Lehigh University
Lehigh Preserve
Fritz Laboratory Reports
Civil and Environmental Engineering
1990
Discretization errors in finite element models of
hyperbolic cooling tower shells, May 1990. 63p
Ashfaque Chowdhury
Celal N. Kostem
Follow this and additional works at: http://preserve.lehigh.edu/engr-civil-environmental-fritz-labreports
Recommended Citation
Chowdhury, Ashfaque and Kostem, Celal N., "Discretization errors in finite element models of hyperbolic cooling tower shells, May
1990. 63p" (1990). Fritz Laboratory Reports. Paper 485.
http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/485
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I[\~iil\iIIijl~ lij~i ~l l ili~[\i1i[IS DISCRETIZATION
39151009428091
ERRORS
IN
FINITE ELEMENT MODELS
OF
HYPERBOLIC COOLING TOWERS
. FRtrz
ENGtNEERlNQ
Lt\SORATORY liBRARY
by
Ashfaque Chowdhury
eelal N. Kostem
Fritz Engineering Laboratory Report No. 400.~O
DISCRETIZATION ERRORS
IN
FINITE ELEMENT MODELS
OF
HYPERBOLIC COOLING TOWER SHELLS
by
Ashfaque ChOWdhury
Celal N. Kostem
Fritz Engineering Laboratory
Department of civil Engineering
Lehigh University
Bethlehem, Pennsylvania 18015
May 1990
Fritz Engineering Laboratory Report No. 400.40
Table of Contents
ABSTRACT
1
2
1. Introduction
1.1 Context
1.2 Literature Survey
1.3 Problem Statement and Scope
2
2
3
2. Methodology
4
2.1 Overview
2.2 Finite Element Models
2.3 Standard Solution
2.4 Error Analysis
4
4
6
6
9
9
11
3. Data from FE Analysis
3.1 Overview
3.2 Analysis Results of Structure A
3.3 Analysis Results of Structure B
3.4 Analysis Results of Structure C
21
31
41
4. Discussion of Results
41
41
46
46
48
4.1 Effect of Mesh Fineness
4.1.1 Axial Direction
4.1.2 Radial Direction
4.2 Effect of Shell Geometry
4.3 Aspect Ratio Variation
4.4 Cost Effectiveness of Analysis
5. Conclusions and Recommendations
5.1 Conclusions
5.2 Recommendations
5.3 Summary
51
52
52
54
54
References
Appendix A. Nomenclature
Appendix B. Standard Solution
56
57
60
B.l Theoretical Background
B.2 Numerical Analysis
60
62
Appendix C. Vita
3
iv
List of Figures
Figure 2-1: Typical Hyperbolic cooling tower and FE Mesh
scheme
Figure 3-1: Shape of Structure A
Figure 3-2: Results for Structure A : Mesh size 24 * 6
Figure 3-3: Results for Structure A: Mesh size 24 * 10
Figure 3-4: Results for Structure A : Mesh size 24 * 12
Figure 3-5: Results for Structure A : Mesh size 24 * 15
Figure 3-6: Results for Structure A : Mesh size 12 * 12
Figure 3-7: Results for Structure A: Mesh size 16 * 12
Figure 3-8: Results for Structure A : Mesh size 20 * 12
Figure 3-9: Results for Structure A : Mesh size 30 * 12
Figure 3-10: Shape of Structure B
Figure 3-11: Results for Structure B : Mesh size 24 * 6
Figure 3-12: Results for Structure B : Mesh size 24 * 10
Figure 3-13: Results for Structure B : Mesh size 24 * 12
Figure 3-14: Result$ for Structure B : Mesh size 24 * 15
Figure 3-15: Results for Structure B : Mesh size 12 * 12
Figure 3-16: Results for Structure B : Mesh size 16 * 12
Figure 3-17: Results for Structure B : Mesh size 20 * 12
Figure 3-18: Results for Structure B : Mesh size 30 * 12
Figure 3-19: Shape of Structure C
Figure 3-20: Results for Structure C : Mesh size 24 * 6
Figure 3-21: Results for Structure C : Mesh size 24 * 10
Figure 3-22: Results for Structure C : Mesh size 24 * 12
Figure 3-23: Results for Structure C : Mesh size 24 * 15
'Figure 3-24: Results for Structure C : Mesh size 12 * 12
Figure 3-25: Results for Structure C : Mesh size 16 * 12
Figure 3-26: Results for Structure C : Mesh size 20 * 12
Figure 3-27: Results for Structure C : Mesh size 30 * 12
Figure 4-1: Relationship between Normalized RMS error in
Longitudinal Direction and Longitudinal Mesh
Fineness
Figure 4-2: Relationship between Normalized RMS error in
Meridianal Direction and Longitudinal Mesh
Fineness
Figure 4-3: Relationship between Normalized RMS error in
Longitudinal Direction and Meridiana! Mesh
Fineness
Figure 4-4: Relationship between Normalized RMS error in
Meridianal Direction and Meridiana! Mesh
Fineness
Figure B-1: Typical Shell Element and Membrane Forces
v
7
11
13
14
15
16
17
18
19
20
21
23
24
25
26
27
28
29
30
31
33
34
35
36
37
38
39
40
42
43
44
45
61
List of Tables
Table
Table
Table
Table
Table
Table
3-1:
3-2:
3-3:
4-1:
4-2:
4-3:
Summary of Results for Structure A
Summary of Results for Structure B
Summary of Results for Structure C
Comparison of Geometric Parameters
Maximum Aspect Ratio of Shell Elements
Average Aspect Ratio of Shell Elements
vi
12
22
32
47
49
50
ABSTRACT
The effect of mesh fmeness of fmite element models on accuracy of finite
element analysis solution is investigated for hyperbolic cooling tower shells.
The sensitivity of the errors to geometry of the shell is also examined. Elastic
material model and static gravity loading is used.
The study fmdings indicate that under symmetric state of stress,
inadequate mesh fmeness may cause significant error in the shell force estimate
from finite element analysis. The accuracy of longitudingal shell force estimate
is fairly insensitive to the geometry of shell. However even small changes in the
shape may drastically affect the accuracy ofmeridianal shell force estimate.
1
Chapter 1
Introduction
1.1 Context
Finite Element (FE) mesh characteristics for thin shell continuum
problems have traditionally been based on engineering judgement. The effect of
mesh fineness and different geometries on the accuracy of the solution is not
known. Whether these errors, arising from modelling of the shell are significant
or not is an open question.
This study tries to quantify the error in the estimate of shell forces in FE
analysis for different mesh refinement levels for the case of a hyperbolic cooling
tower shell. The sensitivity of these errors, due to gravity loading, to geometry
in this class of shells is also investigated. The material model is elastic and
analysis is linear. Error in only gravity loading is considered.
