Elastic-Plastic Behavior of an Ideal Cylinder Subject to Mechanical
and Thermal Loads
by
Peter P. Poworoznek
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF MECHANICAL ENGINEERING
Approved:
_________________________________________
Professor Ernesto Gutierrez-Miravete, Project Advisor
Rensselaer Polytechnic Institute
Hartford, CT
December, 2008
i
© Copyright 2008
by
Peter P. Poworoznek
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
ABSTRACT .................................................................................................................... vii
PROGRESS REPORT .................................................................................................... viii
1. INTRODUCTION/BACKGROUND .......................................................................... 1
2. ELASTIC RESPONSE ................................................................................................ 2
2.1
2.2
2.3
Pressure Loading ................................................................................................ 2
2.1.1
Thin-Walled vs. Thick-Walled............................................................... 2
2.1.2
Analytical Solution ................................................................................ 2
2.1.3
Finite-Element Solution ......................................................................... 7
2.1.4
Comparison of Results ........................................................................... 9
Thermal Loading .............................................................................................. 12
2.2.1
Analytical Solution .............................................................................. 12
2.2.2
Finite-Element Solution ....................................................................... 16
2.2.3
Comparison of Results ......................................................................... 18
Combined Pressure and Thermal Loading ....................................................... 19
2.3.1
Analytical Solution .............................................................................. 20
2.3.2
Finite-Element Solution ....................................................................... 20
2.3.3
Comparison of Results ......................................................................... 20
3. ELASTIC-PLASTIC RESPONSE ............................................................................ 22
3.1
3.2
Pressure Loading .............................................................................................. 22
3.1.1
Analytical Solution .............................................................................. 22
3.1.2
Finite-Element Solution ....................................................................... 22
Thermal Loading .............................................................................................. 22
3.2.1
Analytical Solution .............................................................................. 22
3.2.2
Finite-Element Solution ....................................................................... 22
iii
3.3
Combined Pressure and Thermal Loading ....................................................... 22
3.3.1
Analytical Solution .............................................................................. 22
3.3.2
Finite-Element Solution ....................................................................... 22
4. DISCUSSION ............................................................................................................ 23
5. BIBLIOGRAPHY...................................................................................................... 24
APPENDIX A – SAMPLE ABAQUS FILES ................................................................. 25
APPENDIX B – ADDITIONAL PLOTS ........................................................................ 33
iv
LIST OF TABLES
Table 1 – Thin-Walled Cylinder Plane Stress Results....................................................... 4
Table 2 – Thin-Walled Cylinder Plane Strain Results....................................................... 4
Table 3 – Mesh Size vs. Solution Convergence ................................................................ 8
v
LIST OF FIGURES
Figure 1 – Exact Hoop Stress (Pressure Load) .................................................................. 6
Figure 2 – Exact Radial Displacement (Pressure Load) .................................................... 7
Figure 3 – ABAQUS Hoop Stress (Pressure Load)........................................................... 9
Figure 4 – ABAQUS Radial Displacement (Pressure Load) ............................................. 9
Figure 5 – ABAQUS vs. Exact Hoop Stresses as a Function of r/t Ratios...................... 10
Figure 6 – ABAQUS vs. Exact Radial Displacements as a Function of r/t Ratios ......... 10
Figure 7 – ABAQUS vs. Exact Hoop Stresses as a Function of r/t Ratios...................... 11
Figure 8 – ABAQUS vs. Exact Radial Displacements as a Function of r/t Ratios ......... 12
Figure 9 – Exact Hoop Stress (Thermal Load) ................................................................ 16
Figure 10 – Exact Radial Displacement (Thermal Load) ................................................ 16
Figure 11 – ABAQUS Hoop Stress (Thermal Load)....................................................... 17
Figure 12 – ABAQUS Radial Displacement (Thermal Load)......................................... 18
Figure 13 – Exact vs. ABAQUS Temperature Distribution (Thermal Load).................. 18
Figure 14 – Exact vs. ABAQUS Hoop Stress (Thermal Load) ....................................... 19
Figure 15 – Exact vs. ABAQUS Radial Displacement (Thermal Load) ......................... 19
Figure 16 – Exact vs. ABAQUS Hoop Stress (Combined Load) .................................... 21
Figure 17 – Exact vs. ABAQUS Radial Displacement (Combined Load) ...................... 21
vi
ABSTRACT
This project examines the elastic-plastic behavior of an ideal cylinder with both
plane-stress and plane-strain end conditions subject to axisymmetric mechanical
(pressure) and thermal loading. Both analytical methods and finite-element models are
used to predict the stress and strain levels and radial displacements.
Initially the elastic solution for an infinitely long cylinder subject to an internal
pressure is discussed. Although the majority of this project focuses on thick-walled
cylinders, thin-walled cylinders are addressed for the linear elastic/pressure case as is the
boundary between what constitutes thin and thick walls. Then the effects of a thermal
load on the cylinder are examined; both by itself and in combination with a pressure
load.
Next the pressure loads are increased to induce plasticity in the cylinder for both a
perfectly plastic and a strain-hardening material. Finally thermal effects are looked at
(both by themselves and in addition to the pressure loads) to complete the elastic-plastic
analysis.
vii
PROGRESS REPORT
The elastic portion of this project is complete. Analytical solutions were derived to
predict the principal stresses, strain, and radial displacements for cylinders with both
plane-stress and plane-strain conditions subject to both pressure loads, thermal loads,
and combined pressure/thermal loads. Then finite-element models were built, using
ABAQUS, which showed excellent correlation with the analytical solutions.
The elastic-plastic phase of the project is underway. A number of sources have been
located to help in this analysis, and I am now in the process of gathering the key
concepts and equations for inclusion in the report. The project should be completed by
the assigned date.
viii
1. INTRODUCTION/BACKGROUND
This section will give a general introduction to the project.
This project will look at an ideal cylinder subject to both plane-stress and planestrain end conditions. Plane-strain conditions are typical for a cylinder where the length
is much larger than its radius (i.e. a fluid filled pipe). Typically plane-stress conditions
are used when the length is smaller than the radius; the most common example being a
rotating disk where the pressure is really a form of centrifugal force.
2. ELASTIC RESPONSE
2.1 Pressure Loading
2.1.1
Thin-Walled vs. Thick-Walled
The most common definition of a thin-walled cylinder is one where the ratio of the
radius to the wall thickness is greater than ten-to-one [1], although some texts
recommend ratios from as low as five-to-one to as high as twenty-to-one. This is done so
that the “assumption of constant stress across the wall results in negligible error.” [2]
The next sections will examine the linear elastic stresses and strains in cylinders
with a range of radius-to-wall-thicknesses subject to pressure loading. The results will be
used to justify the ten-to-one ratio.
2.1.2
Analytical Solution
2.1.2.1 Thin-Walled Cylinder
For an open-ended, unconstrained (plane-stress) thin-walled infinite cylinder of
thickness (t) and radius (r) subject to either an internal or external pressure (p), the only
stresses present are the radial stress and the hoop stress. The radial stress is assumed to
be constant and is equal to the negative of the applied pressure.
r  p
(1)
The hoop stress can be readily found by examining the free body diagram of a halfcylinder and is given by the formula [1]:
 
p r
t
(2)
From Hooke’s law, the strains are calculated using:
r 
1
 
1
z 
1
E

E
E

 r      z 


(3)
     r  z 




(4)


(5)
 z    r  

2
In this case, the longitudinal (σz) stress is zero, therefore:
r 

E 
p
 
z 
 1 
p
E
 
r
t
 p
 r 
(6)
  
(7)

  1 
r


E


t
(8)
t
In terms of displacement, the circumference of the cylinder will grow by 2πrεθ for a
positive (internal) pressure and small displacements. Therefore the change in radius is:
u r 
u r  r 
p r
E
 
r
t
  

(9)
If the ends are constrained (plane-strain), then there are radial, hoop, and
longitudinal stresses. The radial and hoop stresses are the same as in plane stress, but the
longitudinal stress is found by using:
z 
1
E

 z    r  


z    r  

0

(10)
(11)
The radial stress is constant (-p), therefore:
z    p   1 

r

(12)
t
The hoop and radial strains, using the same equations as in plane stress are:

