CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
DYNAHIC
RESPONSE OF CYLINDRICAL
,,
SHELLS WITH INTERNAL LIQUID
A graduate project submitted in partial
satisfaction of the requir'ements for the
degree of Master of Science in
EBgineering
h'-'
-.)
Samuel Lee Angelino
June, 1977
The Graduate Project of Samuel Lee Angelino is approved:
California ·state University, Northr·idge
t-~.C KNO~JL EDG~1ENT
The author is grateful to his advisor, Or. Roger
Di ·Julio, and the faculty members for· the val uab 1e
guidance given during the completion of this project.
Special thanks are also given to his colleagues
a~
the Atomics International Division of Rockwell
Internationa.1.
iii
I
- -
•• -·- --··--· --·····-
----
--·--··-·--~-·-··
- - - ··--------- •.• -··---·.- -·· -· .. -····- ·-
- --- .. --·- -··-·-·. --- ····--·
TABLE OF CONTENTS
Page
Abstract
I.
...
....
f.
B.
r
7
....
2.
....
Finite E1ement Methods .
Thin Shell Vibration . . . . .
3.
Liquid Sloshing Behavior
....
MODELING
FUNDAfvlENTAL
11
• .J.
21
Fraction Parametric Study
37
Liquid
3.
Thin Shell
40
Liquid Sloshing Effects
....
Approx·imations
A.
SAP
B.
LIQUID W\SS FRACTION
40
43
... .
DbCUSSlON
.J
21
2.
IV
1'-
t~ode l
SAP IV Finite Element
RESULTS
7
7
.. .. ... .
1.
r~a.ss
1
4
AND PROCEDURES
REVIEW OF LITERATURE
1.
IV.
....
THEORET! CAL SIMULATlON TECHN I,QUES
A.
III.
1
PURPOSE AND OBJECTIVE
B. METHOD
iii
X
. ...
INTRODUCTION
A.
-L
..........
.
Acknowledgment
52
..
r·mDEL
PAR/\~1ETRIC
STUDY
.... .. ..........
iv
...
52
76
83
Page
V.
SUMMARY AND CONCLUSIONS
86
VI.
REFERENCES .
89
VII.
APPENDIXES .
91
A.
SAP IV Finite Element Computer Model
91
B.
Liquid Mass Parametric Model . . •
109
C.
SAP IV Finite Element Mode Shapes
113
;
! .
v
~---·
---~---~-
·--------~-
···----··-------·-·- ·-----·--·----·-·-----·---
~·
------· ----·------ -·-·----------
·---~---~---
-·---·----------·-
-~
-----------------
LIST OF TABLES
H.ble
....
1
Summary of Natural Frequencies
the Seismic
fOl~
Models of the Structural Shell and Support (Liquid
~iass Contents = 0~~) .••• e ~..........................
2.
Su:nmary of Natura·! Frequencies for the Seismic
Models (Liquid Mass Contents= 100%) •••• "" ••••.,..
3.
Locations of t,1aximurn
r~ode 1s •••
0
•••••••
0
Stt~ess
61
in Fi ll'ite Element
•••••••••••• '
•••
r
•
•
•
•
•
•
•
•
•
•
•
•
•
•
4.
Summary of Primar·y
~fi.embrane
Stt"eSs" ••••••
5.
Summa.ry of Pri rr.a ry
~lembrane
and Bending Stress .•.
5.
Summary of Ground Reaction Loads
fat~
e ........
67
70
72
Fixed Saddle
Base Due to Seismic Spectral Excitation..........
7.
53
74
Summary of Ground Reaction Loads for Longitudinally Free Saddle Base Due to Seismic Spectral
Ex c i ta t i on • ~ • ~ o ~
8.
• • ,_, • • • u • • .. • • • • • • • • • • • • • • • "' • v . . . . . ,.
75
Summary of Natural Frequencies ftorn the Liquid
Mass Fraction Parametric Model •.•••••••.••••.••••
vi
79
--··~---~--------~-
--------.------------------------- ----
----
·--·-·-·-··-·----·----·-·-·
·--··--------·---~
I
LIST OF FIGURES
Pa~
1
....
Cylindrical Containment Vessel
Sti~ucture .•••••••
23
2.
Cylindrical Vessel Dimensions •••••••••.•••••••••
24
3.
Finite Element Mesh for Cylindrical Vessel
Shell...........................................
4.
Finite Element Mesh for Large Fixed Saddle and
Wearplate.......................................
5.
Finite Element
29
for Circumferential Stiffen-,
~iesh
ing Rings.......................................
30
7.
Sadd1e Baseplate Boundary Elements..............
31
8.
Seismic Response Acceleration Spectra...........
36
9.
Dynamic Model for Liquid Mass-Fraction Parametric
Study, ......... e
10.
First Mode
• • " • • • • o • • • • • • • • • • • ,.
Shap~-Vessel
~
•
1(1
••••••
~
,
•
•
11.
e e e • e
fOi
9
C·
>a
lt
p
e e e e e e
<ll
e e • e e e • •
0
•
0
•
Q
•
oJ
0
~
-..
~ ~
9
e •• e . . . . . . . . . . . . . . .
~
• • • • a • • ., • •
57
Second Mode Shape-Vessel Structure - Vertical
Simulation......................................
-.-----~
56
Second Mode Shape-Vessel Structure - Transverse
si mu 1at i 0 n••••••••
13.
55
First Mode Shape-Vessel Structure - Vertical
si Jnu 1at i 0 n
12.
39
Structure - Transverse
Simulation......................................
--
28
Finite Element Mesh for Small Free Saddle and
Wearplate.......................................
6.
26
-~
---------- --·---------------+---- -----+----· --------------- ------
vii
58
r·------ ---------------------------------------------------------------------------------------- -----------:-·----------------------- -1
I
1
LIST OF FIGURES
j
(continued)
I
i
'1
I
i
i
Fig~~
14.
Pa~
First Mode Shape-Vessel Structure (Full Condition)
-Vertical Simulation............................
15.
63
Second Mode Shape-Vessel Structure (Full Condition) -Vertical Simulation ••••••.••••.• ,........
16.
Third Mode Shape-Vessel Structure (Full Condition)
17.
Frequency Ratios vs Liquid Mass Fractions for the
64
First Four Vessel Parametric Model Modal ReSJlOnses •••••••••••
18.
tl
•••••••••••••
4
•••••••••••
-a....
First Mode Shape-Vessel and Liquid Mass Parametric
r~odel............................................
19.
0
•••••••••• "
•
•
•
•
•
•
•
...
114
Second Mode Shape-Vessel Structure (Full Condi-
. ) - ·T ransverse
t 1on
23.
82
First Mode Shape-Vessel Structure (Full Condition)
-Transverse Simulation..........................
22.
81
Third Mode Shape-Vessel and Liquid Mass Parametric
Mode 1 • , • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
21.
80
Second Mode Shape-Vessel and Liquid Mass Parametric f. 1ode 1 ~ .... , ...............
20.
77
s·1mu.a~1on
, +. ••••••••••••••••••••
115
Third Mode Shape-Vessel Structure {Full Condition)
- Transverse Simu'Jation ••••••••••••••• ~..........
116
i
I
i
24. First Mode Shape-Vess~l Structure (Full Condition)
I__ --- . . . __- --- --------- -------~---- ------- --- ---- --- -- ----------- ---·--------------------- ----- ________ j
viii
I
~
--~---------
~---~~-----~-. --~--~-~---~~
-~
----~~-- --·-~------~~------~-~
------~
-~-----~·--·----·
--
~----·-----·----·-·····
--·--
LIST OF FIGURES
(continued)
Fiqure
Page
---'-'·-
-Longitudinal Simulation ••••.••••••••.••••••.•••
25.
Second Mode Shape-Vessel Structure (Full Conditic~)
26.
117
-Longitudinal Simulation..................
118
Third Mode Shape-Vessel Structure (Full Condition)
-Longitudinal Simulation........................
'------------·-------·------·-----------·---- ------
- - - - .-------------- --
x·ix
-. -- -· --- ------
119
-- -- --- ---
ABSTRACT
DYNAMIC RESPONSE OF CYLINDRICAL
SHELLS WITH INTERNAL LIQUID
by
Samuel Lee Angelino
Master of Science in Engineering
This report presents the theory, modeling techniques ctnd r·e-
sults of dynamic simulation of cylindrical she!l vessel struci.:t.wes
which contain intetnal liquid.
selected for the study.
The finite
eleme~t
method was
A liquid containment vessel was modeled
usir1g a structural analysis computer program.
A concise presenta-
tion of the theories has been made for the finite element method of
dynamics, thin shell vibration and liquid sloshing.
The goveniir.g
equations for liquid sloshing have been derived in general form.
Techr1iques from the literature were
~sed
to assess the classical
thin shell vibration with liquid sloshing effects.
A liquid mass
fraction paramo::tric study v1as conducted to charcc:terize the response
of pad;fa11y filled vessels subjected to an arbitrary forcing
function.
A discussion of the results has been included to guide
in the seismic design and analysis of vessel structur'es whiCh contain
internal r!uid.
X
r-- _________, _________________________________..:..._ ----- -- --------------·
I
I
I
--~----
-
-------~--------
·-------------~-----
------ ---·-·.
----~-----
---·
The finite element method resulted in an accurate simulation of
the cylindrical shell vessel structure.
The technique of modeling
I
i
·I one half of the vessel with adjusted boundary conditions at the
plane of symmetry resulted in savings in engineering and computer
soiuUon time.
The dynamic simulat·ion indicated that the full 1iq-
uid condition for the vessel was the most critical from the standpoint of engineering design.
Critical stress levels and seismic
embedment loads reached their maximum values fur this case.
The iiquid mass p<Irametric study indicated that the fundamental
fr~quency
of the system decreased in a monotonic manner as the
amount of liquid mass in the system was increased.
Results from
the pat'ametric study have been presented in genera·i form to allow
the evaluation of a vessel structure of arbitr·ary size.
The effects
of interrtnl liquid sloshing for a vessel of this geometry were not
significant.
Dynamic coupling of the liquid and solid systems did
not result and the liquid slosh pressure loads were secondary in
magn·itude compared to the seismic response loads.
xi
INTRODUCTION
SECTION I.
A.
PURPOSE AND OBJECTIVE
This report investigates the dynamic simulation of thin shell
vessel structural systems that contain internal liquids.
One
widely used current solution technique, the finite element method
is presented in detail.
The results of dynamic modeling of a
large structural plate and shell system using a computer program
are i nc1 udcd.
The dynamic behavior of a typical str·ucture encountered in
engineering practice has been investigated.
The structure is a
large horizontal liquid containment vessel subjected to a response
spectrum form of seismic excitation.
These vessels are constructed
with large cylindrical shell bodies and are supported on a saddle
dnd wearplate structure.
They may contain an arbitrary amount :rf
internal fluid, which complicates the accurate prediction of their
seismic response.
Various techniques have been used for numerical simulation of
these systems and experimental studies have been conducted.
This
report treats the problem from a purely theoretical viev:point and
uses computer simulation finite element techniques for the detailed investigation, as well as the generalizations that are
drawn.
Several detailed fin"ite element models have been employed
to characterize the dynamic responses of the thin shell vessel
1
structure.
In addition, hand calculations and techniques from the
literature have been used to assess the classical thin shell vibration and liquid sloshing effects.
Generalizations are drawn concerning the dynamic response of
the vessel structure which contained a varying amount of internal
fluid.
Smaller simulation models have been developed and a
liquid mass parametric study has been conducted.
character-lzat·ion of the dynamic
vessel
subjt:~cted
t~esponse
This a11ows
of the partially filled
to an arbitrary forcing function.
Frequency ratio
plots for the modal responses of the partially filled vessel
illustrate the effect of 'the internal liquid mass.
Some of the basic theory is presented, in concise form: for
the dynamic simulation of liquid and shell structural systems.
Techniques and fundamentals of finite element dynamics are discussed.
The vibfation theory of one type of thin cylindrical shell is ineluded as well as the governing equations for liquid sloshing behavior.
The considerations that must be made prior to and during
the evaluation of this type of structural system are discussed in
general form.
Problems of this type have been posed in various fields of
engineering for many years.
