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Poworoznek, Peter
Elastic-Plastic Behavior of a 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
© Copyright 2008
by
Peter P. Poworoznek
All Rights Reserved
2
CONTENTS
LIST OF TABLES ............................................................................................................. 4
LIST OF FIGURES ........................................................................................................... 5
NOMENCLATURE........................................................................................................... 6
ACKNOWLEDGEMENT ................................................................................................. 7
ABSTRACT ....................................................................................................................... 8
1. INTRODUCTION/BACKGROUND .......................................................................... 9
2. LINEAR ELASTIC CYLINDER .............................................................................. 10
2.1
Pressure Loading .............................................................................................. 10
2.2
Thermal Loading .............................................................................................. 23
2.3
Combined Pressure and Thermal Loading ....................................................... 29
3. ELASTIC PERFECTLY-PLASTIC CYLINDER ..................................................... 31
3.1
Analytical Solution ........................................................................................... 32
3.2
Finite-Element Model ....................................................................................... 45
3.3
Comparison of Results ..................................................................................... 45
4. BIBLIOGRAPHY ...................................................................................................... 48
APPENDIX A – SAMPLE ABAQUS FILES ................................................................. 49
APPENDIX B – ADDITIONAL PLOTS ........................................................................ 60
3
LIST OF TABLES
Table 1 – Thin-Walled Cylinder Plane Stress Results ..................................................... 13
Table 2 – Thin-Walled Cylinder Plane Strain Results ..................................................... 13
Table 3 – Thick-Walled Cylinder Test Case Properties .................................................. 15
Table 4 – Mesh Size vs. Solution Convergence............................................................... 18
4
LIST OF FIGURES
Figure 1 – Typical Cylinder ............................................................................................. 11
Figure 2 – Exact Hoop Stress (Pressure Load) ................................................................ 16
Figure 3 – Exact Radial Displacement (Pressure Load) .................................................. 16
Figure 4 – ABAQUS Model and Mesh ............................................................................ 19
Figure 5 – ABAQUS Hoop Stress (Pressure Load) ......................................................... 19
Figure 6 – ABAQUS Radial Displacement (Pressure Load) ........................................... 20
Figure 7 – ABAQUS vs. Exact Hoop Stresses as a Function of r/t Ratios ...................... 21
Figure 8 – ABAQUS vs. Exact Radial Displacements as a Function of r/t Ratios .......... 21
Figure 9 – ABAQUS vs. Exact Hoop Stresses as a Function of r/t Ratios ...................... 22
Figure 10 – ABAQUS vs. Exact Radial Displacements as a Function of r/t Ratios ........ 23
Figure 11 – Exact vs. ABAQUS Temperature Distribution (Thermal Load) .................. 27
Figure 12 – Exact vs. ABAQUS Hoop Stress (Thermal Load) ....................................... 28
Figure 13 – Exact vs. ABAQUS Radial Displacement (Thermal Load) ......................... 28
Figure 14 – Exact vs. ABAQUS Hoop Stress (Combined Load) .................................... 30
Figure 15 – Exact vs. ABAQUS Radial Displacement (Combined Load) ...................... 30
Figure 16 – Stress-Strain Curve for an Elastic-Perfectly Plastic Material ....................... 31
Figure 17 – Yield Pressure vs. Ratio of b/a ..................................................................... 35
Figure 18 – A Partially-Plastic Thick-Walled Cylinder................................................... 36
Figure 19 – Exact vs. ABAQUS Hoop Stresses (Plastic - Pressure Load) ...................... 46
Figure 20 – Exact vs. ABAQUS Radial Displ. (Plastic - Pressure Load)........................ 46
Figure 21 – ABAQUS Yield Pressure vs. Time Increment – (Fully Plastic) .................. 47
5
NOMENCLATURE
r, θ, z = Radial, hoop, and longitudinal directions in a cylindrical coordinate system
I, II, III= Principal directions
σ
= Stress (psi)
ε
= Strain
p
= Pressure (psi)
t
= Thickness (in.)
T
= Temperature (°F)
r
= Radius (in.)
E
= Young’s modulus (psi)
ν
= Poisson’s ratio
u
= Displacement (in.)
a
= Inner radius of cylinder (in.)
b
= Outer radius of cylinder (in.)
c
= Radius of the elastic-plastic boundary
r/t
= Ratio of inner radius of cylinder to wall thickness
b/a
= Ratio of outer radius of cylinder to inner radius of cylinder
i,o
= Subscripts denoting inner or outer surfaces of cylinder