1.2 Literature Survey
There has been substantial research (Rheinboldt, 1985) on error
estimation of FE analysis results using different meshes for a given structure,
especially for nonlinear problems. However the purpose of these studies were to
develop computer programs for automatically generating and adapting finite
element meshes, i.e.:'Adaptive FE mesh generation".
focused on specific problems in thin shell analysis.
None of the research
How the different
parameters of the mesh itself effect the accuracy of the solution for a particular
shell structure was not investigated.
Most studies on FE analysis of hyperbolic cooling tower shells have
2
concentrated either on the seismic response modelling (Grant, 1980) or on the
stresses at the shell-column junction (lyer, 1990). Though publications on FE
modelling (Meyer, 1987) discuss the effects of aspect ratio of elements, little is
said about the effects of overall mesh fineness in thin shell problems. The effect
of modelling decisions by the analyst (e.g. pattern and fineness of FE mesh,
choice of element ete.) on the quality of the solution over the entire shell surface
has remained largely unknown.
1.3 ProbleDl StateDtent and Scope
This study investigates the effect of fmenese of FE mesh on accuracy of
solution of hyperbolic cooling tower shells of different curvatures. So that more
informed decisions about choice of mesh fmenese may be made for shells of this
type. The error analysis was based on an elastic and static FE model under
gravity loading. The scope of this study is limited to examining the relationship
of mesh fineness and accuracy of solution in a chosen range of hyperbolic cooling
tower shells. Effect of choosing different elements or other ranges of geometric
parameters is not investigated. The scope is also limited to elastic material
models and symmetric states of stress generated by gravity loading.
Number of divisions in FE mesh in longitudinal (height) and meridiana!
(radial) directions are taken as the primary variables. Error distribution of FE
results for different combinations of these variables are analyzed.
Different
structural geometries, in this class of shells, are also introduced as a variable.
Cost effectiveness of the analysis is also discussed. Based on these results
guidelines for FE modelling of hyperbolic cooling tower shells is sought.
3
som~
Chapter 2
Methodology
2.1 Overview
For an accurate estimation of errors introduced in FE analysis, a
consistent methodology has to be followed. This means a set of defInition of
stresses and errors has to be established and followed rigorously for each set of
analysis results, so that they can be compared.
The following sections each deal with a particular stage of analysis and
define the pertinent methods that were used at that stage.
2.2 Finite EleDlent Models
. !
Finite element software package ADINAl was used for all the FE analysis
in this study. All the FE models used to analyze the hyperbolic cooling tower
shells had a number of generic qualities so that the results based on these
models could be compared.
Also by defming these guidelines a computer
program 'CREATE' (Chowdhury, 1990) could be developed, so that the input
files required for the FE analysis could be generated automatically.
The
program is written in C programming language. The program prepares the
input files, for the preprocessor ADINA-IN provided with the software package,
as well as post-processing and batch job processing files as required, for an
entire series of shells.
U sage of the input program not only reduced the manual effort necessary
lA.riINA is a proprietary software distributed by ADINA R&D, Inc., MA.
4
for each set of FE analysis, but also ensured that the input files for each case
were uniform and free of oversight error.
The characteristics which were
common to all FE models were as follows.
• Divisions in both radial and height direction of the shells were
uniform. The spacing between the divisions were not varied over the
entire shell. (See Fig.2-1)
• The bottom of the shell was considered hinged while the top was
modelled as free.
• Four noded shell elements were used to model the shell.
• All the nodes in the shell were modelled with 5 degrees of freedom
(DOF) except when the surface of the shell elements joining there
had an angle exceeding 20° .
• A constant shell thickness of 5 inches was assumed for all the shells.
• Uniform gra~ty load of 150 psf was considered to be acting.
Uniform divisions in both the directions are used in every FE mesh. So
meshes with a higher number of divisions represent a fmer mesh. Throughout
this study FE mesh sizes are presented as,
No. of radial divisions * No. of axial divisions
The cost index was taken as the cost of executing only the FE solution part
(ADINA) of the FE package, in batch mode at off hours on a CDC-CYBER
computer at Lehigh University Computing Center.
5
The variation of stresses along the height of the structure was found from
the FE analysis and stored for comparison. The results were included in a
spreadsheet and the appropriate parts outputted into an ASCII file. Manual
re-entering of the data was not necessary at any step, thereby reducing the risk
of introducing accidental errors.
2.3 Standard Solution
Classical Solutions for gravity loading on hyperbolic cooling tower shells
are available in literature (Soare, 1967) (Ramaswamy, 1986). A numerical
integration scheme of this standard solution (See Appendix B) was used as the
'benchmark' solution for estimating errors in the FE solutions.
Membrane
theory solution which mainly considers the in-plane action of the shell was
adopted. Since the structure under gravity load primarily displays an in-plane
response, membrane theory provides a very accurate solution. At the ends of
the shell where there is some bending action the membrane theory solution may
deviate marginally from the actual stresses, but over most of the shell surface
the solution may be taken as very close to an 'exact' solution.
2.4 Error Analysis
In the context of this study, error is estimated as the difference between
shell forces from membrane theory model and FE solution. Errors are found
and plotted along the height for each FE analysis for both axial and radial
direction shell forces. (See Appendix A for nomenclature)
eep = Nep,FE
Nep, std
ee = Ne,FE - Ne, std
A computer program 'ERROR', in PASCAL programming language,
6
I
I
I --------1-------I
I
I --------~-------I
I
I --------~-------I
I
--------t-------,
,
Uniform
Axial
Divisions
,
I
I
-------~-------t
I
--------r-------II
I
I
----------t--------t-
Elevation
View
-~-------~-------~-
-~~--------l---------~II
;
I,
-------------~--------~--II
I
I
----Tr--------i--------tr--II
I,
I,
I,
"I,
Uniform
Radial
Divisions
"
I
I
I
I
I
I
I
Plan View
~
Figure 2-1: Typical Hyperbolic cooling tower and FE Mesh scheme
7
(Chowdhury, 1990) was developed and used to handle the error calculations.
Root mean squared (RMS) value of the error was used to compare errors
between different meshes.
2
1
n
e'f'
"e~,'t' l'
th
rms = -n £..i
i~1
2
1
n
2
ee ,rms =-n ""
£..i ee , l'
i=1
For comparison between hyperbolic cooling tower shells of different
curvature the actual value of the error is normalized by dividing it Vlith RMS
value of appropriate shell force over the height. RMS value of the shell forces
are calculated from standard solution, using equations similar to the ones for
errors.
n
~e ,rms =~n £..i
~ ~e ,std
'
,l
i=1
ea =
ea, nns
X 100
(%)
Ne,rms
8
Chapter 3
Data from FE Analysis
3.1 Overview
This study involved analysis and presentation of large amounts of data.