E 
p
r 
 
p
E
2
 1   
 r
t


  1   




r
2
    1      1   
t


(13)
(14)
And the change in radius is:
ur 
r
2
    1      1   
E 
t

p r
3
(15)
For the range of cylinders to be discussed in Section 2.1.4, the “exact” analytical
values calculated using the formulas above are shown in Table 1 and Table 2 (all are
based on an outer radius of 10.0, all units are in inches & psi, ν=0.3, E=30.0E6).
Wall
Thick.
2.000
1.500
1.000
0.750
0.500
0.250
0.125
r/t
4.0
5.7
9.0
12.3
19.0
39.0
79.0
psi
7019
5323
3577
2690
1797
900
450
σr
-7019
-5323
-3577
-2690
-1797
-900
-450
σθ
28706
30164
32193
33177
34143
35100
35550
σz
0
0
0
0
0
0
0
Plane Stress
εr
-0.00051
-0.00048
-0.00044
-0.00042
-0.0004
-0.00038
-0.00037
εθ
0.00100
0.00106
0.00111
0.00113
0.00116
0.00118
0.00119
εz
-0.00021
-0.00025
-0.00029
-0.0003
-0.00032
-0.00034
-0.00035
ur
0.00804
0.009
0.00998
0.01048
0.01098
0.0115
0.01175
Table 1 – Thin-Walled Cylinder Plane Stress Results
Wall
Thick.
2.000
1.500
1.000
0.750
0.500
0.250
0.125
r/t
4.0
5.7
9.0
12.3
19.0
39.0
79.0
psi
7482
5768
3949
3001
2026
1026
516
σr
-7482
-5768
-3949
-3001
-2026
-1026
-516
Plane Strain
σz
εr
6734
-0.00062
8075
-0.0006
9478
-0.00058
10203
-0.00057
10940
-0.00056
11696
-0.00055
12074
-0.00055
σθ
29928
32685
35541
37012
38494
40014
40764
εθ
0.00101
0.00107
0.00113
0.00116
0.00119
0.00123
0.00124
εz
0
0
0
0
0
0
0
ur
0.00804
0.00906
0.01016
0.01075
0.01134
0.01196
0.01228
Table 2 – Thin-Walled Cylinder Plane Strain Results
2.1.2.2 Thick-Walled Cylinder
For a thick-walled cylinder of inner radius (a), outer radius (b), inner pressure (pi),
and outer pressure (po) equations for the hoop stress and radial stress were developed by
Lamé in the early 19th century [4]. In general form, they are:
2
r 
2
a pi  b  po
2
b a
2
 
2

2
b a
2
2
2
r  b a
2
a pi  b  po
pi  po a2 b2


2
pi  po a2 b2
2

2
r  b a

(16)
(17)
2
The following calculations will assume that the pressure on the cylinder is an
internal pressure only (po = 0), however they can be similarly derived for a purely
external pressure or a pressure gradient across the cylinder.
For a strictly internal pressure (pi = p), equations (16) and (17) reduce to:
4
p a

2
 1 
2


r 
(18)
2

b 

 
 1
2
2 
2 
b a 
r 
(19)
r 
b

b a 
2
2
p a
2
2
From equation (19), the hoop stress will be the largest at the inner radius (r is the
smallest) and smallest at the outer radius (r is the largest). The ratio of the largest to the
smallest hoop stresses is given by:
_max
_min
2

2
a b
2 a
(20)
2
Thus for b = 1.1a (radius/wall thickness ratio of about ten to one), the difference
between the maximum and minimum hoop stresses is about ten percent. This is the basis
for the classic definition of a thin-walled cylinder.
For the plane-stress case, the longitudinal stress (σz) is zero, and the strains are
calculated using Hooke’s law as follows:
r 
 
p a
2
E b  a
p a
2

2
 1   
 
2
2
 1   
 
2
z 
b
2
2
E b  a
2

  1   
2
r
2   p  a
2
2
r

2
E b  a
b


(21)

  1   



(22)
(23)
And the change in radius (rεθ) is:
u r 
p a
2

  1    
2
2 
E  b  a  
b
2
2
r

  1     r


(24)
For plane-strain, the longitudinal strain is zero, and following the procedure used for
the thin-walled cylinder, the longitudinal stress, strains and displacements become:
5
z 
2   p  a
2
b a
2
(25)
2
2

b
2

r 
 1 
  1     2  
2
2 
2

E b  a 
r


p a
2
p a
 


2
2
E b  a
u r 
1 
E

 1   
 
2
p a
b
2
2
  1     2 
r

2
  1  2  

b a 
2
2
2



(27)
2
r

r 
b
(26)
(28)
2
For a 10.0-inch outer radius and 7.0-inch inner radius cylinder (r/t = 2.3) with an
internal pressure of 10199 psi (material properties are the same as above) the hoop stress
and radial displacements are shown in Figure 1 and Figure 2. Longitudinal strain is a
constant at 0.00019598, see Appendix B for plots of the other quantities.
Hoop Stress - Plane-Stress
29000.00
Hoop Stress (psi)
27000.00
25000.00
23000.00
21000.00
Exact
19000.00
17000.00
15000.00
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
Radius (in)
Figure 1 – Exact Hoop Stress (Pressure Load)
6
9.5
9.75
10
Radial Displacement - Plane-Stress
0.00780000
Radial Displacement (in)
0.00760000
0.00740000
0.00720000
0.00700000
Exact
0.00680000
0.00660000
0.00640000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure 2 – Exact Radial Displacement (Pressure Load)
2.1.3
Finite-Element Solution
The finite element code ABAQUS [5] was used for the numerical solutions. A
parameterized input file was used to generate 2D cylinders of different cross-sections
(outer radius and wall thickness), element types (plane-stress vs. plane-strain), and
loading conditions (internal vs. external pressure). A sample input file is listed in
Appendix A.
A one-sixteenth section (22.5 degrees) of the full cylinder was modeled. Symmetry
boundary conditions (circumferential displacement equal to zero, the elements chosen do
not have nodal rotation DOFs) were applied at the ends to ensure that the behavior of the
full cylinder was represented. ABAQUS element types CPS4R (plane-stress) and
CPE4R (plane-strain) were used. Both are solid continuum “4-node bi-linear, reduced
integration with hourglass control” [5] elements. The plane-stress element (CPS4R) does
not calculate longitudinal strains directly as “the thickness direction is computed based
on section properties rather than at the material level,” [5] so the longitudinal strains
were calculated using Hooke’s law similar to equation (5) by creating an additional
output field:
z 