T, VM = Subscripts denoting Tresca or Von-Mises criteria
DOF = Degree of freedom
S__ = ABAQUS stress in the radial (11), hoop (22), or longitudinal (33) direction
E__ = ABAQUS strain in the radial (11), hoop (22), or longitudinal (33) direction
α
= Coefficient of thermal expansion (in./in./°F)
k
= Yield strength in shear (psi) for stress/displacement applications
Y
= Yield strength is tension (psi)
y
= Subscript denoting yield
φ
= Auxiliary variable
6
ACKNOWLEDGEMENT
I wish to thank Professor Ernesto Gutierrez-Miravete for his help and guidance in
preparation of this project. I also wish to thank my wife and daughter for putting up with
me while I worked on this.
7
ABSTRACT
This project examines the thermo-mechanical behavior of an ideal cylinder with
both plane-stress and plane-strain end conditions. Both analytical methods and finiteelement models are used to predict the stress and strain levels and radial displacements,
and the results compared.
Initially the elastic solution for a cylinder subject to an internal pressure is
discussed. Although the majority of the report focuses on thick-walled cylinders, thinwalled 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 temperature gradient
across the cylinder are examined; both by itself and in combination with a pressure load.
Next the pressure loads are increased to induce plastic behavior in the cylinder for a
perfectly-plastic material. Partially and fully plastic thick-walled cylinders are examined
using Tresca and Von-Mises yield criteria for both plane-strain and plane-stress end
conditions.
For the elastic domain, 2D finite-element models of sufficient mesh density provide
excellent correlation with classic analytical solutions. In the plastic domain, the
correlation is not as close; analytical solutions usually employ some simplification of the
material model for solvability or ease of use and finite-element models also require
approximations of the material models. The choice of element and mesh size is critical,
but if care is taken both the analytical solutions and the finite-element models can be
used satisfactorily.
8
1. INTRODUCTION/BACKGROUND
This project looks at the stresses, strains, and displacements in cylinders subject to
mechanical loads such as a high internal pressure, and thermal loads such as a
temperature gradient across the thickness of the cylinder. Both plane-stress and planestrain end conditions are analyzed. Plane-strain conditions are typical for a cylinder
where the length is much larger than its radius (i.e. a fluid filled pipe), and plane-stress
conditions are used when the length is smaller than the radius; such as a thin ring or disk.
Historically problems involving cylinders have been extensively studied due to their
practical importance. Understanding how tubes react to high pressures and temperatures
has aided the design of things such as pressure vessels used in steam generators to the
gun barrels employed during wartime. By understanding the forces at work, engineers
have been able to optimize designs to improve the safety and reliability of their products.
In the purely elastic domain, solutions with great accuracy have existed since the
19th century. But the addition of material yield and plastic deformation greatly increased
the difficulty. Each material demonstrates slightly different characteristics in the plastic
domain which do not lend themselves to simple modeling. And certain quantities depend
on the rate at which the deformation occurs, not just on the forces involved. As such,
multiple theories have emerged, each tailored to a specific material model or application.
Sometimes simplification of the material behavior leads to solutions that agree well with
experimentation, other time a more rigorous approach is warranted.
This project examines the behavior of a cylinder in both the elastic and plastic
domains. For the elastic domain, both thin-walled and thick-walled cylinders are
analyzed and the results used to justify classic thin-walled tube theory. Then an elastic
perfectly-plastic material model is used to examine the plastic behavior of a cylinder
subject to high pressures.
9
2. LINEAR ELASTIC CYLINDER
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 examine the linear elastic stresses and strains in cylinders with a
range of radius-to-wall-thicknesses subject to pressure loading. The results are used to
justify the ten-to-one ratio.
2.1.2
Analytical Solution
In a cylinder that is loaded axisymmetrically and uniformly along its length, “no
shearing stresses will be transmitted along any co-axial cylindrical surface or any plane
which is perpendicular to the axis.” [10] .Thus by using a cylindrical coordinate system,
all of the stresses in the r, θ, and z directions are principal stresses and they depend only
on the radius of the point in question from the axis of the cylinder.
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 (see Figure 1). The radial stress
is assumed to be constant and is equal to the negative of the applied pressure.
σr := −p
(1)
The hoop stress can be readily found by examining the free body diagram of a halfcylinder and is given by the equation [1]:
σθ :=
10
p⋅ r
t
(2)
Figure 1 – Typical Cylinder
From Hooke’s law, the strains are calculated using:
εr :=
1
εθ :=
1
εz :=
1
E
⋅ σr − ν ⋅ σθ + σz 