In particular, the effect of liquid
sloshing and its ·influence on the overall dynamic r'esponse of a
structure has received much attention.
Earlier investigators ln
the nuclear ·industry (Reference 1) attributed part of the contuined
:
I fluid mass as impulsive, or wall lumped in nature, and part
a~
L_ ________ ---- .------ --- . ------ ------------------------- --------------- ---- --------- ------------- ------------------ -- --- -- - -- ""
3
["---------------------·----
--·--------·----·----------·-
-------- --·--------------·--·-
------------~-----··--·------·-
----:
I
,
I convective, or internally oscillating in behavior. This investigation has been based on the complete fluid mass modeled as wall
lumped to the vessel shell which has resulted in a modified system
inertia matrix.
In the aerospace industry, the problems of in-
ternal propellant fuel sloshing led to much theoretical work
(Reference 2) to characterize the effects for various geometries.
Also explored were the effects of v.,rall elasticity and possible
structural coupling, damping, and nonlinear effects due to large
amplitude excitation or irregular geometries.
These phenomena
affected the wing vibration flutter mode characteristics as well
as the dynamic stability ·control system characteristics.
The civil engineering discipline has concentrated research
studies in the areas of safety of dam design {Reference 3) to insure that the designs will withstand additional seismic loading
_,
\
due to the forced liquid motion.
The literature
pl~ovides
a summary
of the ever-evolving modeling techniques as well as current
methods for predicting the behavior of these phenomena such that
safe engineering designs can be realized.
This report emphasizes
the methods employed in the mechanical engineering discipline to
insure adequate engineering design and analysis of cylindrical
vessels.
It is believed that this report may prove useful to those who
wish to acquaint themselves with some of the various modeling
techniques available.
It may be useful in the initial phases of
similar engineering problems to suggest a solution procedure.
I
[______________ ..________
--- ·----·-·- ·----------- ---- -·----·- ---·-------- ________ --- ·---------------·--------·----,
4
though the finite element form of modeling formed the basis of this
investigation, other techniques may soon be available.
A descrip-
tion of the pertinent prot:::edures, as well as delimitations, for
this study is made in the section which follows.
B. METHODS AND PROCEDURES
The finite element method of solution has been selected for
the detailed structural modeling of thE horizontal containment
vessel.
With this technique, the overall structural geometry is
discretized into various elements and an approximation to the actual
solid continuum is made.
program,
11
A popular and current computer analysis
A Structural Analysis Program for Static and Dynam·ic
Response of L-inear Systems, 11 SAP IV, (Reference 4) is chosen for
the study.
Initial computer solutions have been made for the vessel shell
structure, only, with no inter·nal liquid contents.
Then the mass
lumping techniques are used for simulation of the full vessel condition.
Since the models are large, bandwidth minimization has
been employed to minimize computer solution time.
The original
models contain 2748 degrees of freedom and are condensed by the
pteprogr·ammed
minimization routine to 270 equations
(half-band~;Jidth).
The first three frequencies and mode shapes for the dynamic models
are summarized.
Three models are employed, one each for· the ver-
tical, transverse, and longitudinal dynamic responses.
The size
·5
- - - - - - - - - · - - - - - - - - - - - - - - - - - - - - - - - - - ··----·-- ···-·-···-- ·-· ·------ -· ···-- ·----·-··--------·--·· ----·······-···-··--··-····
·····-------·--- ----·--·-····1
!!
and symmetry of the shell structure permit modeling of only half
of the shell and plate assemblage about the plane of symmetry.
The boundary conditions and effective mass lumping
have
con~tants
been adjusted to provide an accurate simulation for each of the
three orthogonal response directions.
Pertinent results from the dynamic models are summarized in
later sections of the report.
The amount of data was large, so in
addition to frequency and plotted mode shape information, only a
summat·y of stress levels in critical regions of the vessel is presented.
Ground reaction loads are also tabulated.
To emphasize
the significance of the solution for the engineering industry, a
comparison of stress quantities with ASME Boiler and Pressure
Vesse1 Code (Reference 5) allowable stress limits is made.
For the generalizations concerning the frequency response of
the vessel \'lith an arbitrary fraction of internal fluid, a much
smaller finite element model has been constructed.
With this
model, a parameter study is conducted and the effects of the
varying quantity of ·internal fluid are determined.
A- parameter plot
has been constructed which presents the effects of various liquid
mass fractions on the frequency response of the structure.
For
the parameter study, var·ious fractions of the liquid mass have been
considered as totally impulsive and are thus lumped on to the
shell structure.
This is an assumption since some smaller fraction
of the liquid may behave convectively and oscillate internally as
a multidegree of freedom system, while the remainder may behave im-
6
~-----------------·-·--------------
1
I1
-------.
-~---·
-·--- ---·--·--. ----· - · - -------- -·- ··--·- -------- ---·· --·-. ---· ------------
pulsively and contribute to the total structural response due to the
added mass effect.
However, this assumption allows the simulation
I models to be less complicated and yield better results. Also, this
I
I is !'lelieved to be conservative since the addition of the total
I
i
I
i
mass will result in the lowest, or most fundamental, structural
:
system frequencies and the largest dynamic loads and stresses.
i
Thus,
it becomes an effective technique for both the parameter study and
the en9ineering analysis.
Various hand calculations have been made to
ass~ss
the agree-
ment with more classical techniques for thin shell vibration and
predicted liquid sloshing behavior.
A consistent geometry is
er.1-
played and the techniques for the dynamic response of thin shells
(Reference
6)
acterization.
are used for frequency prediction
and
modal char-
Liquid sloshing effects are deter-·mined in a
simila\~
manner, using parameter charts (Reference 2) for calculation of
liquid slosh frequencies.
The possibility of dynamic coupling of
the liquid and solid shell systems chosen for this study is also
investigated.
!__ --------- --·- ·-- -
-· -· -··
-- ---~---~---------------- ··------- ---·-- ------------- - - - - · -~-----------------------· -----~----------
-------- ·------ _______ j
------
-------------·--·-
- ·---- -----· ------- --------- -
SECTION II.
A.
··----·----------------~-----
---·---··--
-------····-·--------- ------·--- -·.-----------
~-
-··------·-------·-··-··
THEORETICAL SIMULATION TECHNIQUES
REVIEW OF LITERATURE
The available theories in the literature that are used to simu-
late cylindrical shell vessel structures \'lhich contain an interr.a1
liquid have been reviewed for this study.
The study has been
divided into three parts, the dynamic simulation of vessel structures,
using finite element computer techniques, the assessment of the
classical thin shell vibration, and the investigation of the effects
of liquid sloshing.
A concise presentation of each of the theories
used in this study is made, with emphasis on the general approach
employed in each.
1.
Finite Element Methods
"----'--·------------
The finite element method is a powerful, numerical solution tool.
which may be used when dealing with complex engineering problems.
problem with an infinite
numbe~
A
of degrees of freedom may be re-
duced to one with finite degrees of freedom by discretizinga the con-;
11
tinuum and applying the numerical method.
!
Any structure may be con- '
sidered as a series of elements that have known load-deformation
characteristics and are interconnected at a finite number of nodes.
The
~onditions
structure.
of equilibrium and compatibility are applied to the
The direct stiffness method is then employed to formulate.
the element stiffness matrices.
I
L._____
The direct stiffness method assumes that a function character.:C------··· ---- ---- -·-- ·--------------·------ - - - - - - - - - - -----·---- -- ------·---------------- -----------. ----- ----- ----- -----
7
8
izing the internal displacements of an element at any point can be
uniquely determined by the nodal displacements of the element.
Then
the internal strains, and the internal stresses using Hooke•s law,
are defined in matrix form.
The principle of virtual work may be
applied to equate the internal and external work done on the element
during a virtual displacement.
From the load-deflection relation-
ship for any element, the element stiffness matrix can then be obtained.
The system stiffness matrix is then assembled using the
techniques of matrix structural analysis.
The details of the method
are illustrated in the literature (References 7 and 8).
The thin plate and shell element of SAP IV is used for the
finite element discretization of the cylindrical vessel shell
structure into a series of elements.
This thin shell element is a
quadrilatel'al of arbitrar·y geometry formed ft·om four compatible
triangles.
The element simulates membrane plane stress and bending
behavior.
A central node is located at the average of the coordin-
ates of the four corner nodes.
Th~
six interior degrees of freedom
of the element are eliminated prior to matrix assembly.
The final
quadl'ilatel"al element r-esults in twenty-four degrees of ft·eedom in
the global coordinate system.
In addition to the static stiffness representation of the
structure, the finite element method may be used to characterize the
frequencies and modes of the system oscillations.
The loading on
the system may be complex, such as that of an earthquake shock.
vantages of the numerical solution by high speed computer rather
Ad-
9
- --- ------- ----------
--··-
~~--
----------- --------------
-- -------
than by classical techniques such as the Stodola or Myklestad methods include the ability to extract selective eigenvalues and the
greater precision and speed involved.
The principles of solution of the dynamics problems are described in general form.
The equations of motion for a system of
structural elements may be written in matrix form,
[M]{u} + [C]{u} + [K]{u} = {P}
( 1)
where [M], [C], and [K],are the assembled system mass, damping, and
stiffness matrices, respectively.
The system mass matrix [M] is assembled from the element mass
j].
matrix, [m 1
The element mass matrix is defined as,
[mu] "/[N;] \ [Nj] dV
(2)
v
The matrices [N 1J are the element shape functions that are selected such that the appropriate nodal displacements can be calculated based on the nodal coordinates. · The shape functions themselves
are functions of the coordinates and are obtained by minimizing an
arbitrary function a 1.
A similar procedure is sometimes employed for the consistent
1
l
I
1
damping matrix [C].
However the damping matrix is usually constructed
i
from test da.ta.
I
I
------------------ ·--- ------- ------------------- -·--· - - · - - - - - - - - - - - - - - - - - - - - - - .---------------------------------- ·----···-····-·_]
10
r·---------·-·-·-··---------- -----------· ------·--------·-------- ···-···. -----------1
I
For the free vibration condition, a periodic response may be
assumed, that is u = u0 coswt.
The substitution leads to the eigen-
va 1 ue prob 1en1,
(cKJ
or
-w 2
[K]{~}
[MJ}
{u } =
0
o
( 3)
= w2 [M]{¢}
For determination of the natural frequencies, the frequency determinant is evaluated,
(4)
Solution for the values of w yields the natural frequencies of
the system.
I
1
For a computer solution of the problem, a rearrangement to a
I specialized form is made. This avoids the sometimes impractical
I polynomial expansion of the frequency dete~inant.
1
I
The special eigen-
value problem is, with A = l/w 2
I
I!
[ KJ- 1
[
M] { u
I
I
1
I
This may be
0
}
= >d u 0 }
~onverted
( 5)
to triangular form using the lower tri-
angular matr!x [L], and the formulations,
[K]=[L][L]T and
[K]-L=
(cLJT)- 1 [L]- 1
(6)
L----~ .. --- ~------~ -------·- ----- -----------~------ -----------------------~--;--·--· --------------------· ---------.-------------J
11
r-- -------------- -------·---------,------·-·-·-··--·--- -·- ·- -- ---··-· ----·-------· -·-- -- ·---·- -· --·--·-- -----···---·-----
!
to arrive at a final form,
[HJ{x}=Jdx}
(7)
in which
[H]
and
=
[L]- 1 [MJ{[LJ-
1
T
)
{x}=[LJT{u }
l'8'I
0
The [H] matrix is symmetric and the eigenvalues and modes of
{x}
and thus of
{u }
0
may be obtained.
has then been solved.
The free vibration problem
A detailed illustration of the methods of
finite element dynamics is given in the literature (References 9
and 10).
2.
Thin Shell Vibration
A shell is a three-dimensional body bounded by tvm closely-
spaced curved surfaces.
If the distance between these surfaces is
small in comparison with other dimensions, it is a thin shell.
The middle surface of the shell is defined and possesses a character- •
istic curvature.
This curvature differentiates shell behavior from
that of the classical plate theory.
i
The theory of shells is governed by an eighth order system of
I partial
I
I
differe~tial
equations of motion, whereas plate bending
theol'Y is governed by a fourth order set.
