ xn is ingevoerd, welke afhangt van
de respectieve factoren, het aantal golven (in omtreksrichting) van de belasting en de
parameter ηx ; deze is op zijn beurt vooral beschreven door de geometrische
eigenschappen van de cilinder en de verhouding tussen de axiale elastische
ondersteuning en de elasticiteitsmodulus.
Uit een vergelijking met numerieke resultaten is het toepassingsgebied van de
verbeterde en nieuwe ontwerpformules verkregen. De formules zijn van toepassing op
cilinders waarvoor de karakteristieke lengte l2 groter dan of gelijk aan de straal is. De
formule voor ring-verstijfde cilinders is van toepassing voor cilinders met een
ringafstand korter dan de helft van de invloedslengte van de lange golf in de oplossing;
bedoeld is de invloedslengte voor de belastingscomponent met twee golven in
omtreksrichting ( n = 2 ).
Numerieke studie van korte cirkelcilindrische schalen (tanks)
Voor korte cirkelcilindrische schalen (lengtes 0,5 tot 3 maal de straal) zijn numerieke
oplossingen gepresenteerd om de geschiktheid van het programma te demonstreren
voor het modelleren van de schaalwand van grote opslagtanks. Daarnaast is inzicht
verkregen in de reactie van dergelijke tankwanden onder de beschouwde drie
specifieke belastingstoestanden. Dit is bereikt op basis van verscheidene berekeningen
en door vergelijking met het inzicht dat verkregen is voor het gedrag van de lange
cilinder. Voor de berekeningen is gewerkt met de belasting ten gevolge van de
tankinhoud of winddruk; ook is de response op een variërende zakking langs de
volledige omtrek onderzocht.
xvi
Conclusies
Dit onderzoek heeft zich gericht op een grondige analyse van het gedrag van
cirkelcilindrische schalen met de volgende resultaten:
o De eerste-orde benaderingstheorie voor dunne schalen en de verscheidenheid
in aanpak in de literatuur zijn in een terugblik geëvalueerd, en een consistente
set van dunne schaalvergelijkingen is voorgesteld. Op basis van deze set is de
Morley-Koiter vergelijking geïdentificeerd als de meest geschikte
differentiaalvergelijking voor het afleiden van oplossingen in gesloten vorm.
o Op basis van deze oplossingen is een snel PC-georiënteerd
computerprogramma ontwikkeld voor een eerste ontwerp van lange en korte
cirkelcilindrische schalen zoals bijvoorbeeld schoorstenen en tanks.
o In de literatuur bestaat een ontwerpformule voor de spanningsverhoging aan
de voet van een schoorsteen. Deze is tot stand gekomen door het combineren
van een oplossing op basis van de (niet nauwkeurige) Donnell theorie en
EEM-berekeningen. De formule is bevestigd met de Morley-Koiter theorie.
Het voordeel van de nieuwe oplossing is dat de ontwerpformule
veralgemeniseerd is met betrekking tot de winddrukverdeling rond de
schoorsteen. Hij geldt ook voor andere verdelingen dan gebruikt in deze
studie.
o Bovengenoemde ontwerpformule is uitgebreid voor de invloed van een
elastische ondersteuning aan de voet van de schoorsteen.
o De ontwerpformule is ook uitgebreid voor de invloed van verstijvingsringen
langs de schoorsteen (ringeigenschappen en onderlinge afstand).
o Het toepassingsgebied van de formules is overtuigend en doelmatig verkregen
door vergelijking met resultaten van het computerprogramma.
xvii
xviii
List of symbols
indices written as subscript with a specific range
in case of a single quantity with two indices, the following applies:
( i, j, k ) = (1,2,3)
first index denotes fibre orientation or surface,
second index denotes direction of subject quantity
in case of a single quantity with two indices, the following applies:
( α, β ) = (1, 2 )
first index denotes fibre orientation or surface,
second index denotes direction of subject quantity
generic notation
A
a
A, A −1
a
da
δa
a
a∗
ah
ai
â
ac
ae
an
a′
a...0
a...1
a...n
quantity within the shell space
quantity on the reference surface or boundary line
matrix or vector with components A and its inverse, respectively
vector with components a
differential increment of quantity a
virtual variation of quantity a
quantity a at an edge
adjoint of quantity a
homogeneous solution for quantity a
inhomogeneous solution for quantity a
amplitude of quantity a
continuous expression for quantity a within the element
expression for quantity a at the edges of the element
expression for quantity a at the nodes connecting the elements
quantity a in the deformed state
quantity a for mode number n = 0
quantity a for mode number n = 1
quantity a for mode numbers n > 1
specific notation (in order of introduction)
Chapter 2
S
xi , ξi
ξ1 , ξ 2
ζ
R
r
n
(arbitrary or total boundary) surface
rectangular and curvilinear coordinate system, respectively
orthogonal curvilinear coordinates of the reference surface
coordinate in the thickness direction, viz. normal to the reference
surface
position vector within the shell space
position vector on the reference surface
unit normal vector of the reference surface
xix
( ds )
2
Po , P
g ii
Ai
α1 , α 2
R1 , R2
V
ds1 , ds2
dS1 , dS 2
dV
f
∆
Ui
eii , eij
Uζ
ψ1 , ψ 2
ϖn
ui
ϕ1 , ϕ2
uζ
ε11 , ε 22
ε12 , ε 21
ε1ζ , ε 2ζ
β11 , β22
β12 , β21
σii , σij
E
υ
G
E3 , υ3 , G3
σ11 , σ 22
σ12 , σ21
xx
line element on the reference surface
point within the shell space and infinitesimal close point, respectively
metric coefficients along the orthogonal parametric lines
scale factors
Lamé parameters of the reference surface
principal radii of curvature at the point on the reference surface
volume
differential lengths of arc of the edge of an infinitesimal element
differential areas of a strip on the edge of an infinitesimal element
differential volume of a layer within an infinitesimal element
scalar field
Laplace operator
displacements in the direction normal to the coordinate surfaces ξi
extension and shear components of the strain tensor, respectively
displacement in the thickness direction, viz. normal to the reference
surface
rotation in the ξ 2 -direction of a fibre along the ξ1 -direction and
rotation in the ξ1 -direction of a fibre along the ξ 2 -direction,
respectively
rigid body rotation about the normal to the reference surface
displacement components at the reference surface
rotation of a normal to the reference surface in the direction of the
parametric lines ξ1 and ξ 2 , respectively
displacement components of the reference surface in the thickness
direction
normal strains of the reference surface
longitudinal shearing strains of the reference surface
transverse shearing strains
changes of rotation of the normal to the reference surface
torsion of the normal to the reference surface
normal stress and shearing stress components, respectively
modulus of elasticity, Young’s modulus
Poisson’s ratio
shear modulus
elastic constants specifically in the direction normal to the reference
surface
normal stresses
longitudinal shearing stresses
σ1ζ , σ 2ζ
transverse shearing stresses
n11 , n22
normal stress resultants
longitudinal shearing stress resultants
transverse shearing stress resultants
bending stress couples
torsional stress couples
finite thickness of the thin shell
surface force vector per unit area of the reference surface
resultant components of the surface force vector
n12 , n21
v1 , v2
m11 , m22
m12 , m21
t
p
p1 , p2 , pζ
m1 , m2
f
S f , Su
u
Ep
Es
WP
WF
Es′
Pi
Fi
ξ1(1) , ξ1( 2)
ξ(21) , ξ(22)
f1 , f 2 , f ζ
t1 , t2
Rn
e
s
p
B
B∗
Bij , Bij∗
D
γ12 , ρ12
couple components of the surface force vector
edge force vector per unit length of the boundary lines
part of the boundary surface where the edge forces and edge
displacements are known, respectively
displacement vector
potential energy
strain energy
work done by the surface force vector
work done by the edge force vector
strain energy density function
components per unit volume of the external force vector
components per unit area of the boundary surface of the external force
vector
pair of edges of constant ξ1
pair of edges of constant ξ 2
resultant components of the edge force vector
couple components of the edge force vector
point load at the corner of and in the direction normal to the reference
surface
strain vector
stress vector
load vector, viz. equal to the external surface force vector
differential operator matrix
transpose of the matrix B where the components are the adjoint
operators
components of the differential operator matrix and its adjoint,
respectively
rigidity matrix
alternative shearing strain angle quantities; shear strain and torsion of
the reference surface, respectively
xxi
n12 , m12
κ11 , κ 22
n11 , m11
n22 , m22
κ12 , κ 21
Dm , Db
v1 , v2
n12∗ , v1∗
alternative longitudinal shearing stress quantities; longitudinal
shearing stress resultant and torsional stress couple, respectively
changes of curvature, alternative deformation quantities for β11 , β22
alternative stress quantities for n11 , m11
alternative stress quantities for n22 , m22
alternative deformation quantities for β12 , β21
extensional (membrane) rigidity and flexural (bending) rigidity,
respectively
alternative transverse shearing stress resultants for v1 , v2
combined internal stress resultants; the latter is similar to Kirchhoff’s
effective shearing stress resultant
Chapter 3
ξ, θ, ζ
n
u h , ui
Ch
c
uˆ ( ξ )
c
continuous displacement vector
uˆ ( ξ )
i
A ( ξ)
nˆ ( ξ )
c
nˆ ( ξ )
B (ξ)
uˆ , uˆ
c
c
continuous stress quantity vector
inhomogeneous part of the continuous stress quantity vector
c
continuous stress quantity matrix
i;e
Ae
fˆ e , fˆ i;e
Be
fˆ prim;e
fˆ tot ;e
Ke
fˆ ext ;n
xxii
inhomogeneous part of the continuous displacement vector
continuous displacement matrix
c
i
e
orthogonal coordinate system for a shell of revolution, viz.
meridional, circumferential and normal to the reference surface,
respectively
also used as index for load, stress and strain quantities and rotations
of a shell of revolution
mode number equal to the number of whole waves of a trigonometric
quantity in circumferential direction
also used as (additional) index to denote parameters typically
depending on the mode number
homogeneous and inhomogeneous displacement solutions,
respectively
arbitrary constant of the homogeneous solution, h = (1, 2,3,...,8 )
vector containing the constants of the homogeneous solution
element displacement vector and its inhomogeneous part, respectively
element displacement matrix
element force vector and its inhomogeneous part, respectively
element stress quantity matrix
element primary load vector
total element load vector
element stiffness matrix
external nodal load vector
fˆ prim;e;n
fˆ tot ; n
fˆ e; n
K
fˆ tot
nodal primary load vector
total nodal load vector
nodal force vector
global stiffness matrix
global load vector
Chapter 4
Lij
radius of a circular cylindrical reference surface
orthogonal coordinates of a circular cylindrical reference surface
coordinate in the thickness direction of a circular cylindrical shell
displacements at the reference surface of a circular cylindrical shell
normal strains of a circular cylindrical shell
changes of curvature of a circular cylindrical shell
shear strain and torsion of a circular cylindrical shell, respectively
normal stress resultants of a circular cylindrical shell
bending stress couples of a circular cylindrical shell
longitudinal shearing stress resultant and torsional stress couple of a
circular cylindrical shell, respectively
transverse shearing stress resultants of a circular cylindrical shell
surface forces at the reference surface of a circular cylindrical shell
normal stress of a circular cylindrical shell; axial and circumferential,
respectively
longitudinal shearing stress of a circular cylindrical shell
resultants of the edge forces at a circular cylindrical reference surface
couple of the edge forces at the circular edge of a circular cylindrical
shell
combined internal stress resultants of a circular cylindrical shell;
similar to Kirchhoff’s effective shearing stress resultant
rotation of a normal to the circular cylindrical reference surface in the
x -direction and θ -direction, respectively
components of a differential operator matrix
k
dimensionless parameter used to describe the components Lij
β
dimensionless parameter used to describe differential equations of a
circular cylindrical shell
dimensionless parameters of the homogeneous solution for n = 0
dimensionless parameter used to describe a0 , b0
dimensionless parameters of the homogeneous solution for n = 1
dimensionless parameter used to describe a1 , b1
a
x, θ
z
u x , uθ , u z
ε xx , εθθ
κ xx , κ θθ
γ xθ , ρ xθ
nxx , nθθ
mxx , mθθ
nx θ , m x θ
vx , vθ
p x , pθ , p z
σ xx , σθθ
σ xθ
f x , fθ , f z
tx
v∗x
ϕ x , ϕθ
a0 , b0
γ0
a1 , b1
γ1
a1n , bn1
dimensionless parameters of the homogeneous solution for n > 1
describing the short edge disturbance
xxiii
an2 , bn2
ηn , γ n
lc
li
l
lc ,1 , lc ,2
li ,1 , li ,2
dimensionless parameters of the homogeneous solution for n > 1
describing the long edge disturbance
dimensionless parameters used to describe a1n , bn1 , an2 and bn2
characteristic length of an edge disturbance
influence length of an edge disturbance
length of a circular cylindrical shell
characteristic length ( n > 1) of the short edge disturbance and the long
edge disturbance, respectively
influence length ( n > 1) of the short edge disturbance and the long
edge disturbance, respectively
Chapter 5
pw
α 0 ,.., α5
σ 2xx≤ n ≤5
σ0xx≤,nt ≤ 5
axial stress at the base of a circular cylindrical shell due to the mode
numbers 2 ≤ n ≤ 5 , i.e. the self-balancing terms of the specified wind
load
axial stress at the base of a circular cylindrical shell due to the mode
numbers 0 ≤ n ≤ 5 , i.e. all terms of the specified wind load
tensile axial stress at the base of a circular cylindrical shell
σ0xx≤,nc≤ 5
compressive axial stress at the base of a circular cylindrical shell
σ nxx=1
axial stress at the base of a circular cylindrical shell due to the mode
number n = 1 , i.e. the “beam term” of the specified wind load
characteristic lengths of a circular cylindrical shell introduced to
describe the axial stress ratio of the self-balancing terms to the “beam
term”
influence length of the long edge disturbance specifically for n = 2
σ 0xx≤ n ≤ 5
l1 , l2
lin,2= 2
Ar , Sr , I r
ηring
anSMC , bnSMC
γ SMC
n
Db ,mod
lr
kmod , βmod
λr
xxiv
wind stagnation pressure
factors per mode number of the wind load distribution
ring cross-sectional properties
ring parameter of the long edge disturbance
dimensionless parameters of the homogeneous solution for n > 1
describing the long edge disturbance within the SMC approach
dimensionless parameter used to describe anSMC , bnSMC
modified bending stiffness, viz. the bending stiffness of the stiffening
rings is “smeared out” along the bending stiffness of the circular
cylinder
ring spacing, along which the ring bending stiffness is “smeared out”
modified dimensionless parameters
stiffness ratio of bending stiffness of the circular cylindrical shell only
to the modified bending stiffness of the shell
b, h
bf
width and height of a stiffening ring
width of the flange of a beam cross section
beff
effective width of the flange of a curved beam
leff
effective length of the circular cylindrical shell in axial direction
er
eccentricity of the ring centre of gravity to the middle plane of the
cylinder
axial, circumferential and radial spring stiffness, respectively
rotational spring stiffness
k x , kθ , k z
kϕ
ηx
ηϕ
ηθz
η x ,mod
λ xn
ηθz ,mod
λ θzn
axial elastic support parameter
rotational elastic support parameter
combined circumferential and radial elastic support parameter
modified axial elastic support parameter
normalised stress ratio to account for influence of axial elastic support
modified combined circumferential and radial elastic support
parameter
normalised stress ratio to account for influence of planar elastic
support
Chapter 6
γw
h
λg
density of water
height of shell course in a tank wall
ratio of bending rigidity of the wind girder itself to the tank wall
hg , t g
wind girder dimensions; plate width and thickness, respectively
Ig
wind girder circumferential bending rigidity
u s ,max
maximum circumferential settlement of a tank shell
auxiliaries
L, L∗
u, v
µ1 , µ 2
operator and its adjoint, respectively
vectors
factors that account for the curvature of the parametric lines
λi , λi
q( x)
factors in differential operator matrices and their adjoint factors,
respectively
order of differential equation, identification of a cylindrical
subdomain
identifier for opposite circular edges
scalar function on the reference surface of a circular cylindrical shell
scalar functions for n = 0 , n = 1 and n > 1 , respectively
alternative surface load on a circular cylindrical shell
∆1 , ∆ n
Laplace operator for n = 1 and n > 1 , respectively
i
a, b
φ
φ0 , φ1 , φn
xxv
δ1 , δ 2
ω1 , ω2
ω, γ, η
r
r0 , r1
ε
s0 , s1
Sh
parameters used to describe a1n , bn1 , an2 and bn2
parameters used to describe δ1 and δ 2
parameters used to describe ω1 and ω2
root in trial solution to characteristic equation
expansions of the large roots in case of parameter perturbation
small parameter in case of parameter perturbation
expansions of the small roots in case of parameter perturbation
arbitrary constants in case of a rewritten homogeneous solution,
h = (1, 2,3,...,8 )
Ψh
phase angle, arbitrary constants in case of a rewritten homogeneous
solution, h = (1,2,3,4 )
dc
distance to the centre across the profile of a circular
density
gravitational acceleration
ρ
g
xxvi
1 Introduction
1 Introduction
1.1 Motive and scope of the research
In the field of structural mechanics the word shell refers to a spatial, curved structural
member. The enormous structural and architectural potential of shell structures is used
in various fields of civil, architectural, mechanical, aeronautical and marine
engineering. The strength of the (doubly) curved structure is efficiently and
economically used, for example to cover large areas without supporting columns. In
addition to the mechanical advantages, the use of shell structures leads to aesthetic
architectural appearance.
Examples of shells used in civil and architectural engineering are: shell roofs,
liquid storage tanks, silos, cooling towers, containment shells of nuclear power plants,
arch dams, et cetera. Piping systems, curved panels, pressure vessels, bottles, buckets,
parts of cars, et cetera are examples of shells used in mechanical engineering. In
aeronautical and marine engineering, shells are used in aircrafts, spacecrafts, missiles,
ships, submarines, et cetera.
Because of the spatial shape of the structure the behaviour of shell structures is
different from the behaviour of beam and plate structures. The external loads are
carried by both membrane and bending responses. As a result, the mathematical
description of the properties of the shell is much more elaborate than for beam and
plate structures. Therefore, many engineers and architects are unacquainted with the
aspects of shell behaviour and design.
In practice, many shell structures are single or combined shells of revolution (also
referred to as axisymmetric shells) and often they are stiffened by rings. The research
in this thesis focuses on the analyses of these shell structures, which find their
application in industries involved with structures like, for example, pipelines, liquid
storage tanks, chimneys and cooling towers.
The considerable effort in the development of rigorous shell theories dates back to the
early twentieth century. These shell theories reduce a basically three-dimensional
problem to a two-dimensional one. Nevertheless, the analysis of shells with the aid of
such theories involves complicated differential equations, which either cannot be
solved at all, or whose solution requires the use of high-level mathematics unfamiliar
to structural engineers. Therefore many approximate shell theories have been
developed, mainly on the assumption that the shell is thin, and to obtain generic
analysis tools obviously some accuracy had to be traded for convenience and
simplicity.
Hence, it is not surprising that the development of the numerical formulations
since the 1950’s has led to a gradual cessation of attempts to find closed-form solutions
to rigorous formulations. But, with today’s availability of greatly increased computing
power (also since the mid twentieth century), completeness rather than simplicity is
given more emphasis.
1
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The drawback of the numerical methods is that they do not provide generic
knowledge of the shell behaviour and the prevailing parameters. Also the foundations
of the formulations that are used and thus their justification and validity are often not
completely understood, which has resulted in numerous finite element formulations
that work quite well for certain problems but do not work well in other problems. This
results from the sensitivity of the problem to the geometry and support conditions,
which characterises the complicated behaviour of shell structures under various loading
types. For the use of numerical programs and to understand and validate the results,
some basic knowledge of the underlying theories and the mechanical behaviour of the
structure is obviously essential.
These observations give rise to a need for a study that is not based on blunt
computer power but on the rigorous shell formulations obtained by the classic
approach. But, due to its highly mathematical character, this reappraisal is only useful
if this approach is combined with modern methods for handling complicated boundary
and transition equations in a stiffness method approach. Hereby a generic study of the
shell behaviour can be conducted by evaluating the solution to the general equations as
well as the output of the computer program.
1.2 Research objective and strategy
This research project intends to combine the classic shell theories with the
contemporary numerical approach. The goal is to derive and employ a consistent and
reliable theory of shells of revolution and to present that theory in the context of
modern computational mechanics.
The contemplated set of equations concentrates on physically as well as
geometrically linear behaviour under static loading. A lot of basic and necessary
knowledge of this static and linear behaviour is lacking or not well understood and it is
this incomprehension that obstructs the shell analyst of gaining valuable insight into
the general shell behaviour.
This research not only focuses on the axisymmetric loading, but also on nonaxisymmetric loading, which means that for example a quasi-static wind load or nonuniform settlements can be studied. The results from the studies of both bending and
membrane dominated responses will enable a better evaluation and interpretation of the
results from finite element studies regarding the same and the more complete
behaviour.
With the proper set of equations as a starting point, the following successive steps are
performed. For cylindrical shells with circular boundaries, which are the most
frequently used in structural application, it is possible to obtain a closed-form solution
or at least an approximate solution (within the assumptions of the theory) to the
rigorous shell formulations. Already from these solutions, valuable insight is gained
into the type of response to each type of load and the prevailing parameters describing
this response. By reshaping the precise formulation of the classic approach into the
well-known direct stiffness approach of the displacement method, the valuable
knowledge of the classic approach is preserved. The aim of the project is to derive a
fast PC-oriented computer program for that. This is done using the Fortran-package in
2
1 Introduction
combination with graphical software and has resulted in a stable and well-working tool
that can be used by structural analysts for rational first-estimate design of shells of
revolution.
The approach of the displacement method enables the calculation of combinations
of elements and type of elements, which makes the use of an electronic calculation
device more sensible in view of the increasing number of equations. Next to that, it is
fairly simple to implement stiffening rings in the formulation and hereby the influence
of the number and size of these members on the shell behaviour can be studied.
Similarly, the elastic supports and prescribed displacements can be easily implemented
and various load types can be described. Combined with the generic knowledge from
the closed-form solutions, appropriate design tables, graphs and formulas are properly
presented using the suitable parameters.
1.3 Outline of the thesis
Chapter 2 deals with the fundamentals of the theory, the results by former authors and
the proposed set of equations. In chapter 3, the numerical solution procedure for this
set is introduced and this not often applied procedure is clarified. The formulations for
circular cylindrical shells that are implemented in this computational method are
derived in chapter 4. The combination of the generic knowledge from these two
chapters with the numerical results from the computer program enables a parametric
study of the geometrical properties of the shell types. These numerical results and
parametric study for long circular cylindrical shells (such as industrial chimneys) are
presented in chapter 5, while chapter 6 presents the numerical study for short circular
cylindrical shells (such as storage tanks). The conclusions from this study and
recommendation for further application of the proposed method are discussed in
chapter 7.
Introduction
CH1
General
part on
shell
theory
CH2
Conclusions
CH7
Computational
method CH3
Chimney
CH5
Tank
CH6
Circular
cylindrical
shells CH4
3
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
1.4 Short review of the existing work within the scope
In 2001, Van Bentum started a graduation project, which embodied a part of the tasks
of the present research. The main goal of that project was to show that, on basis of the
closed-form solutions to the Donnell equation for circular cylindrical shells, an exact
(within the theory) stiffness matrix could be synthesized. The resulting report was
published 2002 [1]. As Donnell’s solution is only applicable to the load-deformation
behaviour for circumferential modes with at least two whole waves in circumferential
direction, the solution for the axisymmetric and beam mode were implemented using
an alternative solution. For the response to axisymmetric loads, a simplified Donnell
solution was adopted using the displacement normal to the middle surface as the only
degree of freedom. For the response to beam loads, the membrane solution was
employed. In a successive step, the possible incompatibility between this membrane
solution and the requirements at the edges was compensated by an edge disturbance
congruent with the solution for the axisymmetric mode.
Although these solutions were successfully implemented and the result for the
study of rather long cylinders subject to a wind load were very satisfactory, the
following drawbacks can be noticed. Firstly, the axisymmetric mode can be better
described by using two independent degrees of freedom by taking into account the
longitudinal displacement in axial direction. Secondly, the approach for the beam mode
is only valid for a cylinder with rather large length-to-radius-ratio. For shorter
cylinders, the membrane behaviour and the edge disturbance resulting from the
complete differential equation should be described simultaneously. Thirdly, as it is well
known that Donnell’s solution does not describe the ring-bending behaviour, a better
description in circumferential direction should be adopted for the lower mode numbers
of the self-balancing modes (the modes with at least two whole waves in
circumferential direction).
The present study is restricted to closed circular cylindrical shells like long
industrial chimneys and storage tanks. The differential equations also facilitate
calculating cylindrical roof shells, but this study refrains from this type of structure.
Substantial research in this domain was performed by A.L. Bouma, H.W. Loof and H.
van Koten in The Netherlands, which was reported in [2]. This research was based on
the Donnell equation that sufficiently accurately describes the behaviour of this
structural type.
The concept of generating the stiffness matrix on basis of the closed-form solution
was already proposed as early as 1964 by Loof [3]. A number of systematically and
efficiently structured calculation schemes were developed, be it restricted to certain
load-deformation cases per shell structure due to the state of the programmable
electronic machines and available programming procedures of that period.
A literature study showed that Bhatia and Sekhon [4] recently applied the method
to axisymmetric structures. In their first paper of a series, the method is introduced and
applied to an annular plate element. Three follow-up papers [5-7] focus on the
generation of exact stiffness matrixes for a cylindrical, a conical and a spherical shell
element, respectively. However, Bhatia and Sekhon did only employ the method to
axisymmetric structures subject to loads that are also axisymmetric with respect to the
axis of symmetry of the structure. Hereby, the problem is reduced considerably, but the
application is rather limited and important engineering problems cannot be modelled.
4
1 Introduction
To study the influence of, e.g., elastic supports, stiffening rings and various load
types on the behaviour of circular cylindrical shells, these can be implemented into a
computer program as described above. With the same objective, Melerski [8] derived
solutions for beams, circular plates and cylindrical tanks, especially on elastic
foundations, and included a diskette with the resulting software. However, for circular
plates and cylindrical tanks the application of the in other aspects general approach is
limited to axisymmetric load cases.
Another interesting approach, which has the objective to obtain insight into the
load carrying behaviour of cylindrical shell structures, is the semi-membrane concept,
which is able to deal with non-axisymmetric load cases. The semi-membrane concept
assumes that, to simplify the initial equilibrium equations, the circumferential strain as
well as both the axial and torsional bending stiffness may be equated to zero. The
resulting equation exactly describes the ring-bending behaviour, but it can only be
applied to self-balancing modes. As shown by Pircher, Guggenberger and Greiner [9],
this concept can be applied to, e.g., a radial wind load, an axial elastic support and an
axial support displacement. However, not all load cases or support conditions can be
described. Moreover, the semi-membrane concept is only applicable to certain loaddeformation behaviours of cylindrical shell structures. Closely related to the
simplifications, it should be allowed to neglect the influence of the part of the solution
described by the short influence length in comparison to the part described by the long
influence length. In other words, the cylinder should be sufficiently long in comparison
to its radius and the boundary effects should mainly influence the more distant
material.
The present research overcomes the above-mentioned drawbacks of the solutions
used by Van Bentum and extends the results of that and the other mentioned research,
which is limited to either axisymmetric or non-axisymmetric load-deformation
behaviour. Instead of the Donnell equation, the Morley-Koiter equation is employed in
the present research. This equation is probably the best alternative, as it overcomes the
inaccuracy of Donnell’s simplifications in its inability to describe rigid-body modes but
preserves its elegance and simplicity.
The Morley-Koiter equation can be derived by using a so-called first-order
approximation theory. To understand the assumptions and simplifications, which are
introduced to obtain such an equation for a thin elastic shell, the set of equations
resulting from a fundamental derivation for thin elastic shells are reproduced. Since
these are well established, similar derivations can be found in many textbooks, which
are referenced in the text. However, the derivation in this research is set up as a more
integrated treatment of concepts by various authors. The objective of this treatment is
to correctly and consistently introduce the assumptions and simplifications throughout
the derivation of (i) the differential equations and boundary conditions, (ii) the single
differential equation and its solution and (iii) the expressions for all quantities obtained
by back substitution of this solution.
5
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
6
2 General part on shell theory
2 General part on shell theory
This chapter deals with the fundamentals of the theory. The geometry of a thin elastic
shell is treated briefly and the equations that describe the shell behaviour are derived.
The formulations for thin, shallow, non-linear and cylindrical shells by some former
authors are discussed and as a result of the comparison a set of equations is proposed.
This set comprises kinematical and constitutive relations that are complemented by the
equilibrium relation and boundary conditions, which are derived by making use of the
principle of virtual work.
2.1 Introduction to the structural analysis of a solid
shell
2.1.1 Geometrical interpretation
The primary purpose of a structure is to carry the applied external loading. Every
particle of this structure is a three-dimensional object on its own. In spite of this,
structural engineers (almost) never use the three-dimensional theory of elasticity, but
they model the structural elements as lines with a finite cross-sectional area, which has
become customary in the theory of structures.
The structural purpose of shell elements is to span a finite space. As a result of this,
a description of the structural element by one line is not possible and the stress analysis
has to be established with the concept of a “physical surface”. An important difference
has to be made to this: plates refer to flat surfaces and shells refer to curved surfaces.
Describing it, the shell element is interpreted as a materialisation of a curved
surface. This definition implies that the shell problem is reduced to the study of the
displacements of the reference (or middle) surface and that the thickness of the shell is
small in comparison to its other dimensions. The geometry of the shell is thus
completely described by the curved shape of the middle surface and the thickness of
the shell. In structural mechanics this geometrical description corresponds to the one of
the beam with a rectangular cross-section; the course of the middle axis in combination
with the accompanying cross-section. The shell thickness is henceforth kept constant
for convenience, but the analysis method and considerations are also applicable to
shells with a varying thickness.
The above-mentioned schematisation does not require that the shell be made of an
elastic material. Since most shells are made of a solid material, it will further be
assumed that the material behaves linear elastic conform Hooke’s law.
2.1.2 Generalised Hooke’s law
The first rough law of proportionality between the forces and displacements was
published by Hooke. The generalisation of Hooke’s law assumes that at each point of
the medium the strain components are linear functions of the stress components and
that it is possible to invoke the principle of superposition of effects. For many
engineering materials, the relation between stress and strain is indeed linear and the
7
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
deformation disappears during unloading. Obviously any material has its elastic limit,
viz. the greatest stress that can be applied and removed without permanent
deformation. Beyond this limit, which is nearly equal to the proportional limit, the
material behaves both elastic and plastic.
In this thesis, it will further be assumed that the material behaves conform a
generalised Hooke’s law, because we are interested in the general behaviour of shell
structures, especially since for rational first-estimate design it is naturally not advised
to rely on the plastic range of the structural capacity.
The assumption of homogeneity and isotropy of the material seems plausible for
most structural materials since we are interested in the global behaviour of an entire
body. It is not our objective to study the very small portions of material, which must be
regarded as orthotropic, but the chaotically distribution of the orthotropy over the entire
body allows the natural interpretation of a homogeneous and isotropic medium.
2.1.3 Mechanical behaviour of elastic shells
This research focuses on thin elastic shells. A thin shell has a very small thickness-tominimal-radius ratio, often smaller than 1 50 . Due to its initial curvature, a shell is able
to carry an applied load by in-plane as well as out-of-plane actions. Similar to the
behaviour of plates and beams, the resistance of a shell structure is optimally used if
bending actions are minimised as much as possible. A thin shell therefore mainly
produces in-plane actions, which are called membrane forces. These membrane forces
are actually resultants of the normal stresses and the in-plane shear stresses that are
uniformly distributed across the thickness. The corresponding theory of this membrane
behaviour is called the membrane theory.
However, the membrane theory does not satisfy all equilibrium and/or
displacement requirements in case of:
• Boundary conditions and deformation constraints that are incompatible with
the requirements of a pure membrane field, (b) and (c);
• Concentrated loads (d); and
• Change in the shell geometry (e).
(a)
Membrane compatible
8
(b)
Membrane incompatible
(c)
Deformation constraint
2 General part on shell theory
(d)
(e)
Concentrated load
Change in the geometry
In the regions where the membrane theory will not hold, some (or all) of the
bending field components are produced to compensate the shortcomings of the
membrane field in the disturbed zone. These disturbances have to be described by a
more complete analysis, which will lead to a bending theory of thin elastic shells.
If the bending field components are developed, it often has a local range of
influence. Theoretical calculations and experiments show that the required bending
field components attenuate and mostly this effect is confined to the vicinity of the
origin of the membrane nonconformity. In many cases, the bending behaviour is
restricted to an edge disturbance. Therefore, the undisturbed and major part of the shell
behaves like a true membrane. This unique property of shells is a result of the curvature
of the spatial structure. The efficient structural performance is responsible for the
widespread appearance of shells in nature.
2.1.4 Static-linear analysis of shells of revolution
Many shell theories have been developed to analyse the mechanical behaviour of shell
structures. To overcome the complexity of an exact theory assumptions are made
wherein the membrane theory is the most appealing. Because of its simplicity, the
membrane theory gives a direct insight into the structural behaviour and the order of
magnitude of the expected response without elaborate computations. But in the cases
where the membrane behaviour is not the dominant type of response, use is often made
of finite element packages.
The usefulness of the finite element approach for the initial design and analysis is
however doubtful and an intermediate approach between the contemporary and the
classic approach is recommendable. This intermediate approach is thus the main focus
of this study.
For shells of revolution with circular boundaries, which are the most frequently
used in structural application, the rigorous shell formulations have been well
established. Keeping in mind the objective of employing closed-form solutions,
attempting to investigate the linear models first seems to be the natural strategy. Hence,
the starting point is the analysis of the small deformation behaviour of shells of
revolution under static loading.
9
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
2.2 Fundamental theory of thin elastic shells
The set of equations resulting from a fundamental derivation for thin elastic shells are
well established. Consequently, the expressions derived in this section are probably
well known but they are stated without accurate reference here for further use. Similar
derivations can be found, e.g., in the books by Kraus [10] and Leissa [11] and the
report by Hildebrand, Reissner and Thomas [12]. However, the following derivation is
set up as an integrated treatment and complement of concepts by various authors.
2.2.1 Kirchhoff-Love assumptions
On the basis of the assumptions Kirchhoff introduced with the purpose of deriving a
theory of a thin plate, Love [13] was the first to derive a set of basic equations which
describe the behaviour of a thin elastic shell. Generally referred to as Love’s first
approximation this classic small deformation theory of a thin shell is based on the
following postulates, which are also known as the Kirchhoff-Love assumptions:
1. The shell is thin.
2. Strains and displacements are sufficiently small so that the quantities of
second- and higher-order magnitude in the strain-displacements relations may
be neglected in comparison to the first-order terms.
3. The transverse normal stress is small in comparison to the other normal stress
components and may be neglected.
4. A normal to the reference surface before deformation remains straight and
normal to the deformed reference surface and suffers no extension.
Before utilising these assumptions, it is useful to discuss their implications
individually.
The assumption that the shell is thin is inevitable for the other assumptions as these
are only appropriate if the thickness of the shell is small in comparison to the other
dimensions. The thinness of a shell is often characterised by the ratio of the thickness
to the radius of curvature, but no precise definition is available and suggestions differ
largely. For the present discussion, the thinness will be such that the ratio mentioned is
negligible in comparison to unity.
The second assumption is necessary to keep the equations linear and to be allowed
to describe all resulting equations in the initial configuration. This assumption also
implies that the first derivatives of all displacements are negligible in comparison to
unity.
The assumption that the transverse normal stress is negligible seems plausible for a
thin shell except in the vicinity of highly localised loading.
The last assumption is a continuation of the well-known Bernoulli-Euler
hypothesis and implies that not only the transverse shear deformation but also the strain
components in the direction of the normal to the reference surface can be neglected.
Flügge [14] states that conclusions drawn from the last two assumptions can only
be exact if the shell be made of a non-existent anisotropic material for which the
modulus of elasticity in the direction normal to the reference surface and the shear
10
2 General part on shell theory
modulus for the transverse shearing strains are infinite, whereas two of the Poisson’s
ratios (that take into account the lateral contraction of a material) are equal to zero.
However, it is obvious that for a thin shell the assumptions are acceptable so that
whatever happens in the direction normal to the reference surface of the shell, stress or
strain, is of no significance to the solution.
2.2.2 Mathematical description of a shell surface
To describe the curved reference surface of a shell it is natural to use a curvilinear
coordinate system that coincides with the lines of principal curvature, which can be
shown to be orthogonal. The derivation and proof of this feature and all the other
expressions in this subsection are exemplified in Appendix A, which contains parts of
the well-documented study of the differential geometry of surfaces especially when
applied to the mathematical description of a shell surface.
A surface S in the rectangular coordinate system x1 , x2 , x3 can be written as a function
of two parameters; viz. ξ1 , ξ 2 , which are the curvilinear coordinates of the reference
surface. To describe the location of an arbitrary point within the two outer surfaces of
the shell a third coordinate ζ is introduced in the thickness direction. The position
vector R to this arbitrary point is described by
R ( ξ1 , ξ 2 , ζ ) = r ( ξ1 , ξ 2 ) + ζn ( ξ1 , ξ2 )
where r is the position vector of the corresponding point on the reference surface and
n is the unit normal vector.
2
The line element ( ds ) is calculated by taking the dot product of the differential
change dR in the position vector from a point Po to an infinitesimal close point P
within the shell space and hence is expressed by
2
2
2
2
(2.1)
( ds ) = dR ⋅ dR = g11 ( d ξ1 ) + g 22 ( d ξ2 ) + g33 ( d ζ )
where gii ( i = 1,2,3) are the metric coefficients along the orthogonal parametric lines.
These coefficients are defined by


ζ 
ζ 
A1 = g11 = α1 1 +  , A2 = g 22 = α 2  1 +  , A3 = g33 = 1
(2.2)
 R1 
 R2 
where Ai are the scale factors, α1 and α 2 are the so-called Lamé parameters of the
reference surface and R1 and R2 are the principal radii of curvature at the point on the
reference surface corresponding to point Po . The Lamé parameters and the principal
radii are related to the position vector and the unit normal vector by
α12 =
∂r ∂r
⋅
∂ξ1 ∂ξ1
1
1 ∂r ∂n
=
⋅
R1 α12 ∂ξ1 ∂ξ1
α22 =
∂r ∂r
⋅
∂ξ2 ∂ξ2
1
1 ∂r ∂n
=
⋅
R2 α 2 2 ∂ξ 2 ∂ξ 2
11
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
There are three differential equations relating the parameters of the reference surface.
The two equations, which are known as the Codazzi conditions, are
1 ∂α1
∂  α1 
=
  ,
R2 ∂ξ 2 ∂ξ2  R1 
1 ∂α 2
∂  α2 
=
 
R1 ∂ξ1 ∂ξ1  R2 
(2.3)
and the third is known as the Gauss condition, which is given by:
∂  1 ∂α 2  ∂  1 ∂α1 
α1α 2

+

=−
R1R2
∂ξ1  α1 ∂ξ1  ∂ξ 2  α 2 ∂ξ2 
An infinitesimal element within the volume V of the thin shell is obtained by making
four cuts perpendicular to the reference surface, which coincide with a pair of
differentially spaced parametric lines of the reference surface, and the space that is then
limited by two surfaces that are dζ apart (at distance ζ from the reference surface) is
the infinitesimal element. By evaluating the expressions for a line element (2.1), it is
obvious that the differential lengths of arc of the edges of the element are

ζ 
ds1 ( ξ1 , ξ2 , ζ ) = α1  1 +  d ξ1
R

1

ζ 
ds2 ( ξ1 , ξ 2 , ζ ) = α 2  1 +  d ξ 2
R

2 
,
(2.4)
and that the differential areas of a strip on the faces of the element are

ζ 
dS1 ( ξ1 , ξ 2 , ζ ) = α1  1 +  d ξ1d ζ
R

1
,

ζ 
dS 2 ( ξ1 , ξ 2 , ζ ) = α 2 1 +  d ξ2 d ζ
R

2 
(2.5)
Hence, the differential volume of a layer of the element bounded by these strips is

ζ 
ζ 
dV ( ξ1 , ξ2 , ζ ) = α1α 2  1 + 1 +  d ξ1d ξ 2 d ζ
 R1  R2 
(2.6)
Finally, the Laplace-Beltrami operator of a scalar field f described in an orthogonal
curvilinear coordinate system is a scalar differential operator defined by
∆f =
1  ∂  α 2α 3 ∂f  ∂  α1α3 ∂f  ∂  α1α 2 ∂f  


+

+


α1α 2α 3  ∂ξ1  α1 ∂ξ1  ∂ξ 2  α 2 ∂ξ 2  ∂ξ3  α3 ∂ξ3  
as derived, for example, by Borisenko and Tarapov [15]. For the scalar field f that
acts on the reference surface within a shell space described by (2.2), the LaplaceBeltrami operator, which is further referred to as the Laplace operator, is given by
∆f =
1  ∂  α 2 ∂f  ∂  α1 ∂f  


+


α1α 2  ∂ξ1  α1 ∂ξ1  ∂ξ 2  α 2 ∂ξ2  
(2.7)
2.2.3 Kinematical relation
For a curvilinear coordinate system determined by the coordinate lines ξi , which are
assumed to be orthogonal, the metric coefficients along these parametric lines are
denoted by gii as shown in Appendix A. The displacements in the direction normal to
the coordinate surfaces ξ1 , ξ2 , ξ3 are represented by U1 , U 2 , U 3 respectively. By
applying the assumptions of infinitesimal deformations in this curvilinear coordinate
12
2 General part on shell theory
system as shown in Appendix B, the extension and shear components of the strain
tensor, eii and eij respectively, are obtained in the form
∂  Ui 
1 3 ∂gii U k
,
i = 1,2,3

+
∑
∂ξi  gii  2 g ii k =1 ∂ξ k g kk

∂  Ui 
∂  U j 
1
 gii


eij =
,

 + g jj
( i, j ) = (1,2,3) , if i ≠ j
∂ξi  g jj  
2 gii g jj  ∂ξ j  gii 



Hereby the extension eii is defined as the relative elongation in the ξi -direction of a
eii =
fibre in the ξi -direction and the shear component eij is defined as half of the angle with
which the originally perpendicular ξi - and ξ j -directions decreases.
By substituting gii = ( Ai ) from (2.2), we get:
2
e11 =
1  ∂U1 U 2 ∂A1 U 3 ∂A1 
+
+


A1  ∂ξ1 A2 ∂ξ 2 A3 ∂ξ3 
,
2e12 =
A1 ∂  U1  A2 ∂  U 2 
 +
 
A2 ∂ξ2  A1  A1 ∂ξ1  A2 
e22 =
1  U1 ∂A2 ∂U 2 U 3 ∂A2 
+
+


A2  A1 ∂ξ1 ∂ξ 2 A3 ∂ξ3 
,
2e13 =
A1 ∂  U1  A3 ∂  U 3 
 
 +
A3 ∂ξ3  A1  A1 ∂ξ1  A3 
e33 =
1  U1 ∂A3 U 2 ∂A3 ∂U 3 
+
+


A3  A1 ∂ξ1 A2 ∂ξ 2 ∂ξ3 
,
2e23 =
A2 ∂  U 2  A3 ∂  U 3 
 
 +
A3 ∂ξ3  A2  A2 ∂ξ2  A3 
(2.8)
In the case of the adopted coordinate system, the substitutions ξ3 = ζ for the coordinate
and U 3 = U ζ for the displacement in the direction of the normal to the reference surface
are made.
By definition the in-plane shear angle 2e12 is defined by:
2e12 = e12 + e21 = ψ1 − ϖ n + ψ 2 + ϖ n
Hereby the angle ψ1 is the rotation in the ξ 2 -direction of a fibre along the ξ1 -direction
and the angle ψ 2 is defined correspondingly. The angle ϖ n is the rigid body rotation
about the normal to the reference surface, which is taken positive according to the
right-hand rule.
The introduction of the rotation ϖ n is similar to the procedure that is well known
for a plate element. For that geometry, the shear strain is found by describing two
changes of the straight angle in the respective directions. These changes are then split
in a symmetric part (the shear strain) and a skew-symmetric part (the rigid body
rotation). This is exactly the procedure that is applied above.
Therefore, it is remarkable that this procedure is not widely applied in describing
the deformation of a shell element. Sanders [16] does introduce the rigid body rotation
ϖ n , but on a reverse consideration, which is discussed in subsection 2.6.3.
13
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Following the above-mentioned procedure the shear angle 2e12 expressed by (2.8)
is also described by
2e12 =
A1 ∂  U1  A2 ∂  U 2 
 +
 
A2 ∂ξ 2  A1  A1 ∂ξ1  A2 
U ∂A1 1 ∂U 2
1 ∂U1 U 2 ∂A2
=− 1
+
− ϖn +
−
+ ϖn
A1 A2 ∂ξ2 A1 ∂ξ1
A2 ∂ξ2 A1 A2 ∂ξ1
Hence, it follows that
e12 = ψ1 − ϖ n = −
e21 = ψ 2 + ϖ n =
U1 ∂A1 1 ∂U 2
+
− ϖn
A1 A2 ∂ξ 2 A1 ∂ξ1
1 ∂U1 U 2 ∂A2
−
+ ϖn
A2 ∂ξ 2 A1 A2 ∂ξ1
From the definition e12 = e21 it is obtained that the rigid body rotation ϖ n is equal to
ϖn =
∂A U 
1  ∂AU
1 1
+ 2 2
−
2 A1 A2  ∂ξ 2
∂ξ1 
which is also shown in Appendix B.
To relate all components of strain to quantities of the reference surface the fourth
assumption of Love’s postulates has to be employed.
The first part of that assumption which requires that a normal remains straight is
satisfied when the displacements are linearly distributed through the thickness of the
shell. Hence, the displacement components are represented by
U i ( ξ1 , ξ2 , ζ ) = ui ( ξ1 , ξ 2 ) + ζ
∂U i
( ξ1 , ξ2 ,0 )
∂ζ
where ui is the respective displacement component at the reference surface and
∂U i
is
∂ζ
the change of the displacement component in the normal direction.
The second part of the fourth assumption requires inextensibility of a normal to the
reference surface, which implies that normal strain vanishes. By substituting A3 = 1
from (2.2) into (2.8) for the normal strain, we get
e33 =
∂U 3 ∂U ζ
=
∂ξ3
∂ζ
and hence
∂U ζ
∂ζ
( ξ1, ξ2 , ζ ) = 0 to disregard the normal strain.
Since a normal to the reference surface remains straight, the derivatives
∂U1
and
∂ζ
∂U 2
are equal to the respective rotations of the normal from its initial position to its
∂ζ
position after deformation. So, the rotations ϕ1 and ϕ2 are introduced, which denote
the rotations of a normal to the reference surface in the direction of the parametric lines
ξ1 and ξ 2 , respectively.
14
2 General part on shell theory
As a consequence of the above, the displacement components are represented by
U1 = u1 + ζϕ1
U 2 = u2 + ζϕ2
(2.9)
U ζ = uζ
To relate the strain components to the displacements of the reference surface, the scale
factors (2.2) and the representation of the displacement components (2.9) are
substituted into (2.8). Making use of the Codazzi conditions (2.3) we arrive at the
following six expressions of the strain components related to ten deformation
quantities.
e11 =
e12 =
1
ζ
1+
R1
1
ζ
1+
R1
2e1ζ =
( ε11 + ζβ11 )
e22 =
( ε12 + ζβ12 )
e21 =
1
ζ
1+
R1
( 2ε )
1ζ
1
1+
ζ
R2
1
1+
2e2ζ =
ζ
R2
( ε 22 + ζβ22 )
( ε 21 + ζβ21 )
1
1+
ζ
R2
(2.10)
( 2ε )
2ζ
The ten deformation quantities are separated in four strains of the reference surface
denoted by ε11 , ε 22 , ε12 and ε 21 , in four changes of rotation of the normal to the
reference surface denoted by β11 , β22 , β12 and β21 , and in two transverse shearing
strains denoted by ε1ζ and ε 2ζ . The ten deformation quantities of the kinematical
relation are related to the reference surface displacements by
1  ∂u1 u2 ∂α1  uζ
+

+
α1  ∂ξ1 α 2 ∂ξ 2  R1
u ∂α 2  uζ
1  ∂u
ε 22 =  2 + 1
+
α 2  ∂ξ2 α1 ∂ξ1  R2
ε11 =
ε12 =
1  ∂u2 u1 ∂α1 
−

 − ϖn
α1  ∂ξ1 α 2 ∂ξ 2 
1  ∂u1 u2 ∂α 2 
−

 + ϖn
α 2  ∂ξ 2 α1 ∂ξ1 
1 ∂uζ u1
− + ϕ1
2ε1ζ =
α1 ∂ξ1 R1
ε 21 =
1  ∂ϕ1 ϕ2 ∂α1 
+


α1  ∂ξ1 α 2 ∂ξ2 
ϕ ∂α 
1  ∂ϕ
β22 =  2 + 1 2 
α 2  ∂ξ2 α1 ∂ξ1 
β11 =
β12 =
1  ∂ϕ2 ϕ1 ∂α1  ϖ n
−

−
α1  ∂ξ1 α 2 ∂ξ2  R1
(2.11)
1  ∂ϕ1 ϕ2 ∂α 2  ϖ n
−

+
α 2  ∂ξ2 α1 ∂ξ1  R2
1 ∂uζ u2
−
+ ϕ2
2ε 2 ζ =
α 2 ∂ξ 2 R2
β21 =
The fourth assumption also requires that a normal remains a normal to the reference
surface, which implies that the transverse shear deformations are neglected. By setting
the expressions for the transverse shearing strains ε1ζ and ε 2ζ equal to zero, we arrive
at the expressions for the rotations related to the other displacements of the reference
surface, which become
15
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
u1 1 ∂uζ
−
R1 α1 ∂ξ1
ϕ1 =
(2.12)
u
1 ∂uζ
ϕ2 = 2 −
R2 α 2 ∂ξ 2
2.2.4 Constitutive relation
As stated in subsection 2.1.2 homogeneity and isotropy of the material is assumed and
the objective of the Kirchhoff-Love assumptions for a thin shell is to relate all
expressions to the behaviour of the reference surface within this material. Next to this,
whatever happens in the direction normal to this reference surface, stress or strain, is
assumed to be negligible in order to simplify the analysis. So, assuming homogeneity
of the material, isotropy with respect to the reference surface and elastic symmetry with
respect to the normal to that surface, the generalisation of Hooke’s law for a thin shell
is set up starting from
e11 =
1
υ
( σ11 − υσ 22 ) − 3 σ33
E
E3
e22 =
υ
1
( σ22 − υσ11 ) − 3 σ33
E
E3
e33 =
1
υ
σ33 − 3 ( σ11 + σ22 )
E3
E3
2e12 =
1
σ12
G
; 2e13 =
1
σ13
G3
; 2e23 =
1
σ23
G3
where the normal strains eii and the shearing strains eij are related to the normal
stresses σii and the shearing stresses σij . The material properties in the directions of
the parametric lines on the reference surface are represented by Young’s modulus E
and Poisson’s ratio υ , which describe the linear elasticity and the lateral contraction,
respectively. The shear modulus G is related to these material properties by
G=
E
2 (1 + υ)
The elastic constants with the single index 3 relate to the material properties in the
direction normal to the reference surface.
To fulfil both that σ33 = 0 (see the third assumption of subsection 2.2.1) and that e33 = 0
(see the fourth assumption) is not possible unless it is assumed that, in the direction
normal to the reference surface, the modulus of elasticity E3 is infinite and the
Poisson’s ratio υ3 is equal to zero. As stated in subsection 2.2.1, these assumptions are
acceptable if the shell is thin.
Also based on the fourth assumption, the transverse shearing strains are set equal
to zero for a thin elastic shell, but the transverse shearing stresses are not necessarily
zero. This implies that G3 is assumed to be infinite so that these stresses do not
produce any deformation.
16
2 General part on shell theory
Introducing the above-mentioned assumptions, the two-dimensional Hooke’s law
for thin elastic shells becomes
1
( σ11 − υσ22 )
E
1
e22 = ( σ22 − υσ11 )
E
1
2e12 = σ12
G
e11 =
or equally in an inverse form
E
( e11 + υe22 )
1 − υ2
E
σ 22 =
( e22 + υe11 )
1 − υ2
σ12 = 2Ge12
σ11 =
(2.13)
which describes a plane stress state as a reduced form of the constitutive relation.
The stress resultants and the stress couples of these stresses are the normal stress
resultants n11 and n22 , the longitudinal shearing stress resultants n12 and n21 , the
transverse shearing stress resultants v1 and v2 , the bending stress couples m11 and m22 ,
and the twisting stress couples m12 and m21 . It is convenient to express these resultants
and couples as area integrals of the stresses acting on the faces of an infinitesimal shell
element, and since we are treating a surface, these are resultants and couples per unit
length of arc on the reference surface. In section 2.3 it is shown that these stress
resultants are the ones that correspond to these deformation quantities defined in the
previous subsection.
In a section ξ1 = constant of the infinitesimal element, the resultant of the normal
stress σ11 acting in the ξ1 -direction is by definition equal to n11ds2 ( ξ1 , ξ2 ,0 ) and
similarly the couple of the normal stress is equal to m11ds2 ( ξ1 , ξ2 ,0 ) . Variations of ds2
can be neglected since its value is already of differential magnitude. However, the
variation of the stress across the thickness of the shell has to be considered. It is
therefore necessary to consider a strip with differential area dS 2 ( ξ1 , ξ 2 , ζ ) given by
expression (2.5). Hence, the resultant and the couple on this face of the differential
element are given by
n11 =

ζ 
1
σ11dS 2 ( ξ1 , ξ 2 , ζ ) = ∫ σ11 1 +  d ζ
ds2 ( ξ1 , ξ2 ,0 ) ∫ζ
R
2 

ζ

1
ζ 
ζσ11dS 2 ( ξ1 , ξ 2 , ζ ) = ∫ σ11 1 +  ζd ζ
ds2 ( ξ1 , ξ 2 ,0 ) ∫ζ
R

2 
ζ
since ds2 ( ξ1 , ξ2 ,0 ) = α 2 d ξ 2 , which follows from expression (2.4).
m11 =
Note that the resulting expressions are independent of the stress distribution
through the thickness. By sign convention, a stress resultant is thus defined as positive
in case of a positive direction of the normal to the face of the differential element in
combination with the corresponding stress acting in the positive direction of the
17
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
parametric lines. For a positive stress couple, the distance ζ from the reference surface
must also be positive. Obviously, a resultant or couple is also positive in case of a
negative direction of the normal to the face and the stress acting in the negative
direction. Hence, a positive stress resultant represents tension and a positive stress
couple represents tension at the upper side and compression at the lower side.
In the same way, the longitudinal shearing stress σ12 and the transverse shearing
stress σ1ζ are integrated to obtain the longitudinal shearing stress resultant n12 and the
transverse shearing stress resultant v1 , respectively. The same reasoning holds for the
side surface with ξ 2 = constant , while bringing into account the fact that the line
element ds1 has another radius of curvature, and hence all stress resultants and couples
are given by

ζ 
n11 = ∫ σ11 1 +  d ζ
ζ
 R2 

ζ
n22 = ∫ σ22 1 +  d ζ
ζ
 R1 

ζ 
n12 = ∫ σ12 1 +  d ζ
ζ
R

2 

ζ
n21 = ∫ σ 21 1 +  d ζ
ζ
R

1

ζ 
v1 = ∫ σ1ζ  1 +  d ζ
ζ
R

2 

ζ 
v2 = ∫ σ2 ζ 1 +  d ζ
ζ
R

1 

ζ 
m11 = ∫ σ11  1 +  ζd ζ
ζ
 R2 

ζ 
m12 = ∫ σ12  1 +  ζd ζ
ζ
 R2 

ζ 
m22 = ∫ σ22  1 +  ζd ζ
ζ
 R1 

ζ 
m21 = ∫ σ21  1 +  ζd ζ
ζ
 R1 
(2.14)
By comparing the expression for the longitudinal shearing stress resultant n12 with the
one for n21 , it is observed that the equality of the longitudinal shearing stresses,
σ12 = σ21 , not implies the equality of the longitudinal shearing stress resultants. This
difference disappears if R1 = R2 (a sphere or a plate) or when the reference surface is
the middle surface and the shearing stress σ12 does not depend on the ordinate ζ . The
equality of the longitudinal shearing stresses does also not imply the equality of the
twisting stress couples m12 and m21 . But both differences may often be neglected
because the thickness is small in comparison to the radii.
As a result of the factors (1 + ζ R1 ) and (1 + ζ R2 ) , the stress couples are not equal
to zero for a uniformly distributed stress. This is a result of the curvature of the shell.
These factors represent the fact that the faces of a shell element are not rectangular but
trapezoidal and that their centre of gravity is not exactly situated on the middle surface.
The transverse shearing stresses σ1ζ and σ2ζ do not lead to stress couples because their
moment arms are of differential length.
18
2 General part on shell theory
Having found the expressions for the stress resultants and stress couples, the
constitutive relation is obtained by substituting Hooke’s law. On the basis of the
Kirchhoff-Love assumptions for a thin shell theory, Hooke’s law is reduced to the
expressions (2.13). Herein the transverse shearing stresses σ1ζ and σ2ζ are not
described since the transverse shearing strains are neglected. However, this cannot
imply that the corresponding stresses are also set equal to zero since their resultants
cannot be zero if the stress couples can vary along the shell.
The integrations are not performed in this subsection for reasons that become
apparent in section 2.6 where the differences between the analyses by several former
authors are comparatively discussed. This comparison is only possible with the proper
equilibrium equations as complementation of the set of fundamental relations.
2.2.5 Equilibrium relation
Having defined the stress resultants and the stress couples in the previous subsection by
integrations of the stresses through the thickness, it is henceforth assumed that the
fundamental element of the thin shell has a finite thickness t . The internal stress
resultants and couples thus act upon the edges of this fundamental element, which are
of differential lengths of arc ds1 ( ξ1 , ξ 2 ,0 ) = α1d ξ1 and ds2 ( ξ1 , ξ2 ,0 ) = α 2 d ξ 2 , respectively.
These internal forces must balance the external forces that consist of both surface
forces and body forces. The surface forces act upon the inner and outer surface of the
element and the body forces act over the volume of that element. To maintain the
representation of the shell by quantities defined with respect to the reference surface,
these external forces should accordingly be replaced by statically equivalent forces that
act upon the reference surface. Recalling that Love’s first approximation is postulated
to derive a small deformation theory of a thin shell, it is natural to make the additional
assumption that the surface and body forces induce negligibly small couples with
respect to the reference surface. Moreover, it will be shown that as the transverse
shearing strain is neglected, it is no longer allowed to induce these external moments
on the reference surface, but for the sake of completeness, the couples will be retained.
The components of the surface force vector p per unit area of the reference
surface are denoted by the resultants p1 , p2 and pζ , which act along the two
parametric lines ξ1 and ξ 2 and along the normal (in ζ -direction), respectively, and by
the couples m1 and m2 . Since we are allowed to describe all resulting equations in the
initial configuration (see subsection 2.2.1), we arrive at six linear equations of which
three are equilibriums of stress resultants and three are equilibriums of stress couples.
The derivation shown in Appendix C results in the following six equations of the
equilibrium relation
19
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
∂n11α 2
∂α ∂n α
∂α
αα
+ n12 1 + 21 1 − n22 2 + v1 1 2 + p1α1α 2 = 0
∂ξ1
∂ξ2
∂ξ2
∂ξ1
R1
∂n22α1
∂α
∂n α
∂α
αα
+ n21 2 + 12 2 − n11 1 + v2 1 2 + p2α1α 2 = 0
∂ξ2
∂ξ1
∂ξ1
∂ξ2
R2
n
∂v1α 2 ∂v2α1
n 
+
− α1α 2  11 + 22  + pζ α1α 2 = 0
∂ξ1
∂ξ 2
 R1 R2 
∂m11α 2
∂α ∂m α
∂α
+ m12 1 + 21 1 − m22 2 − v1α1α 2 + m1α1α 2 = 0
∂ξ1
∂ξ2
∂ξ 2
∂ξ1
(2.15)
∂m22α1
∂α
∂m α
∂α
+ m21 2 + 12 2 − m11 1 − v2α1α 2 + m2α1α 2 = 0
∂ξ 2
∂ξ1
∂ξ1
∂ξ 2
n12 − n21 +
m12 m21
−
=0
R1
R2
The sixth equilibrium equation is in fact an identity, which is in accordance with the
fact that there are only five independent displacements of the reference surface. The
identity is easily obtained by substitution of the definitions (2.14) for the stress
resultants and stress couples, which by observing the symmetry of the shearing stresses
results in the equality
∫ (σ
ζ
12

ζ 
ζ 
− σ21 )  1 +  1 +  d ζ = 0
 R2  R1 
If the transverse shearing strain is neglected, there are not five but only three
independent displacements. This implies that in this case there are three equilibrium
equations for three external forces in the direction of those displacements and hence
that the influence of external moments can no longer be taken into account. The sought
equations are obtained by eliminating the transverse shearing stress resultants from
(2.15), which results in
∂n11α 2
∂α ∂n α
∂α
+ n12 1 + 21 1 − n22 2
∂ξ1
∂ξ2
∂ξ2
∂ξ1
+
1  ∂m11α 2
∂α ∂m α
∂α 
+ m12 1 + 21 1 − m22 2  + p1α1α 2 = 0

R1  ∂ξ1
∂ξ 2
∂ξ 2
∂ξ1 
∂n22α1
∂α
∂n α
∂α
+ n21 2 + 12 2 − n11 1
∂ξ2
∂ξ1
∂ξ1
∂ξ2
+
1  ∂m22α1
∂α
∂m α
∂α 
+ m21 2 + 12 2 − m11 1  + p2α1α 2 = 0

R2  ∂ξ2
∂ξ1
∂ξ1
∂ξ2 
∂α ∂m α
∂α  
∂  1  ∂m11α 2
+ m12 1 + 21 1 − m22 2  
 
∂ξ1  α1  ∂ξ1
∂ξ2
∂ξ2
∂ξ1  
+
n
∂  1  ∂m22α1
∂α
∂m α
∂α  
n 
+ m21 2 + 12 2 − m11 1   − α1α 2  11 + 22  + pζ α1α 2 = 0
 
∂ξ2  α 2  ∂ξ 2
∂ξ1
∂ξ1
∂ξ 2  
 R1 R2 
(2.16)
20
2 General part on shell theory
2.3 Principle of virtual work
In this section, the principle of virtual work is employed to a thin elastic shell by
utilising the kinematical and constitutive relations derived in subsections 2.2.3 and
2.2.4. The virtual work equation is in this way applied to obtain the equilibrium
relation expressed in the appropriate stress resultants and couples that correspond to the
chosen deformation quantities of the kinematical relation. The fact that these
correspond is observed when the expression of the internal strain energy, which is
formulated in quantities with respect to the reference surface, is evaluated. As a result,
the constitutive relation will be symmetric.
Having assessed the correspondence of the internal quantities, the resulting
equilibrium relation will be such that an elegant similarity exists between the
equilibrium relation and the kinematical relation. This similarity again assures that
after successive substitution, the resulting set of differential equations expressed in the
displacements and the external loads are symmetric.
The elaboration of the virtual work equation not only shows that a consistent set of
internal shell quantities have been chosen, but it also gives, in a simple and elegant
manner, the natural boundary conditions that complement the three sets of equations.
Consider an elastic body under a specified body force vector and a boundary surface
force vector. For a thin shell the body force vector and that part of the boundary surface
force vector that acts upon the inner and outer surface of the element are replaced by
the statically equivalent surface force vector p that acts upon the reference surface as
described in subsection 2.2.5. The other part of the boundary surface force vector,
denoted by the edge force vector f , acts on the boundary surfaces that are
perpendicular to the reference surface and thus collects the resultants of the edge
stresses. Generally, these edge forces are known over a portion of the boundary surface
(which is denoted by S f ) while the displacements are known over the remainder of the
boundary surface (which is denoted by Su ) so that the total boundary surface S of the
shell body is S = S f + Su .
For the shell body we assume that in a certain state it is in equilibrium under the
specified force vectors p and f . Having a displacement vector u at equilibrium, we
consider an arbitrary displacement vector u + δu . Note that over the portion Su of the
surface, δu must vanish since u is prescribed there. Over the rest of the surface, δu is
arbitrary and these components are known as the virtual displacements.
For a steady process as is considered here, the kinetic energy is equal to zero and
in the absence of non-conservative loads (which means that the total amount of energy
is constant) the principle of minimum potential energy can be applied which is
formulated by
δE p = 0 ,
E p = minimum
(2.17)
The quantity E p is known as the potential energy and is given by
E p = Es − W p − W f
Herein is Es the strain energy (or deformation energy), which is present in the body as
potential energy, WP is the work done by the surface force vector p and WF is the
21
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
work done by the edge force vector f , respectively. Equation (2.17) thus states that at
equilibrium the value of the potential energy is a minimum and that this value is not
changed by a virtual variation to a perturbed equilibrium state. In other words, the
work done by the external surface and body forces along the virtual displacements is
balanced by the work done by the internal stresses along the virtual strains. This
principle is known as the principle of virtual work and for this case given by the
following equation that is referred to as the virtual work equation
δE p = δEs − δW p − δW f = 0
(2.18)
The strain energy is defined in terms of a strain energy density function as
Es = ∫ Es′dV
V
where the differential volume dV is given by (2.6) and the strain energy density
function, which represents work per unit volume, without the effect of thermal
expansion, is given by
Es′ = ∫ σij deij
,
e
( i, j ) = (1, 2, ζ )
Inversely the last relation leads to the conclusion that
σij =
∂Es′
∂eij
which gives as an expression for the virtual work per unit volume by the stresses σij
δEs′ =
∂Es′
δeij = σij δeij
∂eij
The variation of the strain energy along the virtual strains, which are in correspondence
with the strain description (2.10), is thus given by
δEs = ∫ ∫ ∫ ( σ11δe11 + σ22δe22 + σ12δe12 + σ21δe21
ζ ξ 2 ξ1
+2σ1ζ δe1ζ + 2σ2 ζ δe2 ζ ) α1α 2 (1 + ζ R1 )(1 + ζ R2 ) d ξ1d ξ 2d ζ
In the first term of this expression δe11 can be related to quantities of the reference
surface making use of expression (2.10), which for δe11 results in
δe11 =
1
( δε11 + ζδβ11 )
1 + ζ R1
and substitution of this variation the first term gives
∫ ∫ ∫ ( σ δε
1
11
+ ζσ1δβ11 ) α1α 2 (1 + ζ R2 ) d ξ1d ξ2 d ζ
ζ ξ2 ξ1
By making use of the definitions (2.14) of the stress resultants and stress couples the
integrations can be easily carried out and proceeding in the same manner, we obtain for
the expression of the variation of the strain energy in reference surface quantities
δEs =
∫ ∫ (n
δε11 + m11δβ11 + n12δε12 + m12δβ12 + v1δ2ε1ζ
11
ξ2 ξ1
+ n21δε21 + m21δβ21 + n22δε 22 + m22δβ22 + v2δ2ε 2ζ ) α1α 2 d ξ1d ξ 2
(2.19)
This result assures that the stress resultants and stress couples of (2.14) correspond to
the deformation quantities of (2.11).
22
2 General part on shell theory
The work done by the surface force vector p on the reference surface must be equal to
W p = ∫ ( p ⋅ u ) dV = ∫ ∫ ∫ ( PU
1 1 + PU
2 2 + PU
ζ ζ ) α1α 2 (1 + ζ R1 )(1 + ζ R2 ) d ξ1d ξ 2 d ζ
ζ ξ2 ξ1
V
where the force Pi is a component of the external force vector per unit volume at a
specific point within the shell space. By making use of the definition (2.9) of the
displacement components, the expression becomes
W p = ∫ ∫ ∫  P1 ( u1 + ζϕ1 ) + P2 ( u2 + ζϕ2 ) + Pζuζ  α1α 2 (1 + ζ R1 )(1 + ζ R2 ) d ξ1d ξ 2d ζ
ζ ξ2 ξ1
On the reference surface, the resultants p1 , p2 and pζ and the couples m1 and m2 of
the body forces are now introduced in the same manner as was done in subsection 2.2.4
for the stress resultants and the stress couples. These resultants are thus expressed as
p1 = ∫ P1 (1 + ζ R1 )(1 + ζ R2 ) d ζ
m1 = ∫ P1 (1 + ζ R1 )(1 + ζ R2 ) ζd ζ
p2 = ∫ P2 (1 + ζ R1 )(1 + ζ R2 ) d ζ
m2 = ∫ P2 (1 + ζ R1 )(1 + ζ R2 ) ζd ζ
ζ
ζ
ζ
ζ
pζ = ∫ Pζ (1 + ζ R1 )(1 + ζ R2 ) d ζ
ζ
where surface forces on the outer and inner surface can easily be added. The integral
for the work done by the surface force vector p hereby becomes
Wp =
∫ ∫( pu
+ p2u2 + pζuζ + m1ϕ1 + m2ϕ2 ) α1α 2 d ξ1d ξ2
1 1
ξ 2 ξ1
The work done by these forces along the virtual displacements is thus given by
δWp = ∫ ∫ ( p1δu1 + p2δu2 + pζuζ + m1δϕ1 + m2δϕ2 ) α1α 2 d ξ1d ξ 2
(2.20)
ξ2 ξ1
The work done by the edge force vector f on the boundary lines of the boundary
surface S is equal to the work done by the components Fi of the external force vector
per unit area of the boundary surface. The components of the external force vector act
in the same direction as the displacement components. Hence, the work done by the
edge force vector on the two pairs of edges of constant ξ1 and ξ 2 , respectively, which
are denoted by ξ1(1) , ξ1( 2) , ξ(21) and ξ(22) , can be written as
W f = ∫ f ⋅ udS =
S
∫ ∫ ( FU
1
ξ2 ζ
1
+ F2U 2 + FζU ζ ) α 2 (1 + ζ R2 ) d ζd ξ2
(1)
( 2)
ξ1 =ξ1 , ξ1
+ ∫ ∫ ( FU
1 1 + F2U 2 + FζU ζ ) α1 (1 + ζ R1 ) d ζ d ξ1
ξ1 ζ
(1)
( 2)
ξ2 =ξ2 , ξ 2
Introducing the definition (2.9) of the displacement components the integrals become
Wf =
∫ ∫ ( F (u
1
ξ2 ζ
1
+ ζϕ1 ) + F2 ( u2 + ζϕ2 ) + Fζ uζ ) α 2 (1 + ζ R2 ) d ζd ξ 2
(1)
+ ∫ ∫ ( F1 ( u1 + ζϕ1 ) + F2 ( u2 + ζϕ2 ) + Fζ uζ ) α1 (1 + ζ R1 ) d ζd ξ1
ξ1 ζ
( 2)
ξ1 =ξ1 , ξ1
(1)
( 2)
ξ 2 =ξ 2 , ξ 2
23
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The displacement components are referred to the reference surface and thus
independent of the coordinate ζ and by making use of the definitions (2.14) of the
stress resultants and stress couples the integrations with respect to dζ can be easily
carried out. Introducing the edge forces f1 , f 2 and f ζ and edge couples t1 and t2 , our
final result is obtained as
∫( f u
Wf =
+ f 2u2 + f ζ uζ + t1ϕ1 + t2ϕ2 ) α 2 d ξ2
1 1
ξ2
(1)
( 2)
ξ1 =ξ1 , ξ1
+ ∫ ( f1u1 + f 2u2 + f ζ uζ + t1ϕ1 + t2ϕ2 ) α1d ξ1
ξ1
(1)
( 2)
ξ 2 =ξ 2 , ξ2
The work done by the edge forces and edge couples along the virtual displacements is
thus given by
∫ ( f δu
δW f =
1
+ f 2δu2 + f ζ δuζ + t1δϕ1 + t2δϕ2 ) α 2 d ξ2
1
ξ2
+ ∫ ( f1δu1 + f 2δu2 + f ζ δuζ + t1δϕ1 + t2δϕ2 ) α1d ξ1
ξ1
(1)
( 2)
ξ1 =ξ1 , ξ1
(2.21)
(1)
( 2)
ξ2 =ξ2 , ξ 2
All terms of the virtual work equation (2.18) have now been given either in virtual
strains (for the internal work quantities) or in virtual displacements (for the external
work quantities) and all terms are referred to by quantities of the reference surface. A
natural step is to obtain the internal work only in terms of the virtual displacements to
be able to elaborate further towards the equilibrium equations and the natural boundary
conditions.
The first term from expression (2.19) is, after substitution of kinematical relation
(2.11) and noting that derivative operations and variation are commutative, given by
∫ ∫ (n
 1 ∂δu1 δu2 ∂α1 δuζ 
α1α 2 ) 
+
+
 d ξ1d ξ 2
 α1 ∂ξ1 α1α 2 ∂ξ 2 R1 
11
ξ 2 ξ1
By integration by parts the derivatives of the virtual displacements are removed and we
obtain for the first term
∫ [n
ξ2
( 2)
∂  n11α1α 2 