(
)
(3)
⋅ σθ − ν ⋅ σr + σz 


(
)
(4)
(
)
(5)
E
E
⋅ σz − ν ⋅ σr + σθ

In this case, the longitudinal (σz) stress is zero, therefore:
εr :=

E 
−p
εθ :=
εz :=
⋅ 1 +
p
E
⋅ 
ν ⋅p
E
11

r
t
ν ⋅r 
(6)
+ ν 
(7)
r
(8)
⋅  1 −



t


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 := r⋅ εθ
u r :=
p⋅ r
⋅ 
r
t
E
(9)
+ ν 
(10)

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:
εz :=
1
E
(
⋅ σz − ν ⋅ σr + σθ

(
σz := ν ⋅ σr + σθ
)
0
)
(11)
(12)
The radial stress is constant (-p), therefore:
σz := −ν ⋅ p ⋅  1 −

r

(13)
t
The hoop and radial strains, using the same equations as in plane stress are:
εr :=
εθ :=
−p
E


2
⋅ 1 − ν +
ν ⋅r
t


⋅ ( 1 + ν )
(
)
(
)
2
r
⋅ ν ⋅ ( 1 + ν ) + ⋅ 1 − ν 
E
t

p
(14)
(15)
And the change in radius is:
u r :=
2
r
⋅ ν ⋅ ( 1 + ν ) + ⋅ 1 − ν 
E 
t

p⋅ r
(16)
For the range of cylinders to be discussed in Section 2.1.4, the exact analytical
values calculated using the equations above are shown in Table 1 and Table 2 (all are
based on a 10.0-inch outer radius, all units are in inches & psi, ν=0.3, E=30.0E6 psi).
The pressures chosen come from [6] and are meant to bring the cylinder to near yield
stress.
12
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
A typical thick-walled cylinder of inner radius (a), outer radius (b), inner pressure
(pi), and outer pressure (po) is shown in Figure 1. Equations for the hoop stress and radial
stress in a thick-walled cylinder were developed by Lamé in the early 19th century [4]. In
general form, they are:
2
σr :=
2
a pi − b ⋅ po
2
b −a
2
σθ :=
2
−
2
a pi − b ⋅ p o
2
b −a
2
13
+
(pi − po)⋅ a2⋅ b2
2
2
r ⋅ (b − a )
(17)
2
(pi − po)⋅ a2⋅ b2
2
(
2
r ⋅ b −a
2
)
(18)
The following calculations 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 (17) and (18) reduce to:
p⋅ a
σr :=

2
⋅ 1 −

b −a 
2
σθ :=
2
p⋅ a
⋅ 1 +
2
2


r 
b

b −a 
2
(19)
2

2
2


r 
b
(20)
2
From equation (20), the hoop stress is 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:
σθ_max
σθ_min
2
:=
a +b
2⋅ a
2
(21)
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:
εr :=
εθ :=
p⋅ a

2
(2
E⋅ b − a
p⋅ a
⋅ 1 − ν −
b
) 
2
2
(2
E⋅ b − a
εz :=
⋅ 1 − ν +
) 
2


b
2
2
2
E⋅ b − a
2
(22)

⋅ ( 1 + ν )
r
−2⋅ ν ⋅ p ⋅ a
(2

⋅ ( 1 + ν )
r

2
2


)
(23)
(24)
And the change in radius (rεθ) is:
2

b

(
)
u r :=
⋅ 1−ν +
⋅ (1 +
2
2 
2
E⋅ b − a 
r
(
p⋅ a
2
)
14

ν ) ⋅ r


(25)
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:
2⋅ ν ⋅ p⋅ a
σz :=
εr :=
p⋅ a
2
b −a

2
⋅ 1 − ν −
2
2 
E⋅ ( b − a ) 
εθ :=
u r :=
p⋅ a
(2
E⋅ b − a
1+ ν
E
⋅
⋅ 1 − ν +
) 
2
p⋅ a
2
2
⋅ ( 1 + ν ) − 2⋅ ν