For clarity, the results of FE analysis are presented in graphical form. For each
set of FE analysis, actual values of forces (F ep and F 9)' as well as, actual values of
errors (e<p and e s) are plotted along the height.
Three different hyperbolic cooling tower shells are analyzed using
different FE mesh sizes.
These structures are similar in all ways except
curvature. All three structures have equal height, radius and location of neck,
and thickness. Only the shell parameter 'b' is varied. It is increased by 20% for
a less curved and reduced by 20% for a more curved hyperbolic cooling tower
shell.
Structure A :
Typical hyperbolic cooling tower shell.
Structure B :
Less curved cooling tower.
Structure C :
More curved cooling tower.
Results for each of the structures is presented in a different section of this
chapter. Along with FE results, corresponding standard solution is also plotted
using dotted lines. Maximum numerical value (without sign) and RMS value of
errors are found and shown on the figures.
'The common data in the analysis of all the shells are given below.
9
=30 ft.
Radius at origin (neck)
a
Coordinate at top of the shell
Ztop
Coordinate at foot of the shell
Zbottom
Shell Thickness
0= 5 in.
Gravity Loading
q =150 psf
Young's Modulus
E
= 519~19.5 kip/ft 2
Poisson's Ratio
V
= 0.15
10
=30 ft.
=- 100 ft.
3.2 Analysis Results of Structure A
= 70 ft.
b
,,
,,\,,
.,
\t \,
\
\
I
\,
,
\,
t
\,
,
\,
,
\,
\
\,
,
\,
,
\,
,
\
Figure 3·1: Shape of Structure A
RMS value of shell forces for this structure is found (See Sec 2.4) as,
Nell. rms = 9.542167 kip/ft2
N e, rms
= 2.278598 kip/ft2
11
Table 3-1: Summary of Results for Structure A
Mesh
Size
kip/ft
percent
eep
percent
Cost
Index
0.550
1.258
24.1
13.2
0.52
24 * 10
0.414
0.752
18.2
7.9
1.60
24 * 12
0.372
0.580
16.3
6.1
2.32
24 * 15
0.330
0.497
14.5
5.2
3.24
* 12
0.380
0.614
16.7
6.4
0.46
16 * 12
0.370
0.593
16.2
6.2
0.86
* 12
0.370
0.585
16.2
6.1
1.49
24 * 12
0.372
0.580
16.3
6.1
2.32
* 12
0.375
0.576
16.5
6.0
3.36
24
12
20
30
ee,rms
eep,rms
kip/ft
*6
ee
12
120
100
..-J
C+-I
80
Fig. A
~60
b.O
.,....oj
Shell Forces
Q)
== 40
Along Height
-10
o
-5
Force kip/ft
5
120
e
100
...,.J
80
(J
e8 ,rms=
'+-t
~60
b.O
.......
e
Fig. B
0.550
=-1.388
8,max
Error
Q)
== 40
Distribution
Along Height
e rms = 1.258
¢t
e ¢,max = -1.624
20
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Error kip/ft
Figure 3-2: Results for Structure A : Mesh size 24 * 6
13
120
100
~
80
Fig. A
'+-4
:db.D 60
Shell Forces
Along Height
.~
= 40
Q)
20
-10
o
-5
Force kip/ft
5
120
ee
100
~
80
Fig. B
ee
= 0.414 Error
ee,max= -1.318 Distribution
:dt:tO 60
.,...,.
t
Q,)
= 40
rms
Along Height
e¢,rms = 0.752
e¢,max = -0.998
20
0
-2
-1.5
-1
-0.5
0.5
0
1
1.5
2
Error kip/ft
Figure 3-3: Results for Structure A: Mesh size 24 * 10
14
120
100
80
~
Fig. A
~
~ 60
b.D
.,....
Shell Forces
Along Height
Q)
:::=
40
20
O.....-------------...I-~--.a.-----.....I
5
-5
o
-15
-10
Force kip/ft
120
I
I
I
~ee
100
I
I
I
I
~
I
I
I
80
I
I
I
~ 60
I
.,..04
I
b.O
I
I
I
Q)
I
==
Fig. B
I
~
ee ,rIDS,=
0.372
e e,max=-1.297
I
I
40
I
= 0.580
=-0.831
~tmax
e
~trms
e
I
I
I
I
I
I
20
Error
Distribution
Along Height
0
-2
-1.5
-1
-0,5
0
0.5
1
1.5
2
Error kip/ft
Figure 3-4: Results for Structure A : Mesh size 24 * 12
15
120
100
Fig. A
Shell Forces
Along Height
20
0 .....- - - - - - - - - - -.........-...-----'---------'
-5
o
-15
5
-10
Force kip/ft
120
100
+J
e....,.
80
Fig. B
ell(] ,rms= 0.330
ee,max=-1.272
:dbLl 60
.,.....
Q)
~ 40
Distribution
Along Height
e'fI,rms
At.
= 0.497
e ¢,max =-0.666
20
0
-2
Error
-1.5
-1
-0.5
0
0.5
1
1.5
2
Error kip/ft
Figure 3-5: Results for Structure A : Mesh size 24 * 15
16
120
,tfI'
100
N¢~ ,/""",
80
."..,
"
"
,'""
;"
t.-l
60
~
btl
"
.1"""'1
=
QJ
/
,
,
I
/
,/
/
I
Shell Forces
.Along Height
I
!
/
I
I
,/
20
o 'I
e-----........J'
,~~
,/
40
Fig. A
I.
," N
--'
......
"
"
I
/
I
I
I
'
,
-15
-10
a
-5
5
Force kip/ft
120
I
I
e
I
I
I
J--- eo
100
-.:l
Fig. B
80
t.+-I
...,
,.d
b.O
• ...-.4
eo ,rms= 0.380
e
=-1.204
9,max
60
QJ
==
40
Error
. Distribution
Along Height
eq, ,rms =0.614
e¢ ,max =-0.839'
20
0
-2
-1.5
-1
0.5
Error kip/ft
-0.5
0
1
1.5
2
Figure 3-6: Results for Structure A : Mesh size 12 * 12
17
120
100
Fig. A
:db.O 60
.......
C1J
Shell Forces
==
Along Height
40
20
0 .......-----------.-..---....----------'
-15
-10
-5
o
5
Force kip/ft
120
e
¢
e
100
..,
e
80
'+-4
Fig. B
ee = 0.370
,rms
eo ,max=-1.255
:db.D 60
.~
Error
QJ
==
40
Distribution
Along Height
e ¢,rms = 0.593
e ¢,max =-0.835
20
0
-2
-1.5
-1
-0.5
0.5
0
1
1.5
2
Error kip/it
Figure 3-7: Results for Structure A: Mesh size 16 * 12
18
120
100
80
I.