This is due to the fact
~---·-··----·-·--·------·--·--·-··----------··-··-------··------·----·---------·-----·----------------__j
12
,--------
• - - - - - - - - - - - - - - - - - - - - - - - · - - -- -- - - - · - - - - - - - - - - - - .. .
.:~:.,
1
I
• -- . -- - - - - - - - - - - - - - - -. - - - - - - - - --,
that the characteristic bending of shells cannot be separated from
the stretching.
In shell vibration, the frequencies tend to be
more closely spaced and less easily identified than those of plates,
which further complicates their study.
P..dditional boundary condi-
tions and par·ameters such as length-to-radius and thickness-toradius ratios are specified in the thin shell vibration problem.
The principles of the vibration theory of thin circular cylindrical shells are described in general form.
Many thin shell
theories have been derived in the literature, with the differences
being
·j
n the assumptions used in each.
will be used here.
The Donne ll-~lushta ri Theory
The equations of motion for the shell may be
written in matrix form,
[DJ {ui}
= {0}
(9)
The displacement vector {ui} is defined,
{ u.}
(10)
1
and a
ch~nge
of coordinate and thickness parameter are used,
I
.I
I
i
Ii
i
I
l
1
!
········-·-----·--··-· ·- ...•.... · · - - - · - - · - - · · - - · · · · · · - - - · · · - - - - - - - - - - - - - - - · - · · - - - - - -- .• ··-·--·--·---·--·-------····--·--··· .... 1
13
,-------------- ---·--·-··-- -----·---------------·-------·--- -------- ----·--···--·-··--·------ --·· -------··- ·------------··-l
I
I
!
i
=
1
S
1
k = h 2 /12R 2
I
I
I
x/R
(11)
w
Th:e [D] operator is the sum of two operators and serves to differentiate the various shell theories,
(12)
with the [D 0J being the differential operator £orresponding to the
Donnell-·rljushtari theory, and [DM] being an operator 111hich changes
the basic theory to yield another shell theory.
[OM] is assumed to
be identically zero for the Donnell-r>'lushtari theory.
The differential operator for the Donnell-Mushtari theory is,
·----~---------------- ---------·---------------------:---------------------~-----------------··--·
14
·----
------~-----------
--------------"---------·--- ··--------.. ------------
l+v a2
\
2
as
+
L
2
ae 2
-p(l-v~R a
2
---E -
vasa
I
··---·------- --i
a
vas
-2-asaa(1-v L
--------~-----------_-----------
a
ae
I
l
2
~t2
a
(13)
ae
)
1 + kv"+
(1-v...:lR:. -a2-
p --
E
at2
_j
For a closed circular cylindrical shell of infinite length, the
displacements can be expressed as,
u
= A cos
AS
cos ne cos wt
v
~
B sin
AS
sin ne cos wt
w = C sin
AS
cos ne cos wt
(14)
with A, B, C, and A as undetermined modal constants.
This form of
displacement solution assumes that the spacial and time variables
are separable and normal modes result with the same period and phase
of motion for all points on the shell.
Substituting these displace-
······-·-··--·-··-··--------- --- ·-·- · ----------.----------------·---------------·--···-··--·.J
15
~----------------------------,--------------
------------------------ -----------------------------·----------1
I.
I
i
I
ment functions into Equation 9 and using the previous shell theory
I
a set of homogeneous equations results,
(15)
This leads to an eigenvalue problem, such that for a given
~
one or more values of the frequency parameter p(l-v 2 )R 2 w2 /E exists.
The determinant of the coefficients wi 11 then vanish, and the frequencies and modes of the shell vibration are determined.
The
details of the method may be applied for various boundary conditions
and modifying shell theories.
Many of the solutions are p1·esented
in the form of parameter charts.
Illustrations of the methods and
solutions for various configurations and conditions are given in the
literature (References 6 and 11).
3.
Liquid Sloshing Behavior
Liquid sloshing is a phenomena which occurs when a liquid con-
tained in a vessel or other structure oscillates with its own
characteristic frequencies and modal patterns.
Motion of the
vessel excites characteristic frequencies of the liquid mass and the
liquid free surface begins to oscillate.
observed at the liquid free surface.
Movement is most easily
However, during sloshing in a
vessel, the lower· part of the liquid may also oscillate as if it were
l
i
a rigid body) while the surface liquid moves independently.
~-
-------- -·-- ----------- ----------------------------------------
This
___________
_,
1,.
10
I·
surface motion will permeate less deeply into the liquid as the frequency of oscillation of the free surface increases.
The sloshing
will become independent of vessel depth for height to depth ratios
greater than one.
Important considerations for a liquid system which exhibits
sloshing, in addition to the liquid system frequencies and modal
patterns, are the possible effects on the containment structure
itself.
The sloshing may result in pressure loading of the struc-
ture, which in some instances may form an appreciable part of the
total load.
Interaction effects between the liquid slosh and
structural frequencies may occur and dynamic coupling of the systems
may result.
This could lead to a new set of characteristic fre-
quencies for the coupled system.
For thes2 reasons, the sloshing
phenomena of internal liquids in structures is an important part of
the field of continuum vibrations.
Simplifying assumpt·i ons must be made to employ a theoreti ca 1
development for liquid sloshing.
Exact solutions are difficult,
and the assumption that the liquid is of constant density with no
viscosity is made.
The flm'l field is assumed to be irr·otational,
without sources or sinks.
Small displacement theory is assumed to
apply and the vessel structure is initially assumed to be rigid.
The theory of the liquid sloshing phenomena, with the previous
assumptions, is described in general form.
The governing equations
for the behavior of the liquid are the momentum and continuity
equations.
The momentum equation may be written in vector fcrm (Re-
17
~i
------------------------------- ----- - - ··------·--·-·-·-··--------··--------··-----,
I
ference 12) ,
I
(16)
where
~t
is the substantial time derivative, V· is the diver·gence,
and v is the gradient.
The continuity equation may be written as,
v--v = a
(17}
The liquid is assumed incompressible and inviscid.
The stress tensor
for newtonian f"J ui ds may be written in cylindrical coordinates as,
'[ rr = -ll
•ee
= -ll
-rzz _=
Tre =
-jl
f2
avr
la;
~]
- 32 (v·v) .
[2(_L_3_+
vr)
r
ae
r
[2 -av,.
az
, ar =
-11
-
~
(v·VJ
J
2 -]
- - (v·v)
3
[r --a (va- ) + -1 -J
avr l
a.r
r
-rez
= -rze = -p
[•vaz8. + :. avaszJ
Tzr
= 'rz = -ll
r•v,
+ avr l
ar
az
r ae
(18)
L
·
Using the continuity equation and substituting the stress
i
I
L-·--~-------
i
I
- ---------- --· ----------------- ----------------------------------------~J
18
~----··-
-------··-· --·· -·--·--·· -· ·-· ··--·-----------· ·. -· ··-·-· ·---·---· - -·---· ····-·-····------·-----·-·-l·
I
tensor into the general
j
tion results,
~ment~
1
equation, the Navier-Stokes equa-
The viscous effects may be considered as negligible, that is,
(v·•]
=o
(20}
This assumption implies that the shear stress terms are negligible compared to the other terms in the momentum equation.
The
rate of momentum gain by viscous transfer per unit volume is sma1i
and the fluid can exert no shear stresses.
These assumptions will
be realistic especially for sloshing phenomena in larger geometries.
Using these assumptions, the Euler equation results and may be
written,
p
Dv = -VP +
Dt
(21)
pg
The assumption of irrotational flow insures the existence of
a single valued velocity potential
~
(r, e, z, t) from which the
velocity field c~n be obtained by taking the gradient,
........
I
l__
v
= IJ<f>
(22)
l
I
7 _________ - - - - - - - -
- - - - · · · - - · - - - - · - · . ······-·· ··-··--.··----------·-·····-j
19
Substituting this into Equation 17, the result is that
<1>
must
satisfy the Laplace equation,
(23)
Since the flow is irrotational and incompressible, Euler's
equations of motion can be integrated and linearized to obtain
Bernoulli 1 s law which may be written as,
3$
(24)
p
at + -p· + gz = o
Therefore, in the solution to the liquid sloshing problem, the
, veiocity potent·ial
,
priat<~
<1>
is determined ft·om Equation 23 using the appro-
boundary conditions.
The velocity components are obtained
from Equation 22 and the press11re distribution from Equation 24.
The slosh forces and moments acting on the solid structure may be
found by integration of the pressure function.
The vibrations portion of the liquid sloshing phenomena may
nmt
be formulated.
The boundary conditions for the free vibration
of the liquid system are; a·t the structure walls,
•
{25)
vn as the common velocitvJ of the fluid and structure at the
20
~---~-·
!
j
·-------·--· ···-·-----·-- ---------·-----·------·--·--·---~---------· --
interface.
------·-----,
The free surface condition is,
I
II
Il
!
D.
act>
"t"- + ge -az
a
.L
{26)
- 0
with ge the total effective acceleration, and the surface pressure
assumed as constant.
The eigenvalue problem for the vibration characteristics, that
is the slosh frequencies and mode shapes, may be formulated by
assuming solutions of the form ~ = <f>o (r, e, z)eiwt.
Using Equa-
tions 25 and 26, the eigenfrequencies wmn and eigenfunctions of the
form,
(27)
may be obtained.
The governing system of equations leads to a boundary value
problem which may be solved by separation of vadables, integral
equation techniques or principles of the calculus of variations or
extremum of functions.
As for the case of thin shell vibration,
many of the solutions have been presented in the literature in
graphical form.
Illustrations of the method of solution and results
for various conf·j gurations may be found in the 1 iterature (References
I
I 2, 13~ and 14). ·
i
II
I!________________: _____________________. -~-.- - - - - -
I
:.--~~
... --:--.. ____________________ _ j
21
r ----·- --·-··-·------··---- ------·--·-··-----------·- ·-------------------··---------------- -- -·----··------·:--- --------I
--~
,
II
B.
FUNDAMENTAL MODELING
I
SAP IV Finite Element
t~odels
I
1.
i
simulation of the dynamic behavior of the prototype vessel shell
The finite element technique was selected for the computer
structure.
The computer program used for the investigation was
SAP IV (Reference 4).
linear~
This program has the capability to handle
static, and dynamic finite element analysis of structures
composed of a number of different element types. ·The simulation
models used here wet·e developed with a nodal geometry, material
properties, section properties, and mass distributions typical of
the actual structural system.
From these, the computer program
·assembled the system stiffness and mass matrices.
The solution
technique was relatively complex and highly automated.
The actual shell structure whose dynamic response was investigated was a horizontally mounted cylindrical vessel designed for
I
containment of a liquid.
The vessel itself was constructed of mild
·I carbon steel with an 18-ft outside diameter, 3/4-in. shell wall
I
I
i
I
I
thickness, with standard ASME flanged torispherical heads.
The
vessel was supported on two saddle and wearplate assemblies, constructed of 1-1/2 in. thick material.
The large saddle was fully
-·
anchored in all three directions at the base, while the small saddle
i
was designed as longitudinally free at the baseplate to allow for
I
relative expansion.
Anchor bolts were used to secure the structure
! baseplates to ground.
Four 1-1/2 in. thick stiffening rings were
I
L_ -----·--------·----- -·----------------------------"--··----------·--·-- ------------·--~----·_1
22
r---~--
------
------~----------~--------------~---
j
I
------------ - ------------ --------------------- ---· -----------------------------
-
1
employed as an extension of the saddle plate uprights to lend
J
stiffness to the shell structure.
A photograph of a small scale
model used in the study is shown as Figure 1.
I
I
I
This
v~ssel ~onfiguration
was selected to simulate an engineer-
I ing component frequently encountered in practice, e.g., a large
she1"1 type vessel used for liquid containment.
The gross internal
capacity for each vessel was 52,500 gallons of liquid.
The liquid
used in this vessel was sodium \'lhich had a density of 56.13 pounds
per cubic foot.
The empty vessel weight was
107~270
pounds while
the full vessel weight was calculated at 501,270 pounds.
Basic
dimensions for the cylindrical vessel are provided in Figure 2.
Two element types were used for the finite element models.
The vessel was modeled using the thin shell or Type 6 element and
the boundary or Type 7 element varieties . . The thin shell elements
were used to simulate the cylindrical shell, torispherical heads,
saddle support structure, and the circumferential stiffening rings.