E
7
 ( S11  S22)
(29)
where S11 and S22 are the radial and hoop stresses in the ABAQUS output database.
Material properties typical of steel, Young’s Modulus (E) = 30.0E6 & Poisson’s
Ratio (ν) = 0.3, were used.
Mesh convergence – In order to set a mesh size for use in the remainder of this
project, several different mesh sizes for a typical plane-strain thick-walled cylinder (10”
outer radius, 2” wall thickness, 1000 psi internal pressure) were analyzed and the results
compared to the analytical solution. As there is not much variation expected
circumferentially, eight elements in that direction were judged to be adequate and the
variation in mesh density accomplished radially. Table 3 below shows the results for
hoop stress at the inner radius and radial displacement at the outer radius for different
sized meshes.
σθ_a
Nodes
Radially
FEA
% Err
FEA
% Err
3
4243.93
-6.84%
0.00107851
-0.0008345%
5
4389
-3.66%
0.00107852
9.272E-05%
4468.13
-1.92%
0.00107852
9.272E-05%
4503.53
-1.14%
0.00107852
9.272E-05%
20
4516.18
-0.87%
0.00107852
9.272E-05%
25
4523.6
-0.70%
0.00107852
9.272E-05%
9
15
Exact
ur_b
4555.6
Exact
0.001078519
Table 3 – Mesh Size vs. Solution Convergence
As it should have been expected, the displacement solution converged rapidly even
with a coarse mesh, but the stress solution took longer. A radial mesh of twenty elements
was sufficient to produce less than 1% error and it will be used for the remainder of the
analyses.
For the same typical thick-walled cylinder discussed in section 2.1.2.2, the hoop
stress and radial displacements are shown in Figure 3 and Figure 4. See Appendix B for
plots of the other quantities.
8
Hoop Stress - Plane-Stress
29000
Hoop Stress (psi)
27000
25000
23000
21000
ABAQUS
19000
17000
15000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure 3 – ABAQUS Hoop Stress (Pressure Load)
Radial Displacement - Plane-Stress
0.0078
Radial Displacement (in)
0.0076
0.0074
0.0072
0.007
ABAQUS
0.0068
0.0066
0.0064
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure 4 – ABAQUS Radial Displacement (Pressure Load)
2.1.4
Comparison of Results
Thin-Walled Cylinders - To examine the classical definition of a thin-walled
cylinder, a series of models were run using the same outer diameter (ten-inches) and
differing wall thicknesses to produce a range of radius/wall-thickness (r/t) ratios (from 4
to 79). The pressures chosen for each case were taken from [6] and meant to produce
near-yield stresses in the cylinders.
9
The following plots show normalized hoop stresses vs. normalized thickness (Figure
5) and normalized radial displacement vs. normalized thickness (Figure 6) for a range of
radius-to-wall-thickness (r/t) ratios, both using plane-stress assumptions. The normalized
quantities are the ABAQUS value (i.e. S22 for the hoop stress) divided by the “exact”
value (equation (2) for plane-stress hoop stress). The normalized thickness runs the
range from zero (for the inner radius) to one (for the outer radius), regardless of the
actual thickness. See Appendix B for plots of other quantities.
Hoop Stress - Plane-Stress
1.15
1.1
Normalized Stress
1.05
1
r/t = 4.0
r/t = 5.7
r/t = 9.0
0.95
r/t = 12.3
r/t = 19.0
0.9
r/t = 39.0
r/t = 79.0
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure 5 – ABAQUS vs. Exact Hoop Stresses as a Function of r/t Ratios
Radial Displacement - Plane-Stress
1.14
1.12
Normalized Displacement
1.1
1.08
r/t = 4.0
1.06
r/t = 5.7
r/t = 9.0
r/t = 12.3
1.04
r/t = 19.0
r/t = 39.0
1.02
r/t = 79.0
1
0.98
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure 6 – ABAQUS vs. Exact Radial Displacements as a Function of r/t Ratios
10
For most quantities, once the r/t ratio was greater then five, the calculated values
were within ten-percent of the exact values. The radial stresses and strains were a major
exception to this rule; however this is due to the assumption that the radial stress is
constant across the thickness. In reality it is at a maximum at the point of pressure
application and falls off to zero on the other side. The longitudinal stresses, longitudinal
strains, and hoop strains did not come within ten-percent of the expected value until r/t
reached 9.0, but this is within the ten-to-one ratio recommended by most texts. Therefore
for most non-radial quantities, a minimum radius-to-wall-thickness ratio of ten-to-one is
sufficient to provide answers accurate within ten-percent.
Thick-Walled Cylinders – When the formulas for stresses and strain in thick-walled
cylinders, equations (16) through (28), were used, the results from the finite-element
analyses were much closer regardless of the radius and wall thickness. Figure 7 and
Figure 8 show normalized hoop stresses vs. normalized thickness (Figure 7) and
normalized radial displacement vs. normalized thickness (Figure 8) for a range of radiusto-wall-thickness (r/t) ratios, both using plane-stress assumptions. See Appendix B for
plots of other quantities.
Hoop Stress - Plane-Stress
1.05
Normalized Stress
1
0.95
r/t = 4.0
0.9
r/t = 2.3
r/t = 1.5
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure 7 – ABAQUS vs. Exact Hoop Stresses as a Function of r/t Ratios
11
Radial Displacement - Plane-Stress
1.01
Normalized Displacement
1.005
r/t = 4.0
1
r/t = 2.3
r/t = 1.5
0.995
0.99
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure 8 – ABAQUS vs. Exact Radial Displacements as a Function of r/t Ratios
For most quantities the ABAQUS solution was within a few percent of the actual
solution. The radial stresses showed a small amount of error (less than five-percent) near
the inner radius and a much greater error near the outer radius - but this was because at
the inner radius the exact solution was zero, leading to infinitely large ratios (which
Excel plots as going to zero). The hoop stresses, radial strain, and hoop strains were
within a few percent at either edge and almost exact through most of the thickness. The
longitudinal stresses, longitudinal strains, and radial displacements were nearly exact –
within a fraction of a percent. On the whole, the finite-element solution was an excellent
representation of the exact solution.
For the remainder of the elastic portion of this project, only the typical thick-walled
cylinder discussed above will be analyzed. It is assumed that the solutions are consistent
enough that multiple wall thicknesses do not need to be addressed.
2.2 Thermal Loading
2.2.1
Analytical Solution
When a long cylinder is subject to different constant temperatures on both the inside
walls and the outside walls, thermal stresses develop due to the uneven expansion.
Timoshenko [4] presented a solution for this steady-state based on methods similar to
that used for the stresses in a thick-walled cylinder subject to internal pressure.
12
For the plane stress case, the radial stress is given by:
b
2
2
 1  r


r a

r    E
  t r dr 
  t r d r
 2 a

2 2
2 a
r  b a
r



(30)
and the hoop stress can be found by the relationship:
d  
r
 dr 
(31)
  r  r 
which in turn gives:
2
2
1 


r a
    E    t r d r 
  t r d r  t
 2 a

2 2
2 a
r  b a
r

r
b


(32)
If the inside surface of the cylinder is subject to a constant temperature t i, with the
outside surface held at a temperature of zero, the temperature distribution inside the
walls of the cylinder is given by:
t 
ti
b
ln  
a
 ln 
b

r
(33)
Any other temperature distribution can be analyzed assuming a uniform heating or
cooling which does not produce additional stresses. Substituting this into equations (30)
and (32) and integrating gives:
r 
 
E   ti

 ln 
b

b  r
2 ln   
a
E   ti

 1  ln 

2
2
b a
b

 

2 ln   
a
b
a


r
a
2
 1 



  ln b 
2 
a 
r   
b

2
 1 

b a 
2
2
2

  ln b 
2 
a 
r   
b
2
(34)
(35)
For the plane-stress case, the longitudinal stress (σz) is zero, and the strains are once
again found using Hooke’s Law with the addition of a uniform thermal expansion term:
r 
 
r
E


E

E



    z    t

E


 r  z    t
13
(36)
r 
r
E


E


    z    t


      r  z    t
  E  E  r  z    t
E
E
z 
z  z    r      t
z  E  E  r      t
E
E








(37)
(38)
The resulting strains are:
2 
2

b   a
b   b 










r 
   1    ln   
 1  1  
 ln  
b 
2
a 
 r   b2  a2  
r   
2 ln   
a
  ti
 

  ti
2

b   b 







1



1



 ln  

2   a 
 r   b2  a2  
r 

 1   1     ln 
b 
2 ln   
a
z 
  ti
b


a
2

   2  1     ln 
b
b

r

 
a
2 ln 
(39)

2   a
2
2
b a
2
 ln 
b 

 a 
(40)
(41)
Of interest is that unlike the pressure-only case where the longitudinal strain is
constant, under a thermal load the longitudinal strain is a function of the radius. The
radial displacement is calculated by:
ur  r 
u r 
  ti

2

b   b 







1



1



 ln    r

2   a 
 r   b2  a2  
r 

 1   1     ln 
b 
2 ln   
a
b


a
(42)
2
(43)
For the plane-strain case, the longitudinal strain (εz) is zero, and the radial and hoop
stresses are similar to the plane-stress case with the addition of one term (the (1-ν) in the
denominator):
r 
2
2

a
b   b 
 1 
 ln  

b   r 
2
2 
2   a 



b

a
r
2 1    ln   



a
E   ti

 ln 
b
2
2


b
a
b   b 



 
 1  ln   
 1
 ln  
b 
2 
a 
 r  b2  a2 
r   
2  1     ln   
a
 
E   ti
14
(44)
(45)
The longitudinal stress is found using the equation:


z    r      E t
(46)
which results in:
E   ti
z 

    2 ln 
b

r
b 
2  1     ln   
a

2   a
2
2
b a
2
 ln 
b  

 a  
(47)
The radial and hoop strains become:
r 

  ti
2
2

a
b
2
b 
  1      1    
 2    ln  

2
  a  (48)
 r  b2  a2 
r


      1     ln 
2
b 
 
a
2  1     ln 
 

  ti
2  1     ln 

 