N
()~
P
Fig. A
Shell Forces
Along Height
20
O......-----------'""'--~-----I~-------J
-15
-10
o
-5
5
Force kip/ft
."
120
e
¢
e
100
e
80
ee = 0.370
,rms
ee,max=-1.281
~
~
-+-> 60
,.Q
taD
'1""'"'4
~ 40
etP
,rms = 0.585
e
=-0.833
Fig. B
Error
Distribution
Along Height
¢,max
20
0 ......-----..-::;;;;;;;"""",....--.........- - - ' - - - - - - - - - - -.........
-0.5
0
0.5
-1.5
-1
1
1.5
2
-2
Error kip/ft
Figure 3-8: Results for Structure A : Mesh size 20 * 12
19
120
100
-...,.J
80
Fig. A
~
..,..J
,.c:=
60
QD
Shell Forces
Along Height
.,..-l
Q)
~
40
20
o
-5
-10
5
Error kip/ft
120
I
I
I
J.--- ee
~
100
....,
....,
I
e
80
~
,.Q
b.O
eB ,rms=
60
0.375
eB,max= -1.310
......
(1)
=:t::1
Fig. B
40
Error
Distribution
Along Height
e ¢ ,rms = 0.576
e ¢,max = -0.830
20
O~----=:;;;;;'---'-""-------------_----I
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Force kiP/ft
Figure 3-9: Results for Structure A : Mesh size 30 * 12
20
3.3 Analysis Results of Structure B
,,
I
b
=84 ft.
Figure 3·10: Shape of Structure B
RMS value of shell forces for this structure is found (See Sec 2.4) as,
N•• rms = 9.751214 kip/ft2
N e, rms
= 1.700731 kip/ft2
21
Table 3-2: Summary of Results for Structure B
Mesh
Size
e9,rms
eq>,rms
ee
kip/ft
kip/ft
percent
ecp
percent
Cost
Index
24 * 6
0.234
1.302
13.8
13.4
0.52
24
* 10
0.168
0.780
9.9
8.0
1.54
24
* 12
0.145
0.604
8.5
6.2
2.24
24 * 15
0.128
0.518
7.5
5.3
3.22
* 12
0.187
0.623
11.0
6.4
0.47
16 * 12
0.160
0.612
9.4
6.3
0.86
20
* 12
0.150
0.607
8.8
6.2
1.50
24
* 12
0.145
0.604
8.5
6.2
2.24
30 * 12
0.142
0.602
8.3
6.2
3.39
12
22
120
100
80
:d
....
t:U)
Fig. A
60
=
Shell Forces
Along Height
Q)
40
20
. o. . . _ --_.. .
O~.-.-_------_..a.-_~---_
-10
-15
-5
5
Force kip/ft
120
100
e
¢
......, 80
.......
:d
QJ)
......
cu
==
Fig. B
e8,rms= 1.141
60
Error
e8,max=-2.939
40
Distribution
Along Height
= 1.223
=-1.590
¢.max
e¥J,rms
,i
e
20
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Error kip/ft
Figure 3-11: Results for Structure B : Mesh size 24 * 6
23
120
100
+J
80
Fig. A
~
:db.O 60
Shell Forces
Along Height
• ...-4
Q)
~ 40
20
, 0 .......
o--.Ia..--_
-15
___.--""'---------'
o
-5
-10
Force kip/ft
5
120
100
....,J
80
Fig. B
~
:dbJJ 60
ee ,rms= 0.864ee,max=-2.822
• ...-4
(1)
== 40
e''fJ,rms
"
20
Error
Distribution
Along Height
= 0.727
e¢,max =-1.005
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Error kip/ft
Figure 3-12: Results for Structure B : Mesh size 24 * 10
24
120
100
80
~
Fig. A
~
~ 60
b.O
Shell Forces
.~
Q)
==
Along Height
40
20
0
-10
-15
-5
0
Force kip/ft
5
120
e¢
100
c=
80
Fig. B
ee ,fIDS= 0.780
~ 60
b.O
.""""
cu
~ 40
e~,rms
e
20
0
-2
Error
ee,max=-2.783
~,max
-1.5
-1
-0.5
0
0.5
1
Distribution
Along Height
= 0.561
=-0.832
1.5
2
Error kip/ft
Figure 3-13: Results for Structure B : Mesh size 24
25
* 12
120
100
~
80
Fig. A
C+-t
~60
bD
......
Q)
Shell Forces
Along Height
=40
20
IJ
Li
0
-15
-10
o
-5
5
Force kip/ft
120
100
Fig. B
ee ,rIDS= 1.046
ef) ,max=-2.832
Error
Distribution
Along Height
= 0.486
e¢,max =-0.732
e ¢,rms
20
0-....----------.
. . 0. . - ---------..
-0.5
0.5
-1
1.5
2
-1.5
1
-2
Error kip/ft
Figure 3-14: Results for Structure B : Mesh size 24 * 15
26
"",'
120
,,'
/'
N~
///
~
100
/'
,,,'
",,'
...)
~
~
,..,....
""""
.....
80
//
60
Q.D
/
Q)
==
~'
~'
/'
Fig. A
/
Shell Forces
/1
40
Along Height
II
II
,I
20
/
II
0
-15
-10
-5
Force kip/ft
0
5
120
100
~
80
Fig. B
'--'t
.....,.J
,..c=
60
eB rms-- 0.750 ,Error
e(} :ma~ -2.563 Distribution
Along Height
bIJ
'1lI'IOI4
Q)
=
40
=
e
0.627
¢,rms
e
= -0.854
¢tmax
20
0
-2
-1.5
-1
-0.5
0.5
0
1
1.5
2
Error kip/ft
Figure' 3-15: Results for Structure B : Mesh size 12 III 12
27
120
100
80
Fig. A
:dblJ 60
Shell Forces
Along Height
• ....-1
Q)
~ 40
20
O............J-----~----
-15
-10
.-5. . .---..;:IIll-----------.
....
o
5
Force kip/ft
120
I
I
e
100
I
J.--- ee
~
I
I
I
I
I
~
I
80
Fig. B
I
I
I
~
I
I
I
:dt:tD 60
e8 = 0.757
,rms
e8,max=-2.682
I
I
I
."""""
Q.)
==
40
e
= 0.585
¢,rms
e
=-0.842
20
Error
Distribution
Along Height
¢,max
0
-2
-1.5
-1
-0.5
0.5
0
1
1.5
2
Error kip/ft
Figure 3-16: Results for Structure B : Mesh size 16 * 12
28
120
100
Fig. A
~ 60
b()
. ,.....