2


b
2
2
⋅ ( 1 + ν ) − 2⋅ ν

2
r

2
⋅  1 − 2⋅ ν +

b −a 
2
(26)
2
r

2
b
2
2


(28)
2
⋅r

r 
b
(27)
2
(29)
For many of the analyses in the project, a typical thick-walled cylinder is examined
as a test case. Unless otherwise specified, the parameters shown in Table 3 are used.
Parameter
Value
a (inner radius)
7.0 inches
b (outer radius)
10.0 inches
E (Young’s Modulus)
30.0E6 psi
ν (Poisson’s Ratio)
0.3
α (thermal expansion)
7.3E-06 in/in/°F
Y (yield strength)
36000 psi
Table 3 – Thick-Walled Cylinder Test Case Properties
For the cylinder described in Table 3 (r/t = 2.3) with an internal pressure of 10199
psi the hoop stress and radial displacements are shown in Figure 2 and Figure 3.
Longitudinal strain is a constant at 0.00019598; see Appendix B for plots of the other
quantities.
15
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
10
Radius (in)
Figure 2 – Exact Hoop Stress (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 3 – Exact Radial Displacement (Pressure Load)
2.1.3
Finite-Element Model
The finite element code ABAQUS [5] is used for the numerical models. A
parameterized input file is employed to generate 2D cylinders of different cross-sections
(outer radius and wall thickness), element types (plane-stress vs. plane-strain), and
16
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 is modeled. Symmetry
boundary conditions (circumferential displacement equal to zero, the elements chosen do
not have nodal rotation DOFs) are applied at the ends to ensure that the behavior of the
full cylinder is represented. ABAQUS element types CPS4R (plane-stress) and CPE4R
(plane-strain) are 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 are
calculated using Hooke’s law similar to equation (5) by creating an additional output
field:
εz :=
−ν
E
⋅ ( S11 + S22)
(30)
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 psi & Poisson’s
Ratio (ν) = 0.3, are 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-stress thick-walled cylinder (10-inch outer radius, 3-inch
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 (nine nodes circumferentially) are judged to be adequate and
the variation in mesh density is accomplished radially. Table 4 below shows the results
for hoop stress at the inner radius and radial displacement at the outer radius for several
different sized meshes.
17
σθ_a
Nodes
Radially
Exact
ur_b
FEA
% Err
3
2573.59
5
Exact
FEA
% Err
-11.91
0.000640518
-7.81E-04
2729.68
-6.56
0.000640521
-3.12E-04
9
2819.62
-3.49
0.000640522
-1.56E-04
15
2861.03
-2.07
0.000640523
0.00
20
2921.569 2876.00
-1.56
0.000640523 0.000640523
0.00
25
2884.85
-1.26
0.000640523
0.00
30
2890.68
-1.06
0.000640523
0.00
35
2894.83
-0.92
0.000640523
0.00
40
2897.92
-0.81
0.000640523
0.00
Table 4 – Mesh Size vs. Solution Convergence
As it should have been expected, the displacements converge rapidly even with a
coarse mesh, but the stresses take longer. A radial mesh of thirty-four elements (thirtyfive nodes) is sufficient to produce less than 1% error and it is used for the remainder of
the analyses for this typical thick-walled cylinder. See Figure 4 for a plot of this model
and mesh.
18
Figure 4 – ABAQUS Model and Mesh
For the thick-walled cylinder test case shown in Table 3, the hoop stress and radial
displacements are shown in Figure 5 and Figure 6. See Appendix B for plots of the other
quantities.
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 5 – ABAQUS Hoop Stress (Pressure Load)
19
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 6 – 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 are taken from [6] and meant to produce near-yield stresses in the
cylinders.
The following plots show the normalized hoop stresses vs. normalized thickness
(Figure 7) and normalized radial displacement vs. normalized thickness (Figure 8) 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.
20
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 7 – 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 8 – ABAQUS vs. Exact Radial Displacements as a Function of r/t Ratios
For most quantities, once the r/t ratio is greater then five, the ABAQUS values are
within ten-percent of the exact values. The radial stresses and strains are 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 do not come within ten-percent of the expected value until the
21
r/t ratio 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 equations for stresses and strain in thick-walled cylinders, equations (17)
through (29), are used, the results from the finite-element analyses are much closer
regardless of the radius and wall thickness. Figure 9 and Figure 10 show normalized
hoop stresses vs. normalized thickness (Figure 9) and normalized radial displacement vs.
normalized thickness (Figure 10) for a range of radius-to-wall-thickness (r/t) ratios, both
using plane-stress assumptions. See Appendix B for plots of other quantities.
Hoop Stress - Plane-Stress
1.01
Normalized Stress
1
0.99
r/t = 4.0
0.98
r/t = 2.3
0.97
r/t = 1.5
0.96
0.95
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 9 – ABAQUS vs. Exact Hoop Stresses as a Function of r/t Ratios
22
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 10 – ABAQUS vs. Exact Radial Displacements as a Function of r/t Ratios
For most quantities the ABAQUS model is within a few percent of the exact
solution. The radial stresses show a small amount of error (less than five-percent) near
the inner radius and a much greater error near the outer radius - but this is because at the
inner radius the exact solution is zero, leading to infinitely large ratios (which Excel
plots as going to zero). The hoop stresses, radial strain, and hoop strains are within a few
percent at either edge and almost exact through most of the thickness. The longitudinal
stresses, longitudinal strains, and radial displacements are nearly exact – within a
fraction of a percent. On the whole, the finite-element model is 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 is 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.
23
For the plane stress case, the radial stress is given by:
b
2
2

 −1 ⌠ r
⌠
r −a
⋅  T⋅ r dr +
⋅  T⋅ r dr
 2 ⌡a

2 2
2 ⌡a
r ⋅ b −a
r

σr := α ⋅ E⋅ 
(
)
(31)
and the hoop stress can be found by the relationship:
d σ 
r
 dr 
(32)
σθ := σ r + r⋅ 
which in turn gives:
2
2