 δu1d ξ1d ξ 2
∂ξ1  α1 
ξ 2 ξ1
α 2δu1 ]ξ1 =ξ1(1) d ξ2 − ∫ ∫
11
ξ =ξ
1
1
Here the parameters α1α 2 have deliberately not been cancelled where appropriate for
later purposes of evaluation. Proceeding in the same manner for all terms of (2.19) we
obtain
24
2 General part on shell theory
ξ1 =ξ
( 2)
1
δEs = ∫ ( n11δu1 + n12δu2 + v1δuζ + m11δϕ1 + m12δϕ2 ) α 2  (1) d ξ2
ξ1 =ξ1
ξ2
+ ∫ ( n21δu1 + n22δu2 + v2δuζ + m21δϕ1 + m22δϕ2 ) α1 
ξ1
( 2)
ξ 2 =ξ2
(1)
ξ 2 =ξ2
d ξ1
  ∂  n11α1α 2  n12α1α 2 ∂α1
∂  n21α1α 2  n22α1α 2 ∂α 2 v1α1α 2 
− ∫ ∫ 
+
+
 δu1

+

−
α1α 2 ∂ξ2 ∂ξ2  α 2 
α1α 2 ∂ξ1
R1 
ξ 2 ξ1 
  ∂ξ1  α1 
 ∂  n22α1α 2  n21α1α 2 ∂α 2
∂  n12α1α 2  n11α1α 2 ∂α1 v2α1α 2 
+
+
+
 δu2

+

−
α1α 2 ∂ξ1 ∂ξ1  α1  α1α 2 ∂ξ 2
R2 
 ∂ξ2  α 2 
 ∂  v1α1α 2  ∂  v2α1α 2  n11α1α 2 n22α1α 2 
+
−
 δuζ

+

−
R1
R2 
 ∂ξ1  α1  ∂ξ 2  α 2 
(2.22)
 ∂  m11α1α 2  m12α1α 2 ∂α1

∂  m21α1α 2  m22α1α 2 ∂α 2
+
+
− v1α1α 2  δϕ1

+

−
α1α 2 ∂ξ2 ∂ξ2  α 2 
α1α 2 ∂ξ1
 ∂ξ1  α1 

 ∂  m22α1α 2  m21α1α 2 ∂α 2

∂  m12α1α 2  m11α1α 2 ∂α1
+
+
− v2α1α 2  δϕ2

+

−
α1α 2 ∂ξ1 ∂ξ1  α1 
α1α 2 ∂ξ 2
 ∂ξ2  α 2 



m αα
m αα 
+  n12α1α 2 − n21α1α 2 + 12 1 2 − 21 1 2  δϖ n  d ξ1d ξ 2
R
R

1
2


If the sum of all variations (2.18) is set equal to zero, two sets of equations are
obtained, i.e., one for the double integral over the reference surface and one for the
integral over the boundary lines. As stated previously, the variations of the
displacements are arbitrary and non-zero, so the sets of equations can only vanish if
each coefficient of the variations vanishes individually. From the set for the double
integral over the reference surface of (2.22) and (2.20), six equilibrium equations are
obtained, which read
∂  n11α1α 2  ∂  n21α1α 2  n12α1α 2 ∂α1 n22α1α 2 ∂α 2 v1α1α 2
−
+
+ p1α1α 2 = 0

+

+
∂ξ1  α1  ∂ξ 2  α 2 
α1α 2 ∂ξ 2
α1α 2 ∂ξ1
R1
∂  n22α1α 2  ∂  n12α1α 2  n21α1α 2 ∂α 2 n11α1α 2 ∂α1 v2α1α 2
−
+
+ p2α1α 2 = 0

+

+
∂ξ2  α 2  ∂ξ1  α1 
α1α 2 ∂ξ1
α1α 2 ∂ξ2
R2
∂  v1α1α 2  ∂  v2α1α 2  n11α1α 2 n22α1α 2
−
+ pζ α1α 2 = 0

+

−
∂ξ1  α1  ∂ξ 2  α 2 
R1
R2
∂  m21α1α 2  ∂  m11α1α 2  m22α1α 2 ∂α 2 m12α1α 2 ∂α1
+
− v1α1α 2 + m1α1α 2 = 0

+

−
∂ξ2  α 2  ∂ξ1  α1 
α1α 2 ∂ξ1
α1α 2 ∂ξ 2
(2.23)
∂  m12α1α 2  ∂  m22α1α 2  m11α1α 2 ∂α1 m21α1α 2 ∂α 2
+
− v2α1α 2 + m2α1α 2 = 0

+

−
∂ξ1  α1  ∂ξ 2  α 2 
α1α 2 ∂ξ2
α1α 2 ∂ξ1
n12α1α 2 − n21α1α 2 +
m12α1α 2 m21α1α 2
−
=0
R1
R2
25
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The set for the integrals over the boundary lines of (2.22) and (2.21) reads
∫ {( f
1
}
− n11 ) δu1 + ( f 2 − n12 ) δu2 + ( f ζ − v1 ) δuζ + ( t1 − m11 ) δϕ1 + ( t2 − m12 ) δϕ2 α 2 d ξ2
ξ2
( 2)
ξ1 =ξ1
+∫
{( f + n
) δu1 + ( f2 + n12 ) δu2 + ( fζ + v1 ) δuζ + ( t1 + m11 ) δϕ1 + ( t2 + m12 ) δϕ2} α 2 d ξ2 ξ =ξ( )
+∫
{( f − n
) δu1 + ( f 2 − n22 ) δu2 + ( fζ − v2 ) δuζ + ( t1 − m21 ) δϕ1 + ( t2 − m22 ) δϕ2} α1d ξ1 ξ =ξ( )
+∫
{( f + n
) δu1 + ( f 2 + n22 ) δu2 + ( fζ + v2 ) δuζ + ( t1 + m21 ) δϕ1 + ( t2 + m22 ) δϕ2} α1d ξ1 ξ =ξ( )
1
11
1
ξ2
1
21
2
ξ1
1
ξ1
21
2
1
1
(2.24)
2
2
1
2
Obviously, the equilibrium equations (2.23) are identical to the set (2.15), which shows
that the principle of virtual work is a straightforward and elegant approach to obtain
consistent equation sets. Also, the fact that the sixth equilibrium equation is an identity
is reflected by the fact that it is obtained by a virtual rigid body rotation δωn about the
normal to the reference surface. The set (2.24) is the subject of the next section.
2.4 Boundary conditions
The set (2.24) is the complete set for the five independent displacements and states
that, per variation of each displacement over the surface S f , each of the internal stress
measures (three stress resultants and two stress couples) must be balanced by aligned
external stress measures. If u is prescribed over the surface Su , on which in
consequence the virtual displacement δu vanishes, each displacement must be equal to
the prescribed displacement at that surface. Hence, at each edge either the stress
resultant or the corresponding displacement must be equal to the known edge force or
prescribed edge displacement.
So, for the edge ξ1 = constant the boundary conditions are
f1 = −n11
or
f 2 = − n12
f ζ = −v1
or
or
u1 = u1 
f1 = n11 or u1 = u1 
u2 = u2 
f 2 = n12 or u2 = u2 


uζ = uζ  ξ1 = ξ1(1) and
f ζ = v1 or uζ = uζ  ξ1 = ξ1( 2)
t1 = − m11 or ϕ 1 = ϕ1 
t1 = m11 or ϕ 1 = ϕ1 


t2 = − m12 or ϕ 2 = ϕ2 
t2 = m12 or ϕ 2 = ϕ2 
and equally for the edges ξ 2 = constant the boundary conditions are
f1 = −n21
or
f1 = n21
or
f 2 = − n22
f ζ = −v2
or
or
f 2 = n22
f ζ = v2
or
or
t1 = − m21
t2 = − m22
u1 = u1 
u2 = u2 

uζ = uζ  ξ2 = ξ(21)
or ϕ 1 = ϕ1 

or ϕ 2 = ϕ2 
and
t1 = m21
t2 = m22
u1 = u1 
u2 = u2 

uζ = uζ  ξ 2 = ξ(22)
or ϕ 1 = ϕ1 

or ϕ 2 = ϕ2 
where the tilde indicates the prescribed edge displacement.
26
2 General part on shell theory
However, if the transverse shear strains are neglected, the rotations ϕi are no longer
independent displacements since then these are related to the displacements of the
reference surface by expression (2.12). The first integral of the set (2.24) after
substitution of expression (2.12) becomes
∫ {( f
− n11 ) δu1 + ( f 2 − n12 ) δu2 + ( f ζ − v1 ) δuζ
1
ξ2
u
 u2 1 ∂uζ  
1 ∂uζ 
+ ( t1 − m11 ) δ  1 −
 + ( t2 − m12 ) δ  −
  α 2 d ξ2
( 2)
 R1 α1 ∂ξ1 
 R2 α 2 ∂ξ 2  
ξ1 =ξ1
Integration by parts of the term originating from ϕ2 , gives
( 2)
ξ 2 =ξ 2
 ∂uζ 
= − ( t2 − m12 ) δuζ  ξ =ξ(1)
 d ξ2
2
2
( 2)
 ∂ξ 2 
ξ =ξ
∫ ( t2 − m12 ) δ  −
ξ2
1
1
( 2)
ξ1 =ξ1
∂
( t2 − m12 ) δuζ d ξ2
∂ξ2
( 2)
ξ2
ξ =ξ
+∫
1
1
and by substitution of this result and rearrangement per virtual displacement, the
boundary integral is rewritten to

f
∫ 

ξ2
1

− n11 +



1
1
( t1 − m11 )  δu1 +  f 2 − n12 + ( t2 − m12 )  δu2
R1
R
1



( 2)
ξ 2 =ξ 2


 1 ∂uζ  
1 ∂
+  f ζ − v1 +
− ( t2 − m12 ) δuζ  (1)
( t2 − m12 )  δuζ + ( t1 − m11 ) δ  −
 α 2d ξ2
ξ 2 =ξ 2
α 2 ∂ξ2
( 2)


 α1 ∂ξ1  
ξ1 =ξ1
( 2)
ξ1 =ξ1
or equally to

∫ ( f
ξ2

1


1
− n11 ) δu1 +  f 2 − n12 + ( t2 − m12 )  δu2
R1




u
1 ∂
1 ∂uζ  
+  f ζ − v1 +
( t2 − m12 )  δuζ + ( t1 − m11 ) δ  1 −
  α 2d ξ2
α 2 ∂ξ2
( 2)


 R1 α1 ∂ξ1  
ξ1 =ξ1
( 2)
ξ2 =ξ 2
− ( t2 − m12 ) δuζ  ξ
(1)
2 =ξ 2
( 2)
ξ1 =ξ1
where the virtual rotation δϕ1 has been employed according to expression (2.12).
In these expressions, we observe Kirchhoff’s effective shearing stress resultant as
the boundary edge resultant in the direction of uζ . But next to this it is noticed that also
the in-plane stress resultants are combinations of the internal stress quantities. Due to
the neglect of the transverse shear deformation, not five boundary edge resultants can
be assigned but only four.
Since the rotation of the normal ϕ1 seems to have a more physical interpretation in
describing the boundary conditions at a specific edge, this choice is described by the
second of the possible expressions for the boundary integral. Hence, the total set of
four stress quantities at the boundary corresponding to the four displacement measures
becomes for the edge ξ1 = ξ1( 2)
27
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
f1 = n11
f 2 = n12 +
f ζ = v1 +
1
m12
R2
1 ∂m12
α 2 ∂ξ 2
t1 = m11
or u1 = u1 

or u2 = u2 

( 2)
 ξ1 = ξ1
or uζ = uζ 


or ϕ 1 = ϕ1 
(2.25)
with the additional condition that
( 2)
[t2 ]ξ =ξ( )
ξ 2 =ξ 2
2
1
2
ξ =ξ
( 2)
ξ1 =ξ1
( 2)
= [ m12 ]ξ2 =ξ(21)
2
or
( 2)
2
ξ1 =ξ1
uζ = uζ
Similarly, for the edge ξ1 = ξ1(1)
f1 = − n11
or u1 = u1 



1
f 2 = −  n12 + m12  or u2 = u2 

R2



(1)
 ξ1 = ξ1

1 ∂m12 

f ζ = −  v1 +
 or uζ = uζ 
α 2 ∂ξ2 


t1 = − m11
or ϕ 1 = ϕ1 
(2.26)
with the additional condition that
( 2)
[t2 ]ξ =ξ( )
ξ 2 =ξ 2
2
1
2
ξ =ξ
( 2)
ξ1 =ξ1
( 2)
= − [ m12 ]ξ2 =ξ(21)
2
2
( 2)
ξ1 =ξ1
or
uζ = uζ
For the edges ξ 2 = constant , equivalent expressions can be obtained where the indices
denoting the parametric lines are interchanged where applicable.
If the edge curve on one of the two coordinate lines is a closed curve, the
additional condition is identically satisfied and the boundary conditions on the other
coordinate line are replaced by continuity conditions for all quantities. Otherwise, the
additional conditions describe four point loads at the corners of the reference surface
with the magnitude Rn ( ξ1 , ξ2 ) = m12 ( ξ1 , ξ2 ) + m21 ( ξ1 , ξ 2 ) .
2.5 Synthesis
In this section the kinematical relation and equilibrium relation are presented in such a
way that the well-known analogy between these relations becomes patently obvious.
The analogy comprises that a derivative in the differential operator matrix for the
kinematical relation is also present in the differential operator matrix for the
equilibrium relation, but then as the adjoint operator at the transposed position.
Next to this analogy, the corresponding matrices for the two relations needed to
obtain the constitutive relation are given in what follows. These two relations are the
strain distribution across the thickness of the shell expressed in the deformation
quantities of the reference surface and the relation of the stress resultants and stress
couples at the reference surface as expressions of the stress distribution across the
thickness.
28
2 General part on shell theory
The following four vectors are used, where u is the displacement vector, e is the
strain vector, s is the stress vector and p is the load vector.
u = u1 u2
uζ
ϕ1 ϕ2
ωn 
e =  ε11 ε 22
ε12
ε 21 2ε1ζ
s = [ n11 n22
n12
n21 v1 v2
p =  p1
pζ
m1 m2
p2
T
2ε 2 ζ
0 
β11 β22 β12 β21 
m11 m22
m12
T
m21 ]
T
T
The kinematical relation (2.11) is notated symbolically by e = Bu where B is the
“differential operator matrix” that relates the displacement vector u to the strain vector
e . By using temporarily the sign convention that an operator within the curled brackets
does not apply to the vector on which the operator matrix acts, the kinematical relation
is presented by

1 ∂
1  ∂α1 
1



α
∂ξ
α
α
∂ξ
R
1
1
1 2 
2 
1


1 ∂
1
 1  ∂α 2 
 α1α 2  ∂ξ1 
α 2 ∂ξ 2
R2

 1  ∂α1 
1 ∂
0


−
α α ∂ξ
α1 ∂ξ1
 ε11   1 2  2 
ε  
1 ∂
1  ∂α 2 
−
0
 22  


α1α 2  ∂ξ1 
 ε12   α 2 ∂ξ 2

 
1
1 ∂
 ε 21  
−
0
 2ε  
α
R
1
1 ∂ξ1
 1ζ  = 
 2ε 2ζ  
1
1 ∂
0
−
β  
α
R
11

2
2 ∂ξ 2


 β22  

 
0
0
0
 β12  

 β21 

 
0
0
0




0
0
0



0
0
0



0 


0
0
0 



0
0
−1 


0
0
1 
u 
 1 
 u
1
0
0  2 
u 
 ζ 
  ϕ1 
0
1
0  
  ϕ2 
  ωn 
 
1 ∂
1  ∂α1 
0 


α1 ∂ξ1
α1α 2  ∂ξ 2 


1  ∂α 2 
1 ∂
0 


α1α 2  ∂ξ1 
α 2 ∂ξ2


1  ∂α1 
1 ∂
1
−
−


α1α 2  ∂ξ 2 
α1 ∂ξ1
R1 

1 ∂
1  ∂α 2 
1 
−



α 2 ∂ξ2
α1α 2  ∂ξ1  R2 
0
0
29
where
where
(..) → n11α1α 2
(..) → u1
30
To obtain corresponding differential operators in the two
relations the stress resultants and stress couples are thus
multiplied by the factor α1α 2 .
∂  .. 
 
∂ξ1  α1 
1 ∂ (..)
α1 ∂ξ1
B11∗ (..) = −
B11 (..) =
B∗s = p , which are given by
relation e = Bu that corresponds to the term B11∗ in the relation
The analogy is observed by, for example, the term B11 in the
 ∂  1 
1  ∂α 2 
1  ∂α1 
∂  1 
1
−
0
−

 −

 −
 
 
∂ξ
α
α
α
∂ξ
α
α
∂ξ
∂ξ
α
R








1
1
1
2
1
1
2
2
2
2
1


∂  1 
1  ∂α 2 
1
 1  ∂α1  − ∂  1 
−
−
0
−


 
 α1α 2  ∂ξ2 
∂ξ 2  α 2 
∂ξ1  α1 
α1α 2  ∂ξ1 
R2


1
1
∂  1 
∂  1 
−
0
0
  −
 

R
R
∂ξ
α
∂ξ


1
2
1
1
2  α2 



0
0
0
0
1
0



0
0
0
0
0
1



0
0
−1
1
0
0


The equilibrium relation (2.15) is notated symbolically by
B∗s = p where s is the stress vector, p is the load vector, and
B∗ is the “differential operator matrix” that relates the two
0
0
Since by definition the operator L∗ is the adjoint of the
operator L if for the inner product of two vectors, say
u • v = v • u the relation holds that u • Lv = L∗u • v . Hence, per
partial integration the sign changes and the sequence of the
terms switches round. This is exactly what is observed in
section 2.3 when deriving the equilibrium equations from the
expressions for the work done by the virtual strains. Hereby
the rule of thumb (if e = Bu and B∗s = p ) is that an even
differential operator does not change sign and that an uneven
differential operator does change sign and that, for a
curvilinear coordinate system, the sequence of the terms in the
operator changes as indicated.
0


  n11α1α 2 


  n22α1α 2 
0
0
0
0
  n α α   p1α1α 2 
  12 1 2  

  n21α1α 2   p2α1α 2 
0
0
0
0
 v α α   p α α 
 1 1 2  =  ζ 1 2
∂  1 
1  ∂α 2 
1  ∂α1 
∂  1    v2α1α 2   m1α1α 2 
−
 

 −

 −

 
  
∂ξ1  α1  α1α 2  ∂ξ1 
α1α 2  ∂ξ 2 
∂ξ2  α 2    m11α1α 2   m2α1α 2 
 m α α   0 

1  ∂α1 
∂  1 
∂  1 
1  ∂α 2   22 1 2  
−
−

 −

  m12α1α 2 
 
 
α1α 2  ∂ξ 2 
∂ξ2  α 2 
∂ξ1  α1 
α1α 2  ∂ξ1 
 m α α 
  21 1 2 
1
1
0
0
−

R1
R2

0
quantities. Using the temporary sign convention, this relation
is presented by
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
2 General part on shell theory
One of the two relations needed to obtain the constitutive relation is the strain
distribution across the thickness of the shell expressed in the deformation quantities of
the reference surface. This strain distribution is given by relation (2.10) and reads
notated in matrix form
 e11  
e  
 22  
 e12  

=
 e21  
 2e  
 1ζ  
 2e2 ζ  
µ1
0
0
0
0
0
ζµ1
0
0
0
0
0
µ2
0
0
0
µ1
0
0
0
µ2
0
0
0
0
0
0
0
0
0
ζµ 2
0
0
0
ζµ1
0
0
0
0
0
0
0
0
0
µ1
0
0
µ2
0
0
0
0
0
0
 ε11 
ε 
 22 
0   ε12 


0   ε21 
0   2ε1ζ 


ζµ 2   2ε 2 ζ 
0   β11 

0   β22 


 β12 
 β21 


where the factors µ1 and µ 2 that account for the curvature of the parametric lines are
µ1 =
1
ζ
1+
R1
µ2 =
;
1
1+
ζ
R2
The other one of the two relations relates the stress resultants and stress couples at the
reference surface to the stress distribution across the thickness. These integrals are
given by relation (2.14) and reads notated in matrix form
0
 n11 
 µ1
n 
 0
µ2
 22 

 n12 
 0
0
 

0
 n21 
 0
v 
 0
0
 1  = ∫
0
 v2  ζ  0
m 
ζµ
0
 11 
 1
 m22 
 0 ζµ 2
 

0
 m12 
 0
 m21 
 0
0
0
0
0
0
0
0
µ1
0
0
µ2
0
0
0
0
0
0
µ1
0
0
0
0
0
0
0
ζµ1 0
0 ζµ 2
0
0
0
0 
0   σ11 
 
0   σ22 
0   σ12  
ζ 
ζ 
    1 + 1 +  d ζ
µ 2   σ 21  R1  R2 
0   σ1ζ 
 
0   σ2 ζ 

0
0 
where the same factors are used to account for the curvature.
The fact that this matrix for the stress distribution is the transpose of the one
presented above for the strain distribution guarantees the symmetry of the constitutive
relation between the stress vector s and the strain vector e , which can be symbolically
presented by s = De where D is the rigidity matrix.
31
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
2.6 Analysis by former authors
2.6.1 General
Love [13] was the first to derive a set of basic equations which describe the behaviour
of a thin elastic shell. Two drawbacks of the equations resulting from Love’s first
approximation exist:
a. The torsion does not meet the requirements that strains resulting from rigid
body motion must vanish; and
b. The expression for the corresponding “twisting moment” is described by
neglecting certain non-negligible term.
Love himself [17] already attempted to improve his pioneering work and many others
attempted likewise. Flügge [14] for the circular cylindrical shell and independently
Byrne in lines of curvature coordinates tried to obtain a better approximation as a first
order theory by attempting a more careful omission of terms of higher order by using
the series expansion of quotients of the type 1 (1 + ζ Rα ) ( α = 1, 2 ) , but under the same
assumptions. However, the improvement of Love’s first approximation by retaining
2
terms as ( ζ Rα ) in the series expansion leads to correction terms that are of the same
order as the terms that would be introduced if the shear deformation were taken into
account (for a homogeneous, isotropic material). This is shown by, for example,
Reissner [18] and Koiter [19] and Reissner therefore states that if corrections to Love’s
first approximation were desired these should be obtained by simultaneously
abandoning all assumptions except the second of the assumptions stated in subsection
2.2.1.
Despite the fact that Love’s first approximation is more than a century old and
many attempts have been made at arriving at a better theory, the subject is still open to
discussion. The difference in most theories lies in the simplifications made to obtain a
good approximation of the constitutive relation within the assumptions of the theory.
A full set of equations for the static response of a linear elastic body to external
loading should at least possess a number of qualities, which are:
1. The constitutive relation must be symmetric.
2. The kinematical relation must be such that the strains resulting from rigid body
motion vanish.
3. The kinematical and equilibrium equation must be each others adjoint as
illustrated in section 2.5.
4. The boundary conditions must be congruent and in accordance with the
number of independent degrees of freedom.
These qualities will ensure that the resulting set of differential equations expressed in
the displacements is symmetric and that no further small errors are introduced than
those which are already introduced by the Kirchhoff-Love assumptions.
Next to the abovementioned qualities, it seems advantageous to switch over from
two resultants and two couples of the in-plane shearing stress σ12 to only one resultant
and one couple. This is possible by combining these stress quantities but a restriction
should be that the “sixth equilibrium equation” should not be violated, because this
would introduce new small errors.
32
2 General part on shell theory
To obtain these combinations, a reduction of the four corresponding strain
measures to one for the shear angle and one for the torsion might be a suitable first
step, which is deduced from the original description of subsection 2.2.3 by the
following.
By definition, the shear angle over the thickness of the shell is given by 2e12 = e12 + e21
where the respective angles are given by (2.10). Taking the sum of those gives
2e12 = e12 + e21 =
1
ζ
1+
R1
( ε12 + ζβ12 ) +
1
1+
ζ
R2
( ε 21 + ζβ21 )
in which the deformation quantities are given by expressions (2.11). These are thus
quantities that are referred to the reference surface. The expression can be rewritten to
2e12 =



ζ2 
ζ 
ε 
ζ 
ε 
ε12 + ε 21 ) + ζ 1 +  β12 + 21  + ζ  1 +  β 21 + 12   (2.27)
1−
(



ζ
ζ  R1R2 
R1 
R2  
 R2 
 R1 
1+ 1+
R1
R2
1
1
By means of comparing the term β12 +
ε 21
ε
with the term β21 + 12 by substituting the
R1
R2
respective expressions (2.11) it can be deduced that their difference is equal to

ε21  
ε12  1  ∂ϕ2 ϕ1 ∂α1  1  1 ∂u1
u ∂α 2 
−
− 2
 β12 +
 −  β 21 +
= 
+ 

R1  
R2  α1  ∂ξ1 α 2 ∂ξ2  α 2  R1 ∂ξ2 R1α1 ∂ξ1 

1  ∂ϕ1 ϕ2 ∂α 2  1  1 ∂u2
u ∂α1 
−
− 1

− 

α 2  ∂ξ2 α1 ∂ξ1  α1  R2 ∂ξ1 R2α 2 ∂ξ2 
1  ∂α 2ϕ2 ∂α1ϕ1 
1  ∂  α1  ∂  α 2  
=
−


+
 u1  −
 u2  
α1α 2  ∂ξ1
∂ξ2  α1α 2  ∂ξ2  R1  ∂ξ1  R2  
−
where, to perform the last step, use is made of the Codazzi conditions (2.3). By
rewriting and making use of expressions (2.11) for the transverse shearing strains, it is
obtained that

 ∂  α2

ε  
ε 
∂  α1
α1α 2  β12 + 21  −  β21 + 12   =
 u1 − α1ϕ1  −
 u2 − α 2 ϕ2 
R1  
R2   ∂ξ2  R1
 ∂ξ1  R2


=
∂uζ  ∂ 
∂uζ 
∂ 
 α1 2ε1ζ −
−
 α 2 2ε 2 ζ −

∂ξ2 
∂ξ1  ∂ξ1 
∂ξ2 
=
∂
∂
α1 2ε1ζ ) −
(
( α 2 2ε2ζ )
∂ξ2
∂ξ1
where the partial derivatives of uζ cancel out because the function for uζ and its
partial derivatives shall exist and must be continuous. The result hereby obtained
means that if the transverse shearing strains are neglected according to (2.12) and thus
equal to zero in the expression above the equality holds that
β12 +
ε 21
ε
= β 21 + 12
R1
R2
(2.28)
33
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The equality (2.28) is for example derived in different ways by Sanders [16] and
Novozhilov [20], but is worthwhile to note that this equality only holds if the
transverse shearing strains are set equal to zero.
Having derived the equality (2.28), a natural step seems to be the introduction of two
alternative deformation quantities
γ12 = ε12 + ε 21
ρ12 = β12 + β21 +
ε21 ε12
+
R1 R2
(2.29)
into the expression for the shearing strain angle (2.27), which is hereby written as
2e12 =


ζ2 
ζ
ζ  
1−
γ12 + ζ  1 +
+



 ρ12 
ζ
ζ  R1R2 
 2 R1 2 R2  
1+ 1+
R1
R2
1
1
(2.30)
Conveniently, in the new deformation quantity ρ12 , the rigid body rotation about the
normal to the reference surface is cancelled out. The alternative deformation quantities
expressed in the displacements are thus given by
γ12 =
α1 ∂  u1  α 2 ∂  u2 
 +
 
α 2 ∂ξ2  α1  α1 ∂ξ1  α 2 
α1 ∂  ϕ1  α 2 ∂  ϕ2 
1  ∂u1 u2 ∂α 2 
1  ∂u2 u1 ∂α1 
−
−
 +
 +

+


α 2 ∂ξ2  α1  α1 ∂ξ1  α 2  R1α 2  ∂ξ2 α1 ∂ξ1  R2α1  ∂ξ1 α 2 ∂ξ 2 
2  1 ∂u1
u ∂α1  2  1 ∂u2
u ∂α 2 
= 
− 1
− 2
+ 

R1  α 2 ∂ξ2 α1α 2 ∂ξ2  R2  α1 ∂ξ1 α1α 2 ∂ξ1 
ρ12 =
+
(2.31)
1  1 ∂α1 ∂uζ 1 ∂α 2 ∂uζ  1 ∂  1 ∂uζ  1 ∂  1 ∂uζ 
+

−

−


α1α 2  α1 ∂ξ2 ∂ξ1 α 2 ∂ξ1 ∂ξ 2  α1 ∂ξ1  α 2 ∂ξ2  α 2 ∂ξ2  α1 ∂ξ1 
The latter deformation quantity is often presented as
ρ12 =
α1 ∂  ϕ1  α 2 ∂  ϕ2 
1  ∂u1 u2 ∂α 2 
1  ∂u2 u1 ∂α1 
−
−
 +
 +

+


α 2 ∂ξ2  α1  α1 ∂ξ1  α 2  R1α 2  ∂ξ2 α1 ∂ξ1  R2α1  ∂ξ1 α 2 ∂ξ 2 
2  1 ∂u1
u ∂α1  2  1 ∂u2
u ∂α 2 
− 1
− 2

+ 

R1  α 2 ∂ξ2 α1α 2 ∂ξ2  R2  α1 ∂ξ1 α1α 2 ∂ξ1 
∂ 2u ζ 
2  1 ∂α1 ∂uζ 1 ∂α 2 ∂uζ
+
+
−


α1α 2  α1 ∂ξ2 ∂ξ1 α 2 ∂ξ1 ∂ξ 2 ∂ξ1∂ξ 2 
=
but this seems inconvenient for later purposes of presenting the synthesis of all sets of
equations similar to what is shown in section 2.5 and hence for arriving at expressions
for the boundary conditions and interpreting them as is to be shown in section 2.7.
34
2 General part on shell theory
If the transverse shearing strain are to be neglected, the expressions for changes of
rotation corresponding to the normal strain distribution are obtained by substituting
expressions (2.12) into (2.11), which for these changes of rotation results in
1 ∂  u1 
u2 ∂α1 1 ∂  1 ∂uζ 
1  ∂α1  ∂uζ
−


 +

−
α1 ∂ξ1  R1  R2α1α 2 ∂ξ2 α1 ∂ξ1  α1 ∂ξ1  α1α 22  ∂ξ2  ∂ξ 2
β11 =
u1 ∂α 2 1 ∂  u2  1 ∂  1 ∂uζ 
1  ∂α 2  ∂uζ
β22 =
+

 −

− 2 
R1α1α 2 ∂ξ1 α 2 ∂ξ 2  R2  α 2 ∂ξ2  α 2 ∂ξ2  α1 α 2  ∂ξ1  ∂ξ1
(2.32)
while the two corresponding strains of the reference surface as described by (2.11)
remain unaltered.
2.6.2
Flügge-Byrne
Although corrections to Love’s first approximation by retaining terms like ( ζ Rα ) will
be redundant, the constitutive relation by such an analysis is given here for later
reference. Flügge [14] roughly performed the following analysis while not making use
of the rigid body rotation in the expressions for the shear strain distribution.
To derive the expressions for the stress resultants and stress couples in terms of the
deformation quantities, the expressions for the strain distribution (2.10) across the
thickness of the shell are substituted into Hooke’s law (2.13). The resulting expressions
are substituted into the definitions (2.14), where expansions of the type
2
(1 + ζ
Rα ) 1 − ζ Rα + ( ζ Rα ) − ... , α = (1,2 )
−1
2
are carried out to perform the integrations. By omitting terms of higher order than the
2
third in the thickness, it is required that ( ζ Rα ) < 1 , which is not as restrictive as
Love’s thinness assumption. The additional terms introduced are hence meaningless as
long as the other assumptions are not relaxed simultaneously as argued in the previous
subsection.
However, integration of the stress distribution across the thickness expressed in the
deformation quantities results in
n11 =
Et 
t2  1
1 
ε 
ε + υε 22 ) +  −  β11 − 11  
2 ( 11
R1  
1 − υ 
12  R2 R1 
m11 =

 1
Et 3
1 
β + υβ 22 +  −  ε11 
2  11
12 (1 − υ ) 
 R2 R1  
n22 =
Et 
t2  1
1 
ε 
ε + υε11 ) +  −  β22 − 22  
2 ( 22
1 − υ 
12  R1 R2 
R2  
m22 =

1
Et 3
1  
β + υβ11 +  −  ε 22 
2  22
12 (1 − υ ) 
 R1 R2  
n12 =
Et 1 − ν 
t2  1
1 
ε12  
ε12 + ε 21 +  −  β12 −

2
1 − ν 2 
12  R2 R1 
R1  
m12 =
 1
Et 3
1− ν 
1 
β12 + β21 +  −  ε12 
2
12 (1 − ν ) 2 
 R2 R1  
n21 =
Et 1 − ν 
t2  1
1 
ε 21  
ε12 + ε 21 +  −  β21 −

2
1 − ν 2 
12  R1 R2 
R2  
m21 =
1
Et 3
1− ν 
1  
β12 + β21 +  −  ε 21 
2
12 (1 − ν ) 2 
 R1 R2  
where it is interesting to note that the stress resultants n12 and n21 and stress couples
m12 and m21 fulfil the sixth equilibrium equation.
These resulting equations are elegant, symmetric and the deformation measures
fulfil the requirement that these vanish for a rigid body motion, but based on the
observation mentioned in subsection 2.6.1 these equations are rejected.
35
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
2.6.3 Koiter-Sanders
Sanders [16] and Koiter [19] use identical strain measures to Love’s first
approximation. Using (2.11) but neglecting terms ζ Rα in comparison to unity for the
expressions (2.10) of the strain distribution, gives
e11 = ε11 + ζβ11
e22 = ε22 + ζβ22
2e12 = γ12 + ζρ12
where ρ12 = β12 + β21 is slightly different from the expression for the torsion as defined
above.
Although these deformation strain measures meet the requirement, this
introduction means that, Sanders finds, for example for a cylinder with radius a and
coordinates ξ1 = x and ξ1 = θ of the reference surface and ζ = z for the normal
direction (which are introduced without further explanation), for the torsion
ρ12 = β xθ + βθx =
1 ∂uθ 2 ∂ 2u z ϖ n
−
+
a ∂x a ∂x∂θ a
where
1  ∂u 1 ∂u x 
ϖn =  θ −

2  ∂x a ∂θ 
which is obtained from the equality by definition ε xθ = εθx .
The expression for the torsion hence becomes
ρ12 = −
1 1 ∂u x 3 1 ∂uθ 2 ∂ 2u z
+
−
2 a 2 ∂θ 2 a ∂x a ∂x∂θ
So, a rigid body rotation about the normal to the middle surface is needed to obtain a
strain measure for the torsion that meets the requirements of describing rigid body
modes without strain. This is rather contradictory and results in unfamiliar equilibrium
equations that correspond to the expressions of the kinematical relation.
Before neglecting terms ζ Rα in comparison to unity, Sanders derives the equilibrium
equation from his deformation measures with the aid of the principle of virtual work.
By using the equality (2.28) and the equality ε12 = ε21 he then suggest the introduction
of the shearing stress resultant and couple as
1
1 1
1
( n12 + n21 ) +  −  ( m12 − m21 )
2
4  R2 R1 
1
m12 = ( m12 + m21 )
2
n12 =
and then neglects the contribution of the stress couples in the stress resultant n12 .
Hereby, the “sixth equilibrium equation” is no longer satisfied but these small errors
are accepted by Sanders.
36
2 General part on shell theory
Substitution of the strain distribution suggested by Sanders and Koiter into (2.14), but
also here neglecting terms ζ Rα in comparison to unity, yields after integration
n11 =
Et
( ε11 + υε 22 )
1 − υ2
m11 =
Et 3
( β11 + υβ22 )
12 (1 − υ2 )
n22 =
Et
( ε 22 + υε11 )
1 − υ2
m22 =
Et 3
( β22 + υβ11 )
12 (1 − υ2 )
n12 = n21 =
Et 1 − υ
( ε12 + ε21 )
1 − υ2 2
m12 = m21 =
Et 3
1− υ
( β12 + β21 )
2
12 (1 − υ ) 2
which is an elegant approximation but the expression for the torsion remains
questionable.
2.6.4 Novozhilov
Novozhilov [20] introduces the following terms on the basis of the identity of the sixth
equilibrium equation (2.15)
n12 = n12 −
m12 =
m21
m
= n21 − 12
R2
R1
1
( m12 + m21 )
2
(2.33)
and derives the deformation measures introduced in subsection 2.6.1, which read
γ12 = ε12 + ε 21
1
ε
ε
ρ12 = β12 + 21 = β 21 + 12
2
R1
R2
By rewriting the corresponding part of the virtual work equation to
n12δε12 + n21δε21 + m12δβ12 + m21δβ21




ε 
ε 
m 
m 
=  n12 − 21  δε12 +  n21 − 12  δε21 + m12δ  β12 + 21  + m21δ  β 21 + 12 
R
R
R
R2 




2 
1 
1 
ρ 
= n12δ ( ε12 + ε 21 ) + ( m12 + m21 ) δ  12 
 2 
= n12δγ12 + m12δρ12
(2.34)
it is shown that the stress resultant and the stress couple correspond to the chosen
deformation measures for the shearing strain and the torsion. So all drawbacks are
overcome and six strain measures and six stress resultants are formulated to describe
the equations.
The symmetry of the constitutive relation is also guaranteed, which is shown by
the two following relations, where the second “curvature matrix” is the transpose of the
first.
37
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
 1

ζ
1 +
R1

 e11  
e = 0
 22  
 2e12  


 0


0
1
0
ζ
1+
R2
 1

ζ
 1+
R1


 0



 n11 

n 
 0
22
 

 n12 

  = ∫
 m11  ζ ζ 1
m 
ζ

 22 
 1+ R
1
 m12 


 0



 0



ζ
0
ζ
0
ζ
R2
0
0
1
1+
0
ζ
R2






0



1
1 
ζ2  
1−
 
ζ
ζ 
1 + 1 +  R1R2    σ11 
R1
R2
  
ζ 
ζ 
  σ22   1 + 1 +  d ζ
  σ   R1  R2 
0
  12 



0




1
1 
ζ
ζ 
1+
ζ
+



ζ
ζ
1 + 1 +  2 R1 2 R2  
R1
R2

0
1
1+
0

  ε11 
 
  ε 22 
 γ 
  12 
0
  β11 
 β 
  22 
1
1 
ζ
ζ    ρ12 
ζ
1+
+

ζ
ζ 
1 + 1 +  2 R1 2 R2  
R1
R2

0
1
1+
1 
ζ2 
1−
ζ
ζ  R1R2 
1+ 1+
R1
R2
ζ
0
ζ
1+
R1
0
1
0
1
ζ
R2
0
Using the strain measures introduced above, Novozhilov derived the corresponding
equations of equilibrium using the principle of virtual work. Strangely, he did not
derive the natural boundary conditions via this principle, but on the basis of
geometrical considerations. However, he arrives at four conditions, which are equal to
the conditions (2.25) and (2.26).
Using (2.33) and (2.15) for v1 where m1 is set equal to zero since the transverse
shear deformation is neglected, he rewrites the second and third condition for the
boundary stress resultants at the edge ξ1 = ξ1( 2) to
f 2 = n12 +
1
2
m12 = n12 + m12
R2
R2

1 ∂m12
1  ∂α1m12 ∂α 2m11 ∂α 2
f ζ = v1 +
=
+
−
m22 
2
α 2 ∂ξ 2 α1α 2  ∂ξ 2
∂ξ1
∂ξ1

38
(2.35)
2 General part on shell theory
where especially the second expression seems rather peculiar, which is due to the use
of the original expression for v1 .
The boundary conditions resulting from the application of the principle of virtual
work is given in section 2.7 where the correctness of the expressions derived above is
shown.
2.6.5 Authors on nonlinear, shallow and cylindrical shell equations
Donnell [21] derived an approximate shell equation for buckling of cylinders. Amongst
many others, Vlasov [22] and Mushtari [23] generalised the derivation to large
deflections of thin elastic shells of arbitrary curvature, which were also reproduced by
Donnell [24].
The nonlinear Donnell-Mushtari-Vlasov theory (DMV-theory) for large
deflections of isotropic thin elastic shells holds for arbitrary curvature. The equilibrium
equations for this theory are given by two coupled nonlinear fourth-order partial
differential equations, which contain two independent variables (the coordinates on the
middle surface), and two dependent variables, which are the displacement normal to
the middle surface and Airy’s stress function. As an approximation of this theory, a
special case can be obtained: the equilibrium equations for so-called shallow shells.
According to Flügge [14], Marguerre [25] formulated a general theory of shallow
shells, which has been further developed and applied to many problems in various
papers by E. Reissner. Hence, the equations of the shallow shell theory are also known
as Marguerre’s equations for large deflection of plates with small initial curvature,
which follow from Marguerre’s shell theory. As a special case, for plates with no
curvature, the well-known Von Kármán equations for large deflection of plates are
identically described and therefore other authors refer to the shallow shell equations as
the (generalised) Von Kármán-Donnell equations.
The first rigorous development of the theory of circular cylindrical shells is presented
by Flügge (in a first edition of [14]). The equilibrium equations for this theory are
given by three coupled partial differential equations, which contain two independent
variables (the coordinates on the middle surface), and three dependent variables, which
are the two displacements on the middle surface and the displacement normal to the
middle surface. As stated in subsection 2.6.1, Flügge’s approach in retaining secondorder terms, which do not exceed the accuracy of the initial assumptions, is rather
meaningless. However, the solution to Flügge’s equations is often used as a standard to
which approximated or simplified solutions are compared.
As an approximation to Flügge’s equations, a special case are the equilibrium
equations obtained by the so-called semi-membrane concept. According to Zingoni
[26], Finsterwalder [27] and [27] was of one of the first to derive such an approximate
bending theory by neglecting the flexural rigidity in axial direction and torsional
rigidity. Derived shortly after Flügge, e.g., Schorer [28] assumed that not only the
flexural rigidity in axial direction and torsional rigidity may be equated to zero, but
also that the circumferential strain and the shear strain are both small in comparison to
the axial strain. However, as observed by Moe [29] by studying several characteristic
equations, the applicability of such equations is limited to long shells, which means
that the length in axial direction is sufficiently longer than the radius. As mentioned,
39
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Donnell [21] derived approximate equations for buckling of cylinders, which are a
simplified form of Flügge’s equations and are more general than Schorer’s
approximate equations. Von Kármán, together with Tsien, [30], extended the theory for
non-linear behaviour to study wrinkling of cylindrical shells. Jenkins [31] was the first
to apply a similar equation to study the static behaviour of shell roofs. Hence, the
abovementioned approach is sometimes referred to as the Donnell-Von KármánJenkins theory (DKJ-theory). Based on this theory, Bouma and Van Koten [32],
derived exact and approximate solutions for cylindrical shells with circular edges. Such
a solution procedure is mainly possible due to the simplifications that form the basis of
the DKJ-theory. Because of its reduced form, this theory, often referred to as the
simplified Donnell theory, remains to be the most popular.
The equations of Flügge and Donnell for cylindrical shells are reproduced by
Hoefakker [33]. Herein, it is shown that the Donnell theory does not fulfil the
requirement that zero strains occur at rigid body motion. Since the shallow shell theory
is derived using the same simplifications as Donnell’s theory of cylindrical shells, the
shallow shell theory does also not fulfil the requirement that strains resulting from rigid
body motion vanish.
To interpret this drawback of Donnell’s equation, Hoff [34] studied the accuracy
by comparing the roots of Donnell’s equation with the ones obtained by Flügge’s
equation. As shown by Hoff, significant errors can occur if Donnell’s equation is used.
Based on reasoning and judgement, Morley [35] suggested an equation, which was
later derived more explicitly by Koiter [36] and Niordson [37], that overcomes both the
completeness of Flügge’s approach in retaining second-order terms, which do not
exceed the accuracy of the initial assumptions, and the inaccuracy of Donnell’s
simplifications in its inability to describe rigid-body modes, but preserves its elegance
and simplicity. Morley compared the roots obtained from this equation with
approximated roots obtained from Flügge’s equation and with the roots obtained from
Donnell’s equation. Morley showed that, for the solutions that describe an edge
disturbance originating at a straight edge, Donnell’s equation leads to significant errors
if the straight edge of the cylindrical shell is long in comparison to its radius, while
Morley’s equation is in agreement with Flügge’s equation for any significant shell
geometry. For the solutions that describe an edge disturbance originating at a closed
curved edge, Morley showed that, both Morley’s and Donnell’s equation closely agree
with Flügge’s. However, roots for deformation modes with only one whole wave in
circumferential direction, cannot be obtained satisfactorily from Donnell’s equation.
Although the roots further closely agree, it is also shown that for the inhomogeneous
solution for a cylindrical shell subject to surface load, significant errors occur for the
Donnell solution for the lower values of the mode number in circumferential direction.
Conveniently, Morley’s solution closely agrees with Flügge’s solution for any shell
geometry and deformation mode.
Many other authors, investigated roots of characteristic equations that are derived
either as being exact or as, e.g., approximated, simplified and improved. Houghton and
Johns [38] compared the roots of various characteristic equations, which are derived
for circular cylindrical shells, by representing the deformation in a Fourier series in the
circumferential direction. The roots are obtained for the characteristic equations of
40
2 General part on shell theory
Flügge [14], Vlasov [22], Novozhilov [20], Morley [35], Donnell [21] and others, of
which Donnell’s equations is the most approximate equation. It is shown that, for thin
shells and especially large values of the circumferential mode number, the roots of the
equations closely agree and that for lower mode numbers small differences occur,
which diminish with increasing thinness. Houghton and Johns state that there is some
advantage in using the special simplified equations such as those by Donnell and
Morley, since it is possible to determine a solution without resort to a quartic equation.
However, Donnell’s equation should not be used for the lower values of the
circumferential mode number.
Seide [39, 40] compared the roots of the complete Flügge equation, the simplified
Flügge equation and the Morley equation for large values of the circumferential mode
number (ranging from 0 to 300) and for a thick cylindrical shell with a radius-tothickness-ratio of 10. The simplified Flügge equation is the equation that is obtained if
second-order terms, which do exceed the accuracy of the initial assumptions, are
omitted. Seide showed that the roots of the complete and simplified Flügge equations
are in excellent agreement and the roots of the Morley equation closely agree for a
large range of the circumferential mode number. Since, for large circumferential mode
numbers and thick cylindrical shells, the validity of the applicability of the KirchhoffLove assumptions is questionable, the roots of Morley’s equations are always a good
approximation within these assumptions.
The above-mentioned equations are obtained within the assumptions of the KirchhoffLove assumptions. Shirakawa [41] presented a method for finding the roots of the
characteristic equation in the theory of circular cylindrical shells which contains the
effect of shear deformation (the Mirsky-Herrmann theory). In this paper, the numerical
values are shown for the roots of the characteristic equation in the axial and the
circumferential direction. These numerical values are compared with the values that are
obtained by an improved theory by Shirakawa and Flügge’s theory. It is shown that
Flügge’s solutions are very accurate in the case of thin shells and that, for shells with a
radius-to-thickness-ratio smaller than 20, the effect of shear deformation should be
accounted for. Although Flügge’s solutions show a slight discrepancy, the results
become inaccurate.
Hence, it is concluded that the accuracy of Morley’s equation is assessed. As the
equation proposed by Morley is later derived by Koiter, this equation will be further
referred to as the Morley-Koiter equation.
For later reference, the (simplified) Flügge equation, the Morley-Koiter equation and
the Donnell equation for thin elastic circular cylindrical shells are reproduced. In
accordance with the used notation x , θ and z are associated with ξ1 , ξ 2 and ζ ,
respectively. The straight generator in x -direction has an infinite radius and therefore
its curvature is equal to zero. The radius in θ -direction is already mentioned and equal
to a . Here the dimensionless parameter β and the flexural rigidity Db are introduced
without further explanation, which are defined by
41
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
a
β4 = 3 (1 − υ2 )  
t
2
;
Db =
Et 3
12 (1 − υ2 )
The (simplified) Flügge equation reads
2
4
4
1 

 β  ∂ uz
∆∆  ∆ + 2  u z + 4  
4
a 

 a  ∂x
−2 (1 − υ )
=
1  ∂4
1 ∂ 4  ∂ 2u z
1  1 + υ ∂2
∂ 2  ∂ 2u z
−
+
2
1
−
υ
3
+
(
)




a 2  ∂x 4 a 4 ∂θ4  ∂x 2
a 4  2 ∂x 2 a 2∂θ2  ∂x 2
1 
1 ∂ 3 pθ
1 ∂ 3 pθ
1 ∂ 3 px 1 ∂ 3 px 
+υ
−
 ∆∆pz + (2 + υ) 2 2 + 4

3
Db 
a ∂x ∂θ a ∂θ
a ∂x3 a 3 ∂x∂θ2 
The Morley-Koiter equation reads
2
4
4
1 
1 
1 ∂ 3 pθ
1 ∂ 3 pθ
1 ∂ 3 p x 1 ∂ 3 px 

 β  ∂ uz
∆∆  ∆ + 2  u z + 4  
=
+υ
−
 ∆∆pz + (2 + υ) 2 2 + 4

4
3
a 
Db 
a ∂x ∂θ a ∂θ
a ∂x3 a 3 ∂x∂θ2 

 a  ∂x
The Donnell equation reads
4
4
1 
1 ∂ 3 pθ
1 ∂ 3 pθ
1 ∂ 3 px 1 ∂ 3 px 
 β  ∂ uz
∆∆∆∆u z + 4  
=
+υ
−
 ∆∆pz + (2 + υ) 2 2 + 4

4
3
Db 
a ∂x ∂θ a ∂θ
a ∂x3 a 3 ∂x∂θ2 
 a  ∂x
2.7 Proposed theory
2.7.1 Alternative description of the strain distribution
Novozhilov [20], Vlasov [22] and others also expanded the strain description in a
series expression with respect to the coordinate ζ . Hoefakker [33] applied a somewhat
similar idea to the strain distribution of a circular cylindrical shell by rewriting the
strain description. Especially for the deformation quantities in the curved direction of
the cylinder, this rewritten description has the advantage that the strain description after
neglecting terms ζ Rα in comparison to unity, remains such that the stress couple in
this curved direction is not altered by this approximation. As a result of this feature, the
so-called ring bending action is still perfectly described. The derivation of the
expressions for a circular ring and the approximation of those expressions are given in
Appendix E and Appendix F, respectively. Based on these results, it is observed that,
as a result of the difference between the more conventional description with the
changes of rotation and the description with the alternative deformation quantities,
which can be interpreted as a changes of curvature, the so-called ring bending solution
is not only satisfied by the single differential equation but also by the set of equilibrium
equations if the mentioned expansion is performed. Also based on the discussions in
section 2.6, it is therefore concluded that the expansion of the strain description, which
adopts the changes of curvature, should be preferably considered for a reliable and
consistent theory of shells of revolution.
42
2 General part on shell theory
Adopting the procedure proposed by Hoefakker, but now only to the normal strains e11
and e22 , their expressions (2.10) can be identically represented by
e11 =
1
ζ
1+
R1
( ε11 + ζβ11 ) = ε11 +

ε 
ζ
β − 11 = ε +
κ
ζ  11 R1  11
ζ 11
1+
1+
R1
R1
ζ

ε 
ζ
e22 =
( ε + ζβ22 ) = ε 22 + ζ  β22 − 22  = ε 22 + ζ κ22
ζ 22
R
2 
1+
1+ 
1+
R2
R2
R2
ζ
1
(2.36)
where two alternative deformation quantities are introduced, which are a combination
of the change of rotation and the normal strain in that respective direction.
Hence, the relation between the alternative deformation quantities and the former
quantities is given by
κ11 = β11 −
ε11
R1
κ 22 = β22 −
,
ε22
R2
(2.37)
The alternative deformation quantity represents, corresponding to the strain description
for a plate by a normal strain plus a curvature times the coordinate in thickness
direction, the change of curvature.
By rewriting the corresponding part of the virtual work equation to
n11δε11 + n22δε 22 + m11δβ11 + m22δβ 22




ε 
ε 
m 
m 
=  n11 + 11  δε11 +  n22 + 22  δε 22 + m11δ  β11 − 11  + m22δ  β22 − 22 
R
R
R
R2 




1 
2 
1 
= n11δε11 + n22δε 22 + m11δκ11 + m22δκ 22
(2.38)
it is shown that the alternative stress resultants and stress couples correspond to
n11 = n11 +
m11
R1
,
m11 = m11
n22 = n22 +
m22
R2
,
m22 = m22
These alternative strain and stress quantities for the normal strain and stress
distributions can be complemented by the alternative strain and stress quantities for the
shear strain and stress distribution as introduced in subsection 2.6.1. If this were
desired, the following relations can be collected for the stress resultants and the
changes of curvature
n11 = n11 +
m11
R1
,
κ11 = β11 −
ε11
R1
n22 = n22 +
m22
R2
,
κ 22 = β22 −
ε 22
R2
n12 = n12 −
m21
R2
,
κ12 = β12 +
ε21
R1
n21 = n21 −
m12
R1
,
κ 21 = β 21 +
ε12
R2
43
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The complete set of relations can be compactly represented by
εαβ =
1
( εαβ + εβα )
2
,
κ αβ = βαβ − ( −1)
α +β
εβα
,
Rα
( α, β ) = (1,2 )
for the deformation quantities, while the stress quantities can be represented by
nαβ = nαβ + ( −1)
α +β
mβα
Rβ
,
mαβ =
1
( mαβ + mβα )
2
,
( α, β ) = (1,2 )
where the notation above corresponds to the deformation measures that are introduced
to describe the shear strain distribution by the relation
γ αβ = 2εαβ
,
ραβ = 2 κ αβ
,
α≠β
2.7.2 Kinematical relation
As a starting point for a kinematical relation, the set (2.11) is used but the transverse
shearing strains are set equal to zero. The neglect of these strains is justified in
subsection 2.6.1 on the argument by Reissner and thus the rotations can be described
by the expressions (2.12). However, for the normal strain distributions the description
by the set (2.36) is proposed. Next to this, the reduction of the four in-plane shear
deformation quantities to one for the shear angle and one for the torsion as given by
expression (2.29) is applied.
The expressions for the kinematical relation can now be obtained by substituting
expressions (2.12) into (2.11) and (2.29). The expressions that are altered are already
given by (2.31) and (2.32). However, the expressions (2.32) have to be substituted into
the expressions (2.37) for the changes of curvature.
The resulting kinematical relation is presented by using temporarily the sign
convention that an operator within the curled brackets does not apply to the vector on
which the operator matrix acts and hence becomes

1 ∂

α
1 ∂ξ1


1  ∂α 2 




α1α 2  ∂ξ1 
 ε11  
ε   1 ∂
1  ∂α1 
−
 22  


 γ12   α 2 ∂ξ2 α1α 2  ∂ξ2 
 =
1  ∂  1  
 κ11  

 
κ  
α1  ∂ξ1  R1  
 22  
 ρ12    1
1  1  ∂α 2 
 − 


  R1 R2  α1α 2  ∂ξ1 

 2  ∂ − 1  ∂α1  
 R α  ∂ξ α  ∂ξ  
1
2 
 1 2 2
1  ∂α1 


α1α 2  ∂ξ2 
1 ∂
α 2 ∂ξ2
1 ∂
1  ∂α 2 
−


α1 ∂ξ1 α1α 2  ∂ξ1 
 1
1  1  ∂α1 


 − 
 R2 R1  α1α 2  ∂ξ2 
1  ∂  1  

 
α 2  ∂ξ2  R2  
2  ∂
1  ∂α  
−  2  

R2α1  ∂ξ1 α 2  ∂ξ1  
where the three factors in the third column are given by
44
1

R1 

1
R2 


0  u 
1
 
u
 2
λ1  uζ 



λ2 


λ3 


(2.39)
2 General part on shell theory
λ1 = −
1 ∂  1 ∂ 
1  ∂α1  ∂
1
−



−
α1 ∂ξ1  α1 ∂ξ1  α1α 22  ∂ξ2  ∂ξ2 R12
λ2 = −
1 ∂  1 ∂ 
1  ∂α 2  ∂
1
− 2


− 2 
α 2 ∂ξ2  α 2 ∂ξ 2  α1 α 2  ∂ξ1  ∂ξ1 R2
λ3 =
1  ∂α1  ∂
1  ∂α 2  ∂
1 ∂  1 ∂  1 ∂  1 ∂ 
+
−





−


α12α 2  ∂ξ2  ∂ξ1 α1α 22  ∂ξ1  ∂ξ 2 α1 ∂ξ1  α 2 ∂ξ2  α 2 ∂ξ2  α1 ∂ξ1 
As mentioned in subsection 2.6.1 the rigid body rotation about the normal to the
reference surface is cancelled out. In subsection 2.6.4 and subsection 2.7.1, the stress
resultants and stress couples that correspond to the chosen deformation quantities are
given and these fulfil the sixth equilibrium equation of (2.15). This means that by using
the kinematical relation given above, a proper set of equilibrium equation with the
corresponding boundary conditions can be derived. The last desired quality mentioned
in subsection 2.6.1 concerns the symmetry of the constitutive relation.
2.7.3 Constitutive relation
Except for the Flügge-Byrne derivation, the derivations discussed in section 2.6 obtain
constitutive equations by neglecting terms ζ Rα in comparison to unity. This is based
on Love’s observation that the strain energy can be split in a two independent parts;
one that represents the potential energy of extension and shear and one that represents
the potential energy of bending and torsion.
Using (2.11) but neglecting terms ζ Rα in comparison to unity for the expressions
(2.10) and (2.30) of the strain distributions, gives
 ε11 
ε 
22
e
1
0
0
ζ
0
0
 11  
 

γ
 e  = 0 1 0 0 ζ 0  12 
 22  
 κ 
 2e12  0 0 1 0 0 ζ   11 
κ 
 22 
 ρ12 
These linear strains are related to their respective stresses by (2.13), which reads
0 
1 υ
 σ11 
υ 1
  e11 
E
0
σ  =

  e22 
 22  1 − υ2 

1− υ 
 σ12 
0 0
  2e12 

2 
Corresponding to the expressions for the strain distributions expressed in the
deformation quantities, the stress resultants are described by the following integrals
 n11 
1
n 
0
 22 

 n12 
0
  = ∫
m
 11  ζ ζ
m 
0
 22 

 0
 m12 
0 0
1 0 
 σ11 
0 1  
 σ22 d ζ
0 0  
 σ12 
ζ 0  

0 ζ 
45
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
which is a simplification of (2.14) by neglecting the terms ζ Rα in comparison to
unity. This guarantees the symmetry since the neglect is simultaneously imposed on the
strain and stress distribution across the thickness.
It is assumed that the reference is the middle surface and the thickness t is
constant. Hereby, subsequent substitution of the three sets of expressions given above
results in
 Dm
 n11   υDm
n  
 22   0
 n12  
 =
 m11   0
m   0
 22  
 m12   0


υDm
0
0
0
Dm
0
0
0
0
0
0
1− υ
Dm
2
0
Db
υDb
0
0
υDb
Db
0
0
0
0
0

0   ε11 
 
 ε 22
0   
γ
  12 
0   κ11 
0   κ 22 
 
1 − υ   ρ12 
Db

2 
0
(2.40)
which is the constitutive relation between the stress resultants and stress couples and
the deformation quantities of the reference surface. Herein the quantities Dm and Db
are the extensional (membrane) rigidity and flexural (bending) rigidity, respectively,
which are given by
Dm =
Et
1 − υ2
;
Db =
Et 3
12 (1 − υ2 )
(2.41)
As a result of the simplification above, a linear description of the stress distribution
across the thickness has been assumed. The respective normal stress and shearing stress
components can be conveniently obtained from the following relations
n11
12m
+ ζ 3 11
t
t
n22
12m22
σ 22 =
+ζ 3
t
t
n12
12m12
σ12 =
+ζ 3
t
t
σ11 =
(2.42)
2.7.4 Equilibrium relation
The equilibrium equations are derived by using the principle of virtual work as
described in section 2.3 but by substituting expression (2.34) for the virtual work done
by the shear angle and the torsion, by substituting expression (2.38) for the virtual
work done by the normal strains and the changes of curvature, and by neglecting the
contributions of the transverse shear strains. The variation of the strain energy
described by (2.19) now becomes
δEs =
∫ ∫ (n
δε11 + m11δκ11 + n22δε 22 + m22δκ 22 + n12δγ12 + m12δρ12 ) α1α 2 d ξ1d ξ 2
11
ξ2 ξ1
Proceeding in the same manner as in section 2.3 results in the relation (2.43) presented
on the next page where the temporary sign convention for the curled brackets
employed for presenting the kinematical relation is used.
46
∂  1 ∂  1   ∂  1  ∂α1   1



 −
  +
∂ξ1  α1 ∂ξ1  α1   ∂ξ2  α1α 22  ∂ξ 2   R12
∂  1 ∂  1   ∂  1  ∂α 2   1



 −
  +
∂ξ 2  α 2 ∂ξ2  α 2   ∂ξ1  α12α 2  ∂ξ1   R22
∂  1  ∂α1   ∂  1  ∂α 2   ∂  1 ∂  1   ∂  1 ∂  1  



 −

 −


  −
 
∂ξ1  α12α 2  ∂ξ2   ∂ξ 2  α1α 22  ∂ξ1   ∂ξ2  α 2 ∂ξ1  α1   ∂ξ1  α1 ∂ξ 2  α 2  
λ1 = −
λ2 = −
λ3 = −
where the three factors in the third row are given by
47
(2.43)
 ∂  1 
1
1  ∂α 2 
∂  1 
1  ∂α1 
1  ∂  1  
1  1  ∂α 2 
∂  2 
2  ∂α1    n α α 
−

 −




 −

  11 1 2
 
 −
 − 

−
 


∂ξ
α
α
α
∂ξ
∂ξ
α
α
α
∂ξ
α
∂ξ
R
R
R
α
α
∂ξ
∂ξ
R
α
R
α
 1  1  

1 1
1 2 
1 
2 
2 
1 2 
2 
1
 1
2  1 2 
1 
2  1 2 
1 1α 2  ∂ξ 2   n α α
22 1 2 
 p1α1α 2 


1  ∂  1  
∂  2 
2  ∂α 2   n12α1α 2  

 1  ∂α1  − ∂  1  − ∂  1  − 1  ∂α 2   1 − 1  1  ∂α1 
−
−
p2α1α 2 
=



 








 
 
 
 α α  ∂ξ 
α 2  ∂ξ2  R2  
∂ξ1  R2α1  R2α1α 2  ∂ξ1   m11α1α 2  
∂ξ2  α 2 
∂ξ1  α1  α1α 2  ∂ξ1   R2 R1  α1α 2  ∂ξ2 
1 2 
2
 pζ α1α 2 




  m22α1α 2  
1
1
1
2
3
0
λ
λ
λ

 m α α 
 12 1 2 
R1
R2


2 General part on shell theory
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The third equation for the equilibrium in the normal direction thus reads
∂  1  ∂  m11α1α 2  m22α1α 2 ∂α 2 m12α1α 2 ∂α1
∂  m12α1α 2   
−
+
+
 



 
∂ξ1  α1  ∂ξ1  α1 
α1α 2 ∂ξ1
α1α 2 ∂ξ2 ∂ξ2  α 2   
+
∂  1  ∂  m22α1α 2  m11α1α 2 ∂α1 m12α1α 2 ∂α 2
∂  m12α1α 2   
+
+
 

−

 

∂ξ 2  α 2  ∂ξ2  α 2 
α1α 2 ∂ξ2
α1α 2 ∂ξ1 ∂ξ1  α1   
−
n11α1α 2 m11α1α 2 n22α1α 2 m22α1α 2
+
−
+
+ pζ α1α 2 = 0
R1
R12
R2
R21
By comparing this result with the original equation (2.23), which reads
∂  v1α1α 2  ∂  v2α1α 2  n11α1α 2 n22α1α 2
−
+ pζ α1α 2 = 0