The major advantage for the selection of this element was the
dua·l capability of modeling in-plane membrane and out-of-plane
bending effects.
The element could have either a triangular or a
quadrilateral geometry.
The shell element used four· compatible,
constant strain triangles with a central node, whose degrees of freei
I
I
!
dom were
eliminat~d.prior
to the system stiffness matrix assembly.
This resulted in six degrees of freedom per node in the global co-
I ordinate system.
I
The boundary or Type 7 elements were used to simulate the
[___________________________ ----·----------------------------------·--------------------------------·
23
FIGURE 1
CYLINDRICAL CONTAINMENT VESSEL STRUCTURE
?.4
I
I
I
I
I
A
Overall Vessel length Vessel Outer Diameter B
Distance to Vessel Center c
D
Vessel t to Large Saddle t Large Saddle to Small Saddle E
F
Saddle Base to Vessel ~ lateral Span Between Baseplates - G
Estimated Distances to C.G.
(Vessel Empty}
L
H
v
31 ft
18 ft
15.5 ft
7.75 ft
16.5 ft
11 ft
9.5 ft.
I
::::7 in.
::::27 in.
I
J
.t
~--G~
FIGURE 2
CYLINDRICAL VESSEL DIMENSIONS
25
ground attachment points on the structural model saddle baseplates.
This provided an idealization of an external elastic support at the
attachment nodes.
A one dimensional axial stiffness was specified
and this was entered directly into the total stiffness matrix.
One basic geometry was used for the finite element models.
Each model consisted of 488 nodes, 499 thin shell elements, and 150
boundary elements.
The torispherical heads consisted of shell ele-
ments Numbers 1 through 41 and 263 through 303.
The cylindrical
vessel shell consisted of Shell Elements 42 through 262.
The finite
element mesh for this region is shown as Figure 3.
The large saddle
an~
wearplate have been modeled as Elements
340 through 419, and the small saddle and wearplate, which are
longitudinally free to deform in a guided fashion, are modeled by
Elements 420 through 499.
Figures 4 and 5 present the finite ele-
ment meshes for these regions.
The circumferential stiffening rings were modeled as Elements
304 through 339 and are shown in Figure 6.
Finally, the bounda·ry
elements, which were included to obtain values for seismic embedment loads,
ar~e
shown in Figure 7.
Boundary Elements 1 through 52
were included on the large saddle, and Elements 53 through 92 were
on the small saddle.
Therefore, Figures 1 through 7, inclusively, illustrate the
finite element geometry used for the fundamental modeling.
Each
figure illustrates regions of the structure shown detached from
one another for clarity.
'·---·----~-··--
. -·- ---
. ··- -·-·---·- ···-··-·--·----·-
In actuality, Elements 340 through 359
-·-·- --·-·-------------
·-··---·----·-----·--- ·- ·- -
--·· --~ -·
-----
··~-·
·------ ·------ -·-·--···--
26
FIGURE 3
FINITE ELEMENT MESH FOR
CYLINDRICAL VESSEL SHELL
2/
28
FIGURE 4
FINITE
~L.FI XED SADDLE
t:..
d·1ENT MEC ' FOR LARGE
,C.NO vJEARPLATE
.):i
29
FIGURE 5
FINITE ELEM~NT
FREE SADDLE A~6SHWEARPLATE.
FOR S~1Al L
30
STIFFENING RINGS (2 AT EACH SADDLE)_
~~
L~RGE
FIXED SADDLE
_J
~
I
OUT!\0.1\RD
lONG:=lY FREE
FIGURE 6
FINITE ELEMENT MESH FOR CIRCUMFERENTIAL
STIFFENING RINGS
SADDl~ I
J
31
BOUNDARY ELEM'.IH ORIENTATION
!
II
I
!
POSITIVE REACTIONS
AT NODE N
/
! / . .~~~'{..~
I
I
SHALL LONGITUDINALLY FREE
SADDLE BASE PLATES
Vet·tical Boundary Elements are shown in Figure
Transverse Elements (X d1rection)
Fixed Saddle, tlu;nbers 31-60
Free Saddle, Numbers 121-150
Longitudinal Elerncnr.s (Z direction)
fixed Saddle only, Numbers 60-90
Numbering Order of Transverse and Longitudinal
Elements same as the Verti ca 1.
FIGURE 7
I
I
SADDLE BASEPLATE BOUNDARY ELEMENTS
.!.----------- ··---··-· --· -----· -··--·---·---·----···--- -----------··----------··----· -------- ---·------· -----·----····------!
32
!_____ _-
I
-----------·-------------- ----------------
----~----·-------------.
---·
---~·--·
----------
~---~------- -------------------~--
overlay on Elements 68 tht·ough 72, 81 through 85, 94 through 98,
and 107 through 111, respectively of the cylindrical shell.
Simi-
larly, Elements 420 through 439 of the saddles overlay on Elements
185 through 189, 198 through 202, 211 through 215, and 224 through
228 of the shell.
The stiffening ring elements were connected to
the shell in the circumferential direction at four nodal series locations~
Nodes 116, 184, 246, and 271 along the length of the model. ·
A complete listing of the entire finite element model may be found
in Section VII as Appendix A.
Five different finite element models were employed in the
analytical investigation.· Two models were used to simulate the
empty vessel and support structure.
The vessel shell with total
liquid contents was simulated with the other three models.
Since the vessel was symmetric about a vertical midplane, only
one-half of the structure was modeled in an effort to minimize the
size and cost of the computer analyses.
The physical boundary con-
ditions and effective mass lumping constants were adjusted to provide an accurate simulation for each of the three orthogonal response directions.
The models used in this investigation were defined for the
vertical, transverse, and longitudinal simulations as follows:
i
Vertical
' a
'~
I
.
Sim~lation
Vessel Shell and Support Structure Only- denoted as Model I
l. . . . ~he bo~nd:ryconditi:s
Vessel Shell with Total Liquid Contents - Model III
t~~displac~ments. and forces in
for ..
33
~--
I
------ -------
i.
- - - - - -- --- ----- ------- ---- -- - ----- -- ---··-··. ··-·---··· -·-·-··------- ·----------------- --- ·-·-·--·-- ·-·--·-1
I
terms of the finite element model global axes, were,
I
1
I
L
At the plare of
symmetry,
X
I
-=
--~--
FZ
1
(28)
I
I
I
ez = o
= .Fy = MX = 0
r_s-Pl ane of
symmetry
I
The liquid mass distribution was made by lumping one-half of
the total liquid mass on the accessible half of the vessel cylindrical shell.
L
b.
X
Radial
distribution
of liquid mass.
Longitudinal Simulation
Vessel Shell and Support Structure Only
I
Mode'J I
I
Vessel Shell with Total Liquid Contents - Model V
)
The boundary conditions were the same as those of Model I for
I
l the vertical simulation.
I
------· · -
------- ------ --------·--------- -------- -
j'
------------------
--·------------------------·--·-·-·------
34
,~·---
---- ---·---------
··-
·---~
------·--- --- -·
~---
-- --- ---------·-·-··-····-·- -·--· ···------··-- --
----·---~-
.
----- --- ----·· ---· ----------------------.--------
-------------~
I
The mass distribution was made by lumping one-half of the total
liquid mass on the accessible half of the vessel torispherical head
near the smaller free saddle support.
z
~X
c.
Transverse Simulation
Vessel Shell and Support Structure Only - Model II
Vessel Shell with Total Liquid Contents - Model IV
The
boundat~y
conditions for the displacements and forces in
terms of the finite element model global axes were,
r
·r····t
At the plane of
symmetry,
-~----..:
oz
:--~-;
= oy
:::
(29)
ex = 0
Mz = My = Fx
=0
.,
I ___ JI
The mass distributi6n was the same as the vertical simulation,
I
L____________ ---- ______ _: _____ -·- ----·--·- --· ·-· ·-----····--· ----------- --·--· --------· ·------··- ---------·--· ··--------- -------- _____________ j
35
Model III.
For the static analysis, SAP IV solved the system displacements
by a Gaussian elimination of the equilibrium equations.
The element
displacements were calculated from the system displacements, and
the element stresses were obtained from these.
For the dynamic
portion of the analysis, SAP IV determined the eigenvalues and
eigenvectors from the coupled equations of motion.
These eigen-
values and eigenvectors were used to decouple the equations of
motion, resulting in a set of uncoupled second order linear differential equations.
These modal equations were then solved independ-
ently for the modal responses.
The solution techniques are discussed
in detail in the literature (References 15 and 16).
This completed
the time invariant portion of the analysis, that is, the characterization of the dynamic system.
Eigenvalues for the system frequencies
and eigenvectors for defining the mode shapes of vibration were
computed.
For the forced response, the response spectrum method of analysis with the SAP IV program was used.
Spectral acceleration curves,
shown in Figure 8, were used as the system forcing function.
These
spectra are typical of an operating basis earthquake response spectra with magnitude of acceleration g-levels as a function of frequency in hertz.
They are a form of ground base excitation with an
assumed value of 2% constant damping.
Techniques of earthquake an-
alysis of structures are discussed in the literature (Reference 17).
The seismic response spectra were of significance because they
.36
------- -.----- ----------. ----------------- ~-------------~-------------- -------·-···---------------------- -- -·-
3i'
allowed the computation of maximum modal response quantities.
The
maximum modal displacements of the system for acceleration in each
of the three orthogonal directions was ·computed using the generalized mass, the normalized eigenvector, and the spectral accelerations
for each given frequency and direction.
\'Jere
The directional effects
combined for a given mode by summation of absolute values of
the three modal responses.
Then, the total displacement response
was calculated as the square root of the sum cf the squares of the
modal responses.
Finally, the stress levels and reaction loads
were computed on an element level for th2
2.
~ntire
system.
Liquid Nass Fraction Parametric Study
The mass ft·action parameter study was conducted to characterize
the effects of various fractions of internal liquid mass on the
dynamic response of the complete vessel shell system.
While the
larger SAP IV model of the entire vessel system used the total mass
of the contained liquid, it was believed that a partially filled
vessel would have a different dynamic response.
It was expected
that the natural frequencies of the fluid shell system would be
altered as the amount of internal fluid was varied.
The most useful characterization of this response was a graphical technique to allow prediction of the ratio of system frequencies
for a partially filled vessel to those for the completely full
vessel.
The amount of internal liquid could arbitrarily range from
the empty vessel to that for the fu11 vessel condition.
In this
38
manner, an assessment could be made of the response of the system
with any intermediate fluid level.
Important values of the fluid
shell system frequencies could be determined by inspection of the
frequency ratio plots.
Graphical results from the parametric study
have been included in Section III. B. as Figure 17.
A smaller simulation model was developed to conduct the liquid
mass parametric study.
The model consisted of 14 elements that re-
presented the vessel shell and support structure.
A geometry and
configuration similar to the large vessel shell finite element model
was employed.
The dynamic model for the parametric study is shown
as Figure 9.
To minimize the complexity of the parametric model, it was constructed with the use of a beam-type structural analysis computer
program, rather· than with SAP IV.
This allowed representation of
the vessel shell by 12 elements instead of the large number that
were used in the more detailed finite element models.
For the para-
metric model, the fluid mass was included with the vessel shell
and varied to generate data for the parametric curves.
Each support
member was modeled with one beam element and one rigid link.
The
computer program utilized was one developed at Rockwell International.
for dynamic structural analysis of beam-type three-dimensional frame
structures
compo~ed
of straight, curved, and rigid members (Ref-
erence 18).
Various computer solutions were made using values of internal
liquid mass ranging from the empty to the full vessel condition.
~-·--·---------- ------·-·----·---~--------------------~-------------------~--------~-----------------------------·--·-J
'~(!
._,_,
legend,
0 •
0 "
0 "
llorle
Element
Rigid Link
FIGURE 9
DYNAtltlC
~10DEL
FQR LIQUID MASS--FRACTION
PARAt~ETRIC
STUDY
40
------------- ···------ . --------- ----------- ---- -------------
-------- . -.--- ·---- -··· --
--
------------------------------- ---------------- ---------1
!.
The effect of fluid sloshing on the frequency of the system was
s1nall.
The fluid mass was lumped to the vessel shell wall by using
a uniform mass distribution.
The lumped fluid mass resulted in a
modified system inertia matrix.