1 ⌠
⌠
r −a
σθ := α ⋅ E⋅  ⋅  T⋅ r dr +
⋅  T⋅ r dr − T
 2 ⌡a

2 2
2 ⌡a
r ⋅ b −a
r

r
b
(
)
(33)
If the inside surface of the cylinder is subject to a constant temperature Ti, with the
outside surface held at a temperature of zero, the temperature distribution inside the
walls of the cylinder is given by:
Ti
T :=
b
ln 
a
⋅ ln
b

r
(34)
Any other temperature distribution can be analyzed assuming a uniform heating or
cooling which does not produce additional stresses. Substituting this into equations (31)
and (33) and integrating gives:
σr :=
σθ :=

E⋅ α ⋅ Ti
⋅ −ln
b
  r 


a
2⋅ ln
b
E⋅ α ⋅ Ti
b
2⋅ ln 
a

⋅ 1 − ln

2
2
b −a
b

r


a
−
−
a
2
⋅ 1 −



 ⋅ ln b 
2 
a 
r   
b

2
⋅ 1 +

b −a 
2
2
2

 ⋅ ln b 
2 
a 
r   
b
2
(35)
(36)
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:
εr :=
σr
E
−
ν
E
(
)
⋅ σθ + σz + α ⋅ T
24
(37)
σθ
εθ :=
εz :=
E
σz
E
−
−
ν
E
ν
E
⋅ σr + σz + α ⋅ T
(
)
(38)
(
)
(39)
⋅ σr + σθ + α ⋅ T
The resulting strains are:

b
εr :=
⋅ −ν + ( 1 + ν ) ⋅ ln 
b 
r
2⋅ ln  
a
 
α ⋅ Ti
εθ :=
α ⋅ Ti

2

b   b 


(
)
(
)
⋅
1
−
ν
+
1
+
ν
⋅
⋅ ln 

2   a 
 r   b 2 − a2  
r 

⋅ 1 + ( 1 + ν ) ⋅ ln
b 
2⋅ ln  
a
εz :=
2
 a2  
b   b 



(
)
(
)
−
⋅ 1−ν − 1+ ν ⋅
⋅ ln 
 b 2 − a2  
2
a 
r   


α ⋅ Ti
b

−
a
2

⋅ −ν + 2⋅ ( 1 + ν ) ⋅ ln
b
b

r



a
2⋅ ln
(40)
+
2⋅ ν ⋅ a
2
2
b −a
2
⋅ ln
b 

 a 
(41)
(42)
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:
(43)
u r := r⋅ εθ
u r :=
α ⋅ Ti
2

b   b 


(
)
(
)
⋅
1
−
ν
+
1
+
ν
⋅
⋅ ln  ⋅ r

2   a 
 r   b 2 − a2  
r 


⋅ 1 + ( 1 + ν ) ⋅ ln
b 
2⋅ ln  
a
b

−
a
2
(44)
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):
2
2
 b

a
b   b 


σr :=
⋅ −ln  −
⋅ 1−
⋅ ln 
2
2 
2 
b  r
a 
b −a 
r   
2⋅ ( 1 − ν ) ⋅ ln  
a
 
E⋅ α ⋅ Ti
σθ :=
E⋅ α ⋅ Ti

⋅ 1 − ln



a
2⋅ ln
b
b

r
−
25
a
2

2
b −a
2
⋅ 1 +



 ⋅ ln b 
2 
a 
r   
b
(45)
2
(46)
The longitudinal stress is found using the equation:
(
)
σz := ν ⋅ σr + σθ − α ⋅ E⋅ T
(47)
which results in:

E⋅ α ⋅ Ti
σz :=
⋅  ν − 2⋅ ln

r



a
2⋅ ( 1 − ν ) ⋅ ln
b
b
−
2⋅ ν ⋅ a
2
2
b −a
2
⋅ ln
b  

 a  
(48)
The radial and hoop strains become:
εr :=

α ⋅ Ti
2
2

a
b
2
b 
⋅ ( 1 − ν ) − ( 1 + ν ) ⋅
− 2⋅ ν  ⋅ ln 
−
2
  a  (49)
 r  b 2 − a2 
r



⋅ −ν − ν + ( 1 + ν ) ⋅ ln
2

 
a
2⋅ ( 1 − ν ) ⋅ ln
εθ :=
b

α ⋅ Ti
b 
 
a
2⋅ ( 1 − ν ) ⋅ ln
2
2

a
b
2
b 
⋅ ( 1 − ν ) + ( 1 + ν ) ⋅
− 2⋅ ν  ⋅ ln 
−
2
  a  (50)
 r  b 2 − a2 
r



⋅ 1 − ν + ( 1 + ν ) ⋅ ln
2
b
b
And the radial displacement:
u r :=

α ⋅ Ti
2

 
a
2⋅ ( 1 − ν ) ⋅ ln
b
2
2

a
b
2
b 
⋅ ( 1 − ν ) + ( 1 + ν ) ⋅
− 2⋅ ν  ⋅ ln  ⋅ r
−
2
  a  (51)
 r  b 2 − a2 
r