b
a
2
2

a
b
2
b 
  1      1    
 2    ln  

2
  a  (49)
 r  b2  a2 
r


 1     1     ln 
2
b
b
And the radial displacement:
u r 

  ti

 
a
2  1     ln 
b
2
2

a
b
2
b 
  1      1    
 2    ln    r

2
  a  (50)
 r  b2  a2 
r


 1     1     ln 
2
b
Once again the typical thick-walled cylinder discussed above (10.0-inch outer
radius, 7.0-inch inner radius, plane-stress conditions, E = 30.0E6, ν = 0.3) will be used as
an example. It is assumed that it has a constant coefficient of linear thermal expansion of
α = 7.3E-06 in/in/°F (typical for steel). A constant temperature of 200°F is applied at the
inside surface while the outer surface is held at 0°F. The hoop stress and radial
displacements are shown in Figure 9 and Figure 10. See Appendix B for plots of the
other quantities.
15
Hoop Stress - Plane-Stress
25000.00
20000.00
15000.00
Hoop Stress (psi)
10000.00
5000.00
0.00
7.00
-5000.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Exact
-10000.00
-15000.00
-20000.00
-25000.00
Radius (in)
Figure 9 – Exact Hoop Stress (Thermal Load)
Radial Displacement - Plane-Stress
0.00700000
Radial Displacement (in)
0.00600000
0.00500000
0.00400000
0.00300000
Exact
0.00200000
0.00100000
0.00000000
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Radius (in)
Figure 10 – Exact Radial Displacement (Thermal Load)
2.2.2
Finite-Element Solution
The finite element code ABAQUS was again used for the numerical solutions. Two
input files were required for each model; one for the steady-state heat transfer part, and
one for the stress/displacement part. A parameterized input file was used for each part to
generate 2D cylinders of different cross-sections (outer radius and wall thickness),
element types (plane-stress vs. plane-strain for the stress/displacement phase), and
16
loading conditions (internal vs. external temperature and pressure). Sample input files
are listed in Appendix A.
For the heat transfer part, element type DC2D4 (4-noded linear heat transfer and
mass diffusion element) was used. The parameterized input file for the pressure section
was modified to produce the same mesh for this analysis. A thermal conductivity of
6.944E-04 BTU/s-in-°F was used, but in reality as this analysis is steady-state the exact
value is not critical. The desired temperature fields were applied as boundary conditions
and the steady-state nodal temperatures saved in an output file to feed the static phase.
For the static phase, the parameterized file used for the pressure analysis was
modified slightly to include the thermal effects. The nodal temperature file is read in and
used as initial conditions. A coefficient of thermal expansion of α = 7.3E-06 in/in/°F
(typical for steel) was added. The input file retains the ability the include pressure
effects, this will be used for the combined analysis. For other details on the finiteelement models, see Appendix A and section 2.1.3.
For the same typical thick-walled cylinder discussed in section 2.2.1, the hoop stress
and radial displacements are shown in Figure 11 and Figure 12. See Appendix B for
plots of the other quantities.
Hoop Stress - Plane-Stress
25000.00
20000.00
15000.00
Hoop Stress (psi)
10000.00
5000.00
0.00
7.00
-5000.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
r/t = 2.3
-10000.00
-15000.00
-20000.00
-25000.00
Radius (in)
Figure 11 – ABAQUS Hoop Stress (Thermal Load)
17
Radial Displacement - Plane-Stress
0.00700000
Radial Displacement (in)
0.00600000
0.00500000
0.00400000
0.00300000
r/t = 2.3
0.00200000
0.00100000
0.00000000
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Radius (in)
Figure 12 – ABAQUS Radial Displacement (Thermal Load)
2.2.3
Comparison of Results
Temperature Distribution – Figure 13 shows a plot comparing the steady-state
temperature distributions calculated by equation (33) and the ABAQUS heat transfer
model. They are identical.
Temperature Distribution
200.00
180.00
160.00
Temperature (°F)
140.00
120.00
Exact
ABAQUS
100.00
80.00
60.00
40.00
20.00
0.00
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75
10.00
Radius (in)
Figure 13 – Exact vs. ABAQUS Temperature Distribution (Thermal Load)
Figure 14 and Figure 15 show the hoop stress and radial displacements for both the
exact solution and the ABAQUS finite-element model for the typical thick-walled
18
cylinder discussed above. Apart from a small error at the inside and outside edges in the
hoop stress, the curves are nearly co-linear. See Appendix B for plots of the other
quantities.
Hoop Stress - Plane-Stress
25000
20000
15000
Hoop Stress (psi)
10000
5000
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-5000
Exact
-10000
ABAQUS
-15000
-20000
-25000
-30000
Radius (in)
Figure 14 – Exact vs. ABAQUS Hoop Stress (Thermal Load)
Radial Displacement - Plane-Stress
0.007
Radial Displacement (in)
0.006
0.005
0.004
Exact
0.003
ABAQUS
0.002
0.001
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure 15 – Exact vs. ABAQUS Radial Displacement (Thermal Load)
2.3 Combined Pressure and Thermal Loading
19
2.3.1
Analytical Solution
For the elastic domain the stresses, strains, and displacements resulting from a
combination of pressure and thermal loads may be found by simple superposition. For
the plane-stress case, the radial stress will be a combination of equations (18) and (34).
r  r_pressure  r_thermal
r 
p a