The results of the study were
normalized on the basis of the full vessel fundamental frequency
and are presented in Section III.B.
3.
Thin Shell A£Q!9ximations
The horizontal saddle supported vessel was represented as a
thin cylindrical shell.
The approximation was made that the shell
will respond as a closed shell with a shear diaphragm at both ends.
This assumed that the vessel stiffening rings and integral saddle
support structure result in considerable stiffness in their own
planes restraining the vertical and lateral components of shell
displacement at the common boundaries.
However, with their relative-:
ly large flexibility in the longitudinal direction
transvers~
to
their planes, they would generate negligible bending moment Mx and
longitudinal membrane force Nx in the shell as the shell deforms.
The rings and saddle support would supply shearing forces Nxe to the
she 11.
The shear diaphragm boundary conditions were,
j
i
:
I
I
!
I
!---------·----------- - - -·----------------------------- --------------·-------------------·-------------------~--------.J
I
41
I
w = MX = NX = v = 0
(30)
at X = 0, L
(shell ends)
This circular cylindrical shell supported at both ends by shear
diaphragms has received much attention in the literature.
Primarily,
because one simple form of the solution to the eighth order differential equations of motion \'tas also capable of satisfying the shear
:
I
I
i
diaphragm boundary conditions (at both ends) exactly.
This solution
could be t'epresented as,
i
I
I
~:
u = A cos
A
s cos n
8
coswt
v = B sin
A
s sin n
8
coswt
w = C sin
A
s cos n
8
coswt
(31)
With proper selection of A = mTIR/L, the original boundary conditions can be satisfied exactly.
Substitution of the displacement
functiori into the equations of motion yields the characteristic,
ft~equency,
determinant.
Expansion of this will yield the char-
)
ot
I
acteristic equation, the roots of which will be the nondimensional
I
I
I
frequency parameter ei genva 1ues.
This is i1lustrated with applicable
parameter charts in the 1iter a ture (Reference 6).
t -·------~----· ----------.- --·- ------ -·------- ---~-----------~----------.- - - - - · · - - - - - - - - ---------------------------------------------------------------- -------
I
For calculation of the natural frequency (fundamental) for this
system the parameters of importance were considered,
R = shell radius
=
108 in.
( 32)
h - shell thickness = 0.75 in.
L
= length between shear diaphragms = 180 in.
m = parameter for fundamental frequency = 1
Then the pertinent
L/R
ratios were computed,
= 1.6667
( 33)
L/mR = 1. 666 7
R/h
==
144.
Using Figure 2.12 from Reference 6 for these parameters, the
nondimensionai frequency parameter
Q
was,
n ""' 0.15
(34)
circumfer~ntial)
n modal
parameter - 5 to 6 ·
( 35)
with,
(
The frequency parameter was defined as,
ll"
'1· .j
r·
-~~~
---~
-- - - - - - -------- - - - - --~~--
0
~--
-----~~
~
------- - - ---~--~
~~
-~---~-----~-
- - - --
~-- -~--
-~-
- ----
-~---~~--- ~---
-
~-
~-
= wR~-v 2 )/E
( 36)
1
I
Solution for the frequency, w(rad/sec), yielded,
I
I
(37)
w ::::
I
!
Substitution of values for
n
v. and E, resulted
in~
0.15_ _
w ::;:
f
p,
( 38)
= 43.01 Hz.
This represented the fundamental longitudinal modal response
for the thin shell system.
4.
Li~)jd
S~!oshing
Effects_
The horizontal cylindrical vessel shell represented the problem
of the behavior of a contained liquid in a circu·lat' "cana1.
11
The
syrnmetr-·ic modes of liquid oscillation would not be ·induced by
transverse motion of the canal, therefore, only the antisymmetric
modes were investigated.
Since, with symmetry of the liquid cross
section, a net vertical motion of the canal will not produce horizontal slosh forces, nor will rotation of the canal about its
44
center produce sloshing, these effects will hot be sigriificant.
The resultant fluid pressures must act in the radial
and
direction~
the net slosh force acting on the shell must pass through the
center of the section.
The effects of both transverse and longitudinal liquid sloshing
have been assessed.
Available parameter charts from the literature
were used in calculation of the fundamentill frequencies for liquid
sloshing.
First the parameters of importance for the transverse
fluid sloshing mode were
evaluated~
Motion
Vessel inside radius of 107.25
inches
Assumed gas
Vessel Cross Section
freeboard height of 18 in.
(arbitrary)
(39)
eR - 107.25 - 18
= 89.25 in.
eR
89.25
e=rc=l07.25
= 0.832
(40)
I
I
I
From the literature (Reference 19), the fundamental slosh fre-
1
l
! quency parameter A was defined as,
/
I
I
[ __ ·-····-·--·- - ---· -- ···-- . -· . -·--···· -···· ··-··---·-----------··· ---··-···---------·-····-······ ·-····· - - ·-----.-- ---·······-··· -···-·----·-·····. ---- .J
45
A1
(41)
= _0.Q2_'!_
V 1-e 2
:\l = Z~09.1__
1-{ .832) 2
The fundamental frequency was related to the slosh frequency
parameter ass
(01
lj}
1
=\
(42)
{.fdl
VR
=,/i 38QlLU.il
i
107.25
-- 3.685 rad/sec
f
1
= 0.587 Hz.
For calculation of the second and third modes, Figure 9 of
Reference 19 was employed.
Fo·r n
= 2 {second mode),
(43)
..• f 2 =0.936 Hz
46
For n = 3 (third mode),
(44)
:. f 3 = 1.147 Hz
These transverse fluid slosh frequencies were sufficiently far
removed from the structural shell frequency that dynamic coupling
of the systems will not result.
Now the parameters of importCl.nce for the longitudinal fluid
sloshing modes were considered,
{45)
-r-'
h
. ___L_
Liquid height h = 196.5 in.
Vessel inside radius, a : 107.25 in.
Free vessel length = 282 in.
Vessel head depth = 45 in.
(1/3 of which was effective)
Total vessel length = L =
282 + 2(1/3)(45) = 312 in.
h/2a = 0.916
From the literature (Reference 2), the fundamental slosh frequency ·parameter was defined as, using Figure 2.24,
(46)
The fundamental frequency was related to the slosh frequency
parameter
as~
·--
yl
(47)
\1V (l;;) tanh1rh
_1_~
l
2.05
=
w.l
= 2.237 rad/sec
f
==
1
0. 356 Hz
For calculation of the second and third modes, a parameter
chart (Reference 2) was used,
For n = 2 (second mode)
~
y2
.&.
f
(48)
3.3
2 = 0.573 Hz
For n = 3 (third mode)
(49)
:. f 3
= 0. 729
Hz
Since these slosh frequencies were removed from the shell frequencies, dynamic coupling will not result.
48
Now the effect of the additional liquid mass on the dynamic
behavior of the fluid and solid system was considered.
The system
was approximated as one with a single degree of freedom for the
initial calculation of natural frequency.
Although it was under-
stood that the vessel shell system had ·multi-degrees of freedom,
the fundamental could be approximated in this ma.nner
and
the pr-ox-
imHy of the liquid and solid frequencies investigated further.
The hand-calculated value for the fundamental vertical response
of the vessel shell structure, using a thin shell approximation
{Refer·ence Section II.B.3) was 43.01 hertz.
From this value, using
the empty vessel mass of 107,270 lb., the relative stiffness, i.e.,
spring constant, was calculated as,
M
(50)
g,...
\...
(10],270)
386
K - 2.0295
X
10 7 lb/in.
This represented the stiffness of the vessel structure without
the liquid contents.
386,059 lb.
lb.
The additional fluid mass to be included was
This would increase the
tot~l
system mass to 493,329
The single degree of freedom shell and liquid frequency in-
eluding the impulsive mass was calculated as,
49
r------------------------------------ ------------------------------------------------ ---I·
fn = 1 1/
2-rrV
I
I
K
M
(51)
=1
21T
fn
= 20.06
Hz.
This demonstrated that the frequency of the vessel structure
system~
when the liquid mass contents were included, was removed
by at least one order of magnitude from the fundamental liquid slosh
frequencies.
Therefore, dynamic coupling of the internal liquid
and the shell structure should not tesult.
The additional convective fluid loading on the vessel structure
was calcula.ted by techniques in the liter·atut'e (Reference 1).
acce 1erati on of the vesse 1
~o~;ou1
The
d generate hydrodynamic forces
acting outward on one side of the vessel and inward on the other.
Although the mathematical procedure Vl'as very complex, the correlations simplified for the convective slosh force.
The time variation
of the periodic force was given as,
(52)
with,
H
1
=oscillatory convective mass of internal liquid
I
I
I
l
---- -------
wl
·1
w= o. 527 n tanh
------------- -- ---- ----- --
---
h
( 1. 58 T)
-- --- ----- --- ~- - ---- --- ------ - - - - - - - - - - - -- -
50
6
h = 1 · 58
;1
tanh (1.58
~)
A1 =maximum displacement of the convective mass .
..
(Was approxhnated by Y max. from the spectral curves)
w2
h and 1
wer~
as previously defined.
For the transverse and longitudinal directions, the maximum
convective slosh force was calculated as,
(53)
The values cf eh and A1 were obtained by us·ing the seismic response spectra curves for the transverse and longitudinal directions
(Reference Section II.B.l), respectively.
Results were:
Al = 5.675 inches (transverse at 0.587 hz)
=
(54)
9.643 inches (longitudinal at 0.356 hz)
eh - 0.0218 (transverse)
- 0.0371 ( 1ongitudi na l )
The slosh forces were evaluated as
Ps
= 4073 lb (transverse)
- 6932 lb (longitudinal)
(55)
51
Using the concept of projected areas for the two directions,
these fluid convective slosh forces were equated to a net pressure
resultant.
For the 216 inch diameter vessel, of effective length
of 312 inches, this resulted in pressure resultants of 0.06 psi
transverse pressure and 0.19 psi for the longitudinal direction.
Pressure resultants of this magnitude would not appreciably affect
the forced system response compared to the response spectrum excitation.
Therefore, only the additional mass of the liquid has been
included in the simulation model and convective effects have been
considered as secondary in magnitude.
SECTION III.
A.
RESULTS
SAP IV MODEL
The results of the five finite element models used for simula-
tion of the structural response of the cylindrical vessel are presented in this section.
Natural frequencies and mode shapes for
the characterization of the dynamic response of the vessel are included.
These results have been obtained for the vessel shell and
support only, without liquid contents, as well as for a full vessel.
Separate dynamic models have been used for each seismic re~ponse
direction.
The ldcations of maximum stress in the static
and the seismic models have been identified.
Stress levels have
been tabulated by region for the various conditions and design marg·ins of safety ca-lculated using the techniques of the
Vessel Code.
AS~1E
Pressure
The ground reaction or embedment loads at the base of
the vessel have been determined.
The results have been presented first for the vessel and shell
structure on-ly, that is, the vessel in the empty condition without
the effect of internal liquid.
Since the amount of data obtained
from the finite element models was extensive only the significant
results have been included.
A summary of the natural frequencies for the structural models
, of the vessel shell and support structure in the empty condition is
included as Table 1.
The lowest natural frequency is 23.63 hertz
for the transverse dynamic response.
52
In the vertical direction
TASLE 1
SUMMARY OF NATURAL FREQUENCIES
AND MODE SHAPES FOR TilE SEIS~IIC MODELS
OF THE STRUCTURAL SI!ELL AND SUPPORT ONLY
(LIQUID MASS CONTENTS = 0)
r-·
Seismic Model
!.
Vert1ca1
II.
I
1
Mode
,
-Number
--
I
Eigenvalue
Mode
E·!genvalue
Shape
(Hz)
t=·
(Hz}
1
1
L
Transverse
longitudinal
------y------------·------~
t·1 d
--·
2
.
~
________[_
!
39.66
46.84
Ref
Fig
II
Ref: Fig: 13
2~.~3
•"9
46
T=====
... o e
. .~~.
Ref. F1i.
F~ • 11~
I
--·
Same '' "•del I
,_ _ _ _ _L _ _
U"!
e...:;
54
natural frequencies are higher, beginning at 39.66 hertz.
The
longitudinal response was a duplicate of the vertical since it was
believed that the simulation for the empty vessel condition could
be identical for these two situations.