⋅ 1 − ν + ( 1 + ν ) ⋅ ln
b
Once again the typical thick-walled cylinder shown in Table 3 is used as an
example. 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 12
and Figure 13. See Appendix B for plots of the other quantities.
2.2.2
Finite-Element Model
The finite element code ABAQUS is again used for the numerical models. Two
input files are required for each model; one for the steady-state heat transfer part, and
one for the stress/displacement part. A parameterized input file is 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
loading conditions (internal vs. external temperature and pressure). Sample input files
are listed in Appendix A.
26
For the heat transfer part, element type DC2D4 (4-noded linear heat transfer and
mass diffusion element) are used. The parameterized input file for the pressure section is
modified to produce the same mesh for this analysis. A thermal conductivity of 6.944E04 BTU/s-in-°F is used, but in reality as this analysis is steady-state the exact value is
not critical. The desired temperature gradients are 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 is modified
slightly to include the thermal effects. The nodal temperature file is read in and used as
initial conditions. The input file also retains the ability the include pressure effects; this
will be used for the combined analysis. For other details on the finite-element models,
see Appendix A and section 2.1.3.
For the typical thick-walled cylinder test case shown in Table 3, the hoop stress and
radial displacements are shown in Figure 12 and Figure 13. See Appendix B for plots of
the other quantities.
2.2.3
Comparison of Results
Figure 11 shows a plot comparing the steady-state temperature distributions
calculated by equation (34) 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 11 – Exact vs. ABAQUS Temperature Distribution (Thermal Load)
27
Figure 12 and Figure 13 show the hoop stress and radial displacements for both the
exact solution and the ABAQUS finite-element model for the typical thick-walled
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 12 – 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 13 – Exact vs. ABAQUS Radial Displacement (Thermal Load)
28
2.3 Combined Pressure and Thermal Loading
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 is a combination of equations (19) and (35).
σr := σr_pressure + σr_thermal
σr :=
p⋅ a