2
 1 

b a 
2
2
2
2
E   ti 

b
a
b   b 

 ln   
 1 
 ln  
2 
b   r 
2
2 
2   a 

r  2 ln   
b a 
r 

a
b
2
(51)
(52)
Similar combinations for the other quantities can also be done. Plots of the
combined quantities may be found in section 2.3.3.
2.3.2
Finite-Element Solution
The ABAQUS models used for this solution are the same as in the thermal analysis,
except for this case the pressure is non-zero. See Appendix A for a listing of the
ABAQUS input files and the next section for plots of the stresses, strains, and
displacements.
2.3.3
Comparison of Results
Figure 16 and Figure 17 show the exact and ABAQUS hoop stress and radial
displacement for a combined pressure and thermal load. As can be seen, the two
solutions are nearly identical. See Appendix B for plots of the other quantities.
20
Hoop Stress - Plane-Stress
45000
40000
Hoop Stress (psi)
35000
30000
25000
Exact
20000
15000
ABAQUS
10000
5000
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure 16 – Exact vs. ABAQUS Hoop Stress (Combined Load)
Radial Displacement - Plane-Stress
0.0134
Radial Displacement (in)
0.0132
0.013
0.0128
Exact
0.0126
ABAQUS
0.0124
0.0122
0.012
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure 17 – Exact vs. ABAQUS Radial Displacement (Combined Load)
21
3. ELASTIC-PLASTIC RESPONSE
The elastic-plastic solutions go here.
3.1 Pressure Loading
3.1.1
Analytical Solution
3.1.2
Finite-Element Solution
3.2 Thermal Loading
3.2.1
Analytical Solution
3.2.2
Finite-Element Solution
3.3 Combined Pressure and Thermal Loading
3.3.1
Analytical Solution
3.3.2
Finite-Element Solution
22
4. DISCUSSION
23
5. BIBLIOGRAPHY
[1] Young, W.C., 1989, Roark’s Formulas for Stress & Strain, 6th Edition, McGrawHill, New York, NY.
[2] Avalone, E.A. & Baumeister (III), T, 1987, Marks’ Standard Handbook for
Mechanical Engineers, 9th Edition, McGraw-Hill, New York, NY.
[3] Case, J, 1999, Strength of Materials and Structures, 4th Edition, John Wiley & Sons
Inc., New York, NY.
[4] Timoshenko, S., 1956, Strength of Material Part II, Advanced Theory and Problems,
3rd Edition, D. Van Nostrand Company Inc., Princeton, NJ.
[5] ABAQUS, v6.7-2, DSS Simulia, Providence, RI.
[6] Hojjarti, M.H. & Hassani, A., 2006, “Theoretical and finite-element modeling of
autofrettage process in strain-hardening thick-walled cylinders,” International
Journal of Pressure Vessels and Piping, 84 (2007) 310-319.
24
APPENDIX A – SAMPLE ABAQUS FILES
1) Sample ABAQUS input file (.inp) for the elastic pressure-only case.
*heading
10-Inch OD, 2.0-Inch Wall Thickness, Plane-Strain, 7482 psi internal
pressure
*parameter
#
# geometric/load parameters,
# radius is the outside radius
# thickness is the thickness of the shell
# press_type is either 'int' for internal or 'ext' for external
#
radius = 10.000
thickness = 2.000
pressure = 7482
press_type = 'int'
#
# elastic material properties
#
young = 30e+06
poisson = 0.3
#
# mesh parameters (can be modified)
# elem_type = PE for plane-strain, PS for plane-stress
# node_circum = nodes around 1/16 circumference
# node_radial = nodes through the thickness (minimum 2)
#
elem_type = 'PE'
node_circum = 9
node_radial = 20
##
## dependent parameters (do not modify)
##
node_circum4 = (node_circum-1)*4
node_ang = 22.5/float(node_circum)
node_tot = node_circum4*node_radial
iradius = radius-thickness
node_int = node_radial-1
node_circum0 = node_circum-1
node_circum40 = node_circum4-1
node_circum1 = node_circum4+1
node_circum2 = node_circum4+2
node_circum3 = node_tot-node_circum4+1
node_tot1 = node_circum3+node_circum-1
elem = 'C' + elem_type + '4R'
load_surf = press_type + '_surf'
chn = node_tot-2*node_circum4+1
chn1 = node_tot-2*node_circum4+node_circum-1
#
#end of parameter list
#
**
** define nodes around outer circumference
**
25
*node,system=c
1,<radius>,33.75,0.0
<node_circum>,<radius>,56.25,0.0
*ngen,line=c,nset=outside
1,<node_circum>,1,,0.0,0.0,0.0,0.0,0.0,1.0
**
** define nodes around inner circumference
**
*node,system=c
<node_circum3>,<iradius>,33.75,0.0
<node_tot1>,<iradius>,56.25,0.0
*ngen,line=c,nset=inside
<node_circum3>,<node_tot1>,1,,0.0,0.0,0.0,0.0,0.0,1.0
**
** generate the interior nodes
**
*nfill
outside,inside,<node_int>,<node_circum4>
**
** define node set for boundary conditions, transformation,
transformation CS
**
*nset, nset=ends, generate
1,<node_circum3>,<node_circum4>
<node_circum>,<node_tot1>,<node_circum4>
*nset, nset=allnodes, generate
1, <node_tot1>
*transform, nset=allnodes, type=C
0.0,0.0,0.0,0.0,0.0,1.0
**
** define first element on outer ring and element type
**
*element,type=<elem>
1,1,2,<node_circum2>,<node_circum1>
**
** generate remainder of elements
**
*elgen,elset=cylinder
1,<node_circum0>,1,1,<node_int>,<node_circum4>,<node_circum4>
**
** define load surfaces
**
*elset, elset=int, generate
<chn>, <chn1>
*elset, elset=ext, generate
1, <node_circum0>
*surface, type=element, name=int_surf
int, S3
*surface, type=element, name=ext_surf
ext, S1
**
** define section properties, unit out-of-plane thickenss assumed
**
*solid section, elset=cylinder, material=steel
1.0,
**
** define material
26
and
**
*material,name=steel
*elastic
<young>,<poisson>
**
** define boundary conditions
**
*boundary
ends,2,2
ends,6,6
**
** define pressure load step
**
*step, name=Pressure_Load
*static
*dsload
<load_surf>, P, <pressure>
**
** Output variable requests
**
*output,field, variable=preselect
*output, history, variable=preselect
*end step
2) Sample ABAQUS input file (.inp) for the steady-state heat transfer analysis.
*heading
10-Inch OD, 3.0-Inch Wall Thickness, Heat Transfer, 200F internal temp
*parameter
#
# geometric/load parameters,
# radius is the outside radius
# thickness is the thickness of the shell
# int_temp is the internal temperature
# ext_temp is the external temperature
#
radius = 10.000
thickness = 3.000
int_temp = 200
ext_temp = 0
#
# elastic/thermal material properties
#
k is the thermal conductivity
#
young = 30e+06
poisson = 0.3
k = 6.944E-04
#
# mesh parameters (can be modified)
# node_circum = nodes around 1/16 circumference
# node_radial = nodes through the thickness (minimum 2)
#
node_circum = 9
node_radial = 20
##
## dependent parameters (do not modify)
27
##
node_circum4 = (node_circum-1)*4
node_ang = 22.5/float(node_circum)
node_tot = node_circum4*node_radial
iradius = radius-thickness
node_int = node_radial-1
node_circum0 = node_circum-1
node_circum40 = node_circum4-1
node_circum1 = node_circum4+1
node_circum2 = node_circum4+2
node_circum3 = node_tot-node_circum4+1
node_tot1 = node_circum3+node_circum-1
elem = 'DC2D4'
chn = node_tot-2*node_circum4+1
chn1 = node_tot-2*node_circum4+node_circum-1
#
#end of parameter list
#
**
** define nodes around outer circumference
**
*node,system=c
1,<radius>,33.75,0.0
<node_circum>,<radius>,56.25,0.0
*ngen,line=c,nset=outside
1,<node_circum>,1,,0.0,0.0,0.0,0.0,0.0,1.0
**
** define nodes around inner circumference
**
*node,system=c
<node_circum3>,<iradius>,33.75,0.0
<node_tot1>,<iradius>,56.25,0.0
*ngen,line=c,nset=inside
<node_circum3>,<node_tot1>,1,,0.0,0.0,0.0,0.0,0.0,1.0
**
** generate the interior nodes
**
*nfill
outside,inside,<node_int>,<node_circum4>
**
** define node set for boundary conditions, transformation,
transformation CS
**
*nset, nset=ends, generate
1,<node_circum3>,<node_circum4>
<node_circum>,<node_tot1>,<node_circum4>
*nset, nset=allnodes, generate
1, <node_tot1>
*transform, nset=allnodes, type=C
0.0,0.0,0.0,0.0,0.0,1.0
**
** define first element on outer ring and element type
**
*element,type=<elem>
1,1,2,<node_circum2>,<node_circum1>
**
** generate remainder of elements
28
and
**
*elgen,elset=cylinder
1,<node_circum0>,1,1,<node_int>,<node_circum4>,<node_circum4>
**
** define load surfaces
**
*elset, elset=int, generate
<chn>, <chn1>
*elset, elset=ext, generate
1, <node_circum0>
*surface, type=element, name=int_surf
int, S3
*surface, type=element, name=ext_surf
ext, S1
**
** define section properties, unit out-of-plane thickenss assumed
**
*solid section, elset=cylinder, material=steel