The listed eigenvalues form
a· summary of the system natural frequencies for the vessel shell
structur·e alone v:ithout the effect of contained liquid.
Characterization of the modal response for the vessel and
shell structure is provided by Figures 10 through 13.
These modal
response curves were hand-plotted using the computer output data
for the system eigenvectors for the simulation models.
Figure 10
is representative in format of the type in which all the modal shape
plots are presented.
Sections have been taken through critical zones of the vessel
to graphically illustrate the modal response in a somewhat brief
manner.
Longitudinal planes have been passed vertically and hori-
zontally through the cylindrical shell to ascertain the overall
motion of the shell and support structure.
Finally, three trans-
verse sections5 A-A and C-C at the zone of saddle support and stiffening rings and B-B at the vessel center, were used to demonstrate
local modal behavior of the cylindrical shell.
The modal behavior of the cylindrical shell structure as evidenced by the finite element results was typical of that predicted
by the more classical techniques.
The fundamental mode exhibited
was a transverse rolling motion of the vessel shell above the
supports with no apparent axial or circumferential nodal pattern.
55
I
·-----1-
1
FIGURE 10
FIRST MODE SHAPE
VESSEL STRUCTURE
TRANSVERSE SH1ULAiiON
56
J
c
l
I
I
I
---I-
I
\\
IH
..\ ' J:.S
'\\~t7"
'-~_n;
I
I
1
'~~~;?j'
!.fZ 1$5
..... _
L --- __j_
.
---~-
•
.
s£.C\\()~ c- c
I
L
____
_ _ __j
FIGURE 11
FIRST MODE SHAPE
VESSEL STRUCTURE
VERTICAL SIMULATION
S7
FIGURE 12
SECOND MODE SHAPE
VESSEL STRUCTURE
TRANSVERSE SIMULATION
58
.,......----------------------
l
L-----~-~--~-~----------·------=-~--~-=--~-=--~-~---·------
-----·----
----------·--------~
I.
--'-
____
.,._
1
FIGURE 13
SECOND MODE SHAPE
VESSEL STRUCTURE
VERTICAL SIMULATION
59
Local effects were observed near the saddle supports and stiffening
ring region of the shell.
This was shown in Figure 10.
For the higher modes, the vessel shell exhibited a pitching
type of motion in the longitudinal direction above the saddle
supports.
Also observed was the beginning of very definite circum-
ferential and axial nodal patterns in the shell itself.
The axial
nodal pattern is evident in the central section of each of the
figures,,
The circumferential nodal pattern is very clearly de-
picted in Section B-B of the modal response plots.
These mode
shapes are characteristic of the dynamic response to be expected
from a thin shell vessel.
Results are now presented
the full condition.
fOl~
the vessel shell structure in
This represents the more critical condition
with respect to seismic design and analysis.
The system natural frequencies will be lower and will be more
likely to be within the range of most common seismic response
spectra curves.
Values of maximum stresses will be larger due to
the additional system mass and the larger seismic excitation levels
for the response spectra used.
larger for similar reasons.
Ground reaction loads will be
Therefore, the results for the full
vessel condition are more critical than those for the empty vessel
condition and deserve close examination.
A summaty of the natural frequencies is included as Table 2
for the three dynamic models employed for the vessel structures
with total liquid contents modeled by the mass lumping technique.
60
The fundamental or lowest natural frequency for this condition is
9.43 hertz for the transverse dynamic response.
The response in
the longitudinal direction results in a somewhat larger value of
natural frequency beginning at 12.62 hertz.
Vertical simulation
natural frequencies begin at a value of 16.13 hertz.
All the frequency values as an entity in Table 2 form the
fundaf!!ental and the higher harmonics for the system.
The fundamen-
tal frequency for the system would be 9.43 hertz, and the remaining
values taken in ascending order would form the higher harmonics.
It is of interest to note that these frequencies as a whole are
much lower and much more·closely spaced than for the empty vessel
cond'ition.
The results of the tht·ee simulation models have been summarized
individually to emphasize the variation in the expected system response from the three distinct simulation models.
Resultant stress
levels and ground reaction dynamic loading may be correlated using
results from the individual dynamic models and by examination of
the seismic response spectra curves.
It would be expected that a
larger percentage of the total dynamic load and stress would result
from the response direction that possessed predominant modes near
the peak zone of the seismic spectra.
As illustrated in Section 11.8.1, the basic finite element mesh
used for the structural model included only one-half of the structure
on one side of the vertical plane of symmetry.
reduce the size of the rnode·l.
This was done to
A decrease in time expenditure for
TABLE 2
SUM!1ARY OF NATURAL FREQUENCIES
AND MODE SHAPES FOR THE SEISMIC MOllELS
(LIQUID t4ASS CONTENTS = 100%)
"
Se'!sm1c Model
III.
Mode
Number
I E1 genva 1ue
(Hz)
v.
IV. Transverse
Vertical
Eigenvnlue
Mode
Shape
(Hz)
.
Mode
Shape
Longitudinal
Eigenvalue
(Hz)
'-----
1
16.13
Ref. Fig. 14
2
16.90
Ref. Fig. 15
3
17.32
Ref. Fig. 16
4
18.62
18.53
22.79
5
18.78
19.90
26.30
6
19.33
20.70
27.49
7
19.98
21.23
I
Mode
Shape
9.43
Ref. Fig. 21
12.62 Ref. Fig. 24
16.74
Ref. Fig. 22
15.86 Ref. Fig. 25
17.79
Ref. Fig. 23
18.53 Ref, Fig. 26
-
I
_j
28.81
I
0')
!-'
62
the actual modeling process as well as a large decrease in computer
time for the eigensolution process for the models were direct results.
However, this did entail adjustment of the physical boundary
conditions at the plane of symmetry and variation of the mass
lumping constants to provide accurate simulation for all conditions.
This resulted in two unique mode 1s for the empty vessel shell
structure and three for the full vessel simulation.
Modal behavior for the vessel shell structure with total liquid
contents was similar in many respects to that for the empty vesse'l
condition.
The fundamental mode was again a transverse rolling
motion of the vessel shell with the absence of an axial or circumferential nodal pattern.
For the higher modes, the definite circum-
ferential and axial nodal patterns again become evident.
The modal
response is that to be expected of a thin shell vessel.
The plotted mode shapes are presented in groups for the vertical~
transverse, and longitudinal simulation directions.
allows the results of the individual
models~
This
in accordance with the
tabulated frequencies, to be more readily examined.
The plots for
the vertical simulations are shown as Figures 14 through 16.
The
transverse modes in Figures 21 through 23, and the longitudinal
characteri-stic modes in Figures 24 through 26, are located in Section
VII as Appendix C.
Some differences in the modal responses between the vessel
shell structure without internal liquid and that with full mass
lumping should be noted.
The longitudinal simulation for the full
63
--------j
r-
.
~Iff
,.i!i1
., IJSS
'1'151
I.
I
--~
-· -- -,
I
II
-~I
I
J
FIGURE 14
FIRST MODE SHAPE
VESSEL STRUCTURE (FULL CONDITION)
VERTICAL SH~UUHION
54
L
- .,zl·
=-T
+Xf
----------------------
--------·----.------------,.-----------1'1
.
,
~~
·x__l /.:_'}5J
~-~;,
·-·
~
Zz8
~:Ul
~
.
/,
ISS
Zv5
1
2!rC
1 .2SZ
I.
--~
---1--
I
FIGURE 15
SECOND MODE SHAPE
VESSEL STRUCTURE (FULL CONDITION)
VERTICAL SIMULATION
I
I
----1-·
65
------------~------------------------·--------~
I.
~--~
I
I
s,C,\0~
• FIGURE 16
THIRD MODE SHAPE
VESSEL STRUCTURE (FULL CONDITION)
VERTICAL. SIMULATION
c-c
66
vessel reveals a local
diaphl~agm
type behavior of the vessel head.
This is due in large part to the additional mass lumped in that
region.
The effect can be observed in Figure 26.
Local effects
in general are much more evident due to the additional lumped mass
in the system.
The number of calculated values of stress resultants for all
the finite element models was large.
Five di.fferent models were
employed, each with 499 thin shell elements.
For each element,
thr'ee primary membrane and three secondary bending stress values
were computed.
In addition, the boundary element forces were cal-
culated.
The locations of the maximum stresses in the models were determined by examination of the computer results.
The models were
divided into six regions, vessel shell, vessel heads, shell-toradius transition, wearplates, saddle supports, and stiffening
rings.
The location of these zones by model element number is pro-
vided in Table 3.
Shown are the element numbers within each zone
for which the stress levels reach a maximum value.
As an illustration for the vessel shell, modeled by Elements
42-262, the maximum stressed element was Number 160 for the static
pressure condition.
This condition included an internal pressure
of 50 psig and the hydrostatic liquid and vessel deadweight loading.
Consulting the finite element mesh of Figure 3, this element is
located on the lower girth of the vessel shell between the vessel
center and the smaller sliding saddle support.
-
.. i
r-.
.0 I
TABLE 3
LOCATIONS OF NAXll'i!JM
STRESS HI FINITE ELEl':Pn W)DELS
58
For the vertical seismic simulation) the maximum stressed element was Number 158.
This element is located on the uppermost
portion of the shell on a transverse midplane at the vessel center.
The
transvers~
simulation resulted in a maximum stress value at a
location on the lower quarter of shell near the head transition on
the large fixed saddle side of the vessel.
44.
The element number was
Finally, for the longitudinal model, the
element in the vessel shell was Number 257.
maximu~
stressed
The element was located
on the upper quarter of the shell in the proximity of the vessel
shell to head transition on the small sliding saddle end of the
vessel.
Similar summaries were conducted for other zones of the vessel,
support structure, and ring stiffeners.
Stress levels of interest
from the models were of two categories, primary membrane, and bending stt·esses.
A summary of the primary membrane str·esses was
initially made for the models.
The membrane stresses were those
that result in the shell and plate elements due to the edge membrane
and shear force resultants that were required for dynamic equilibrium
of the vessel shell structure (Reference 20).
Membrane stresses
were classed by the ASME Code as the component of norma 1 str·ess
which is uniformly distributed and equal to the average value of
stress across
th~
thickness of the section under consideration.
Primary stresses \vere classed by the
AS~1E
Code as those developed
by the imposed loading which were necessary to satisfy the laws of
equilibrium.
They were not self-limiting ·and would result in gross
69
distortion if the yield strength was exceeded.
The general primar-y
membrane stresses were those that were distributed in the structure
such that no redistribution of load would occur as a result of
yielding.
Results for the primary membrane stresses have been presented
for the static pressure condition only, the combined seismic responses, and the final summation due to all combined effects. The
results have been included for each of the six major regions of
the models.
The principal stresses were calculated and a design
margin of safety computed using the techniques and stress allowables
.of the ASME Pressure Vessel Code Section III Requirements for Class
III Nuclear Power Plant Components (Reference 5).
These results
are shown in Table 4.
Examination of the table indicated the minimum design margin
of safety for the entit'e vessel shell structure
vessel sheil to head radius zone.
occun~ed
at the
The principal stress for this
zone was 12841 psi, and the design margin of safety computed at a
value of 0.17.
The next most critical regions were the vessel shell
with a design margin of 0.40 and the vessel heads with margins of
0.65.
The saddle support structure, wearplates, and stiffening
rings all had margins of safety greater than 1.0.
It was not ex-
pected that they would be critically stressed in the final analysis;
rather, their purpose was to provide rigidity to the vessel shell
to optimize its seismic response.
Results for the
pt'imal~y
membrane with the addition of the
TABLE 4
SUN~lARY
·
~-------···--:--·-----
___ _ j __ St<l_t.l~.---
f;;~,~~·--t ~x
G·~ncral
, l.
Vessel
5h.:1:
2
'
'
=-~'
I.
~:~~~, ~
6,857
"Y_ :_':"11
1.208
2.129
886
5,653
955
989
2,058
339
----·
H
l
"l
1\,;dLic-
!i
~
4,00/
10,380
292
l! 1.664
,--u- - ~L
-~
!
4.
1
I1 5.