+

−
R1
R2
∂ξ1  α1  ∂ξ 2  α 2 
while keeping in mind that the relations
n11 = n11 −
m11
R1
; n22 = n22 −
m22
R2
hold, it can be observed that the transverse shearing stress resultants are now described
by
v1α1α 2 =
∂  m11α1α 2  m22α1α 2 ∂α 2 m12α1α 2 ∂α1
∂  m12α1α 2 
+
+

−


∂ξ1  α1 
α1α 2 ∂ξ1
α1α 2 ∂ξ2 ∂ξ2  α 2 
∂  m22α1α 2  m11α1α 2 ∂α1 m12α1α 2 ∂α 2
∂  m12α1α 2 
v2α1α 2 =
+
+

−


∂ξ 2  α 2 
α1α 2 ∂ξ2
α1α 2 ∂ξ1 ∂ξ1  α1 
(2.44)
where the bar indicates that these are not the usual stress resultants but the ones that
correspond to the alternative shearing stress resultant and couple introduced by
Novozhilov as described in subsection 2.6.4.
2.7.5 Boundary conditions
Using the principle of virtual work as described in the previous subsection, the total set
of four stress quantities at the boundary corresponding to the four displacement
measures as given in section 2.4 can be obtained. For the edges ξ1 = constant , this set
becomes
f1 = n11
or u1 = u1 
or u2 = u2 
( 2)
 ξ1 = ξ1 ,
or uζ = uζ 
or ϕ 1 = ϕ1 
∗
12
∗
1
f2 = n
fζ = v
t1 = m11
f1 = −n11
∗
12
∗
1
f 2 = −n
f ζ = −v
t1 = −m11
or u1 = u1 
or u2 = u2 
(1)
 ξ1 = ξ1
or uζ = uζ 
or ϕ 1 = ϕ1 
with the additional conditions that
( 2)
[t2 ]ξ =ξ( )
2
1
2
( 2)
(1)
= − [ m12 ]ξ
ξ =ξ
ξ1 =ξ1
( 2)
[t2 ]ξ =ξ( )
ξ 2 =ξ2
2
1
2
2
( 2)
2
ξ1 =ξ1
n11 = n11 −
1
m11
R1
or
uζ = uζ ,
ξ1 = ξ1( 2)
or
uζ = uζ ,
ξ1 = ξ1(1)
ξ1 =ξ1
( 2)
ξ2 =ξ2
where the relation
48
( 2)
= [ m12 ]ξ2 =ξ(21)
ξ 2 =ξ2
(1)
2
=ξ2
(1)
ξ1 =ξ1
(2.45)
2 General part on shell theory
is utilised and the combined internal stress resultants denoted by the superscript ∗ are
defined by
n12∗ = n12 +
2
m12
R2
(2.46)
1 ∂m12
v = v1 +
α 2 ∂ξ2
∗
1
However, the rotations are not displacement measures that can be independently
varied, since by neglecting the shear deformation the rotations are described by the
expressions (2.12), which read
ϕ1 =
u1 1 ∂uζ
−
R1 α1 ∂ξ1
(2.47)
1 ∂uζ
u
ϕ2 = 2 −
R2 α 2 ∂ξ 2
Therefore, at the edge ξ1 = constant , not the rotation ϕ1 , but the angle
1 ∂uζ
is the
α1 ∂ξ1
independent displacement that should be considered. Hereby, the total set of four stress
quantities at the boundary corresponding to the four displacement measures becomes
for the edges ξ1 = constant
f1 = n11
f 2 = n12∗
f ζ = v1∗
t1 = m11
or u1 = u1


or u2 = u2

( 2)
or uζ = uζ
 ξ1 = ξ1
1 ∂uζ 1 ∂uζ 
=
or
α1 ∂ξ1 α1 ∂ξ1 
f1 = −n11
f 2 = − n12∗
,
f ζ = −v1∗
t1 = − m11
or u1 = u1


or u2 = u2

(1)
or uζ = uζ
 ξ1 = ξ1
1 ∂uζ 1 ∂uζ 
=
or
α1 ∂ξ1 α1 ∂ξ1 
(2.48)
with the additional conditions that
( 2)
[t2 ]ξ =ξ( )
2
1
2
( 2)
(1)
= − [ m12 ]ξ
ξ =ξ
ξ1 =ξ1
( 2)
[t2 ]ξ =ξ( )
ξ 2 =ξ 2
2
1
2
( 2)
= [ m12 ]ξ2 =ξ(21)
ξ 2 =ξ 2
2
( 2)
2
uζ = uζ ,
ξ1 = ξ1( 2)
or
uζ = uζ ,
ξ1 = ξ1(1)
ξ1 =ξ1
( 2)
ξ 2 =ξ 2
ξ1 =ξ1
or
(1)
2
=ξ 2
(1)
ξ1 =ξ1
where the combined internal stress resultants denoted by the superscript ∗ are defined
by (2.46).
Rewriting the expression for f ζ at the edge ξ1 = ξ1( 2) of conditions (2.48) by
making use of (2.44) gives
f ζ = v1 +

1 ∂m12
1  ∂α1m12 ∂α 2m11 ∂α 2
=
+
−
m22 
2
α 2 ∂ξ 2 α1α 2  ∂ξ 2
∂ξ1
∂ξ1

which shows that the stress quantities (2.35) as derived by Novozhilov are identical to
the derived quantities.
For the edge ξ 2 = constant , equivalent expressions can be obtained where the
indices denoting the parametric lines are interchanged where applicable.
49
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The derivation of these boundary conditions is elaborated independently from the
approximation that is introduced in the constitutive relation in subsection 2.7.3. As a
result of this approximation, the strain distribution is linear and hereby the stress
resultants are only described by the normal and shear strains, while the stress couples
are only described by the changes of curvature and the torsion. Hence, by making use
of the combined internal stress resultant n12∗ , additional terms are introduced in the
boundary conditions, which are of the same order of the terms that are neglected in the
constitutive relation. Therefore, while simultaneously approximating the constitutive
relation, it is allowed to approximate the (combined) internal stress resultants n11 and
n12∗ according to
n11 = n11 −
1
m11 ≈ n11
R1
,
n12∗ = n12 +
2
m12 ≈ n12
R2
The boundary conditions (2.48) can thus be rewritten to
f1 = n11
f 2 = n12
f ζ = v1∗
t1 = m11
or u1 = u1 
f1 = − n11

or u2 = u2 
f 2 = − n12
( 2)
 ξ1 = ξ1 ,
or uζ = uζ 
f ζ = −v1∗
or ϕ 1 = ϕ1 
t1 = − m11
or u1 = u1 
or u2 = u2 
(1)
 ξ1 = ξ1
or uζ = uζ 
or ϕ 1 = ϕ1 
(2.49)
where for convenience the rotation ϕ1 is used.
Another method for deriving the boundary conditions that are consistent with the
equilibrium equations is the application of Hamilton’s principle. The potential energy
is formulated on basis of the kinematical relation as well as the constitutive relation.
However, for the linear elastic and geometrical linear equations, the application of this
principle results in the same equilibrium equations and boundary conditions as
obtained by application of the principle of virtual work. Hence, the inconsistency
between the natural boundary conditions and the approximation of the constitutive
relation cannot easily be remedied.
50
3 Computational method and analysis method
3 Computational method and analysis method
The use of a digital computer for the solution of problems in the linear theory of thin
elastic shells of revolution under static loading is described. The described numerical
procedure is integrated into the well-known direct stiffness approach of the
displacement method. The considerations that have to be made for shells of revolution
are presented in the calculation scheme and the back substitution.
3.1 Introduction to the numerical techniques for a
solid shell
The theory of elastic plates and shells subject to static loading has been treated
extensively and the analysis of thin shells of revolution has attracted much attention.
Some of these works are of basic nature and deal with what can be referred to as
analytical models. The exact analytical solutions derived in these classical works have
been found only for the more simple problems, e.g. for axisymmetric loading in the
case of shells of revolution. Approximate solutions have been obtained by, among
other methods, series expansions of the exact solution, perturbation methods and the
method of asymptotic integration, but mainly for special loading cases and boundary or
transition conditions. This restrictive feature of the analytical models is due to the
highly mathematical character of the equations leading to inconvenient and involved
solutions. Therefore the major part of the works on shells of revolution and, even more
general, plates and shells fall back on what can be referred to as numerical models. As
mentioned in the introductory chapter 1, this change of focus is largely related to
today’s availability of greatly increased computing power.
The common numerical techniques for analysis of shells are the method of
stepwise integration, the finite difference method, the boundary integral method and
the finite element method. The finite element method, which contains a procedure of
approximation to continuous problems, is probably the most widely used and
innumerable finite shell elements have been proposed.
In the finite element method, the continuous structure is divided into a finite
number of parts (which are called elements) that are connected by nodes. The
behaviour of the elements is specified by a finite number of parameters, which are the
element displacements and element forces related to one another by an element matrix.
The element matrices per part of the structure are combined into a global matrix by
enforcing continuity of the element displacements at the nodal points, which are called
the nodal displacements. The element forces are combined into a global vector by
enforcing equilibrium of the element (and external) forces at the nodal points, which
are called the nodal forces.
The key step in any finite element procedure is the generation of the element
matrix for each element. The large variety of element formulations can be roughly
classified into conventional elements, which are mainly based on assumed
displacement, strain or stress fields, and super elements, which require analytical
solutions of governing equations.
51
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
From the several conventional approaches that exist for the formulation of finite
elements, the displacement method is the prevailing one. As the name suggest, the
nodal displacements are the primary unknowns, which are used to describe the
displacement field within the element. This displacement field is usually expressed by
imposed displacement functions, which are called shape functions, and these assumed
functions are of an approximate character.
The accuracy and method of approximation that is employed differs widely and
hence many different types of shell elements have been proposed. Without being
complete, the variety of assumed displacement fields have led to constant strain, linear
and higher order elements and, by also using shape functions to approximate the
geometry, isoparametric elements. The large variety also finds its origin in the shape of
the element as flat triangular, quadrilateral or rectangular, shallow shell, cylindrical
shell as well as doubly curved elements have been developed.
One of the attempts at seeking a method for minimising the number of elements
needed to model a given problem domain is the finite strip method. In this method, the
structure is divided into strip domains in which one opposite pair of sides or faces
coincide with the boundaries of the structure. Within these strip domains, use is made
of polynomials in some direction and continuously differentiable series in other
directions so that the boundary conditions at the ends of the strip domains are satisfied.
Somewhat similar methods use combinations of shape functions of finite elements with
finite strips, combinations of trigonometric and hyperbolic functions, and spline
functions.
Such methods enable accurate descriptions of rather large substructures and,
instead of the term conventional element, the term super element seems more
appropriate for such an element.
The asset of super elements lies in the degree of approximation employed in deriving a
stiffness matrix for each element. If exact analytical solutions to governing equations
are available, an exact stiffness matrix can be obtained. Obviously, this is ideally
desirable owing to the advantage that only one single element is sufficient to account
for a complete part of a structure (within which geometry and load are continuous). A
tremendous advantage of employing an exact solution for the response of a shell
structure lies in the fact that the edge disturbances are described in a superior manner
whereby the attenuating bending field components are easily captured, which is
especially valuable for the reproduction of the short influence length components.
This is in sharp contrast to the necessary high degree of mesh refinement for
conventional elements that is usually required to obtain acceptable results, which is
related to the much higher degree of approximation used in deriving a stiffness matrix.
As a result, it is difficult to identify which shell elements are the most effective
elements currently available. Especially since various types of elements have varying
degrees of convergence rate and accuracy, can show sensitivity to the geometry and
support conditions, and possibly suffer from somewhat unexpected sorts of locking.
Unfortunately, super elements do not lack in general of the above-mentioned
shortcomings since exact solutions to governing equations cannot always be obtained.
Especially for shell models, the determination of the stiffness matrix coefficients is not
only complicated and cumbersome, but the resulting expressions are also lengthy and
52
3 Computational method and analysis method
inconvenient. Next to this, exact solutions to, for example, approximate governing
equations are surplus to requirements and might therefore be approximated to the same
degree to avoid the appearance of a wider range of applicability.
However, for such cases the employment of approximate solutions or approximate
equations will often lead to accurate stiffness relations for the practical range of the
considered geometry. Solutions of this kind are obviously bound by a number of
restrictions for the load distribution and the type of response, simplifications for the
constitutive relation, etc, but from practical point of view, such solutions are not a
drawback. Moreover, the super element derived by such means will decrease the
computational costs (due to the minimised number of degrees of freedom) and, for
example, reduce the data preparation effort. This seems to be useful if computational
effort is limited, expensive or not accessible. Ultimately, the leading feature of the
super elements is the expected computational time gain when calculating series of
variation in geometry, load or boundary conditions in order to conduct parameter
studies for the problem at hand.
3.2 The super element approach
To conduct parametric studies of the response of shells of revolution a method for
deriving a super element for these shells is suggested. As referred to in the previous
section, this approach avoids the shortcomings of most existing element stiffness
matrices and attempts to minimise the number of elements needed to model a given
problem domain. Similar to the conventional method, the first and crucial step is to
compute the element stiffness matrix but for the super element, this is synthesized on
the basis of an analytical solution to the governing equation.
As the starting point, a proper set of differential equations governing the elastic
behaviour of thin shells of revolution under distributed surface and line loads is
selected. For cylindrical shells (and similarly for conical and spherical shells) with
circular boundaries, which are the most frequently used in structural application, it is
possible to obtain a closed-form solution (within the assumptions of the theory) to
these rigorous shell formulations. The precise formulation of the classic approach is
reshaped into the well-known direct stiffness approach of the displacement method
enabling the calculation of combinations of elements and type of elements, which
makes the use of an electronic calculation device more sensible in view of the
increasing number of equations. The implementation of this direct stiffness approach
into an expeditious PC-oriented computer program is accomplished by using the
Fortran-package in combination with graphical software. The discussion of the
successive steps is the topic of this section.
3.2.1 Substructures for a shell of revolution
A shell of revolution is a body of which the middle surface is a surface of revolution. A
surface of revolution is generated by the rotation of a plane curve about an axis in its
plane. This generating curve is called a meridian. An arbitrary point on the middle
surface of the shell is described by specifying the particular meridian on which it is
positioned and by indicating a second coordinate that varies along the meridian but is
53
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
constant on a circle around the axis of rotation of the shell. This circle is called a
parallel or latitude circle. The meridian is indicated by the angular distance θ of its
plane from that of the datum meridian. The other coordinate is specified by the angle φ
between a normal to the shell surface and its axis of revolution. At any angle φ the
geometry does not vary along the parallel circle, i.e. the surface parameters are constant
in circumferential direction.
In general, any applied load can quite easily and very conveniently be transformed
in a Fourier series with respect to the circumferential coordinate θ on the parallel
circle. Due to the symmetry of the surface and the probable continuity in
circumferential direction of any surface and line loads, such a series will be a
trigonometric series. Moreover, most of the practical loads on a shell of revolution will
anyhow have of a trigonometric character in circumferential direction.
When considering a structure that consists of several shells of revolution, which
are joined at their parallel circular edges, the above-mentioned observations enable the
division of the structure into substructures per single type of shell of revolution. Within
the edges of such a circumferentially closed substructure, the load and the shell
geometry should be continuous in meridional direction. This means that a substructure
is bounded by (i) a support or free edge, (ii) a transition of two shells, (iii) a stiffening
ring, (iv) a transition of the intensity of the surface load, or (v) a circumferential line
load. Hereby an analytical solution to the governing equations can be employed that
consists of the above-mentioned trigonometric series and functions with arbitrary
constants in meridional direction. Similarly to the finite strip method, use is thus made
of continuously differentiable series in the circumferential direction, i.e. the
trigonometric series, to satisfy the continuity conditions of the circumferentially closed
substructure. The meridional part of the analytical solution contains functions with the
arbitrary constants that are unknown and this part serves as the shape functions of the
displacement field. At every parallel circle, a displacement is thus expressed as the
trigonometric series multiplied by the amplitude per respective parallel circle and this
amplitude is equal to the magnitude of the shape function. By substituting the
meridional coordinates of the two parallel circular edges of the substructure, the two
edge amplitudes per displacement are obtained. Each substructure is thus captured by
only one element with two nodes and the number of the degrees of freedom per
element is equal to the number of arbitrary constants.
In general, a differential equation for the elastic response of thin shells of revolution
(and thus neglecting the influence of the shear deformation) is of the i th order and
hence a solution will contain i arbitrary constants. These have to be determined by i
boundary conditions that can be formulated for two opposite edges. Since it is not
possible to prescribe both an edge displacement as well as an edge force in
corresponding direction simultaneously, a maximal number of i 2 edge displacements
and a maximal number of i 2 edge forces can be prescribed per edge. A shell element
based on such a solution thus has i 2 degrees of freedom per node or in other words
i 2 generalised displacements at either side. In keeping with the number of degrees of
freedom, an equal number of generalised forces with corresponding directions may act
at each node.
54
3 Computational method and analysis method
3.2.2 Application of the suggested approach
To discuss the application of the suggested approach in the form of a super element
method, from here onward the proposed set of equations given in section 2.7 is
considered. To obtain this set the influence of transverse shear deformation is neglected
and because of these three independent displacements, viz. u1 , u2 and uζ , can be
identified. Hence, three simultaneous differential equations can be obtained that
express the equilibrium equations for p1 , p2 and pζ in terms of the independent
displacements. By elimination of these displacements, a single differential equation for
one of the displacements can be obtained and this equation will be of the eighth order.
The general solution to this single differential equation has two parts, the
homogeneous solution and the inhomogeneous solution. The homogeneous solution
contains the eight arbitrary constants and depends on the boundary conditions since
this solution accounts for the effects of edge loads on the distribution of the stress and
strain quantities of the reference surface. The inhomogeneous solution is independent
of the boundary conditions and takes care of any distributed load that acts on the
surface.
Henceforth the coordinate ξ1 will be associated with the meridional direction and
ξ 2 with the circumferential direction. The shape of the meridian is arbitrary but at each
point on the meridian, the parallel circle has a constant radius in circumferential
direction whereby it is useful to introduce an angular coordinate. Since it is customary
to denote this circumferential coordinate by θ , the coordinates ( ξ1 , ξ2 , ζ ) are in this
chapter henceforth replaced by ( ξ, θ, ζ ) .
Assuming not only continuity but also symmetry of the load in circumferential
direction and choosing this line of symmetry to be indicated by θ = 0 , it is observed
that the loads pξ and pζ are even periodic functions with period 2π with respect to
that line of symmetry and that the load pθ is an odd periodic function. The Fourier
series of any even or odd function consists only of the even trigonometric functions
cos ( nθ ) or odd trigonometric functions sin ( nθ ) , respectively, and a constant term [42].
Therefore, the three load components can be described by a Fourier trigonometric
series expressed by
∞
pξ ( ξ, θ ) = ∑ pξn ( ξ ) cos nθ
n=0
∞
pθ ( ξ, θ ) = ∑ pθn ( ξ ) sin nθ
n =0
∞
pζ ( ξ, θ ) = ∑ pζn ( ξ ) cos nθ
n =0
where n is the mode number and represents the number of whole waves in
circumferential direction.
In keeping with the trigonometric series load a trial solution to the reduced
differential equation will be of the trigonometric series form. Obviously, not only the
homogeneous solution u h is to be described by a congruent form to the load
distribution but also the inhomogeneous solution u i will have the same circumferential
55
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
distribution. So, in correspondence with the distribution of the load components, the
general solution for the displacements is of the congruent form
∞
uξ ( ξ, θ ) = ∑ Chuξh ( ξ ) + uξi ( ξ )  cos nθ
n =0
∞
uθ ( ξ, θ ) = ∑ Chuθh ( ξ ) + uθi ( ξ )  sin nθ
(3.1)
n=0
∞
uζ ( ξ, θ ) = ∑ Chuζh ( ξ ) + uζi ( ξ )  cos nθ
n=0
where Ch represents the eight arbitrary constants per circumferential mode number.
On the basis of the same consideration and by inspecting the sets of equations of
section 2.7 it can now be concluded that the rotations, strain and stress quantities are
described by functions of the form
ϕξ ,
εξξ , εθθ , κξξ , κθθ ,
nξξ , nθθ , vξ , mξξ , mθθ
⇒ cos nθ
ϕθ ,
γ ξθ , ρξθ ,
nξθ , vθ , mξθ
⇒ sin nθ
On the basis of these arbitrary solutions to the differential equation of the eighth order
at hand, an element stiffness matrix has to be synthesized. As explained in the previous
subsection, four edge displacements (degrees of freedom) and four edge forces should
hence be described. However, three displacements and two rotations seem to be
available and therefore one of these is redundant. This is result of the neglect of the
influence of shear deformation, due to which the rotations are no longer independent
displacements but are related to the displacements of the reference surface. That the
rotation ϕθ is not a degree of freedom is easily observed when substituting the general
solution (3.1) for the displacements into the expressions for the rotations, which are
given in subsection 2.7.5. Hereby it is obtained that at the edge, in contrast to the
rotation ϕ ξ , the rotation ϕ θ is a linear combination of the independent displacements.
This means that if the displacements uθ and uζ are prescribed at a certain edge
ξ = constant , the rotation ϕ θ along the edge is explicitly prescribed, while the rotation
ϕ ξ not only depends on the magnitude of the displacements but also on the distribution
in the direction normal to the edge.
Therefore, in correspondence with the boundary conditions formulated in
subsection 2.7.5 for an edge ξ = constant , the edge displacements are associated with
uξ , uθ , uζ and ϕ ξ , and the edge forces are associated with f ξ , f θ , f ζ and tξ .
According to the relation between these edge forces and the internal stress quantities,
the edge quantities are distributed along the edge by functions of the form
f ξ , uξ
⇒ cos nθ
f θ , uθ
⇒ sin nθ
f ζ , uζ
⇒ cos nθ
tξ , ϕ 1ξ
⇒ cos nθ
56
3 Computational method and analysis method
The considerations described here are exemplified for a load that is symmetric to a
certain axis, but can easily be extended to an asymmetric load. Then the Fourier series
will be described by combinations of sine and cosine series per load term, which can
both simultaneously but in fact separately be treated as described above. Therefore, the
choice of a symmetric load does not degenerate the generality of the approach.
In the above, a suitable coordinate system has been chosen to represent the reference
surface of the element. For each node, the degrees of freedom and the generalised
forces are identified. The sign convention of the degrees of freedom is identical to the
sign convention that is assigned in chapter 2 for the displacements. Hence, the edge
displacements uξ , uθ and uζ are positive when acting in the positive direction of their
respective coordinate line and the edge rotation ϕ ξ is positive when rotating a point
with positive ζ coordinate in positive ξ -direction. The sign convention for the
direction in which the generalised forces act is thus identical at both sides of the
element.
Assuming that the governing equations are obtained, the general solution can be
derived and hence be put in matrix form. Having shown that the edge forces have the
same distribution in circumferential direction as the corresponding edge displacements,
it can be concluded that the relation between these quantities only depends on the
meridional distribution. In other words, a stiffness relation for the element depends on
the amplitude of the circumferential distribution (which can depend on the
circumferential mode number n ) but the trigonometric distribution needs not to be
taken into account. Hence, for every possible mode number n the general solution for
the degrees of freedom can be represented by
i
 uˆξ ( ξ ) 
 C1   uˆξ ( ξ ) 


  A11 ( ξ ) A18 ( ξ )     i
    uˆθ ( ξ ) 
 uˆθ ( ξ )  =  +
    uˆ i ( ξ ) 
 uˆζ ( ξ )  
ζ


  A41 ( ξ ) A48 ( ξ )     i
ϕˆ ξ ( ξ ) 
C8   ϕˆ ξ ( ξ ) 
(3.2)
or briefly as
uˆ ( ξ ) = A ( ξ ) c + uˆ i ( ξ )
c
c
c
where A ( ξ ) is a rectangular matrix of size 4 × 8 of which the coefficients depend on
the element geometry, the material properties and the mode number n . The hat
notation refers to amplitude and the superscript c represents that the matrix equation
refers to continuous quantities.
The general solution for the stress resultants and stress couples can be obtained by
successive substitution of the general solution for the displacements in the expressions
of the deformation quantities and the stress quantities. With the objective of
formulating expressions for the edge forces in mind, the internal stress quantities have
to be transformed into suitable quantities according to the boundary conditions
formulated in subsection 2.7.5. Performing the above-mentioned substitutions and
transformation, the general solution for the internal stress quantities is obtained which
can be represented by
c
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
i
 nˆξξ ( ξ ) 
 C1   nˆξξ ( ξ ) 

 ∗
  B11 ( ξ ) B18 ( ξ )     ∗i
    nˆξθ ( ξ ) 
 nˆξθ ( ξ )  =  +
    vˆ∗i ( ξ ) 
 vˆξ∗ ( ξ )  
ξ


  B41 ( ξ ) B48 ( ξ )     i
m
ξ
C
m
ˆ
ˆ
(
)
 ξξ 
 8   ξξ ( ξ ) 
(3.3)
or briefly as
nˆ ( ξ ) = B ( ξ ) c + nˆ i ( ξ )
c
c
c
where the matrix B ( ξ ) is also a rectangular matrix of size 4 × 8 of which the
coefficients depend on the element geometry, the material properties and the mode
number n .
The general solutions (3.2) and (3.3) can easily be transformed in expressions for
the edge displacements and edge forces of the element by substituting the nodal
coordinates of the parallel edge lines. While formulating the expressions for the edge
forces it is necessary to take into account that internal stress quantities on the negative
side of the element act in negative coordinate direction and thus in opposite direction to
the positive direction of the edge forces. Identifying one edge with ξ = a and the other
with ξ = b , the expressions for the element displacements and element forces become
c
 uˆξ ( a ) 


 uˆθ ( a )   A11 ( a )
 uˆζ ( a )   

ϕˆ ξ ( a )   A41 ( a )
 uˆ ( b )  =  A b
11 ( )
 ξ

 uˆθ ( b )   u b  
 ˆζ ( )   A41 ( b )
 ϕˆ ξ ( b ) 
i
 C1   uˆξ ( a ) 
 i



A18 ( a )  C2
 uˆθ ( a ) 



  C3   uˆζi ( a ) 

  
A48 ( a )  C4   ϕˆ iξ ( a ) 

+ i
A18 ( b )  C5   uˆξ ( b ) 
i
 
 C6   uˆθ ( b ) 
 i



A48 ( b )  C7   uˆζ ( b ) 
 C8   ϕˆ iξ ( b ) 
(3.4)
for the element displacements and
 fˆξ ( a ) 
ˆ

 f θ ( a )   B11 ( a )
 fˆζ ( a )   
 
 tˆξ ( a )   B41 ( a )
 fˆ ( b )  =  B b
 ξ
  11 ( )
ˆ
 fθ ( b )   ˆ
 
 f ζ ( b )   B41 ( b )
 tˆξ ( b ) 


 C1   fˆξi ( a ) 


B18 ( a )  C2   fˆθi ( a ) 
    ˆi
  C3 
f ( a )
 ζi



B48 ( a )  C4   tˆξ ( a ) 

+ i
B18 ( b )  C5   fˆξ ( b ) 
i
 
 C6   fˆθ ( b ) 

   i
B48 ( b )  C7   fˆζ ( b ) 
i
 C8   tˆξ ( b ) 
(3.5)
for the element forces. Notated briefly these two equations become
uˆ e = A ec + uˆ i ;e
(3.6)
and
fˆ e = B ec + fˆ i;e
(3.7)
where Ae and B e are square matrices of size 8 × 8 of which the coefficients depend on
the element geometry, the material properties and the mode number n . The hat
58
3 Computational method and analysis method
notation refers to amplitude and the superscript e represents that the matrix equation
refers to element quantities.
The element stiffness matrix relates the element displacements in (3.6) to the
element forces in (3.7). Therefore, the constants should be eliminated, which is done by
first rearranging expression (3.6) to
c = A -1;e ( uˆ e - uˆ i ;e )
(3.8)
and by substituting this expression into (3.7) resulting in
fˆ e = B e A -1;e ( uˆ e - uˆ i ;e ) + fˆ i ;e
(3.9)
From this equation, the so-called fixed edge forces can be obtained by setting the
element displacement uˆ e equal to zero. The result is denoted by the so-called primary
load vector fˆ prim;e per element, which can thus be computed by
fˆ prim;e = -B e A -1;euˆ i;e + fˆ i;e
(3.10)
Equation (3.9) can be rearranged into
fˆ e - fˆ i ;e + B e A -1;euˆ i;e = B e A -1;euˆ e
which, by using relation (3.10) for the primary forces, can be rewritten as
fˆ tot ;e = K euˆ e
(3.11)
where the element total load vector fˆ tot ;e and the element stiffness matrix K e are
introduced which can be computed by
fˆ tot ;e = fˆ e - fˆ prim;e
K e = B e A −1;e
(3.12)
Equations (3.12) determine per element the stiffness matrix and the load vector on the
element edges that correspond with the nodes. At such a node, an external force can be
applied and if more elements are to be calculated, two elements share one common
node. Introducing the external nodal load vector fˆ ext ;n , where the superscript n refers
to a nodal quantity, the load vector at a node fˆ tot ; n is given by the external load and the
primary load vector. Since a primary load acts on the element, it acts in opposite
direction on the node. Therefore, the contribution of a positive primary load to the
nodal load vector is in the negative direction and the expression for fˆ ext ;n becomes
fˆ tot ; n = fˆ ext ;n - ∑ fˆ prim;e; n
(3.13)
e
Comparing equation (3.12) for the element total load vector with equation (3.13) for
the nodal load vector it can be concluded that
fˆ ext ; n - ∑ fˆ e;n = 0
e
which represents that at a node the external load must be in equilibrium with the
element forces. In other words, the internal forces at the element edges are in
equilibrium with the applied load.
To correctly assemble the separate elements, care must be taken that at each node the
conditions of equilibrium of the loads and compatibility of the displacements are met.
The process of the assembly resulting in the global matrix equation, the incorporation
of the prescribed displacements and the solution of the resulting reduced global matrix
equation are to be done according to the common procedure of the stiffness method.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Consequently, incorporation of, for example, an elastic support and stiffening ring
elements or assembly of different geometries can be easily taken care of whereby,
when applicable, the transition to a global coordinate system can be plainly guaranteed.
The solution of the reduced global matrix yields the magnitude of the nodal
displacements and since these are equal to the element displacements uˆ e , the constants
c per element can be computed by relation (3.8). Having obtained the constants the
continuous distribution of the displacements and stress quantities within the element
can be computed by expressions (3.2) and (3.3), respectively. Finally, element forces
and support forces can be determined from these solutions.
3.3 Calculation scheme
The following steps are thus performed by a finite element program that is suited for
super elements:
1. Read number of elements and nodes;
2. Read geometry and material properties of each element and nodal coordinates;
3. Read initial displacements;
4. Read distributed forces on the element and external forces on the nodes;
5. Compute matrices A e and B e ;
6. Generate the element stiffness matrix K e according to relation (3.12);
7. Assemble the global stiffness matrix K via a location procedure;
8. Compute the primary load vector fˆ prim;e per element according to relation
(3.10);
9. Generate the load vector at a node fˆ tot ; n according to relation (3.12);
10. Assemble the global load vector fˆ tot via a location procedure;
11. Compose the system of equations and incorporate the prescribed
displacements;
12. Solve the system of equations to obtain the nodal displacements;
13. Compute per element the constants c according to relation (3.8);
14. Obtain the continuous distribution of the displacements and stress quantities of
each element according to expressions similar to (3.2) and (3.3), respectively;
15. Solve element and support forces.
A flow chart of such a program is given in Figure 3-1.
3.4 Introduction to the program CShell
An expeditious PC-oriented computer program, which is called CShell, is written to
calculate the elastostatic response of stiffened and non-stiffened circular cylindrical
shell structures. The program is based on the method presented the previous sections.
The calculation is performed by evaluating super elements that span a large subdomain
of the whole structure. Only one such element is needed to calculate the response of a
cylinder with a constant geometry and material properties, which is subject to linearly
distributed surface loads, nodal line loads and point loads.
60
All elements
3 Computational method and analysis method
INPUT
All elements
Ke
INPUT
fe
All elements
Ce
All load terms
u
Ke
Assemble global stiffness matrix
K
Compute element load vector
fe
Assemble global load vector
f
Incorporate prescribed displacements
INPUT
Red. K, f
Compute element stiffness matrix
Red. K, f
Solve system of equations
u
Solve coefficients
Ce
Compute continuous element
displacements and forces
ue, fe
fe
Determine continuous internal stresses
σe
fe
Determine support forces
F0
ue, fe, σe
Add all terms
∑ u, f, σ
Figure 3-1 Flow chart of the FEM program with super elements
61
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Figure 3-2 Example of a shell structure modelled by super elements.
3.4.1 Structure, supports and loading
The program CShell can be applied to calculate the response of thin shell structures,
that:
Consist of circular cylindrical elements and ring stiffening elements (see
Figure 3-2); extension and bending of the stiffener is taken into account while
torsion is neglected;
Are subject to static distributed surface loads, circumferential line loads and
point loads; these loads are symmetric with respect to the axis θ = 0 and are
developed in trigonometric series in circumferential direction (in θ -direction,
e.g., modes pn cos ( nθ ) for n = 0, 1, 2, ... ); results of the calculation appear as
one or all terms of the trigonometric series;
Have constant linear elastic material properties per element;
Have constant geometrical properties per element; and
Are supported by fixed and/or elastic supports.
3.4.2 Shell theory
Two theories for thin shells have been implemented, the theory of Morley-Koiter and
the theory of Donnell. The theory of Morley-Koiter is considered to be the most exact
one because it uses more appropriate kinematical relations for the changes of curvature.
The Donnell theory is less accurate for lower modes and clearly fails for mode n = 2 .
On the other hand, to obtain the solution to the Morley-Koiter theory an approximation
was introduced which limits its applicability for higher modes. Therefore, MorleyKoiter is used for lower modes and is compulsory for mode n = 2 . For higher modes,
Donnell gives more reliable results. The user indicates at which mode the switch
between Morley-Koiter and Donnell should be made. Default, Donnell is adopted for
n = 6 and higher.
62
3 Computational method and analysis method
The formulations resulting from the Morley-Koiter equation that are used in the
program CShell are derived in chapter 4. The exact homogeneous solution and the
inhomogeneous solution are given for mode number n = 0 in subsection 4.4.3, for n = 1
in subsection 4.4.4, and for n ≥ 2 in subsection 4.4.5. As mentioned above, the
approximate expressions for the homogeneous solution presented in section 4.5 are
used in the program CShell to derive the stiffness matrix of a circular cylindrical shell
element.
The formulations that are derived for the ring element stiffness matrices are
reproduced in Appendix E, which is based on the solution implemented by Van
Bentum [1]. The analysis herein presented is largely based on the same set of relations
as for a circular cylindrical shell on basis of the Morley-Koiter theory, but all quantities
in axial direction are omitted. The result is thus a stiffness relation between the
displacements of the ring element and the load on the ring element, viz. the forces at
the circular edges of the cylinder and the external ring loading.
3.4.3 Output
The following automated output is available:
Line plots in axial as well as circumferential direction using pre-defined
Grapher files;
The deformation of a circular profile as well as the whole structure by using a
pre-set Maple worksheet; and
A data file that can be addressed to select the quantities of interest and their
respective location.
3.4.4 Verification of the program CShell
At an early stage of the development and by using the packages available within the
research group, verification of the program for long circular cylindrical shells has been
performed. The results obtained by the super element program and the finite element
packages showed a close agreement and revealed that only improvement with respect
to small terms needed to be considered to accurately synthesize the stiffness matrix and
perform the back substitutions for the stress and displacement quantities.
Upon completion of the main program, the correctness of the implemented numerical
solution method has been verified versus finite element modelling using the wellknown ANSYS package. For this verification, a short circular cylinder with a radiusto-thickness ratio of 100 and a length-to-radius of 2 has been modelled. This short
cylinder represented a steel tank with a fixed bottom and with either a free end or an
eccentric ring at the top. For the verification purpose, the ring has been modelled as a
thick annular plate with a thickness equal to two times the wall thickness of the shell
and a width equal to the radius divided by 16. Hence, the structural super elements that
are implemented in the program are congruently modelled in the finite element package
to verify their performance.
To model the cylindrical shell in the ANSYS package, the SHELL281 element has
been adopted. The element is based on Mindlin-Reissner shell theory and is suitable for
analysing thin to moderately thick shell structures. The quadrilateral shaped element
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
has eight nodes with six degrees of freedom at each node (viz. three translations and
three rotations) and three integration points located through the thickness are
designated.
To model the ring beam in the ANSYS package, the BEAM189 element has been
adopted in view of compatibility with the SHELL281 element. The element is based on
Timoshenko beam theory and is suitable for analysing slender to moderately
stubby/thick beam structures. The quadratic three-node beam element in 3D with
unrestrained warping has six degrees of freedom at each node (viz. three translations
and three rotations) and employs a two point Gaussian integration.
The adopted mesh comprises a single element layer across the thickness and
element lengths in circumferential and axial direction have been automatically
generated based on the short influence length (refer to section 4.6). In circumferential
direction, the number of elements per quadrant has been generated based on a
maximum of half of the short influence length. The element length in axial direction is
generated based on an initial mesh of 4 elements in axial direction along the short
influence length, which is refined in the area of the element boundary while a larger
element axial length has been adopted in between the short influence lengths.
The verification of the developed program CShell by finite element modelling using
the well-known ANSYS package revealed an excellent agreement with respect to
displacement and deformed shape of the models. For the stresses, axial and shear
stresses are accurately predicted with respect to magnitude and shape. However, the
circumferential stresses (mainly the membrane component of those stresses) are less
accurate, which is closely related to the simplifications introduced to arrive at the
Morley-Koiter equation (refer to section 4.3). Only negligible numerical differences
could be detected between the respective results for the above-mentioned models.
Based on these observations, the numerical capability of the developed program and
the tremendous benefit of the super element approach for rational first-estimate design
are conclusively demonstrated.
3.5 Overview of the analysed structures
In this thesis, the following structures are studied with the aid of the developed
program:
Chimneys, which are supported at the bottom, with or without stiffening rings
and elastic supports (chapter 5); and
Tanks, which are supported at the bottom, with or without a roof or stiffening
ring at the top and under full circumferential settlement (chapter 6).
The chimneys are all loaded by a wind load. As described in section 5.1, this wind
load is developed into a quasi-static load series. The advantage is that each possible
load-deformation behaviour (as described in subsection 4.4.2) is present. Hence, the
different response for the same geometry enables the interpretation and enlarges the
understanding of the phenomena that occur per mode number.
The tanks are loaded by a content or wind load or subject to a full circumferential
settlement. These cases represent the three main load-deformation conditions that can
be identified for the overall response of the tank wall.
64
4 Circular cylindrical shells
4 Circular cylindrical shells
The analysis of circular cylindrical shells subject to static loading is carried out by an
exact method. The set of equations proposed in chapter 2 is formulated for circular
cylindrical shells with circular boundaries and the solution for three different loaddeformation behaviours is derived. For thin elastic shells, an approximation of this
exact solution is given and this approximate solution is compared with the solution
obtained by a perturbation technique. The resulting formulations for the displacements
and stress resultants can be readily presented in the context of the computational
method and solution procedure as explained in chapter 3. To enable understanding of
the shell behaviour and the prevailing parameters, characteristic and influence length
for the different load-deformation behaviours are discussed.
4.1 Introduction
4.1.1 Geometry
For a circular cylindrical shell, it is convenient to apply a polar coordinate system to
the cross-sectional profile with a constant radius a . The directions of the axes are
chosen in longitudinal direction, in circumferential direction and in transverse
direction. In relation to the description of the middle surface of shells of revolution, the
longitudinal direction is the direction along the meridian, the circumferential direction
is the direction along the parallel circle and the transverse direction is along the normal
to the reference surface.
An infinitesimal element has thus sides with length of arc, measured on the
reference surface, dx in longitudinal direction and adθ in circumferential direction.
The constant thickness of the element is denoted by t within which an infinitesimal
layer has a thickness dz in normal direction to the reference surface. The three positive
directions of the displacements ( u x , uθ , u z ) are taken corresponding to the three positive
coordinate directions ( x, θ, z ) .
4.1.2 Coordinate system
The straight generator in x -direction has an infinite radius and therefore its curvature
is equal to zero. The radius in θ -direction is already mentioned and equal to a . The
expression of the line element in Appendix A can now be given by
2
z

ds 2 = dx 2 + a 2  1 +  d θ2 + dz 2
 a
where x , θ and z are associated with ξ1 , ξ 2 and ζ , respectively.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Measured on the reference surface, the line element is thus equal to ds 2 = dx 2 + a 2 d θ2 .
This means that the following substitution can be made if the proposed theory of
section 2.7 is used as a starting point of our analysis
ξ1 = x
α1 = 1
R1 = ∞
ξ2 = θ
α2 = a
R2 = a
(4.1)
and that the Laplace operator defined by (2.7) becomes
∆=
∂2
1 ∂2
+
∂x 2 a 2 ∂θ2
(4.2)
4.2 Sets of equations
For the geometry and coordinate system as introduced in the previous section and
substituted accordingly in the proposed theory of section 2.7, the following vectors are
adopted to describe the kinematical, constitutive and equilibrium relations
u = [u x
uθ u z ]
e = [ ε xx
εθθ
γ xθ
κ xx
κ θθ ρ xθ ]
s = [ nxx
nθθ
nx θ
mxx
mθθ
p = [ px
pθ
pz ]
m xθ ]
(4.3)
4.2.1 Kinematical relation
The kinematical relation (2.39) is rewritten using the description (4.1) of the reference
surface of the circular cylindrical shell resulting in
 ∂
 ∂x

 0
 ε xx  
ε  
 θθ   1 ∂
 γ xθ   a ∂θ
 =
 κ xx   0
κ  
 θθ  
 ρ xθ   0


 0

0
1 ∂
a ∂θ
∂
∂x
0
0
2 ∂
a ∂x



1


a

 u 
0
 x
  uθ 
2
∂
 u 
− 2
 z
∂x

2
1 ∂
1
− 2 2 − 2
a ∂θ a 

2 ∂2

−
a ∂x∂θ 
0
4.2.2 Constitutive relation
The constitutive relation is given by (2.40) but with the new indices reads
66
(4.4)
4 Circular cylindrical shells
 Dm
 nxx   υDm
n  
 θθ   0
 n xθ  
 =
 mxx   0
m   0
 θθ  
 mxθ   0


υDm
0
0
0
Dm
0
0
0
0
0
0
1− υ
Dm
2
0
Db
υDb
0
0
υDb
Db
0
0
0
0
0

0   ε xx 
 
  εθθ 
0  
γ
  xθ 
0   κ xx 
0   κθθ 
 
1 − υ   ρ xθ 
Db

2 
0
(4.5)
where the quantities Dm and Db are the extensional (membrane) rigidity and flexural
(bending) rigidity, respectively, which are given by
Dm =
Et
1 − υ2
Db =
;
Et 3
12 (1 − υ2 )
(4.6)
The normal stresses σ xx and σθθ and the longitudinal shearing stress σ xθ can be
conveniently obtained from the relations given by (2.42) but with the new indices these
read
nxx
12m
+ z 3 xx
t
t
nθθ
12mθθ
σθθ =
+z 3
t
t
nx θ
12mxθ
σ xθ =
+z 3
t
t
σ xx =
(4.7)
4.2.3 Equilibrium relation
The equilibrium relation (2.43) is rewritten using the description (4.1) resulting in
 ∂
 − ∂x

 0


 0

0
−
1 ∂
a ∂θ
1
a
1 ∂
a ∂θ
∂
−
∂x
−
−
0
0
0
0
0
∂2
∂x 2
−
1 ∂2
1
− 2
2
2
a ∂θ a
  nxx a 
  nθθ a 
  p a

x
2 ∂   nx θ a  

−
=
p


θa 
a ∂x   mxx a  
 p a

2 ∂ 2   mθθ a   z 
−


a ∂x∂θ   mxθ a 
0
(4.8)
and the transverse shearing stress resultants are described by (2.44) and become
vx =
∂mxx 1 ∂mxθ
+
∂x
a ∂θ
;
vθ =
1 ∂mθθ ∂mxθ
+
a ∂θ
∂x
(4.9)
4.2.4 Boundary conditions
The boundary conditions (2.49) are rewritten using the description (4.1) resulting in
f x = nxx
f θ = nx θ
f z = vx∗
t x = mxx
or u x = u x
or uθ = uθ
or u z = u z





or ϕ x = ϕ x 
f x = −nxx
x = x( 2) ,
f θ = − nx θ
f z = −vx∗
t x = − mxx
or u x = u x
or uθ = uθ
or u z = u z



(1)
 x=x

or ϕ x = ϕ x 
(4.10)
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
where, by making use of (2.46), the combined internal stress resultant v∗x has become
v∗x = vx +
1 ∂mxθ ∂mxx 2 ∂mxθ
=
+
a ∂θ
∂x
a ∂θ
(4.11)
and, for completeness, both rotations defined by (2.47) have become
ϕx = −
∂u z
∂x
(4.12)
u 1 ∂u z
ϕθ = θ −
a a ∂θ
The additional condition for the torsion couple is identically satisfied since a circular
cylindrical shell with full circumferential boundaries is the subject of the further
analysis.
4.3 The resulting differential equations
4.3.1 The differential equations for the displacements
Up to this point, no additional simplifications or assumptions have been introduced. To
obtain convenient differential equations for the displacements, it is assumed that the
parameters describing the material properties and the cross-sectional geometry, i.e.
E , υ and a, t respectively, are constant for the whole circular cylindrical shell.
Substitution of the kinematical relation (4.4) into the constitutive relation (4.5)
results in what is sometimes referred to as the “elastic law”, which reads
1 ∂uθ
u 
 ∂u
nxx = Dm  x + υ
+υ z 
a ∂θ
a
 ∂x
 ∂ 2u
1 ∂ 2u z
u 
mxx = − Db  2z + υ 2
+ υ 2z 
2
a ∂θ
a 
 ∂x
 ∂u 1 ∂uθ u z 
nθθ = Dm  υ x +
+ 
 ∂x a ∂θ a 
 ∂ 2u
1 ∂ 2u z u z 
mθθ = − Db  υ 2z + 2
+ 
a ∂θ2 a 2 
 ∂x
nxθ = Dm
1 − υ  1 ∂u x ∂uθ 
+


2  a ∂θ ∂x 
(4.13)
 1 ∂uθ 1 ∂ 2u z 
mxθ = − Db (1 − υ)  −
+

 a ∂x a ∂x∂θ 
Substitution of this elastic law into (4.8) yields the following three differential
equations for the displacements
−
−
υ
∂ 2u x 1 − υ 1 ∂ 2u x 1 + υ 1 ∂ 2u θ
1 ∂u z
p
−
−
−υ
= x
∂x 2
2 a 2 ∂θ2
2 a ∂x∂θ
a ∂x Dm
1 + υ 1 ∂ 2u x 1 − υ ∂ 2uθ 1 ∂ 2uθ 1 ∂u z
−
−
−
2 a ∂x∂θ
2 ∂x 2 a 2 ∂θ2 a 2 ∂θ
D
∂ 2u
D
∂ 3u
p
− b 2 2 (1 − υ ) 2θ + b 2 2 (1 − υ ) 2 z = θ
Dm a
∂x
Dm a
∂x ∂θ Dm
1 ∂u x 1 ∂uθ 1
D
∂ 3u
+ 2
+ 2 u z − b 2 2 (1 − υ) 2 θ
a ∂x a ∂θ a
Dm a
∂x ∂θ
+
68
Db  ∂ 4u z 2 ∂ 4u z
1 ∂ 4u z 2 υ ∂ 2 u z 2 ∂ 2u z u z  p z
+
+
+ =
 4 + 2 2 2+ 4
Dm  ∂x
a ∂x ∂θ a ∂θ4 a 2 ∂x 2 a 4 ∂θ2 a 4  Dm
(4.14)
4 Circular cylindrical shells
The three differential equations are symbolically described by
 L11
L
 21
 L31
L13  u x 
 px 
1  



L23  uθ  =
pθ
Dm  
 pz 
L32 L33  u z 
The operators L11 up to and including L33 form a differential operator matrix, in which
L12
L22
the operators are
1 + υ 1 ∂2
2 a 2 ∂θ2
∂2
1 + υ ∂2
L22 = −∆ +
−
k
−
υ
2
1
(
)
2 ∂x 2
∂x 2
L11 = −∆ +
1 + υ 1 ∂2
2 a ∂x∂θ
1 ∂
L13 = − L31 = −υ
a ∂x
L12 = L21 = −
2
1
1 
1 ∂
∂2
∂3
2
2
ka
2
ka
1
L
L
2
k
1
+
∆
+
−
−
υ
=
−
=
−
+
−
υ
(
)
(
)
23
32


a2
a2 
a 2 ∂θ
∂x 2
∂x 2∂θ

Here the Laplace operator ∆ is defined by (4.2) and the dimensionless parameter k is
L33 =
introduced, which is defined by
k=
Db
t2
=
2
Dm a 12a 2
(4.15)
Hence, it is noted that for a thin shell where t < a it follows that the parameter k is
negligibly small in comparison to unity ( k 1) .
4.3.2 The single differential equations
By eliminating u x from the first and second equation, the differential equation
describing the relation between uθ and u z is obtained. Equivalently, uθ is eliminated
from the first and second equation to obtain a relation between u x and u z . The
resulting equations symbolically read, respectively
( L11L22 − L21L12 ) uθ + ( L11L23 − L21L13 ) u z =
1
( L11 pθ − L21 px )
Dm
( L22 L11 − L12 L21 ) u x + ( L22 L13 − L12 L23 ) u z =
1
( L22 px − L12 pθ )
Dm
This operation is only possible if the operators on a scalar function φ are commutative,
which means that for example ( L21L11 − L11L12 ) φ = 0 .
By substituting these two relations into the third equation, the single differential
equation for the displacement un is obtained, which symbolically reads
 L31 ( L12 L23 − L22 L13 ) + L32 ( L21L13 − L11L23 ) + L33 ( L22 L11 − L12 L21 )  u z
1
=
( L22 L11 − L12 L21 ) p z + ( L31L12 − L32 L11 ) pθ + ( L32 L21 − L31L22 ) px 
Dm 
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
In this multiplication of the derivatives in the proper manner, the terms with the square
of the parameter k are neglected in comparison to unity ( k 2 1) . The single
differential equation is then obtained as
2
4
4
1 

 β  ∂ uz
∆∆  ∆ + 2  u z + 4  
4
a 

 a  ∂x
1  ∂4
1 ∂4
1 ∂ 4  ∂ 2u
− 2 (1 + υ ) 2 2 2 − 4 4  2z
2
4
a  ∂x
a ∂x ∂θ a ∂θ  ∂x
4
1 ∂u
+4 (1 − υ2 ) 4 4z
a ∂x
1 
1 ∂ 3 pθ
1 ∂ 3 pθ
1 ∂ 3 px 1 ∂ 3 px 
=
+υ
−
 ∆∆pz + (2 + υ) 2 2 + 4

3
Db 
a ∂x ∂θ a ∂θ
a ∂x3 a 3 ∂x∂θ2 
−2 (1 − υ )
(4.16)
Here the dimensionless parameter β is introduced, which is defined by
4β4 =
1 − υ2
a
= 12 (1 − υ2 )  
k
t
2
(4.17)
and hence it can be concluded that, since t < a for any circular cylinder, β > 1 .
Moreover, in case that the fourth derivative with respect to x of the function u z exists,
the one that is multiplied with the parameter β4 will be a leading term since β4 1 .
For the edge disturbances described by the homogenous solution it is apparent that
the highest derivatives with respect to x may account for the rapid variations in axial
direction of the displacement field while the lowest derivatives may account for the
slow variations of this field. The magnitude of the fourth derivative with respect to x
that is multiplied with the parameter β4 is sufficiently large to neglect the other fourth
derivatives with respect to x , which also holds for the probably small contributions of
sixth and second derivatives. Hence, purely for convenience a simplified and from
mathematical point of view considerably more elegant differential equation can be set
up with practically the same numerical accuracy.
Small differences between an exact solution to equation (4.16) and a solution to an
approximated equation will however exist, but these differences are of the same order
as introduced by neglecting the terms ζ Rα in comparison to unity in order to obtain
the constitutive relation (2.40). Hence, emphasizing these differences is meaningless
unless the transverse shear deformation and the deformation due to strain in the
direction normal to the reference surface are also taken into account.
In accordance with the above-mentioned considerations, the single differential equation
is approximated by neglecting the derivatives that do not contribute substantially in
comparison to the derivative multiplied by the parameter β4 . For the other two
equations relating u x and uθ to u z , a similar observation leads to the neglect of small
terms, which are multiplied by the parameter k . In this way, three differential equation
are obtained that read
70
4 Circular cylindrical shells
1− υ 
1 ∂ 3u z
1 ∂ 3u z  1 1 + υ 1 ∂ 2 p x
1 + υ 1 ∂ 2 pθ 
=
−
∆
p
+
 ∆∆uθ + ( 2 + υ ) 2 2 + 4



θ
2 
2 a 2 ∂θ2 
a ∂x ∂θ a ∂θ3  Dm  2 a ∂x∂θ
1− υ 
1 ∂ 3u z 1 ∂ 3u z  1 
1 + υ ∂ 2 p x 1 + υ 1 ∂ 2 pθ 
− 3
=
+
 ∆∆u x + υ
 −∆px +

3
2
2 
a ∂x
a ∂x∂θ  Dm 
2 ∂x 2
2 a ∂x∂θ 
2
4
4
1 

 β  ∂ uz
∆∆  ∆ + 2  u z + 4  
=
4
a 

 a  ∂x
(4.18)
1 
1 ∂ 3 pθ
1 ∂ 3 pθ
1 ∂3 px 1 ∂3 px 
+
υ
−
 ∆∆p z + (2 + υ) 2 2 + 4

a ∂x ∂θ a ∂θ3
a ∂x 3 a 3 ∂x∂θ2 
Db 
where the last is the well-known Morley-Koiter equation.
This equation, suggested by Morley [35] and later derived by Koiter [36] and Niordson
[37], overcomes both the completeness of Flügge’s approach in retaining second-order
terms, which do not exceed the accuracy of the initial assumptions, and the inaccuracy
of Donnell’s simplifications in its inability to describe rigid-body modes but preserves
its elegance and simplicity.
Strangely, the first two equations of (4.18) are widely accepted but the many variations
for the single differential equation (similar to the third equation of (4.18)) indicate that
general consensus has not yet been obtained. A discussion on the variety of proposed
equations is presented in section 2.6. In subsection 4.5.2 the Morley-Koiter equation
and some of the suggested equations are listed. In that subsection, the somewhat forced
simplification and its implication is exemplified by means of the respective solutions to
the homogeneous equation. Since the homogeneous solution to the Morley-Koiter
equation is mathematically the most suitable for substitution with the same accuracy,
this equation is considered in the further treatment of circular cylindrical shells.
4.4 Full circular
boundaries
cylindrical
shell
with
curved
4.4.1 Load as infinite trigonometric series
As stated in subsection 3.2.2, continuity and symmetry of the load in circumferential
direction is assumed. By choosing this line of symmetry to be indicated by θ = 0 , the
three load components can be described by a Fourier trigonometric series expressed by
∞
p x ( x, θ ) = ∑ pxn ( x ) cos nθ
n =0
∞
pθ ( x, θ ) = ∑ pθn ( x ) sin nθ
(4.19)
n=0
∞
p z ( x, θ ) = ∑ pzn ( x ) cos nθ
n=0
where n is the circumferential mode number and represents the number of whole
waves in circumferential direction.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The following considerations are derived for a load that is symmetric to a certain
axis, but can easily be extended to an asymmetric load by describing combinations of
sine and cosine series per load term. These can be treated separately with congruent
resulting expressions, whereby the choice of a symmetric load does not degenerate the
generality of the approach.
4.4.2 Three load-deformation behaviours
The response of a cylinder to all possible loads indicated by a different mode number
n can be subdivided into three different load-deformation behaviours. Consider a
(long) circular cylinder without restricting boundary conditions. The response of such a
cylinder to any load is obviously equal to the response of a ring to that load. In Figure
4-1 the load and the corresponding deformation for four terms is displayed for a
circular ring. Actually, Figure 4-1 only gives the response to the loads pθ ( θ ) and
p z ( θ ) , but it is obvious that a load px on a cylinder gives a congruent behaviour per
mode number n .
Figure 4-1 Four load terms and their corresponding deformation for a circular ring
4.4.2.1 Axisymmetric mode
The mode indicated by n = 0 (left in Figure 4-1) is generally known as the
axisymmetric mode and describes a constant behaviour in circumferential direction.
Such a load leads, in principle, to a change in the radius of the cylinder with circular
edges. Any quantity φ must be constant in circumferential direction; in other words,
∂φ
= 0 is to be made in the governing equations. Moreover, a constant
∂θ
displacement in circumferential direction uθ only represents a rigid body rotation of the
the substitution
cylinder, which does not lead to any strain, and because of symmetry considerations,
the rotation ϕθ should be zero. The same applies to the longitudinal shearing strain
γ xθ , the torsion ρ xθ , the corresponding longitudinal shearing stress σ xθ and its resultant
nxθ and couple mxθ , which also can be concluded by inspecting the governing
equations.
72
4 Circular cylindrical shells
Not being able to describe uθ leaves only two independent displacements to
describe ( u x and u z ), which implies that only two differential equations expressed in
the displacements can be obtained (while pθ is equal to zero). Hence, the resulting
differential equation will not be of the eighth order, but from inspecting the system
(4.14) of the sixth order since the differential equations for px and pz are of the
second order for u x and of the fourth order for u z , respectively.
To remain consistent the axisymmetric mode will be studied with the MorleyKoiter equation, while making the necessary substitutions, but also the sets of
equations given in section 4.2 are used as the starting point.
4.4.2.2 Beam mode
The mode indicated by n = 1 (second left in Figure 4-1) is generally known as the beam
mode and describes the response of the circular cylinder that is obtained if it were
treated as a beam with a circular cross-section. In other words, the lateral deflection of
the circular cylinder is caused by the resultant of such a load term. However, using the
expressions derived for three independent shell displacements, viz. u x , uθ and u z ,
results in an inherent description corresponding to a beam with flexural as well as shear
rigidity. Moreover, the governing equation is an eighth order differential equation and
hence not only the fourth order polynomial solution representing the beam type of
behaviour is described, but also a solution is obtained that takes care of the
nonconforming deformation states that can be obtained at the circular boundary.
Obviously, this part of the solution describes an edge disturbance that mainly originates
from the cross-sectional deformation that can be prohibited.
The behaviour described above is excellently described by the Morley-Koiter equation,
where for n = 1 all quantities can be expressed as functions of the type
φ ( x, θ ) = φ1 ( x ) cos θ and φ ( x, θ ) = φ1 ( x ) sin θ depending on the axis of symmetry of the
quantity under consideration.
4.4.2.3 Self-balancing modes
The modes indicated by n = 2,3,4,... ( n = 2 and n = 3 are depicted at the right-hand side
in Figure 4-1) are generally known as the self-balancing modes. Obviously, the load
has as many symmetry axes as the mode number n , where these axes cross each other
at the middle point of the circle, which also holds for n anti-symmetry axes. The
response of a ring to such a load is fully described by a deformation of the circular
shape without displacing the middle point of that circle since the resultant of the load is
equal to zero.
The response of a full cylinder without restriction to the deformation at its circular
boundaries will be equal to the response of a ring with the circular profile. However, if
this response behaviour is restricted at any circle, not only bending in circumferential
direction will occur, but also both bending and membrane straining in axial direction.
The behaviour described above is excellently described by the Morley-Koiter
equation, where for the mode numbers n > 1 all quantities can be expressed as
73
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
functions of the type φ ( x, θ ) = φn ( x ) cos nθ and φ ( x, θ ) = φn ( x ) sin nθ depending on the
axis of symmetry of the quantity under consideration.
4.4.3
Solution for the axisymmetric load ( n = 0 )
As stated in subsection 4.4.2.1, all quantities are constant in circumferential direction
for the axisymmetric mode. Hence, the following substitutions for the load and
displacements can be made
p x ( x, θ ) = p x 0 ( x ) ,
u x ( x, θ ) = u x 0 ( x )
pθ ( x, θ ) = 0
uθ ( x, θ ) = 0
,
p z ( x, θ ) = p z 0 ( x ) ,
u z ( x, θ ) = u z 0 ( x )
 ∂φ

Obviously, the derivative with respect to θ is equal to zero  = 0  for all quantities.
 ∂θ

4.4.3.1
Differential equation
∂φ
= 0 , the set (4.14) can thus be written as
∂θ
∂ 2u
1 ∂u z
p
− 2x − υ
= x
∂x
a ∂x Dm
By setting
D  ∂ 4u 2 υ ∂ 2u z u z  p z
1 ∂u x 1
υ
+ 2 u z + b  4z + 2
+ =
a ∂x a
Dm  ∂x
a ∂x 2 a 4  Dm
(4.20)
By substitution of the load and displacement functions given above, the single
differential equation (4.18) becomes
2
4
4
4
3
 d 2
1 
1 d px 0 ( x ) 
 β   d u z 0 ( x ) 1  d pz 0 ( x )
 2 + 2  + 4   
=
+
υ


4
dx
a 
Db  dx 4
a dx 3 
 a   dx

The equation presented here is of the eighth order, but this is due to the fact that
equation (4.18) is derived by eliminating the displacement uθ , which is zero for the
axisymmetric case under consideration.
Integrating thrice with respect to coordinate x yields the sought equation, which
reads
2
4
 d 2

1 
1
 β   du ( x ) 1  dpz 0 ( x )
 2 + 2  + 4    z 0
=
+ υ px 0 ( x ) 

dx
a 
Db  dx
a
 a   dx


(4.21)
Applying a similar procedure to the second equation of the set (4.18), results in
∂ 2u x 0 ( x )
∂x 2
+υ
1 ∂u z 0 ( x )
1
px 0 ( x )
=
a ∂x
Dm
(4.22)
Obviously, equation (4.22) is equal to the first equation of the set (4.20) and an
equation similar to (4.21) could be directly obtained from this set. Doing so, the single
differential equation for u z reads
 Db  d 4
 du z
2 d2
1  1
1  dpz
1 
2
=
+ υ px 


 4 + υ 2 2 + 4  + 2 (1 − υ ) 
a dx
a  a
a 
 Dm  dx
 dx Dm  dx
74
4 Circular cylindrical shells
A single differential equation for u x can also be obtained, which reads
 D  d4
 d 2u
2 d2
1  1
1   1 Db d 4 
1 dp z 
−  b  4 + υ 2 2 + 4  + 2 (1 − υ2 )  2x =
p +υ
 2 +

4 x
a dx
a  a
Dm   a
Dm dx 
a dx 
 Dm  dx
 dx
These two equations show that the solution for u x will contain one more constant (i.e.
six) than the solution for u z (i.e. five constants).
Equation (4.21), which is obtained from the single differential equation (4.18), is
slightly different from the equation obtained from the set (4.20). However, the
difference between the solutions to these equations is small and since it is allowed to
neglect this difference, equation (4.21) will be adopted in the further analysis.
4.4.3.2 Homogeneous solution
The general solution consists of a homogeneous and an inhomogeneous part. By
inspecting the differential equation (4.21), it is observed that the homogeneous part can
be separated in a polynomial part and a non-polynomial part. The latter can be obtained
by solving the following homogeneous equation
2
4
 d 2
1 
β 
 2 + 2  + 4    u z 0 ( x ) = 0
a 
 a  
 dx
for which the solution is given by (see also Appendix H)
uz0 ( x ) = e
− a0 β
x
a

x
x   a0β a 
x
x 




C1 cos  b0β a  + C2 sin  b0β a   + e
C3 cos  b0β a  + C4 sin  b0β a   (4.23)










x
where the dimensionless parameters a0 and b0 are defined by
1
1
1

2
a0 = (1 + γ 02 ) 2 + γ 0 


1

2
; b0 = (1 + γ 02 ) 2 − γ 0 


in which γ 0 = −
1
2β2
(4.24)
The corresponding part for the axial displacement u x is obtained by solving the second
equation of the set (4.18), which results for the independent displacement u x in
1
uz 0 ( x ) dx
a∫
in which u z 0 ( x ) is thus given by expression (4.23).
u x 0 ( x ) = −υ
4.4.3.3 Inhomogeneous solution
Assuming a constant load px and a linear load pz , the inhomogeneous solution to the
single differential equation (4.21) reads
Etu z 0 ( x ) = a 2 p z 0 ( x ) + υa ∫ px 0 ( x ) dx
and by substitution of this result into the second equation of the set (4.18) the
inhomogeneous solution for the axial displacement u x can be obtained, which gives
Etu x 0 ( x ) = −υa ∫ pz 0 ( x ) dx − ∫∫ px 0 ( x ) dxdx
The two constants that will arise are in fact part of the homogeneous solution but can
be presented here for convenience allowing for a better insight into the solution.
75
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
4.4.3.4 Complete solution
Describing the loads px and pz by the forms
p x 0 ( x ) = px 0
p z 0 ( x ) = p z( 20)
x
+ pz(10)
l
the complete solution for the independent displacement u z reads
uz ( x ) = e
− a0 β
+
x
a
x

x
x   a0 β a 
x
x 




C1 cos  b0β a  + C2 sin  b0β a   + e
C3 cos  b0β a  + C4 sin  b0β a  











1  2  ( 2) x

a  p z 0 + pz(10)  + υa ( p x x + C5 ) 
Et  
l


(4.25)
and the complete solution for the independent displacement u x reads
ux ( x ) = υ
x
1
1
x
x 
 − a0 β a 
e
( a0C1 + b0C2 ) cos  b0β  + ( −b0C1 + a0C2 ) sin  b0β  


2
2
2β ( a0 ) + ( b0 ) 
a
a 



+e
−
a0 β
x
a

x
x   


( − a0C3 + b0C4 ) cos  b0β a  + ( −b0C3 − a0C4 ) sin  b0β a   



  

(4.26)

1   1 p z( 20) 2
1
1 
x + pz( 0) x  + px x 2 + C5 x + C6 
 υa 
Et   2 l
 2

By substitution of these expressions for the independent displacements into expressions
(4.9), (4.11) and (4.13), the complete solution for all nontrivial quantities can be
obtained as exemplified in Appendix I.
For the axisymmetric mode, a further approximation can be adopted by neglecting of
β−2 in comparison to unity to provide more insight into the response of the circular
cylindrical shell.
For n = 0 , the dimensionless parameters a0 and b0 (4.24) become equal to unity. If
only the loading normal to the shell surface is considered, i.e. pz = q ( x ) and px = 0 , the
full solution is then described by
uz0 ( x ) = e
−β
x
a

 x
 x  β a 
 x
 x  q ( x ) a
C1 cos  β a  + C2 sin  β a   + e C3 cos  β a  + C4 sin  β a   + Et










x
2
It is readily verified that this approximated solution would be the exact solution to
the following differential equation
Db
d 4u z 0 ( x ) Et
+ 2 uz0 ( x ) = q ( x )
dx 4
a
which is the corresponding approximation of differential equation (4.21).
The above differential equation is identical to the one for a beam on an elastic
foundation if the modulus of subgrade is taken as
Et
and the flexural rigidity of the
a2
beam is described by the flexural rigidity of a (curved) plate. Hence, it is observed that
76
4 Circular cylindrical shells
the circular cylinder under axisymmetric loading behaves as a flexural strip that is
elastically supported by the membrane action of a ring.
4.4.4
Solution for the beam load ( n = 1)
As stated in subsection 4.4.2.2, all quantities for the beam mode can be described by
functions of the type φ ( x, θ ) = φ1 ( x ) cos θ and φ ( x, θ ) = φ1 ( x ) sin θ depending on the axis
of symmetry of the quantity under consideration. Hence, the following substitutions for
the load and displacements can be made
p x ( x, θ ) = px1 ( x ) cos θ
,
u x ( x, θ ) = u x1 ( x ) cos θ
pθ ( x, θ ) = pθ1 ( x ) sin θ
,
uθ ( x, θ ) = uθ1 ( x ) sin θ
p z ( x, θ ) = p z1 ( x ) cos θ
,
u z ( x, θ ) = u z1 ( x ) cos θ
while for the derivates with respect to the circumferential coordinate θ and
consequently for the Laplace operator (4.2) substitutions can be made of the form
∂φ ( x, θ ) ∂φ1 ( x ) cos θ
=
= −φ1 ( x ) sin θ
∂θ
∂θ
 d2
1 
∆φ ( x, θ ) = ∆1φ1 ( x ) cos θ =  2 − 2  φ1 ( x ) cos θ
a 
 dx
(4.27)
for quantities generally described by φ ( x, θ ) = φ1 ( x ) cos θ and similarly for the quantities
generally described by φ ( x, θ ) = φ1 ( x ) sin θ .
4.4.4.1 Differential equation
By substituting the load and displacement functions given above, the single differential
equation (4.18) becomes an ordinary differential equation and by omitting the cosine
function for the circumferential distribution, the governing differential equation is
reduced to
4
4
2