1.0,
**
** define material
**
*material,name=steel
*elastic
<young>,<poisson>
*conductivity
<k>,
**
** define thermal load step
**
*step, name=Thermal_Load
*heat transfer, steady state
**
** define boundary conditions
**
*boundary
inside, 11, 11, <int_temp>
outside, 11, 11, <ext_temp>
**
** Output variable requests
**
*node file
nt,
*output, field
*node output
nt,
*end step
3) Sample ABAQUS input file (.inp) for the stress/displacement phase of the thermal
and combined pressure/thermal analyses.
*heading
10-Inch OD, 3.0-Inch Wall Thickness, Plane-Stress, 200F internal temp
*parameter
#
29
# heat transfer results file name
#
ht_file = '10OD_3.0WTDC'
#
# geometric/load parameters,
# radius is the outside radius
# thickness is the thickness of the shell
# press_type is either 'int' for internal or 'ext' for external
#
radius = 10.000
thickness = 3.000
pressure = 0.0
press_type = 'int'
#
# elastic/thermal material properties
#
alpha is the thermal expansion
#
young = 30e+06
poisson = 0.3
alpha = 7.3e-06
#
# mesh parameters (can be modified)
# elem_type = PE for plane-strain, PS for plane-stress
# node_circum = nodes around 1/16 circumference
# node_radial = nodes through the thickness (minimum 2)
#
elem_type = 'PS'
node_circum = 9
node_radial = 20
##
## dependent parameters (do not modify)
##
node_circum4 = (node_circum-1)*4
node_ang = 22.5/float(node_circum)
node_tot = node_circum4*node_radial
iradius = radius-thickness
node_int = node_radial-1
node_circum0 = node_circum-1
node_circum40 = node_circum4-1
node_circum1 = node_circum4+1
node_circum2 = node_circum4+2
node_circum3 = node_tot-node_circum4+1
node_tot1 = node_circum3+node_circum-1
elem = 'C' + elem_type + '4R'
load_surf = press_type + '_surf'
chn = node_tot-2*node_circum4+1
chn1 = node_tot-2*node_circum4+node_circum-1
#
#end of parameter list
#
**
** define nodes around outer circumference
**
*node,system=c
1,<radius>,33.75,0.0
<node_circum>,<radius>,56.25,0.0
*ngen,line=c,nset=outside
30
1,<node_circum>,1,,0.0,0.0,0.0,0.0,0.0,1.0
**
** define nodes around inner circumference
**
*node,system=c
<node_circum3>,<iradius>,33.75,0.0
<node_tot1>,<iradius>,56.25,0.0
*ngen,line=c,nset=inside
<node_circum3>,<node_tot1>,1,,0.0,0.0,0.0,0.0,0.0,1.0
**
** generate the interior nodes
**
*nfill
outside,inside,<node_int>,<node_circum4>
**
** define node set for boundary conditions, transformation,
transformation CS
**
*nset, nset=ends, generate
1,<node_circum3>,<node_circum4>
<node_circum>,<node_tot1>,<node_circum4>
*nset, nset=allnodes, generate
1, <node_tot1>
*transform, nset=allnodes, type=C
0.0,0.0,0.0,0.0,0.0,1.0
**
** define first element on outer ring and element type
**
*element,type=<elem>
1,1,2,<node_circum2>,<node_circum1>
**
** generate remainder of elements
**
*elgen,elset=cylinder
1,<node_circum0>,1,1,<node_int>,<node_circum4>,<node_circum4>
**
** define load surfaces
**
*elset, elset=int, generate
<chn>, <chn1>
*elset, elset=ext, generate
1, <node_circum0>
*surface, type=element, name=int_surf
int, S3
*surface, type=element, name=ext_surf
ext, S1
**
** define section properties, unit out-of-plane thickenss assumed
**
*solid section, elset=cylinder, material=steel
1.0,
**
** define material
**
*material,name=steel
*elastic
<young>,<poisson>
31
and
*expansion
<alpha>,
**
** define boundary conditions
**
*boundary
ends,2,2
**
** define thermal load step
**
*step, name=Thermal Load
*static
*temperature, file=<ht_file>
*dsload
<load_surf>, P, <pressure>
**
** Output variable requests
**
*output,field, variable=preselect
*output, history, variable=preselect
*end step
32
APPENDIX B – ADDITIONAL PLOTS
Thick-Walled Cylinder Under Internal Pressure (Exact Solution)
The following plots show the exact solution for a typical thick-walled cylinder,
10.0-inches outer radius, 7.0-inches inner radius, 10199 psi internal pressure with planestress conditions. Material properties are ν = 0.3, E = 30.0E6.
Radial Stress - Plane-Stress
0.00
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-1000.00
-2000.00
Radial Stress (psi)
-3000.00
-4000.00
-5000.00
-6000.00
Exact
-7000.00
-8000.00
-9000.00
-10000.00
-11000.00
Radius (in)
Figure A1– Exact Radial Stress (Pressure Load)
Hoop Stress - Plane-Stress
29000.00
Hoop Stress (psi)
27000.00
25000.00
23000.00
21000.00
Exact
19000.00
17000.00
15000.00
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
Radius (in)
Figure A2 – Exact Hoop Stress (Pressure Load)
33
10
Radial Strain - Plane-Stress
0.00000000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-0.00010000
Radial Strain
-0.00020000
-0.00030000
-0.00040000
Exact
-0.00050000
-0.00060000
-0.00070000
Radius (in)
Figure A3 – Exact Radial Strain (Pressure Load)
Hoop Strain - Plane-Stress
0.00120000
0.00100000
Hoop Strain
0.00080000
0.00060000
Exact
0.00040000
0.00020000
0.00000000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
Radius (in)
Figure A4 – Exact Hoop Strain (Pressure Load)
34
10
Longitudinal Strain - Plane-Stress
0.00000000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Longitudinal Strain
-0.00005000
-0.00010000
-0.00015000
Exact
-0.00020000
-0.00025000
Radius (in)
Figure A5 – Exact Longitudinal Strain (Pressure Load)
Radial Displacement - Plane-Stress
0.00780000
Radial Displacement (in)
0.00760000
0.00740000
0.00720000
0.00700000
Exact
0.00680000
0.00660000
0.00640000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A6 – Exact Radial Displacement (Pressure Load)
Thick-Walled Cylinder Under Internal Pressure (ABAQUS Solution)
The following plots show the ABAQUS solution for the typical thick-walled
cylinder discussed above.
35
Radial Stress - Plane-Stress
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-1000
Radial Stress (psi)
-3000
-5000
ABAQUS
-7000
-9000
-11000
Radius (in)
Figure A7 – ABAQUS Radial Stress (Pressure Load)
Hoop Stress - Plane-Stress
29000
Hoop Stress (psi)
27000
25000
23000
21000
ABAQUS
19000
17000
15000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A8 – ABAQUS Hoop Stress (Pressure Load)
36
Radial Strain - Plane-Stress
0.00000000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-0.00010000
Radial Strain
-0.00020000
-0.00030000
-0.00040000
ABAQUS
-0.00050000
-0.00060000
-0.00070000
Radius (in)
Figure A9 – ABAQUS Radial Strain (Pressure Load)
Hoop Strain - Plane-Stress
0.00120000
0.00100000
Hoop Strain
0.00080000
0.00060000
ABAQUS
0.00040000
0.00020000
0.00000000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A10 – ABAQUS Hoop Strain (Pressure Load)
37
Longitudinal Strain - Plane-Stress
0.00000000
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Longitudinal Strain
-0.00005000
-0.00010000
-0.00015000
ABAQUS
-0.00020000
-0.00025000
Radius (in)
Figure A11 – ABAQUS Longitudinal Strain (Pressure Load)
Radial Displacement - Plane-Stress
0.0078
Radial Displacement (in)
0.0076
0.0074
0.0072
0.007
ABAQUS
0.0068
0.0066
0.0064
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A12 – ABAQUS Radial Displacement (Pressure Load)
Thin-Walled Cylinder Discussion
Below are additional plots detailing the correlation between the exact and finiteelement solutions for a series of thick-to-thin-walled cylinders as the radius-to-wallthickness (r/t) ratio is increased.
38
Radial Stress - Plane-Stress
1.1
1
0.9
Normalized Stress
0.8
0.7
0.6
r/t = 4.0
0.5
r/t = 5.7
0.4
r/t = 9.0
r/t = 12.3
0.3
r/t = 19.0
0.2
r/t = 39.0
r/t = 79.0
0.1
0
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A13 – Radial Stress vs. r/t Ratios for Plane-Stress
Radial Stress - Plane-Strain
1.1
1
0.9
Normalized Stress
0.8
0.7
0.6
r/t = 4.0
0.5
r/t = 5.7
0.4
r/t = 9.0
r/t = 12.3
0.3
r/t = 19.0
0.2
r/t = 39.0
r/t = 79.0
0.1
0
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A14 – Radial Stress vs. r/t Ratios for Plane-Strain
39
Hoop Stress - Plane-Stress
1.15
1.1
Normalized Stress
1.05
1
r/t = 4.0
r/t = 5.7
r/t = 9.0
0.95
r/t = 12.3
r/t = 19.0
0.9
r/t = 39.0
r/t = 79.0