6.
i
"'"''''" ~ 2,a91
474
o-y
Ii 4,914
i
10.587
7,856
7,711
l
~
5,591
'"
434
i 4,o93
897
Stiffening
I
11
338
1,262
I
3,042
206
I
5~_e;i
92711:242
4,180 1,420
l
3,390
954
l_,f·~~M!I
6.432
1,170
2,097
4.776
----
k
'
I
0.40
0.65
6,487115,000
,
5,510
l
'
1,B4z 1 15,ooo
I
·
7,166
15,000
~-
~54 __d
I
lS,OOO
I
I
2,066
:
904 2,898
10,726
1,295 I 9,081
,
1
I
~------,
1
Allo1·1ab1e Margin
·
St>·m, s 1 of Safety
"2
+I
m • .m
-+l
I
Code
Stress
765 112,841
'
1.m 276,1,202
TXY
ASME
~-- I
__
12,760
~--
1
1
Strudu"
R·ings
2,380
o-,
-1;
I
She 1 1
Saddle
Support
_ TXl'
8,458
r-----+ne;1·
ll 3....To--He:1d
("'
~
Sun,.nation
I
"Y
I
-~---Priri'cipal
-
L
______ se·lc-rl·ic
·
!i 3,706
~--~----n·-
i
Of PRIM/\R'f NEWJRANE STRESS {psi)
I
0.17
,.47
J=r=-
437
15,000 I
2.16
15,000
1.09
J-
......
0
71
------- --------- ------ ---·- ----- -------- ----
bending stress terms were obtained and are shown in Table 5.
The
results are presented in the same general format as those for the
primary membrane stress condition.
The bending stresses were those
that resulted in the shell and plate elements due to the edge
bending and twisting moments required for dynamic equilibrium of
the vessel.
The bending could be classified as one of either two types,
primary or secondary, according to the guidelines provided by the
ASME Pressure Vessel Code (Reference 5).
Primar·y bending stresses
were those due to bending across a full section or in the central
region of a head or shell'.
Secondary bending stresses were those
bending stresses at a gross structural discontinu-ity such as the
junction of a vessel head to a shell.
The summary table indicated the minimum design margin of
safety for the membrane plus bending condition occurred in the
vessel shell.
The principal stress for this zone was 11002 psi.
and the design margin of safety was computed at a value of 0.36.
The next most critical region was the vessel heads with a design
margin calculated at 0.50.
Although the stress levels were rela-
tively high in the vessel shell to head radius zone, the bending
stresses were classed as secondary in this r·egion.
With the inclu-
: sian of secondary stresses, the ASME Code allowable stress became
3S, rather than the allowable S, which resulted in a margin of
; safety greater than those of the other shell regions.
support
stl~ucture,
The saddle
wearplate, and stiffening rings once again had
f ----------------------------------------------------------
---------------------------------~
I
i
TABLE 5
I
I
I
SUMMARY OF PRIMARY MEMBRANE
AND DENOTNG STRESS (psi)
!
I
~
General
1o-x
Region
I
1.
2.
"Y
I,
Vessel
Shell
~ 3,834
-~~
""d'rl.
Vessel
Stat~-
3. She 11To-Head
Radius
8.533
TXY
a-x
a-y
5,529 10,855
956 1,076 2.143
339
9,191
897
11,002
5,382 15,000
0.36
o,us
5,727
7,870 1,295
9,984
7,077 15,000
0.50
6.
Saddlt!
Support
Structure
I
I
I
l
3,515
3.712 276 11,498
779
501
--
St1ffenir.g
Rings
13.338
1,322 493
989 3,072
766 15,889 13,253 45,000(a)
474 13,499 15,643
434
5,013
710
5,509
3,996114
4,394 1,420
4,973
911,22,500(b)
1,280 2,097
7,497
573 1 1s,ooo
4,491
•·
5.
TXY
ASME
Code
Allowc:b1e Margin
Stress, S of Safety
-
11,004 12,364 292 2.495 3,279
Uearplates
a-y
886
I
4. Shell
a-x
TXY
1,695 2,322
11
Principal
Stress
(Tl
a-2
Summation
Seismic
927! 1,490
-
--
!
225 528 3.452 1,055
1,56~-L 6,790
-
(a) In the vessel-to-head radius region. the bending stt•ess is a secondary.
(b) The saddle support structure is govel·ned
by Sect·ion l'lf
-
Q,
I
1.83
J.72
3.52
1.00
stress and the allowable becomes 3S.
of tha ASME B&PV code.
·--- -·--·- ---·---·
----·-·--·-·-----------·
- -·-- ----~---····-· - · - · J
"--1
,._.,
73
margins of safety greater than one.
The final results from the finite element models are the ground
reaction or embedment loads.
These loads are important for the
foundation design or the seismic design of structural attachments
at the base of the vessel.
The loads were computed for each of
the boundary elements at the base of the vessel.
The net loads
presented represent the combined effect for the three seismic response directions.
Force components, such as the vertical, include
the effect of dynamic system excitation in the transverse and longitudinal as well as vertical directions.
The reaction loads at
the large fixed saddle base are shown in Table 6 while those at the
smaller sliding saddle base are shown in Table 7.
'
The vertical reaction loads were the largest of the three force
'
components on both the fixed and sliding saddle base locations.
This
is primarily due to the vertical offset of the vessel center of
gravity and the net overturning effect produced by transverse and
longitudinal dynamic forces.
This becomes an important consideration
in the seismic design of large vessel structures.
Finally, it is of interest to note the effect of relative
member stiffness on the dynamic load distributions.
The higher mag-
nitude loads at the boundary elements near the stiff saddle support
uprights is evident from examination of the tables.
An example
would be the loading at Elements 8 through 11 for the fixed saddle
base.
These boundary elements are near the vertical uprights of the
saddle support oriented transverse to the vessel shell.
Over 50%
74
TABLE 6
SUMMARY OF GROUHD REliCTION LOADS (1 b)
FOR FIXED SADDLE BASE Dlit. TO SEISMIC SPECTR."L EXCITATON
(LIQUID MASS CONTENT = 100~)
r
Boundary
Spring Ele:nent:
Numbers
I
I
-+-
..
1, 31. 61
2 I
3
4
5
6
7
8
9
!0
11
.
I
I
l
12
13
14
15, 45, 75
16
17
13
19
20
21
~
22
23
24
25
26
27
28
. 29 1
30, 60, 90
TOTAL LOAD
I .
.Force
Vertical
54.
129.
306.
216 •.
64.
37.
124.
40,759.
66,817.
75,577.
67,384.
97.
179.
402.
12,752.
23,742.
178.
45.
100.
19,072.
33,355.
59,458.
40,794.
49.
31.
62.
240.
287.
137.
12.
443,969.
I
l __--:-·--·--· ····-··-------······-·-------·---
Cor~.ponent
Latera1
605.
857.
1. 911.
674,
200.
138.
3,2•U.
11,.165.
17,661.
6,866.
27,099.
537.
707.
231.
643.
833.
270.
416.
1,457.
5,201.
6,279.
5,019.
7,032.
1,034.
268.
317.
477.
1,016.
706.
323.
103,533.
-
Longitudinal
405.
982.
3,385.
3,188.
471.
108.
198.
1,089.
14,889.
14,622.
1,099.
93.
103.
1,572.
23,735.
25,618.
1,871.
110.
175.
1,053.
14,185.
15,827.
1,131.
112.
184.
465.
3,342.
3,682.
176 •
246.
134,716.
I
I
75
r---------------------4-------
-~----------------~
---------------- ------- -------------- _ . ___ --------- ----------------
-----------------~
'
TABLE 7
OF GROUND REACTION LOADS (l b)
FOR LONGI7UOINALLY FREE SAOOLE BASE
SUM~1ARY
DUE TO SEISI1TC SPECTRAL EXCITATIOri
(LIQUID MASS CONTENT = 100%)
Boundary
Spl"ing
E1er.~ent
~------------~F~or~c=e C~,pcnent____________~
Vertical
Numbers
Laterai
longitudi!'lal
~=======+=-=--===========
19, 121
92
93
94
95
96
97
98
99
100
101
102
I
103
i
104
t
105, 135
105
107
108
109
110
111
I
112
113
i
i.14
I
116
117
118
119
I
I
I
I
us
120, 150
TOTAL LOAD
63.
279.
438.
419.
446.
41.
192.
372.
2,750.
3,308.
91.
891.
25,511.
30,672.
39,071.
49,308.
72.
154.
512.
15,096.
21,594.
669.
37.
213.
41,145.
32,105.
63,123.
72.309.
128.
120.
196.
163.
570.
351.
68.
394,964.
0.
I
374.
1~3.
6,572.
8,754.
5,247.
4,830.
261.
751.
731.
3,005.
I
-I
2,772.
887.
49l..
1,587.
11,362.
22,404.
17.37-1.
9,430.
545.
364.
1,314.
8,378.
7 ,891.
1,443.
284.
124,695.
o.
I
I
I
II
i
!
i
L__________________ ---------_________· --------------------------------------------------·---------·-- - - ____...i
-
76
of the seismic load is reacted through these four elements.
This
would be an important consideration for design of the vessel
support pads.
B.
LIQUID MASS FRACTION PARAMETRIC STUDY
Numerical results obtained from the parametric study model
demonstrated the effect of liquid mass on the dynamic response of
the system.
As the amount of liquid mass in the system was in-
creased, the fundamental frequency of the system decreased in a
steady manner until the lowest frequency was reached for the full
vessel condition.
This behavior was consistent throughout the first
four modal responses examined.
The effect of the liquid lumped mass on the frequency response
of the system is illustrated in Figure 17.
The ratio of the fre-
quency of interest to the fundamental frequency of the liquid-shell
system has been denoted wi/wf.
These have been plotted versus the
fraction of liquid mass in the vessel which is labeled ~1 L/MTL"
The plot represents a normalization of all frequencies to the
fundamental frequency for the full vessel (ML/MTL = 1.).
The con-
sistent decrease of the frequency ratio to that of the full vessel
fundamental can be observed.
t~odal
responses for arbitrary frac-
tions of liquid contents of the vessel can be determined by examination of the frequency ratio plot.
Aithough the primar·y results of the parametric study were the
variation of frequency versus liquid mass fraction, the representa-
...
77
:r
-~~
4
3'.
~
0!
~!
3
l; f .
a:
ail
·::>•
ol
w'
a:
I
__I
U.'
o.:
0.2
0.4
0.6
0.8
1.0
LIQUID MMS FRACTION, ML/MrL
FIGURE 17
FREQUENCY RATIOS VS LIQUID MASS FRACTIONS FOR THE
FIRST FOUR VESSEL PARAMETRIC MODEL MODAL RESPONSES
78
tive frequencies and mode shapes for the parametric models have
been included.
A summary of the natural frequencies for the first
ten modes is included as Table 8.
The frequencies ranged from
18.54 to 2086.39 hz depending on what fraction of liquid was
modeled throughout the ten modal responses.
The first three mode shapes for the vessel have been included.
The mode shapes reveal the characteristic behavior of the vessel
shell and support structure dynamic responses.
Vertical and trans-
verse modal deformation of the vessel shell were shown as was the
longitudinal distortion of the shell and saddle support structure .
. The mode shapes are shown as Figures 18 thrtiugh 20.
79
,-----------·- --------- ----------·..,.--- --------·- ---- ---------
i
i
L_______ ---~-----------------------
··---
------- -----
-------- ....
·---·-----·· ···--· ---------·--··· --·-··-·
80
. !
- - - - - - __ :..._-- - - - - - - - - - - ____ J ________ _
----=-rr-----
-. --=-==----------~----!I
---
---------I
1
\
.,
.
1
tlI
.
I
I
'
-i
FIGURE 18
FIRST MODE SHAPE
VESSEL AND LIQUID MASS PARAMETRIC MODEL
...... ·-··-·
. -------- ---·-- -- ·-
. --· -·-··-· ···-·-·- ------·---~----~----·-- --·
-----------~
. -· -- ··-··- ~ ·----·-·
--- --~-- -----
81
--------------- -
..
I
---------~
.....
_ .... __
----- ... -
-------------------r--------
------~---~-~-~-~-~
~
I
I
I
I
I
I
I
I
I
I
FIGURE ·19
SECOND MODE SHAPE
VESSEL AND LIQUID MASS PARAMETRIC MODEL
-···
----·-··------·
---··· ·---···------·--
--
---···-------
-----·--·-----.