1 d pθ1 ( x ) 1
 β   d u z1 ( x ) 1 
p
x
(2
)
=
∆
∆
+
+
υ
− 4 pθ1 ( x )
(
)
 ∆1∆1 + 4   

1
1
1
z
4
Db 
a 2 dx 2
a
 a   dx

3
1 d p x1 ( x ) 1 dpx1 ( x ) 
+υ
+ 3

a dx3
a
dx 
(4.28)
in which the Laplace operator ∆1 for n = 1 is defined by (4.27)
4.4.4.2 Homogeneous solution
The general solution consists of a homogeneous and an inhomogeneous part. By
inspecting the differential equation (4.28), it is observed that the homogeneous part can
be separated in a polynomial part and a non-polynomial part. The latter can be obtained
by solving the following homogeneous equation
4

β 
 ∆1∆1 + 4    u z1 ( x ) = 0
 a  

for which the solution can be written as
77
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
4
1 a 
u z1 ( x ) = −   ∆1∆1u z1 ( x )
4β 
(4.29)
The solution to the homogeneous equation is given by (see also Appendix H)

x
x   a1β a 
x
x 




C1 cos  b1β a  + C2 sin  b1β a   + e C3 cos  b1β a  + C4 sin  b1β a   (4.30)











where the dimensionless parameters a1 and b1 are defined by
u z1 ( x ) = e
− a1β
x
a
x
1
1
1

2
a1 = (1 + γ12 ) 2 + γ1 


1

2
1
; b1 = (1 + γ12 ) 2 − γ1  in which γ1 = 2
2
β


(4.31)
The homogeneous solution for the independent displacements u x and uθ can be
obtained by solving the first two equations of the set (4.18) for which the homogeneous
equations read
1 ∂ 3u z
1 ∂ 3u z
− 4
2
2
a ∂x ∂θ a ∂θ3
1 ∂ 3u z 1 ∂ 3u z
∆∆u x = −υ
+
a ∂x3 a 3 ∂x∂θ2
∆∆uθ = − ( 2 + υ )
By substitution of the displacement functions given above, these become ordinary
differential equations in which the sine function (for uθ ) and the cosine function (for
u x ) can be omitted. By substituting the representation (4.29) for u z1 ( x ) the following
equations are obtained
4
1 a  
1 d2
1
∆1∆1uθ1 ( x ) = −   ( 2 + υ) 2 2 − 4  ∆1∆1u z1 ( x )
4β  
a dx
a 
1 a 
∆1∆1u x1 ( x ) =  
4 β 
4
 1 d3
1 d 
− 3  ∆1∆1u z1 ( x )
 −υ
3
a
dx
a
dx 

and by omitting the Laplace operators the differential equations reduce to
2

1 a  
1 d u z1 ( x ) 1
uθ1 ( x ) = −   ( 2 + υ ) 2
− 4 u z1 ( x ) 
4β  
a
dx 2
a

4
3
1  a   1 d u z1 ( x ) 1 du z1 ( x ) 
u x1 ( x ) =    υ
+ 3

4  β   a dx3
a
dx 
4
4.4.4.3 Inhomogeneous solution
Assuming linear loads px , pθ and pz , the inhomogeneous equation of (4.28) reduces
to
2
4
4
2
2
1 
1 d pθ1 ( x ) 1
 β  d u z1 ( x ) 1  d
=
− 4 pθ1 ( x )
4 
 2 − 2  p z1 ( x ) + (2 + υ) 2
4
dx
Db  dx
a 
a
dx 2
a
a

3
1 d p x1 ( x ) 1 dpx1 ( x ) 
+υ
+ 3

a dx3
a
dx 
since β4 1 . By subsequent integration and rearrangement of the expressions, the
inhomogeneous solution is obtained as
78
4 Circular cylindrical shells
u z1 ( x ) =
a2  1
2 
2+υ


p x − pθ1 ( x ) ) dxdxdxdx − 2 ∫∫  pz1 ( x ) −
pθ1 ( x )  dxdx + pz1 ( x ) 

4 ∫∫∫ ∫ ( z1 ( )
Et  a
a
2



+
a 1

p x1 ( x ) dxdxdx + υ∫ p x1 ( x ) dx 
Et  a 2 ∫∫∫

The four constants that will arise are in fact part of the homogeneous solution but can
be presented here for convenience allowing for a better insight into the solution.
Obviously, this part of the solution is in fact the membrane solution. A formal way of
obtaining expressions for the other independent displacement would be to substitute the
solution for u z1 and subsequently neglecting small terms. However, a more natural way
is to determine the membrane solution to the equilibrium equations.
As shown by in [43], the membrane solution to the set of equilibrium equations (4.8)
reads
nθθ ( x, θ ) = a cos θ pz1 ( x ) 
1

nxθ ( x, θ ) = a sin θ  ∫ ( p z1 ( x ) − pθ1 ( x ) ) dx 
a


1
 1

nxx ( x, θ ) = a cos θ  − 2 ∫∫ ( pz1 ( x ) − pθ1 ( x ) ) dxdx − ∫ px1 ( x ) dx 
a
 a

Using the “elastic law” (4.13), the displacements can be obtained by successive
determination, which yields
u x ( x, θ ) =
a2
1
 1
cos θ  − 3 ∫∫∫ ( p z1 ( x ) − pθ1 ( x ) ) dxdxdx − υ ∫ pz1 ( x ) dx
Et
a
 a
−
uθ ( x , θ ) =
1

p x1 ( x ) dxdx 
a 2 ∫∫

a2
 1
sin θ  − 4 ∫∫∫∫ ( p z1 ( x ) − pθ1 ( x ) ) dxdxdxdx
Et
 a
+ ( 2 + υ)
−
u z ( x, θ ) =

2 (1 + υ )
1 
pθ1 ( x )  dxdx
 p z1 ( x ) −
a 2 ∫∫ 
2+υ

1

p x dxdxdx 
3 ∫∫∫ x1 ( )
a

a2
1
cos θ  4 ∫ ∫∫∫ ( pz1 ( x ) − pθ1 ( x ) ) dxdxdxdx
Et
a
−2
1 
2+υ

p x −
pθ1 ( x )  dxdx + pz1 ( x )
2 ∫∫  z1 ( )
a
2


1
1

p x dxdxdx + υ ∫ px1 ( x ) dx 
3 ∫∫∫ x1 ( )
a
a

where the expression for u z ( x, θ ) is exactly equal to the solution to the inhomogeneous
+
equation presented above. Moreover, if these expressions are substituted into the first
two equations of the set (4.18), these equations are identically satisfied.
79
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
By substitution of the expression for u z ( x, θ ) the rotations become
ϕ x ( x, θ ) = −
du z1 ( x )
cos θ
dx
;
ϕθ ( x, θ ) = u z1 ( x ) sin θ
and all other quantities are zero for the membrane solution, which is thus equal to the
inhomogeneous solution.
4.4.4.4 Complete solution
Describing the loads px , pθ and pz by the forms
p x1 ( x ) = px1
pθ1 ( x ) = pθ( 12)
p z1 ( x ) = p z(12)
x
+ pθ(11)
l
x
+ p z(11)
l
the complete solution for the independent displacement u z reads
x
x 
 − a1β 
 x

u z ( x, θ ) = cos θ e a C1 cos  b1β  + C2 sin  b1β  
a
a






+e
+
a1β
x
a

 x
 x   
C3 cos  b1β a  + C4 sin  b1β a   



  

 p ( 2)  1 5 1 2 3
1

 1

x − a x + a 4 x  + pz(11)  x 4 − a 2 x 2 + a 4 
cos θ  z1 
2
Eta
l
120
3
24





p( 2)  1 5 2 + υ 2 3 
2+υ 2 2
 1
− θ1 
x −
a x  − pθ(11)  x 4 −
a x 
l  120
6
24
2




1 3

+ apx1  x + υa 2 x  
6

(4.32)
3
 1  x 2

x
x
 1  x 

+ cos θ    − 2  C5 +    − 2  C6 + C7 + C8 
 6 a 



a
a

 2 a 


Similar expressions for the independent displacements uθ and u x are obtained by the
appropriate substitutions.
By substitution of the expressions for the independent displacements into the
expressions (4.9), (4.11) and (4.13), the complete solution for all nontrivial quantities
can be obtained as exemplified in Appendix I.
4.4.5
Solution for the self-balancing loads ( n > 1)
As stated in subsection 4.4.2.3, all quantities for the self-balancing modes can be
described by functions of the type φ ( x, θ ) = φn ( x ) cos nθ and φ ( x, θ ) = φn ( x ) sin nθ
depending on the axis of symmetry of the quantity under consideration. Hence, the
following substitutions for the loads and displacements can be made
80
4 Circular cylindrical shells
p x ( x, θ ) = pxn ( x ) cos nθ
;
u x ( x, θ ) = u xn ( x ) cos nθ
pθ ( x, θ ) = pθn ( x ) sin nθ
;
uθ ( x, θ ) = uθn ( x ) sin nθ
p z ( x, θ ) = p zn ( x ) cos nθ
;
u z ( x, θ ) = u zn ( x ) cos nθ
while for the derivates with respect to the circumferential coordinate θ and
consequently for the Laplace operator (4.2) substitutions can be made of the form
∂φ ( x, θ ) ∂φn ( x ) cos nθ
=
= −nφn ( x ) sin nθ
∂θ
∂θ
 d 2 n2 
∆φ ( x, θ ) = ∆ nφn ( x ) cos nθ =  2 − 2  φn ( x ) cos nθ
a 
 dx
(4.33)
for quantities generally described by φ ( x, θ ) = φn ( x ) cos nθ and similarly for the
quantities generally described by φ ( x, θ ) = φn ( x ) sin nθ .
4.4.5.1 Differential equation
By substitution of the load and displacement functions given above, the single
differential equation (4.18) becomes an ordinary differential equation and by omitting
the cosine function for the circumferential distribution, the governing differential
equation is reduced to
2
4
4

1 
1

β d 
u x =
 ∆ n ∆ n pzn ( x )
∆n∆n  ∆n + 2  + 4 
4  zn ( )
a 
Db 

 a  dx 

2
3
n d pθn ( x ) n3
1 d pxn ( x ) n 2 dpxn ( x ) 
+ (2 + υ) 2
−
p
x
+
υ
+ 3
(
)

θn
a
dx 2
a4
a dx 3
a
dx 
(4.34)
in which the Laplace operator ∆ n for n > 1 is defined by (4.33).
4.4.5.2 Homogeneous solution
The general solution to a differential equation consists of a homogeneous and an
inhomogeneous part. By inspecting the differential equation (4.34), it is observed that
the homogeneous part cannot be separated in a polynomial part and a non-polynomial
part.
The homogeneous equation is given by
2
4
4

1 

β d 
u x =0
∆n∆n  ∆n + 2  + 4 
4  zn ( )
a 

 a  dx 

for which the solution can be written as
1a
u zn ( x ) = −  
4β
4
2
1 

∫∫∫∫ ∆ n ∆ n  ∆ n + a 2  uzn ( x ) dxdxdxdx
(4.35)
81
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The solution to the homogeneous equation is given by (see also also Appendix H)
u zn ( x ) = e
− a1n β
+e
x
a
x
 n
 1 x
 1 x   a1n β a  n
 1 x
 1 x 
n
n
C1 cos  bnβ a  + C2 sin  bnβ a   + e
C3 cos  bnβ a  + C4 sin  bnβ a  











− an2 β
x
a
x
 n
 2 x
 2 x   an2β a  n
 2 x
 2 x 
n
n
C5 cos  bn β a  + C6 sin  bn β a   + e
C7 cos  bn β a  + C8 sin  bn β a  










(4.36)
where the dimensionless parameters a1n , an2 , bn1 and bn2 are defined by
1
1
1 
ω1 − 1  2
a =
δ1 + γ +
 ,
2 
2 
1 
ω1 − 1  2
b =
 δ1 − γ −

2 
2 
1
n
an2 =
1
n
1
2
1 
ω1 − 1 
 δ2 + γ −
 ,
2 
2 
bn2 =
1 
ω1 − 1 
 δ2 − γ +

2 
2 
(4.37)
1
2
in which
1
1

2
δ1 =  γ + ( ω1 + ω2 ) + 1 + γ 2 ( ω1 − 1) + 2 ( ω2 + 1) 
2


,
ω1 = ω + γ 2 − η2
,
ω2 = ω − γ 2 + η2
1
1

2
δ 2 =  γ + ( ω1 + ω2 ) + 1 − γ 2 ( ω1 − 1) − 2 ( ω2 + 1) 
2


and
ω = 1 + 2 ( γ 2 + η2 ) + ( γ 2 − η2 )
1

γ =  n 2 −  β−2
2

2
(4.38)
1
, η = n ( n 2 − 1) 2 β−2
Obviously, the roots (4.37) are surplus to requirements and approximations can be
made for several load-deformation regimes (see e.g. the next section), but the presented
solution is a unification of former results by other authors and the exact solution for
n = 0 and n = 1 is still retained. For these two values of n , the parameter η (4.38) is
equal to zero and the eight roots are calculated with the reduced parameters
1
1
n = 0,1
1

2
= (1 + γ 2n = 0,1 ) 2 + γ n = 0,1 


1
n = 0,1
1

2
= (1 + γ 2n = 0,1 ) 2 − γ n = 0,1 


a
,
an2= 0,1 = 0
,
bn2= 0,1 = 0
1
b
which are in agreement with the solutions (4.24) and (4.31) for the axisymmetric and
beam mode, respectively.
The presented roots are an exact solution to the homogeneous differential equation
and similar, but not equal, to the exact solution by Niordson [37]. However, the slight
difference between Niordson’s solution and the presented solution originates
presumably from setting the dimensionless parameters γ and η equal to one another
by Niordson. Due to this small adjustment (which is clearly admissible), the exact
solution for n = 0 and n = 1 is no longer retained, which is in contrast to the solution
presented above.
82
4 Circular cylindrical shells
The homogeneous solution for the displacements u x and uθ can be obtained by solving
the first two equations of the set (4.18) for which the homogeneous equations read
1 ∂ 3u z
1 ∂ 3u z
− 4
2
2
a ∂x ∂θ a ∂θ3
1 ∂ 3u z 1 ∂ 3u z
∆∆u x = −υ
+
a ∂x3 a 3 ∂x∂θ2
∆∆uθ = − ( 2 + υ )
By substitution of the displacement functions given above, these become ordinary
differential equations in which the sine function (for uθ ) and the cosine function (for
u x ) can be omitted. By substituting the representation (4.35) for u zn ( x ) the following
equations are obtained
4
2
1 a  
n d 2 n3 
1 

∆ n ∆ nuθn ( x ) = −   ( 2 + υ ) 2 2 − 4  ∫∫ ∫∫ ∆ n ∆ n  ∆ n + 2  u zn ( x ) dxdxdxdx
4 β  
a dx a 
a 

4
2
1  a   1 d 3 n2 d 
1 

∆ n ∆ nu xn ( x ) =    υ
+ 3  ∫∫∫∫ ∆ n ∆ n  ∆ n + 2  u zn ( x ) dxdxdxdx
3
a 
4  β   a dx a dx 

and by omitting the Laplace operators the differential equations reduce to
u θn ( x ) = −
u xn ( x ) =
1 n a
 
4 a2  β 
4
2
2


1 
n2
1 


( 2 + υ ) ∫∫  ∆ n + 2  u zn ( x ) dxdx − 2 ∫∫∫∫  ∆ n + 2  u zn ( x ) dxdxdxdx 
a 
a
a 




4
2
2

1 1 a  
1 
n2
1 

υ
∆
+
u
x
dx
+
∆
+
u zn ( x ) dxdxdx 
(
)

 
zn
 n
2 
2 ∫∫∫  n
2 
∫
4 a  β   
a 
a
a 


4.4.5.3 Inhomogeneous solution
Assuming linear loads px , pθ and pz , the inhomogeneous equation of (4.34) reduces
to
n4  n2 − 1 
1  n4
n3
n 2 dpxn ( x ) 
 4 p zn ( x ) − 4 pθn ( x ) + 3

 u zn ( x ) =
4 
2
a  a 
Db  a
a
a
dx 
2
by omitting all second and higher derivatives with respect to x . Hereby the
inhomogeneous solution is obtained as
1  a2  
1
a dpxn ( x ) 

 2   p zn ( x ) − pθn ( x ) + 2
Db  n − 1  
n
n
dx 
2
u zn ( x ) =
(4.39)
By substituting this result into the first two equations of the set (4.18), the
inhomogeneous solution for the circumferential displacement uθ and the axial
displacement u x can be obtained. If the second and higher derivatives with respect to
x are omitted, these differential equations become
1 − υ  1 ∂ 4uθ 1 ∂ 3u z  1 1 + υ 1 ∂ 2 p x 1 − υ 1 ∂ 2 pθ 
+
−

=


2  a 4 ∂θ4 a 4 ∂θ3  Dm  2 a ∂x∂θ
2 a 2 ∂θ2 
1 − υ  1 ∂ 4u x 1 ∂ 3u z  1  1 ∂ 2 p x 1 + υ 1 ∂ 2 pθ 
−
+

=
−

2  a 4 ∂θ4 a 3 ∂x∂θ2  Dm  a 2 ∂θ2
2 a ∂x∂θ 
and by substituting the displacement and load functions given above, these equations
can be rewritten and omitting the cosine and sine terms, the equations become
83
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
1
1  a2
1 + υ a 3 dpxn ( x ) 
uθn ( x ) = − u zn ( x ) +
 2 pθn ( x ) −

n
Dm  n
1 − υ n3 dx 

a du ( x ) 1 1 + υ a 3 dpθn ( x ) a 2 2
u xn ( x ) = − 2 zn
+
+ 2
pxn ( x ) 

3
n
dx
Dm 1 − υ n
dx
n 1− υ

into which the solution (4.39) can be substituted.
4.4.5.4 Complete solution
Describing the loads px , pθ and pz by the forms
p xn ( x ) = p xn
pθn ( x ) = pθ( 2n)
p zn ( x ) = pzn( 2)
x
+ pθ(1n)
l
x
+ pzn(1)
l
the complete solution for the independent displacement u z reads
 − a1nβ x 
x
x 


u z ( x, θ ) = cos nθ e a C1n cos  bn1β  + C2n sin  bn1β  
a
a 




+e
+e
+e
a1n β
x
a
− an2 β
an2 β
x
a
 n
 1 x
 1 x 
n
C3 cos  bnβ a  + C4 sin  bnβ a  





x
a
 n
 2 x
 2 x 
n
C5 cos  bn β a  + C6 sin  bn β a  





(4.40)
 n
 2 x
 2 x   
n
C7 cos  bn β a  + C8 sin  bn β a   



  

2
+
 ( 2) 1 ( 2)  x
1  a2 
(1) 1 (1) 

 cos θ  pzn − pθn  + pzn − pθn 
Db  n 2 − 1 
n
l
n



Similar expressions for the independent displacements uθ and u x are obtained by the
appropriate substitutions.
By substitution of the expressions for the independent displacements into the
expressions (4.9), (4.11) and (4.13), the complete solution for all nontrivial quantities
can be obtained, which are given in Appendix I.
4.5 Approximation of the homogeneous solution
4.5.1 Approximation of the exact solution
To express the eight roots (4.37), the parameters γ and η (4.38) have been introduced.
By definition (4.17) β−2 is a small value for the usual thickness-over-radius-ratio t a .
For the static behaviour of thin shells under the usual loading cases, only the first and
lower values of the mode number n are important (say n = 1,...,5 ) and hereby γ and η
84
4 Circular cylindrical shells
are small in comparison to unity. This enables a tremendous reduction of the
expressions for the eight roots by expanding these into a series development and then
breaking them down after the second term since γ 2 ≈ η2 1 .
For n = 0 and n = 1 , parameter η = 0 and by employing the abovementioned
development, the following approximate expressions are obtained
n = 0:
n = 1:
1
4β2
1
a11 = 1 + 2
4β
a10 = 1 −
,
,
1
4β2
1
b11 = 1 − 2
4β
b01 = 1 +
,
a02 = b02 = 0
,
a =b =0
(4.41)
2
1
2
1
where the subscripts denote the mode number n , which would also be obtained from
the same approximation of (4.24) for n = 0 and (4.31) for n = 1 .
For n > 1 and small values of γ and η the following approximate expressions are
obtained
1
a1n = 1 + γ n
2
1
b = 1 − γn
2
1
n
,
1  1 
an2 = ηn 1 + γ n 
2  2 
.
1  1 
b = ηn 1 + γ n 
2  2 
(4.42)
2
n
from which the solution for n = 0 and n = 1 is still traceable. The parameters γ n and ηn
are identical to the parameters γ and η (4.38), but the subscript n is further adopted to
indicate the mode numbers n > 1 . For n > 1 and larger values of γ n and ηn , the exact
solution or the solution to Donnell’s equation has to be used.
4.5.2 Solution obtained by a perturbation technique
Another approach to obtain an approximate solution to the characteristic equation of a
differential equation (or rather, a solution accurate within the assumptions and
simplifications postulated to derive that differential equation) is investigated in this
subsection for several differential equations. These differential equations have been
introduced in subsection 2.6.5 and represent different accuracies within the first-order
approximation theory for circular cylindrical shells.
The approach employed in this subsection (as, amongst others, shown by Nayfeh
[44]) has the objective to obtain approximate solutions to algebraic equations; e.g.
those as obtained by substitution of the trial solution into the differential equations as
presented in this subsection. The solution is represented as an asymptotic expansion in
terms of the small parameter, which is called parameter perturbation.
The method of parameter perturbation for the mode numbers n > 1 is preformed
below on the following three differential equations: the (simplified) Flügge equation
(refer to subsection 2.6.5), equation (4.16) derived from the set of equations proposed
in chapter 2, the Morley-Koiter equation (4.18) and the Donnell equation (refer to
subsection 2.6.5).
85
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The (simplified) Flügge equation reads
2
4
4
1 

 β  ∂ uz
∆∆  ∆ + 2  u z + 4  
4
a 

 a  ∂x
1  ∂4
1 ∂ 4  ∂ 2u
−2 (1 − υ ) 2  4 − 4 4  2z
a  ∂x
a ∂θ  ∂x
+2 (1 − υ )
=
1  1 + υ ∂2
∂ 2  ∂ 2u
3
+ 2 2  2z
4
2
a  2 ∂x
a ∂θ  ∂x
1 
1 ∂ 3 pθ
1 ∂ 3 pθ
1 ∂ 3 px 1 ∂ 3 px 
+
υ
−
 ∆∆pz + (2 + υ) 2 2 + 4

Db 
a ∂x ∂θ a ∂θ3
a ∂x3 a 3 ∂x∂θ2 
The equation derived from the proposed set of equations reads
2
4
4
1 

 β  ∂ uz
∆∆  ∆ + 2  u z + 4  
4
a 

 a  ∂x
1  ∂4
1 ∂4
1 ∂ 4  ∂ 2u z
−
2
1
+
υ
−
(
)


a 2  ∂x 4
a 2 ∂x 2∂θ2 a 4 ∂θ4  ∂x 2
1 ∂ 4u
+4 (1 − υ2 ) 4 4z
a ∂x
1 
1 ∂ 3 pθ
1 ∂ 3 pθ
1 ∂ 3 px 1 ∂ 3 px 
=
+υ
−
 ∆∆pz + (2 + υ) 2 2 + 4

3
Db 
a ∂x ∂θ a ∂θ
a ∂x3 a 3 ∂x∂θ2 
−2 (1 − υ )
The Morley-Koiter equation reads
2
4
4
1 
1 
1 ∂ 3 pθ
1 ∂ 3 pθ
1 ∂ 3 p x 1 ∂ 3 px 

 β  ∂ uz
∆∆  ∆ + 2  u z + 4  
=
+υ
−
 ∆∆pz + (2 + υ) 2 2 + 4

4
3
a 
Db 
a ∂x ∂θ a ∂θ
a ∂x3 a 3 ∂x∂θ2 

 a  ∂x
The Donnell equation reads
4
4
1 
1 ∂ 3 pθ
1 ∂ 3 pθ
1 ∂ 3 px 1 ∂ 3 px 
 β  ∂ uz
∆∆∆∆u z + 4  
=
∆∆
p
+
(2
+
υ
)
+
+
υ
−


z
4
Db 
a 2 ∂x 2∂θ a 4 ∂θ3
a ∂x3 a 3 ∂x∂θ2 
 a  ∂x
The following trial solution is introduced in the above differential equations
u z ( x, θ ) = C n e
rn β
x
a
cos nθ
The (simplified) Flügge characteristic equation becomes
2
2
4
2

 n n2 − 1   4
n 2 − 12 6   n 2 − 12 
n 2 − 12  n n 2 − 1  2  n n 2 − 1 

 r −4 2 
 r +
 + 4r 4
r − 4 2 r + 4 2  + 2
 β2
 
 β2

 β2

  β 
β
β

 





8
2
2
3
2  n n2 − 1  2
 r =0
− (1 − υ ) 2 r 6 + (1 − υ2 ) 4 r 4 + (1 − υ ) 2 

β
β
β  β2

86
4 Circular cylindrical shells
The “proposed” characteristic equation becomes
r8 − 4
2
2
4
2

 n n2 − 1   4
n 2 − 12 6   n 2 − 12 
n 2 − 12  n n 2 − 1  2  n n 2 − 1 

r
+
4
+
2


r
−
4


r
+


+ 4r 4


2
 β2
 
 β2

 β2

  β2 
β2
β

 





4 ( n − 1) 4
2 6
2n 4
r − (1 − υ2 )
r + (1 − υ ) 6 r 2 = 0
2
4
β
β
β
2
− (1 − υ )
The Morley-Koiter characteristic equation becomes
2
2
4
2

 n n2 − 1   4
n 2 − 12 6   n 2 − 12 
n 2 − 12  n n 2 − 1  2  n n 2 − 1 

 r −4 2 
 r +
 + 4r 4 = 0
r − 4 2 r + 4 2  + 2
 β2
 
 β2

 β2

  β 
β
β

 





8
The Donnell characteristic equation becomes
r8 − 4
n2 6
n4 4
n6 2 n8
r
+
6
r
−
4
r + 8 + 4r 4 = 0
β2
β4
β6
β
It is noted that for large n , which means n 2 1 , all equations above transform into the
Donnell characteristic equation.
For small n , which means n2 β2 , approximate solutions to the algebraic equations
are obtained below in accordance with the adopted approach of parameter perturbation.
Solutions for the large roots are obtained by assuming that the roots have expansions of
the form r = r0 + εr1 , in which ε is the small parameter of the order 1 β2 . Hence, the
second-order term is neglected, which is justified within the Kirchhoff-Love
assumptions.
The perturbed characteristic equation for both the (simplified) Flügge and the
“proposed” equation then read
1 

r 8 − 4  n 2 − υ  εr 6 + O ( ε 2 ) + 4 r 4 = 0
2 

The perturbed Morley-Koiter characteristic equation then reads
1

r 8 − 4  n 2 −  εr 6 + O ( ε 2 ) + 4r 4 = 0
2

The perturbed Donnell characteristic equation then reads
r 8 − 4n 2εr 6 + O ( ε 2 ) + 4r 4 = 0
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
For all the equations above, the solution can be represented by r = ± ( a1n ± ibn1 ) , in which
1
1
a1n = 1 + γ n and bn1 = 1 − γ n . The parameter γ n is different for the four equations and
2
2
reads, respectively,
γn =
n2 − 12 υ
β2
for the (simplified) Flügge and the “proposed” equation,
γn =
n2 − 12
β2
for the Morley-Koiter equation, and
γn =
n2
for the Donnell equation.
β2
Solutions for the small roots are obtained by assuming that the roots have expansions
of the form r = εs0 + ε 2 s1 , in which ε is the small parameter of the order 1 β2 .
The perturbed (simplified) Flügge characteristic equation then reads
(
)
(
)
4 s0 4 + 16εs1s03 − 4 ( n 2 − 1 + 12 υ ) n n 2 − 1 s0 2 + n n 2 − 1 + O ( ε 2 ) + = 0
2
4
The perturbed “proposed” characteristic equation then reads
(
)
(
)
2
4


4 s0 4 + 16εs1s03 − 4 ( n 2 − 12 ) n n 2 − 1 − 12 (1 − υ ) n 4  s0 2 + n n 2 − 1 + O ( ε 2 ) + = 0


The perturbed Morley-Koiter characteristic equation then reads
(
)
(
)
4 s0 4 + 16εs1s03 − 4 ( n 2 − 12 ) n n 2 − 1 s0 2 + n n 2 − 1 + O ( ε 2 ) + = 0
2
4
The perturbed Donnell characteristic equation then reads
4 s0 4 + 16εs1s03 − 4n 6 s0 2 + n8 + O ( ε 2 ) + = 0
For all the equation above, the solution can be represented by r = ± ( an2 ± ibn2 ) , in which
1  1 
1  1 
an2 = ηn  1 + γ n  and bn2 = ηn  1 − γ n  . The parameters ηn and γ n are different for
2  2 
2  2 
the four equations and read, respectively,
ηn =
n n2 − 1
β2
ηn =
n n −1
β2
, γn ≈
ηn =
n n2 − 1
β2
, γn =
, γn =
2
ηn = γ n =
88
n2
β2
n 2 − 1 + 12 υ
β2
n 2 − 12 − 12 (1 − υ )
β2
n 2 − 12
β2
for the (simplified) Flügge equation,
n2
n −1
2
for the “proposed” equation,
for the Morley-Koiter equation, and
for the Donnell equation.
4 Circular cylindrical shells
By comparing these results for the obtained roots of the Morley-Koiter equation by
parameter perturbation and the approximated roots of this equation as derived in the
previous subsection, it is easily observed that these are identical. This shows that
parameter perturbation can be conveniently adopted if an exact solution to the
differential equation cannot be easily obtained.
Furthermore, the envisaged small difference between the roots of the (simplified)
Flügge equation, the “proposed” equation and those of the Morley-Koiter equation is
apparent whereas the previously described accuracy of the solution of the Donnell
equation is once more obvious as the roots an2 and bn2 to this equation are in
considerable error for the lower mode numbers.
Moreover, if other differential equations resulting from a first-order approximation
theory are considered, such as those as listed in section 2.6, similar roots are obtained.
For small deflections, the shallow shell equations all attain to Donnell solution. The
solution to equation (4.16) and the equations referred to in section 2.6, such as the
complete Flügge equation, the Koiter-Sanders equation, Novozhilov’s equations and
the other comparable equations, all provide no significant improvement over the
solution to the Morley-Koiter equation solution as discussed in more detail in
subsection 2.6.5. Hence, it is once more assessed that the Morley-Koiter equation
accurately describes the behaviour of thin circular cylindrical shells as solutions to the
other equations are not considered as improved results or approximated results with a
higher accuracy within the simplifications and assumptions of the first-order
approximation theory for thin shells.
4.6 Characteristic and influence length
4.6.1 Axisymmetric mode
In expression (4.23), the terms multiplied with C1 and C2 are oscillating functions of
the ordinate x that decrease exponentially with increasing x . The terms multiplied
with C3 and C4 are also damped oscillations but these decrease exponentially with
decreasing x .
We introduce the characteristic length lc by
lc =
a
a0β
The influence of an edge disturbance, as a result of the exponential damping of the
wave, is not negligible for a length that is (approximately) equal to π times the
x
of the exponential function is than equal to π
a
and the value of the function in the order of e −π ≈ 0.05 . This influence length li is thus
characteristic length. The power a0β
equal to
li = πlc =
πa
a0β
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
By using the approximate value for a0 (4.41) the influence length can be approximated
by
π at
li ≈
4
3 (1 − υ2 )
in which also β−2 has been neglected in comparison to unity for convenience.
To exemplify the influence of an edge disturbance we compare the influence
length with the radius of the cylinder. This influence-length-to-radius ratio reads:
li
π
at
t
≈
≈ 2.4
2
a
a
4 3 1− υ
( ) a
Since the thickness-to-radius ratio of a thin shell is smaller than 1 50 , the influence
length li of the edge disturbance is (much) shorter than the radius of the cylinder (e.g.
0.24a in case of a t = 100 ).
The length in x -direction of the circular cylinder is denoted by l . If the cylinder is
− a0 β
l
long enough to reduce e a to a small quantity that is negligible in comparison to
unity, the influence of the term in (4.23) with constants C3 and C4 on the solution for
x = 0 will also be negligible. In other words, the length of the cylinder is in those cases
larger than the influence length li . Hence, it is useful to rewrite the solution (4.23) into
the form
uz0 ( x ) = e
− a0 β
+e
x
a

x
x 


 S1 cos  b0β a  + S 2 sin  b0β a  





− a0 β
l−x
a

x
x 


 S3 cos  b0β a  + S 4 sin  b0β a  





(4.43)
Moreover, this transformation is highly desirable from a mathematical point of view
since the value of the constants will be more or less of the same order. Obviously, the
converse applies to the factors with which the constants are multiplied, which ensures
the accuracy and stability of the determination of the stiffness matrix by the super
element approach. This is mainly due to the inversion of the element displacement
matrix A e indicated by expression (3.8).
In some cases, it is convenient to apply an alternative ordinate x′ in negative x direction. With x′ = l − x and thus x′ = 0 for x = l , the alternative ordinate has its origin
at the boundary at x = l . Then, by using another set of free constants, the solution is
rewritten to
u z 0 ( x ) = C1e
− a0 β
x
a
′
x
− a0 β
x
x′




sin  b0β + ψ1  + C2e a sin  b0β + ψ 2 
a
a




(4.44)
The new constants can be determined independently: C1 and ψ1 with the aid of the
boundary conditions for x = 0 and C2 and ψ 2 with the aid of the boundary conditions
for x′ = 0 . This presentation of the solution is convenient for simple cases for which the
phase angle can be determined immediately by the boundary conditions. The other
90
4 Circular cylindrical shells
presentations (4.23) and (4.43) are suitable when a pair of linear equations for the
constants is derived, where, as mentioned above, (4.43) seems advisable when
synthesizing an element stiffness matrix within the direct stiffness approach.
4.6.2 Beam mode
In expression (4.30), the terms multiplied with C1 and C2 are oscillating functions of
the ordinate x that decrease exponentially with increasing x . The terms multiplied
with C3 and C4 are also damped oscillations but these decrease exponentially with
decreasing x . Obviously, the characteristic and influence lengths for this part of the
solution are approximately equal to the characteristic and influence lengths of the
homogeneous solution for the axisymmetric load and can be calculated by
lc =
a
a1β
li = πlc =
πa
a1β
By using the approximate value for a1 (4.41) and neglecting β−2 in comparison to
unity, the influence length can be approximated by
π
li ≈
4
3 (1 − υ2 )
at ≈ 2.4 at
which is equal to the influence length obtained for the axisymmetric behaviour. Since
the thickness-to-radius ratio of a thin shell is smaller than 1 50 , the influence length li
of the edge disturbance is (much) shorter than the radius of the cylinder.
4.6.3
Self-balancing modes
In expression (4.36), the terms multiplied with C1n , C2n , C5n and C6n are oscillating
functions of the ordinate x that decrease exponentially with increasing x . The terms
multiplied with C3n , C4n , C7n and C8n are also damped oscillations but these decrease
exponentially with decreasing x . Obviously, the characteristic and influence lengths
for the terms multiplied with the first four constants are approximately equal to the
characteristic and influence lengths of the homogeneous solution for the axisymmetric
load.
The characteristic and influence lengths for the terms multiplied with the first four
constants can be calculated by
lc ,1 =
a
a1nβ
li ,1 = πlc ,1 =
πa
a1nβ
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
By using the approximate value for a1n (4.42) and neglecting β−2 in comparison to
unity, this short influence length can be approximated by
π
li ,1 ≈
4
3 (1 − υ2 )
at ≈ 2.4 at
The characteristic and influence lengths for the terms multiplied with the other four
constants can be calculated by
lc ,2 =
a
an2β
li ,2 = πlc ,2 =
πa
an2β
By using the approximate value for an2 (4.42) and neglecting β−2 in comparison to
unity, this long influence length can be approximated by
2πa
1
a
8.1 a
= 2π 4 3 (1 − υ2 )
at ≈
at
2
ηnβ
t
n n −1
n n2 − 1 t
which depends on the mode number n . Obviously, these terms describe a far-reaching
a
influence (roughly
times the short influence length li ,1 ), but that their influence
t
length decreases rapidly with increasing n .
In expression (4.40), the terms multiplied with the constants C1n , C2n , C3n and C4n
li ,2 ≈
represent the part of the solution describing the edge disturbance with the short
influence length, which is further referred to as the short-wave solution. Similarly, the
terms multiplied with the constants C5n , C6n , C7n and C8n represent the part of the
solution describing the edge disturbance with the long influence length, which is
further referred to as the long-wave solution.
4.7 Concluding remarks
In this chapter, the solutions to the Morley-Koiter equation (as an approximation of the
exact equation for thin elastic shells within the first-order approximation theory) are
given for the respective load-deformation behaviours. The approximate solution for the
self-balancing mode is compared with several solutions obtained by parameter
perturbation, which confirmed that the Morley-Koiter equation accurately describes the
behaviour of thin circular cylindrical shells. The characteristic and influence lengths
have been derived for the axisymmetric mode, the beam mode and the self-balancing
modes.
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5 Chimney – Numerical results and parametric study
5 Chimney – Numerical results and parametric
study
Solutions obtained by a computer program based on the method presented in chapter 3
are given for long circular cylindrical shell structures. The formulations that are used in
this program are derived in chapter 4. The generic knowledge from that chapter in
combination with the results presented in this chapter provides the basis of a parametric
study of the stiffened and non-stiffened shell geometry, support conditions and loading
on its behaviour and interaction. The conclusions of this study and the applicability of
the computational method for long circular cylindrical shells are given in chapter 7.
5.1 Wind load
The distribution of the wind load around a circular cylindrical chimney has a maximal
value at the windward meridian (denoted by θ=0) equal to the stagnation pressure and a
small pressure at the leeward meridian. The sides in between are subjected to suction,
which in absolute value is even larger than the stagnation pressure (see Figure 5-1 for a
typical distribution).
pw
σbeam
σ total
Figure 5-1 Typical distribution of the wind load (left) and axial stress at the base (right).
Because of the choice of the coordinate system and the symmetry of the load, the wind
load (constant in axial direction) can be developed in a Fourier cosine series for the
circumferential direction. By sign convention, the positive direction of the load is taken
in the positive direction of the coordinate z, which is from inside to outside of the
circular profile. For a quasi-static load series, only the lower mode numbers have to be
taken into account to accurately describe the wind load. Hence, the distribution
exemplified in Figure 5-1 is given by
pz ( x, θ ) = pw [ α 0 + α1 cos θ + α 2 cos 2θ + α3 cos3θ + α 4 cos 4θ + α 5 cos5θ]
(5.1)
in which
α 0 = 0.823 ; α1 = −0.448 ; α 2 = −1.115 ; α 3 = −0.400 ; α 4 = 0.113 ; α5 = 0.027
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
where pw is set equal to 1 kN m 2 , which value is a good reference value for the wind
stagnation pressure in north-western Europe. The shape of the circumferential
distribution of the wind load depends roughly on the geometry of the chimney and
varies from code to code but has the common characteristic that only a part of the
circumference, the so-called stagnation zone, is under circumferential compression,
while the remainder is under suction. The values presented above are taken from the
reports by Van Koten [45] and Turner [46] The normalised distribution of the wind
load (5.1) across the profile of the cylinder is depicted in Figure 5-2 where the distance
to the centre dc across the profile is calculated by dc = a cos θ . For clarity and
reference, the linear distribution of a beam load ( n = 1) is also shown. A negative value
of the load-to-stagnation-pressure-ratio pz pw denotes pressure, while a positive value
denotes suction.
Figure 5-2 Distribution of the wind load (5.1) across the cylinder
5.2 Behaviour for a fixed base and free end
5.2.1 Closed-form solution
In a paper by Hoefakker [47], the closed-form solution is derived for the
circumferential distribution of the axial membrane stress resultant nxx at the clamped
base of a long circular cylinder (for example an industrial, steel chimney) under the
wind load described in section 5.1. The axial stress distribution at the base of such a
long chimney is mainly described by the beam action. However, the large suction at the
sides of the chimney leads to an additional out of roundness of the cross-section, e.g.
94
5 Chimney – Numerical results and parametric study
for n = 2 the circular cross-section deforms to an oval shape. To withstand this out of
roundness at the base additional axial stresses are generated as shown in Figure 5-1. At
the base (denoted by x = 0 ) the chimney is typically clamped and at the top (denoted
by x = l ) the chimney often has a free edge. Over the distance l between these two
edges, the geometrical and material properties are assumed to be constant. This means
that the response to the wind load can be calculated by the solution to the differential
equation (4.18). This solution has to be complemented by the appropriate boundary
conditions that are given by
x =0;
clamped:
u x = u x = 0 ; uθ = uθ = 0 ; u z = u z = 0 ; ϕ x = ϕ x = 0
x=l;
free:
f x = nxx = 0 ;
f θ = n xθ = 0 ;
f z = v∗x = 0 ; t x = mxx = 0
where v∗x is Kirchhoff’s effective shearing stress resultant.
The first term ( n = 0 ) of the series development for the wind load (5.1) is constant in
circumferential direction and represents axisymmetric loading. It leads to a small
circumferential tension in the chimney and due to the clamped edge to a short edge
disturbance. However, the resulting stresses and displacements are known to be
negligible in comparison with the response to the other terms of the wind load.
The second term ( n = 1) describes a varying load that has a negative peak value at the
windward meridian and a positive peak value at the leeward meridian. This is the only
load term that is not self-balancing: i.e. it has a resultant in the wind direction. If the
chimney is long, the stresses and deformations due to this load might be calculated by
the membrane theory. Hence, not all boundary conditions can be fulfilled since there
are more conditions than quantities but the necessary edge disturbance will be
represented by a small influence over a short length. The same result can be obtained
by elementary beam theory if the shear deformation is accounted for. In fact, the
solution to this term is also well known and by solving the boundary conditions for the
membrane stress resultants at x = l the following expression for the axial membrane
stress resultant nxx is obtained, which is quadratic with respect to the axial coordinate
nxx ( x, θ ) = −
pwα1
2
( l − x ) cos θ
2a
However, if the more complete solution as derived in subsection 4.4.4 is employed, it
is shown that the common assumption that the membrane solution is accurate is
slightly in error if the lateral contraction is accounted for. Due to the then arising
incompatibility at the clamped edge, a small but evident edge disturbance is produced
and the resulting bending stress couple mxx does contribute to a certain extent to the
axial stress at the base.
First solving the boundary conditions for the stress resultants at the free edge
( x = l ) , four constants are obtained that read
C3 = 0 ; C4 = 0 ;
C5 = −
pz1la
Et
; C6 =
1 pz1l 2
2 Et
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
in which pz1 = pwα1 . The boundary conditions for the clamped edge ( x = 0 ) can now be
solved, which for a long chimney ( l a 5 ) results in
C1 = −
υ p z1l 2
2 Et
υ p z1l 2
p la
2 + υ pz1l 2
C 7 = υ z1
;
; C8 =
2 Et
2 Et
Et
and mxx is obtained by back substitution and by introducing
; C2 = −
The solution for nxx
p z1 = pwα1 for the wind load. The solution at x = 0 reads
nxx ( 0, θ ) = −
pwα1l 2
cos θ
2a
;
mxx ( 0, θ ) = −υ
pwα1l 2
cos θ
4β2
Hence, the effect of the bending stress couple is mainly limited to the short influence
length but, while accounting for β defined equation (4.17), certainly not negligible at
the base of the chimney. The corresponding axial stress at the base x = 0 due to the
“beam term”, as obtained by relation (4.7), is equal to
nxx 2 z 6mxx
p l 2  2z
υ
+
= − w 1 +
3
2
t
t t
2at 
t
1 − υ2
Note that the load factor α1 is negative for the
σ nxx=1 ( 0, θ, z ) =

 α1 cos θ

(5.2)
current wind load. Hence, the axial
stress is positive (tension) at the windward meridian ( θ = 0 ) and negative
(compression) at the leeward meridian ( θ = π ) . Although the additional bending stress
is only present over a short influence length, the contribution can be quite substantial.
For the outer or inner surface ( z = ± t 2 ) of, e.g., steel with υ = 0.3 , the term between
the brackets becomes 1 ± 3
0.3
≈ 1 ± 0.5 . Hence, a prediction by the membrane stress
0.91
resultants only, might be in a rather large error for such a material (in this case an error
of 50%).
The third term ( n = 2 ) describes a double symmetric and hence self-balancing term
with two waves about the circumference, which results in a pressure at the windward
and the leeward meridian and a suction at the sides. The response to this load is
calculated by using the solution as presented in Appendix I, which is complemented by
the boundary conditions at hand.
The higher-order terms of the development ( n > 2 ) are also self-balancing and
therefore analysed with the same solution procedure as for n = 2 , however, with their
respective value of the circumferential wave number.
As shown in Appendix I, the inhomogeneous solution can be obtained omitting all
derivatives with respect to the axial coordinate x . For the present load
pz ( x, θ ) = ∑ pzn cos nθ
n
96
5 Chimney – Numerical results and parametric study
an inhomogeneous solution for n>1 reads
u z ( x, θ ) = u zn cos nθ =
1 ∞
a4
pzn cos nθ
∑
Db n = 2 ( n 2 − 1)2
1
uθ ( x, θ ) = uθn sin nθ = − u zn sin nθ
n
∞
a2
mθθ ( x, θ ) = mθθn cos nθ = ∑ 2
pzn cos nθ
n=2 n − 1
mxx ( x, θ ) = mxxn cos nθ = υmθθn cos nθ
∞
na
pzn sin nθ
n=2 n − 1
vθ ( x, θ ) = vθn sin nθ = − ∑
2
where the other quantities are equal to zero. Obviously, the inhomogeneous solution
for n>1 is the ring-bending solution.
The inhomogeneous solution for n > 1 shows that the displacements u z and uθ are
not equal to zero. The boundary conditions at the clamped edge ( x = 0 ) are therefore
not fulfilled and an edge disturbance that originates from this edge is necessary. Due to
the largely deformed cross-sectional profile, the resulting edge disturbance has a farreaching influence. The boundary conditions at x = l are also not fulfilled but only due
to a non-zero change of curvature in circumferential direction that is multiplied by
Poisson’s ratio υ . It can be concluded that this fact alone leads to a short edge
disturbance that originates from this free edge with a mainly local effect and a small
influence on the response of the cylinder.
From the abovementioned arguments, it can be concluded that for a chimney with
a length larger than the long influence length only the boundary conditions at the base
are necessary to describe the overall response to the wind load. Hence, the constants in
the homogeneous solution of the edge disturbance that originates from the free edge
can safely be equated to zero. The expressions for the four quantities, which have to be
described at the clamped edge, are derived by back substitution as shown in Appendix
I. The boundary conditions for this edge can now be formulated by adding the
inhomogeneous solution to the expressions for the homogeneous solution at x = 0 ,
which gives four equations with four unknown constants. Making use of the fact that
terms multiplied by β−4 are negligibly small in comparison to unity (for the lower
values of n under consideration) and the solution to these equations for υ = 0 is
C1n = 0 ; C2n = 0 ;
C5n = −uˆ zn
(
)
; C6n = − 1 − ( n 2 − 32 ) β −2 uˆ zn
n
z
where uˆ is equal to uzn as presented in the inhomogeneous solution above. For the
case that Poisson’s ratio is not zero, the solution is
C1n = C2n = −
υ n2 − 1 n
uˆ z
2 β2
 υ n2 − 1  n
; C5n = −  1 −
 uˆ z
2
 2 β 
 
3 1
3
1
C6n = − 1 −  n 2 −  2 − υ  ( n 2 − 1) − υn n 2 − 1  2  uˆ zn
2 β
2
β 
 
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The constants C1n and C2n (which are equal to zero if Poisson’s ratio is equal to zero)
represent the short-wave solution. Additionally, it can be verified that the long-wave
solution (represented by the constants C5n and C6n ) is mainly described by membrane
stress resultants in the axial direction while the loading leads to bending stress
resultants in circumferential direction.
For the free edge at x = l , a similar procedure to obtain the other four constants can
be applied. As described the boundary conditions at this edge are only not met by a
bending stress couple, which occurs if the lateral contraction, described by Poisson’s
ratio υ , is taken into account. For convenience, the solution is obtained at an edge
x = 0 to cancel out the length in the expressions. Solving the four equations for the
boundary conditions, the four constants become
C1n =
υ n2 − 1 n
uˆ z
2 β2
; C2n = −
υ n2 − 1 n
uˆ z
2 β2
; C5n =
υ n2 n
uˆ z
2 β2
; C6n = −
υ n2 n
uˆ z
2 β2
which indeed shows that the long-wave solution is hardly activated since these
constants are of the order O ( υβ −2 ) . The fact that the inhomogeneous solution is
incompatible with the boundary conditions for the free edge is compensated by an edge
disturbance that is described by a small short-wave and equally small long-wave
solution.
On basis of these observations, it is obvious that the influence of the
incompatibility at the free edge is negligible when calculating any quantity at the base
of a sufficiently long cylinder. Additionally, the influence of the bending stress couple
mxx at the base on the axial stress distribution at the base is not negligible if the lateral
contraction is accounted for. However, similar to the stress distribution for the “beam
action”, the contribution can be added to the membrane stress resultant nxx . Moreover,
the addition of the effects gives an identical ratio of the bending stress to the membrane
stress. The expressions for the stress resultant nxx and the stress couple mxx are found
by back substitution of the homogeneous solution. Substitution of the constants and
addition of the inhomogeneous solution results in the expressions
5
nxx ( 0, θ ) = −∑ 2 3(1 − υ2 )
n=2
5
mxx ( 0, θ ) = −∑ a 2
n=2
a 2 pzn  n 2 − 1 υ   2 3 

2
 1 − 2 − 2  2  n −  − υn n − 1   cos nθ
t n2 − 1 
4
β
β  

pzn   2n 2 − 1 υ n 2 − 1  1 n 2 
+
 υ 1 −
 cos nθ
+
n − 1  
2 β2  2 β2 
β2
2
Hence, the expression for the axial stress σ xx ( x, θ, z ) at the base x = 0 due to the terms
( n > 1) is finally obtained by addition of the membrane and bending stress by
σ 2xx≤ n ≤5 ( 0, θ, z ) =
2
5
nxx 2 z 6mxx
p zn  2 z
υ 
2 a
+
=
−
2
3
1
−
υ
3
(
)
1 +
 cos nθ (5.3)
∑
2
2
2
t
t t
t n − 1
t
n=2
1 − υ2 
in which the terms of the order O ( β−2 ) are neglected for convenience.
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5 Chimney – Numerical results and parametric study
Van Koten [45] derived a similar expression for the stress distribution at the base
and at the middle surface ( z = 0 ) on basis of Donnell’s equation. The important
difference between his result given by
5
σ 2xx≤ n ≤5 ( 0, θ,0 ) = −∑ 2 3 (1 − υ2 )
n=2
a 2 pzn
cos nθ
t 2 n2
and the presented solution on basis of the Morley-Koiter equation is obviously the
difference in the inhomogeneous solution that describes the ring-bending action. It is
well known that this part of the full solution is more accurately described by the
Morley-Koiter equation, which gives a considerable improvement of the displacements
(especially for the case n = 2 ). The ratio of the solutions is hence equal to
σMK
n2
xx ( 0, θ,0 )
≈
σ Dxx ( 0, θ.0 ) n 2 − 1
( for 2 ≤ n ≤ 5)
Having found the response of the long chimney to the separate terms of the wind load,
a useful design formula can be derived for the stress distribution at the base. For the
long chimney longer than the long influence length, it is readily verified that the only
non-balancing term ( n = 1) is the leading term of the full response and conveniently, its
response is most easily found by a membrane solution or beam analysis. The other
contributing terms are the self-balancing terms ( n = 2,...,5 ) . The response to these load
terms (5.3) has to be calculated by a more laborious solution and therefore it is
convenient to express their influence by their ratio to the response to the “beam term”
(5.2). This results in an expression for the axial stress at the base x = 0 , which is
composed as
5


σ 2xx≤ n ≤5 ( 0, θ, z ) 
∑


σ0xx≤ n ≤5 ( 0, θ, z ) = σ nxx=1 ( 0, θ, z ) 1 + n = 2 n =1
σ xx ( 0, θ, z ) 



Since the ratio of the bending-to-membrane stress for the non-balancing terms and the
“beam term” are multiplied by the same factor, the formula is further simplified to
5


2≤ n≤5
 ∑ σ xx ( 0, θ,0 )  
υ 
 1 ± 3
σ0xx≤ n ≤5 ( 0, θ, ± t 2 ) = σ nxx=1 ( 0, θ,0 ) 1 + n = 2 n =1

σ
0,
θ
,0
) 

1 − υ2 
xx (


For the maximal tensile stress at θ = 0 (the windward meridian) it reads
2
5
l2 
1 αn  
υ 
2 a a
σ0xx≤,nt ≤5 ( z = t 2 ) = − pwα1
1 + 4 3 (1 − υ )   ∑ 2
 1 + 3

2at 
 l  t n = 2 n − 1 α1 
1 − υ2 

(5.4)
The formula for the maximal tensile stress at the clamped edge is obtained at the
location of the windward meridian ( θ = 0 ) and by substituting the wind load (5.1) this
expression reads
σ0xx≤,nt ≤5 ( z = t 2 ) = 0.224
2

l2
υ 
 a  a 
pw 1 + 6.39 1 − υ2    1 + 3

at 
 l  t  
1 − υ2 
(5.5)
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The formula for the compressive stress at the middle surface ( z = 0 ) and the leeward
meridian ( θ = π ) reads
σ0xx≤,nc≤5 ( z = 0 ) = −0.224
2

l2
 a  a
pw 1 − 4.88 1 − υ2   
at 
 l  t 
which does not necessarily indicate the maximal compressive stress. The location of
this maximum depends on the dimensions of the cylindrical shell and on the constants
in the wind load.
Between the straight brackets of the formula (5.5) for the tensile stress we
recognize the inverse dimensionless parameters l a and t a . Similar to Van Koten
[45] and Turner [46], a plot of the term between the straight brackets is presented in
Figure 5-3 with those dimensionless parameters on the axes and the respective term is
depicted for a Poisson’s ratio equal to zero ( υ = 0 ) to allow comparison with the graph
show in [45]. Note that, to obtain the stress on the outer or inner surface, the term
between the round brackets has additionally to be taken into account. The practical
range for long chimneys extends up to a value of around l a = 60 , which further
depends on the thickness of the cylinder.
The above-mentioned term only depends on the value of Poisson’s ratio. The term
as presented in Figure 5-3 consists of the dimensionless parameters l a and t a which
are multiplied by a factor. This factor depends on the constants of the wind load and is
given by 6.39 if υ = 0 . Van Koten’s formula, as presented in [45] for υ = 0 , is obtained
by adopting the solution to Donnell’s equation and yields 4.87 for that factor, which
shows that adopting the solution to the Morley-Koiter equation gives a tremendous
improvement over the solution based on Donnell’s equation. Turner, using a finite
element analysis in [46], sets the factor to 6.05 to obtain sufficient agreement between
the application of the formula and his range of finite element results (at the maximum
0.5% difference). The value of 6.05 can be obtained from formula (5.5) if υ = 0.32 is
used for Poisson’s ratio, which is a good value for the lateral contraction of steel.
Turner’s results are based on chimneys made of steel, which shows that formula (5.5)
is in excellent agreement with Turner’s finite element results.
Formula (5.5) may be presented in an alternative way. By introduction of the
characteristic lengths l1 and l2 for the present case of a long circular cylinder under
wind load, which are defined by
l1 = at
l2 = 4 atl 2
formula (5.5) for the maximal tensile stress at the clamped edge may be rewritten to
σ
0≤ n ≤5
xx , t
100
2
4

l 
υ 
2  a 
( z = t 2 ) = 0.224 pw   1 + 6.39 1 − υ    1 + 3

1 − υ2 
 l1  
 l2   
(5.6)
5 Chimney – Numerical results and parametric study
0.02
400
0.0175
1.1
0.0125
1.15
1.2
1.4 1.3
1.6
2.0
2.5
0.01
0.0075
0.005
3.0
3.0
300
a/t
t/a
0.015
0.0025
350
1.05
2.5 2.0
1.6
250
1.4
200
1.3
150
1.2
1.15
100
1.1 1.05
50
20 30 40 50 60 70 80 90
20 30 40 50 60 70 80 90
l/a
l/a
2
Figure 5-3 Closed-form multiplier 1 + 6.39 ( a l ) ( a t )  for υ = 0 to the “beam solution”


to obtain the maximal axial tensile stress σ xx at the base of a long, one-sided clamped
chimney with (left) t a and (right) a t on the vertical axis.
Similar to Figure 5-3, a plot of the term between the straight brackets of formula (5.6)
is represented in Figure 5-4 for υ = 0.3 against the dimensionless parameter l2 a . For
the practical range, the dimensionless ratios as employed in formula (5.5) are
10 < l a < 60 and 50 < a t < 400 , i.e. 0.7 < l2 a < 3 and 70 < l l1 < 1200 as employed in
formula (5.6). Note that the largest value for l2 a is obtained for the thickest and
longest chimneys, i.e. smallest a t in combination with largest l a .
The stress calculated by formula (5.5) and (5.6) for υ = 0.3 , and normalized to the
stagnation pressure of the wind load pw , is graphically represented in Figure 5-5
against the dimensionless parameters l a and t a for the abovementioned ranges.
Both figures show that only for a considerable length-to-radius ratio, the stress at
the base of the chimney is dominated by the beam behaviour as supported by the ratios
as depicted in Figure 5-3. Hence, the stress at the base varies not merely quadratic with
the length as might have been expected based on the expression for the beam stress, but
is largely dominated by the non-balancing terms for shorter chimneys.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
( l2 a )
4
Figure 5-4 Closed-form multiplier 1 + 6.39 1 − υ2 ( a l2 )  for υ = 0.3 according to


formula (5.6).
Hinged versus clamped support
The above closed-form solutions are obtained for a fully rigid support at the base of the
chimney for which not only the displacement of the cross-section is prohibited, but also
the rotation of the wall of the chimney is fully withstood, i.e. the clamped edge. If the
support of the chimney allows free rotation, the moment should be zero at the base.
The solution for such a “hinged-wall” edge ( u x = uθ = u z = 0, mxx = 0 ) is almost equal to
the solution for the clamped edge. The change in the long edge disturbance is
negligible, but the short edge disturbance is somewhat different. However, this
difference is not of any importance with respect to the global solution for the stresses at
the edge. The four constants for the “hinged-wall” edge are given by
C1n = −
υ n2 − 1 n
uˆ z
2 β2
; C2n = 0 ; C5n = C6n = −uˆ zn
if the terms multiplied by β−2 are neglected in comparison to unity.
102
5 Chimney – Numerical results and parametric study
a/t=400
a/t=350
a/t=300
a/t=250
a/t=200
a/t=150
a/t=100
a/t=50
Figure 5-5 Tensile membrane stress σ xx ( υ = 0.3 ) according to formula (5.5).
Upon inspection and back substitution of these constants, it is observed that the
membrane stresses for the “hinged-wall” edge are identical to those for the clamped
edge, and that, in the absence of the bending stresses, the stress distribution across the
thickness slightly differs. The formula for the maximal tensile stress at the “hingedwall” edge is thus similar to the formula for the clamped edge (5.5), but with the
difference that the stress distribution across the thickness is only given by the
membrane stress resultant nxx . Situated at the windward meridian ( θ = 0 ) , the
expression reads
σ0xx≤,nt ≤5 ( − t 2 ≤ z ≤ t 2 ) = 0.224
2

l2
 a  a
pw 1 + 6.39 1 − υ2   
at 
 l  t 
The tensile membrane stress at the base of a long chimney having either a clamped
edge or a “hinged-wall” edge can thus be obtained equating the product of the ‘beam
theory stress’ with the multiplier for this ‘beam theory stress’ presented within the
straight brackets of formula (5.5) and as shown in Figure 5-3 for υ = 0 . Alternatively,
the membrane stress is described by formula (5.6) and the multiplier is as shown in
Figure 5-4 for υ = 0.3 . To obtain the maximum tensile stress for the clamped edged, the
term between the round brackets of formulas (5.5) and (5.6) has additionally to be
taken into account, which only depends on the value of Poisson’s ratio.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Discussion and evaluation
In this subsection, the solution to the Morley-Koiter equation is used to obtain a
suitable formula for the stress distribution at the fixed base of a long chimney under
wind loading. Mainly because the inhomogeneous solution for the self-balancing terms
( n > 1) accurately describes the ring-bending action of the cylinder, the result is a
substantial refinement of the formula that is found by using Donnell’s equation for
these terms and shows better agreement with finite element results.
The ratio of the total membrane stress to the ‘beam theory stress’ depends
completely on the geometry of the chimney, the circumferential distribution of the
wind load and to a lesser extend on the lateral contraction of the material. The
influence of the additional stress, due to the higher-order terms of the wind loads,
manifests itself in a long-wave solution. The shell behaviour in the part of the cylinder
where the long-wave solution does not exert influence is in accordance with the ringbending action. The long-wave solution represents the additional membrane action of
the shell to meet the boundary conditions.
Additionally, the stress distribution through the thickness at the base of the
clamped cylinder is derived. It is shown that, if the lateral contraction is taken into
account, a considerable contribution must be incorporated in the maximal tensile and
compressive stress at the base. For steel with Poisson’s ratio equal to υ = 0.3 , the
bending stress is roundabout 50% of the membrane stress. As can be observed from the
solution of the constants, this rather large increase is subdivided into two
approximately equal parts: a part that produces the short edge disturbance and a part
that produces a long edge disturbance that contributes in a comparatively minor extent
to the long edge disturbance produced by the membrane action.
It is noted that the result is obtained under the assumption that the length of the
chimney is at least larger than the long influence length. For shorter cylinders, the
solution cannot be obtained solely on the boundary conditions at the clamped base,
since the long edge disturbance will produce stresses that are incompatible with the
boundary conditions at the free end. Hence, a compensating long edge disturbance will
originate from the free edge that might be of influence to the axial stress distribution.
The range of application of the derived formula is the subject of the next subsection.
5.2.2 Applicability range of formulas
The objective of this subsection is to show the range of application of the formulas
(5.5) and (5.6) derived in the previous subsection. These formulas predict the tensile
axial stress at the base and the windward side of a long clamped chimney subject to
wind load and only differ in the different dimensionless parameters that are adopted.
The formulas describe the stresses at the middle surface and at the outer surface.
The range of application of these formulas is determined by comparison with
results obtained by the program CShell, which applies for short and long cylindrical
shells. As this program is based on the closed-form solution, it is obvious that for
chimneys much longer than the influence length an identical result is obtained. For
chimneys shorter than the influence length, the program is more accurate since the
formulas do not include the effect of the edge disturbance that originates at the free
edge.
104
5 Chimney – Numerical results and parametric study
To investigate the range of application, calculations have been made with a lengthto-radius-ratio ranging from 10 to 30 and a radius-to-thickness-ratio ranging form 50 to
400. To compare the results for the specified range, the multiplication factor for the
middle fibre stress obtained by the formula and the program is plotted in Figure 5-6
against the dimensionless parameter l2 a . The ratio obtained by the formula shown in
Figure 5-6 is thus based on formula (5.6) and represents the term between the straight
brackets. This plot is thus identical to the plot in Figure 5-4.
The agreement between the plot obtained by the formula and the plot obtained with
the program is extremely good up to multiplication factor of about 7 and obviously
even smaller differences will be observed for greater length-to-radius ratios. For a ratio
larger than 7 the formula-to-program-ratio precipitously increases, viz. the formula
predicts a much higher stress than the program, which is conservative but not accurate.
The ratio larger than 7 has also been identified as a limit of applicability by, amongst
others, Schneider and Zahlten [48]. Furthermore, a larger factor seems not practical
from the design point of view.
Figure 5-6 Stress ratio for υ = 0.3 at the middle fibre obtained by formula (5.6) and the
program CShell.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
In Figure 5-7, the multiplication factor for the outer fibre stress is plotted for the ratio
obtained by the formula and for the ratio obtained by the program. The factor shown in
Figure 5-7 is thus based on all terms of formula (5.6). Obviously, the agreement
between the plot obtained by the formula and the plot obtained with the program is
extremely good up to a higher multiplication factor as the term between the round
brackets of formulas (5.5) and (5.6) is additionally taken into account.
Comparison of Figure 5-6 for the middle fibre stress with Figure 5-7 for the outer fibre
stress shows that the range of application is related to the geometry of the cylinder, viz.
directly related to dimensionless parameter l2 a . The figures indicate that the formula
is applicable if the dimensionless parameter l2 a is larger than or equal to unity.
To interpret the dependency on this parameter, it is recalled that, besides the
dependency on the wind load factors, the increase of the beam stress is attributed to the
long-wave solution of the self-balancing terms of the wind load ( n ≥ 2 ) . The long
influence length li ,2 for mode numbers n ≥ 2 is described by the expression derived in
subsection 4.6.3, which can be written as
li ,2
a
≈
a
n n −1 t
8.1
2
It is mode number n = 2 that has the longest influence length and dominates the
difference between the beam stress and the total stress resulting from all terms of the
wind load. For n = 2 , the long influence length is approximately equal to
l in,2= 2 ≈ 2a
a
t
As indicated above, the formula is correct if the dimensionless parameter l2 a is
roughly larger than or equal to unity, which can be written as
l2 4 atl 2 4 t l
=
=
≥1
a
a
a a
and hence it is concluded that formula (5.6) (and thus formulas (5.5) as well) is correct
if the length fulfils the inequality
l≥a
a
t
By substituting the dominating long influence length ( n = 2 ) , the following range of
application is obtained
1
l ≥ l in,2= 2
2
It is thus tentatively concluded that formulas (5.5) and (5.6) are correct for a length
larger than the half influence length for n = 2 .
106
5 Chimney – Numerical results and parametric study
Figure 5-7 Stress ratio for υ = 0.3 at the outer fibre obtained by formula (5.6) and the
program CShell.
To further investigate and show the dependency on the geometry, the multiplication
factor is calculated with the program for a chimney with different radius-to-thicknessratio of 50, 100, 200 and 400 and a varying length-to-radius-ratio taken such that the
length varies from 0.1 up to 1.1 times the long influence for n = 2 . The multiplication
factor is compared with the multiplication factor as calculated by the formula. The
multiplication factor is calculated for the stress at the middle surface and for the stress
at the outer surface and both without and with the lateral contraction.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The results are presented in the next four figures, which show the ratio of the
multiplication factor calculated by the program to the multiplication factor calculated
by the formula for the considered radius-to-thickness-ratios, i.e. σ0xx≤,nt ≤5 ( z ) σnxx=,1t ( 0 ) . On
the horizontal axis the length-to-(half influence length for n = 2 )-ratio is shown. The
four figures show the result for the following radius-to-thickness-ratios:
Figure 5-8 for a t = 50 ( l n = 2 ≈ 16.5a ) ,
i ,2
Figure 5-9 for a t = 100 ( l n = 2 ≈ 23.3a ) ,
Figure 5-10 for a t = 200 ( l n = 2 ≈ 33.0a ) , and
Figure 5-11 for a t = 400 ( l n = 2 ≈ 46.6a ) .
i ,2
i ,2
i ,2
Not surprisingly, the figures show that with increasing thinness of the chimney, the
accuracy of the formula increases. For all radius-to-thickness ratios, the agreement
between the stress calculated by the formula and the stress calculated by the program is
excellent for infinitely long chimneys up to a length equal to half of the influence
length for n = 2 . Additionally, the ratio for the middle surface stress and the ratio for
the outer surface stress are both with approximately the same accuracy predicted by the
formula, whether the lateral contraction is accounted for or not. Herewith the range of
application is conclusively determined. A discussion of the observed relation is
included in the next subsection.
Program-to-formula-ratio
1.25
1.00
0.75
middle fibre, poisson=0
outer fibre, poisson=0
middle fibre, poisson=0.3
outer fibre, poisson=0.3
0.50
0.25
0.00
0.2
0.5
0.7
1.0
1.2
1.5
1.7
length-to-(half influence length) ratio
Figure 5-8 Program-to-formula-ratio of multiplication factor for a t = 50 .
108
2.0
2.2
5 Chimney – Numerical results and parametric study
Program-to-formula-ratio
1.00
0.75
middle fibre, poisson=0
outer fibre, poisson=0
middle fibre, poisson=0.3
outer fibre, poisson=0.3
0.50
0.25
0.00
0.2
0.4
0.7
0.9
1.1
1.3
1.5
1.7
2.0
2.2
length-to-(half influence length) ratio
Figure 5-9 Program-to-formula-ratio of multiplication factor for a t = 100 .
Program-to-formula-ratio
1.00
0.75
middle fibre, poisson=0
outer fibre, poisson=0
middle fibre, poisson=0.3
outer fibre, poisson=0.3
0.50
0.25
0.00
0.2
0.5
0.7
1.0
1.2
1.5
1.7
2.0
2.2
length-to-(half influence length) ratio
Figure 5-10 Program-to-formula-ratio of multiplication factor for a t = 200 .
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Program-to-formula-ratio
1.00
0.75
0.50
middle fibre, poisson=0
outer fibre, poisson=0
middle fibre, poisson=0.3
0.25
outer fibre, poisson=0.3
0.00
0.2
0.4
0.6
0.9
1.1
1.3
1.5
1.7
1.9
2.2
length-to-(half influence length) ratio
Figure 5-11 Program-to-formula-ratio of multiplication factor for a t = 400 .
5.2.3 Discussion of results for a fixed base and free end
In this section, the behaviour of a long circular cylinder (for example an industrial,
steel chimney) with a fixed base and a free end has been studied. The presented closedform solution (as obtained for such a long circular cylinder under the wind load
described in section 5.1) and the range of application (as extracted from the previous
subsection) are summarised here for convenience and discussion.
The circumferential distribution of the axial membrane stress resultant nxx and the
bending stress couple mxx are obtained at the clamped base for the three loaddeformation behaviours. The resulting stresses and displacements from the first term
( n = 0 ) of the series development for the wind load (i.e. the axisymmetric loading) are
known to be negligible in comparison with the response to the other terms of the wind
load and are hence discarded. A useful design formula is derived for the stress
distribution at the clamped base of the long cylinder by expressing the influence of the
self-balancing terms ( n = 2,...,5 ) by their ratio to the response to the “beam term”
( n = 1) . This resulted in expressions (5.5) and (5.6) for the maximum tensile stress at
the windward meridian ( θ = 0 ) , which respectively read
110
5 Chimney – Numerical results and parametric study
σ0xx≤,nt ≤5 ( z = t 2 ) = 0.224
2

l2
υ 
 a  a 
pw 1 + 6.39 1 − υ2    1 + 3

at 
l
t
   
1 − υ2 
l
σ0xx≤,nt ≤5 ( z = t 2 ) = 0.224 pw  
 l1 
2
4

 a  
υ 
1 + 6.39 1 − υ2    1 + 3


1 − υ2 
 l2   
The stress ratio between the total membrane stress and the ‘beam theory stress’ is thus
given by the term within the straight brackets in the above formulas. This membrane
stress ratio has been compared with the stress ratio as obtained from the program
results. For long circular cylindrical shells with υ = 0.3 having a length-to-radius-ratio
ranging from 10 to 30 and a radius-to-thickness-ratio ranging from 50 to 400, Figure
5-6 for the middle fibre (membrane) stress results and Figure 5-7 for the outer fibre
stress results are found to be in excellent agreement for cylinders longer than half of
the influence length of the long-wave solution for n = 2 .
The figures obtained for shorter cylinders from very short up to about the influence
length (Figure 5-8 through Figure 5-11) revealed that the above design formulas
(including the bending stress) are indeed applicable to cylinders longer than half of the
influence length of the long-wave solution for n = 2 , which can be explained by the
following.
If the cylinder is longer than the long influence length for a certain mode number,
the incompatibility with the boundary conditions for the free edge is compensated by
an edge disturbance that is described by a small short-wave and equally small longwave solution which are both of the order O ( υβ −2 ) , when compared with the edge
disturbance originating from the clamped edge. Moreover, these edge disturbances
originating from the free edge do not influence the clamped edge.
However, for a cylinder shorter that the long influence length of a certain mode
number, the long-wave edge disturbance originating from the clamped edge for that
mode number is notable and significant at the free edge, i.e. of the same order but of
smaller magnitude. The incompatibility with the boundary conditions at the free edge is
compensated by a long-wave edge disturbance from the free edge, which in turn is
notable at the clamped edge. This provides an additional incompatibility at the clamped
edge to be compensated by an additional edge disturbance, which provides the main
difference between the actual stress at the base and the stress as predicted by the design
formula.
For the cylinder equal to the half influence length, the edge disturbance is of the
order e
−
π
2
≈ 0.21 at the free edge, which thus results in an additional edge disturbance at