2
⋅ 1 −

b −a 
2
2
2
2
E⋅ α ⋅ Ti 

b
a
b   b 
+
⋅ −ln  −
⋅ 1 −
⋅ ln 
b   r 
2 
2
2 
2   a 

r  2⋅ ln  
b −a 
r 

a
b
(52)
2
(53)
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 Model
The ABAQUS models used for this case 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 See Appendix B for a full set of plots showing the stresses, strains,
and displacements, and the next section for plots of the hoop stresses and radial
displacements.
2.3.3
Comparison of Results
Figure 14 and Figure 15 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.
29
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 14 – 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 15 – Exact vs. ABAQUS Radial Displacement (Combined Load)
30
3. ELASTIC PERFECTLY-PLASTIC CYLINDER
When a material no longer follows the constitutive laws of elasticity, that material is
said to undergo inelastic deformation. Inelastic deformations which results from the
mechanisms of slip and lead to permanent dimensional changes are known as plastic
deformations [7]. To fully describe the elastic-plastic behavior of a material, four things
must be known [8]:
-
The stress-strain relation for the elastic range
-
The yield criterion
-
The stress-strain relation for the plastic range (flow rule)
-
The hardening rule
For this project, the elastic range is assumed to be linear and the relationship
between stress and strain is given by Young’s modulus (E). It is also assumed that the
material is elastic-perfectly plastic (i.e. no hardening occurs) with a yield strength of 36
ksi (typical for ASTM-A36 steel), see Figure 16 for a plot of the stress-curve strain. The
specific yield criterion and flow rule are addressed in the sections below for each
analysis. This project also assumes that the yield stress in compression is that same as
the yield stress in tension (i.e. the Bauschinger effect will be ignored).
Elastic-Perfectly Plastic Stress-Strain Curve
for ASTM-A36 Steel
40000
35000
30000
25000
ASTM-A36 Steel
20000
15000
10000
5000
0
0
0.002
0.004
0.006
0.008
0.01
0.012
Figure 16 – Stress-Strain Curve for an Elastic-Perfectly Plastic Material
31
The elastic-plastic response of a thick-walled cylinder subject to a high internal
pressure is probably one of the most studied cases in plasticity. It has many practical
uses such as pressure vessel design and the autofrettage process used to improve the
fatigue life of pressure vessels and gun barrels.
3.1 Analytical Solution
3.1.1
Initial Yielding
Yield Criteria
To determine the pressure required to initiate yielding, there are two different
yielding criteria that must be considered, each giving a slightly different answer. The
first of these is the Tresca yield criterion, also known as a maximum shear stress
criterion. It asserts that yielding occurs when the maximum shear stress reaches a
prescribed constant (C). In the case of principal stresses, the maximum shear stress is
one-half of the difference between the largest and smallest principal stresses (as can be
seen easily using Mohr’s circle). For a state of pure tension (for a generic element), the
only principal stress is σI, the other two being zero. Thus the maximum shear stress is
σI/2. At yield, σI is equal to the tensile yield strength (Y), therefore the Tresca constant is
equal to Y/2. For yield in a state of pure shear, the maximum shear stress is equal to the
yield strength in shear (k), σI is equal to the positive (k), and σIII is equal to negative (k).
Thus the Tresca constant is equal to (k). Therefore for the Tresca yield criteria:
k :=
Y
2
(54)
or
(55)
Y := 2⋅ k
For a thick-walled cylinder under internal pressure, the cylindrical stresses are the
principal stresses. The largest elastic stress will always be the hoop stress and the
smallest will always be the radial stress (see the elastic section of this project for
equations and plots). The longitudinal elastic stress will always fall in between the other
two. Therefore the Tresca yield criterion can be written:
σ1 − σ3
2
:= k
(57)
or
(56)
32
σ1 − σ3 := Y
The other yield criterion is the Von-Mises yield criterion. It asserts that yielding will
occur when the second deviatoric stress invariant (J2) reaches a critical value. Using
comparisons to the pure tension and pure shear cases, the Von-Mises criteria can be
written as either:
(σ1 − σ2) + (σ2 − σ3) + (σ1 − σ3)
2
2
2
2
:= 2⋅ Y
(58)
or
(σ1 − σ2) + (σ2 − σ3) + (σ1 − σ3)
2
2
2
2
:= 6⋅ k
(59)
With the subsequent relationship that with Von-Mises yielding:
k :=
Y
3
or
Y := 3⋅ k
(61)
(60)
From this it can be seen that the yield shear strength in the Von-Mises criterion is:
kVM :=
2
3
kT
(62)
or approximately 15% greater then the yield shear strength in the Tresca criterion.
Yield Pressure
Hill [9] outlines an approach to determine the yield stress in a thick-walled cylinder
using the Tresca criterion:
Y
σθ − σr :=
2
(63)
For both plane-strain and plane-stress, the equations for the radial stresses, equation
(19), and hoop stresses, equation (20), are the same, so the yield pressure will be the
same for both.
33
2⋅ p ⋅ b
r
σθ − σr :=
b
a
2
2
(64)
2
2
−1
Equation (64) is the largest when r is the smallest, at r = a, therefore setting equation
(64) equal to equation (63) and solving for p:
Y
p :=
2

⋅ 1 −




2
b 
a
2
(65)
For the test case shown in Table 3, this gives a yield pressure of 9180 psi.
For the Von-Mises criterion:
(σr − σθ ) + (σθ − σz) + (σz − σr)
2
2
2
2
:= 2⋅ Y
(66)
or
(σr − σθ )2 + (σθ − σz)2 + (σz − σr)2 :=
4 2
6⋅ b ⋅ p
2
 b2  4
 − 1 ⋅ r
 a2



+ 2⋅ 
p
2
 b −1
 2
a
− σz
2



(67)
Once again, this is the greatest when r = a. Using this and setting equation (67)
equal to equation (66):
4 2
2
3⋅ b ⋅ p
+
2
 b2  4
 − 1 ⋅ r
 a2



 p − σ  := Y2
 2
z
 b −1

 2

a

(68)
For the plane-stress condition, σz is zero, and for plane-strain it is given by equation
(26). Solving for p gives:
Y
p y :=
3