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A15 – Hoop Stress vs. r/t Ratios for Plane-Stress
Hoop Stress - Plane-Strain
1.15
1.1
Normalized Stress
1.05
1
r/t = 4.0
r/t = 5.7
r/t = 9.0
0.95
r/t = 12.3
r/t = 19.0
0.9
r/t = 39.0
r/t = 79.0
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A16 – Hoop Stress vs. r/t Ratios for Plane-Strain
40
Longitudinal Strain - Plane-Stress
1.2
1.18
1.16
Normalized Strain
1.14
1.12
r/t = 4.0
1.1
r/t = 5.7
1.08
r/t = 9.0
r/t = 12.3
1.06
r/t = 19.0
1.04
r/t = 39.0
r/t = 79.0
1.02
1
0.98
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A17 – Longitudinal Strain vs. r/t Ratios for Plane-Stress
Longitudinal Stress - Plane-Strain
1.2
1.18
1.16
Normalized Stress
1.14
1.12
r/t = 4.0
1.1
r/t = 5.7
1.08
r/t = 9.0
r/t = 12.3
1.06
r/t = 19.0
1.04
r/t = 39.0
r/t = 79.0
1.02
1
0.98
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A18 – Longitudinal Stress vs. r/t Ratios for Plane-Strain
41
Radial Strain - Plane-Stress
1.1
1
Normalized Strain
0.9
0.8
r/t = 4.0
r/t = 5.7
r/t = 9.0
0.7
r/t = 12.3
r/t = 19.0
0.6
r/t = 39.0
r/t = 79.0
0.5
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A19 – Radial Strain vs. r/t Ratios for Plane-Stress
Radial Strain - Plane Strain
1.1
1
Normalized Strain
0.9
0.8
r/t = 4.0
r/t = 5.7
r/t = 9.0
0.7
r/t = 12.3
r/t = 19.0
0.6
r/t = 39.0
r/t = 79.0
0.5
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A20 – Radial Strain vs. r/t Ratios for Plane-Strain
42
Hoop Strain - Plane-Stress
1.2
1.15
Normalized Strain
1.1
1.05
r/t = 4.0
1
r/t = 5.7
r/t = 9.0
r/t = 12.3
0.95
r/t = 19.0
r/t = 39.0
0.9
r/t = 79.0
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A21 – Hoop Strain vs. r/t Ratios for Plane-Stress
Hoop Strain - Plane-Strain
1.2
1.15
Normalized Strain
1.1
1.05
r/t = 4.0
1
r/t = 5.7
r/t = 9.0
r/t = 12.3
0.95
r/t = 19.0
r/t = 39.0
0.9
r/t = 79.0
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A22 – Hoop Strain vs. r/t Ratios for Plane-Strain
43
Radial Displacement - Plane-Stress
1.14
1.12
Normalized Displacement
1.1
1.08
r/t = 4.0
1.06
r/t = 5.7
r/t = 9.0
r/t = 12.3
1.04
r/t = 19.0
r/t = 39.0
1.02
r/t = 79.0
1
0.98
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A23 – Radial Displacement vs. r/t Ratios for Plane-Stress
Radial Displacement - Plane-Strain
1.14
1.12
Normalized Displacement
1.1
1.08
r/t = 4.0
1.06
r/t = 5.7
r/t = 9.0
r/t = 12.3
1.04
r/t = 19.0
r/t = 39.0
1.02
r/t = 79.0
1
0.98
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A24 – Radial Displacement vs. r/t Ratios for Plane-Strain
Thick-Walled Cylinder Under Pressure Discussion
Below are additional plots detailing the correlation between the exact and finiteelement solutions for a series of thick-walled cylinders as the radius-to-wall-thickness
(r/t) ratio is decreased.
44
Radial Stress - Plane-Stress
1.1
1
0.9
Normalized Stress
0.8
0.7
0.6
r/t = 4.0
0.5
r/t = 2.3
0.4
0.3
r/t = 1.5
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A25 – Radial Stress vs. r/t Ratios for Plane-Stress
Radial Stress - Plane-Strain
1.1
1
0.9
Normalized Stress
0.8
0.7
0.6
r/t = 4.0
0.5
0.4
r/t = 2.3
0.3
r/t = 1.5
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A26 – Radial Stress vs. r/t Ratios for Plane-Strain
45
Hoop Stress - Plane-Stress
1.05
Normalized Stress
1
0.95
r/t = 4.0
0.9
r/t = 2.3
r/t = 1.5
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A27 – Hoop Stress vs. r/t Ratios for Plane-Stress
Hoop Stress - Plane-Strain
1.15
1.1
Normalized Stress
1.05
1
r/t = 4.0
0.95
r/t = 2.3
0.9
r/t = 1.5
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A28 – Hoop Stress vs. r/t Ratios for Plane-Strain
46
Longitudinal Strain - Plane-Stress
1.001
1.0008
1.0006
Normalized Strain
1.0004
1.0002
1
r/t = 4.0
0.9998
r/t = 2.3
0.9996
r/t = 1.5
0.9994
0.9992
0.999
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A29 – Longitudinal Strain vs. r/t Ratios for Plane-Stress
Longitudinal Stress - Plane-Strain
1.001
1.0008
1.0006
Normalized Stress
1.0004
1.0002
1
r/t = 4.0
0.9998
r/t = 2.3
0.9996
r/t = 1.5
0.9994
0.9992
0.999
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A30 – Longitudinal Stress vs. r/t Ratios for Plane-Strain
47
Radial Strain - Plane-Stress
1.1
1
Normalized Strain
0.9
0.8
r/t = 4.0
0.7
r/t = 2.3
0.6
r/t = 1.5
0.5
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A31 – Radial Strain vs. r/t Ratios for Plane-Stress
Radial Strain - Plane Strain
1.1
1
Normalized Strain
0.9
0.8
r/t = 4.0
0.7
r/t = 2.3
0.6
r/t = 1.5
0.5
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A32 – Radial Strain vs. r/t Ratios for Plane-Strain
48
Hoop Strain - Plane-Stress
1.2
1.15
Normalized Strain
1.1
1.05
1
r/t = 4.0
0.95
r/t = 2.3
0.9
r/t = 1.5
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A33 – Hoop Strain vs. r/t Ratios for Plane-Stress
Hoop Strain - Plane-Strain
1.2
1.15
Normalized Strain
1.1
1.05
1
r/t = 4.0
0.95
r/t = 2.3
0.9
r/t = 1.5
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A34 – Hoop Strain vs. r/t Ratios for Plane-Strain
49
Radial Displacement - Plane-Stress
1.01
Normalized Displacement
1.005
r/t = 4.0
1
r/t = 2.3
r/t = 1.5
0.995
0.99
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A35 – Radial Displacement vs. r/t Ratios for Plane-Stress
Radial Displacement - Plane-Strain
1.01
Normalized Displacement
1.005
r/t = 4.0
1
r/t = 2.3
r/t = 1.5
0.995
0.99
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Distance Through Thickness
Figure A36 – Radial Displacement vs. r/t Ratios for Plane-Strain
Thick-Walled Cylinder Under Thermal Load (Exact Solution)
The following plots show the exact solution for a typical thick-walled cylinder, 10.0inches outer radius, 7.0-inches inner radius, 200°F at inner surface, 0°F at outer surface
with plane-stress conditions. Material properties are ν = 0.3, E = 30.0E6, α=7.3E-06.
50
Radial Stress - Plane-Stress
0.00
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Radial Stress (psi)
-500.00
-1000.00
-1500.00
Exact
-2000.00
-2500.00
Radius (in)
Figure A37 – Exact Radial Stress (Thermal Load)
Hoop Stress - Plane-Stress
25000.00
20000.00
15000.00
Hoop Stress (psi)
10000.00
5000.00
0.00
7.00
-5000.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Exact
-10000.00
-15000.00
-20000.00
-25000.00
Radius (in)
Figure A38 – Exact Hoop Stress (Thermal Load)
51
Radial Strain - Plane-Stress
0.00200000
Radial Strain
0.00150000
0.00100000
0.00050000
0.00000000
7.00
Exact
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
-0.00050000
Radius (in)
Figure A39 – Exact Radial Strain (Thermal Load)
Hoop Strain - Plane-Stress
0.00074000
0.00073000
0.00072000
0.00071000
Hoop Strain
0.00070000
0.00069000
0.00068000
Exact
0.00067000
0.00066000
0.00065000
0.00064000
0.00063000
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Radius (in)
Figure A40 – Exact Hoop Strain (Thermal Load)
52
Longitudinal Strain - Plane-Stress
0.00200000
Longitudinal Strain
0.00150000
0.00100000
0.00050000
0.00000000
7.00
Exact
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
-0.00050000
Radius (in)
Figure A41 – Exact Longitudinal Strain (Thermal Load)
Radial Displacement - Plane-Stress
0.00700000
Radial Displacement (in)
0.00600000
0.00500000
0.00400000
0.00300000
Exact
0.00200000
0.00100000
0.00000000
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Radius (in)
Figure A42 – Exact Radial Displacement (Thermal Load)
Thick-Walled Cylinder Under Thermal Load (ABAQUS Solution)
The following plots show the exact solution for a typical thick-walled cylinder, 10.0inches outer radius, 7.0-inches inner radius, 200°F at inner surface, 0°F at outer surface
with plane-stress conditions. Material properties are ν = 0.3, E = 30.0E6, α=7.3E-06.
53
Radial Stress - Plane-Stress
0.00
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Radial Stress (psi)
-500.00
-1000.00
-1500.00
r/t = 2.3
-2000.00
-2500.00
Radius (in)
Figure A43 – ABAQUS Radial Stress (Thermal Load)
Hoop Stress - Plane-Stress
25000.00
20000.00
15000.00
Hoop Stress (psi)
10000.00
5000.00
0.00
7.00
-5000.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