−
π

2
the clamped edge in the order of  e 2  ≈ 0.043 . For a cylinder shorter than the half


influence length, the difference quadratically and exponentially increases, which
conclusively explains the graphs as presented in the previous subsection.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
5.3 Influence of stiffening rings
The subject of this section is to investigate the influence of stiffening rings on the
behaviour of the long chimney. Additionally, this influence can be captured in a
closed-form solution and the range of application is identified by computational results.
In section 5.2, it is shown, as a description of the behaviour, that the stress at the
fixed base of a long cylinder under wind load can be conveniently related to the beam
mode ( n = 1) . The deformation and stress for the axisymmetric mode ( n = 0 ) are of no
importance on the overall behaviour. It is expected that mainly the response to the
higher modes ( n ≥ 2 ) is altered by the presence of a stiffening ring in comparison with
the response of a cylinder without rings.
For these higher modes, the normal stress resultant nxx at the fixed base is directly
related to the induced out-of-roundness (“ovalisation”) of the cylinder, which cannot
occur at the base. As a consequence, the cross-section intends to warp at the base. The
normal stresses are needed to withstand this warping, in other words: to keep this
section plain. As the presence of stiffening rings within reasonable distance of the fixed
base will reduce the ovalisation and hence the warping that needs to be counteracted,
nxx is reduced accordingly.
The first objective of the next subsection is to define an influence measure for the
resistance to ovalisation based on the situations with and without stiffening rings. To
arrive at such a measure, closed-form solutions will be developed for a number of cases
to determine the governing parameters. The second and successive objective is to
extract a useful formula describing the influence of the stiffening rings on the stress at
the base of a long cylinder.
5.3.1 Closed-form solution (full solution)
Deriving a closed-form solution for a cylinder stiffened by rings involves the
determination of the boundary conditions at both edges of the cylinder and the
transitional conditions at the locations of the stiffening rings between those edges. As
per each edge four conditions and per each stiffening ring in between those edges eight
conditions must be identified, the total set of equations to be formulated can become
rather big. Hence, a closed-form solution will become too cumbersome for
interpretation if not impossible to obtain. The objective of this subsection is to derive
closed-form solutions for a number of basic cases from which insight is gained in the
stiffening effect of the ring on the cylinder and general expressions can be formulated
based on the extracted governing parameters.
The first two cases describe an infinitely long cylinder with only one stiffening
ring and a semi-infinitely long cylinder (a cylinder of infinite length with one free
edge) with one stiffening ring at its edge, respectively. From these, relative simple,
cases, the governing parameters are obtained that describe the interaction of the
stiffening ring with the cylinder. Moreover, a relevant simplification can be introduced
based on these two cases and making use of that simplification, the infinitely long
cylinder with equidistant stiffening rings is investigated as a third case to arrive at the
sought formula describing the influence of stiffening rings on the reduction of the
ovalisation and other quantities accordingly.
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5 Chimney – Numerical results and parametric study
To simplify the analysis, only stiffening rings with their centre of gravity located at the
middle surface of the cylinder are considered. Hence, the relation between the loads on
the ring and the ring displacements is substantially simplified. Based on the relation
(E.7) of Appendix E, the simplified description of the stiffening ring behaviour
becomes
 fˆθ 
 
 fˆz 
ring
 EAr 2
 a2 n
=
 EAr n
 a 2
EAr

n
 uˆθ  ring
a2
 
2
EAr EI r 2
1
+
n
−
( )  uˆz 
a2
a4
(5.7)
in which the combined cross-sectional properties Ar and I r for rings symmetric to the
middle surface of the cylinder are given by the elementary integrals
Ar = ∫ dA , I r = ∫ z 2 dA
A
A
which for a (single) ring of rectangular cross-section with width b and height h
become
Ar = bh , I r =
1 3
bh
12
At the location of the ring in between two cylindrical parts denoted by i and i + 1 , the
following systems of equations, from which the relevant boundary conditions can be
extracted, is formulated for the modes n ≥ 2 .
(i )
 fˆx ( li ) 
 fˆx ( 0 ) 
ˆ

ˆ

 f θ ( li )  +  f θ ( 0 ) 
 fˆ ( l ) 
 fˆ ( 0 ) 
 z i 
 z

ˆ
t
l
 x ( i ) 
 tˆx ( 0 ) 
( i +1)
0
 fˆ 
+  θ
 fˆz 
 
0
( ring )
 fˆx 
ˆ
f
=  θ
 fˆ 
 z
 tˆx 
( ext )
 uˆ x ( li ) 
 uˆ x ( 0 ) 




u
l
ˆ
(
)
 θ i  =  uˆθ ( 0 ) 
 uˆ z ( li ) 
 uˆ z ( 0 ) 




ϕˆ x ( 0 ) 
 ϕˆ x ( li ) 
(i )
and
( i +1)
 uˆ x 
u 
ˆ
= θ
 uˆ z 
 
ϕˆ x 
( ring )
To further simplify the analysis, it is tentatively assumed that the spacing between the
rings or spacing between the ring and the (stiffened or not stiffened) edge is such that
the boundary or transitional conditions can be described for the edge or ring location
only. In other words, the spacing between a considered ring or edge and the adjacent
rings or edges is greater that the long influence length of the cylinder.
Based on the assumption above, two base cases are identified, for which the
closed-form solution will be obtained. The first case is a ring in an infinitely long
cylinder and the second case is a semi-infinitely long cylinder with a ring present at a
free edge.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Infinitely long cylinder with one ring
The first case, a ring in an infinitely long cylinder, represents thus the case that the ring
is located such that, at both sides, the adjacent rings or edges are located further away
than the length of the long edge disturbance. Hence and based on symmetry
considerations, the four boundary conditions are described by
 uˆ x ( 0 ) 
ˆ

 fθ ( 0 ) 
 fˆz ( 0 ) 


ϕˆ x ( 0 ) 
( cylinder )
 0 
 − 1 fˆ 
=  21 θ 
 − 2 fˆz 


 0 
( ring )
(5.8)
where, for the sake of simplicity, the external load on the ring is assumed to be zero.
To solve this system, terms multiplied by β−2 are neglected in comparison to unity.
As a reference, the solution for a clamped base given in section 5.2 is recalled and
reads
C1n = C2n = −
υ n2 − 1 n
uˆ z
2 β2
; C5n = C6n = −uˆ zn
which can thus be obtained from the system (5.8) by equating, the ring extensional and
flexural rigidities to infinity ( EAr = EI r = ∞ ) .
Based on this solution, the following assumption is introduced. For a ring with a
very low extensional and flexural rigidity, the ovalisation of the cylinder will fully
develop under the wind load and all constants are then computed equal to zero. For a
ring with a very high extensional and flexural rigidity, the constants are equal to the
solution as given for the clamped base. Hence, it is assumed that the constants C1n and
C2n are of the order O ( β−2 ) when compared with the constants C5n and C6n .
The neglect of small terms and the assumption presented above further facilitate
obtaining a solution to the four boundary conditions given by the system (5.8) and
reads
C1n = C2n = O ( β−2 ) ⋅ uˆ zn
; C5n = C6n = −
ηring
ηring + 1
uˆ zn
in which the parameter ηring is introduced as
ηring =
1 I r n n 2 − 1 β bh3 n n 2 − 1 1 − υ2
=
β2
β
4 Db
a at 3
4
(5.9)
This parameter is thus described by the ratio of the moment of inertia of the ring to
both the moment of inertia of the cylinder (if taken as the cross-sectional beam
property) as well as the geometrical properties of the cylinder. By back substitution, the
displacement at the ring location is obtained as
5 
5

η
1
u z2 ≤ n ≤ 5 ( 0, θ ) = ∑ 1 − ring  uˆ zn cos nθ =∑
uˆ zn cos nθ


η
+
1
η
+
1
n =2 
n = 2 ring
ring

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5 Chimney – Numerical results and parametric study
Semi-infinitely long cylinder with one ring at its end
The second case, a semi-infinitely long cylinder with a ring present at a free edge,
represents thus the case that, on one side, the adjacent rings or edges are located further
away than the length of the long edge disturbance. Hence, and based on symmetry
considerations, the four boundary conditions are described by
 fˆx ( 0 ) 
ˆ

 fθ ( 0 )
 fˆ ( 0 ) 
 z

 tˆx ( 0 ) 
( cylinder )
 0 
 − fˆ 
=  θ
 − fˆz 


 0 
( ring )
(5.10)
where, for the sake of simplicity, the external load on the ring is assumed to be zero.
Identical to the solution as presented for the ring in an infinitely long cylinder, the
terms multiplied by β−2 are neglected in comparison to unity and it is assumed that the
constants C1n and C2n are of the order O ( β−2 ) when compared with the constants C5n
and C6n .
To further facilitate obtaining a solution to the four boundary conditions given by
the system (5.10), it is assumed that the ring geometrical properties are such that the
following relations hold
t<h,
2
h2
( n2 − 1) 1
12a 2
which seems plausible for a normally sized stiffening ring. However, the solution will
be limited to rings of which the height is larger than the thickness of the cylinder but
smaller than the radius of that cylinder.
Making use of all the simplifications and assumptions as mentioned above, the
solution becomes
C1n = C2n = O ( β−2 ) ⋅ uˆ zn
; C5n = −
ηring
ηring + 1
; C6n = O ( β−2 ) ⋅ uˆ zn
uˆ zn
in which the parameter ηring is introduced above (5.9). By back substitution, the
displacement at the ring location is thus equally described as for the case with the
infinitely long cylinder.
Conclusion from the above cases
From the solution as presented for the two cases above and especially from the
parameter ηring that captures the influence of the stiffening ring on the cylinder, it can
be concluded that, for a sufficiently long and thin ( β−2 1) cylinder, the extensional
rigidity of the ring has a negligible influence on the reduction of the ovalisation in
comparison with the influence of the flexural rigidity. This is easily understood as it
can be expected that not the global deformation of the cross-section of the cylinder is
changed due to the presence of the stiffening ring, but that the amplitudes of the
displacements uθ and u z are reduced within the long influence length originating from
the location of the stiffening ring. Hence, identical to the inhomogeneous solution for
1
n
the displacements, the relation uθn = − u zn also holds for the displacements of the
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
stiffening ring. Introducing this relation into the relation (5.7) between the loads on the
stiffening ring and its displacements, results in
 fˆθ 
 
 fˆz 
ring
0



=  EI r 2
2


−
n
1
u
ˆ
(
)
z
 a 4

ring
(5.11)
which is in line with the assumption that only the flexural rigidity of the ring influences
the behaviour of the cylinder for the modes n ≥ 2 .
If the re-formulated relation (5.11) for the behaviour of the ring is adopted in, e.g.,
the system (5.8) for the first case, the solution to that system is identical to the solution
presented for that case.
Infinitely long cylinder with equidistant rings
Based on the observations above, a third case is analysed which comprises an infinitely
long cylinder with equidistant stiffening rings. The rings are spaced such that the longwave edge disturbance originating from a ring is notable and significant at the adjacent
ring, but that the long-wave edge disturbance from the adjacent ring induced by this
effect is negligible at the subject ring. Based on symmetry considerations, only one
cylinder between two identical rings can be analysed for which the eight boundary
conditions are described by
 uˆ x ( 0 ) 
ˆ

 fθ ( 0) 
 fˆz ( 0 ) 


ϕˆ x ( 0 ) 
 uˆ ( l ) 
 x 
 fˆθ ( l ) 
 ˆ

 f z (l ) 
 ϕˆ x ( l ) 
( cylinder )

− 1
 2
 − 12

=

 1
− 2
− 1
 2

0

 0 
 0 
ˆf θ ( ring 1) 



ring
(
1)

 −uˆ z ( ring 1) 
fˆz



0
 = 1 EI r n 2 − 1 2  0 
) 0 
 2 a4 (
0



fˆθ( ring 2) 
 0 
 −uˆ ( ring 2) 
fˆz ( ring 2) 

 z


 0 
0
(5.12)
where relation (5.11) between the loads on the ring and its displacement u z is
employed and, for the sake of simplicity, the external load on the ring is assumed to be
zero.
Obviously, the solution to these equations describes a symmetric response with
respect to the mid-section of the cylinder length between the two stiffening rings. To
solve this system, the same simplifications are introduced as for the first case. The
terms multiplied by β−2 are neglected in comparison to unity and it is assumed that the
constants C1n and C2n are of the order O ( β−2 ) when compared with the constants C5n
and C6n . The same applies to the constants C3n and C4n which are of the order O ( β−2 )
when compared with the constants C7n and C8n . Moreover, the constants C5n and C6n
are of the same order as the constants C7n and C8n .
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5 Chimney – Numerical results and parametric study
To obtain full symmetry, the displacement (4.40) is rewritten similar to the
proposed representation of (4.43) to
x
x
x 
 − a1n β 


u z ( x, θ ) = cos θ e a C1n cos  bn1β  + C2n sin  bn1β  
a
a






+e
+e
+e
a1n β
x −l
a
− an2 β
an2 β
x
a
 n
 1 x−l 
 1 x − l 
n
C3 cos  bnβ a  + C4 sin  bnβ a  





 n
 2 x
 2 x 
n
C5 cos  bn β a  + C6 sin  bn β a  





x −l
a
 n
 2 x −l 
 2 x − l   
n
C7 cos  bn β a  + C8 sin  bn β a   



  

2
1  a2 
 ( 2) 1 ( 2)  x
(1) 1 (1) 
+  2
 cos θ  pzn − pθn  + p zn − pθn 
Db  n − 1 
n
n
l


Correspondingly, all other quantities describing displacements, stress resultants and
stress couples as given in Appendix I are rewritten by applying the same
transformation for the edge disturbances originating form the edge at x = l .
To solve the system (5.12), the observation that e
been employed to introduce the simplification that
2
 − an2 β al 
 e
 1 and


e
− a1n β
l
a
− a1n β
l
a
<e
− an2 β
l
a
and e
− an2 β
l
a
< 1 has
1
and hence these exponential terms are neglected in comparison to unity.
The solution to the system (5.12) for the eight constants then reads
C1n = C3n = −
l

− an2 β 

n n 2 − 1 n 2 − 1 ηring
 2 l
 2 l    n
a
1
+
η
1
−
2
e


 cos  bn β  + sin  bn β     uˆ z
ring
2
2
2

β
n (1 + ηring ) 
a
a    




C2n = −C4n = C1n
C5n = C7n = −
ηring
(1 + η )
ring
C6n = −C8n = −
2
l
− an2 β


 2 l
 2 l   n
1 + ηring + e a (1 − ηring )  cos  bn β  + sin  bn β    uˆ z
a
a  





l
− an2 β 

 2 l
 2 l   n
a
+
η
+
1
e
 (1 − ηring ) cos  bn β  − (1 + 3ηring ) sin  bn β    uˆ z
ring
2 
a
a  



(1 + ηring ) 
ηring
By back substitution of these constants in the expression for the displacement u z at
x = 0 , this expression reads
l
5

− an2 β 

1
 2 l
 2 l    n
a
u z2 ≤ n ≤ 5 ( 0, θ ) = ∑
1
+
η
1
−
2
e


 cos  bn β  + sin  bn β     uˆ z cos nθ
ring 
2
a
a    


n = 2 (1 + η



ring ) 
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
It is easily observed that, if the rings are spaced with a distance such that e
the solution for the eight constants becomes
C1n = C2n = C3n = −C4n = −
C5n = C6n = C7n = −C8n = −
− an2 β
l
a
1,
n n 2 − 1 n 2 − 1 ηring n
uˆ z
β2
n 2 1 + ηring
ηring
1 + ηring
uˆ zn
which is identical to the solution as presented above for the first case.
Furthermore, the constants C1n , C2n , C3n and C4n of the short edge disturbance are
indeed of the order O ( β−2 ) if compared to the constants C5n , C6n , C7n and C8n of the
long edge disturbance.
Interpretation and governing parameters
In the cases above, the influence of the stiffening rings on the behaviour of the circular
cylinder is analysed. It is shown that, for these cases, the influence is fully captured by
the parameter ηring (5.9), which reads
ηring =
1 I r n n2 − 1 β
4 Db
β2
a
Upon inspection of the description (4.42) for the governing parameters of the longedge disturbance, these parameters can be further approximated by
1
1 n n2 − 1
an2 ≈ bn2 ≈ ηn =
2
2 β2
where the simplification is introduced that terms multiplied by β−2 are neglected in
comparison to unity as performed in section 5.2. Hence, the governing parameter
introduced to describe the influence of the stiffening rings on the behaviour of the
circular cylinder can be rewritten to
ηring ≈
1 Ir 1 β
ηn
2 Db 2 a
In other words, this parameter comprises the ratio of the bending stiffness of the
ring to the wall bending stiffness of the cylinder multiplied with the constants of the
argument of long-influence attenuating terms.
Moreover, it is shown that the extensional rigidity of the ring has a negligible
influence on the reduction of the out of roundness in comparison with the influence of
the flexural rigidity and, as a result, the relation u zn = −nuθn for the amplitudes of the
displacements uθ and u z can be adopted. Although the constants related to the short
edge disturbance are present while being of the order O ( β−2 ) if compared to those of
the long edge disturbance, it can be easily observed from the expressions for the
displacements as given in Appendix I that the presence of the short edge disturbance
has a negligible impact on the displacements in comparison with the terms related to
the long edge disturbance. Hence, the difference between the ring displacements and
the more distant shell material is reduced within the long influence length originating
from the location of the stiffening ring.
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5 Chimney – Numerical results and parametric study
Alternative approach
With the objective to investigate the influence of stiffening rings on the behaviour of
the long chimney, a solution similar to the ones as presented above is too cumbersome
to obtain, e.g., a suitable formula for the stress distribution at the base of a long, ringstiffened chimney under wind loading. Typically, such a chimney is circumferentially
stiffened by multiple, equidistant rings against the ovalisation due to the wind load.
These multiple stiffening rings provide a multitude of the number of equations to be
solved in closed-from disabling such an approach. Alternatively, a resolution might be
found in “smearing out” the relevant ring properties along the circular cylinder. If such
an approach is incorporated into the Morley-Koiter equation, orthotropic relations need
to be accounted for with which its elegance is lost making this approach not feasible.
However, a novel approach is suggested which overcomes the abovementioned
complications.
The observations above for the more rigorous solution indicate that the
simplifications introduced for the semi-membrane concept as developed for circular
cylindrical shells might be adopted for such shells that are stiffened by rings. In this
novel approach, the relevant ring property, i.e. the bending stiffness, is “smeared out”
along the circular cylindrical shell surface, which is further elaborated upon in the next
subsection.
5.3.2 Closed-form solution (SMC)
The semi-membrane concept (SMC), as referred to in section 1.4, and its application to
circular cylindrical shells are described in Appendix G. This concept is applicable to
non-axisymmetric load cases of circular cylindrical shells provided that the cylinder is
sufficiently long in comparison to its radius and that the boundary effects mainly
influence the more distant material. Correspondingly, two main simplifications are
introduced. Firstly, the circumferential strain εθθ is equal to zero and hence u zn = −nuθn .
Secondly, the bending moments about the circumferential axis and torsion axis are zero
and hence mxx = 0 mxθ = 0 and consequently vx = 0 .
As presented in Appendix G, the differential equation for the semi-membrane concept
reads
2
2
  β 4 ∂ 4
 
1
∂2  ∂2  ∂2
2 ∂
4 
+
2
1
+
υ
a
+
+
1
)
 (


  uz
4
a8 
∂x 2 ∂θ2  ∂θ2  ∂θ2
  a  ∂x
 
3
3
3

1 
1 ∂
1 ∂   ∂pz
 1 ∂
px 
=
+ pθ  − 3
 2 (1 + υ ) 2 2 + 4 3  
2
Db 
a ∂x ∂θ a ∂θ   ∂θ
 a ∂x∂θ

where the dimensionless parameter β is introduced in (4.17) and the solution for
u z ( x, θ ) to this equation becomes
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
x
 − anSMC β a  n
 SMC x 
 SMC x  
n
u z ( x, θ ) = cos nθ e
C1 cos  bn β a  + C2 sin  bn β a  






+e
anSMC β
x
a
 n
 SMC x 
 SMC x   
n
C3 cos  bn β a  + C4 sin  bn β a   



  

(5.13)
2
1  a2 
 ( 2) 1 ( 2)  x
(1) 1 (1) 
+  2  cos nθ  pzn
− pθn  + pzn
− pθn 
Db  n − 1 
n
n
l


in which the approximated dimensionless parameters anSMC and bnSMC are defined by
1  1
1  1


anSMC = ηn  1 + γ nSMC 
bnSMC = ηn  1 − γ nSMC 
,
2  2
2  2


4
SMC
for small values of γ n and ηn for β 1 , where the dimensionless parameters ηn is
introduced in (4.42) and γ SMC
is equal to
n
γ nSMC =
1 + υ n2 − 1
2 β2
The main difference between this SMC solution (5.13) and the solution (4.40) to the
Morley-Koiter equation is that the short edge disturbance is not described by the SMC,
which is inherent to the introduced simplifications. Furthermore, the only small
difference between the two solutions is observed in the arguments of the exponential
and trigonometric terms of the long edge disturbance. Hence, it can be proposed to
adopt only the leading term in the SMC, which is similar to the approximation and
interpretation as adopted in the previous subsection, resulting in
1
1 n n2 − 1
an2 ≈ bn2 ≈ ηn =
2
2 β2
In other words, the contributions of the order β−2 in comparison to unity are not
described, which is clearly admissible when the main simplifications of the SMC can
be adopted.
As the inhomogeneous solution is correctly described by the SMC, it is concluded
that the expressions for all quantities can be adequately adopted to obtain a closed-form
solution for engineering purposes.
Stiffening rings with their centre of gravity located at the middle surface of the cylinder
To show that the mentioned differences hardly impact the results, the third case of the
previous subsection, the “Infinitely long cylinder with equidistant rings”, is analysed
by the SMC solution. Analysing only one cylinder between two identical rings, the four
boundary conditions that can be described become
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5 Chimney – Numerical results and parametric study

u x ( 0, θ )



 f ( 0, θ ) + ∂f z ( 0, θ ) 
 θ
∂θ 


ux (l, θ)



∂f z ( l , θ ) 
 fθ ( l , θ) +

∂θ 

( cylinder )
( ring 1)


ux ( θ)


( ring 1)
 1

1 ∂f z ( θ )
( ring 1)
−
− fθ ( θ)

2
2
∂θ

=
( ring 2)


ux ( θ)


( ring 1)
 1

1 ∂f z ( θ )
( ring 1)
−
− fθ ( θ)

2
∂θ
 2

which upon substitution of the appropriate cosine and sine functions results in

uˆ x ( 0 )

ˆ

ˆ
 f θ ( 0 ) − nf z ( 0 ) 


uˆ x ( l )
 ˆ

ˆ
 f θ ( l ) − nf z ( l ) 
( cylinder )
0


 0 
 − fˆ ( ring 1) + nfˆ ( ring 1) 
 ( ring 1) 
1
z
 = EI r n ( n 2 − 1)2  uˆ zz

=  θ
4
 a 2
 0 
0
2
 ˆ ( ring 2)

 ( ring 2) 
+ nfˆz ( ring 2) 
uˆ z

− fθ
where relation (5.11) between the loads on the ring and its displacement u z is
employed.
The solution to these equations becomes
C1n = C3n = −
ηring
(1 + η )
ring
C2n = −C4n = −
2
1
l
− ηn β


l
l   n
1
1
1 + ηring + e 2 a (1 − ηring )  cos  ηnβ  + sin  ηnβ    uˆ z
2
a
2
a



 


1
l
− ηn β 

l
l   n
1
1
a
2
+
η
+
1
e

 (1 − ηring ) cos  ηnβ  − (1 + 3ηring ) sin  ηnβ    uˆ z
ring
2
a
a  
2
2

(1 + ηring ) 
ηring
and from inspection of this solution in comparison with the solution presented in the
previous subsection, it is concluded that the expressions for all quantities as derived by
an SMC approach indeed can be adequately adopted to obtain a closed-form solution
for engineering purposes.
In the SMC approach, the bending stiffness of the shell is only adopted for the
circumferential bending moment. As the ring behaviour can be adequately described by
the bending action of the ring only, it is proposed to “smear out” the bending stiffness
of the rings along the bending stiffness of the cylinder resulting in the following
modified bending stiffness
Db,mod = Db +
EI r
lr
where lr denotes the spacing between the rings. Hence, the difference between the
solution for a long cylinder with multiple equidistant stiffening rings and the solution
for a long cylinder without these rings can be captured by a modified parameter βmod
only, which becomes
4β4mod
1 − υ2
=
kmod
where
kmod =
Db ,mod
Dm a 2
EI r
lr
Dm a 2
Db +
=
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The modified parameter kmod can be rewritten to
kmod =
Db
λ r −1 where
Dm a 2
λr =
Db
EI
Db + r
lr
(5.14)
in which the stiffness ratio λ r represents the ratio of the bending stiffness of the
circular cylindrical shell only to the modified bending stiffness of the shell (with the
contribution of the ring stiffness per spacing). Hence, the following relation exists
between the original and the modified formulations of the modified parameters
kmod = k λ r −1
;
βmod = β 4 λ r
For the special case of rectangular stiffening rings with width b and height h
located at the middle surface of the circular cylindrical shell, the stiffness ratio λ r
becomes
λr =
lr t 3
lr t 3 + bh3 (1 − υ2 )
Similar to subsection 5.2.1, a useful design formula can be derived for the stress
distribution at the base of the long chimney stiffened by equidistant rings by
calculating the response to the wind load. Fully in line with the approach for the long
chimney without stiffening rings, the contribution of the self-balancing terms
( n = 2,...,5 ) can be expressed by their ratio to the response to the “beam term” (5.2).
In order to obtain this ratio, first the appropriate boundary conditions need to be
solved for the self-balancing terms as described in the SMC solution, which are given
by
x =0;
clamped:
u x = u x = 0 ; uθ = uθ = 0
x=l;
free:
f x = nxx = 0 ;
fθ +
∂f z
= nx θ = 0
∂θ
to which, if the chimney is long enough that the edge disturbance originating from the
free edge does not influence that disturbance at the clamped edge, the solution for the
constant becomes
 3
n2 − 1  n
C2n = − 1 − (1 + υ)
 uˆ z
βmod 2 
 2
C1n = −uˆ zn ;
;
C3n = C4n = 0
where for the sake of comparison, the contributions of the order β−2 are retained.
Substitution of the constants into the expression for nxx as presented in Appendix I,
results in the expression for nxx at the clamped edge
5
nxx ( 0, θ ) = −∑ 2 3(1 − υ2 )
n =2
a2
p 
n2 − 1 
λ r 2 zn 1 − (1 + υ)
 cos nθ
β mod 2 
t
n −1
Having found the response for these self-balancing terms ( n = 2,...,5 ) , the useful design
formula can be derived for the stress distribution at the base. This results in an
expression for the axial stress at the base x = 0 , which reads
 σ2 ≤ n ≤ 5 ( 0, θ, z ) 
σ0xx≤ n ≤5 ( 0, θ, z ) = σ nxx=1 ( 0, θ, z ) 1 + xxn =1

σ xx ( 0, θ, z ) 

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5 Chimney – Numerical results and parametric study
It can be assumed that the ratio of the bending-to-membrane stress for the nonbalancing terms is not altered although not described by the SMC solution and remains
identical to ratio as obtained for the “beam term”. Hence, it is tentatively proposed that
the formula for the stress distribution at the base can be further simplified to
 σ 2≤ n ≤5 ( 0, θ,0 )  
υ 
σ0xx≤ n ≤5 ( 0, θ, ± t 2 ) = σ nxx=1 ( 0, θ,0 ) 1 + xxn =1
 1 ± 3

σ
θ
0,
,0
)  
1 − υ2 
xx (

The formula for the maximal tensile stress at the clamped edge at the windward
meridian ( θ = 0 ) for the long chimney without stiffening rings (5.5) is obtained by
substituting the wind load (5.1). Performing the same substitutions for the long
chimney stiffened by rings, this expression becomes approximately
σ0xx≤,nt ≤5 ( z = t 2 ) = 0.224
2

l2
υ 
 a  a 
pw 1 + 6.39 1 − υ2 λ r     1 + 3

at 
 l  t 
1 − υ2 

in which the only change compared with (5.5) is the addition of the factor
(5.15)
λ r within
the straight brackets. The stiffness ratio λ r is defined by (5.14).
Eccentric stiffening rings to the middle plane of the cylinder
For an eccentric ring, the relation between the loads f θ and f z on the ring and its
displacement uθ and u z is slightly more involved resulting in a too complicated set of
equations for deriving a closed-form solution on basis of the full solution to the
Morley-Koiter equation. However, a similar modification as followed above for the
symmetric ring on basis of the SMC approach can be easily employed.
The behaviour of the stiffening ring is described by relation (E.7) of Appendix E,
which reads
EAr
ES

n + 3r n ( n 2 − 1)
 uˆθ  ring
a2
a
(5.16)
 
2
EAr
ESr 2
EI r 2
uˆ z 


+ 2 3 ( n − 1) + 4 ( n − 1)

a2
a
a
in which the combined cross-sectional properties Ar , Sr and I r for rings asymmetric
 fˆθ 
 
 fˆz 
ring
EAr 2

n

a2
=
 EAr n + ESr n n 2 − 1
( )
 a 2
a3
to the middle surface of the cylinder are given by the integrals
Ar = ∫ dA + a −1 ∫ zdA
A
A
S r = ∫ zdA
A
I r = ∫ z 2dA − a −1 ∫ z 3dA
A
A
which are evaluated with respect to the middle surface of the cylinder in the program
CShell.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
1
n
If the simplification uθn = − u zn is introduced in relation (5.16) between the loads
on the stiffening ring and its displacements, this relation becomes
 fˆθ 
 
 fˆz 
ring
ES r


n ( n 2 − 1) uˆ z


a3
=

 ES r n 2 − 1 uˆ + EI r n 2 − 1 2 uˆ 
) z a 4 ( ) z 
 a 3 (
ring
and further by combining the ring loads as performed within the SMC approach, the
relation reads
∂f z 

 fθ +

∂θ 

ring
= −n sin nθ
2
EI r 2
( n − 1) ( uˆz )ring
a4
which is, identical to the symmetric ring case, in line with the assumption that only the
flexural rigidity of the ring influences the behaviour of the cylinder for the modes
n≥2.
1
n
The above depends on the validity of the simplification uθn = − u zn for an
eccentric ring, which has not, similar to the case for a symmetric ring, been concluded
on basis of closed-form solutions on basis of the full solution to the Morley-Koiter
1
n
equation. If the assumption that the simplification uθn = − u zn is also valid for the
eccentric ring, formula (5.15) for the stress distribution at the base of the long chimney
stiffened by equidistant rings holds for both symmetric stiffening rings and eccentric
stiffening rings. To obtain the stiffness ratio λ r as defined by (5.14), the flexural
rigidity of the ring should then be taken as
I r = ∫ z 2dA − a −1 ∫ z 3dA
(5.17)
A
A
which is evaluated with respect to the middle surface of the cylinder in the program
CShell..
It is however envisaged that the determination of the flexural rigidity of eccentric
stiffening rings by expression (5.17) will result in an overestimation of the stiffness
ratio λ r (5.14) if used in conjunction with formula (5.18). The overestimation of the
ring stiffness can be explained by the fact that at the intersection of the cylinder wall
with the web of the eccentric ring forces are transferred from the ring into the shell.
This transfer results in an introduction of nθθ , nxθ and vx into the shell. For the ring
with its neutral axis on the middle plane of the cylinder, these stress resultants are
either negligible (as for nθθ and vx ) or distributed along the cylinder according to the
long wave solution only (as for nxθ ). For the eccentric ring, the resulting (additional)
distribution of these quantities along the cylinder is not negligible and confined to the
vicinity of the ring location, i.e. described by the short-wave solution. This additional
short-wave shell-ring interaction is not accounted for in the SMC and hence the shell
action of that part of the cylinder (with a length comparable to the short influence
length around the ring location) acting with the ring is not considered. The equivalent
bending stiffness is presently obtained by “smearing out” the ring stiffness over the
spacing between the stiffeners. To properly account for the shell membrane action at
124
5 Chimney – Numerical results and parametric study
the location of the ring, the additional stress transfer might be accounted for by
modifying the ring stiffness and by adopting this modified stiffness in the equivalent
bending stiffness. Fully in line with the approaches to determine the stress distribution
in flanges of curved beams and the critical buckling pressure of ring-stiffened shells,
the effective width concept could be adopted. The inclusion of a certain shell length
acting as a flange on the inner side of the eccentric ring replaces the section of the ring
by a combined section that should be evaluated with respect to its centre of gravity.
This would most likely results in a lower flexural rigidity of the ring in comparison
with the ring that is evaluated with respect to the middle surface of the cylinder.
As discussed and shown by Bleich [49], the stress distribution in the flanges of
curved T- and I-beam cross sections differs from the distribution in solid cross sections
as the assumed invariability of the shape of the cross section is not fulfilled by curved
beams with such a non-solid cross-section. Bleich analysed the variation of the
longitudinal stress in the curved flange resulting from the cross-sectional deformation,
viz. the longitudinal stress decreases as the deflection of the flange increases with
increasing distance from the web. The longitudinal stress is thus maximal directly
above the web and decreases towards the ends of the flange. By replacing the flange of
width b f by a narrower flange in which the maximal prevails everywhere and taking
the replacing width in such a manner that the total force in the beam remains
unchanged, Bleich obtained a formula for this effective width beff of symmetrical
flanges with respect to the web of the beam under extension and bending and provided
some remarks on terms and effects for further improvement.
Similar to the above approach, it is obvious that for a ring-stiffened cylinder,
equivalent formulas can be derived for the length of the shell section acting effectively
with the stiffening ring. This so-called effective length leff is then accounted for while
determining the relevant properties of the combined stiffener and shell section, such as
the cross-sectional area, the location of the neutral axis and the moment of inertia.
There are several accepted methods of determining the effective length. According to
Pegg and Smith [50] and MacKay [51], the simplest method to determine the
associated effective length of the cylinder is to take 75 percent of the shell length
between the stiffeners as suggested by Faulkner [52] for the design of submarines.
Especially in case of long circular cylindrical shells stiffened by rings, the spacing
between the rings might be as long as several times the flange width of the stiffener,
which shows that this approach is not likely applicable for the present purpose.
Moreover, the effective length is a function of the shell geometry in its deformed state
and therefore a function of the shell radius and thickness, the stiffener spacing, the ring
dimensions and eccentricity as noted by Hutchinson and Amazigo [53]; which can be
also identified by inspection of the closed-form solutions presented in subsection 5.3.1.
Furthermore and in case of non-axisymmetric loading, the geometric properties
and the wave number are further influencing the effective length. A rather
straightforward equation that accounts for these effects has been presented by Bijlaard
[54] and reads
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
leff
1
l
l
−
cosh β r − cos β r 
4 2
2
 2
2a
n
t
n
t
a
a  1+

=
+
2 a2
β sinh β lr + sin β lr 
3 a 
a
a
of which the expression without the term within the brackets is in fact an improvement
of Bleich’s equation. The equation without the term within the brackets represents in
fact the effective length for stiffened cylinders under axisymmetric and beam loading.
The equation as presented by Bijlaard (with and without the term within the brackets)
has been adopted in many standards, codes and textbooks to account for the effective
length of the cylinder acting with a stiffening member.
Tables with effective length values are also presented in PD5500:2009
“Specification for unfired fusion welded pressure vessels” published by the British
Standards Institution. These tables provide the effective length for different wave
numbers, ring spacing-to-radius-ratios and thickness-to-radius ratios and the code
indicates that equation as presented by Bijlaard for the effective length might be used
for large radius-to-thickness -ratios (larger than approximately 30), i.e. in the range of
the present investigation.
Pegg and Smith [50] have listed a comparison of the tabulated values and the
values as calculated by the equation, which shows that comparable effective length
values are obtained for different thickness-to-radius-ratios and wave numbers.
For the present purpose, Bijlaard’s equation is investigated in more detail.
Obviously, the term within the brackets accounts for a small decrease of the effective
length for the typical wave numbers and thickness-to-radius-ratios. Only for unlikely
high wave numbers and relatively thick cylinders, which are outside the present
investigation, the term provides a marked decrease that is not negligible. Furthermore,
the term dependent on the distance between the rings lr only has a limited and
negligible influence on the effective length based on the envisaged distance for the
long chimneys under consideration. Even for the thickest cylinders with a radius-tothickness-ratio of 50, the reducing effect of the term dependent on lr is negligible for a
distance between the rings larger than about 0.25 times the radius. In other words, this
term rapidly becomes unity for small distances. Hence, a simplified representation of
Bijlaard’s equation for relatively largely spaced stiffening rings and thin cylinders
reads
leff
2a 
n2 t 
≈ 1 +

β 
3 a
−
1
2

n2 t 
1+
≈


4 3 1 − υ2 
( ) 2 3 a
2 at
−1
It is well known that the effective length is directly related to the characteristic
length. The characteristic length for the short-wave solution for the self-balancing
modes is given in subsection 4.6.3 and by adopting for a1n the approximation of the
solution to Donnell’s equation as presented in subsection 4.5.2 the characteristic length
becomes
−1


a
a  1 n2 
at
n2
t
1 +
lc ,1 = 1 =  1 +
=
2 
a nβ β  2 β 
4 3 1 − υ2 
( )  2 3(1 − υ2 ) a 
126
−1
5 Chimney – Numerical results and parametric study
which shows that Bijlaard’s equation is based on the solution to Donnell’s equation.
Hence, the small terms with respect to the thickness-to-radius-ratio as described by
Bijlaard’s equation are not only surplus to requirements, but that the expression for the
effective length can also be slightly improved by adopting for a1n the approximation of
the solution to the Morley-Koiter equation as presented in subsection 4.5.1.
Based on the above and for the present purpose, the following equation is
tentatively considered for the effective length of the cylinder wall acting together with
a stiffening member at its location
−1
−1


n 2 − 12
2a  1 n 2 − 12 
2 at 
t
leff ≈ 1 +
=
1
+

β  2 β2 
4 3 1 − υ2 
( )  2 3(1 − υ2 ) a 
For the typical value of υ = 0.3 for Poisson’s ratio of steel, the effective length is
approximately equated to 1.56 at in which the term within the brackets is set to unity
as an approximation for relatively thin shells.
It is however reiterated that the effective length is not only a function of the shell
radius and thickness and wave number of the loading, but also a function of the
stiffener spacing, the ring dimensions and eccentricity.
The actual effective length to be accounted for in case of eccentric stiffening rings can
be obtained by performing a range of calculations by, e.g., the program CShell, as
closed-form solutions are too complex and involved to be considered. The range of
application of the derived formula for ring-stiffened long cylinders with relatively large
spacing between either symmetric rings or eccentric rings is the subject of the next
subsection.
5.3.3 Applicability range of formulas
The objective of this subsection is to show the range of application of formula (5.15)
derived in the previous subsection based on the closed-form solution. Similar to section
5.2, the formula predicts the tensile axial stress at the base and the windward side of a
long clamped chimney subject to wind load. However, the influence of distributed
stiffening rings is incorporated into the formula.
The range of application of this formula is determined by comparison with results
obtained by the program CShell, which applies for short and long cylindrical shells and
allows accurate modelling of stiffening rings. As this program is based on the closedform solution, it is expected that, for a chimney with closely spaced stiffening rings,
the formula predicts an accurate value of the stress at the base. For chimneys shorter
than, say, the influence length and/or for chimneys with a more uneven distribution of
the ring stiffness, the program is more accurate than the formula since the formula does
not include the effect of the edge disturbance that originates at the free edge and as the
formula is based on a constant distribution of the ring stiffness along the length of the
chimney.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Stiffening rings with their centre of gravity located at the middle surface of the cylinder
To investigate the range of application of formula (5.15) to chimneys with symmetric
stiffening rings, calculations have been made for a radius-to-thickness-ratio of 100,
with length-to-radius-ratios of 10, 20 and 30 and with 2, 3, 4 and 5 equally spaced
stiffening rings per length-to radius ratio. The cylinder is clamped at the base and
stiffened by a ring at the top and by rings evenly distributed in between these edges.
The neutral line and the centre of gravity of the rings are located at the middle plane of
the circular cylindrical shell. To present unambiguous and concise results, only the
response to the mode numbers n = 2 and n = 1 of the wind load (5.1) have been
calculated for a first assessment of the range of application.
Figure 5-12 represents the total-stress-to-beam-stress-ratio with varying amount of
distributed ring stiffness and number of rings for the length-to-radius-ratios of 10. The
vertical axis thus represents the term between the straight brackets of formula (5.15).
The value of the distributed ring stiffness is indicated by the value of the factor λ r on
the horizontal axis, which is the square root of the stiffness ratio λ r as defined by
(5.14). Hence, the values as calculated by the program should be a straight line
between the following limit points. For the case λ r = 1 , no stiffening rings are added
and for the theoretical case λ r = 0 , the total stress is equal to the beam stress. The stress
obtained for case λ r = 1 should thus be accurately predicted by formula (5.5) within its
range of applicability, while the theoretical case λ r = 0 represents the case where
infinitely stiff rings are added that fully withstand the higher order terms of the wind
load allowing the chimney to act as a beam under lateral load. In Figure 5-12, not only
the calculated lines for 2, 3, 4 and 5 equally spaced stiffening rings, but also the
predicted line by formula (5.15) is shown.
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5 Chimney – Numerical results and parametric study
Figure 5-12 Stress ratio obtained by the program CShell and formula (5.15) for l a = 10
and a t = 100 .
Similar to Figure 5-12, Figure 5-13 and Figure 5-14 represent the total-stress-to-beamstress-ratio with varying amount of distributed ring stiffness and number of rings for
the length-to-radius-ratios of 20 and 30, respectively.
For λ r = 1 (the case without stiffening rings), a difference between the value
predicted by formula (5.15) and the value obtained by the program is observed in the
three figures. This difference is mainly related to the neglect of the small terms of the
order O ( β−2 ) in arriving at formula (5.15), which is further increased by the relative
shortness of the modelled chimney. The largest relative difference is namely observed
in Figure 5-12 for the lowest length-to-radius-ratio of 10 (and hence a length shorter
than half of the long influence length).
For the calculated length-to-radius-ratios, the program results obtained for more
than five stiffening rings are nearly identical to the cases with five rings.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Figure 5-13 Stress ratio obtained by the program CShell and formula (5.15) for l / a = 20
and a t = 100 .
For the cylinders with stiffening rings, the line through the values obtained by the
program is fairly in line with the line predicted by formula (5.15) unless the spacing
between the stiffening rings is chosen too large. For stiffening rings with a spacing
roundabout equal to and larger than half of the long influence length, the difference
between the program results and the values predicted by the formula increase with
increasing ring stiffness, i.e. decreasing stiffness ratio λ r . The difference between the
values predicted by the formula and the values obtained by the program is small for the
cases with closely spaced stiffening rings, i.e. with a spacing shorter than half of the
long influence length.
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5 Chimney – Numerical results and parametric study
Figure 5-14 Stress ratio obtained by the program CShell and formula (5.15) for l / a = 30
and a t = 100 .
Eccentric stiffening rings to the middle plane of the cylinder
To investigate the range of application of formula (5.15) to chimneys with eccentric
stiffening rings, calculations have been made for a radius-to-thickness-ratio of 100 and
200 and with 3 and 5 equally spaced stiffening rings per length-to-radius-ratio. For the
radius-to-thickness-ratio of 100, the length-to-radius-ratios of 10, 20 and 30 have been
considered and for the radius-to-thickness-ratio of 200, the length-to-radius-ratios of
15, 30 and 45 have been considered. For both radius-to-thickness-ratios, these
respective length-to-radius-ratios approximately match with a 0.5, 1 and 1.5 times the
influence length of the long-wave solution. Similar to the case with the symmetric
stiffening rings, the cylinder is clamped at the base and stiffened by a ring at the top
and by rings evenly distributed in between these edges. The centre of gravity of the
rings is located outside the middle plane of the circular cylindrical shell. To present
unambiguous and concise results, only n = 2 and n = 1 are calculated for a first
assessment of the range of application.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
For the radius-to-thickness-ratio of 100 and 200, Figure 5-15, Figure 5-16 and
Figure 5-17 represent the ratio of the total-stress-to-beam-stress with varying amount
of distributed ring stiffness and number of rings and for a length-to-radius-ratios
approximately match with a 0.5, 1 and 1.5 times the influence length of the long-wave
solution, respectively. Similar to the figures for the symmetric rings, the vertical axis
represents the term between the straight brackets of formula (5.15). The value of the
distributed ring stiffness is indicated by the value of the factor λ r on the horizontal
axis, which is the square root of the stiffness ratio λ r as defined by (5.14). Not only the
calculated lines for 3 and 5 equally spaced stiffening rings, but also the predicted line
by formula (5.15) is shown.
Figure 5-15 Stress ratio for (left) a t = 100 , l / a = 10 and (right) a t = 200 , l / a = 15 .
Figure 5-16 Stress ratio for (left) a t = 100 , l / a = 20 and (right) a t = 200 , l / a = 30 .
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5 Chimney – Numerical results and parametric study
Figure 5-17 Stress ratio for (left) a t = 100 , l / a = 30 and (right) a t = 200 , l / a = 45 .
It is striking that in the above figures the course of the stress ratio for a t = 100 with
varying stiffness ratio λ r is almost identical to the course of that ratio for a t = 200 .
Based on the figures and especially Figure 5-17, it is can be readily observed that for
the case with stiffening rings spaced at a distance equal to or larger than half of the
long influence length, the effectiveness of the rings is reduced compared to adding
more and smaller rings with the same stiffness ratio λ r . However, the main conclusion
is that the difference between the values predicted by the formula and the values
obtained by the program is increased in comparison with the symmetric ring cases.
Moreover, the formula provides a larger reduction of the total-to-beam stress than that
is actually obtained by the application of the stiffening rings (as calculated by the
program). In other words, the formula overestimates the stiffness of the rings. This
stiffness is calculated by equation (5.17) based on the ring area only, which is
evaluated with respect to the middle surface of the cylinder. The resulting
overestimation of the ring stiffness in formula (5.15) is in full accordance with the
observation and the approach envisaged in the previous subsection.
Based on the above observations, it is proposed to assess the applicability of the
formula (5.15) while adopting a flexural rigidity of the combined ring and the effective
shell length in accordance with
I r , c = ∫ z 2 dAc − ac −1 ∫ z 3dAc
(5.19)
Ac
Ac
which is evaluated with respect to the centre of gravity of the combined section of the
eccentric ring and the effective shell length and the subscript c denotes these
combined quantities.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
For this purpose, calculations have been made for a radius-to-thickness-ratio of 50,
100 and 200 and with a varying number of equally spaced stiffening rings per length-to
radius ratio. For the radius-to-thickness-ratios, the respective length-to-radius-ratios
approximately match with a 0.5, 1 and 1.5 times the influence length of the long-wave
solution. The maximum number of stiffening rings has been chosen such to achieve a
minimum spacing of about 0.2 times the influence length of the long wave solution
while the minimum number of stiffening rings that has been considered is two. The
considered rings are T beams that are bend with the stem inside matching with the
curvature of the shell. The cross-sectional dimensions have been based on practical
considerations related to the thickness of the shell and typical requirements as
prescribed in relevant codes and standards. Three different cross-sections have been
considered to study the impact of this variation with the following generic properties:
a) the web height equal to the flange width,
b) the web height larger than the flange width of the previous case, and
c) the flange width larger that the web height of the first case.
The relevant input data and results of these calculations is summarised in Appendix J.
The following conclusions can be readily drawn from the results of the abovementioned calculations.
The stress ratio between the axial stress at the base due to the self-balancing terms
( n = 2,...,5 ) and the axial stress at the base due to the “beam term” is rather independent
of the length-to-radius ratio ( l a ) and determined by the distance between the ring
stiffeners lr versus the influence length lin,2= 2 of the long-wave solution for the
respective thickness-to-radius-ratios ( t a ) . Hence, for a certain thickness-to-radiusratio and ring stiffener geometry, the effectiveness of the rings is mainly governed by
the ratio between lr and lin,2= 2 .
For a distance between the ring stiffeners lr larger than a quarter of the influence
length lin,2= 2 of the long-wave solution, the effectiveness of the stiffening rings is limited
as shown by the reduction of the stress ratio between the axial stress at the base due to
the self-balancing terms ( n = 2,...,5 ) and the axial stress at the base due to the “beam
term”.
To obtain a linear relation between λ r and the stress ratio, a certain effective
shell length has to be accounted for, which has been determined for the abovementioned cases. These theoretical effective shell lengths to be adopted for the
determination of λ r are presented in Appendix J. The theoretical effective shell
lengths are (much) shorter than as given by the equations presented in subsection 5.3.2
and the referred tabulated vales in that subsection. In other words, the determined
effective lengths are (much) shorter than 1.56 at .
The target root of the stiffness ratios λ r and determined effective shell lengths
leff indicate dependence on the stiffener spacing lr , ring dimensions and eccentricity er
of the ring centre of gravity to the middle plane of the cylinder.
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5 Chimney – Numerical results and parametric study
A suitable formula for the effective shell length could be based on an improved
equation that accounts for the mentioned ring properties. The moment of inertia of the
ring stiffener is accounted for by the resulting λ r . The following trends are observed
for the effective shell length from the result as presented in Appendix J:
For an increasing distance between the ring stiffeners lr , the effective length
decreases (refer to all cases),
For an increasing shell thickness t , the effective length decreases (refer to all
cases),
For an increasing ring area Ar with approximately the same ring eccentricity
er , the effective length increases slightly (refer to increased width of flange
case)
For an increasing ring area Ar with an increasing ring eccentricity er , the
effective length decreases (refer to increased height web case).
Based on the above general conclusion and a parametric assessment of the sensitivities,
the following relation for the effective length is proposed
−
1
2
1  lr t  er   2
1 +   
ψ  Ar  a  
in which ψ is an additional factor required to obtain sufficient agreement between the
calculated effective length and the length determined from the program results. To this
end, the value of the additional factor should be taken as ψ = 2 which might be further
dependent on the number of waves as the current result have been obtained for the
combination of the mode numbers n = 2 and n = 1 of the wind load. Hence, a proposed
formula for the effective length of the shell is
−1
leff
1


2 −
n 2 − 12
t  1  lr t  er   2

1+
≈
1 +   
4 3 1 − υ2 
( )  2 3(1 − υ2 ) a  ψ  Ar  a  
2 at
(5.20)
The values as presented in Appendix J for the effective length differ to a certain extent
from the values as obtained with the proposed formula (especially for the radius-tothickness-ratio of 50). However, this difference is limited considering the impact on the
combined stiffness on the ring and the marked improvement over the much larger
difference as presented in Figure 5-15, Figure 5-16 and Figure 5-17 for λ r based on
the stiffening ring properties only, i.e. without the combined properties of the ring and
the effective shell length. Furthermore, the prediction by the proposed formula shows
proper agreement in the relevant and practical range with a reduction to 10% - 30% of
the axial stress at the base due to the self-balancing terms ( n = 2,...,5 ) .
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5.3.4 Discussion of results for stiffening rings
In this section, the influence of stiffening rings on the behaviour of the long chimney
with a fixed base and a free end has been studied. The presented closed-form solution
(as obtained for such a ring-stiffened long circular cylinder under the wind load
described in section 5.1) and the range of application (as extracted from the previous
subsection) are summarised here for convenience and discussion.
To define the influence measure for the resistance to ovalisation based on the situations
with and without stiffening rings, closed-form solutions are developed for a number of
cases to determine the governing parameters. The cases are investigated based on the
closed-form solutions to the Morley-Koiter equation and comprised the following:
1. A ring in an infinitely long cylinder,
2. A semi-infinitely long cylinder with a ring present at a free edge, and
3. An infinitely long cylinder with equidistant rings.
Based on the solutions for these cases, it is determined that the influence of the
stiffening rings on the behaviour of the cylinder is fully captures by the parameter ηring
(5.9), which reads
ηring ≈
1 Ir 1 β
ηn
2 Db 2 a
Hence, it is concluded that the extensional rigidity of the ring has a negligible influence
on the reduction of the ovalisation in comparison with the influence of the flexural
rigidity and that the relation u zn = −nuθn for the amplitudes of the displacements uθ and
u z can be adopted. Furthermore, the difference between the ring displacements and the
more distant shell material is reduced within the long influence length originating from
the location of the stiffening ring.
Based on these observations, the SMC approach is proposed for further analysis and
the suitability of this approach is confirmed by a verification of the third case as
analysed with the solutions to the Morley-Koiter equation.
Within the SMC approach, a novel approach is suggested that comprised the
proposal to “smear out” the bending stiffness of the rings along the bending stiffness of
the cylinder resulting in a modified bending stiffness. The resulting design formula
(5.15) for the maximal tensile stress reads
2

l2
υ 
 a  a 
pw 1 + 6.39 1 − υ2 λ r     1 + 3

at 
 l  t 
1 − υ2 
in which the stiffness ratio λ r (5.14) represents the ratio of the bending stiffness of the
σ0xx≤,nt ≤5 ( z = t 2 ) = 0.224
circular cylindrical shell only to the modified bending stiffness of the shell (with the
contribution of the ring stiffness per spacing). The root of this factor thus represents the
influence of the stiffening rings on the stress distribution at the base of the long circular
cylindrical shells.
For stiffening rings with their centre of gravity located at the middle surface of the
cylinder, the design formula is verified with the program CShell (with a range of shell
geometries and ring spacing). The stress ratio between the stress due to the mode
136
5 Chimney – Numerical results and parametric study
numbers n = 1 and n = 2 and the stress due to the “beam term” is obtained by the
program. The calculated stress ratio is fairly in line with the stress ratio predicted by
formula (5.15) unless the spacing between the stiffening rings is chosen too large. For
stiffening rings with a spacing roundabout equal to and larger than half of the long
influence length, the difference between the program results and the values predicted
by the formula increase with increasing ring stiffness, i.e. decreasing stiffness ratio λ r .
The difference between the values predicted by the formula and the values obtained by
the program is small for the cases with closely spaced stiffening rings, i.e. with a
spacing shorter than half of the long influence length.
For eccentric stiffening rings, the envisaged necessity to account for a certain
effective shell length to determine the equivalent ring stiffness within the SMC
approach is confirmed by the program results (with a range of shell geometries, ring
spacing and ring geometries). Based on these program results, it is shown that the
determined effective lengths are (much) shorter than the existing formulation for the
effective shell length, i.e. 1.56 at . Furthermore, it is shown that the effective shell
length to be accounted for also depends on the stiffener spacing, ring dimensions and
eccentricity.
To match with the results of the program, a preliminary proposal for the effective
length is provided based on the observations above. As a conclusive result could not be
obtained, it is proposed to conservatively take the effective shell length equal to half of
the existing formulation. Considering the applicability of the design formula, a marked
improvement is already achieved by inclusion of a certain effective length and the need
for more improvement within the practical ranges is considered to be unnecessary for
rational first-estimate design of ring-stiffened circular cylindrical shells.
5.4 Influence of elastic supports
The subject of this section is to investigate the influence of elastic supports on the
behaviour of the long chimney. Additionally, this influence can be captured in a
closed-form solution and the range of application is identified by computational results.
In section 5.2, it is shown that the behaviour can be conveniently related to the
beam mode ( n = 1) . The deformation and stress for the axisymmetric mode ( n = 0 ) are
of no importance on the overall behaviour. It is expected that mainly the response to
the higher modes ( n ≥ 2 ) is altered by the presence of an elastic support in comparison
with a cylinder with a clamped support.
For these higher modes, the normal stress resultant nxx at the elastically supported
base is directly related to the induced out-of-roundness (“ovalisation”) of the cylinder,
which is partly withstood at the base by the planar (circumferential and radial) elastic
supports. As a consequence, the cross-section intends to warp at the base, which in
turn is partly withstood by the axial elastic support and results in the normal stresses at
the base. As the presence of planar elastic supports at the base will reduce the
ovalisation and hence the warping and as the axial elastic support will reduce the
warping that can be counteracted, nxx is reduced accordingly.
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The first objective of the next subsection is to define an influence measure for the
resistance to ovalisation based on the situations with and without elastic supports. To
arrive at such a measure, closed-form solutions will be developed for a number of cases
to determine the governing parameters. The second and successive objective is to
extract a useful formula describing the influence of the stiffening rings on the stress at
the base of a long cylinder.
5.4.1 Closed-form solution
For a completely elastic supported edge, the following system of equations for the
boundary conditions at the base ( x = 0 ) is obtained for the modes n ≥ 2 .
 k xu x 
k u 
 θ θ
 k zu z 


 kϕϕx 
x =0
 − nxx 
 −n 
=  x∗θ 
 − vx 


 − mxx 
x=0
(5.21)
in which the spring stiffnesses k x (axial), kθ (circumferential), k z (radial) and kϕ
(rotational) are introduced.
To solve this system, with the objective to obtain a formula for the stress at the base of
the chimney, terms multiplied by β−2 are neglected in comparison to unity. However,
such a solution is too cumbersome for practical use. Hence, some particular cases are
investigated. As a reference, the results of section 5.2 are recalled.
The solution for a clamped base is recalled and reads
C1n = C2n = −
υ n2 − 1 n
uˆ z
2 β2
; C5n = C6n = −uˆ zn
which can thus be obtained from the system (5.21) by equating each spring stiffness to
infinity ( k x = kθ = k z = kϕ = ∞ ) .
The solution for the “hinged-wall” edge ( u x = uθ = uz = 0, t x = 0 ) is also recalled. It is
almost equal to the solution for a clamped edge and reads
C1n = −
υ n2 − 1 n
uˆ z
2 β2
; C2n = 0 ; C5n = C6n = −uˆ zn
which can thus be obtained from the system (5.21) by equating the each extensional
spring stiffness to infinity and the rotational spring stiffness to zero
( k x = k θ = k z = ∞, k ϕ = 0 ) .
Various cases of elastic supports are considered for a long chimney, which is
elastically supported at the base and free at the top. Firstly, the presence of an axial
elastic support is considered. Secondly, the influence of both an axial and rotational
elastic support is analysed. Finally, the influence of the combination of a
circumferential and radial elastic support is analysed.
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5 Chimney – Numerical results and parametric study
For the first case, it is assumed that an axial elastic support k x is present and that the
wall of the cylinder is free to rotate. The displacements in the circular plane ( θz -plane)
are supposed to be fixed.
The solution for this elastic supported edge ( k xux = f x , uθ = uz = 0, t x = 0 ) reads
C1n = −
υ n 2 − 1 ηx n
uˆ z
2 β2 η x + 1
; C2n = 0 ; C5n = −uˆ zn
; C6n = −
ηx n
uˆ z
ηx + 1
in which the parameter ηx is introduced as
ηx =
kx a 1
k a
β
= x
E t ηnβ E t n n 2 − 1
(5.22)
This parameter is thus mainly described by the geometrical properties of the cylinder
and the ratio of the axial elastic support to the modulus of elasticity of the cylinder.
By back substitution, the stress resultant nxx and the stress couple mxx are obtained
as
5
nxx2 ≤ n ≤ 5 ( 0, θ ) ≈ −∑ 2 3(1 − υ2 )
n=2
a 2 pn
ηx
cos nθ
2
t n − 1 ηx + 1
;
mxx2 ≤ n ≤ 5 ( 0, θ ) = 0
which finally gives for the axial stress distribution at the base
5
σ 2xx≤ n ≤5 ( 0, θ, z ) = − ∑ 2 3 (1 − υ2 )
n=2
a 2 pn
ηx
cos nθ
t 2 n 2 − 1 ηx + 1
For the second case, it is assumed that, next to the axial elastic support k x , a rotational
elastic support kϕ is also present. The displacements in the circular plane ( θz -plane)
are supposed to be fixed, which is equal to the previous case.
The solution for this elastic supported edge ( k xux = f x , uθ = uz = 0, kϕϕx = t x ) reads
C1n = −
υ n 2 − 1 ηx n
uˆ z
2 β2 η x + 1
; C5n = −uˆ zn
ηϕ  υ n 2 − 1 ηx
1 n n2 − 1 
ηx   n
+

1 −
  uˆ z
2
2
ηϕ + 1  2 β ηx + 1 2 β
 ηx + 1  
in which the parameters ηx and ηϕ are introduced as
C2n = −
ηx =
kx a
β
E t n n2 − 1
; ηϕ =
; C6n = −
ηx n
uˆ z
ηx + 1
kϕ a 2
2β
Ea 2 t
By back substitution, the stress resultant nxx and the stress couple mxx are obtained as
5
nxx2 ≤ n ≤ 5 ( 0, θ ) ≈ −∑ 2 3 (1 − υ2 )
n=2
5
mxx2≤ n ≤5 ( 0, θ ) ≈ −∑ a 2
n=2
a 2 pn
ηx
cos nθ
2
t n − 1 ηx + 1
ηϕ  ηx

pn
n2
ηx  
υ
+

1 −
  cos nθ
2
2
n − 1 ηϕ + 1  ηx + 1 n n − 1  ηx + 1  
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
If the parameter ηx is large and thus the factor
ηx
close to unity, the stress
ηx + 1
distribution at the base can be obtained by
σ 2xx≤ n ≤5 ( 0, θ, z ) =
5
ηϕ 
nxx 2 z 6mxx
a2 p
ηx  2 z
υ
+
= −∑ 2 3 (1 − υ2 ) 2 2 n
3
 1 +
 cos nθ
2
t
t t
t n − 1 ηx + 1 
t
n=2
1 − υ2 ηϕ + 1 
If the parameter ηx is not large, the parameter ηϕ is probably small in the practical
cases and hence the stress couple mxx is almost zero.
The stress at the middle surface is for all cases described by
5
σ 2xx≤ n ≤5 ( 0, θ,0 ) = −∑ 2 3(1 − υ2 )
n=2
a 2 pn
ηx
cos nθ
2
2
t n − 1 ηx + 1
(5.23)
For the third case, it is assumed that, both a circumferential elastic support kθ and a
radial elastic support k z are present. The displacement in axial direction is supposed to
be fixed, while the wall of the cylinder is free to rotate.
The solution for this elastic supported edge ( u x = 0, kθuθ = f θ , k z uz = f z , t x = 0 ) reads
C1n ≈ C2n ≈ O ( β−2 ) ; C5n ≈ C6n ≈ −
ηθz n
uˆ z
ηθz + 1
in which the parameter ηθz is introduced as
ηθz = 2
n 2 k z + kθ a β 1
n 2 k z + kθ a
β
β2
=2
2
2
E
t ηn n − 1
E
t n n2 − 1 n − 1
(5.24)
This parameter is thus mainly described by the geometrical properties of the cylinder
and the ratio of the combined elastic support to the modulus of elasticity of the
cylinder.
The approximate solution above is accurate if the parameter ηθz is not small, since
then the stress couple mxx is almost zero and does not exert influence on the stress
distribution at the base.
By back substitution, the stress resultant nxx and the stress couple mxx are obtained
as
5
nxx2 ≤ n ≤ 5 ( 0, θ ) ≈ −∑ 2 3(1 − υ2 )
n=2
a 2 p zn ηθz
cos nθ
t n 2 − 1 ηθz + 1
;
mxx2 ≤ n ≤ 5 ( 0, θ ) ≈ 0
which finally gives for the axial stress distribution at the base
5
σ 2xx≤ n ≤5 ( 0, θ, z ) = − ∑ 2 3 (1 − υ2 )
n=2
a 2 p zn ηθz
cos nθ
t 2 n 2 − 1 ηθz + 1
(5.25)
5.4.2 Applicability range of formulas
The objective of this subsection is to show the range of application of the formulas
(5.23) and (5.25) based on the closed-form solution. Similar to section 5.2, these
formulas predict the tensile axial stress at the base and the windward side of a long
chimney subject to wind load. However, the influence of elastic supports is
incorporated into the formulas.
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5 Chimney – Numerical results and parametric study
The range of application of these formulas is determined by comparison with
results obtained by the program CShell, which applies for short and long cylindrical
shells and allows accurate modelling of elastic supports. As this program is based on
the closed-form solution, it is obvious that for chimneys much longer than the
influence length an identical result is obtained. For chimneys shorter than the influence
length, the program is more accurate since the formulas do not include the effect of the
edge disturbance that originates at the free edge.
To investigate the range of application of the formula (5.23) to chimneys with an axial
elastic support k x , calculations have been made for a constant radius-to-thickness-ratio
equal to 100 and length-to-radius-ratios of 5, 10, 20 and 40. This range has been
chosen based on the dominating long influence length ( n = 2 ) for the radius-tothickness-ratio equal to 100, which is approximately equal to 20 times the radius. The
parameter ηx (5.22) has been varied from practically infinity to zero in combination
with a “hinged wall”, i.e. kϕ equal to zero and the total-stress-to-beam-stress-ratio has
been investigated. The total-stress-to-beam-stress-ratio for a hinged edge
( k x = kθ = k z = ∞, kϕ = 0 ) can thus be used as a reference. Obviously, if the parameter ηx
is large, this “hinged-wall” solution is obtained and, if the parameter ηx approaches
zero, the total-stress-to-beam-stress-ratio is equal to unity.
As observed from the closed-form solution, the variation of the parameter ηx has
an identical influence on the course of the total stress-to-beam stress for the different
length-to-radius-ratios. Naturally, the total stress-to-beam stress value differs much as
the “hinged-wall” solution has a different value for each length-to-radius-ratio. If first
the membrane stress is deducted from the total stress for both the spring stiffness
solution and the “hinged-wall” solution and then their ratio is taken to obtain a
normalised stress ratio, this normalised stress ratio λ xn is defined by
∞
λ xn =
( ηx ) − σ
(η = ∞) − σ
0≤ n ≤ 5
xx
0≤ n≤5
xx
x
σ
σ
n =1
xx
n =1
xx
=
∑ σ2xx≤ n≤5 ( ηx )
n=2
∞
∑ σ2xx≤ n≤5
n=2
 αn
ηx 