Shell Forces
Along Height
Q)
::=
40
20
O.......- - - - - - - - -.......-~------------I
-15
-10
-5
5
Force kip/ft
120
I
I
I
I
I
100
J.---
ee
I
I
Fig. B
~ 60
ee,rIDs=
0.769 Error
ee.ma~-2.746 Distribution
bD
.. ....-4
Q)
==
Along Height
40
e¢ ,rms
20
= 0.568
e¢,max =-0.835
01‫ס‬0o--------------....1..------------1
-0.5
0
0.5
-1
1.5
-1.5
-2
1
2
Error kip/ft
Figure 3-17: Results for Structure B : Mesh size 20 * 12
29
120
100
~
80
Fig. A
t.+-t
.....,J
,.Q
60
CD
Shell Forces
Along Height
-,..-t
Q)
::::= 40
20
0
-10
-15
-5
Force kip/ft
5
0
120
100
~
......,J
..d
80
Fig. B
ee ,rms= 0.790
e e ,ma~-2.815
60
b.O
-I"""l
Q)
==
40
Error
Distribution
Along Height
e,h,rms = 0.556
e'r
=-0.829
20
~,max
0
-2
-1.5
-1
-0.5
0.5
0
1
1.5
2
Error kip/ft
Figure 3-18: Results for Structure B : Mesh size 30 * 12
30
3.4 Analysis Results of Structure C
56 ft.
b
I
••
••
I
""
I
II
\,
II
II
I.
II"
I"•
II
I ,
I ,
,II,
I ,
II
I ,
,
,I I •,I
I
I
•
,I
•I
11
"\ \
""
\,\,
,,
\\
,,
\, \
,,
.
\
\
\
\, \\
\
\
\\
,
\,
,
\,
,
••
I
\,
,
,•
•
Figure 3-19: Shape of Structure C
RMS value of shell forces for this structure is found (See Sec 2.4) as,
Nq», rms = 9.412002 kip/ft2
N e, rms
= 3.256484 kip/ft2
31
Table 3-3: Summary of Results for Structure C
Mesh
Size
ee,rms
eq" rms
ee
eq.
kip/ft
kip/ft
percent
percent
Cost
Index
24* 6
1.141
1.223
35.0
13.0
0.52
* 10
0.864
0.727
26.5
7.7
1.57
24 * 12
0.780
0.561
24.0
6.0
2.27
24 * 15
1.046
0.486
32.1
5.2
3.27
12
* 12
0.750
0.627
23.0
6.7
0.48
16
* 12
0.757
0.585
23.2
6.2
0.86
20
* 12
0.769
0.568
23.6
6.0
1.51
24
* 12
0.780
0.561
24.0
6.0
2.27
30 * 12
0.790
0.556
24.3
5.9
3.43
24
32
120
100
.....,)
80
Fig. A
.......
..,.;J
,.d
b.D
.........
60
Shell Forces
QJ
==
Along Height
40
20
0
-5
Force kip/ft
-10
-15
5
0
120
100
e
~
~.
:db..O
80
~
ee ,rms=
Fig. B
0.234
eB,max=-0.461
60
.........
Error
Distribution
Along Height
QJ
==
4-0
e",
= 1.302
'f,rms
e
=-1.641
¢.max
20
O~-----""'_&..-_"""""--------_---I
-2
-1.5
-1
-0.5
a
0.5
1
1.5
2
Error kip/ ft
Figure 3·20: Results for Structure C : Mesh size 24
33
*6
120
100
80
Fig. A
:dbD 60
.......
Shell Forces
Along Height
Q)
:=
40
20
O....-.------------.
. . . ----t-------------'
-10
-5
o
5
-15
Force kip/ft
120
ee
100
......,.>
80
Fig. B
~
ee =
,rIDS
:db.D 60
.......
Q)
==
0.168
e a,max= -0.411
40
e
Error
Distribution
Along Height
= 0.780
¢,rms
e ¢,max = -0.993
20
0
-2
-1.5
-1
-0.5
0.5
0
1
1.5
2
Error kip/ft
Figure 3-21: Results for Structure C : Mesh size 24 * 10
34
120
100
Fig. A
Shell Forces
Along Height
20
. .5.
O--------------~L.---+-----
-15
-10
o
-5
Force kip/ft
120
100
~
80
Fig. B
l:+-4
ee ,rms=
0.145
-0.397
eBtmax=
= 0.604
e ~,max =-0.830
e",,+"rms
20
Error
Distribution
Along Height
0......--------.
. . . . - .0. . . .- --------.-1
-0.5
0.5
1.5
2
-1.5
-1
1
-2
Error kip/ft
Figure 8-22: Results for Structure C : Mesh size 24 * 12
35
120
100
.., 80
Fig. A
~
::=b.C 60
Shell Forces
Along Height
• ....-1
OJ
==
40
20
O------------.
....
-5
-10
-15
. a..-.-........- - - -......
o
5
Force kip/ft
I
120
I
I
I
I
.J...--- ee
100
I
I
I
,I
......,J
I
80
Fig. B
"t-4
ee
::==.0 60
e{] ,max=-0.382
£)
.~
OJ
==
= 0.128
~rms
40
e
= 0.518
~trms
e
=-0.665
20
Error
Distribution
Along Height
~,max
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Error kip/ft
Figure 3-23: Results for Structure C : Mesh size 24 * 15
36
120
100
80
Fig. A
Shell Forces
Along Height
20
0 ......- - - - - - - - - - -.........--+--------'
-15
-10
o
-5
5
Force kip/ft
120
e~
100
.+J
80
Fig. B
~
eo ,rIDS=
:dbD 60
.......
[7
e.8,max
Q.)
::c
0.187
= -0.359
40
e
=
~,rms
e~,max =
20
Error
Distribution
Along Height
0.623
0.806
Oa...-----------.
. . . -~------------'
-2
-1.5
-1
-0.5
0
0.5
1.5
2
1
Error kip/ft
Figure 3-24: Results for Structure C : Mesh size 12 * 12
37
120
100
80
Fig. A
Shell Forces
Along Height
20
OI&o-------------oL.&-----6-------I
-5
o
5
-10
-15
Force kip/ft
120
100
Fig. B
ee ,rInS= 0.160
ee,max=-0.380
40
e ~,rms = 0.612
e ~,max =-0.832
20
Error
Distribution
Along Height
-------...-..&.-.
. .0. . - -----------1
-0.5
0.5
1.5
-1
2
-1.5
1
0 ......
-2
Error kip/ft
Figure 3·25: Results for Structure C : Mesh size 16 * 12
38
120
100
80
Fig. A
Shell Forces
Along Height
20
O.....-.------------.