r/t = 2.3
-10000.00
-15000.00
-20000.00
-25000.00
Radius (in)
Figure A44 – ABAQUS Hoop Stress (Thermal Load)
54
Radial Strain - Plane-Stress
0.00180000
0.00160000
0.00140000
0.00120000
Radial Strain
0.00100000
0.00080000
0.00060000
r/t = 2.3
0.00040000
0.00020000
0.00000000
7.00
-0.00020000
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
-0.00040000
Radius (in)
Figure A45 – ABAQUS Radial Strain (Thermal Load)
Hoop Strain - Plane-Stress
0.00074000
0.00073000
0.00072000
Hoop Strain
0.00071000
0.00070000
0.00069000
0.00068000
r/t = 2.3
0.00067000
0.00066000
0.00065000
0.00064000
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Radius (in)
Figure A46 – ABAQUS Hoop Strain (Thermal Load)
55
Longitudinal Strain - Plane-Stress
0.00200000
Longitudinal Strain
0.00150000
0.00100000
0.00050000
0.00000000
7.00
r/t = 2.3
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
-0.00050000
Radius (in)
Figure A47 – ABAQUS Longitudinal Strain (Thermal Load)
Radial Displacement - Plane-Stress
0.00700000
Radial Displacement (in)
0.00600000
0.00500000
0.00400000
0.00300000
r/t = 2.3
0.00200000
0.00100000
0.00000000
7.00
7.25
7.50
7.75
8.00
8.25
8.50
8.75
9.00
9.25
9.50
9.75 10.00
Radius (in)
Figure A48 – ABAQUS Radial Displacement (Thermal Load)
Thick-Walled Cylinder Under Thermal Load (Comparison)
The following plots show the exact solution vs. the ABAQUS solution for a typical
thick-walled cylinder, 10.0-inches outer radius, 7.0-inches inner radius, 200°F at inner
surface, 0°F at outer surface with plane-stress conditions. Material properties are ν = 0.3,
E = 30.0E6, α=7.3E-06.
56
Radial Stress - Plane-Stress
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radial Stress (psi)
-500
-1000
Exact
-1500
ABAQUS
-2000
-2500
Radius (in)
Figure A49 – Exact vs. ABAQUS Radial Stress – Plane-Stress (Thermal Load)
Radial Stress - Plane-Strain
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-500
Radial Stress (psi)
-1000
-1500
Exact
-2000
ABAQUS
-2500
-3000
Radius (in)
Figure A50 – Exact vs. ABAQUS Radial Stress – Plane-Strain (Thermal Load)
57
Hoop Stress - Plane-Stress
25000
20000
15000
Hoop Stress (psi)
10000
5000
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-5000
Exact
-10000
ABAQUS
-15000
-20000
-25000
-30000
Radius (in)
Figure A51 – Exact vs. ABAQUS Hoop Stress – Plane-Stress (Thermal Load)
Hoop Stress - Plane-Strain
40000
30000
Hoop Stress (psi)
20000
10000
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Exact
-10000
ABAQUS
-20000
-30000
-40000
Radius (in)
Figure A52 – Exact vs. ABAQUS Hoop Stress – Plane-Strain (Thermal Load)
58
Longitudinal Strain - Plane-Stress
0.002
Longitudinal Strain
0.0015
0.001
Exact
0.0005
ABAQUS
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-0.0005
Radius (in)
Figure A53 – Exact vs. ABAQUS Longitudinal Strain – Plane-Stress (Thermal
Load)
Longitudinal Stress - Plane-Strain
20000
10000
Longitudinal Stress (psi)
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-10000
-20000
Exact
-30000
ABAQUS
-40000
-50000
-60000
Radius (in)
Figure A54 – Exact vs. ABAQUS Longitudinal Stress – Plane-Strain (Thermal
Load)
59
Radial Strain - Plane-Stress
0.002
Radial Strain
0.0015
0.001
Exact
0.0005
ABAQUS
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-0.0005
Radius (in)
Figure A55 – Exact vs. ABAQUS Radial Strain – Plane-Stress (Thermal Load)
Radial Strain - Plane Strain
0.0025
0.002
Radial Strain
0.0015
0.001
Exact
0.0005
ABAQUS
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-0.0005
Radius (in)
Figure A56 – Exact vs. ABAQUS Radial Strain – Plane-Strain (Thermal Load)
60
Hoop Strain - Plane-Stress
0.00074
0.00073
0.00072
0.00071
Hoop Strain
0.0007
0.00069
Exact
0.00068
0.00067
ABAQUS
0.00066
0.00065
0.00064
0.00063
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A57 – Exact vs. ABAQUS Hoop Strain – Plane-Stress (Thermal Load)
Hoop Strain - Plane-Strain
0.00098
0.00096
0.00094
Hoop Strain
0.00092
0.0009
Exact
0.00088
ABAQUS
0.00086
0.00084
0.00082
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A58 – Exact vs. ABAQUS Hoop Strain – Plane-Strain (Thermal Load)
61
Radial Displacement - Plane-Stress
0.007
Radial Displacement (in)
0.006
0.005
0.004
Exact
0.003
ABAQUS
0.002
0.001
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A59 – Exact vs. ABAQUS Radial Displ. – Plane-Stress (Thermal Load)
Radial Displacement - Plane-Strain
0.009
0.008
Radial Displacement (in)
0.007
0.006
0.005
Exact
0.004
0.003
ABAQUS
0.002
0.001
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A60 – Exact vs. ABAQUS Radial Displ. – Plane-Strain (Thermal Load)
Thick-Walled Cylinder Under Combined Load (Comparison)
The following plots show the exact solution vs. the ABAQUS solution for a typical
thick-walled cylinder, 10.0-inches outer radius, 7.0-inches inner radius, 200°F at inner
surface, 0°F at outer surface with either a 10199 psi internal pressure (plane-stress) or a
10600 psi internal pressure (plane strain). Material properties are ν = 0.3, E = 30.0E6,
α=7.3E-06.
62
Radial Stress - Plane-Stress
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radial Stress (psi)
-2000
-4000
-6000
Exact
-8000
ABAQUS
-10000
-12000
Radius (in)
Figure A61 – Exact vs. ABAQUS Radial Stress – Plane-Stress (Combined Load)
Radial Stress - Plane-Strain
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-2000
Radial Stress (psi)
-4000
-6000
Exact
-8000
ABAQUS
-10000
-12000
Radius (in)
Figure A62 – Exact vs. ABAQUS Radial Stress – Plane-Strain (Combined Load)
63
Hoop Stress - Plane-Stress
45000
40000
Hoop Stress (psi)
35000
30000
25000
Exact
20000
15000
ABAQUS
10000
5000
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A63 – Exact vs. ABAQUS Hoop Stress – Plane-Stress (Combined Load)
Hoop Stress - Plane-Strain
60000
50000
Hoop Stress (psi)
40000
30000
Exact
20000
ABAQUS
10000
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-10000
Radius (in)
Figure A64 – Exact vs. ABAQUS Hoop Stress – Plane-Strain (Combined Load)
64
Longitudinal Strain - Plane-Stress
0.002
Longitudinal Strain
0.0015
0.001
Exact
0.0005
ABAQUS
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-0.0005
Radius (in)
Figure A65 – Exact vs. ABAQUS Long. Strain – Plane-Stress (Combined Load)
Longitudinal Stress - Plane-Strain
20000
10000
Longitudinal Stress (psi)
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-10000
-20000
Exact
-30000
ABAQUS
-40000
-50000
-60000
Radius (in)
Figure A66 – Exact vs. ABAQUS Long. Stress – Plane-Strain (Combined Load)
65
Radial Strain - Plane-Stress
0.0012
0.001
0.0008
Radial Strain
0.0006
0.0004
Exact
0.0002
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
ABAQUS
-0.0002
-0.0004
-0.0006
Radius (in)
Figure A66 – Exact vs. ABAQUS Radial Strain – Plane-Stress (Combined Load)
Radial Strain - Plane Strain
0.002
0.0015
Radial Strain
0.001
0.0005
Exact
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
ABAQUS
-0.0005
-0.001
Radius (in)
Figure A67 – Exact vs. ABAQUS Radial Strain – Plane-Strain (Combined Load)
66
Hoop Strain - Plane-Stress
0.002
0.0018
0.0016
Hoop Strain
0.0014
0.0012
0.001
Exact
0.0008
0.0006
ABAQUS
0.0004
0.0002
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A68 – Exact vs. ABAQUS Hoop Strain – Plane-Stress (Combined Load)
Hoop Strain - Plane-Strain
0.0025
Hoop Strain
0.002
0.0015
Exact
0.001
ABAQUS
0.0005
0
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A69 – Exact vs. ABAQUS Hoop Strain – Plane-Strain (Combined Load)
67
Radial Displacement - Plane-Stress
0.0134
Radial Displacement (in)
0.0132
0.013
0.0128
Exact
0.0126
ABAQUS
0.0124
0.0122
0.012
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A70 – Exact vs. ABAQUS Radial Displ. – Plane-Stress (Combined Load)
Radial Displacement - Plane-Strain
0.015
0.0148
Radial Displacement (in)
0.0146
0.0144
0.0142
Exact
0.014
0.0138
ABAQUS
0.0136
0.0134
0.0132
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
Radius (in)
Figure A71 – Exact vs. ABAQUS Radial Displ. – Plane-Strain (Combined Load)
68