2
− 1 ηx + 1 
n=2
= 5
 αn 
∑
 2

n=2  n − 1 
5
∑ n
(5.26)
The value of this ratio ranges from zero to unity for all length-to-radius-ratios and is
used as the quantity to plot on the vertical axis against varying spring stiffness. Figure
5-18 shows these normalised curves for the considered four ratios as obtained by the
program CShell and the theoretical curve as obtained by formula (5.23). On the
horizontal axis, the modified parameter ηx ,mod has been adopted, which is in fact a
reduction of the parameter ηx (5.22) according to
n n2 − 1
ηx ,mod = ηx
4
3 (1 − υ
2
)
=
kx a a
E t t
This modified parameter is thus independent of the mode number n , while the
dependency on the radius-to-thickness-ratio and the elastic properties of the chimney is
preserved.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
From Figure 5-18, it is observed that the three curves for the length-to-radius-ratios
of 10, 20 and 40 almost coincide with the curve predicted by the formula, while the
curve for the smallest length-to-radius ratio, i.e. l a = 5 , markedly differs from the
other curves.
As shown in Figure 5-18, the agreement between the theoretical factor and the
factors calculated by the program CShell is very good and excellent for the higher
length-to-radius ratios. However, it seems that if the length-to-radius-ratio is smaller
than half of the influence length-to-radius-ratio (here lin,2= 2 2a ≈ 10 for a t = 100 ), the
closed-form solution is no longer applicable.
1.2
normalised stress ratio λxn
1.0
0.8
0.6
l/a=5
l/a=10
l/a=20
l/a=40
formula
0.4
0.2
0.0
0.01
0.1
1
10
parameter ηx, mod (a/t =100)
100
1000
Figure 5-18 Theoretical factor compared with factor for several length-to-radius-ratios.
To investigate the above-mentioned observation, another set of calculations has been
made for a constant radius-to-thickness-ratio equal to 200 and length-to-radius-ratios of
7.5, 15, 30 and 60, respectively. This range has been chosen based on the influence
length for the radius-to-thickness-ratio equal to 200, which is approximately equal to
30 times the radius.
The results are similar to the results for a radius-to-thickness-ratio equal to 100 as
the variation of the parameter ηx has an identical influence on the course of the total
stress-to-beam stress. Identical to the presentation of Figure 5-18, the total-to-beamstress-ratio has been normalised against the “hinged-wall” solution and the plotted
value thus ranges from zero to unity for all length-to-radius-ratios. Figure 5-19 shows
these normalised curves for three considered ratios as obtained by the program CShell
142
5 Chimney – Numerical results and parametric study
and the theoretical curve as obtained by formula (5.23). The curve for l a = 60 is
omitted for clarity as it fully coincides with the theoretical curve. Similar to Figure
5-18 for a t = 100 , only the curve for the smallest length-to-radius ratio, i.e. l a = 7.5 ,
markedly differs from the other curves.
As shown in Figure 5-19, the agreement between the theoretical factor and the
factors calculated by the program CShell is very good and excellent for the higher
length-to-radius ratios. However, it seems that if the length-to-radius-ratio is smaller
than half of the influence length (here lin,2= 2 2a ≈ 15 for a t = 200 ), the closed-form
solution is no longer applicable.
1.2
normalised stress ratio λxn
1.0
0.8
0.6
0.4
l/a=7.5
l/a=15
l/a=30
formula
0.2
0.0
0.01
0.1
1
10
100
1000
parameter ηx, mod (a/t =200)
Figure 5-19 Theoretical factor compared with factor for several length-to-radius-ratios.
Figure 5-18 and Figure 5-19 show that almost identical results are obtained for the
factor, which can be applied to the formula for the stress distribution at the base of a
long chimney, for different radius-to-thickness-ratios. Hence, the validity of the
theoretical formula (5.23) has been verified. The closed-form solution is thus
applicable for any value of the parameter ηx , which expresses the stiffness of the axial
elastic support, and if the length of the chimney is longer than half of the influence
length for mode number n = 2 , which coincides with the range of application of
formulas (5.5) and (5.6) for a fixed base. In other words, formula (5.23) that
additionally accounts for the presence of an axial elastic support has the same range of
application as the formula without this additional term.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
In the above, it is shown that formula (5.23) based on the closed-form solution is valid
for several radius-to-thickness-ratios and the range of application is specified. Another
formula derived on basis of the closed-form solution is formula (5.25), which describes
the influence of both a circumferential and a radial elastic support at the base of a long
chimney.
To investigate the influence of these elastic supports by varying the parameter ηθz
(5.24), which is a ratio of the combined spring stiffness to the properties of the
cylinder, a radius-to-thickness-ratio has been chosen equal to 100. To determine the
parameter ηθz , it has been assumed that the circumferential spring stiffness kθ is equal
to the radial spring stiffness k z and, as such, a planar elastic support is provided.
Calculations have been made for length-to-radius-ratios of 2.5, 5, 10 and 20,
respectively. This range has been chosen based on the influence length for the radiusto-thickness-ratio equal to 100, which is approximately equal to 20 times the radius.
Longer chimneys are not considered since, from the previous results for the axial
elastic support described with the parameter ηx , it can be concluded that the formula is
valid for chimneys longer than the influence length.
The results are similar to the results for the variation of the parameter ηx . To adopt
the presentation of Figure 5-18 and Figure 5-19, the vertical axis is correspondingly
normalised against the “hinged-wall” solution. The normalised stress ratio λ θzn is
introduced, which is defined by
∞
λ θzn =
σ
σ
(η ) − σ
(η = ∞) − σ
0≤ n≤5
xx
θz
0≤ n ≤5
xx
θz
n =1
xx
n =1
xx
=
∑ σ2xx≤ n≤5 ( ηθz )
n=2
∞
∑σ
2≤ n≤5
xx
n=2
 αn
ηθz 

2
− 1 ηθz + 1 
n=2
= 5
 αn 
∑
 2 
n=2  n − 1 
5
∑ n
(5.27)
The value of this ratio ranges from zero to unity for all length-to-radius-ratios. Figure
5-20 shows these normalised curves for three considered ratios as obtained by the
program CShell and the theoretical curve as obtained by formula (5.25). On the
horizontal axis, the modified parameter ηθz ,mod has been adopted, which is modified of
the parameter ηθz (5.24) according to
ηθz ,mod = ηθz
1
2 ( n + 1)
2
n2 − 1
ηn
4
3 (1 − υ
2
)
=
k θz a a
E t t
in which kθz = kθ = k z . The parameter ηθz is thus modified corresponding to the
modification of the parameter ηx to shown the relative influence of the parameters.
The curve for l a = 20 is omitted for clarity as it fully coincides with the
theoretical curve. In comparison with the figures for the influence of an axial elastic
support, it is observed the agreement between the theoretical factor and the factors
calculated by the program CShell is even better than for the variation of the parameter
ηx . The agreement is even quite good for a length-to-radius-ratio of 2,5, which is much
less than half of the influence length (here lin,2= 2 2a ≈ 10 for a t = 100 ).
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5 Chimney – Numerical results and parametric study
Since the formula for the stress at the base of a chimney is not accurate for a length
smaller than the half influence length, it is remarkable that the closed form solution for
the influence of an elastic supported edge is even more accurate. Additionally, smaller
values than 2,5 for the length over the radius are not practical from an engineering
point of view. However, the range of application for the total stress at the elastic
supported base of a chimney loaded by the wind load is governed by the limitations of
the formula for the clamped or hinged supported base.
1.2
normalised stress ratio λθzn
1.0
0.8
0.6
l/a=2.5
l/a=5
l/a=10
l/a=20
formula
0.4
0.2
0.0
0.0001
0.001
0.01
0.1
1
10
parameter ηθz, mod (a/t =100)
Figure 5-20 Theoretical factor compared with factor for several length-to-radius-ratios.
5.4.3 Discussion of results for elastic supports
In this section, the influence of elastic supports on the behaviour of the long chimney
with such a support at the base and a free end has been studied. The presented closedform solution (as obtained for such a ring-stiffened long circular cylinder under the
wind load described in section 5.1) and the range of application (as extracted from the
previous subsection) are summarised here for convenience and discussion.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
New design formulas that describe the stress distribution at the elastically supported
base of long cylindrical shells subject to wind load representing a chimney with the
following elastic support conditions are derived:
1. Axial elastic support only,
2. Combination of axial and rotational elastic supports, and
3. Combination of circumferential and radial elastic supports.
For the first case with only the axial elastic support described by the axial spring
stiffness k x , the formula for the maximal tensile stress at the middle surface reads
σ0xx≤,nt ≤5 ( z = 0 ) = 0.224
2

l2
 a  a
pw 1 + 6.39 1 − υ2 λ xn   
at 
 l  t 
in which the normalised stress ratio λ xn (5.26) is introduced, which is defined by
 αn
ηx 

2
− 1 ηx + 1 
n=2 
λ xn =
5
 αn 
∑
 2 
n=2  n − 1 
5
∑ n
where the parameter ηx (5.22) is introduced, which reads
ηx =
kx a
β
E t n n2 − 1
This parameter is thus mainly described by the geometrical properties of the cylinder
and the ratio of the axial elastic support to the modulus of elasticity of the cylinder.
For the second case with the combination of an axial and a rotational elastic support,
the latter support is described by the rotational spring stiffness kϕ . Similar to parameter
ηx , a parameter ηϕ is introduced to describe the influence of the rotational elastic
support, which reads
ηϕ =
kϕ a 2
2β
Ea 2 t
Based on the closed-form solution and practical considerations, it is assessed that the
additional influence of the rotational support will be limited, as the rotational spring
stiffness kϕ will decrease rapidly with decreasing axial spring stiffness k x .
For the third case with the combination of a circumferential and a radial elastic support,
the support is described by the circumferential spring stiffness kθ and the radial spring
stiffness k z . For this support, the formula for the maximal tensile stress at the middle
surface reads
σ0xx≤,nt ≤5 ( z = 0 ) = 0.224
146
2

l2
 a  a
pw 1 + 6.39 1 − υ2 λ θzn   
at 
 l  t 
5 Chimney – Numerical results and parametric study
in which the normalised parameter λ θzn (5.27) is introduced, which is defined by
 αn
ηθz 

2
− 1 ηθz + 1 
n=2
= 5
 αn 
∑
 2 
n=2  n − 1 
5
λ θzn
∑ n
where the parameter ηθz (5.24) is introduced, which reads
ηθz = 2
n 2 k z + kθ a
β
β2
2
E
t n n2 − 1 n − 1
This parameter is thus mainly described by the geometrical properties of the cylinder
and the ratio of the combined elastic support to the modulus of elasticity of the
cylinder.
For the axial elastic support (described by ηx ) and the planar elastic support (described
by ηθz with assumed equal kθ and k z ), the design formulas are separately verified with
the program CShell (with a range of shell geometries and elastic support properties).
The total-to-beam-stress-ratio for the elastically supported condition is normalised to
the “hinged-wall” solution. The calculated stress ratios are in close agreement with the
stresses predicted by formulas (5.23) and (5.25).
1.2
normalised stress ratio
1.0
0.8
0.6
0.4
0.2
formula axial
formula planar
0.0
0.0001
0.001
0.01
0.1
1
10
100
1000
parameter ηx, mod and ηθz, mod (a/t =100)
Figure 5-21 Theoretical factor for axial elastic support and planar elastic support.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Figure 5-21 shows the plot for the theoretical factor in case of varying parameter ηθz
and for a varying parameter ηx . Similar to the previous plots, the horizontal axis has a
logarithmic distribution. Obviously, the decrease of the latter parameter has more
influence than a decrease of the first. This is clearly related to the ratio of the parameter
ηθz to the parameter ηx , which reads (for the case that kθz = kθ = k z )
ηθz kθz
a n2 + 1
=
2 3(1 − υ2 )
t n2 − 1
ηx k x
In other words, the parameter ηθz is at least
a
times the parameter ηx .
t
The physical interpretation for practical cases seems to be that for the stress
distribution at the base of a long chimney, only the axial spring stiffness has to be taken
into account and that the presented formula is applicable for any value of the parameter
ηx in combination with cylinders longer than half of the influence length of the longwave solution for n = 2 .
148
6 Tank – Numerical study
6 Tank – Numerical study
Solutions obtained by a computer program based on the method presented in chapter 3
are given for short circular cylindrical shell structures. The formulations that are used
in this program are derived in chapter 4. To demonstrate the capability of the
developed program, a numerical study of tanks under the main load-deformation
conditions is performed. The conclusions of this study are given in the chapter 7.
6.1 Introduction
Circular cylindrical tanks are used for storing liquids, gases, solids and mixtures
thereof. Tanks for storing solids are more usually known as silos and these are fitted
with, e.g., a flat top end cap and conical bottom end cap. Liquefied or compressed gas
at substantial pressure is mainly stored in ball tanks or cylindrical tanks with dished
end caps. Such tanks are referred to as pressure vessel, i.e. a closed container designed
to hold gases or liquids at a pressure substantially different from the ambient pressure.
Silos and pressure vessels are not considered in this chapter.
Liquid storage tanks can be horizontal in shape, but large liquid storage tanks for
storing water, oil, fuel, chemicals and other fluids are usually vertical in shape. These
large, thin-walled tanks can be open top and closed top, have fixed roofs, floating roofs
and internal roofs, single walls and double walls, flat bottom, cone bottom, slope
bottom and dish bottom. The functional layout of the tank depends on operational
considerations and the required safety measures and pollution prevention.
A typical large liquid storage tank is obviously much shorter than the long
chimney such that the diameter is of the same order in comparison with the length as
opposed to the chimney. The geometry of such stocky cylinders is typically such that
the diameter is at least equal to the length or that the length can even be much smaller
than the radius, viz. a ratio between radius and length between 0.5 and 3.
For such short lengths between the circular boundaries, the short influence length
has a more marked contribution to the load-deformation behaviour of the cylinder and
the long influence length is much longer than the height of the shell. This feature
prevents that closed-form solution to non-axisymmetric loads similar to those obtained
for the long chimneys in chapter 5 can be readily obtained.
Concrete tanks typically might have a relatively large ratio between radius and
thickness of about 30 - 80, but especially large steel storage tanks are thin-walled such
that the ratio between radius and thickness might even be between 500 and 2,000.
This chapter intends primarily to demonstrate the CShell capability to model the
shell of large vertical liquid storage tanks and additionally to provide tentative insight
into the response of such tank shells to the relevant load and/or deformation conditions,
which is obtained by several calculations with the program CShell and comparison
with the insight as obtained for the behaviour of the long cylinder.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
6.2 General description of large liquid storage tanks
Most liquid storage tanks are constructed of (typically carbon) steel or steel alloys, but
concrete is also used to construct water tanks and tanks with a separate secondary
containment (double wall and full-containment tanks). The choice to adopt concrete for
water tanks is favourable if a longer life and less involved recycling is desired.
Separate concrete outer walls to the inner steel tank walls as secondary containment are
used as a relatively large bunding is not desired or to provide protection to the inner
(stainless) steel inner tank. In these cases, the concrete outer wall functions as catch
basin in case of failure of the inner tank.
Cryogenic liquid storage tanks (for liquefied gases such as LNG, which are
typically stored at very low temperature, but at ambient pressure) typically require fullcontainment tanks that include a primary steel containment tank, a concrete secondary
containment tank that includes a full-vapour barrier and an insulation layer between the
two tanks. The (prestressed) concrete outer tank protects the sensitive inner steel tank
against external hazards and serves as catch basin in case of failure of the inner tank.
Alternatively, a concrete secondary tank with a membrane containment/insulation
system within the concrete secondary tank is adopted. In the latter case, the concrete
secondary tank takes the hydrostatic load. For in ground tanks, the surrounding earth
may be used to provide mechanical support or an in-pit construction is considered in
which the tank is built as a separate unit within the pit that provides containment in
case of leakage and rupture. The combined mechanical behaviour of the tanks in case
of a full-containment concept and, for in ground tanks, the interaction of the soil and
varying internal fluid level pose some more involved analyses which are not
considered in this chapter.
Oil and oil products are most commonly stored in large vertical cylindrical carbon
steel tanks at atmospheric pressure or at low pressure.
A tank might have an open top (e.g. in case of water storage) and, depending on the
type of liquid, a cover to the contents may be provided to reduce evaporation or ingress
of contaminants. Generally and especially for liquid fuel (oil and oil products), the
choice between a fixed roof tank and a floating roof tank depends on the flash point of
the particular fluid.
Fixed roof tanks are used to store liquids with very high flash points (fuel oil,
water, bitumen, etc.). Such tanks typically have cone roofs, dome roofs and umbrella
roofs. Dome roofs are provided to tanks with a slightly higher than ambient storage
pressure. These tanks might be insulated to prevent clogging of the fluid by internal
(bottom) heating.
Floating roof tanks are broadly divided into external floating roof tanks and
internal floating roof tanks. The floating roof rises and falls with the liquid level inside
the tank whereby the vapour space above the liquid level is decreased and consequently
a much smaller risk of internal tank explosion is achieved. In principle, the floating
roof eliminates emission of air pollutants and greatly reduces the evaporative loss of
the stored liquid. External floating roof tanks are used to store medium flash point
liquids (naphtha, kerosene, diesel, crude oil, etc.). These tanks do not have a fixed roof
(i.e. the tank is open at the top). Internal floating roof tanks are used to store liquid with
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6 Tank – Numerical study
low flash points (aviation fuel, gasoline, ethanol, etc.). These tanks have both a fixed
roof at the top and a floating roof inside the tank.
A fixed roof provides an adequate radial (and tangential) restraint to the top of the
tank wall due to its high membrane stiffness. This restraint is considered to be a full
circular restraint to the tank wall. For a floating roof, a flexible seal is normally
provided between the tank shell and the edge of the roof. Hence, the roof provides little
restraint on the tank shell and this influence can be neglected.
The tank floor is generally formed by a thin steel membrane consisting of welded
plates and acts principally as a seal to the tank. These steel bottom plates are laid and
fully supported on a prepared foundation. The pressure of the contents is directly
transmitted to the base (e.g. compact soil foundation, pile-supported, concrete ring or
slab foundation). Alternatively, a reinforced concrete slab can be adopted as also
commonly provided for water tanks. The bottom plates are welded to the shell wall (or
alternatively the shell wall is connected to a reinforced concrete slab) and, due to the
high membrane stiffness of the floor, this shell-to-floor junction provides a full circular
restraint to the shell wall (i.e. no radial and tangential displacement). The tank might be
freely placed on its foundation (unanchored) or anchored (typically by vertical anchor
bolts connected to a pile-supported concrete strip foundation or long prestressed anchor
bolts with grout anchors). The anchorage can thus be modelled as a (rather) stiff soil
spring that acts as a (nearly) rigid support to the shell wall.
This chapter further focuses mainly on large, single wall, concrete or steel, vertical
tanks for the storage of liquids at low or ambient temperatures and with a design
pressure near ambient pressure which are either closed or open at the top. The design of
such tanks can be divided in three major areas; 1) the shell, 2) the bottom, and 3) the
roof. The bottom and roof layout of the tank typically vary with the operational
conditions, preferences and safety requirements. In any case, these provide a rigid
support, no support, or an elastic (intermediate) support to the tank shell. In view of the
capability of the CShell program, the next sections focus on the shell of the tank while
considering the various connections of the shell to the top and bottom.
6.3 Load-deformation conditions and analysed cases
The typical design loads specifically for the shell wall are the following:
Dead load (the weight of the tank or tank component)
Superimposed loads (roof live load, snow, partial internal vacuum, etc)
Stored liquid load (the load due to filling the tank to maximum capacity with
liquid with the design specific gravity)
Wind load (wind pressure on vertical areas and, in case of fixed roof, load
from uplift pressure on the roof)
Seismic load (for specific areas only and not considered for the present
purpose)
External loads and constraints (shell connections/nozzles allowing inlet, outlet
and drainage, and venting, and ladders, stairs, platforms, shell
openings/manholes providing access for inspection and maintenance, etc.)
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Axial stresses due to dead and superimposed load in the shell with a wall designed to
carry the content load are typically of a relatively low stress level due to the high
membrane stiffness of the tank wall. The external loads and constraints affect only the
localized behaviour of the shell and are typically not governing for the overall design.
Three main load-deformation conditions can therefore be identified for the overall
response of the tank wall:
Content load (especially when being filled to maximum capacity)
Wind load (especially for the open top tank and external floating roof tank)
Settlement induced load and/or deformation
6.3.1 Content load
The content load obviously depends on the density and level of the fluid that is stored
in the tank. The relative density of typical fluids, such as crude oil and white oil
products, is less than unity, As the hydrostatic test of tanks for such liquids is normally
performed by filling with water, a minimum density of 1000 kg/m3 is conservatively
adopted and obviously the full capacity of the tank should be considered.
The content provides an axisymmetric hydrostatic internal pressure on the wall of
the vertical cylinder, which induces an increase of the initial diameter and a
corresponding simple hoop tension. In vertical direction, this circumferential tension
varies linearly and directly with the static head of the fluid. Normally, no
circumferential stiffening is required to counteract this action. At the shell-to-bottom,
the full radial restraint of the tank wall induces axial bending stresses along the short
influence length and reduces the radial displacement (to zero at the bottom) and
corresponding circumferential stress in that disturbance length.
6.3.2 Wind load
The wind load is described in section 5.1. The shape of the circumferential distribution
of the wind load depends roughly on the geometry of the chimney and varies from code
to code but has the common characteristic that only a part of the circumference, the socalled stagnation zone, is under circumferential compression, while the remainder is
under suction. For convenience, the wind load distribution as described by expression
(5.1) is adopted.
Overturning and sliding stability of the tank under wind load needs to be evaluated
and, if necessary, the required anchorage should be defined. Anchorage is typically
required for taller tanks with smaller ratios between radius and length.
Due to the wind load, the cross-section of the tank tends to distort into an oval
shape. At the shell-to-bottom, the full circular restraint of the tank wall in combination
with an axial restraint in case of anchorage, induces axial bending stresses along the
long influence length that are resulting from the withstood out-of-roundness as
similarly observed for the chimney under wind load.
In case of closed, fixed roof tanks, the roof provides an adequate restraint to
maintain the roundness of the tank. The wind load is then mainly carried by axial
tensile stresses at the windward side and compressive stresses at the leeward side, i.e.
mainly by beam action of the shell. For open top and external floating roof tanks,
circumferential primary wind girders are normally externally provided at or near the
152
6 Tank – Numerical study
tank top to maintain the roundness and stability of the tank under wind load (especially
while emptying the tank). Especially for tall tanks, secondary wind girders at intervals
in the height of the tank might be required to prevent local buckling.
6.3.3 Settlement induced load and/or deformation
Large cylindrical storage tanks constructed on soft foundations may be subjected to
various types of shell deflections due to settlement of the foundation. The subject,
cause and consequences of these foundation settlements and implication on the
response and requirements to large storage tank has been investigated by, for example,
Malik et al. [55], Marr et al. [56] and Godoy and Sosa [57].
The foundation settlements can be described in terms of three components; 1)
uniform settlement, 2) planar tilt, and 3) circumferential settlement (non-planar or
differential settlement). The uniform settlement and planar tilt cause rigid body
displacement and rotation of the tank and are therefore of relatively little importance
for the overall design.
Even a minimal non-planar settlement leads to serious consequences for the tank
structure. For liquid storage tanks with a floating-cover-system, the non-planar
settlement of the foundation often results in jamming of the cover. It turns out that at
some height of the cylinder the shape of the cross-section becomes elliptic, which is at
first sight the unlikely consequence of the vertical displacements of the bottom. The
explanation of this phenomenon is found in the well-known principle of the inextensional deformations. The feature of the in-extensional deformations is that the
strains of the middle surface are equal to zero. As thin-walled structure, which is much
stiffer in-plane than perpendicular to its plane, a shell has a strong preference for such
deformations.
Take a cylindrical shell, open at the top and with a thin bottom, as shown in Figure
6-1 in which a double sinusoidal settlement ( n = 2 ) is depicted at the bottom. The
shell-to-floor junction prevents the opposite points Q from moving inward and the
opposite points Q′ from moving outward. In order to keep the developed length of the
circumference constant and not to induce any axial strain, the generatrices PQ have to
turn inward and the generatrices P′Q′ have to turn outward such that the areas PQQ′P′
follow the turning over without shear deformation in the plane of the shell. With this it
is made roughly clear that the curving of the bottom causes an increasing ovality with
increasing height. As shown in [43], the depicted in-extensional deformation is
obtained by equating the middle surface strains of the kinematical relation (4.4) equal
to zero. Adopting a sinusoidal settlement with wave number n , the normal
displacement u z is to the axial displacement u x as n 2 x to the radius a where x
denotes the axial coordinate from the bottom of the shell. For a cylinder with height a
and a general sagging of the cylinder ( n = 2 ) , the horizontal displacement at the top is
thus equal to 4 times the vertical displacement.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The settlement induced out-of-roundness thus results in both inward and outward
deflections. Inward deflections may impede the motion of the floating roof. Outward
deflections may cause loss of the flexible seal between the tank shell and the edge of
the roof. In case of an anchorage at the bottom and in case of a fixed roof or wind
girder at the top, the in-extensional deformation as described above cannot be fully
realized, as the restraints at the top and bottom do not necessarily conform to the
settlement of its foundation. Hence, not only the settlement induced out-of-roundness
in the upper parts of the tank, but also high circumferential stress developed in the
primary wind girder and high axial stresses developed at the tank bottom may be
induced by the circumferential settlement.
Figure 6-1 In-extensional deformation of a circular cylindrical shell with a thin bottom and
an open top.
6.3.4 Analysed cases
Based on the main load-deformation conditions for the shell as described in the
previous subsections, the following relevant cases have been identified for the shell of
the tank:
Content load of a fully filled tank,
Wind load on the tank with various top restraints, and
Circumferential settlement of the foundation with various top restraints.
For the hydrostatic load and the wind load both steel tanks and concrete tanks have
been analyzed, while only steel shells have been considered for the settlement analyses.
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6 Tank – Numerical study
6.4 Content load cases
6.4.1 Concrete tank
A concrete tank with radius a , height l and uniform wall thickness t is considered
which is complete filled with water with density γ w as shown in Figure 6-2.
Figure 6-2 Circular cylindrical concrete tank connected to a thick flat plate.
For such a simple case, the solution can be obtained by elementary analysis. The
content provides an axisymmetric hydrostatic internal pressure on the wall of the
vertical cylinder equal to pz = γ w ( l − x ) in which x = 0 at the bottom of the tank. Over
the distance l between the two edges, the geometrical and material properties are taken
as constant. This means that the response to the wind load can be calculated by the
solution to the differential equation (4.21) that is given in subsection 4.4.3. This
solution has to be complemented by the appropriate boundary conditions that are given
by
x =0;
clamped:
u x = u x = 0 ; u z = u z = 0 ; ϕ x = ϕ x = 0
x=l;
free:
f x = nxx = 0 ;
f z = vx∗ = 0 ; t x = mxx = 0
where v∗x is Kirchhoff’s effective shearing stress resultant. The inhomogeneous
solution (refer to subsection 4.4.3) thus reads
a2
(l − x )
Et
a2
ϕx ( x ) = γ w
Et
a 1 
u x ( x ) = −υγ w  l − x  x
Et  2 
uz ( x ) = γ w
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
By solving the equations for the boundary conditions, the constants of the full solution
are obtained. Assuming that the length of the cylinder is larger than the (short)
influence length li , only the edge disturbance originating from the bottom edge has to
be taken into account. By back substitution of these constants in the approximated
expressions, the circumferential stress resultant nθθ and the bending stress resultant mxx
become
 x 

 

1
nθθ = γ w al  1 −  − e −βx  cos βx +  1 −  sin βx  
l
 βl 


 


1
l
mxx = −γ w 2 e−βx 1 −  cos βx − sin β x 
2β
 β l 

which is in fact the membrane solution plus an edge disturbance originating from the
bottom. This edge disturbance will not be correct if the bottom is also unrestrained
such that it fully accommodates a radial displacement.
These quantities and the approximate values at characteristic points are shown in
Figure 6-3 for a thick concrete tank. The lateral contraction of the material is not
accounted for as Poisson’s ratio is chosen as υ = 0 . The density of water is taken
γ w = 10 kN 3 . The adopted dimensions are proportionally chosen as a = 3m , t = 0.3m
m
and l = 4m . The straight line in the diagram of nθθ represents the inhomogeneous
solution and thus also represents the membrane response. The plot of nθθ obviously has
the same course as the plot of the normal displacement u z .
The thickness-to-radius-ratio, which is equal to
t
1
=
, is deliberately chosen large to
a 10
show that, even for this thick shell, the influence length of the edge disturbance for the
axisymmetric behaviour is actually very short. This means that in almost every case the
influence of one edge on the other will be negligible. Furthermore, it means that the
inhomogeneous solution describes the global behaviour of the shell and that a bending
field only disturbs this global behaviour of the shell over a relatively short section of
the shell.
To illustrate the fact that the edge disturbance is indeed very short for a thin shell under
axisymmetric loading, the same calculation is made for a tank with the same length and
radius, but with a thickness t = 0.03m (see Figure 6-4). So the thickness-to-radius ratio
is equal to the rather thin value
t
1
=
and the disturbance therefore influences a
a 100
much shorter length of the shell. This results in a higher peak of the circumferential
stress resultant nθθ , but greatly reduces the peak of the bending stress resultant mxx .
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6 Tank – Numerical study
Figure 6-3 A tank wall ( a = 10t ) rigidly connected to a thick bottom plate.
Figure 6-4 A tank wall ( a = 100t ) rigidly connected to a thick bottom plate.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
6.4.2 Steel tank
For practical reasons, steel tanks are built up from fairly small rectangular pieces of
carbon steel plate, which are curved in a cylindrical shape and joined by butt-welding.
The shell is thus built up in rings (also referred to as courses) and typically the
thickness of the plates varies with the internal pressure, i.e. thicker plates are applied in
the lower courses and thinner plates in the upper courses.
A steel tank (material properties taken as E = 210 × 106 kN m 2 and υ = 0.3 ) with
radius a = 10m , height l = 20m and varying wall thickness is modelled as completely
filled with water (density taken as γ w = 10 kN 3 ). The bottom edge ( x = 0 ) is fully
m
fixed and the top ( x = l ) is free. The thickness of the shell courses has been applied as
follows from the bottom to the top ( h indicates height of the respective courses or
courses with the same thickness)
h = 2.5m × t = 11mm
h = 2.5m × t = 9.5mm
h = 2.5m × t = 8.5mm
h = 5.0m × t = 7.5mm
h = 7.5m × t = 7.0mm
The relevant displacements and stresses are shown in Figure 6-5 and Figure 6-6.
The circumferential stress resultant nθθ varies rather linear with the content level up to
the region near the bottom where the radial displacement is fully restraint. The small
disturbances coincide with the transitions in course thickness. It is obvious that the
course of the circumferential stress σθθ and the displacement u z are identical, as
expected. The shape of the hoop stress diagram is reduced by the increased thickness of
the courses towards the bottom the tank. The smooth changes are clarified by the
stiffening effect of the thicker plate to the thinner plate above, which can be considered
as a partial restraint at the bottom edge of the thinner plate. The axial stress associated
with the bending stress resultant mxx is quite considerable, but in fact an anchored tank
is modelled. If the tank is not anchored, an elastic rotational support is present that
allows some rotation of the shell-to-floor junction, which effectively reduces the
bending stresses at the bottom. Large tanks may alternatively have rounded corners
(transition from vertical wall to bottom profile) to easier withstand these
hydrostatically induced stresses with the additional benefit of reducing localized
stresses in case of planar tilt.
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6 Tank – Numerical study
Figure 6-5 Steel tank with content load, displacement u z and circumferential stress σθθ
along the height.
Figure 6-6 Steel tank with content load, stress resultants nθθ and mxx along the height.
6.5 Wind load cases
6.5.1 Concrete tank
In this subsection the response of two different concrete storage tanks under the wind
load (5.1) is presented. The two cases are:
1. A storage tank, which is clamped at the base, with a free edge at the top; and
2. The same storage tank, but with a fully rigid roof at the top.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The rigid roof is modelled as a ring with a very high modulus of elasticity so that the
ring is non-deformable by in-plane actions and thus provides a circular restraint to the
top. At the bottom an anchored condition is considered, as wind load is typically more
important for taller tanks that are normally anchored against overturning. The
anchorage provides a certain elastic axial and rotational support to the tank shell. For
the present purpose, a full axial and rotational restraint to the shell wall is considered.
If a more realistic elastic support in axial direction (in combination with a congruent
rotational support) is adopted, no considerable impact on the response is envisaged in
view of the high stiffness of the soil springs generated by either the prestressed anchor
bolts or the pile foundation.
The geometrical properties of both concrete shells are the same
( l = 30m, a = 25m, t = 0.3m ) and for the material properties E = 35 × 106 kN m2 and υ = 0.2
are used.
The anchored case is in fact identical to the analyses as presented in a previous
paper by Hoefakker and Blaauwendraad [58], which provided program results for tanks
based on the solution to the Donnell equation for n > 1 . In the current program, the
Morley-Koiter equation has been implemented for all load-deformation behaviours and
a one to one comparison with the Donnell’s solution is hereby available.
From the following figures, it is observed that in view of the magnification factors
the deformation is drastically reduced by the (rigid) stiffening ring and that the axial
stress σ xx for n = 1 is distributed like a clamped-free beam and that for n > 1 this stress
is distributed like a clamped-hinged beam in case of a rigid roof. Interesting is the fact
that the bending stress resultant is mainly described by the short influence length and
that the membrane stress resultant is described by either the polynomial solution (viz.
the membrane solution) or the long influence length.
Furthermore, it is observed that the magnitude of the stress resultants and crosssectional deformation is much smaller in comparison with the values reported in [58]
based on the solution to the Donnell equation for n > 1 . The stress resultant nxx is
reduced by about 20% at the base of the tank with the free edge and, in case of a rigid
roof, the maximum along the tank height is reduced by about 33%. Moreover, the
stress resultant mxx at the base is reduced by 50%. For this particular case, a large
reduction in the outer fibre stress at the base is thus observed. To present similar graphs
for the cross-sectional deformation, the magnification factors in [58] are 5000 for the
free edge and 15000 for the rigid roof. The comparison of the two solutions for this
particular case once more shows that description of the quantities and the shape of the
diagrams are properly described by the Donnell equation but that to predict the
magnitude of these quantities the Morley-Koiter equation should be considered.
Finally, the stress resultant nxx at the base is about 120 N mm under the applied
wind load. This value is even less than the axial stress at the base under the dead
weight of the concrete shell only. Adopting a typical density of 2400kg m3 for
concrete, the dead weight stress at the base becomes ρglt ≈ 210 N mm . In this particular
case, it can thus be concluded that the dead weight virtually provides a full axial
restraint at the bottom of the shell.
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6 Tank – Numerical study
Figure 6-7 Cross sectional deformation of the anchored tank with a free edge (left) at the
top (× 3000) and of the anchored tank with a rigid roof (right) at half the length of the
cylinder (× 18000).
Figure 6-8 Anchored tank with a free edge (left) and with a rigid roof (right), nxx at θ = 0 .
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Figure 6-9 Anchored tank with a free edge (left) and with a rigid roof (right), mxx at θ = 0 .
Figure 6-10 Anchored tank with a free edge (left) and with a rigid roof (right), nxx at
x=0.
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6 Tank – Numerical study
6.5.2 Steel tank
In this subsection the response of two different steel storage tanks (material properties
taken as E = 210 × 106 kN m 2 and υ = 0.3 ) under the wind load (5.1) is presented. The
two cases are:
1. A storage tank, which is clamped at the base, with a free edge at the top; and
2. The same storage tank, but with a (steel) wind girder at the top.
In line with the observations of the previous subsection, the connection at the base is
modelled as a full axial and rotational restraint to the shell wall. These cases are
considered to show the impact of the wind girder on the stress distribution and the
deformation of the tank.
A typical geometry for a steel storage tank with a wind girder is a t = 1000 , l a = 1
and λ g = 20 where the ratio λ g represents the bending rigidity of the wind girder itself
to the tank wall. Hence, a tank with l = a = 10m is considered that is built up from
various courses with varying plate thickness as exemplified in subsection 6.4.2 while
maintaining roughly the typical ratio of a t = 1000 . The thickness of the shell courses
has been applied as follows from the bottom to the top ( h indicates height of the
respective courses or courses with the same thickness)
h = 2.5m × t = 12.5mm
h = 2.5m × t = 10.0mm
h = 5.0m × t = 7.5mm
The corresponding tank wall bending stiffness (viz. shell bending rigidity times the
tank height) is thus equal to
Dbl =
E
∑ ht 3
12 (1 − υ2 )
For the present purpose, the wind girder is conveniently modelled as an eccentric
annular plate with width hg = 250mm and thickness t g = 12.5mm resulting in a
circumferential bending rigidity of the wind girder EI g =
Ehg 3t g
12
with respect to its
centre of gravity.
From the following figures, it is observed that the influence of the modelled ring is
confined to a limited length from the top, viz. only affects the shell behaviour within
the short influence length and does not markedly influence the overall behaviour. Note
that although the total radial displacement u z at the top and θ = 0 is larger with the
wind girder, the maximum radial displacement is slightly smaller with the wind girder
( 0.9mm ) compared to the case with the free edge (1.0mm ) .
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Figure 6-11 Cross-sectional deformation at the top of a steel tank with a free edge (left) (×
2000) and with a wind girder (right) (× 2000).
Figure 6-12 Steel tank with a free edge (left) and with a wind girder (right), u z at θ = 0 .
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6 Tank – Numerical study
Figure 6-13 Steel tank with a free edge (left) and with a wind girder (right), σ xx at θ = 0
and at outer face of the cylinder.
Figure 6-14 Steel tank with a free edge (left) and with a wind girder (right), mθθ at the top.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
6.6 Settlement induced load and/or deformation cases
In this section, the response of the steel tank with the wind girder of the previous
subsection (case 2) under a non-planar settlement is presented. For the present purpose,
the non-planar settlement is conveniently considered to induce a general sagging of the
tank shell, i.e., a full circumferential settlement with mode number n = 2 is considered,
which is described by
ux ( θ) = −
us , max
(1 + cos 2θ )
2
in which us ,max is the maximum settlement along the circumference. Values up to
us , max = 50mm are typically considered. As the constant term only produces a rigid body
motion of the shell, this term is further ignored. Hence, only a settlement with mode
number n = 2 and top value of 25mm is modelled.
The geometry and material properties are taken identical to those of the previous
subsection, but at the bottom the tank is modelled as freely supported in axial and
rotational direction. In other words, at the bottom of the tank a prescribed axial
displacement without rotational constraint is modelled.
As stated in subsection 6.3.3, an assessment based on in-extensional behaviour of a
cylinder with height equal to its radius a subject to a general sagging of the tank
( n = 2 ) revealed that the radial displacement u z at the top of such a cylinder is thus
equal to 4 times the axial displacement u x of the settlement and 2 times the
circumferential displacement uθ along the shell height.
From the following figures, it is observed that the circumferential settlement
indeed mainly induces an in-extensional deformation and corresponding stresses.
Furthermore, it is observed that the influence of the modelled ring is confined to a
limited length from the top, viz. only affects the shell behaviour within the short
influence length and does not markedly influence the overall behaviour.
The step change in the diagram of σ xθ is obviously in line with the height of the
shell courses, but the rather large variations rather misrepresent the behaviour, as the
values of this stress are relatively small. Hence, it is observed that the ring at the top
only influences the circumferential stress and the axial stress and hardly affect the
deformation.
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6 Tank – Numerical study
Figure 6-15 Steel tank with a wind girder under a circumferential settlement, σ xx at θ = 0
and σθθ at θ = 0 (right) and both at outer face of the cylinder.
Figure 6-16 Steel tank with a wind girder under a circumferential settlement, σ xθ at
θ = 45deg and at outer face of the cylinder (left) and u x at θ = 0 (right).
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Figure 6-17 Steel tank with a wind girder under a circumferential settlement, uθ at
θ = 45deg (left) and u z at θ = 0 (right).
Figure 6-18 Steel tank with a wind girder under a circumferential settlement,
circumferential stress σ ring .
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7 Conclusions
7 Conclusions
Review of the first-order approximation theory for thin shells
To understand the assumptions and simplifications, which are introduced to obtain the
appropriate thin shell equations, the set of equations resulting from a fundamental
derivation for thin elastic shells has been reproduced. To arrive at a consistent and
reliable first-order approximation theory of shells of revolution, two expansions have
been explored. The most adopted approach in previous work is the expansion of the
strain description that adopts the changes of rotation, while only few authors have
considered the changes of curvature. In this research, it is shown that the expansion of
the strain description that adopts the changes of curvature should preferably be
considered. Hence, this approximation and simplification is only effected in the
constitutive relation, while the kinematical and equilibrium relations maintain to be
solely based on the adopted assumptions. Mathematically consistent, the boundary
conditions have been derived by making use of the principle of virtual work. It has
been concluded that, while simultaneously approximating the constitutive relation, the
combined internal stress resultants of the boundary conditions need to be congruently
approximated to avoid misleadingly representing a greater accuracy than that is
attributed to expressions of the underlying relations. Hence, a mathematically
consistent set of equations representing a first-order approximation theory cannot be
derived.
Computational method and expeditious PC-oriented computer program
An objective was to develop a computer program for the typical shells of revolution,
i.e. circular cylindrical, conical and spherical shells. Due to required effort identified
during the development of such a super element program for circular cylinders and
upon inspection of the sets of equations for conical and spherical shells, it has been
decided to fully focus on circular cylindrical shells as a first, but complete and
successful step towards more applications.
The implementation of the super element approach into a PC-oriented computer
program (using the Fortran-package in combination with graphical software) has
resulted in an expeditious, stable and well-working tool that can be used by structural
analysts for rational first-estimate design of long and short circular cylindrical shells.
The accuracy and reliability of the developed program is conclusively demonstrated by
the finite element verification for long and short circular cylindrical shells. The
numerical study conducted for large vertical liquid storage tanks demonstrated the
capability and user-friendliness of the program.
General solutions to the circular cylindrical shell equations
The so-called Morley-Koiter equation is an approximation of the exact equation for
circular cylindrical shells. It has been shown that this equation is mathematically the
most suitable equation for substitution if compared to similar equations with the same
accuracy of other authors.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The equation overcomes both the completeness of Flügge’s approach in retaining
second-order terms, which do not exceed the accuracy of the initial assumptions, and
the inaccuracy of Donnell’s simplifications in its inability to describe rigid-body modes
but preserves its elegance and simplicity.
The exact roots to the Morley-Koiter equation have been obtained and, albeit being
surplus to requirements, the presented solution is a unification of former results by
other authors. To facilitate insight in the prevailing parameters of the shell response to
the respective load-deformation conditions, approximate roots have been derived for
the axisymmetric, beam-type, and non-axisymmetric load-deformation conditions.
The approximate solution for the self-balancing modes has been compared with
several similar solutions obtained by parameter perturbation, which conclusively
confirmed that the Morley-Koiter equation accurately describes the behaviour of thin
circular cylindrical shells.
Furthermore, it can be concluded that the perturbation technique is highly suitable
for obtaining roots of the reduced equation, especially in the case that obtaining exact
and closed-form solutions is very involved.
Parametric study of long circular cylindrical shells (chimneys)
The formulations that are implemented in the PC-oriented computer program are the
approximated solutions to the Morley-Koiter equation for circular cylindrical shell.
Design formulas, based on closed-form solutions to the Morley-Koiter equation and an
equation derived by the semi-membrane concept, and numerical solutions obtained by
the developed program are given for long circular cylindrical shell structures, i.e. long
in comparison with their radius (for example industrial, steel chimneys).
Design formula without stiffening rings or elastic supports
The design formula that describes the stress distribution at the fixed base of long
cylindrical shells subject to wind load representing a chimney without stiffening rings
has been derived, which reads for the maximal tensile stress at the outer surface
σ0xx≤,nt ≤5 ( z = t 2 ) = 0.224
2

l2
υ 
 a  a 
pw 1 + 6.39 1 − υ2    1 + 3

at 
 l  t  
1 − υ2 
for the specified wind load. By introduction of the characteristic lengths l1 = at and
l2 = 4 atl 2 , this formula can be alternatively written as
l
σ0xx≤,nt ≤5 ( z = t 2 ) = 0.224 pw  
 l1 
2
4

 a  
υ 
1 + 6.39 1 − υ2    1 + 3

l

1 − υ2 
 2   
The term within the round brackets describes the effect of a full rotational constraint
and should be omitted in case the shell wall is free to rotate at the base.
This design formula is a marked improvement of the existing formula that is based
on the Donnell equation and comprises excellent agreement with existing numerical
simulations over a large range of shell geometries. The design formula expresses the
influence of the self-balancing terms ( n = 2,...,5 ) by their ratio to the response to the
“beam term” ( n = 1) . Furthermore, the range of application (as extracted from the
program results) has been conclusively determined for long circular cylindrical shells
170
7 Conclusions
having a length-to-radius-ratio ranging from 10 to 30 and a radius-to-thickness-ratio
ranging from 50 to 400. The formula is shown to be applicable to cylinders longer than
half of the influence length of the long-wave solution for n = 2 . This range of
application is equivalently determined in terms of the introduced characteristic length,
which provided that the formula is applicable to cylinders with a characteristic length
l2 longer than its radius a .
Design formula with stiffening rings
A useful formula describing the influence of the stiffening rings on the stress at the
base of a long cylinder has been developed. To define the influence measure for the
resistance to ovalisation based on the situations with and without stiffening rings,
closed-form solutions were developed for a number of cases to determine the
governing parameters. It is shown that (i) the extensional rigidity of the ring has
negligible influence on the reduction of the ovalisation in comparison with the
influence of the flexural rigidity and (ii) the difference between the ring displacements
and the more distant shell material is reduced within the long influence length
originating from the location of the stiffening ring. Based on these observations for the
more rigorous approach and a resulting simplification based on the SMC approach, a
novel approach has been suggested that comprised the proposal to “smear out” the
bending stiffness of the rings along the bending stiffness of the cylinder resulting in a
modified bending stiffness. The resulting design formula for the maximal tensile stress
reads
σ0xx≤,nt ≤5 ( z = t 2 ) = 0.224
2

l2
υ 
 a  a 
pw 1 + 6.39 1 − υ2 λ r     1 + 3

at 
 l  t 
1 − υ2 

or alternatively
4

 a  
υ 
2

σ
1 + 6.39 1 − υ λ r    1 + 3

l

1 − υ2 
 2   
in which the stiffness ratio λ r represents the ratio of the bending stiffness of the
0≤ n ≤5
xx , t
l
( z = t 2 ) = 0.224 pw  
 l1 
2
circular cylindrical shell only to the modified bending stiffness of the shell (with the
contribution of the ring stiffness per spacing). The root of this factor thus represents the
influence of the stiffening rings on the stress distribution at the base of the long circular
cylindrical shells.
For stiffening rings with their centre of gravity located at the middle surface of the
cylinder, the design formula has been verified to be applicable for the cases with
closely spaced stiffening rings, i.e. with a spacing shorter than half of the long
influence length for n = 2 (as extracted from the program results with a range of shell
geometries and ring spacing). In this case, the ring stiffness is to be determined based
on the properties of the ring only.
For eccentric stiffening rings, it was envisaged that a certain effective shell length
has to be accounted for to determine the equivalent ring stiffness within the SMC
approach, which was confirmed by the program results (with a range of shell
geometries, ring spacing and ring geometries). Based on these program results, it is
proposed to conservatively take the effective shell length equal to half of the existing
formulation, as a conclusive result could not be obtained. Considering the applicability
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
of the design formula, a marked improvement is already achieved by inclusion of a
certain effective length and the need for more improvement within the practical ranges
is considered to be unnecessary for rational first-estimate design of ring-stiffened
circular cylindrical shells.
Design formula with elastic supports
The design formula that describes the stress distribution at the fixed base of long
cylindrical shells subject to wind load representing a chimney with axial elastic support
has been derived, which reads for the maximal tensile stress at the middle surface
σ0xx≤,nt ≤5 ( z = 0 ) = 0.224
2

l2
 a  a
pw 1 + 6.39 1 − υ2 λ xn   
at 
 l  t 
or alternatively
4

a 
1 + 6.39 1 − υ2 λ xn   

 l2  
in which the normalised stress ratio λ xn is introduced, which depends on the respective
l
σ0xx≤,nt ≤5 ( z = 0 ) = 0.224 pw  
 l1 
2
factors and mode number of the load and the parameter ηx , which in turn is mainly
described by the geometrical properties of the cylinder and the ratio of the axial elastic
support to the modulus of elasticity of the cylinder.
Similarly, the design formula that describes the influence of a combined
circumferential and radial elastic support has been derived, which reads for the
maximal tensile stress at the middle surface
σ0xx≤,nt ≤5 ( z = 0 ) = 0.224
2

l2
 a  a
pw 1 + 6.39 1 − υ2 λ θzn   
at 
 l  t 
or alternatively
4

a 
1 + 6.39 1 − υ2 λ θzn   

 l2  
in which the normalised stress ratio λ θzn is introduced, which depends on the respective
l
σ0xx≤,nt ≤5 ( z = 0 ) = 0.224 pw  
 l1 
2
factors and mode number of the load and the parameter ηθz , which in turn is mainly
described by the geometrical properties of the cylinder and the ratio of the combined
elastic support to the modulus of elasticity of the cylinder.
The factors with the elastic support parameter thus represent the influence of the
respective elastic supports on the stress distribution at the base of the long circular
cylindrical shells. The range of application (as extracted from the program results) of
these new formulas has been conclusively determined for long circular cylindrical
shells having a radius-to-thickness-ratio of 100 and 200, respectively, and a varying
length-to-radius-ratio based on the long influence length of the long-wave solution for
n = 2 . The formulas are shown to be applicable to cylinders for which the characteristic
length l2 is larger or equal to its radius.
172
7 Conclusions
It is concluded that, in case of an elastic support to a long circular cylinder, only
the axial spring stiffness has to be taken into account and that the presented formula is
applicable for any value of the parameter ηx in combination with cylinders longer than
half of the influence length of the long-wave solution for n = 2 .
Reflection on objective and scope of the research
The development of a super element program for circular cylindrical shells and the
derivation of design formulas based on parametric studies could only have been
successfully performed with the aid of the generic knowledge of the shell behaviour in
combination with the presented closed-form solutions. To arrive at practical closedform solution based on first-order approximation theories for thin elastic shells, the
basic and generic knowledge of the shell behaviour, the prevailing parameters and the
underlying theories are essential, which is most apparently demonstrated for the cases
of the ring stiffened and elastically supported circular cylindrical shells.
173
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
174
Appendices
Appendices
175
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
176
Appendices
Appendix A
Results from differential geometry
of a surface
(Kraus [10] Chapter 1 & 2.)
(Struik [59])
Curve
Consider a 3-dimensional space described by the Cartesian coordinates x1 , x2 , x3 . The
parametric representation of a curve γ with respect to the parameter ξ is given by
3
x = ∑ xi ( ξ ) ei .
i =1
Arc length is defined as length along a curve by s ≡ ∫γ dx , so ds = dx where dx is the
differential increment vector along the curve, which for Cartesian coordinates is given
by
3
dx = ∑ dxiei = dx1e1 + dx2e 2 + dx3e3 .
i =1
The line element is defined as ( ds ) = dxidx and it follows that
2
( ds )
2
= ( dx1 ) + ( dx2 ) + ( dx3 ) .
Because
2
2
2
dx
is thus a unit vector in the direction of dx and hereby tangent to the
ds
curve, the unit tangent vector is defined as
t=
dx
.
ds
The curvature along this curve is defined by
dt
= k = − kn
ds
in which n is the unit normal vector in the direction of the principal normal to the
curve, k is the curvature vector, which expresses the rate of change of the tangent
vector along the curve, and k is called the curvature and the reciprocal ( R = k −1 ) is
called the radius of curvature. The minus sign is introduced to reflect that it is assumed
that the parametric curves are arranged in such a manner that the unit normal points
from the concave side to the convex side of the surface.
Consider a set of three independent functions of the Cartesian variables x1 , x2 , x3 given
by
ξi = ξi ( x1 , x2 , x3 ) ,
( i = 1,2,3)
and let the intersections of the surfaces
ξi ( x1 , x2 , x3 ) = constant
,
( i = 1, 2,3)
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
determine the coordinate lines of a curvilinear coordinate system and the intersection of
these coordinate lines determines a point that is labelled ( ξ1 , ξ2 , ξ3 ) . The position vector
x in the rectangular coordinate system as a function of these curvilinear coordinates
ξ1 , ξ2 , ξ3 is represented by
3
x ( ξ1 , ξ 2 , ξ3 ) = ∑ xi ( ξ1 , ξ 2 , ξ3 ) ei = x1 ( ξ1 , ξ 2 , ξ3 ) e1 + x2 ( ξ1 , ξ2 , ξ3 ) e 2 + x3 ( ξ1 , ξ2 , ξ3 ) e3 .
i =1
The differential change dx in the position vector from a point P0 to an infinitesimal
close point P within the space is written with respect to the curvilinear coordinates as
3
dx = ∑ x,i d ξi = x,1d ξ1 + x,2 d ξ 2 + x,3d ξ3
i =1
where the notation x,i =
∂x
is introduced.
∂ξi
Hereby it follows that the line element is calculated by
( ds )
2
3
= dxi dx =
∑ g dξ dξ
ij
i
j
i , j =1
where the magnitudes are defined by
∂xk ∂xk
.
k =1 ∂ξi ∂ξ j
3
gij = x,i ix, j = ∑
The equation for the line element is also known as the first fundamental form and gij is
called the metric tensor. Noting that along a parametric curve ξi the differential length
of arc dsi takes the simplified form
dsi = g ii d ξi = x,i d ξi ,
the unit tangent vectors along the parametric curves can be defined by
ti =
dx , i
dsi
=
x , i d ξi
g ii d ξi
=
x,i
x,i
.
Surface
A surface S in the rectangular coordinate system can be written as a function of the
two parameters ξ1 and ξ 2 , which are the curvilinear coordinates of the surface, and
these parameters determine the parametric curves ξ 2 = constant and ξ1 = constant ,
respectively. The position vector x as a function of these curvilinear coordinates ξ1 , ξ 2
is represented by
3
x ( ξ1 , ξ 2 ) = ∑ xi ( ξ1 , ξ2 ) ei .
i =1
Hence, the differential change dx in the position vector x from a point P0 to an
infinitesimal close point P on the surface is written with respect to the curvilinear
coordinates as
dx = x,1d ξ1 + x,2 d ξ 2 .
178
Appendices
The line element or first fundamental form ( ds ) is now expressed by
2
( ds )
2
= dxidx = E ( d ξ1 ) + 2 Fd ξ1d ξ 2 + G ( d ξ 2 )
2
2
where
E = g11 = x,1 ix,1 , F = g12 = g 21 = x,1 ix,2
, G = g 22 = x,2 ix,2 .
From vector algebra for the dot product, the angle θ12 angle between the vectors x,1
and x,2 along two parametric curves can be found by elaborating
cos θ12 =
x,1 ix,2
x,1 x,2
=
g12
=
g11 g 22
F
EG
which for the sine of this angle results in
EG − F 2
.
EG
sin θ12 = 1 − cos 2 θ12 =
To describe a real surface in a right-handed coordinate system, the sine of the angle θ12
is always positive and hence EG − F 2 > 0 since E and G are always positive. From
the result for the cosine of the angle θ12 it is observed that, if θ12 = π 2 and thus the
parametric curves form an orthogonal net, F = 0 .
A representation of the angle α between two arbitrary directions can be obtained by
taking the dot product of the differential change dx in one direction and δx in another
direction, which are respectively represented by
dx = x,1d ξ1 + x,2 d ξ 2
,
δx = x,1δξ1 + x,2δξ2
which gives for the dot product the expression dxiδx = dx δx cos α . Hence, the
condition for orthogonality of two directions and thus α = π 2 is given by
Ed ξ1δξ1 + F ( d ξ1δξ2 + δξ1d ξ2 ) + Gd ξ2δξ 2 = 0 .
The unit tangent vectors along the parametric lines are thus defined by
t1 =
x,1
x,1
,
t2 =
x,2
x,2
The unit normal vector n is thus parallel to the cross product of the vectors x,1 and x,2
and hereby this unit normal vector can be defined by
n=
x,1 × x,2
x,1 × x,2
.
The curvature along any curve on the surface is defined by
dt
= k = kn + kg
ds
where the vector k , which is called the curvature vector, is thus defined as the rate of
change of the unit tangent vector, k n is the normal curvature vector and k g is the
tangential or geodesic curvature vector. The normal curvature vector is given by
k n = − kn
in which k is called the normal curvature. An expression for k is obtained by taking
the dot product of k with n , which after some manipulation results in
179
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
k =−
dt
dxidn
in =
.
ds
dxidx
Since the differential change in the unit normal vector is expressed by
dn = n,1d ξ1 + n,2d ξ2
the normal curvature at any point of the surface is given by
e ( d ξ1 ) + 2 fd ξ1d ξ2 + g ( d ξ 2 )
2
k=
2
E ( d ξ1 ) + 2 Fd ξ1d ξ 2 + G ( d ξ2 )
2
2
in which the numerator is called the second fundamental form and its magnitudes are
defined by
e = x,1 in,1 , 2 f = ( x,1 in,2 + x,2 in,1 ) , g = x,2 in,2 .
Since x,i in = 0 , the magnitudes can alternatively be expressed by
e = −x ,11 in ,
f = − x,12 in ,
where the notation x,ij =
g = − x,22 in
∂ 2x
is introduced.
∂ξi ∂ξ j
d ξ2
for which the normal curvature is a maximum or a minimum
d ξ1
can be obtained by rewriting the expression for k for those directions to
e + 2 f λ + gλ2
k (λ) =
E + 2 F λ + Gλ 2
dk
and deriving the directions λ for which
= 0 , which after some rearrangement
dλ
The directions λ =
results in
λ1,2 =
− ( Eg − Ge ) ±
( Eg − Ge ) − 4 ( Fg − Gf )( Ef
2 ( Fg − Gf )
2
− Fe )
.
 d ξ2 δξ2 
,
 and using this notation the
 d ξ1 δξ1 
The set {λ1 , λ 2 } corresponds to two directions 
orthogonality condition can be rewritten to E + F ( λ1 + λ 2 ) + Gλ1λ 2 = 0 and since
λ1 + λ 2 =
− ( Eg − Ge )
( Fg − Gf )
,
λ1λ 2 =
− ( Ef − Fe )
( Fg − Gf )
this equation is identically satisfied. This means that the directions of maximum and
minimum normal curvature are orthogonal.
If these lines of curvature are taken as the parametric curves, one curvature direction is
described by
d ξ2
d ξ1
= 0 and the other curvature direction is described by
= 0 . Hence,
d ξ1
d ξ2
the following two relations are obtained
,
( Ef − Fe ) = 0
( Fg − Gf ) = 0 .
Since the parametric curves are orthogonal F = 0 and since E and G are always
positive, it is obtained that when the parametric curves are the lines of curvature
180
Appendices
F =0
f =0.
,
Substituting this condition into the expression for k and by keeping first dξ2 = 0 and
then dξ1 = 0 , the corresponding normal curvatures become
1 e
=
R1 E
k1 =
1
g
= .
R2 G
k2 =
,
The normal curvatures in the curvature directions are called the principal curvatures
and further it is assumed that the parametric curves are the lines of curvature.
For such a surface, the first fundamental form is described by the curvilinear
coordinates ξ1 and ξ 2 and the magnitudes E and G according to
( ds )
2
= E ( d ξ1 ) + G ( d ξ2 ) .
2
2
Three mutually orthogonal unit vectors ( t1 , t 2 , n ) are oriented tangent to the ξ1 direction, ξ 2 -direction and normal to the surface, respectively, and these are given by
t1 =
x,1
,
α1
t2 =
x,2
,
α2
n=
x,1 × x,2
α1α 2
in which α1 = E = x,1 and α 2 = G = x,2 . The principal curvatures are equal to
k1 =
1
1
= 2 x,1 in,1
R1 α1
k2 =
,
1
1
= 2 x,2 in,2 .
R2 α 2
Since the derivative of a unit vector is perpendicular to the unit vector itself, it lies in
the plane formed by the other two unit vectors and it can thus be decomposed into its
components along the latter vectors. By making use of the mutual orthogonality of the
unit vectors and that x,12 = x,21 , the magnitudes of the components can be found by
straightforward taking the dot product of the respective vectors with each other. Doing
so and making use of the expressions for the normal curvatures, the following
derivatives of the unit vectors are obtained.
t1,1 = −
1
α
α1,2t 2 − 1 n
α2
R1
,
t1,2 =
1
α 2,1t 2
α1
t 2,1 =
1
α1,2t1
α2
,
t 2,2 = −
n,1 =
α1
t1
R1
,
n ,2 =
1
α
α 2,1t1 − 2 n
α1
R2
α2
t2
R2
The relation between the four quantities α1 , α 2 , R1 and R2 can be obtained by
substitution of the appropriate derivatives defined above into the equality n,12 = n ,21 .
Hereby, it is obtained that
∂  α1 
1 ∂α1
=
 
R2 ∂ξ 2 ∂ξ 2  R1 
,
∂  α2 
1 ∂α 2
=
 
R1 ∂ξ1 ∂ξ1  R2 
which are known as the Codazzi conditions.
Similarly, the equality t1,12 = t1,21 yields two more equations of which one is the Codazzi
condition that is mentioned first and the other is the Gauss condition that reads
∂  1 ∂α 2  ∂  1 ∂α1 
α1α 2
.