82
-----------------------~-~~-----------------------=-
--- ---------------- --
-------- =-T,·-11
., . . . --- ----:-
----------------~
.
I
I
\
I
I
'J
\
\
I
I
FIGURE 20
THIRD MODE SHAPE
VESSEL AND LIQUID MASS PARAMETRIC MODEL
- -·--
-~--
---
·····--
----·------
---
---·
-------------------- ---------- --------- ·----- ----------·--- ----
··-· -----~------ .. ---
------------
----
-~----·.-
------··
-
SECTION IV.
DISCUSSION
ihe primary purpose of the symmetric
one~half
shell structure
simulation was for economy through reduction in the size of the
required computer models.
Appreciable expense was saved by using
these models, with adjusted boundary conditions and mass lumping
constants, over a full vessel shell simulation.
Each of the current
models required approximately 100 IBM system billing units for computer solution.
Since computer costs are proportional to the number
of billing units, and the required billing units for a dynamic solution are propdrtional to the square of the system bandwidth,
appreciable cost savings resulted from the smaller simulation models.:
; · Further cost savings may be realized by the use of a save tape for
the eigenvalue solution.
These can later be used for additional
forced response cases.
The finite element models required a rather fine simulation
mesh to yield accurate results.
The modal responses and stress
values were predicted in a reasonable manner.
Direct comparison of
the results with a classical closed form solution would be difficult
due to
~he
complexity of the structural geometry involved.
When a
shell structure is discretized, the eigenvalues are upper limit
values.
The system frequencies approach the exact values as the
stiffness of the structure is reduced due to the decrease in the
mesh size used in the idealization process.
I
I
As a further result of the liquid mass fraction parametric
.
!_. __________ -··- ·-·---· ----·--·-········-- - -·-·· -·--- .... ·---·---·-----·----··-·-··-···-·-· - · - - - - - - - - · - ···-·-··----·-· -··-·· . ··-··--··
83
84
study, another important part of the finite element idealization, a
family of curves could be generated which would depict the fractional
mass effect in graphic form.
Various shell geometries with varying
length to radius and radius to thickness ratios could be used and
further parametric study conducted.
The parametric curves would
serve as a design tool to assist a seismic vessel designer.
They
would allow the prediction, for a full range of geometries, of the
fundamental and higher level modal responses for any arbitrary
amount of fluid contents.
Such parametric curves for vessels are
presently unavailable in the literature.
Some methods from the literature were applicable and were used
to investigate the thin shell vibration problem and the effects of
liquid sloshing.
The thin shell vibration technique, as applied to
the vessel structure, could only be considered as approximate due
to the complex geometry involved.
No simple classical technique
could predict the dynamic response accurately.
The calculations
for the effects of sloshing indicated the absence of dynamic
coupling and relatively low slosh loads.
This assumed a linear
vibration response, since for significant nonlinear behavior, of
either the liquid or thin shell, coupling of the systems could
occur even though the shell and liquid frequencies are widely separated.
Many general considerations were made during the seismic modeling
process.
These considerations would apply in general to any
evaluation of a vessel structure which contained liquid.
sei~mic
Of impor-
85
tance would be the seismic criteria to be applied to the structure,
1
either an equivalent static technique such as the Uniform Building
Code or a mode superposition method such as the squar·e root of the
sum of the squares method.
The general vessel geometry and_ the
length to radius and radius to thickness ratios would be considered,
as would the leve1 of liquid contents in the vessel.
vessel support would also be selected.
The method of
Design techniques, such as
added support stiffness in the transverse and longitudinal directions
as well as the vertical, to withstand seismic loading would be used.
For the case of saddle supports, stiffening of the assembly by the
addition of an integral ring and saddle structure would help to
optimize the seismic response.
Many of the methods illustrated in
this report could be applied in a general way to any arbitrary vessel
being evaluated.
Further studies are possible.
Other liquid mass fraction para-
metric investigations could be conducted.
Alternate simulation
models could be constructed using other types of finite elements.
Some of these may be the isoparametric or thick shell variety.
In
particular, the internal liquid could be modeled with the use of a
fluid finite element network.
These fluid finite elements are
' presently being developed and soon will be available to complement
the fluid-solid modeling process.
SECTION V.
SUMMARY AND CONCLUSIONS
ihe finite element method provided a good simulation of the
vessel shell structural assembly.
The discretization of the shell
into a series of flat plate elements resulted in a model which
yielded reasonable results for combined bending and stretching
structural effects.
The finite element model did require a very
fine mesh which consisted of a large number of elements.
The vessel shell model employed represented one-half of the
shell about a vertical plane of symmetry and did provide an accurate
simulation of the dynamic response of a thin shell structut·e.
The
agreement with the hand calculated natural frequency for a thin
shell was fairly close considering the complexity of the system
involved.
Computer simulation resulted in a natural frequency of
46.39 Hz versus a hand calculated value of 43.01 Hz.
This lended
support to the technique of modeling only one half of the vessel
structure with adjusted boundary conditions at the plane of symmetry.
Resulta~t
savings in both engineering time and computer solution
time may be realized in this manner.
The effect of the internal liquid on the dynamic response of
the shell system was demonstrated with the liquid mass parametric
study.
The fundamental frequency of. the system decreased in a
monotonic fashion as the amount of liquid mass in the system was
increased.
This behavior was consistent for the higher modal re-
sponses as well.
Prediction of system frequencies may be made for
86
87
vessels with similar geometry with the use of the frequency ratio
plot.
Should the liquid mass of the vessel be decreased to one-
fifth of the full amount, the fundamental frequency of the system
will double in magnitude.
·The dynamic simulation of the vessel structure indicated that
the full vessel condition was the most critical from the standpoint
of engineering design.
The fundamental modal responses were first
the transverse, then the longitudinal and vertical, in that order.
System natural frequencies were 9.43, 12.62, and 16.13 Hz, respectively for those cases.
Mode shapes were characteristic of those for
a thin shell with the additional lumped mass effects evident.
Critically stressed areas for the full vessel condition were in
the region of the torispherical head to cylindrical shell transition
zone.
The next most highly stressed area was in the vessel sheli.
Ground reaction seismic embedment loads were also largest for the
full vessel condition.
The vertical reaction loads were the largest
in magnitude primarily due to the vertical offset of the vessel
center of gravity and the net overturning effect of the dynamic
forces.
These become important
co~siderations
in the seismic de-
sign and analysis of large vessel structures.
The effects of internal fluid sloshing for a vessel of this
geometry were not significant.
Fluid slosh frequencies at values
less than 1.2 Hz were well removed from the structural
~hell
fre-
quencies which indicated that dynamic coupling of the liqu·id and
solid systems would not result.
This justified the use of the
88
distributed fluid inertia on the shell walls to approximate the
hydrodynamic loading.
Linear behavior of the liquid and solid
dynamic responses was implicitly assumed.
The fluid slosh pressure
loads were secondary in magnitude compared to the seismic response
loadings.
The slosh pressures were less than 0.2 psi.
REFERENCES
SECTION VI.
1.
Holmes and Narver, Inc., comp. and ed., "Nuclear Reactors and
Earthquakes, 11 Technical Report TID 7024, United States Atomic
Energy Corrrnission, Division of Technical Information, 1963
2.
Abramson, H.N., "The Dynamic Behavior of Liquids in fv1oving
Containers," NASA Report SP-106, National Aeronautics and
Space Administration, Washington, D.C., 1966
3.
Back, P.A. and Cassell, A.C., "The Seismic Design Study of a
Double Curvature Ar·ch Dam," Proceedings of the Institute of
C)vil Engineers, Vol. 43, 1969
4.
Bathe, K.J., Wilson, E.L., and Peterson, F.E., SAP IV- A
Structural Analysis Program for Static and Dynamic Response of
Linear Systems, Earthquake Engineering Research Center,
College of Engineering, University of California, Berkeley,
1973
5.
ASME Boiler and Pressure Vessel Code, Section III, Division I,
11
Subs~ction NO, "Class 3 Nuclear Power Plant Components,
The
Americ::.n Society of Mechanical Engineers, United Engineering
Center, New York, July 1974
6.
Leissa, A.W., "Vibration of Shells, 11 NASA Report SP-288,
National Aeronautics and Space Administration, Washington,
D.C., 1973.
7.
Desai, C.S., and Abel, J.F., Introduction to the Finite Element
Method :- A Numeri ca 1 Me~hod for Engineering Analys-is, Von
Nostrand Reinhold Company, 1972
8.
Cook, R.D., Concepts and~2~cations of Finite Element Analysis - A Treatment of the Finite Element Method as Used for
the An~lysis of Displacement, Strain, and Stress., ~~iley and
Sons, New York, 1974
9.
Zienkiewicz, O.C., The Finite Element Me_thod in Engineering
Sci_~nce, McGraw-Hi rr; London, 1971
10.
Hurty, W.C. and Rubinstein, M.F., Dynamics of
Prentice Hall, New Jersey, 1964
11.
Kraus, H.~ Thin Elastic Shells -An Introduction to the Theoretical Foundat10ns ana·the Analysis of Then· Stat1c and
Dvnarnic Behavior, Wiley and Sons, New York, 1967
89
Structure~,
90
···-
12.
Bird, R.B.~ Stewart~ W.E.~ and Lightfoot~ E.N., Transport Phenomena, Wiley and Sons, New·York, 1960
13.
Tayler, C., Patil~ B.S.~ and Zienkiewicz, O.C., 11 Harbour Oscillation: A Numerical Treatment fm~ Undamped Natural Modes, ..
froceedings of the Institute for Civil Engineers, Vol. 43,
1969
14.
McCarty, J.L. and Stephens, D.G., 11 Investigation of the Natural Frequencies of Fluids in Spherical and Cylindrical
Tanks, .. NASA TN D-252, 1960
15.
Bathe, K.J. and Wilson, E.L., Solution ~1ethods for Eigenvalue
Problems in Structural Mechanics, .. International Journal for
Numerical Methods in Engineering, Vol. 6, No. 2, 1973
16.
Bathe, K.J. and Wilson, E.L., 11 Large Eigenvalue Problems in
Dynamic Analysis, .. ASCE Journal of Engineerino Mechanics
Division~ 1972
17.
Clough, R.W~~ 11 Earthquake Analysis by Response Spectrum Superposition, .. Bulletin of the Seismological Society of America,
Vol. 52, No. 3, July 1962
18.
Gelman, A.P., .. Report on the Dynamic Response in Piping Systems
Computer Code, .. Rockwell International Corporation, Atomics
International Division, Canoga Park, California, 1975
19.
Budiansky, B., .. Sloshing of Liquids in Circular Canals and
Spheri ca 1 Tanks, 11 Journa 1 of the Aero/Space Sciences_, Vo 1. 27,
Number 3:t 1960
20.
Timoshenko, S. and Woinowsky-Krieger,
Shells~ McGraw-Hill, New York, 1968
21.
ASME Bo·iler and Pressure Vessel Code, Section III, Division 1,
Subsection NF, 11 Nuclear Power Plant Component Supports, 11 The
American Society of Mechanical Engineers, United Engineering
Center, New York, July 1974
11
-- --.-·-·--· --- --- -----·- -----
--------
-----·----- ------ ----- -
s., Theory of
__________
....,......._
P1ates~nd
------ ----
-~------
---- ---
SECT ION VI I.
APPENDIXES
APPENDIX A
SAP IV Finite Element Computer Model
91
5ENfRA T£t
NODC.l OAU
!IIIUNoARY CONDITION coors
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APPENDIX C
· SAP IV Finite Element Mode Shapes
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FIRST MODE SHAPE ·
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FIGURE 22
SECOND MODE SHAPE
VESSEL STRUCTURE (FULL CONDITION)·
TRANSVERSE SHqlJLAT ION
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THlRD MODE SHAPE
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TRANSVERSE SIMULATION
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FIRST MODE SHAPE
VESSEL STRUCTURE (FULL CONDITION)
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118
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SECOND MODE SHAPE
VESSEL STRUCTURE (FULL CONDITION)
LONGITUDINAL SIMULATION
SctiiON C-C
119
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THIRD MODE SHAPE
VESSEL STRUCTURE (FULL CONDITION)
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Se.c>ION C-C