. . . . .- ---t--------a
-5
o
-15
-10
5
Force kip/ ft
120
100
~ 80
Fig. B
ee ,rms= 0.160
e 8 ,max=-0.380
40
e ¢,rms
20
= 0.612
Error
Distribution
Along Height
e ¢,max =-0.832
0"'---------.
. . . - . .0. .- ------------0.5
0.5
-1.5
1.5
-2
1
2
-1
Error kip/ft
Figure 3-26: Results for Structure C : Mesh size 20 * 12
39
120
100
80
Fig. A
Shell Forces
Along Height
20
O--------------.
. . . . . . -----t~--.....
-10
-5
o
-15
5
Force kip/ft
120
100
~
80
Fig. B
~
ee ,rIDS=
:dbD 60
Error
eB,max= -0.402
.~
cu
==
0.142
40
Distribution
Along Height
e\f,rms
,i
= 0.602
e¢,max =-0.830
20
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Error kip/ft
Figure 3-27: Results for Structure C : Mesh size 30 * 12
40
Chapter 4
. Discussion of Results
4.1 Effect of Mesh Fineness
The data presented in Chapter 3 is summarized in Fig.4-1 to 4-4. The
variation of normalized RMS value of errors (
e4>
and ee ) for different mesh
fineness is shown. Fig.4-1 and 4-2 represent the error variation for different
number of divisions in the fmite element mesh along axial direction, while
number of divisions in radial direction was kept constant at 24.
Similarly
Fig.4-3 and 4-4 represent the error variation for the different number of
divisions along radial direction, while the axial direction divisions is flXed at 12.
4.1.1 Axial Direction
In the axial direction number of divisions in the FE mesh was varied from
6 to 15. Norma1ized RMS error in longitudinal shell force (
e
t1> )
was found to
range from 13.5% at 6 divisions to approximately 6% at 15 divisions ( See
Fig.4-1). As the mesh fmeness in the axial direction was increased e<t> gradually
reduced. Above 12 divisions it remained fairly constant.
There was a wide variation of normalized RMS error in meridiana! shell
forces ( ea). For Structure C,
ee was as high as 35% for low mesh fmeness
(6
div.) while for Structure B, it was as low as 7.5% for high mesh fineness (15 div.)
(See Fig. 4-2). Generally ea appeared to decrease for fmer meshes.
In the hyperbolic cooling tower shells analyzed, 12 or higher number of
divisions along the axial direction appeared to be necessary for an optimal
solution.
41
40,.------------------------_
30
z-s. 20
Cases A, B & C coincide
~
o
~
10
0....-------........- - - - - - -.........---------+
6
9
12
15
No. of Axial Divisions
Figure 4-1: Relationship between Normalized RMS error in
Longitudinal Direction and Longitudinal Mesh Fineness
42
40...-----------------------Case C
30
r
r
Case A
Case B
10
01-----------..----------.-----------.
6
12
9
No. of Axial Divisions
15
Figure 4-2: Relationship between Normalized RMS error in
Meridiana! Direction and Longitudinal Mesh Fineness
43
40,....------------------------.
30
Z -s..
Cases A, B & C coincide
20
10
os - - - - - - - - - - r -I - - - - - - . - 1 - - - - - - +
12
18
24
30
No. of Radial Divisions
Figure 4-3: Relationship between Normalized RMS error in
Longitudinal Direction and Meridianal Mesh Fineness
44
40...------------------------..
30
Case C
...-..
rn
8
~
zr:::t:::J
C+-4
20
Case A
0
~
'-'"
e:::t;)
Case B
IQ)
10
0......--------........- - - - - - -........---------1
12
18
24
30
No. of Radial Divisions
Figure 4-4: Relationship between Normalized RMS error in
Meridiana! Direction and Meridiana! Mesh Fineness
45
4.1.2 Radial Direction
In the radial direction number of divisions in the FE mesh was varied
from 12 to 30. Fineness of mesh in the radial direction appeared to have little
effect on the accuracy of the solution.
Curves showing the variation of
normalized RMS errors for shell forces ( e<t> and
( See Fig.4-3 and 4-4).
ee ) were more or less constant
The level of error was pretty low for e<p • While for ee it
was much higher in some cases.
This insensitivity of the accuracy of the solution to mesh fmeness in the
radial direction may be because there is no shell force gradient in the radial
direction under the loading chosen ( gravity loading, symmetric state of stress ).
For unsymmetric states of stress there would possibly be some difference in
normalized RMS error levels for different mesh fineness in the radial direction.
From the available data on the hyperbolic cooling tower shells analyzed,
18 or higher number of divisions along the radial directions appeared to be a
good choice. Though a lower number of divisions may not substantially raise
the normalized RMS errors under the chosen loading, for a more general case,
considering aspect ratio of the elements and different states of stress, at least 18
divisions in the radial direction is advised.
4.2 Effect of Shell GeoDletry
Three hyperbolic cooling tower shells (A, B and C) of different shapes were
analyzed in this study. In general FE analysis of comparatively flat shells were
more accurate than curved shells. From the results it was clear that
ea was
very sensitive to the geometry. From Fig.4-2 and 4-4 it appears that there is a
variation of 35% to 10% normalized RMS error in the estimate of N e depending
46
on the geometry. On the other hand, in the range of shells analyzed, normalized
RMS error
e
<1>
did not show any dependency on the shell geometry. Nate in
Fig.4-1 and 4-3 the curves for all three structures coincide.
Table 4-1: Comparison of Geometric Parameters
Structure
b
ft.
A
ft.
A
30.0
70.0
66.8°
B
30.0
84.0
70.35°
C
30.0
56.0
61.82°
a
Table 4-1 shows the comparison of different geometric parameters of
different shells. The costants a and b· are used to define the shell generator
curve. The angle A is the angle between the radial axis and the asymptotes of
the meridianal hyperbola. The relations between these parameters can be found
from simple geometry as,
A = tan -1 ( b/a )
From table 4-1 it is evident that comparatively small changes in the, angle
A, can substantially effect the level of normalized RMS error ee' In this range
of shells e<t> appear to be fairly insensitive to changes in the geometry.
Further parametric study is required to establish a numerical estimate of
47
FE mesh fmeness required for a solution with acceptable level of errors for
different geometrIes.
However qualitatively it can be concluded that, for
comparatively curved shells 24 * 12 or finer FE mesh is necessary, while for
typical or flat shells 18
* 10 mesh fmeness would be sufficient.
4.3 Aspect Ratio Variation
It is well known that aspect ratio of the elements may substantially affect
the quality of FE analysis results. Aspect ratios close to unity are optimum for
minimizing errors, while elements with aspect ratios greater than three may
show significant error in analysis. In this study, aspect ratio is taken as,
a
Length of the longest side of element
= Length
of the shortest side of element
. So aspect ratio is always greater than or equal to one.