+

=−
∂ξ1  α1 ∂ξ1  ∂ξ 2  α 2 ∂ξ2 
R1R2
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Obviously, the equality t 2,12 = t 2,21 yields again the Gauss condition and the other
Codazzi condition.
Shell space
A surface S in the rectangular coordinate system x1 , x2 , x3 can be written as a function
of two parameters; viz. ξ1 , ξ 2 , which are the curvilinear coordinates of the reference
sur ζ face. To describe the location of an arbitrary point within the two outer surfaces of
the shell a third coordinate is introduced in the thickness direction. The position vector
R to this arbitrary point is described by
R ( ξ1 , ξ 2 , ζ ) = r ( ξ1 , ξ 2 ) + ζn ( ξ1 , ξ2 )
where r is the position vector of the corresponding point on the reference surface and
n is the unit normal vector at that point. Hence, the differential change dR in the
position vector R from a point P0 to an infinitesimal close point P on the surface is
written as
dR = dr + ζdn + nd ζ
in which dr = r,1d ξ1 + r,2d ξ2 and dn = n,1d ξ1 + n,2d ξ2 .
The line element ( ds ) is calculated by taking the dot product of the differential change
dR in the position vector. Taking into account the orthogonality of the coordinate
system, the expression for the line element becomes
2
2
2
2
( ds ) = ( r,1 ir,1 + 2ζr,1 in,1 + ζ 2n,1 in,1 ) ( d ξ1 ) + ( r,2 ir,2 + 2ζr,2 in,2 + ζ 2n,2 in,2 ) ( d ξ2 ) + ( d ζ ) .
2
Since r,1 = α1t1 it follows that n,1 =
1
1
r,1 and similarly n,2 = r,2 , which after
R1
R2
substitution into the expression above gives
( ds )
2
= g11 ( d ξ1 ) + g 22 ( d ξ 2 ) + g 33 ( d ζ )
2
2
2
where the coefficients gii ( i = 1,2,3) are the metric coefficients along the orthogonal
parametric lines. These coefficients are defined by


ζ 
ζ 
A1 = g11 = α1 1 +  , A2 = g 22 = α 2  1 +  , A3 = g33 = 1
R
R
1 
2 


where Ai are the scale factors, α1 and α 2 are the so-called Lamé parameters of the
reference surface and R1 and R2 are the principal radii of curvature at the point on the
reference surface corresponding to point Po . The Lamé parameters and the principal
radii are related to the position vector and the unit normal vector by
α12 = r,1 ir,1
α 2 2 = r,2 ir,2
182
1
1
= 2 r,1 in,1
R1 α1
,
,
1
1
= 2 r,2 in,2
R2 α 2
Appendices
Appendix B
Kinematical relation in orthogonal
curvilinear coordinates
(Sokolnikoff [60] Chapter 4, §48. pp.177.)
For a line element within a medium referred to by a curvilinear coordinate system
determined by the coordinate lines ξi ( i = 1,2,3) , which are assumed to be orthogonal,
the metric coefficients are denoted by gii calculated by
∂xk ∂xk
k =1 ∂ξi ∂ξi
3
g ii = x,i ix ,i = ∑
where x is the position vector to a point within that medium as shown in Appendix A.
Only infinitesimal deformations are taken into account and the displacement in the
directions normal to the coordinate surfaces ξ1 , ξ2 , ξ3 are represented by U1 , U 2 , U 3
respectively.
Denoting two neighbouring points in an unstrained medium by Po and P , their
positions before deformation are given by ξi and ξi + d ξi respectively. After
deformation the positions of Po′ and P′ are given by ξi + δi and ξi + δi + d ξi + d δi . The
displacements are related to the change in position by the description of the line
element and hence described by U i = gii δi .
The length of the element ds joining Po and P is expressed by the relation
( ds )
2
3
= ∑ gii ( ξ1 , ξ2 , ξ3 )( d ξi )
2
(B.1)
i =1
and the length of the same element after deformation is expressed by
( ds′ )
2
3
= ∑ gii ( ξ1 + δ1 , ξ2 + δ2 , ξ3 + δ3 )( d ξi + d δi ) .
2
(B.2)
i =1
To the order of approximation considered in the linear theory the respective parts are
expressed by
∂gii
δj
j =1 ∂ξ j
3
gii ( ξ1 + δ1 , ξ2 + δ 2 , ξ3 + δ3 ) = gii ( ξ1 , ξ2 , ξ3 ) + ∑
( d ξi + d δi )
2
3
= ( d ξ i ) + 2 d ξ i d δ i + ( d δ i ) ( d ξ i ) + 2∑
2
2
2
j =1
∂δi
d ξ j d ξi
∂ξ j
and hence (B.2) can be written as
( ds′ )
2
3
3
= ∑∑ Gij d ξi d ξ j
(B.3)
i =1 j =1
in which
3
∂δ


∂g
∂δ
(B.4)
Gij = δij  g ii + ∑ ii δk  + gii i + g jj j
∂ξ j
∂ξi
k =1 ∂ξ k


∂δ
where products of δ j and i are neglected and δij denotes the Kronecker delta.
∂ξ j
183
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
According to the expressions (B.1) and (B.3), the length dsi in the direction of one of
the coordinate lines ξi is dsi = gii d ξi and the length after deformation is dsi′ = Gii d ξi .
The extension eii of this element is thus
eii =
Gii d ξi − gii d ξi
g ii d ξi
= 1+
Gii − g ii
1 Gii − g ii
−1 where
g ii
2 g ii
the
nonlinear
terms
are
neglected. Using (B.4) this result becomes
eii 1 Gii − gii ∂δi
1 3 ∂g ii
=
+
∑ δk .
2 g ii
∂ξi 2 g ii k =1 ∂ξk
(B.5)
As the metric coefficients of the unstrained medium are calculated by
∂xk ∂xk
,
k =1 ∂ξi ∂ξi
3
g ii = x,i ix ,i = ∑
the metric coefficients of the deformed medium are calculated by
∂x′k ∂x′k
k =1 ∂ξi ∂ξ j
3
Gij = x′, i ix′, j = ∑
where the vector x′ is the position vector of a point in the deformed medium. Hence,
the angle θij between the vectors x′,i and x′, j along two parametric curves on the
surface can be found by elaborating
cos θij =
x′, i ix′, j
x′,i x′, j
=
Gij
GiiG jj
(i ≠ j ) .
Defining the angle αij by θij =
π
− αij it follows that, for infinitesimal displacements,
2
cos θij = sin α ij αij . The shear components of the strain tensor are defined by the
relation α ij = 2eij , so we get
eij =
1 Gij
1 Gij
2 GiiG jj 2 g ii g jj
(i ≠ j )
(B.6)
since by relation (B.4) it is obtained that
3
3 ∂g
∂δ

∂g
∂δ  
GiiG jj =  gii + ∑ ii δ k + 2 gii i   g jj + ∑ jj δ k + 2 g jj j

∂ξi  
∂ξ j
k =1 ∂ξ k
k =1 ∂ξ k




 ∂δ ∂δ 
 ∂g

∂g
= gii g jj + ∑  jj g ii + ii g jj  δk + 2 g ii g jj  i + j 
 ∂ξi ∂ξ j 
∂ξ k
k =1  ∂ξ k



.
3
The extension and shear components of the strain tensor are obtained by substituting
the relation U i = gii δi into expressions (B.5)and (B.6), which results in
∂  Ui 
1 3 ∂gii U k

+
∑


∂ξi  gii  2 g ii k =1 ∂ξ k g kk

1
∂  Ui 
∂  Uj
 gii

eij =

 + g jj


∂ξi  g jj
2 gii g jj  ∂ξ j  gii 


eii =
184




(B.7)
,
if i ≠ j.
Appendices
Appendix C
Equilibrium
equations
curvilinear coordinates
in
Consider the reference surface of an infinitesimal shell element bounded by two pairs
of normal planes of the coordinate lines ξ1 and ξ 2 , respectively. The equilibrium
conditions of this infinitesimal element under the influence of the internal stress
resultants and stress couples and applied external forces and torques will be
determined. Similar derivations can be found in, e.g., Reissner [61], Novozhilov [20],
Leissa [11], Ventsel and Krauthammer [62].
The external force and torque vectors acting on the reference surface are introduced by
p = p1t1 + p2t 2 + pζ n
m = − m2t1 + m1t 2
(C.1)
The internal stress resultant and stress couple vectors acting on the reference line of a
face ξ1 with its normal in positive direction of the tangent vector t1 are introduced by
n1 = n11t1 + n12t 2 + v1n
m1 = − m12t1 + m11t 2
(C.2)
and on a face ξ 2 with its normal in positive direction of the tangent vector t 2 are
introduced by
n 2 = n21t1 + n22t 2 + v2n
m 2 = − m22t1 + m21t 2
(C.3)
The differential length of the reference line on a face ξ1 is equal to α 2 d ξ 2 , which also
holds for a face ξ1 + d ξ1 since it is already of differential length. By considering
internal stress resultants acting on the faces ξ1 and ξ1 + d ξ1 , it is observed that the
magnitude of the force due to the internal stress resultants on the face ξ1 is equal to
−α 2n1d ξ 2
since the components of the stress resultant vector act in negative direction of the
tangent vector t1 . On the face ξ1 + d ξ1 , the increment of the stress resultants along the
ξ1 coordinate line has to be accounted for, by which the magnitude is equal to


∂α 2n1
d ξ1  d ξ 2
 α 2n1 +
∂ξ1


Corresponding expressions can be formulated for the stress couples on these faces and
for the faces ξ 2 and ξ 2 + d ξ 2 .
185
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
By taking the sum of the forces and the sum of moments on all four faces, the
equilibrium equation are obtained and these read, respectively,




∂α n
∂α n
−α 2n1d ξ 2 +  α 2n1 + 2 1 d ξ1  d ξ2 − α1n 2 d ξ1 +  α1n 2 + 1 2 d ξ 2  d ξ1 + α1α 2pd ξ1d ξ2 = 0
∂ξ
∂ξ

1


2





∂α m
∂α m
−α 2m1d ξ2 +  α 2m1 + 2 1 d ξ1  d ξ 2 − α1m 2d ξ1 +  α1m 2 + 1 2 d ξ2  d ξ1
∂ξ1
∂ξ2




∂r
∂r
+
d ξ1 × α 2n1d ξ2 +
d ξ2 × α1n 2 d ξ1 + α1α 2md ξ1d ξ2 = 0
∂ξ1
∂ξ 2
in which, e.g., the contribution
∂r
d ξ1 × α 2n1d ξ2 has been constructed on basis of
∂ξ1
neglecting terms multiplied by the square of a differential length. Adding up all terms
and dividing by the common factor d ξ1d ξ2 results in the vector equations
∂α 2n1 ∂α1n 2
+
+ α1α 2p = 0
∂ξ1
∂ξ2
∂α 2m1 ∂α1m 2 ∂r
∂r
+
+
× α 2n1 +
× α1n 2 + α1α 2m = 0
∂ξ1
∂ξ2
∂ξ1
∂ξ 2
in which the relations
∂r
∂r
= α1t1 and
= α 2t 2 (Appendix A) can be utilised.
∂ξ1
∂ξ2
Substitution of the expression (C.1) for the applied external forces and torques, and the
expressions (C.2) and (C.3) for the internal stress resultants and stress couples and
utilisation of the expressions derived in Appendix A for the derivatives of the unit
vectors t1 , t 2 and n , resolves in
 ∂α 2n11

∂α ∂α n
∂α
αα v
+ n12 1 + 1 21 − n22 2 + 1 2 1 + α1α 2 p1  t1

R
∂ξ
∂ξ
∂ξ
∂ξ

1
2
2
1
1



∂α ∂α n
∂α
∂α n
αα v
+  − n11 1 + 2 12 + n21 2 + 1 22 + 1 2 2 + α1α 2 p2  t 2
R2
∂ξ2
∂ξ1
∂ξ1
∂ξ2


 αα n

αα n
∂α v ∂α v
+  − 1 2 11 − 1 2 22 + 2 1 + 1 2 + α1α 2 pζ  n = 0
R1
R2
∂ξ1
∂ξ2


for the three equations of the equilibrium of forces and in


∂α1 ∂α 2m12
∂α ∂α m
−
− m21 2 − 1 22 + α1α 2v2 − α1α 2 m2  t1
 m11
∂ξ
∂ξ
∂ξ
∂ξ
2
1
1
2


 ∂α m

∂α ∂α m
∂α
+  2 11 + m12 1 + 1 21 − m22 2 − α1α 2v1 + α1α 2m1  t 2
∂ξ
∂ξ
∂ξ
∂ξ

1
2
2
1


αα m
αα m 
+  α1α 2 n12 − α1α 2 n21 + 1 2 12 − 1 2 21  n = 0
R1
R2 

for the three equations of the equilibrium of moments.
186
Appendices
Appendix D
Strain energy and
Beltrami operator
Laplace-
Strain energy
In section 2.3 the expression for the strain energy of a shell is derived as
Es = ∫ Es′dV
V
where
Es′ = ∫ σij deij
,
e
( i, j ) = (1, 2, ζ )
For a homogeneous and isotropic material obeying the generalisation of Hooke’s law
the integral simplifies to
1
( σ11e11 + σ22e22 + σζζeζζ + 2σ12e12 + 2σ1ζe1ζ + 2σ2ζe2ζ ) dV
2 V∫
Es =
Introducing the Kirchhoff-Love assumptions and the differential volume by (2.6), the
integral reduces to
1
( σ11e11 + σ 22e22 + 2σ12e12 ) α1α 2 (1 + ζ R1 )(1 + ζ R2 ) d ξ1d ξ2 d ζ
2 ∫ζ ξ∫2 ξ∫1
Es =
(D.1)
The stresses are described by the two-dimensional Hooke’s law (2.13), which reads
E
E
E
e + υe22 ) ,
σ22 =
e + υe11 ) ,
σ12 =
e12
2 ( 11
2 ( 22
1− υ
1− υ
1+ υ
The normal strains e11 and e22 are described by (2.10), which read
σ11 =
e11 =
1
ζ
1+
R1
( ε11 + ζβ11 )
,
e22 =
1
1+
ζ
R2
( ε 22 + ζβ22 )
(D.2)
(D.3)
and the shearing strain e12 is described by (2.30), which reads
2e12 =


ζ2 
ζ
ζ  
γ + ζ 1 +
+
1−


 ρ12 
ζ
ζ  R1R2  12
2
R
2
R2  

1
1+ 1+
R1
R2
1
1
which for convenience is rewritten to
e12 =


ζ2 
ζ
ζ  
1−
γ + ζ 1 +
+



 ρ
ζ
ζ  R1R2 
 2 R1 2 R2  
1+ 1+
R1
R2
1
1
(D.4)
in which, by comparing the latter expressions
1
γ = γ12
2
,
1
ρ = ρ12
2
187
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Alternatively, the normal strains e11 and e22 are described by (2.36), which read
e11 = ε11 +
ζ
ζ
1+
R1
κ11 ,
e22 = ε 22 +
ζ
1+
ζ
R2
κ 22
(D.5)
in which, according to (2.37),
κ11 = β11 −
ε11
R1
,
κ 22 = β 22 −
ε 22
R2
(D.6)
By substituting the expressions (D.2), (D.3) and (D.4) into (D.1) and carrying out the
integration with respect to ζ for − t 2 ≤ ζ ≤ t 2 , the strain energy as an integral over the
reference surface becomes
Es =
+
1 Et
( ε11 + ε22 )2 − 2 (1 − υ) ( ε11ε 22 − γ 2 )  α1α 2 d ξ1d ξ 2

2 1 − υ2 ξ∫2 ξ∫1 
1 Et 3
 β + β 22 )2 − 2 (1 − υ ) (β11β22 − ρ2 )
2 ∫ ∫ ( 11
24 1 − υ ξ2 ξ1
1 1 
1
1 
−2  −  ( ε11β11 − ε22β22 ) − 2 (1 − υ )  +  γρ
 R1 R2 
 R1 R2 
(D.7)
 1 1  ε 2 ε 2 
 1
1
1  
+  −  11 − 22  + 2 (1 − υ)  2 −
+ 2  γ 2  α1α 2 d ξ1d ξ2
R2 
 R1 R2  R1
 R1 R1R2 R2  
Alternatively, by substituting (D.5) instead of (D.3), which are related to one another
by (D.6), an equal expression is obtained that becomes
Es =
+
1 Et
( ε11 + ε22 )2 − 2 (1 − υ) ( ε11ε 22 − γ 2 )  α1α 2 d ξ1d ξ 2

2 1 − υ2 ξ∫2 ξ∫1 
1 Et 3
( κ11 + κ 22 )2 − 2 (1 − υ) ( κ11κ 22 − ρ2 )
24 1 − υ2 ξ∫2 ξ∫1 
 ε
κ
1
κ 
ε
1  
+2 ( ε11 + ε22 )  11 + 22  − 2 (1 − υ )  κ11 22 + κ 22 11 +  +  γρ 
R2
R1  R1 R2  
 R2 R1 

+
188
2
 1
1
1
1  
2
ε11ε 22 − γ 2 ) −  −  γ 2   α1α 2 d ξ1d ξ 2
( ε11 + ε 22 ) − 2 (1 − υ) 
(
R1R2
RR
 R1 R2   
 1 2
(D.8)
Appendices
As Novozhilov does, by introducing the parameters
t
′ = κ11 ,
ε11
2
t
ε′22 = κ 22
2
,
t
γ′ = ρ
2
the strain energy (D.8) becomes
Es =
+
1 Et
( ε11 + ε 22 )2 − 2 (1 − υ) ( ε11ε 22 − γ 2 )  α1α 2 d ξ1d ξ 2

2 1 − υ2 ξ∫2 ξ∫1 
1 Et
1
2
1 ′
′ ε′22 − γ′2 )
ε + ε′22 ) − 2 (1 − υ ) ( ε11
2 ∫ ∫  ( 11
2 1 − υ ξ2 ξ1  3
3
t  ε′ ε′ 
t  ε 22
ε 
t 1
1 
′
+ ( ε11 + ε 22 )  11 + 22  − (1 − υ)  ε11
+ ε′22 11  − (1 − υ)  +  γ′γ
3  R2 R1 
3  R2
R1 
3  R1 R2 
+
2
1 t2
1 t2
t2  1
1  
2
ε11ε 22 − γ 2 ) + 2 (1 − υ)  −  γ 2  α1α 2 d ξ1d ξ 2
( ε11 + ε22 ) − 2 (1 − υ)
(
R1R2 12
R1R2 12
12  R1 R2  

The last three terms are obviously negligible in comparison to the first two since they
 t2 
. As a first order approximation, e.g. based on the linearization
2 
R 
are of the order O 
of the strain distribution, the terms of the order O   are also neglected. Using the
R
original strain measures, the strain energy reduces to
t
Es =
+
1 Et
( ε11 + ε22 )2 − 2 (1 − υ) ( ε11ε 22 − γ 2 )  α1α 2 d ξ1d ξ2

2 1 − υ2 ξ∫2 ξ∫1 
1 Et 3
 κ + κ 22 )2 − 2 (1 − υ ) ( κ11κ 22 − ρ 2 )  α1α 2 d ξ1d ξ2
2 ∫ ∫  ( 11

24 1 − υ ξ2 ξ1
Hence, the contribution of the normal and shear strain and the contribution of the
changes of curvature and twist are uncoupled. The approximated expression for the
strain energy can also be directly obtained if the linearization of the strain distribution
as proposed in subsection 2.7.3 is adopted. On basis of this observation, it seems to be
allowed to assume that the constitutive relation can be given according to the relation
presented in subsection 2.7.3.
Derivation of the Laplace-Beltrami operator
The Laplace-Beltrami operator is a differential operator defined as
∆ ≡ ∇ i∇
where ∇ is the differential operator which, when applied to a scalar field f , produces
the vector field grad f as shown by Borisenko and Tarapov [15]. In a system of
orthogonal curvilinear coordinates ξ1 , ξ2 , ξ3 with orthogonal local basis t1 , t 2 , t 3 , a
natural generalisation of the gradient in rectangular coordinates is obtained as
3
∇f ≡ grad f = ∑ t i
i =1
∂f
∂si
where dsi = αi d ξi .
189
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The second derivatives of the position vector to a point are assumed to be continuous,
such that x,ij = x, ji . Since x,i = α i t i , the expression
α i t i , j = α j ,i t j + α j t j ,i − α i , j t i
is obtained and noting that t i , j it i = 0 the relation
α i t i , j it k = α j , i t j it k + α j t j ,i it k − α i , j t i it k
can be written as
α i t i , j it j = α j , i
if k = j and i ≠ j .
The Laplace-Beltrami operator of the field f is constructed by taking
 1 ∂
1 ∂
1 ∂   1 ∂f
1 ∂f
1 ∂f 
∆f ≡ ∇i∇f =  t1
+ t2
+ t3
+ t2
+ t3
i t1

α1 ∂ξ2
α1 ∂ξ3   α1 ∂ξ1
α1 ∂ξ2
α1 ∂ξ3 
 α1 ∂ξ1
and since the three unit vectors are mutual orthogonal, it is obtained that the first three
non-zero terms become
∆f = t1 it1
1 ∂  1 ∂f 
1 ∂f
1 ∂f
+ t1 it 3,1
+ ...

 + t1 it 2,1
α1 ∂ξ1  α1 ∂ξ1 
α1 ∂ξ2
α1 ∂ξ3 1
and by making use of the derived nontrivial relations between the unit vectors and their
respective derivatives, the full expression for the Laplace-Beltrami operator of field f
can be rewritten to
∆f =
190
1  ∂  α 2α 3 ∂f  ∂  α1α3 ∂f  ∂  α1α 2 ∂f  


+

+


α1α 2α 3  ∂ξ1  α1 ∂ξ1  ∂ξ 2  α 2 ∂ξ 2  ∂ξ3  α3 ∂ξ3  
Appendices
Appendix E
Expressions and derivation of the
stiffness
matrix
for
the
elastostatic
behaviour
of
a
circular ring
The formulations that are derived for the ring element stiffness matrices are based on
the solution presented by Van Bentum [1], which is implemented in the precursor of
the present computer program CShell. The objective of the analysis herein presented is
to integrate the ring element in the super element approach as described in chapter 3.
This analysis is largely based on the same set of relations as for a circular cylindrical
shell on basis of the Morley-Koiter theory.
For a ring employed as a stiffener to cylindrical elements described by a super
elements approach, the ring element shall be present at the boundary of the adjacent
elements. Hence, the nodal forces of these elements must be in equilibrium with the
loads acting on the ring and the nodal displacements must be equal to the
displacements of the ring element. In other words, the stiffness relation between the
displacements of and the forces acting on the ring element can be added to the stiffness
matrix for the circular cylindrical structure at its respective nodal position.
It might be assumed that the ring has relatively little resistance against loads acting
out of its circular plane, such as lateral and torsional loads. Hence, the ring is modelled
as a curved beam, which is only able to withstand loads acting in its circular plane.
Additionally, the ring element is connected to the circular shell outer or inner surface.
Therefore, the neutral line of the curved beam is not located at the middle surface of
the shell element, which is chosen as the reference surface. To facilitate connection to
the cylindrical shell description, the directions of the axes are chosen in the
circumferential and transverse directions and the three sets of equations are referenced
to the middle surface of the shell element. A polar coordinate system is thus applied to
this middle surface serving as the reference line of the circular ring. Hereby, an
infinitesimal element of the circular beam has a side with length of arc, measured on
the reference line, adθ in circumferential direction (where a denotes the radius of the
reference line) and dz in normal direction to the reference line. The height of such an
element is not necessarily constant, but assumed to be constant for convenience.
191
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Coordinate system
Based on the choice as described above, the radius of the reference line in θ -direction
is constant and denoted by a . The expression of the line element in Appendix A can
now be given by
2
z

ds 2 = a 2  1 +  d θ2 + dz 2
 a
where θ and z are associated with ξ 2 and ζ , respectively. Measured on the reference
surface the line element is thus equal to ds 2 = a 2 d θ2 . This means that the following
substitution can be made if the proposed theory of section 2.7 is used as a starting point
of our analysis
ξ1 = x
α1 = 0
R1 = ∞
ξ2 = θ
α2 = a
R2 = a
(E.1)
As the curved beam is assumed to be able to withstand only loads acting in its own
plane, it is free to deform in its lateral direction. Hence, changes of the line element in
this direction (the x -direction) are not described and all quantities not acting in the
plane of the circular ring are equated to zero. Therefore, the vectors used with respect
to this coordinate system are
u = [ uθ u z ]
e = [ε κ]
s = [N
M]
f = [ fθ
fz ]
The presented strain ε and change of curvature κ are identical to those for the strain
εθθ and the change of curvature κθθ in the circumferential direction of the shell middle
surface, but for the circular ring the redundant indices can be omitted. The presented
stress resultant N and the stress couple M can be interpreted in the same manner as
the stress resultant nθθ and the stress couple mθθ , but for the circular ring as a force and
bending moment acting on its cross-section instead of a force and bending moment per
unit length. Similarly, the loads f θ and f z are loads per unit length acting on the
reference line instead of loads per unit area of the middle surface.
Kinematical relation
The kinematical relation (2.39) is rewritten using the description (E.1) of the reference
line of the circular ring resulting in
1 d
ε   a dθ
κ = 
   0

192
1

  uθ 
a
 
2
1 d
1  u z 
− 2 2 − 2
a dθ a 
(E.2)
Appendices
Constitutive relation
As the neutral line of the curved beam is not located at the middle surface of the shell
element, the stress resultant and the stress couple cannot be obtained by rewriting an
expression from section 2.7. Based on the expressions (2.14), the stress resultant and
stress couple can be readily defined as
N = ∫ σdA
A
M = ∫ σzdA
A
in which dA is the infinitesimal area on which the circumferential stress σ acts.
However, to adopt the kinematical relation as described above, the alternative stress
resultants according to section 2.7 need to be used for consistency. Hence, the
applicable stress resultants are introduced by
N=N+
M
z

= ∫ σ 1 +  dA
A
a
 a
M = M = ∫ σzdA
A
Figure E-1 Geometry of a typical connection of a ring element to a cylindrical shell
element.
193
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
In Figure E-1, a segment of the connection of the ring beam to the cylindrical wall is
shown facing the cross-section of the ring. The thickness and the radius to the middle
line of the cylinder are denoted by t and a , respectively, and the typical ring element
consists of two parts. The dimensions of the part 1 and part 2 are denoted by t1 and t2
in the radial direction and l1 and l2 in the transverse direction, respectively.
In the figure, two ring parts are shown, which are both connected to the outside of
the cylindrical wall. To facilitate a generic configuration, e.g. also applicable to a ring
on the inside, the radius of the part 1 is indicated by a1 .
To relate the stress resultant and stress couple to quantities acting at the reference line,
the radius to the middle surface of the cylindrical wall is thus adopted and the origin of
the thickness coordinate z is correspondingly chosen at the reference line. For this
coordinate basis and the generic configuration described above, the integration ranges
become
1
z0 = a1 − t1 − a
2
z1 = z0 + t1
z2 = z1 + t2
where the constant width of the part 1 between z0 and z1 is l1 , and the constant width
of part 2 between z1 and z2 is l2 . Hence, the stress integrals are represented by
z1
z2
z
z


N = ∫ σl1  1 +  dz + ∫ σl2 1 +  dz
z0
z1
a
a




z1
z2
z0
z1
M = ∫ σl1zdz + ∫ σl2 zdz
The same assumptions for the elastic behaviour are employed as for the elastic shell.
Hence, the stress-strain relation for the ring beam is adapted from the two-dimensional
Hooke’s law for a thin elastic shell (2.13) and reads
σ = Ee
The strain distribution in the thickness direction is adapted from (2.36), which for the
circumferential direction only reads
z
e=ε+
1+
z
a
κ
where the strain distribution is consistently related to the reference line.
By subsequent substitution of the strain distribution into the stress-strain relation and
this result into the stress integrals, the stress resultant and the stress couple read
(
)
z2 
z1
z2
z
z 
 z1 
N = E ε  l1 ∫  1 +  dz + l2 ∫ 1 +  dz  + E κ l1 ∫ zdz + l2 ∫ zdz
z0
z
z
z
1
0
1
 a
 a 

−1
−1
 z1 

z1
z2
z2
z
z

M = E ε l1 ∫ zdz + l2 ∫ zdz + E κ  l1 ∫ z 2  1 +  dz + l2 ∫ z 2 1 +  dz 
z0
z1
z0
z
1
 a
 a


(
194
)
Appendices
The above formulas show that a number of integrals need to be developed. To facilitate
a convenient solution the factor within the brackets multiplied by z 2 of the formula for
−1
the moment is expanded into (1 + z a ) 1 − z a + ... and truncated as presented. Finally,
the following moments of area are introduced
z1
z2
z0
z1
A0 = ∫ dA = l1 ∫ dz + l2 ∫ dz
A
z1
z2
z0
z1
A1 = ∫ zdA = l1 ∫ zdz + l2 ∫ zdz
A
z1
z2
z0
z1
A2 = ∫ z 2 dA = l1 ∫ z 2 dz + l2 ∫ z 2 dz
A
z1
z2
z0
z1
A3 = ∫ z 3dA = l1 ∫ z 3dz + l2 ∫ z 3dz
A
By adopting this notation for the integrals and making use of the truncated expansion
where applicable, the stress resultant and the stress couple read
A

N = E  A0 + 1  ε + EA1κ
a

A 

M = EA1ε + E  A2 − 3  κ
a 

Finally, the constitutive relation is rewritten to
 N   EAr
  =  ES
M   r
ES r   ε 
EI r   κ 
(E.3)
where the new sectional quantities are introduced as
A1
a
Ar = A0 +
S r = A1
I r = A2 −
A3
a
where the subscript r indicates a combined sectional ring quantity.
Equilibrium relation
The equilibrium relation (2.43) is rewritten using the description (E.1) resulting in
 1 d
− a d θ

 1
 a

  N   fθ 
  =  
2
f
1 d
1 M
− 2 2 − 2    z
a dθ a 
0
(E.4)
and the transverse shearing stress resultant is adapted from (2.44) and becomes
V=
1 dM
a dθ
(E.5)
195
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Stiffness matrix
With the objective to integrate the ring element in the super element approach for shells
of revolution as described in chapter 3, continuity and symmetry of the load in
circumferential direction of the ring is assumed. Hence, all quantities must be
continuous in circumferential direction, which can be interpreted as transitional
conditions.
In accordance with the derivation presented in chapter 3, the two load components
can be described by a Fourier trigonometric series expressed by
∞
f θ ( θ ) = ∑ f θn sin nθ
n=0
∞
f z ( θ ) = ∑ f zn cos nθ
n=0
where n is the mode number and represents the number of whole waves in
circumferential direction. The dependence on the axial coordinate is obviously omitted,
as the ring cannot withstand loads in that direction. So, in correspondence with the
distribution of the load components, the general solution for the displacements is of the
congruent form
∞
uθ ( θ ) = ∑ uθn sin nθ
n=0
∞
u z ( θ ) = ∑ u zn cos nθ
n=0
On the basis of the same consideration and by inspecting the sets of equations above, it
can now be concluded that the relevant strain and stress quantities are described by
trigonometric functions of the form
ε, κ, N , M
⇒ cos nθ
V
⇒ sin nθ
On the basis of these arbitrary solutions describing the behaviour of a ring connected to
a cylindrical surface, an element stiffness matrix has to be synthesized. The
considerations described here are exemplified for a load that is symmetric to a certain
axis, but can easily be extended to an asymmetric load.
As mentioned above, the stiffness relation between the displacements of and the
forces acting on the ring element is to be derived and these quantities are distributed
along the circular reference line by functions of the form
f θ , uθ
⇒ sin nθ
f z , uz
⇒ cos nθ
Having shown that the forces have the same distribution as the corresponding
displacements, it can be concluded that the stiffness relation for the ring element only
depends on the amplitude of the circumferential distribution (which can depend on the
circumferential mode number n ). Such a stiffness relation can thus be presented by
 fˆθ   K θθ
 =
 fˆz   K θz
196
K θz  uˆθ 
K zz  uˆ z 
Appendices
where circumferential distributions of the ring quantities are related to the amplitudes
by the relations
 f θ ( θ )  sin nθ
uθ ( θ )  sin nθ
0   fˆθ 
0  uˆθ 
  , 

=
=

cos nθ   fˆz 
cos nθ  uˆ z 
 f z ( θ )   0
u z ( θ )   0
(E.6)
By substitution of (E.6) for the displacements into the kinematical relation (E.2), the
expressions for the strain and curvature become
 f θ ( θ )  sin nθ
uθ ( θ )  sin nθ
0   fˆθ 
0  uˆθ 
  , 

=
=

ˆ
u
θ
0
cos
0
cos
θ
n
nθ  uˆ z 
(
)
θ
f
(
)
f
  z 
 z
 
 z
 
n
ε ( θ )   a

=
 κ ( θ)  0

1 
a  uˆθ cos nθ 


2
n − 1  uˆ z cos nθ 
a 2 
By substitution of this result into the constitutive relation (E.3), the expressions for the
stress resultant and stress couple become
 EAc
n
 N ( θ)   a

=
 M ( θ )   ESc n
 a
EAc ESc 2

+ 2 ( n − 1) 
uˆθ cos nθ 
a
a

uˆ cos nθ 
ESc EI c 2
+ 2 ( n − 1)   z

a
a
By substitution of this result into the equilibrium relation (E.4), the expressions for the
forces acting on the ring element become
EAc 2

n
 f θ ( θ )  sin nθ
0 
a2
=


 
cos nθ   EAc
ES
 f z ( θ)   0
n + 3c n ( n 2 − 1)
a
 a 2
EAc
ES

n + 3c n ( n 2 − 1)
2
 uˆθ 
a
a
 
2
uˆ
EAc
ES
EI
+ 2 3c ( n 2 − 1) + 4c ( n 2 − 1)   z 
a2
a
a

The stiffness relation is thus obtained and reads
EAc 2

n
 fˆθ  
a2
=

 
 fˆz   EAc n + ESc n ( n 2 − 1)
 a 2
a3
EAc
ES

n + 3c n ( n 2 − 1)
 uˆθ 
a2
a
 
2
uˆ
EAc
ES
EI
+ 2 3c ( n 2 − 1) + 4c ( n 2 − 1)   z 
2

a
a
a

(E.7)
For the sake of completeness, the transverse shearing stress resultant (E.5) is given.
 ES
V ( θ) =  − 2c
 a
−
ESc
EI
 uˆ 
n − 3c n ( n 2 − 1)   θ  sin nθ
2
a
a
 uˆ z 
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198
Appendices
Appendix F
Ring equations comparison
For a circular ring described by a reference line with radius a and circumferential
coordinate θ and with a thickness coordinate z with its origin at this reference line,
the expression for the normal strains (2.36) is rewritten to
e=
1
z
1+
a
( ε + zβ ) = ε +
z 
ε
z
β−  =ε+
κ
z 
z
a
1+
1+
a
a
The strain energy described by the strain ε and the change of rotation β can be
obtained easily from (D.7), which results in
2
Es =
1 Et
1 Et 3 
ε 22 
ε 2 ad θ +
 β22 −
 ad θ
2 ∫ 22
21− υ θ
24 1 − υ2 ∫θ 
a 
An identical expression for the strain energy described by the strain and the change of
curvature κ can be obtained easily from (D.8), which results in
Es =
1 Et
1 Et 3
2
ε
ad
θ
+
κ 22 2 ad θ
22
2 1 − υ2 ∫θ
24 1 − υ2 ∫θ
It is believed that the latter description is more appropriate for a ring. However, this
would be true for a mathematically exact solution, which is not desirable in view of the
assumptions readily introduced to obtain the strain expression. The objective of this
appendix is thus to define the best linearization in the framework of a firstapproximation theory.
As a starting point, the linearization of the normal strain description is presented for
both kinematical quantities β and κ . In line with the approach of subsection 2.7.3,
terms z a will be neglected in comparison to unity, resulting in an expression with the
change of rotation β denoted by superscript Β and one with the change of curvature κ
denoted by superscript Κ , respectively.
eΒ = ε + zβ
eΚ = ε + z κ
For the kinematical quantities based on the description that adopts the change of
curvature, the equations (E.2) are at hand, which are extracted from the kinematical
relation (2.39). In the same manner, but extracted from the kinematical relation (2.11),
the kinematical quantities based on the description that adopts the change of rotation
can be obtained.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Hence, the following kinematical relations are to be considered
1 d
ε
   a dθ
β  = 
  1 d
 a 2 d θ
1 d
ε   a dθ
κ = 
   0


  uθ 
 
2
1 d  u z 
− 2 2
a dθ 
1

  uθ 
a
 
1 d2
1 u
− 2 2 − 2  z
a dθ a 
1
a
In the following, the approach of Appendix E will be adapted per linearization. Both
strain descriptions are substituted into the stress-strain relation σ = Ee . Subsequently,
this is to be substituted into the expressions for the stress resultant and the stress
couple, which read
N = ∫ σdA
A
M = ∫ σzdA
A
These relations hold for both strain descriptions as the neglect of terms z a is to be
simultaneously imposed on the strain and stress distribution.
Performing the substitutions as described above per linearization, the constitutive
relations become
 N Β   EA ES   ε 
 Β = 
 
 M   ES EI  β 
 N Κ   EA ES   ε 
 Κ = 
 
 M   ES EI   κ 
where the following moments of area are introduced
A = ∫ dA
A
S = ∫ zdA
A
I = ∫ z 2dA
A
The expressions for the stress resultants and stress couples as described above are to be
substituted in the corresponding equilibrium equations. For the resultant and couple
that are based on the description that adopts the change of curvature, the equations
(E.4) are at hand, which are extracted from the equilibrium relation (2.43). In the same
manner, but extracted from the equilibrium relation (2.15), the equilibrium equations
based on description that adopts the change of rotation can be obtained.
200
Appendices
Hence, the following equilibrium relations are to be considered
 1 d
− a d θ

 1
 a
 1 d
− a d θ

 1
 a
1 d 
Β
a 2 d θ   N   fθ 
 Β =  
2
1 d  M   fz 
− 2 2 
a dθ 
−

  N Κ   fθ 

= 
1 d2
1  M Κ   fz 
− 2 2 − 2 
a dθ a 
0
By substitution of (E.6) for the displacements into the kinematical relations above, the
expressions for the kinematical quantities become
n
ε   a
β  = 
  n
 a 2
n
ε a
κ = 
  0

1
a  uˆθ cos nθ 


n 2  uˆ z cos nθ 
a 2 
1 
a  uˆθ cos nθ 


2
n − 1  uˆ z cos nθ 
2 
a 
By substitution of these results into the constitutive relations above, the expressions for
the stress resultant and stress couple become
ES
EA ES 2 
 EA
n+ 2 n
+ 2 n 
 N Β ( θ)   a
uˆθ cos nθ 
a
a
a
=


 Β 
ES
EI
ES
EI
uˆ cos nθ 
M
θ
(
)

 
n+ 2 n
+ 2 n2   z
 a

a
a
a
EA ES 2
 EA

n
+ 2 ( n − 1) 
 N Κ ( θ)   a
uˆθ cos nθ 
a
a

 Κ
=
uˆ cos nθ 
ES
ES
EI
M
θ
( )  n

+ 2 ( n 2 − 1)   z
 a

a
a
By substitution of these results into the equilibrium relations above, the expressions for
the forces acting on the ring element become
ES
EI
EA
ES
EI 
 EA 2
n + 2 3 n2 + 4 n2
n + 3 n ( n 2 + 1) + 4 n3 
2
 f θB ( θ )  sin nθ
0   a2
uˆθ 
a
a
a
a
a

 
 B =

cos nθ   EA
ES
EI
EA
ES
EI
uˆ
 f z ( θ)  0
+ 2 3 n2 + 4 n4   z 
n + 3 n ( n 2 + 1) + 4 n3

a
a
a2
a
a
 a 2
EA 2
EA
ES


n
n + 3 n ( n 2 − 1)
 uˆθ 
 f θK ( θ )  sin nθ
0 
a2
a2
a

 
 K
=

2
uˆ
cos nθ   EA
ES
EA
ES 2
EI 2
2
 f z ( θ)  0
n + 3 n ( n − 1)
+ 2 3 ( n − 1) + 4 ( n − 1)   z 
a
a2
a
a
 a 2

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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
By substitution of the combined sectional ring quantities defined in Appendix E, which
are rewritten making use of the definitions above, into the stiffness relation (E.7), the
following is obtained
 EA 2 ES 2
n + 3 n
 fˆθ   a 2
a
 =
ˆ
EA
ES

f
 z 
n + 3 n3
a
 a 2
EA
ES

n + 3 n3
2
 uˆθ 
a
a
 
3
2
2  u
ˆ
EA ES
EI 2
EA 2
2
+ 3 ( 2n − 1) + 4 ( n − 1) − 5 ( n − 1)  z 
2
a
a
a
a

Especially for larger values of the circumferential mode number n , but also for n = 0
and n = 1 , the expressions based on description that adopts the change of rotation are
less accurate than the expressions based on description that adopts the change of
curvature.
Hence, it can be concluded that the description that adopts the change of curvature
is more appropriate.
Moreover, for a reference line that coincides with the neutral line of a ring element,
the stiffness matrix derived on basis of the description that adopts the change of
curvature is identical to the stiffness matrix obtained in Appendix E. This is clearly not
the case if the description that adopts the change of rotation is employed.
Additionally, by substitution of expressions for the kinematical quantities into the
two expressions for the linearization of the normal strain and hence the normal stress,
the following is obtained
 n
1
n 
n2  
eΒ =  + z 2  uˆθ +  + z 2  uˆ z  cos nθ
a 
a  
a
 a
n
1
n2 − 1  
eΚ =  uˆθ +  + z 2  uˆ z  cos nθ
a  
a
a
while by substitution of the (identical) kinematical quantities defined in Appendix E
into the expression for the normal strains (2.36), the following is obtained
1
e=
1+
( ε + zβ ) = ε +
z
a
n
1
a n2 − 1  
κ =  uˆθ +  + z
 uˆ z  cos nθ
z
a + z a2  
a
a
1+
a
z
Obviously, as the stress-strain relation is linear, the expression that adopts the change
of curvature is almost identical for both the strain as the stress distribution across the
thickness. The difference observed is only present for the non-linearly varying part,
which will be negligible for small thickness-to-radius ratio as exemplified in Appendix
D.
202
Appendices
Appendix G
Semi-membrane concept
Introduction
Another interesting simplified approach, which has the objective to obtain insight into
the load carrying behaviour of cylindrical shell structures, is the semi-membrane
concept (SMC), which is able to deal with non-axisymmetric load cases. The semimembrane concept assumes that, to simplify the initial kinematical equations, the
circumferential strain εθθ is equal to zero and that, to simplify the initial equilibrium
equations, the bending moments about the circumferential axis and torsion axis are
zero ( mxx = 0 mxθ = 0 and hence vx = 0 ). The resulting equation exactly describes the
ring-bending behaviour, but it can only be applied to self-balancing modes. As shown
by Pircher, Guggenberger and Greiner [9], this concept can be applied to, e.g., a radial
wind load, an axial elastic support and an axial support displacement. However, not all
load cases or support conditions can be described. Moreover, the semi-membrane
concept is only applicable to certain load-deformation behaviours of cylindrical shell
structures. Closely related to the simplifications, it should be allowed to neglect the
influence of the part of the solution described by the short influence length in
comparison to the part described by the long influence length. In other words, the
cylinder should be sufficiently long compared with its radius and the boundary effects
should mainly influence the more distant material.
Geometry
For the semi-membrane concept, the same polar coordinate system is applied and the
axes are chosen in the same direction as for the circular cylindrical shell. Accordingly,
the three positive directions of the displacements ( u x , uθ , u z ) are taken corresponding to
the three positive coordinate directions ( x, θ, z ) .
Sets of equations
The sets of equations formulated for the circular cylindrical shell are tremendously
simplified by the assumptions of the semi-membrane concept, i.e. the circumferential
strain as well as both the axial and torsional bending moments may be equated to zero.
The vectors (4.3) used with respect to the coordinate system become
u = [u x
uθ u z ]
e = [ ε xx
γ xθ
κθθ ]
s = [ nxx
nx θ
mθθ ]
p = [ px
pθ
pz ]
where it should be noted that aεθθ =
∂uθ
∂u
+ u z = 0 and hence u z = − θ .
∂θ
∂θ
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The kinematical relation (4.4) is rewritten resulting in
 ∂

 ε xx   ∂x
γ  = 1 ∂
 xθ   a ∂θ
 κθθ  
 0



 u x 
 u 
0
 θ
 u 
2
1 ∂
1  z
− 2 2 − 2 
a ∂θ a 
0
0
∂
∂x
0
Making use of the introduced relation u z = −
 ∂

 ε xx   ∂x
γ  = 1 ∂
 xθ   a ∂θ
 κθθ  
 0

∂uθ
, this relation is rewritten to
∂θ



∂
 u x 
 u 
∂x
 θ
1 ∂3
1 ∂
+
a 2 ∂θ3 a 2 ∂θ 
0
(G.1)
in which only the two independent displacements are employed.
The constitutive relation is given by (4.5) but becomes, rewritten for the assumptions
introduced above,
2
 nxx   Dm (1 − υ ) 0
n  =
0
Ds
 xθ  
 mθθ  
0
0

0   ε xx 

0   γ xθ 
(G.2)

Db   κ θθ 

where the quantities Dm , Ds and Db are the extensional (membrane) rigidity, the shear
rigidity and the flexural rigidity, respectively, which are given by
Dm =
Et
1 − υ2
Ds =
;
Et
2 (1 + υ )
;
Db =
Et 3
12 (1 − υ2 )
(G.3)
The equilibrium relation (4.8) is rewritten resulting in
 ∂
 − ∂x

 0


 0

1 ∂
a ∂θ
∂
−
∂x
−
0


  nxx a   p x a 
  n a  =  p a
0
  xθ   θ 
 m a   p a 
1 ∂2
1   θθ   z 
− 2 2 − 2
a ∂θ a 
0
However, this relation is not in line with the analogy as explained in section 2.5. The
analogy comprises that a derivative in the differential operator matrix for the
kinematical relation (G.1) is also present in the differential operator matrix for the
equilibrium relation, but then as the adjoint operator at the transposed position. Hence,
the proposed equilibrium equation becomes
204
Appendices
 ∂
 − ∂x


 0
1 ∂
a ∂θ
∂
−
∂x
−
 n a
px a

  xx  
n a = 


∂
p
xθ 
 p + z  a 
1 ∂3
1 ∂ 
− 2 3 − 2   mθθ a   θ ∂θ  
a ∂θ a ∂θ 
0
(G.4)
The transverse shearing stress resultant is described by (4.9) and becomes
vθ =
1 ∂mθθ
a ∂θ
(G.5)
Principle of virtual work
In this section, the principle of virtual work is employed for the semi-membrane
concept by utilizing the kinematical and constitutive relations derived in the previous
section. The elaboration of the virtual work equation shows that a consistent set of
internal shell quantities has been chosen. Hence, the elaboration confirms the proposed
equilibrium relation (G.4) and provides, in a simple and elegant manner, the natural
boundary conditions that complement the three sets of equations.
The virtual work equation (2.18) is formulated by
δE p = δEs − δW p − δW f = 0
The variation of the strain energy is described by (2.19) and becomes
δEs = ∫∫ ( nxx δε xx + nxθδγ xθ + mθθδκ θθ ) ad θdx
x θ
where
∂u x
∂x
1 ∂u x ∂uθ
γ xθ =
+
a ∂θ ∂x
1 ∂ 2u z u z
1 ∂ 3uθ 1 ∂uθ
κθθ = − 2
−
=
+
a ∂θ2 a 2 a 2 ∂θ3 a 2 ∂θ
ε xx =
and
nxx = Dm (1 − υ2 ) ε xx
nxθ = Ds γ xθ
mθθ = Db κ θθ
The work done by the surface force vector p on the reference surface along the virtual
displacements is formulated by (2.20) and becomes
∂u 

δWp = ∫∫  px δu x + pθδuθ − p z δ θ  ad θdx
∂θ 

x θ
which by integration by parts becomes

∂p 

θ

δWp = ∫∫  px δu x +  pθ + z  δuθ  ad θdx − ∫ [ apz ]θ2 dx
1
∂θ 


x θ
x
where the second part represents a line load along x at θ = θ1 and θ = θ2 . For a closed
circular cylinder, this integral is equal to zero.
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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The final expression becomes

∂p

δWp = ∫∫  p x δu x +  pθ + z
∂θ

x θ


 δuθ  ad θdx


The work done by the edge force vector f on the boundary lines of the boundary
surface S along the virtual displacements is formulated by (2.21) and becomes
∂u 

δW f = ∫  f x δu x + f θδuθ − f z δ θ  ad θ
∂θ 
1
2
θ
x = x( ) , x( )
∂u


+ ∫  f x δu x + f θδuθ − f z δ θ + tθδϕθ  dx
∂θ
 θ=θ(1) , θ( 2)
x
which by integration by parts becomes

∂f 

θ

δW f = ∫  f x δu x +  f θ + z  δuθ  ad θ
− [ f z δuθ ]θ2
1
∂θ
1
2



θ
x = x( ) , x( )
1
2
x = x( ) , x( )
∂u


+ ∫  f x δu x + f θδuθ − f z δ θ + tθδϕθ  dx
∂θ

 θ=θ(1) , θ( 2)
x
where the second part of the first line represents the four point loads at the corners of a
confined surface. For a closed circular cylinder, this integrand and the second integral
are equal to zero. The final expression becomes

∂f 


δW f = ∫  f x δu x +  f θ + z  δuθ  ad θ
∂θ
1
2



θ
x = x( ) , x( )
All terms of the virtual work equation have now been given either in virtual strains (for
the internal work quantities) or in virtual displacements (for the external work
quantities). A natural step is to obtain the internal work only in terms of the virtual
displacements to be able to elaborate further towards the equilibrium equations and the
natural boundary conditions.
After substitution of the expression for the kinematical relation and noting that
derivative operations and variation are commutative, the following expression is
obtained
 ∂δu x
 1 ∂ 3δuθ 1 ∂δuθ  
 1 ∂δu x ∂δuθ 
δEs = ∫∫  nxx
+ nx θ 
+
+
m
+ 2
 2
  ad θdx
θθ

3
∂x
∂x 
a ∂θ  
 a ∂θ

 a ∂θ
x θ
By (consecutive) integration by parts the derivatives of the virtual displacements are
removed where applicable and we obtain
 ∂nxθ 1 ∂ 3mθθ 1 ∂mθθ 
1 ∂nxθ 
 ∂n
δEs = − ∫∫  xx +
δ
u
ad
θ
dx
−
+
+ 2

 δuθ ad θdx
x

∫∫
∂x a ∂θ 
∂x a 2 ∂θ3
a ∂θ 
x θ
x θ
2
x = x( )
+ ∫ [ nxx δu x + nxθδuθ ]x = x(1) ad θ
θ
2
θ=θ( )

 δu 1 ∂ 2δuθ  
1 ∂ 2mθθ
1 ∂mθθ ∂δuθ
+ ∫  nxθδu x +
δuθ −
+ mθθ  θ +
dx

2
a ∂θ
a ∂θ ∂θ
a ∂θ2   θ=θ(1)
 a
x
206
Appendices
To provide insight into the origin of the respective terms above in comparison with the
equilibrium equations (4.8) and the boundary conditions (4.10), the respective terms
that differ are further investigated. The rotation ϕθ related to mθθ is described by (4.12)
and, making use of the introduced relation u z = −
ϕθ =
∂uθ
, this relation is rewritten to
∂θ
u θ 1 ∂ 2u θ
+
a a ∂θ2
which allows the following substitution in the virtual strain energy formulation
 δu 1 ∂ 2δuθ 
mθθ  θ +
 = mθθδϕθ
a ∂θ2 
 a
The transverse shearing stress resultant vθ related to u z (and hence −
∂uθ
) is described
∂θ
by (G.5), which allows the following substitution in the second line integral
1 ∂ 2 mθθ
1 ∂mθθ ∂δuθ ∂vθ
δuθ −
=
δuθ + vθδu z
a ∂θ2
a ∂θ ∂θ
∂θ
and the following substitution in the second surface integral
 ∂nxθ 1 ∂ 3mθθ 1 ∂mθθ 
 ∂nxθ 1 ∂ 2vθ vθ 
+ 2
+ 2
+
+  δuθ

 δuθ = 
3
2
a ∂θ 
a
 ∂x a ∂θ
 ∂x a ∂θ
If being an independent quantity, the stress resultant nθθ is related to uθ . Hence and as
a result from the simplifications above, the following ”equilibrium equation” can be
identified by comparing the result of the last two substitutions with the equilibrium
equations (4.8)
nθθ =
∂vθ 1 ∂ 2 mθθ
=
∂θ a ∂θ2
Making use of these equilibrium equations and the displacement relations, the
expression above for the internal virtual work can be identically described by
1 ∂nxθ 
 ∂n
 ∂nxθ 1 ∂nθθ vθ 
δEs = − ∫∫  xx +
+
+  δuθ ad θdx
 δu x ad θdx − ∫∫ 
x
a
a
∂
∂θ
∂x a ∂θ

x θ
x θ
2
x = x( )
+ ∫ [ nxx δu x + nxθδuθ ]x = x(1) ad θ
θ
2
θ=θ( )
+ ∫ [ nxθδu x + nθθδuθ + vθδu z + mθθδϕθ ]θ=θ(1) dx
x
which is exactly what would be expected if mxx , mxθ and vx are set equal to zero.
If the sum of all variations (internal and external) is set equal to zero, two sets of
equations are obtained, i.e. one for the double integral over the reference surface and
one for the integral over the boundary lines. As stated previously, the variations of the
displacements are arbitrary and non-zero, so the sets of equations can only vanish if
each coefficient of the variations vanishes individually. From the set for the double
integral over the reference surface, two equilibrium equations are obtained, which read
207
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
∂nxx 1 ∂nxθ
+
+ px = 0
∂x a ∂θ
∂nxθ 1 ∂ 3mθθ 1 ∂mθθ
∂p
+
+ 2
+ pθ + z = 0
∂x a 2 ∂θ3
a ∂θ
∂θ
(G.6)
The set for the integrals over the boundary lines x = constant only (for a closed
cylinder) reads

∫ ( f
θ
x
∂f



− nxx ) δu x +  f θ + z − nxθ  δuθ  ad θ
∂θ


2

x = x( )
(G.7)

∂f



+ ∫ ( f x + nxx ) δu x +  f θ + z + nxθ  δuθ  ad θ
=0
∂θ
1



θ
x = x( )
Obviously, the equilibrium equations (G.6) are identical to the set (G.4). The set (G.7)
is the subject of the next section.
Boundary conditions
The set (G.7) is the complete set for the two independent displacements and states that,
per variation of each displacement over the surface S f , each of the internal stress
measures (two stress resultants) must be balanced by aligned external stress measures.
If u is prescribed over the surface Su , on which in consequence the virtual
displacement δu vanishes, each displacement must be equal to the prescribed
displacement at that surface. Hence, at each edge either the stress resultant or the
corresponding displacement must be equal to the known edge force or prescribed edge
displacement. So, for the edges x = constant the boundary conditions are
f x = − nxx
fθ +
∂f z
= − n xθ
∂θ
or u x = u x 


or uθ = uθ 

x=x
(1)
f x = nxx
and
fθ +
∂f z
= n xθ
∂θ
or u x = u x 

( 2)
 x=x
or uθ = uθ 

where the tilde indicates the prescribed edge displacement.
The differential equations for the displacements
Up to this point, no additional simplifications or assumptions have been introduced. To
obtain convenient differential equations for the displacements, it is assumed that the
parameters describing the material properties and the cross-sectional geometry, i.e.
E , ν and a, t respectively, are constant for the whole circular cylindrical shell.
Substitution of the kinematical relation (G.1) into the constitutive relation (G.2)
results in what is sometimes referred to as the “elastic law”, which reads
∂u x
∂x
 1 ∂u x ∂uθ 
nxθ = Ds 
+

 a ∂θ ∂x 
 1 ∂ 3uθ 1 ∂uθ 
mθθ = Db  2
+ 2

3
a ∂θ 
 a ∂θ
nxx = Dm (1 − υ2 )
208
(G.8)
Appendices
Substitution of this elastic law into (G.4) yields the following two differential equations
for the displacements
− (1 − υ2 )
∂ 2u x Ds  1 ∂ 2u x 1 ∂ 2uθ  p x
−
+

=
∂x 2 Dm  a 2 ∂θ2 a ∂x∂θ  Dm
(G.9)
2

D  1 ∂ 2u x ∂ 2uθ  Db 1 ∂ 2  ∂ 2
1 
∂pz 
− s
+
+ 1  uθ =
−

 pθ +

∂θ 
Dm  a ∂x∂θ ∂x 2  Dm a 4 ∂θ2  ∂θ2
D


m
The two differential equations are symbolically described by
 px 
L12  u x 
1 
=
−
∂p 
L22  uθ 
Dm  pθ + z 
∂θ 

The operators L11 up to and including L22 form a differential operator matrix, in which
 L11
L
 21
the operators are
∂2 1 − υ 1 ∂2
+
2 a 2 ∂θ2
∂x 2
2
1− υ 1 ∂
L12 = L21 =
2 a ∂x∂θ
L11 = (1 − υ2 )
L22 =

1 − υ ∂2
k ∂2  ∂2
+
+ 1
2
2
2 
2
2 ∂x
a ∂θ  ∂θ

2
Here the dimensionless parameter k is introduced, which is defined by
k=
Db
t2
=
Dm a 2 12a 2
(G.10)
Hence, it is noted that for a thin shell where t < a it follows that the parameter k is
negligibly small in comparison to unity ( k 1) .
The single differential equation
By eliminating u x from the two equations, the single differential equation for the
displacement uθ is obtained, which symbolically reads
( L11L22 − L21L12 ) uθ =
1 
∂p

L21 px − L11  pθ + z

Dm 
∂θ




This operation is only possible if the operators on a scalar function φ are commutative,
which means that for example ( L21L11 − L11L12 ) φ = 0 .
The single differential equation is then obtained as
2
 ∂4

 
k
∂2
1 ∂2  ∂2  ∂2
 4+ 2
 2 (1 + υ) 2 + 2 2  2  2 + 1  uθ
∂x
∂x
a ∂θ  ∂θ  ∂θ
a (1 − υ2 ) 
 

 1 ∂2

1
∂2
1 ∂2 
∂p
=
px −  2 (1 + υ) 2 + 2 2   pθ + z
2 
∂
x
a
∂θ
∂θ
Dm (1 − υ )  a ∂x∂θ






(G.11)
209
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
To facilitate comparison between the solutions presented herein, it is preferred to solve
the homogeneous equation for the displacement u z as then all quantities for the semimembrane concept can be described similar to those resulting from the solution to the
Morley-Koiter equation. This can be easily accomplished by noting that, as a result of
the simplification that εθθ is equal to zero, the relation u z = −
∂uθ
holds. Hence, by
∂θ
taking the derivative of (G.11) with respect to θ and by rearranging the resulting
equation, the single differential equation for u z becomes
2
2
  β 4 ∂ 4
 
1
∂2  ∂2  ∂2
2 ∂
4 
+ 8  2 (1 + υ ) a
+
+ 1  u z


4
∂x 2 ∂θ2  ∂θ2  ∂θ2
a 
  a  ∂x
 
3
3
3

1 
1 ∂
1 ∂   ∂pz
 1 ∂
px 
=
+ pθ  − 3
 2 (1 + υ ) 2 2 + 4 3  
2
Db 
a ∂x ∂θ a ∂θ   ∂θ
 a ∂x∂θ

Here the dimensionless parameter β is introduced, which is defined by
4β4 =
1 − υ2
a
= 12 (1 − υ2 )  
k
t
(G.12)
2
(G.13)
Load and solution as infinite trigonometric series
As stated in chapter 1, the semi-membrane concept exactly describes the ring-bending
behaviour and can only be applied to self-balancing modes. The modes indicated by
n = 2,3,4,... are generally known as these self-balancing modes.
The following considerations are derived for a load that is symmetric to a certain
axis, but can easily be extended to an asymmetric load by describing combinations of
sine and cosine series per load term. These can be treated separately with congruent
resulting expressions, whereby the choice of a symmetric load does not degenerate the
generality of the approach.
As stated in the introduction, the cylinder should be sufficiently long compared
with its radius and the boundary effects should mainly influence the more distant
material. Then the behaviour described above is excellently described by the
differential equation resulting from the semi-membrane concept, where for the mode
numbers n > 1 all quantities can be expressed as functions of the type
φ ( x, θ ) = φn ( x ) cos nθ and φ ( x, θ ) = φn ( x ) sin nθ depending on the axis of symmetry of the
quantity under consideration. Hence, the substitutions similar to those as presented for
the solution of Morley-Koiter equation in subsection 4.4.5 can be made. Hence, the
following substitutions for the loads and displacements can be made
p x ( x, θ ) = pxn ( x ) cos nθ
;
u x ( x, θ ) = u xn ( x ) cos nθ
pθ ( x, θ ) = pθn ( x ) sin nθ
;
uθ ( x, θ ) = uθn ( x ) sin nθ
p z ( x, θ ) = p zn ( x ) cos nθ
;
u z ( x, θ ) = u zn ( x ) cos nθ
(G.14)
while for the derivates with respect to the circumferential coordinate θ substitutions
can be made of the form
∂φ ( x, θ )
∂ cos nθ
= φn ( x )
= − nφn ( x ) sin nθ
∂θ
∂θ
210
Appendices
for quantities generally described by φ ( x, θ ) = φn ( x ) cos nθ and similarly for the
quantities generally described by φ ( x, θ ) = φn ( x ) sin nθ .
Reduced single differential equation
By substitution of the load and displacement functions (G.14), the single differential
equation (G.12) becomes an ordinary differential equation and by omitting the cosine
function for the circumferential distribution, the governing differential equation is
reduced to
2
  β 4 d 4 n 2 
d 2 n 2  n 2 − 1  
4 
 u zn ( x )
−
2
1
+
υ
−
(
)



4
a2 
dx 2 a 2  a 2  
  a  dx

2
2
4
2
1 
n ∂
n 
1
 n dpxn ( x ) 
=
 −2 (1 + υ ) 2 2 + 4   pzn ( x ) − pθn ( x )  + 3

Db 
a ∂x
a 
n
dx 
 a
(G.15)
Homogeneous solution
The general solution to a differential equation consists of a homogeneous and an
inhomogeneous part. By inspecting the differential equation (G.15), it is observed that
the homogeneous part cannot be separated in a polynomial part and a non-polynomial
part.
The homogeneous equation is given by
2
2
 4
2 n2 − 1
) d 2 + n4 ( n2 − 1)  u ( x ) = 0
 d − 1+ υ n (
zn
 dx 4
2 a2
β4
dx 2 a 4 4β 4 


By introducing the following trial solution similar to the substitution as performed in
subsection 4.5.2
u z ( x, θ ) = u zn ( x ) cos nθ = Cne
rn β
x
a
cos nθ
the characteristic equation reads
2
4
n2 − 1  n n2 − 1  2  n n2 − 1 
 r +
 =0
4r − 2 (1 + υ ) 2 

 β2

β  β 2



4
The solution to the homogeneous equation is given by (see also Appendix H)
u zn ( x ) = e
− anSMC β
+e
x
a
anSMC β
 n
 SMC x 
 SMC x  
n
C1 cos  bn β a  + C2 sin  bn β a  





x
a
 n
 SMC x 
 SMC x  
n
C3 cos  bn β a  + C4 sin  bn β a  





(G.16)
where the dimensionless parameters anSMC and bnSMC are defined by
1
anSMC = ηn 1 + γ nSMC
2
,
1
bnSMC = ηn 1 − γ nSMC
2
in which
γ nSMC = 12 (1 + υ) ( n 2 − 1) β−2
1
, ηn = n ( n 2 − 1) 2 β −2
211
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
For small values of γ SMC
and ηn , the following approximate expressions are obtained
n
1  1

anSMC = ηn  1 + γ nSMC 
2  2

1  1

bnSMC = ηn  1 − γ nSMC 
2  2

,
(G.17)
The homogeneous solution for the displacement uθ can be obtained by solving the
∂uθ
, which resulted from the initial assumption that, to simplify the
∂θ
initial kinematical equations, the circumferential strain εθθ is equal to zero.
relation u z = −
The homogeneous solution for the displacement u x can be obtained by solving the
second equation of the set (G.9) for which the homogeneous equation read
2

∂ 2u 1 + υ 1 ∂ 2  ∂ 2
1 ∂ 2u x
= − 2θ − 4 2 2  2 + 1 uθ
a ∂x∂θ
∂x
2β a ∂θ  ∂θ

(G.18)
By substitution of the displacement functions given above, this becomes an ordinary
differential equation in which the sine function (for uθ ) and the cosine function (for u x )
can be omitted and hence the following equations are obtained
1
uθn ( x ) = − u zn ( x )
n
2
a duθn ( x ) 1 + υ n ( n − 1)
u xn ( x ) =
−
2 a β4
n dx
2
∫ u ( x ) dx
θn
2
a du ( x ) 1 + υ ( n − 1) 1
u zn ( x ) dx
= − 2 zn
+
n
dx
2
a∫
β4
2
Inhomogeneous solution
Assuming linear loads px , pθ and pz , the solution to the inhomogeneous equation of
(G.19) becomes
1  a2  
1
a dpxn ( x ) 
u zn ( x ) =

 2   pzn ( x ) − pθn ( x ) + 2
Db  n − 1  
n
n
dx 
2
by omitting all second and higher derivatives with respect to x .
Similar to the homogeneous solution, the inhomogeneous solution for the
displacement uθ can be obtained by solving the relation u z = −
∂uθ
∂θ
and the
homogeneous solution for the displacement u x can be obtained by solving the first
equation of the set (G.9). If the second derivative with respect x is omitted, the latter
differential equation becomes
1 ∂ 2u x
p x 1 ∂ 2u θ
=
−
−
a 2 ∂θ2
Ds a ∂x∂θ
212
Appendices
and by substituting the displacement and load functions given above, the equations can
be rewritten and omitting the cosine and sine terms, the equations become
1
uθn ( x ) = − u zn ( x )
n
1 a2
a duθn ( x ) 1 a 2
a du ( x )
u xn ( x ) =
pxn ( x ) +
=
pxn ( x ) − 2 zn
2
Ds n
n dx
Ds n 2
n
dx
By substituting of the solution u zn ( x ) to the inhomogeneous equation, the
inhomogeneous solution for the displacements becomes
1  a2  
1
a dpxn ( x ) 


  pzn ( x ) − pθn ( x ) + 2
Db  n 2 − 1  
n
n
dx 
2
u zn ( x ) =
1 1  a2  
1
a dpxn ( x ) 


  pzn ( x ) − pθn ( x ) + 2
Db n  n 2 − 1  
n
n
dx 
2
u θn ( x ) = −
(G.20)
1 a2
a 1 a  a 2   dpzn ( x ) 1 dpθn ( x ) 
u xn ( x ) =
pxn ( x ) − 2
−


 
2
Ds n
n Db n 2  n 2 − 1   dx
n dx 
2
where the second derivative of the loads have been omitted for the assumed linear
loads.
Complete solution
Describing the loads px , pθ and pz by the forms
p xn ( x ) = p xn
pθn ( x ) = pθ( 2n)
p zn ( x ) = pzn( 2)
x
+ pθ(1n)
l
x
+ pzn(1)
l
the (approximated) complete solution for the independent displacement u z reads
x
 − anSMC β a  n
 SMC x 
 SMC x  
n
u z ( x, θ ) = cos nθ e
C1 cos  bn β a  + C2 sin  bn β a  






+e
anSMC β
x
a
 n
 SMC x 
 SMC x   
n
C3 cos  bn β a  + C4 sin  bn β a   



  

(G.21)
2
1  a2 
 ( 2) 1 ( 2)  x
(1) 1 (1) 
 2  cos nθ  pzn − pθn  + pzn − pθn 
Db  n − 1 
n
n
l


Similar expressions for the independent displacements uθ and u x are obtained by the
+
appropriate substitutions.
By substitution of the expressions for the independent displacements into the
expressions (G.8), the complete solution for all nontrivial quantities can be obtained,
which are given in Appendix I.
213
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Remark considering the accuracy
As implicated by the simplifications, the influence of the part of the solution described
by the short influence length is neglected in comparison to the part described by the
long influence length. Consequently, the small terms of the dimensionless parameters
anSMC and bnSMC as presented by (G.17) are identified as superfluous and discarded to not
suggest an accuracy that it not described.
If only the leading terms are retained (i.e. by neglecting β−2 in comparison to
unity), the dimensionless parameters anSMC and bnSMC become equal to
1
ηn . If only the
2
loading normal to the shell surface is considered, i.e. pz + ∫ pθd θ = q ( x ) and px = 0 , the
full solution is then described by
u zn ( x ) = e
1
x
− ηn β
2
a
1
+e 2
ηn β
x
x 
 n
1
1
n
C1 cos  2 ηnβ a  + C2 sin  2 ηnβ a  





x
a
 n
x
x 
1
1
n
C3 cos  2 ηnβ a  + C4 sin  2 ηnβ a  





2
+
1  a2 

 q( x)
Db  n 2 − 1 
and similarly for all other quantities the same approximation can be adopted.
Hence, it is readily verified that this approximated solution would be the exact
solution to the following homogenous differential equation
4
 n2 − 1 
Eta 2 d u zn ( x )
+ Db  2  u zn ( x ) = q ( x )
4
4
n
dx
 a 
2
which is the corresponding approximation of differential equation (G.15).
The above differential equation is similar to the one for a beam on an elastic
2
 n2 − 1 
foundation if the modulus of subgrade is taken as Db  2  and the flexural rigidity
 a 
2
Eta
of the beam is described by 4 . Hence, it is observed that the circular cylinder under
n
non-axisymmetric loading behaves as a curved membrane that is elastically supported
by the so-called ring bending action.
214
Appendices
Appendix H
Solution to
equations
MK
and
SMC
Introduction
This appendix provides the exact homogeneous solution to the Morley-Koiter (MK)
and semi-membrane concept (SMC) differential equations and the general expressions
for all quantities by back substitution. The back substitution and the resulting
expressions are separately provided in Appendix I.
Exact homogeneous solution to the Morley-Koiter equation for
n >1
The third equation of (4.18) is the well-known Morley-Koiter equation. The
homogeneous differential equation is given by
2
4
4
 
1 
β ∂ 
∆∆
∆
+
+
4
u =0
 
 
2 
4 z
a 
 a  ∂x 
 
in which
∆=
∂2
1 ∂2
a
+ 2 2 , β = 4 3 (1 − υ2 )
2
∂x
a ∂θ
t
By substituting, for a closed circular cylindrical shell, the periodic trial function
u z ( x, θ ) = Fn ( x ) cos nθ + Gn ( x ) sin nθ
the differential equation becomes
4
 d 2 n2   d 2 n2 1 
 β  d Fn ( x )
−
−
+
F
x
cos
n
θ
+
4
cos nθ = 0
(
)
 2
 
 n
 
a 2   dx 2 a 2 a 2 
dx 4
a
 dx
and a similar equation for the functions Gn ( x ) .
2
2
4
By assuming an exponential function in the axial direction, according to
Fn ( x ) = e
β
r x
a
the characteristic equation is obtained
2
2
 2 n2   2 n2 − 1 
4
 r − 2   r − 2  + 4r = 0
β
β

 

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Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
This yields two algebraic equations of second degree in r 2
 2 n 2  2 n 2 − 1 
2
 r − 2  r − 2  = ±2ir
β
β



 2 1 
 n − 2  2 n 2 ( n 2 − 1)
r4 − 2
± i r +
=0
2
β4
 β



By introducing the dimensionless parameters
1

γ =  n2 −  β−2
2

1
, η = n ( n2 − 1) 2 β−2
the two characteristic equations become
r 4 − 2 ( γ ± i ) r 2 + η2 = 0
The four roots of the rapid attenuating boundary layers (short-wave solution) read
r(1,2) = a1 ± ib1
r(3,4) = − ( a1 ± ib1 )
where
1
2
1
b1 =
2
a1 =
α12 + β12 + α1
α12 + β12 − α1
in which
α1 = γ + ρ1
β1 = 1 + ρ2
The four roots of the gradual attenuating boundary layers (long-wave solution) read
r(3,4) = a2 ± ib2
r( 7,8) = − ( a2 ± ib2 )
where
1
2
1
b2 =
2
a2 =
in which
α 2 = γ − ρ1
β2 = 1 − ρ2
216
α 2 2 + β2 2 + α 2
α 2 2 + β2 2 − α 2
Appendices
In which for all eight roots the following parameters are used
1
2
1
ρ2 =
2
ρ1 =
ξ2 + ζ 2 + ξ
ξ2 + ζ 2 − ξ
where
ξ = ( γ 2 − η2 − 1)
ζ = 2γ
The exact solution is thus described by
u z ( x, θ ) = Fn ( x ) cos nθ
in which
Fn ( x ) = e
− a1β
+e
x
a
− a2 β
x
x
x 


 x   a1β a 
 x

C1 cos  b1β a  + C2 sin  b1β a   + e C3 cos  b1β a  + C4 sin  b1β a  










x
a
x

x
x   a2 β a 
x
x 




C5 cos  b2β a  + C6 sin  b2β a   + e
C7 cos  b2β a  + C8 sin  b2β a  










By using some alternative parameters, the four expressions can be rewritten to
1
1

2
ω1 − 1 
1 
1
2
γ
+
ω
+
ω
+
1
+
γ
2
ω
−
1
+
2
ω
+
1
+
γ
+
a1 =
(
)
(
)
(
)
1
2
1
2