Tables 4-2 and 4-3 present the aspect ratio variation for different meshes
used in this study. From the tables it can be seen that the maximum aspect
ratio in different meshes varied from 1.31 to 2.77 and was below 2 for most
cases. The average aspect ratio varied from 1.09 to 2.54 and was below 1.5 for
most cases.
An aspect ratio of 1-.0 is the best possible aspect ratio.
These
figures indicate that the elements in the FE meshes used in this analysis had
excellent aspect ratio. So it can be safely assumed that the error analyses based
on the results using these meshes were mostly free from the errors associated
with elements having extreme aspect ratios.
48
Table 4-2: Maximum Aspect Ratio of Shell Elements
Mesh
Size
Struct.
A
Struct.
Struct.
B
C
24 * 6
2.76
2.75
2.77
24* 10
1.66
1.65
1.66
24 * 12
1.38
1.38
1.38
24 * 15
1.49
1.36
1.68
* 12
2.39
2.18
2.7
16 * 12
1.79
1.63
2.02
* 12
1.43
1.31
1.62
24* 12
1.38
1.38
1.38
30 * 12
1.73
1.72
1.73
12
20
49
Table 4-3: Average Aspect Ratio of Shell Elements
Mesh
Size
Struct.
Struct.
Struct.
A
B
C
24 * 6
2.49
2.54
2.43
24* 10
1.47
1.51
1.44
24 * 12
1.26
1.26
1.28
24* 15
1.11
1.09
1.14
12 * 12
1.77
1.69
1.88
16 * 12
1.33
1.27
1.41
* 12
1.18
1.15
1.23
24* 12
1.26
1.26
1.28
30 * 12
1.52
1.56
1.48
20
50
4.4 Cost Effectiveness of Analysis
Cost of solring the problem is taken as the measure of the size of the
model rather than number of nodes or other measures.
Only the cost of
executing the FE solution step is included in the cost estimate. Preprocessing
costs (e.g. cost of running ADINA-IN") are not included.
The cost of running ADINA for each of the meshes analyzed is presented
in Tables 3-1 to 3-3. The cost index is the actual cost in dollars for running
ADINA for that mesh in batch mode and at off hours on a CDC-CYBER
computer. The range of costs is from $0.52 for simpler meshes to $3.43 for the
more sophisticated and rermed meshes. This indicates that an analysis using
finer meshes may cost 5 to 7 times the cost of using a simpler mesh. However
these costs are so low that cost considerations do not play any part in choice of
~ptimum meshes.
This analysis was. limited to elastic material models and simple static
loading.
For nonlinear material models and dynamic analysis substantially
more cost will be involved.
Memory and computational effort (CPU time)
limitations may actually limit the size of the model, rather than sheer cost. The
ratio of costs between different meshsizes may be used as guidelines for
estimating costs for meshes in other situations.
51
Chapter 5
Conclusions and Recommendations
5.1 Conclusions
The following conclusions about fmite element (FE) analysis of hyperbolic
cooling tower shells may be drawn from the data collected in this study.
• There is signicicant amount of error (as high as 20% to 35% in some
cases) in the estimate of shell forces, especially for meridiana!
(hoop) forces-.
• For symmetric states of stress, mesh fmeness in the radial direction
has little or no effect on the accuracy of the solution.
However
rerming the mesh in the axial direction does improve the accuracy of
the solution.
• The accuracy of longitudinal shell force ( N q> ) estimate is quite good
(approximately 10% or less error) in most choices of mesh and it is
also fairly insensitive to the geometry in this class of shells in the
range analyzed.
• The accuracy of meridianal shell force ( N e ) results is, however,
very much dependent on the shape of the structure. Small changes
in shape 'may significantly affect the accuracy ofNe estimate. In the
cases analyzed in this study the accuracy varied from 30% to 10%
depending on the geometry of the shell for the same FE mesh.
• For a hyperbolic cooling tower of typical shape or flatter than typical
shape, a mesh fineness of at least 18
reasonably good solution.
*
10 is necessary for a
However for more curved shells a
52
minimum mesh fineness of 24
* 12 is required.
Even finer meshes
may be necessary, in applications where a higher degree of accuracy
is desired.
• Finite element analysis results typically underestimate ( 30% to 6%
depending on the shell parameters) the compressive shell forces.
N ate that the distribution of errors over the height is not random
and they show a substantial bias to the negative side in all the cases
analyzed. Since most shells in practice are designed such that the
shell surface is mostly in compression, this tendency of FE solutions
produce unconservative estimates of shell forces.
• The aspect ratio of the elements in FE meshes used in this study are
all less than 2.8 and mostly averages below 1.5. Errors arising
from extreme aspect ratios do not substantially affect the
conclusions of this study.
• The cost of analysis is fairly low (typically $0.50 to $3.70 in batch
mode in off hours on a CDC-CYBER machine) for elastic static
analysis, even using fine meshes.
The ratio of cost for analysis
between the finest and crudest meshes considered in this study is
between 5 to 7. Nonlinear material models and dynamic analysis
would increase this estimate substantially.
53
5.2 Reconunendations
To estimate the errors in FE analysis of the shell forces on hyperbolic
cooling tower shells with better accuracy following areas may be researched.
• A thorough parametric study to quantify the relationship between
errors and shape of the shell may be done.
• The choice of elements (e.g. 8 noded shell element) also significantly
effect the accuracy of the solution. This is another possible area of
research.
• To complement this study done under a symmetric state of stress,
similar studies may be done on unsymmetric states of stress to fmd
optimal mesh characteristics for minimizing error.
Similar studies on other shell structures (e.g. Cylindrical shells,
Hyperbolic Paraboloids etc.) may be done to develop further understanding of
the limitations of the accuracy of fmite element method in thin shell structures.
Development of such a knowledge base would ensure a more uniform treatment
of FE analysis of thin shell structures and would prevent analysis errors from
adversely affecting the structural performance.
5.3 SUD1D1ary
Increased dependence on fmite element modelling of structures in practice
has made awareness of error potential of these analyses essential. Specifically
this study indicates that FE analysis of hyperbolic cooling tower shells under
gravity loading, may have substantial error and the analysis results may be
unconservative in some cases.
Error levels of unsymmetric states of stress
would probably be higher.
54
Choice of appropriate FE mesh patterns, fmeness and elements while
forming the model, along with adoption of appropriate safety factors on the
results when applied to design can be done to improve the reliability of FE
results. This concern is more pertinent in case of shell structures, since FE
analysis results of these structures is more error-prone. Further research is
needed to develop guidelines for FE modelling of common structures, so that the
analysis procedure may be more uniform and the results may be more reliable
and error-free.