2
2 
2 



1
1

2
1  2 1
ω1 − 1 
2
b1 =
 γ + ( ω1 + ω2 ) + 1 + γ 2 ( ω1 − 1) + 2 ( ω2 + 1)  − γ −
2
2 
2 



1
1

2
1  2 1
ω1 − 1 
2
γ
+
ω
+
ω
+
1
−
γ
2
ω
−
1
−
2
ω
+
1
+
γ
−
a2 =
(
)
(
)
(
)
1
2
1
2


2
2 
2 



1
1

2
1  2 1
ω1 − 1 
2
b2 =
 γ + ( ω1 + ω2 ) + 1 − γ 2 ( ω1 − 1) − 2 ( ω2 + 1)  − γ +
2
2 
2 



where
ω1 = ω + γ 2 − η2
ω2 = ω − γ 2 + η2
in which
ω = ξ 2 + ζ 2 = 1 + 2 ( γ 2 + η2 ) + ( γ 2 − η2 )
So
2
ω1 + ω2
=ω
2
217
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Exact homogeneous solution to the Morley-Koiter equation for
n = 0 and n = 1
As explained in subsection 4.4.5.2, the exact homogeneous solution for n = 0 and n = 1
is described by
uz ( x ) = e
− a1β
x
a

x
x   a1β a 
x
x 




C1 cos  b1β a  + C2 sin  b1β a   + e C3 cos  b1β a  + C4 sin  b1β a  











x
in which
1
1

2
a1 = (1 + γ 2 ) 2 + γ 


1
1

2
b1 = (1 + γ 2 ) 2 − γ 


in which
γ=−
γ=
1
2β 2
for n = 0 , and
1
2β 2
for n = 1 .
Exact homogeneous solution to the SMC equation
The differential equation resulting from the semi-membrane concept is presented in
Appendix G by equation (G.12). The homogeneous differential equation is given by
2
2
  β 4 ∂ 4
 
1
∂2  ∂2  ∂2
2 ∂
4 
 uz = 0
+
2
1
+
υ
a
+
+
1
(
)




∂x 2 ∂θ2  ∂θ2  ∂θ2
a ∂x 4 a8 
 
  
By substituting, for a closed circular cylindrical shell, the periodic trial function
u z ( x, θ ) = Fn ( x ) cos nθ + Gn ( x ) sin nθ
the differential equation becomes
2
4
4
 n2 
d 2 n 2  n 2 − 1  
 β  d Fn ( x )
 − 2  2 (1 + υ ) 2 − 2  2   Fn ( x ) cos nθ + 4  
cos nθ = 0
dx
a  a  
dx 4
a
 a 

and a similar equation for the functions Gn ( x ) .
By assuming an exponential function in the axial direction, according to
Fn ( x ) = e
β
r x
a
the characteristic equation is obtained
2
4
n2 − 1  n n2 − 1  2  n n2 − 1 
 r +
 =0
4r − 2 (1 + υ ) 2 

 β2

β  β 2



4
By introducing the dimensionless parameters
γ = 12 (1 + υ ) ( n 2 − 1) β −2
218
1
, η = n ( n 2 − 1) 2 β−2
Appendices
the characteristic equation becomes
1
r 4 − γη2 r 2 + η4 = 0
4
The four roots, representing the gradual attenuating boundary layers (long-wave
solution), read
r(1,2) = a ± ib
r(3,4) = − ( a ± ib )
where
1
a= η
2
1
b= η
2
α 2 + β2 + α
α 2 + β2 − α
in which
α=γ
β = 1 − γ2
and hence α 2 + β2 = 1 which gives
1
a = η 1+ γ
2
1
b = η 1− γ
2
General homogeneous solution to the Morley-Koiter equation
The general homogenous solution to the Morley-Koiter equation for n > 1 is provided.
All relevant quantities are related to the displacement u z by adopting the expressions
as presented in chapter 4. For n = 0 and n = 1 , the general homogeneous solutions for
all quantities are a reduction of the solutions for n > 1 , which are therefore not
explicitly presented here, but which have been obtained by adopting a similar approach
as followed below.
A general representation of the homogeneous solution for the displacement u z is
described by (4.35), which in a slightly rewritten form reads
1 a 
uz = −  
4β 
4

2
1 
u z ( x, θ ) dxdxdxdx
2 

∫∫∫∫ ∆∆  ∆ + a
As shown in subsection 4.4.5.2, this solution can be substituted into the first two
equations of the set (4.18) resulting in the following expressions
2
2


1 
1 


∂  ∆ + 2  uz
∂3  ∆ + 2  uz


1 a 
1
a 
a 

uθ = 2   ( 2 + υ ) ∫∫ 
dxdx + 2 ∫∫∫∫ 
dxdxdxdx

4a  β  
∂θ
a
∂θ3
4
2


1 

∂ 2  ∆ + 2  uz

2

1 1a  
1 
1
a 


ux =
υ
∆
+
u
dx
−
dxdxdx
 

 z
4 a  β   ∫ 
a2 
a 2 ∫∫∫
∂θ2

4
219
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The rotation ϕ x is described by equation (4.12) and reads
ϕx = −
∂u z
∂x
The normal stress resultants and the longitudinal shearing stress resultant are described
by equation (4.13) and read
1 ∂uθ
u 
 ∂u
nxx = Dm  x + υ
+υ z 
a ∂θ
a
 ∂x
 ∂u 1 ∂uθ u z 
+ 
nθθ = Dm  υ x +
 ∂x a ∂θ a 
1 − υ  1 ∂u x ∂uθ 
+
nxθ = Dm


2  a ∂θ ∂x 
Upon substitution of the expressions for the displacements above, these expressions
read
2
1 

∂ 2  ∆ + 2  uz
a
nxx = − Db a ∫∫  2 2 
dxdx
a ∂θ
2
1 

nθθ = − Db a  ∆ + 2  u z
a 

2
1 

∂  ∆ + 2  uz
a 
nxθ = Db a ∫ 
dx
a∂θ
The stress couples are described by equation (4.13) and read
 ∂ 2u
1 ∂ 2u z
u 
mxx = − Db  2z + υ 2
+ υ 2z 
2
a ∂θ
a 
 ∂x
 ∂ 2u
1 ∂ 2u z u z 
mθθ = − Db  υ 2z + 2
+ 
a ∂θ2 a 2 
 ∂x
 1 ∂uθ 1 ∂ 2u z 
mxθ = − Db (1 − υ)  −
+

 a ∂x a ∂x∂θ 
Upon substitution of the expression for uθ above, the expression for mxθ reads
2
2


1 
1 


∂  ∆ + 2  uz
∂3  ∆ + 2  uz


1 a
1
a 
a 

mxθ = Db (1 − υ ) 3   ( 2 + υ ) ∫ 
dx + 2 ∫∫∫ 
dxdxdx

4a  β  
∂θ
a
∂θ3
2
1 ∂ uz
− Db (1 − υ)
a ∂x∂θ
4
The transverse shearing stress resultants are described by equation (4.9) and read
∂mxx 1 ∂mxθ
+
∂x
a ∂θ
1 ∂mθθ ∂mxθ
vθ =
+
a ∂θ
∂x
vx =
220
Appendices
Upon substitution of the expressions for the stress couples above, these expressions
read
2
2


1 
1 
2
4
∂
∆
+
∂
∆
+
u
uz


z


2 
2 
1 a
1
a 
a 
vx = Db (1 − υ) 4   ( 2 + υ) ∫ 
dx + 2 ∫∫∫ 
dxdxdx 
2
4

4a  β  
∂θ
a
∂θ
 ∂ 3u
1 ∂ 3u z
1 ∂u 
+ Db  − 3z − 2
−υ 2 z 
2
∂
x
a
∂
x
∂θ
a
∂x 

4
2
2


1 
1 


∂  ∆ + 2  uz
∂3  ∆ + 2  uz


1 a
1
a 
a 

+ 2 ∫∫ 
vθ = Db (1 − υ ) 3   ( 2 + υ ) 
dxdx
4a  β  
∂θ
a
∂θ3

4
1  ∂ 3u
1 ∂ 3u z 1 ∂u z 
+ Db  − 2 z − 2
−

a  ∂x ∂θ a ∂θ3 a 2 ∂θ 
The combined internal stress resultant v∗x is described by equation (4.11) and reads
v∗x =
∂mxx 2 ∂mxθ
+
∂x
a ∂θ
Upon substitution of the expressions for the stress couples above, this expression reads
2
2


1 
1 


∂ 2  ∆ + 2  uz
∂ 4  ∆ + 2  uz




1
a
1
a
a




v∗x = Db (1 − υ) 4   ( 2 + υ ) ∫ 
dx
+
dxdxdx

2a  β  
∂θ2
a 2 ∫∫∫
∂θ4
 ∂ 3u 2 − υ ∂ 3u z
1 ∂u 
+ Db  − 3z − 2
−υ 2 z 
2
∂
∂
∂θ
∂x 
x
a
x
a

4
General homogeneous solution to the SMC equation
The general homogenous solution to the SMC equation for n > 1 is provided. All
relevant quantities are related to the displacement u z by adopting the expressions as
presented in Appendix G.
The homogeneous solution is presented by expression (G.16), which reads
x
x
x 
 − a2β 


u z = cos nθ e a C1 cos  b2β  + C2 sin  b2β  
a
a






+e
a2 β
x
a

x
x   


C3 cos  b2β a  + C4 sin  b2β a   



  

in which
1
a2 = η 1 + γ
2
1
, b2 = η 1 − γ
2
where
1
η = n ( n 2 − 1) 2 β−2 , γ = 12 (1 + υ) ( n 2 − 1) β−2
221
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Resulting form the initial assumption that the circumferential strain εθθ is equal to zero,
the homogeneous solution for the displacement uθ is obtained by
uθ = − ∫ u z d θ
The displacement u x can be obtained by equation (G.18), which is rewritten to
2

∂u
1+ υ 1
∂  ∂2
u x = − ∫ θ ad θ −
 2 + 1 uθdx
4 ∫
∂x
2 aβ ∂θ  ∂θ

Upon substitution of the expressions for the displacement uθ above, this expression
reads
2

1 + υ 1  ∂2
∂u
u x = ∫∫ z ad θd θ +
+ 1 u z dx
4 ∫
2
2 aβ  ∂θ
∂x

The normal stress resultant, the longitudinal shearing stress resultant and the stress
couple are described by equation (G.8) and read
∂u x
∂x
 1 ∂u x ∂uθ 
nxθ = Ds 
+

 a ∂θ ∂x 
 1 ∂ 3uθ 1 ∂uθ 
mθθ = Db  2
+ 2

3
a ∂θ 
 a ∂θ
nxx = Dm (1 − υ2 )
Upon substitution of the expressions for the displacements above, these expressions
read
2
 ∂ 2u
 
1 + υ 1  ∂2
nxx = Dm (1 − υ2 )  ∫∫ 2z ad θd θ +
+
1

 u z 

2 aβ4  ∂θ2
∂x
 

2
1+ υ 1


∂  ∂2
1
nxθ = Ds 
+
u z dx 


2 4 ∫
2
 2 a β ∂θ  ∂θ




mθθ = − Db

1  ∂ 2u z
+ uz 

a 2  ∂θ2

The transverse shearing stress resultant is described by equation (G.5) and reads
vθ =
1 ∂mθθ
a ∂θ
Upon substitution of the expressions for the stress couples above, these expressions
read
vθ = − Db
222
1  ∂ 3u z ∂u z 
+


a 3  ∂θ3
∂θ 
Appendices
Appendix I
Back substitution for MK and SMC
solutions
Homogeneous solution to the Morley-Koiter equation
The expressions for all quantities, which are obtained by back substitution of the exact
homogeneous solution to the Morley-Koiter (MK) differential equation, are provided
for n > 1 . For n = 0 and n = 1 , the expressions obtained by back substitution are a
reduction of the expressions for n > 1 , which are therefore not explicitly presented here,
but which have been obtained by adopting a similar approach as followed below.
A complete representation of the derivation of all expressions is surplus to
requirements in view of the number of quantities and number of terms involved. For a
more elaborate discussion of the successive substitutions, reference is made to the back
substitution of the homogeneous solution to the SMC equation hereafter.
The expressions for the back substitution are extracted from Appendix H. For a
specific inhomogeneous solution, the expressions for all quantities are also provided.
The solution to the differential equation is thus described by
u z ( x, θ ) = Fn ( x ) cos nθ
in which
x
 − a1β x 
x

 x   a1β 
 x
 x 
Fn ( x ) = cos nθ e a C1 cos  b1β  + C2 sin  b1β   + e a C3 cos  b1β  + C4 sin  b1β  
 a
 a 
 a
 a 



+e
− a2 β
x
a
x

x
x   a2 β a 
x
x   




β
+
β
+
C
cos
b
C
sin
b
e
C7 cos  b2β  + C8 sin  b2β   
6
 2 
 2 
 5

a
a 
a
a   






where the approximated parameters (refer to the subsection 4.5.1) read
1
a1 = 1 + γ
2
,
1  1 
b2 = η 1 + γ 
2  2 
1
b1 = 1 − γ
2
,
1  1 
b2 = η 1 + γ 
2  2 
in which
1
η = n ( n 2 − 1) 2 β −2 , γ = ( n 2 − 12 ) β−2
By back substituting the expression for u z ( x, θ ) , the expressions for all quantities can
be derived while introducing the appropriate approximations, viz. neglecting the small
terms in comparison with unity as γ 2 ≈ η2 1 . Without further elaboration, the result is
provided below for all relevant quantities, which is obtained by same procedure as
described for the SMC solution hereafter.
223
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
For the displacement u x , the substitution yields
ux =
 − a1β x 
1
x 
 x

12
12
11
cos nθ e a {u11
x C1 − u x C2 } cos  b1β
 + {u x C1 + u x C2 } sin  b1β  
4β
 a
 a 


+e
+e
+e
a1β
x
a
− a2 β
a2 β
x
a
 11
 x
 x 
12
12
11
{−u x C3 − u x C4 } cos  b1β a  + {u x C3 − u x C4 } sin  b1β a  





x
a
 21
x
x 


22
22
21
{u x C5 − u x C6 } cos  b2β a  + {u x C5 + u x C6 } sin  b2β a  






x
x   


21
22
22
21
{−u x C7 − u x C8 } cos  b2β a  + {u x C7 − u x C8 } sin  b2β a   



  

 n2  1
 3
n2 − 1 
n2 − 1  
1
+
γ
−
−
2
υ
1
−
γ
+
u11




x = −
2
2
β
β 
β 2  
 2
  2
 n n2 − 1  3
n2 
n2 − 1 
u x21 =  2
 1 − γ + 2  − υ 2 η 
2
β 
n
n
 2


 n2  1
 3
n2 − 1 
n2 − 1  
u12
=
−
1
−
γ
+
+
2
υ
1
+
γ
−





x
 2
β2 
β2  
 2
β  2
For the displacement uθ , the substitution yields
 n n2 − 1  3
n2 
n2 − 1 
u x22 =  2
1
+
γ
−
+
υ
η 


n2  2
n2
β2 


uθ =
 − a1β x 
1
 x
 x 
sin nθ e a {uθ11C1 − uθ12C2 } cos  b1β  + {uθ12C1 + uθ11C2 } sin  b1β  
2n
 a
 a 


+e
+e
+e
n
uθ11 =  
β
2
a1β
x
a
− a2 β
a2 β
x
a
 11
x
x 


12
12
11
{uθ C3 + uθ C4 } cos  b1β a  + {−uθ C3 + uθ C4 } sin  b1β a  





x
a
 21
x
x 


22
22
21
{uθ C5 − uθ C6 } cos  b2β a  + {uθ C5 + uθ C6 } sin  b2β a  





 21
x
x  


22
22
21
{uθ C7 + uθ C8} cos  b2β a  + {−uθ C7 + uθ C8 } sin  b2β a   



  

 1  n 2

n2 − 1  
   − ( 2 + υ)  γ − 2  

β  

 2 β 
2
n
uθ12 = − ( 2 + υ )  
β
For the rotation ϕ x , the substitution yields
ϕx =
 n2

n2 − 1
uθ22 = 2  2 − 2 γ  + ( 2 + υ ) 2
β
β

 − a1β x 
1β
 x
 x 
cos nθ e a {ϕ11x C1 − ϕ12x C2 } cos  b1β  + {ϕ12x C1 + ϕ11x C2 } sin  b1β  
2a
a


 a 


+e
+e
+e
224
uθ21 = −2
a1β
x
a
− a2 β
a2 β
x
a

 x
 x 
11
12
12
11
{−ϕ x C3 − ϕ x C4 } cos  b1β a  + {ϕx C3 + ϕ x C4 } sin  b1β a  





x
a
 21
x
x 


22
22
21
{ϕ x C5 − ϕ x C6 } cos  b2β a  + {ϕ x C5 + ϕ x C6 } sin  b2β a  






x
x   


21
22
22
21
{−ϕ x C7 − ϕ x C8 } cos  b2β a  + {ϕ x C7 − ϕ x C8 } sin  b2β a   



  

Appendices
 1 
ϕ21
γ
x = η1 +
 2 
 1 
ϕ22
γ
x = η1 −
 2 
ϕ11x = ( 2 + γ )
ϕ12x = ( 2 − γ )
For the stress resultant nxx , the substitution yields
nxx =
2
 − a1β ax  11
Et  n 
x 
 x

12
12
11
a
cos
n
θ
e
 
{nxxC1 − nxx C2 } cos  b1β a  + {nxx C1 + nxxC2 } sin  b1β a  
4β2  a 






+e
+e
+e
a1β
x
a
− a2 β
a2 β
x
a
 11
 x
 x 
12
12
11
{nxxC3 + nxx C4 } cos  b1β a  + {− nxx C3 + nxxC4 } sin  b1β a  





x
a
 21
x
x 


22
22
21
{nxx C5 − nxx C6 } cos  b2β a  + {nxx C5 + nxx C6 } sin  b2β a  





 21
x
x  


22
22
21
{nxx C7 + nxx C8} cos  b2β a  + {− nxx C7 + nxx C8 } sin  b2β a   



  


n2 − 1 
2
n11
=
γ
−


xx
β2 

nxx21 = 2
n2 − 1 
n2 
γ
−


n2 
β2 
n12
xx = 2
nxx22 = −2
n2 − 1
n2
For the stress resultant nθθ , the substitution yields
nθθ =
 − a1β x 
Et
x 
 x

12
12
11
cos nθ e a {n11
θθC1 − nθθC2 } cos  b1β
 + {nθθC1 + nθθC2 } sin  b1β  
a
a
a






+e
+e
+e
n =1
11
θθ
12
nθθ
=
n2 − 1
− 2γ
β2
a1β
x
a
− a2 β
a2 β
x
a
 11
x
x 


12
12
11
{nθθC3 + nθθC4 } cos  b1β a  + {− nθθC3 + nθθC4 } sin  b1β a  





x
a
 21
x 
x


22
21
22
{nθθC5 − nθθ C6 } cos  b2β a  + {nθθ C5 + nθθC6 } sin  b2β a  





 21
x
x  


22
22
21
{nθθC7 + nθθ C8} cos  b2β a  + {− nθθ C7 + nθθC8 } sin  b2β a   



  

1  n2 − 1 
n =−  2 
4 β 
2
21
θθ
22
nθθ
=0
225
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
For the stress resultant nxθ , the substitution yields
nx θ =
 − a1β x 
Et n
x
x 


12
12
11
sin nθ e a {n11
xθ S1 − nxθ S 2 } cos  b1β
 + {nxθ S1 + nxθ S 2 } sin  b1β  
4β a
 a
 a 


+e
+e
+e
a1β
x
a
− a2 β
a2 β
x
a
 11
x

 x 
12
12
11
{− nxθ S3 − nxθ S 4 } cos  b1β a  + {nxθ S3 − nxθ S 4 } sin  b1β a  





x
a
 21
x
x 


22
22
21
{nxθ S5 − nxθ S6 } cos  b2β a  + {nxθ S5 + nxθ S6 } sin  b2β a  






x
x   


21
22
22
21
{− nxθ S7 − nxθ S8 } cos  b2β a  + {nxθ S 7 − nxθ S8 } sin  b2β a   



  

 3
n2 − 1 
γ+ 2 
n11
xθ = −2  1 −
β 
 2
nx21θ =
n2 − 1  1
n2 
η1 + γ − 2 
2
n
β 
 2
 3
n2 − 1 
n2 − 1  1
n2 
n12
γ− 2 
nx22θ = − 2 η 1 − γ + 2 
xθ = 2  1 +
n
β 
β 
 2
 2
For the stress couple mxx , the substitution yields
2
 − a1β x 
β
 x
 x 
12
12
11
mxx = Db   cos nθ e a {m11
xx C1 − mxx C2 } cos  b1β
 + {mxx C1 + mxxC2 } sin  b1β  
a
 a
 a 


+e
+e
+e
a1β
x
a
− a2 β
a2 β
x
a
 11
 x
 x 
12
12
11
{mxxC3 + mxx C4 } cos  b1β a  + {− mxx C3 + mxxC4 } sin  b1β a  





x
a
x
x 
 21


22
22
21
{mxx C5 − mxx C6 } cos  b2β a  + {mxx C5 + mxx C6 } sin  b2β a  





x
x  
 21


22
22
21
{mxx C7 + mxx C8} cos  b2β a  + {− mxx C7 + mxx C8} sin  b2β a   



  


n2 − 1 
m11
=
−
2
γ
−
υ


xx
β2 

mxx21 = υ
m12
xx = −2
1
mxx22 = − η2
2
n2 − 1
β2
For the stress couple mθθ , the substitution yields
2
 − a1β x  11
β
 x
 x 
12
12
11
mθθ = Db   cos nθ e a {mθθ
C1 − mθθ
C2 } cos  b1β  + {mθθ
C1 + mθθ
C2 } sin  b1β  
a
 a
 a 


+e
+e
+e
226
a1β
x
a
− a2 β
a2 β
x
a
 11
 x
 x 
12
12
11
{mθθC3 + mθθC4 } cos  b1β a  + {− mθθC3 + mθθC4 } sin  b1β a  





x
a
 21
x
x 


22
22
21
{mθθC5 − mθθ C6 } cos  b2β a  + {mθθ C5 + mθθC6 } sin  b2β a  





 21
x
x  


22
22
21
{mθθC7 + mθθ C8} cos  b2β a  + {− mθθ C7 + mθθC8} sin  b2β a   



  

Appendices
 n2 − 1

11
mθθ
=  2 − 2υγ 
 β

12
mθθ = −2υ
21
mθθ
=
n2 − 1
β2
22
mθθ
=0
For the couple resultant mxθ , the substitution yields
mxθ = Db
 − a1β x 
1− υ β n
 x
 x 
12
12
11
sin nθ e a {m11
xθC1 − mxθC2 } cos  b1β
 + {mxθC1 + mxθC2 } sin  b1β  
2 aa
 a
 a 


+e
+e
+e
a1β
x
a
− a2 β
a2 β
x
a

x

 x 
11
12
12
11
{− mxθC3 − mxθC4 } cos  b1β a  + {mxθC3 − mxθC4 } sin  b1β a  





x
a
 21
x
x 


22
22
21
{mxθC5 − mxθ C6 } cos  b2β a  + {mxθ C5 + mxθC6 } sin  b2β a  






x
x   


21
22
22
21
{− mxθC7 − mxθ C8 } cos  b2β a  + {mxθ C7 − mxθC8 } sin  b2β a   



  


1
m11
xθ = −  2 + γ + ( 2 + υ ) 2 
β 

  1
2 + υ n2 − 1 1  1 
n2  
mx21θ = −  η 1 + γ −
− 2 η  2 − 3γ + 2 2  
2
2 
2
n β  2n 
β 
  2

1
m12
xθ = −  2 − γ − ( 2 + υ ) 2 
β 

  1
2 + υ n2 − 1 1  1 
n2  
mx22θ = −  η 1 − γ +
− 2 η  2 + 3γ − 2 2  
2
2 
2
n β  2n 
β 
  2
For the stress resultant vx , the substitution yields
3
x
 − a1β 
β
 x
 x 
12
12
11
vx = Db   cos nθ e a {v11
x C1 − vx C2 } cos  b1β
 + {vx C1 + vx C2 } sin  b1β  
a
 a
 a 


+e
+e
+e
a1β
x
a
− a2 β
a2 β
x
a
 11
 x
 x 
12
12
11
{−vx C3 − vx C4 } cos  b1β a  + {vx C3 − vx C4 } sin  b1β a  





x
a
 21
x
x 


21
22
22
{vx C5 − vx C6 } cos  b2β a  + {vx C5 + vx C6 } sin  b2β a  





 21
x
x   


22
22
21
{−vx C7 − vx C8 } cos  b2β a  + {vx C7 − vx C8 } sin  b2β a   



  


n2 − υ 
v11
=
−
2
−
3
γ
+


x
β2 

 n2 − υ η 1 − υ n n2 − 1 

vx21 = −  2
−
 β 2 β2
2β2 


n2 − υ 
v12

x =  2 + 3γ −
β2 

 n2 − υ η 1 − υ n n2 − 1 

vx22 = −  2
−
 β 2 β2
2β 2 

227
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
For the stress resultant vθ , the substitution yields
2
 − a1β x 
β n
 x
 x 
12
12
11
vθ = Db   sin nθ e a {v11
θ C1 − vθ C2 } cos  b1β
 + {vθ C1 + vθ C2 } sin  b1β  
a a
 a
 a 


+e
+e
+e
a1β
x
a
 11
 x
 x 
12
12
11
{vθ C3 + vθ C4 } cos  b1β a  + {−vθ C3 + vθ C4 } sin  b1β a  





− a2 β
a2 β
x
a
x
 21
x 


22
22
21
{vθ C5 − vθ C6 } cos  b2β a  + {vθ C5 + vθ C6 } sin  b2β a  





 21
x
x  


22
22
21
{vθ C7 + vθ C8} cos  b2β a  + {−vθ C7 + vθ C8} sin  b2β a   



  

x
a

n2 − 1
1− υ 
v11
+ ( 2 + υ) 2 
θ =  2γ −
2
β
β 

12
vθ = 2
vθ21 = −
n2 − 1
β2
vθ22 = 0
For the combined internal stress resultant v∗x , the substitution yields
 − a1β 
x
x 
β


v∗x = Db   cos nθ e a {vx∗11C1 − v∗x 12C2 } cos  b1β  + {v∗x 12C1 + v∗x 11C2 } sin  b1β  
a
 a
 a 


3
x
+e
+e
+e
a1β
x
a
− a2 β
a2 β
x
a
 ∗11
x

 x 
∗12
∗12
∗11
{−vx C3 − vx C4 } cos  b1β a  + {vx C3 − vx C4 } sin  b1β a  





x
a
 ∗21
x
x 


∗22
∗21
∗22
{vx C5 − vx C6 } cos  b2β a  + {vx C5 + vx C6 } sin  b2β a  





 ∗21
x
x   


∗22
∗22
∗21
{−vx C7 − vx C8 } cos  b2β a  + {vx C7 − vx C8 } sin  b2β a   



  


( 2 − υ) n 2 − υ 
v∗x 11 = −  2 − 3γ +

β2


 ( 2 − υ) n2 − υ η 1 − υ n n2 − 1 

−
vx∗21 = − 


2 β2
β2
β2



( 2 − υ) n2 − υ 
v∗x 12 =  2 + 3γ −

β2


 ( 2 − υ) n2 − υ η 1 − υ n n 2 − 1 

vx∗22 = − 
−


β2
2 β2
β2


Inhomogeneous solution to the Morley-Koiter equation
For a load, constant with respect to coordinate x and presented by pz ( x, θ ) = pzn cos nθ ,
the inhomogeneous solution is derived for all relevant quantities. For convenience, the
other load terms are assumed to be zero. For the sake of clarity, this rather simple load
case is considered, but this does degenerate the generality of the approach as it can
easily be extended to more involved load cases.
228
Appendices
The inhomogeneous solution can be obtained by omitting all derivatives with respect to
the axial coordinate x and by omitting the load terms px and pθ in the differential
equation (4.34). The single differential equation then reads
2
1 ∂4  ∂2
1 
1 ∂ 4 pz
n4
+ 2  uz =
=
pzn cos nθ
4
4  2
2
4
4
a ∂θ  a ∂θ a 
Db a ∂θ
Db a 4
The solution to this equation is
uz =
1
a4
pzn cos nθ
Db ( n 2 − 1)2
By substituting this result into the first equations of the set (4.18), the inhomogeneous
solution for the circumferential displacement uθ is obtained, which becomes
uθ = − ∫ u z d θ = −
1
a4
pzn
sin nθ
Db ( n 2 − 1)2 n
The other nontrivial solutions (refer to expressions (4.9) and (4.13)) are
 1 ∂ 2u z u z 
a2
mθθ = − Db  2
+
=
pzn cos nθ

2
a2  n2 − 1
 a ∂θ
mxx = υmθθ
vθ =
1 ∂mθθ
na
=− 2
pzn sin nθ
a ∂θ
n −1
which identically satisfies the equilibrium equations (4.8).
Homogeneous solution to the SMC equation
The expressions for all quantities, which are obtained by back substitution of the exact
homogeneous solution to the SMC differential equation, are provided for n > 1 .
The expressions for the back substitution are extracted from Appendix H. For a
specific inhomogeneous solution, the expressions for all quantities are also provided.
The solution to the differential equation is thus described by
u z ( x, θ ) = Fn ( x ) cos nθ
in which
x
x
x 
 − a2β 


Fn ( x ) = e a C1 cos  b2β  + C2 sin  b2β  
a
a






+e
a2 β
x
a

x
x   


C3 cos  b2β a  + C4 sin  b2β a   



  

where the approximated parameters (refer to the expression (G.17)) read
1  1 
1  1 
a2 = η 1 + γ  , b2 = η 1 − γ 
2  2 
2  2 
in which
1
η = n ( n 2 − 1) 2 β−2 , γ = 12 (1 + υ) ( n 2 − 1) β−2
229
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
The number of derivatives and integrals of Fn ( x ) with respect to the coordinate x that
need to be evaluated are limited in case of the SMC solution and become
dFn ( x )
dx
β  − a2β a 
x
x 


e
( − a2C1 + b2C2 ) cos  b2β a  + ( −b2C1 − a2C2 ) sin  b2β a  
a 





x
=
+e
d 2 Fn ( x )
dx
2
a2 β
x
a

x
x   


( a2C3 + b2C4 ) cos  b2β a  + ( −b2C3 + a2C4 ) sin  b2β a   



  

2
x
 β   − a2β 
=   e a 
 a  

{(( a ) − (b ) ) C + ( −2a b ) C }cos  b β ax 
2
2
2
2
1
{
2 2
2
2
) }
x 
2
2

+ ( 2a2b2 ) C1 + ( a2 ) − ( b2 ) C2 sin  b2β 
a 

+e
a2 β
x
a
(
{(( a ) − (b ) ) C + ( 2a b ) C }cos  b β ax 



2
2
2
2
{
3
2 2
4
2
) }
x  
2
2

+ ( −2a2b2 ) C3 + ( a2 ) − ( b2 ) C4 sin  b2β  
a

 
1
∫ F ( x ) dx = ( a ) + ( b )
n
2
2
2
2
(
x
a  − a2β a 
x
x 


⋅ e
( − a2C1 − b2C2 ) cos  b2β a  + ( b2C1 − a2C2 ) sin  b2β a  
β 





+e
a2 β
x
a

x
x   


( a2C1 − b2C2 ) cos  b2β a  + ( b2C1 + a2C2 ) sin  b2β a   



  

However, introducing the approximated parameters and neglecting the small terms in
comparison with unity, viz. γ 2 ≈ η2 1 , the congruent approximation of the derivatives
and integrals reads
dFn ( x )
dx
η β  − a2β a 
x
x 


e
{−α1C1 + α 2C2 } cos  b2β a  + {−α 2C1 − α1C2 } sin  b2β a  
2 a 





x
≈
+e
d 2 Fn ( x )
dx 2
≈
a2 β
x
a
x
x   



{α1C3 + α 2C4 } cos  b2β a  + {−α 2C3 + α1C4 } sin  b2β a   



  

2
x
η2  β   − a2β a 
x
x 
( γC1 − C2 ) cos  b2β  + ( C1 + γC2 ) sin  b2β  
  e

2  a  
a
a 



+e
∫ Fn ( x ) dx ≈
a2 β
x
a

x
x   


( γC3 + C4 ) cos  b2β a  + ( −C3 + γC4 ) sin  b2β a   



  

1 a  − a2β a 
x
x 
{−α1C1 − α 2C2} cos  b2β  + {α 2C1 − α1C2} sin  b2β  
e

η β 
a
a 



x
+e
a2 β
x
a

x
x  


{α1C3 − α 2C4 } cos  b2β a  + {α 2C3 + α1C4 } sin  b2β a   



 

in which
 1 
α1 =  1 + γ 
 2 
230
 1 
, α 2 = 1 − γ 
 2 
Appendices
By back substituting the expression for u z ( x, θ ) and adopting the derivatives and
integrals above, the expressions for all quantities can be derived while introducing the
appropriate approximations, viz. neglecting the small terms in comparison with unity
as γ 2 ≈ η2 1 .
For the displacement uθ , the expression becomes
 − a2β x 
1
x
x 


uθ = − ∫ u z d θ = sin nθ e a  −C1 cos  b2β  − C2 sin  b2β  
n
a
a 




+e
a2 β
x
a

x
x  


 −C3 cos  b2β a  − C4 sin  b2β a   



  

For the displacement u x , the substitution yields
2
u x = −∫

∂uθ
1 + υ 1 ∂  ∂2
ad θ −
 2 + 1 uθdx
4 ∫
2 aβ ∂θ  ∂θ
∂x

2
2


a dFn ( x ) 1 + υ ( n − 1)


F
x
dx
= cos nθ − 2
+
(
)
n
4
∫
 n

dx
2a
β


=
  a 2 dFn ( x ) 
1 n n2 − 1
 β

cos nθ  − 
 + 2 γ  η ∫ Fn ( x ) dx  
2
2β
n
 a
 
  β η dx 
which results in
ux =
 − a2 β ax 
1 n n2 − 1
x
x 


cos
n
θ
e
( u x1C1 − u x 2C2 ) cos  b2β a  + ( u x 2C1 + u x1C2 ) sin  b2β a  
2β
n2






+e
a2 β
x
a

x
x   


( −u x1C3 − u x 2C4 ) cos  b2β a  + ( u x 2C3 − u x1C4 ) sin  b2β a   



  

in which
3
u x1 = 1 − γ
2
3
, ux2 = 1 + γ
2
For the stress resultant nxx , the same procedure results in
nxx =
2
x
Et n 2 − 1  n 
x
x 
 − a2 β a 


 a cos nθ e
( γC1 + C2 ) cos  b2β a  + ( −C1 + γC2 ) sin  b2β a  
2
2 
2β n  a 






+e
a2 β
x
a
x
x  



( γC3 − C4 ) cos  b2β a  + ( C3 + γC4 ) sin  b2β a   



 

For the stress resultant nxθ , the same procedure results in
nx θ =
 − a2β ax 
Et n2 − 1 n
x
x 
η
sin
n
θ
( nxθ1C1 + nxθ 2C2 ) cos  b2β  + ( −nxθ 2C1 + nxθ1C2 ) sin  b2β  
e

2
4β n
a
a
a 




+e
a2 β
x
a

x
x   


( − nxθ1C3 + nxθ 2C4 ) cos  b2β a  + ( − nxθ 2C3 − nxθ1C4 ) sin  b2β a   



  

in which
1
nxθ1 = 1 + γ
2
1
, n xθ 2 = 1 − γ
2
231
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
For the stress couple mθθ , the expression becomes
mθθ = Db
 − a2β ax 
n2 − 1
x
x 


θ
cos
n
C1 cos  b2β  + C2 sin  b2β  
e

2
a
a
a 




+e
a2 β
x
a

x
x   


C3 cos  b2β a  + C4 sin  b2β a   



  

For the stress resultant vθ , the expression becomes
vθ = Db
 − a2β x 
n2 − 1 n
x
x 


sin nθ e a  −C1 cos  b2β  − C2 sin  b2β  
2
a a
a
a 




+e
a2 β
x
a

x
x   


 −C3 cos  b2β a  − C4 sin  b2β a   



  

Inhomogeneous solution to the SMC equation
For a load, constant with respect to coordinate x and presented by pz ( x, θ ) = pzn cos nθ ,
the inhomogeneous solution is derived for all relevant quantities. For convenience, the
other load terms are assumed to be zero. For the sake of clarity, this rather simple load
case is considered, but this does degenerate the generality of the approach as it can
easily be extended to more involved load cases.
The inhomogeneous solution can be obtained by omitting all derivatives with
respect to the axial coordinate x and by omitting the load terms px and pθ in the set
of equations (G.9). A single differential equation is obtained, which reads
2
−

Db ∂ 2  ∂ 2
∂p
+ 1 uθ = z = − npzn sin nθ

∂θ
a 4 ∂θ2  ∂θ2

The solution to this equation is
uθ = −
1
a4
p zn
sin nθ
2
2
Db ( n − 1) n
and hence
uz = −
∂uθ
1
a4
=
pzn cos nθ
∂θ Db ( n 2 − 1)2
The other nontrivial solutions (refer to expressions (G.8) and (G.5)) are
 1 ∂ 3uθ 1 ∂uθ 
a2
mθθ = Db  2
+
=
p zn cos nθ

3
a 2 ∂θ  n 2 − 1
 a ∂θ
1 ∂mθθ
na
=− 2
vθ =
p zn sin nθ
a ∂θ
n −1
of which the stress couple is identically be obtained from the equilibrium equation
(G.4).
232
Appendices
Appendix J
Program solution for influence of
stiffening rings
Introduction
This appendix summarizes the relevant input data and results of the calculations as
referred to in section 5.3.
Input data
Calculations have been made for a radius-to-thickness-ratio of 50, 100 and 200 and
with a varying number of equally spaced stiffening rings per length-to radius ratio. For
the radius-to-thickness-ratios, the respective length-to-radius-ratios approximately
match with a 0.5, 1 and 1.5 times the influence length of the long-wave solution. The
maximum number of stiffening rings has been chosen such to achieve a minimum
spacing of about 0.2 times the influence length of the long wave solution while the
minimum number of stiffening rings that has been considered is two. The considered
rings are T beams that are bend with the stem inside matching with the curvature of the
shell. The cross-sectional dimensions have been based on practical considerations
related to the thickness of the shell and typical requirements as prescribed in relevant
codes and standards. Three different cross-sections have been considered to study the
impact of this variation with the following generic properties:
a) the web height equal to the flange width (Case 1),
b) the web height larger than the flange width of the previous case (Case 2), and
c) the flange width larger that the web height of the first case (Case 3).
The properties of the considered rings are summarized in Table J-1 for which the
notation is depicted in Figure I-1.
Table J-1 Ring dimensions for the three cases (all with a = 1000 mm ).
Case
1
2
3
at
50
100
200
50
100
200
50
100
200
t1
l1
t2
l2
150 mm
100 mm
75 mm
150 mm
100 mm
75 mm
200 mm
133 mm
100 mm
15 mm
7 mm
4 mm
15 mm
7 mm
4 mm
15 mm
7 mm
4 mm
20 mm
10 mm
5 mm
20 mm
10 mm
5 mm
20 mm
10 mm
5 mm
150 mm
100 mm
75 mm
200 mm
133 mm
100 mm
150 mm
100 mm
75 mm
233
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Figure I-1 Geometry of a typical connection of a ring element to a cylindrical shell
element.
Results
In the tables provided in this section, the results for the calculations as described above
have been collated. The following non-dimensional parameters have been adopted to
provide insight in the results:
4lr
lin,2= 2
ratio of spacing between the rings to a quarter of the influence length
of the long wave solution
(
λr
)
program
stiffness ratio determined to obtain a linear relation between
λ r and
the stress ratio between the axial stress at the base due to the selfbalancing terms ( n = 2,...,5 ) and the axial stress at the base due to the
“beam term”
leff , program
at
leff , formula
at
leff , formula
leff , program
234
non-dimensional effective shell length based on
(
λr
)
program
non-dimensional effective shell length calculated by formula (5.20)
ratio of the above non-dimensional effective shell lengths
Appendices
Table J-2 Constant ring with equal width and height a t = 50
Radius-to-thickness ratio a t = 50
Length-to-radius ratio l a = 7.5
Number of
rings
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.607
0.910
)
program
0.187
0.222
0.287
leff , program
at
0.563
0.480
0.342
leff , formula
at
0.725
0.714
0.694
leff , formula
leff , program
1.29
1.49
2.03
Length-to-radius ratio l a = 15
Number of
rings
8
7
6
5
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.520
0.607
0.728
0.910
1.213
1.820
)
program
0.188
0.202
0.221
0.244
0.277
0.333
0.464
leff , program
at
0.551
0.520
0.492
0.457
0.409
0.314
0.104
leff , formula
at
0.725
0.720
0.714
0.706
0.694
0.676
0.643
leff , formula
leff , program
1.32
1.38
1.45
1.55
1.70
2.15
6.17
Length-to-radius ratio l a = 22.5
Number of
rings
12
11
10
9
8
7
6
5
4
3
2
4lr
lin,2= 2
0.455
0.496
0.546
0.607
0.682
0.780
0.910
1.092
1.365
1.820
2.730
(
λr
)
program
0.184
0.193
0.204
0.216
0.231
0.250
0.273
0.305
0.358
0.466
0.708
leff , program
at
0.605
0.582
0.562
0.540
0.513
0.481
0.443
0.384
0.279
0.101
-0.096
leff , formula
at
0.725
0.722
0.718
0.714
0.709
0.703
0.694
0.683
0.667
0.643
0.602
leff , formula
leff , program
1.20
1.24
1.28
1.32
1.38
1.46
1.57
1.78
2.39
6.34
-6.29
235
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Table J-3 Constant ring with equal width and height a t = 100
Radius-to-thickness ratio a t = 100
Length-to-radius ratio l a = 10
Number of
rings
4
3
2
4lr
lin,2= 2
(
λr
0.429
0.572
0.858
)
program
0.193
0.227
0.288
leff , program
at
0.687
0.621
0.492
leff , formula
at
0.735
0.725
0.705
leff , formula
leff , program
1.07
1.17
1.43
Length-to-radius ratio l a = 20
Number of
rings
8
7
6
5
4
3
2
4lr
lin,2= 2
(
λr
0.429
0.490
0.572
0.686
0.858
1.144
1.716
)
program
0.193
0.207
0.224
0.246
0.276
0.326
0.443
leff , program
at
0.695
0.676
0.656
0.632
0.596
0.506
0.249
leff , formula
at
0.735
0.730
0.725
0.717
0.705
0.687
0.655
leff , formula
leff , program
1.06
1.08
1.11
1.13
1.18
1.36
2.63
Length-to-radius ratio l a = 30
Number of
rings
12
11
10
9
8
7
6
5
4
3
2
236
4lr
lin,2= 2
0.429
0.468
0.515
0.572
0.643
0.735
0.858
1.029
1.287
1.716
2.573
(
λr
)
program
0.188
0.197
0.207
0.219
0.232
0.249
0.270
0.299
0.345
0.439
0.661
leff , program
at
0.782
0.767
0.752
0.735
0.717
0.695
0.664
0.608
0.492
0.262
-0.015
leff , formula
at
0.735
0.732
0.729
0.725
0.719
0.713
0.705
0.694
0.678
0.655
0.614
leff , formula
leff , program
0.94
0.95
0.97
0.99
1.00
1.03
1.06
1.14
1.38
2.50
-40.71
Appendices
Table J-4 Constant ring with equal width and height a t = 200
Radius-to-thickness ratio a t = 200
Length-to-radius ratio l a = 15
Number of
rings
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.607
0.910
)
program
0.183
0.214
0.271
leff , program
at
0.794
0.735
0.592
leff , formula
at
0.739
0.729
0.709
leff , formula
leff , program
0.93
0.99
1.20
Length-to-radius ratio l a = 30
Number of
rings
8
7
6
5
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.520
0.607
0.728
0.910
1.213
1.820
)
program
0.184
0.197
0.213
0.234
0.263
0.314
0.441
leff , program
at
0.768
0.757
0.745
0.724
0.683
0.549
0.216
leff , formula
at
0.739
0.735
0.729
0.720
0.709
0.691
0.658
leff , formula
leff , program
0.96
0.97
0.98
1.00
1.04
1.26
3.04
Length-to-radius ratio l a = 45
Number of
rings
12
11
10
9
8
7
6
5
4
3
2
4lr
lin,2= 2
0.455
0.496
0.546
0.607
0.682
0.780
0.910
1.092
1.365
1.820
2.730
(
λr
)
program
0.180
0.188
0.198
0.209
0.221
0.237
0.258
0.287
0.338
0.447
0.696
leff , program
at
0.854
0.845
0.836
0.825
0.812
0.791
0.752
0.667
0.490
0.194
-0.088
leff , formula
at
0.739
0.736
0.733
0.729
0.723
0.717
0.709
0.698
0.682
0.658
0.617
leff , formula
leff , program
0.87
0.87
0.88
0.88
0.89
0.91
0.94
1.05
1.39
3.39
-6.99
237
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Table J-5 Constant ring with increased width and equal height a t = 50
Radius-to-thickness ratio a t = 50
Length-to-radius ratio l a = 7.5
Number of
rings
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.607
0.910
)
program
0.178
0.211
0.276
leff , program
at
0.541
0.465
0.326
leff , formula
at
0.728
0.718
0.700
leff , formula
leff , program
1.34
1.54
2.15
Length-to-radius ratio l a = 15
Number of
rings
8
7
6
5
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.520
0.607
0.728
0.910
1.213
1.820
)
program
0.179
0.193
0.210
0.233
0.266
0.321
0.453
leff , program
at
0.527
0.499
0.473
0.440
0.390
0.296
0.093
leff , formula
at
0.728
0.724
0.718
0.711
0.700
0.683
0.652
leff , formula
leff , program
1.38
1.45
1.52
1.62
1.80
2.31
7.01
Length-to-radius ratio l a = 22.5
Number of
rings
12
11
10
9
8
7
6
5
4
3
2
238
4lr
lin,2= 2
0.455
0.496
0.546
0.607
0.682
0.780
0.910
1.092
1.365
1.820
2.730
(
λr
)
program
0.175
0.184
0.194
0.206
0.221
0.239
0.261
0.294
0.347
0.457
0.702
leff , program
at
0.577
0.554
0.535
0.515
0.490
0.460
0.421
0.361
0.257
0.085
-0.103
leff , formula
at
0.728
0.725
0.722
0.718
0.713
0.708
0.700
0.689
0.675
0.652
0.613
leff , formula
leff , program
1.26
1.31
1.35
1.40
1.46
1.54
1.66
1.91
2.62
7.65
-5.94
Appendices
Table J-6 Constant ring with increased width and equal height a t = 100
Radius-to-thickness ratio a t = 100
Length-to-radius ratio l a = 10
Number of
rings
4
3
2
4lr
lin,2= 2
(
λr
0.429
0.572
0.858
)
program
0.182
0.214
0.274
leff , program
at
0.678
0.614
0.483
leff , formula
at
0.738
0.728
0.710
leff , formula
leff , program
1.09
1.19
1.47
Length-to-radius ratio l a = 20
Number of
rings
8
7
6
5
4
3
2
4lr
lin,2= 2
(
λr
0.429
0.490
0.572
0.686
0.858
1.144
1.716
)
program
0.182
0.196
0.212
0.233
0.263
0.311
0.428
leff , program
at
0.682
0.663
0.644
0.620
0.584
0.493
0.241
leff , formula
at
0.738
0.734
0.728
0.721
0.710
0.694
0.664
leff , formula
leff , program
1.08
1.11
1.13
1.16
1.22
1.41
2.75
Length-to-radius ratio l a = 30
Number of
rings
12
11
10
9
8
7
6
5
4
3
2
4lr
lin,2= 2
0.429
0.468
0.515
0.572
0.643
0.735
0.858
1.029
1.287
1.716
2.573
(
λr
)
program
0.178
0.186
0.196
0.207
0.220
0.236
0.257
0.286
0.332
0.428
0.654
leff , program
at
0.760
0.746
0.731
0.715
0.698
0.676
0.643
0.586
0.468
0.242
-0.022
leff , formula
at
0.738
0.735
0.732
0.728
0.724
0.718
0.710
0.700
0.686
0.664
0.625
leff , formula
leff , program
0.97
0.99
1.00
1.02
1.04
1.06
1.10
1.20
1.46
2.74
-27.80
239
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Table J-7 Constant ring with increased width and equal height a t = 200
Radius-to-thickness ratio a t = 200
Length-to-radius ratio l a = 15
Number of
rings
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.607
0.910
)
program
0.173
0.202
0.257
leff , program
at
0.788
0.734
0.586
leff , formula
at
0.742
0.732
0.714
leff , formula
leff , program
0.94
1.00
1.22
Length-to-radius ratio l a = 30
Number of
rings
8
7
6
5
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.520
0.607
0.728
0.910
1.213
1.820
)
program
0.174
0.186
0.202
0.221
0.250
0.300
0.428
leff , program
at
0.758
0.747
0.734
0.714
0.668
0.528
0.204
leff , formula
at
0.742
0.738
0.732
0.725
0.714
0.697
0.666
leff , formula
leff , program
0.98
0.99
1.00
1.01
1.07
1.32
3.26
Length-to-radius ratio l a = 45
Number of
rings
12
11
10
9
8
7
6
5
4
3
2
240
4lr
lin,2= 2
0.455
0.496
0.546
0.607
0.682
0.780
0.910
1.092
1.365
1.820
2.730
(
λr
)
program
0.170
0.178
0.187
0.197
0.210
0.225
0.245
0.274
0.325
0.437
0.690
leff , program
at
0.837
0.828
0.819
0.808
0.794
0.772
0.730
0.642
0.465
0.174
-0.096
leff , formula
at
0.742
0.739
0.736
0.732
0.727
0.721
0.714
0.703
0.689
0.666
0.627
leff , formula
leff , program
0.89
0.89
0.90
0.91
0.92
0.93
0.98
1.10
1.48
3.82
-6.53
Appendices
Table J-8 Constant ring with equal width and increased height a t = 50
Radius-to-thickness ratio a t = 50
Length-to-radius ratio l a = 7.5
Number of
rings
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.607
0.910
)
program
0.153
0.183
0.243
leff , program
at
0.380
0.304
0.172
leff , formula
at
0.716
0.703
0.680
leff , formula
leff , program
1.88
2.32
3.96
Length-to-radius ratio l a = 15
Number of
rings
8
7
6
5
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.520
0.607
0.728
0.910
1.213
1.820
)
program
0.154
0.167
0.183
0.204
0.235
0.290
0.425
leff , program
at
0.368
0.338
0.309
0.272
0.219
0.122
-0.050
leff , formula
at
0.716
0.711
0.703
0.694
0.680
0.658
0.620
leff , formula
leff , program
1.95
2.10
2.28
2.55
3.10
5.38
-12.34
Length-to-radius ratio l a = 22.5
Number of
rings
12
11
10
9
8
7
6
5
4
3
2
4lr
lin,2= 2
0.455
0.496
0.546
0.607
0.682
0.780
0.910
1.092
1.365
1.820
2.730
(
λr
)
program
0.151
0.159
0.168
0.179
0.192
0.209
0.231
0.263
0.318
0.433
0.689
leff , program
at
0.410
0.389
0.368
0.346
0.319
0.286
0.245
0.182
0.082
-0.062
-0.200
leff , formula
at
0.716
0.713
0.709
0.703
0.697
0.690
0.680
0.666
0.648
0.620
0.574
leff , formula
leff , program
1.75
1.83
1.92
2.03
2.19
2.41
2.77
3.66
7.87
-10.04
-2.87
241
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Table J-9 Constant ring with equal width and increased height a t = 100
Radius-to-thickness ratio a t = 100
Length-to-radius ratio l a = 10
Number of
rings
12
11
10
4lr
lin,2= 2
(
λr
0.455
0.496
0.546
)
program
0.151
0.159
0.168
leff , program
at
0.410
0.389
0.368
leff , formula
at
0.716
0.713
0.709
leff , formula
leff , program
1.75
1.83
1.92
Length-to-radius ratio l a = 20
Number of
rings
8
7
6
5
4
3
2
4lr
lin,2= 2
(
λr
0.429
0.490
0.572
0.686
0.858
1.144
1.716
)
program
0.154
0.165
0.180
0.199
0.226
0.274
0.393
leff , program
at
0.553
0.531
0.507
0.477
0.428
0.315
0.081
leff , formula
at
0.725
0.720
0.712
0.702
0.688
0.667
0.629
leff , formula
leff , program
1.31
1.35
1.40
1.47
1.61
2.11
7.80
Length-to-radius ratio l a = 30
Number of
rings
12
11
10
9
8
7
6
5
4
3
2
242
4lr
lin,2= 2
0.429
0.468
0.515
0.572
0.643
0.735
0.858
1.029
1.287
1.716
2.573
(
λr
)
program
0.150
0.157
0.165
0.175
0.187
0.202
0.221
0.249
0.296
0.398
0.634
leff , program
at
0.627
0.610
0.593
0.573
0.551
0.522
0.479
0.407
0.276
0.067
-0.132
leff , formula
at
0.725
0.722
0.717
0.712
0.706
0.698
0.688
0.675
0.656
0.629
0.582
leff , formula
leff , program
1.16
1.18
1.21
1.24
1.28
1.34
1.44
1.66
2.38
9.44
-4.42
Appendices
Table J-10 Constant ring with equal width and increased height a t = 200
Radius-to-thickness ratio a t = 200
Length-to-radius ratio l a = 15
Number of
rings
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.607
0.910
)
program
0.143
0.168
0.218
leff , program
at
0.653
0.592
0.424
leff , formula
at
0.729
0.716
0.691
leff , formula
leff , program
1.12
1.21
1.63
Length-to-radius ratio l a = 30
Number of
rings
8
7
6
5
4
3
2
4lr
lin,2= 2
(
λr
0.455
0.520
0.607
0.728
0.910
1.213
1.820
)
program
0.144
0.155
0.168
0.185
0.212
0.264
0.394
leff , program
at
0.630
0.614
0.594
0.561
0.489
0.310
0.018
leff , formula
at
0.729
0.723
0.716
0.705
0.691
0.669
0.630
leff , formula
leff , program
1.16
1.18
1.20
1.26
1.41
2.16
34.78
Length-to-radius ratio l a = 45
Number of
rings
12
11
10
9
8
7
6
5
4
3
2
4lr
lin,2= 2
0.455
0.496
0.546
0.607
0.682
0.780
0.910
1.092
1.365
1.820
2.730
(
λr
)
program
0.141
0.148
0.155
0.164
0.175
0.189
0.208
0.237
0.290
0.410
0.674
leff , program
at
0.705
0.694
0.681
0.665
0.643
0.607
0.544
0.429
0.237
-0.017
-0.211
leff , formula
at
0.729
0.725
0.721
0.716
0.709
0.701
0.691
0.678
0.659
0.630
0.583
leff , formula
leff , program
1.03
1.04
1.06
1.08
1.10
1.16
1.27
1.58
2.77
-36.38
-2.76
243
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
244
Literature
Literature
Numbered list
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Van Bentum, C.A., Derivation and application of circular shell elements.
2002, Delft University of Technology, Faculty of Civil Engineering and
Geosciences, graduation report.
Bouma, A.L., Loof, H.W., Van Koten, H., The analysis of the stress
distribution in circular cylindrical shell roofs according to the DKJ-method
with aid of a calculation scheme (in Dutch). IBBC Mededelingen, 1956. 4(2):
p. 35-101.
Loof, H.W., co-workers, The library of computer programmes for shell
structures at the Stevin-Laboratory. Heron, 1964. 12(2): p. 71-98.
Bahtia, R.S., Sekhon, G.S., A novel method of generating exact stiffness
matrices for axisymmetric thin plate and shell elements with special reference
to an annular plate element. Computers & Structures, 1994. 53(2): p. 305-318.
Bahtia, R.S., Sekhon, G.S., Generation of an exact stiffness matrix for a
cylindrical shell element. Computers & Structures, 1995. 57(1): p. 93-98.
Bahtia, R.S., Sekhon, G.S., Generation of exact stiffness matrix for a conical
shell element. Computers & Structures, 1999. 70: p. 425-435.
Sekhon, G.S., Bahtia, R.S., Generation of exact stiffness matrix for a spherical
shell element. Computers & Structures, 2000. 74: p. 335-349.
Melerski, E.S., Design analysis of beams, circular plates and cylindrical tanks
on elastic foundations: with IBM-compatible software. 2000, Rotterdam:
Balkema.
Pircher, M., Guggenberger, W., Greiner, R., Stresses in Elastically Supported
Cylindrical Shells under Wind Load and Foundation Settlement. Advances in
Structural Engineering, 2001. 4(3): p. 159-167.
Kraus, H., Thin Elastic Shells. 1967, New York: John Wiley & Sons, Inc.
Leissa, A.W., Vibration of Shells. 1973, Washington, D.C.: U.S. Government
Printing Office.
Hildebrand, F.B., Reissner, E., Thomas, G.B., Notes on the Foundations of the
Theory of Small Displacements of Orthotropic Shells, in NACA-TN-1833.
1949.
Love, A.E.H., The Small Free Vibrations and Deformations of a Thin Elastic
Shell. Philosophical Transactions of the Royal Society of London - Series A Mathematical and Physical Sciences, 1888. 179: p. 491 - 549.
Flügge, W., Stresses in Shells. Second Edition ed. 1973: Springer-Verlag
Berlin Heidelberg New York.
Borisenko, A.I., Tarapov, I.E., Vector and tensor analysis with applications.
1979, New York: Dover Publications, Inc.
Sanders, J.L., An Improved First Approximation Theory for Thin Shells.
NASA TR-R24, 1959.
245
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
246
Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity. Fourth
Edition ed. 1944: New York: Dover Publications.
Reissner, E., Stress strain relations in the theory of thin elastic shells. Journal
of Mathematics and Physics, 1952. 31: p. 109-119.
Koiter, W.T. A Consistent First Approximation in the General Theory of Thin
Elastic Shells. in Proceedings of the Symposium on Theory of Thin Elastic
Shells. 1960. Delft, August 1959: Amsterdam: North Holland Publishing
Company.
Novozhilov, V.V., Thin Shell Theory. Second ed. 1964, Groningen, The
Netherlands: P. Noordhoff Ltd.
Donnell, L.H., Stability of thin-walled tubes under torsion. NACA, Report No.
479, 1933.
Vlasov, V.S., Basic Differential Equations in General Theory of Elastic Shells.
NACA TM 1241, 1951.
Mushtari, K.M., Galimov, K.Z., Nonlinear theory of thin elastic shells, in
NASA-TT-F-62. 1957.
Donnell, L.H., Beams, Plates and Shells. 1976, New York: McGraw-Hill, Inc.
Marguerre, K. Zur Theorie der gekrümmten Platte grosser Formänderung. in
Proceedings of the 5th International Congress for Applied Mechanics. 1938.
Cambridge, Massachusetts, 1939: New York: Wiley.
Zingoni, A., Shell Structures in Civil and Mechanical Engineering. 1997,
London: Thomas Telford Publishing.
Finsterwalder, U., Die querversteiften zylindrischen Schalengewölbe mit
kreissegmentförmigem Querschnitt. Ingenieur-Archiv, 1933. 4.
Schorer, H., Line load action on thin cylindrical shells. Proceedings of the
American Society of Civil Engineers, 1935. 61: p. 281-316.
Moe, J., On the theory of cylindrical shells - Explicit solution of the
characteristic equation, and discussion of the accuracy of various shell
theories. Mémoires de l'Association Internationale des Ponts et Charpentes,
1953. 13: p. 283-296.
Von Kármán, T., Tsien, H.S., The buckling of thin cylindrical shells under
axial compression. Journal of the Aeronautical Sciences, 1941. 8(June): p.
303.
Jenkins, R.S., Theory and design of cylindrical shell structures. 1947, Ove
Arup & Partners, London.
Bouma, A.L., Van Koten, H., The analysis of cylindrical shells (in Dutch), in
Report No. BI-58-4. 1958, TNO-IBBC: Delft.
Hoefakker, J.H., Theory for thin circular cylindrical shells (in Dutch). 2000,
Delft University of Technology, Faculty of Civil Engineering and
Geosciences, graduation report CM2000.008.
Hoff, N.J., The accuracy of Donnell's equation. Journal of Applied Mechanics,
1955. 22: p. 329-334.
Morley, L.S.D., An improvement on Donnell's approximation for thin-walled
circular cylinders. Quarterly Journal of Mechanics and Applied Mathematics,
1959. 12: p. 88-99.
Literature
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
Koiter, W.T., Foundations and basic equations of shell theory - A survey of
recent progress. 1968, Laboratory of Engineering Mechanics, Department of
Mechanical Engineering, Technical University, Delft, Report No. 381.
Niordson, F.I., Shell theory. North-Holland series in applied mathematics and
mechanics. 1985, New York: Elsevier Science.
Houghton, D.S., Johns, D.J., A comparison of the characteristic equations in
the theory of circular cylindrical shells. The Aeronautical Quarterly, 1961.
August: p. 228-236.
Seide, P., Roots of the cylindrical shell characteristic equation for harmonic
circumferential edge loading. American Institute of Aeronautics and
Astronautics, 1970. 8(3): p. 452-454.
Seide, P., Errata: "Roots of the cylindrical shell characteristic equation for
harmonic circumferential edge loading". American Institute of Aeronautics
and Astronautics, 1972. 10(9): p. 1263-1264.
Shirakawa, K., Characteristic roots of cylindrical shells base on an improved
theory. Ingenieur-Archiv, 1986. 56: p. 201-208.
Boyce, W.E., DiPrima, R.C., Elementary Differential Equations and Boundary
Value Problems. Fifth ed. 1992, New York: John Wiley & Sons, Inc.
Blaauwendraad, J., Hoefakker, J.H., Theory of Shells. Course notes. 2005:
Delft University of Technology, Faculty of Civil Engineering and
Geosciences.
Nayfeh, A.H., Introduction to Perturbation Techniques. 1981: John Wiley and
Sons.
Van Koten, H., The stress distribution in chimneys due to wind pressure.
CICIND Report, 1994. 11(2): p. 21-23.
Turner, J.G., Wind load stresses in steel chimneys. CICIND Report, 1996.
12(2): p. 16-19.
Hoefakker, J.H. Analytical study of full circular cylindrical shells under wind
loading. in 5th International PhD Symposium in Civil Engineering. 2004.
Delft, June 2004: Leiden: A.A. Balkema Publishers.
Schneider, W., Zahlten, W, Load-bearing behaviour and structural analysis of
slender ring-stiffened cylindrical shells under quasi-static wind load. Journal
of Constructional Steel Research, 2004. 60: p. 125-146.
Bleich, H., Stress distribution in the flanges of curved T and I beams (Die
Spannungsverteilung in den Gurtungen Gekrümmter Stäbe mit T- und IFörmigem Querschnitt). 1950, Navy Department, David W. Taylor Model
Basin, Washington 7, D.C.
Pegg, N.G., Smith, D.R., PRHDEF - Stress and stability analysis of ring
stiffened submarine pressure hulls, DREA TM 87/213. 1987, Defense Research
Establishment Atlantic, Canada.
MacKay, J.R., Structural Analysis and Design of Pressure Hulls: the State of
the Art and Future Trends, DRDC Atlantic TM 2007-188. 2007, Defence
Research and Development Canada.
Faulkner, D. The collapse strength and design of submarines. in RINA
Symposium on Naval Submarines. 1983. London: Royal Institution of Naval
Architects.
247
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
Hutchinson, J.W., Amazigo, J.C., Imperfection-sensitivity of eccentrically
stiffened cylindrical shells. American Institute of Aeronautics and Astonautics,
1967. 5(3): p. 392-401.
Bijlaard, S., Journal of the Royal Aeronautical Society, 1957. 24(6): p. 437.
Malik, Z., Morton, J., Ruiz, C., Ovalization of cylindrical tanks as a result of
foundation settlement. Journal of Strain Analysis, 1977. 12(4): p. 339-348.
Marr, W.A., Ramos, J.A.J., Lambe, T.W., Criteria for settlement of tanks.
Journal of the Geotechnical Engineering Division, Proceedings of the
American Society of Civil Engineers, 1982. 108(GT8): p. 1017-1039.
Godoy, L.A., Sosa, E.M., Localized support settlements of thin-walled storage
tanks. Thin-Walled Structures, 2003. 41(10): p. 941-955.
Hoefakker, J.H., Blaauwendraad, J. Effective and efficient analytical study of
full circular cylindrical shells. in Progress in Structural Engineering,
Mechanics and Computation. 2004. Cape Town, July 2004: Leiden: A.A.
Balkema Publishers.
Struik, D.J., Lectures on Classical Differential Geometry. Second ed. 1961,
New York: Dover Publications, Inc.
Sokolnikoff, I.S., Mathematical Theory of Elasticity. Second ed. 1956, New
York: McGraw-Hill Book Company, Inc.
Reissner, E., A new derivation of the equations for the deformation of elastic
shells. American Journal of Mathematics, 1941. 63(1): p. 177-184.
Ventsel, E., Krauthammer, Th., Thin Plates and Shells. 2001, New York:
Marcel Dekker, Inc.
Alphabetical index to the numbered list
Bahtia, R.S., Sekhon, G.S.
Bijlaard, S.
Blaauwendraad, J., Hoefakker, J.H.
Bleich, H.
Borisenko, A.I; Tarapov, I.E.
Bouma, A.L., Loof, H.W., Van Koten, H.
Bouma, A.L., Van Koten, H.
Boyce, W.E.; DiPrima, R.C.
Donnell, L.H.
Faulkner, D.
Finsterwalder, U.
Flügge, W.
Godoy, L.A., Sosa, E.M.
Hoefakker, J.H.
Hoefakker, J.H., Blaauwendraad, J.
Hoff, N.J.
Houghton, D.S., Johns, D.J.
Hildebrand, F.B., Reissner, E., Thomas, G.B.
Hutchinson, J.W., Amazigo, J.C.
248
04, 05, 06
54
43
49
15
02
32
42
21, 24
52
27
14
57
33, 47
58
34
38
12
53
Literature
Jenkins, R.S.
Koiter, W.T.
Kraus, H.
Leissa, A.W.
Loof, H.W.
Love, A.E.H.
MacKay, J.R.
Malik, Z., Morton, J., Ruiz, C.
Marguerre, K.
Marr, W.A., Ramos, J.A.J., Lambe, T.W.
Melerski, E.S.
Moe, J.
Morley, L.S.D.
Mushtari, K.M., Galimov, K.Z.
Nayfeh, A.H.
Niordson, F.I.
Novozhilov, V.V.
Pegg, N.G., Smith, D.R.
Pircher, M., Guggenberger, W., Greiner, R.
Reissner, E.
Sanders, J.L.
Schorer, H.
Seide, P.
Sekhon, G.S., Bahtia, R.S.
Shirakawa, K.
Sneider, W., Zahlten, W.
Sokolnikoff, I.S.
Struik, D.J.
Turner, J.G.
Van Bentum, C.A.
Van Koten, H.
Ventsel, E., Krauthammer, Th.
Vlasov, V.S.
Von Kármán, T., Tsien, H.S.
Zingoni, A.
31
19, 36
10
11
03
13, 17
51
55
25
56
08
29
35
23
44
37
20
50
09
18, 61
16
28
39, 40
07
41
48
60
59
46
01
45
62
22
30
26
249
Theory Review for Cylindrical Shells and Parametric Study of Chimneys and Tanks
Curriculum Vitae
Jeroen Hoefakker was born in Amersfoort, The Netherlands, on 17 February 1974.
After having graduated in 1992 from Rijksscholengemeenschap Thorbecke in
Amersfoort, he attended the Faculty of Civil Engineering of Delft University of
Technology. In 2000, he gained a Master of Science in Civil Engineering and
Geosciences, specialising in structural mechanics.
In October 2000, he commenced to work on his doctoral research in the Section of
Structural Mechanics of the Faculty of Civil Engineering and Geosciences.
In parallel with the PhD research, he produced the lecture notes “Theory of Shells” for
a Master’s course by Professor Blaauwendraad and assisted as a lecturer from 2001
through 2005 for this course.
In 2004 he was a member of the organising committee of the 5th International PhD
Symposium in Civil Engineering held in Delft.
In March 2006, he joined INTEC Engineering (nowadays INTECSEA) employed as an
offshore pipeline systems and marine terminals engineer working on a range of
projects from feasibility studies to detailed design.
Jeroen is the proud father of two daughters, Minke (2006) and Lieke (2009), born to his
partner Mirjam Veenstra.
250