55
References
Ashfaque Chowdhury 1990
SIGNIFICANCE OF MESH..FINENESS IN ACCURACY OF FE ANALYSIS OF
HYPERBOLIC COOLING TOWER SHELLS.. Technical Report
LEHIGHlFU354.548, Fritz Engineering Laboratory, Lehigh University.
P. Fazio and K Go-wri 1987
STRUCTURAL ANALYSIS SOFTWARE AND THE C PROGRAMMING LANGUAGE.
Computers & Structures Vol. 25(No. 3):pp. 463-465.
Carl S. Grant and T. Y. Yang 1980
REFINED ANALYSIS OF THE SEISMIC RESPONSE OF COLUMN-SUPPORTED
COOLING TOWER. Computers & Structures Vol. 11:pp. 225-231.
Nagesh R. Iyer and T. V. S. R. Appa Rao 1990
STUDIES ON STRESS CONCENTRATION AT SHELIrCOLU:MN JUNCTION OF
HYPERBOLOID COOLING TOWERS. Computers & Structures Vol. 34(No.
2):pp. 191·202.
Christian Meyer 1987
FINITE ELEMENT IDEALIZATION FOR LINEAR STATIC AND DYNAMIC ANALYSIS OF
STRUCTURES IN ENGINEERING PRACTICE. Technical Report, Task
Committee on FE Idealization, ASCE.
G. S. Ramaswamy 1986
DESIGN AND CONSTRUCTION OF CONCRETE SHELL ROOFS.
CBS Publishers &
Distributors, Delhi-32, India.
Werner C. Rheinboldt 1985
ERROR ESTIMATES FOR NONLINEAR FINITE ELEMENT COMPUTATIONS.
Computers & Structures Vol. 20(No. 1·3):pp. 91-98.
Mircea Soare 1967
APPLICATION OF FINITE DIFFERENCE EQUATION TO SHELL ANALYSIS.
Press., Oxford, London, UK
56
Pergamon
Appendix A
Nomenclature
a
Aspect ratio of shell elements
Shell thickness, (inch).
Angle between principal radius rl and axis of revolution.
r
Unit density of material, (kip/ft3).
v
Poisson's ratio.
a
Angle in a plane perpendicular to axis of revolution between
any point and radial axis in the same plane.
a
Shell constant, radius at origin (neck), (ft).
A
Angle between radial axis and asymptotes of meridianaI
hyperbola.
b
Shell constant, (ft).
E
Young's Modulus, (kip/ft2).
Error in longitudinal shell force at a point, (kip/ft).
Normalized root mean square (RMS) error in longitudinal
shell force over the entire shell, (%).
eq>, rms
Root mean squared (RMS) value of error in longitudinal shell
force over entire shell, (kip/ft).
Error in meridiana! shell force at a point, (kip/ft).
Normalized root mean square (RMS) error in meridianal
shell force over the entire shell, (%).
57
ee, rms
Root mean squared (RMS) value of error in meridianal shell
. force over entire shell, (kip/it).
f(4)) or f(z)
Function used for numerical integration.
I(z)
Integrated function.
N</>e
In-plane Shear force in shell, (kip/ft)
Nep
Shell force in the longitudinal direction, (kip/ft).
N</>,FE
Longitudinal shell force found by fmite element (FE)
analysis, (kip/ft).
Nep,rms
Root mean squared (RMS) value of longitudinal shell force
over the entire shell, (kip/ft).
N</>,std
Longitudinal shell force found by standard solution, (kip/ft).
Ne
Shell force in the meridianal direction, (kip/ft).
Ne,FE
Meridianal shell force found by fmite element (FE) analysis,
(kip/it).
Ne,rms
Root mean squared (RMS) value of meridianal shell force
over the entire shell, (kip/ft,).
Ne,std
Meridianal shell force found by standard solution, (kip/ft).
q
Gravity loading on shell, (kip/ft2 ).
Radius of shell at any height, (ft).
Maximum principal radius at any point on the shell, (ft).
Minimum principal radius at any point on the shell, (ft).
x
Body force along global x direction, (kip/ft3 ).
y
Body force along global y direction, (kip/ft3 ).
58
z
Coordinate along global z axis which coincides with the axis
. of revolution, (ft).
z at the bottom of the shell, (ft).
z at the top of the shell, (ft).
Llz
Increment in z used in numerical integration, (ft).
Z
Body force along global z direction, (kip/ft3 ).
59
AppendixB
Standard Solution
B.l Theoretical Background
In membrane theory for a shell of revolution, the differential equations of
equilibrium can be derived as follows. (Soare, 1967) (See Fig. B-1)
a( Nq,9 ro
a<\>
)
a( Nq,'o )
act>
+ Ne(p rlcos<\> + Xrell = 0
- Ne'leos<\> + Y'O'l = 0
-z
For a symmetric loading case, X
=0
and N<t,e =
N9~ =
o.
So the
equations of equilibrium reduces to the following.
d(N$ro)
d<\>
- Ne r1cos<\> + Yr<fl
=0
N(p + N e = -2
r1
r2
From these equations, the solution for Net> is found as,
f~
= refl (Y sin4l + Z cos<\> )
For the case of gravity loading of intensity 'Y 0, ( Y: material density, and 0
: shell thickness) the components of loading are,
60
Taken from (Ramaswamy, 1986) Fig 14-7a, pp.366
Figure B-1: Typical Shell Element and Membrane Forces
61
y =
yo sin<t>
Z =
yo cosq>
Substituting and simplifying in the expression for /(<1» ,
Changing the independent variable from <p to height z , finally the
following expressions may be derived.
I(z) =
JZ f(z) dz
Ztop
N~ =
I(z)
rosintP
The meridiana! forces are found as follows.
B.2 Nunterical Analysis
Dimensions and material properties of the structure being known, the
shape properties of the shell at any given height can be found from geometric
consideration. To fmd the forces, numerical integTation of I(z) along the height
is done. Once I(z) is known finding the forces at that z, is trivial.
From geometrical considerations,
62
'0 = a~ 1 + (z/b)2
= tan -1 (
<t>
b '0
)
a~,o2 - a2
ro
'2
=sin
Z
= yo cos'\>
<1>
For Numerical Integration the shell is divided into n (n
= 180
in this
study) 'slices' of equal height of &. Then following equation derived from the
trapezium rule, is used at m th division.
Using this scheme numerical integration of f(z) over the height is done. The
numbering of divisions starts at the top of the shell and increases towards the
bottom.
Knowing numerical values of material properties and ro, '1' r2'
<1>,
Z and
I(z), numerical values of forces at any given z is found by substituting in the
appropriate equations.
63