⋅ 1 −


1+
a


2
b 
a
2
(69)
4
3⋅ b
4
(plane-stress)
34
Y
3
p y :=

⋅ 1 −




2
b 
a
2
1 + ( 1 − 2⋅ ν ) ⋅
2 a
(70)
4
3⋅ b
4
(plane-strain)
For the test case, the yield pressure for plane-stress is 10199.84 psi and for planestrain it is 10532.93 psi. The plane-stress condition yields first, although the difference
between the two is only 3.26%.
Figure 17 shows a plot of the Tresca yield pressure and the Von-Mises yield
pressures vs. the ratio of the outer radius to the inner radius (b/a). The Tresca yield
pressure is much lower for a given cylinder. The difference between the two Von-Mises
pressures is small to begin with, but gets smaller as b increases. When b/a is equal to 2,
the difference is only 0.86%. Also of note is that the yield pressure increases greatly in
proportion to the outer radius only to a point at which it levels off. This means increasing
the outer radius beyond a certain size will have minimal impact on preventing the inner
surface from yielding.
Yield Pressure vs. Ratio of Radii
25000
Yield Pressure (psi)
20000
15000
Tresca
Von-Mises (Plane-Stress)
Von-Mises (Plane-Strain)
10000
5000
0
0
1
2
3
4
5
6
7
8
9
10
b/a (Outside Radius/Inside Radius)
Figure 17 – Yield Pressure vs. Ratio of b/a
35
3.1.2
The Partially-Plastic Cylinder
When a cylinder subject to high internal pressure begins to yield, there are two
distinct regions: a plastic region on the inside of the cylinder and an elastic region on the
outside (see Figure 18). The boundary between the two is cylindrically shaped and
stresses and displacements at the boundary must be consistent. The stresses in the elastic
region are still similar in form to those in section 2.1.2.2, but with different coefficients.
As the solutions for the plane-strain and plane-stress conditions are fairly complex
and sufficiently unique, they are addressed in two separate sections.
Figure 18 – A Partially-Plastic Thick-Walled Cylinder
3.1.2.1 Plane-Strain End Conditions
Hill [9] gives the basic equations for stresses in the elastic region of a plane-strain
thick-walled cylinder. The stresses are still in a form similar to the purely elastic state:
 b2
σr := −C⋅ 
 r2

 b2
σθ := −C⋅ 
 r2


− 1
(72)

(71)


+ 1


(73)
36
σz := E⋅ ε z + 2⋅ ν ⋅ C
where C is a constant to be found and the longitudinal stress is found from a form of
Hooke’s law, equation (5), rearranged. Hill’s solution assumed that there was no workhardening, that the material immediately on the elastic side of the elastic-plastic
boundary is at the point of yielding, and that the longitudinal stress remains the
intermediate stress. Assuming that the Tresca criteria:
(74)
σθ − σr := Y
holds everywhere in the plastic region and with the longitudinal strain being zero for
plane-strain, the stresses in the elastic region are:

−Y c  b
⋅ ⋅
− 1
2  2

2
b r

(75)

Y c b
σθ := ⋅ ⋅ 
+ 1
2 2  2

b r

b 
(76)
2
σr :=
2
2
2
σz := ν ⋅ Y⋅
c
2
b
(77)
2
where c is the radius of the elastic-plastic boundary.
The elastic strains can be found by applying Hooke’s law and making the
assumption that the strains are small enough that the initial radii a and b do not change
enough to warrant keeping track on the incremental changes.
εr :=
εθ :=
Y⋅ c
2
2⋅ E⋅ b
Y⋅ c
2

b


r

b
⋅ 1 − ν −
2
⋅ 1 − ν +

2⋅ E⋅ b 
2
εz := 0
2
2
⋅ ( 1 + ν ) − 2⋅ ν
2
2

2
⋅ ( 1 + ν ) − 2⋅ ν
r
(plane-strain)


2



(78)
(79)
(80)
And the radial displacement:
u r := r⋅ εθ
37
(81)
2
u r :=
Y⋅ c ⋅ r
2⋅ E⋅ b
2

b


r
⋅ 1 − ν +
2
2
⋅ ( 1 + ν ) − 2⋅ ν
2



(82)
One advantage of using the Tresca criterion is that the radial and hoop stresses are
independent of the end conditions. If Von-Mises yield is considered, this is not the case.
However, Hill [9] makes an argument that if the yield stress (Y) is replaced by:
Y :=
2
Y
(83)
3
then an excellent approximation of the Von-Mises yield criterion is obtained by using
the equations above. The local error introduced by using this substitution “is never
greater than two-percent and the overall all error is much less” [9]. Nadai [10] also
agrees with this substitution, and uses it in his equations for the elastic stress.
In the plastic region, Hill [9] combines the Tresca yield criterion with the equations
of equilibrium to obtain:
d
σr :=
dr
σθ − σr
Y
r
r
(84)
This gives the solutions for the radial and hoop stresses:
σr :=
−Y
2

⋅  1 + 2⋅ ln
c

 r


−


2
b 
c
2
2

c  c 


σθ := ⋅ 1 − 2⋅ ln  +
2 
 r  b 2 

Y
(85)
(86)
And by applying continuity of the radial stress at r = c, the internal pressure required to
produce this yield is:
2

c  c 


p := ⋅ 1 + 2⋅ ln  −
2 
 a  b 2 

Y
(87)
The longitudinal stress is not as easy to predict according to Hill. While using the
Tresca yield criterion allows the radial and stresses to be statically determined and
independent of the end conditions, an accurate formulation of the longitudinal stress
requires the use of the Prandtl-Reuss equations and are therefore dependent on the strain
38
histories. Hill develops a set of differential equations which can be used to predict the
remaining stress and strains, but they are complex and require a numerical solution.
Prager and Hodge [8] also derive a set of partial differential equations for the plastic
stresses and present results from the numerical solution. They simplify the solution by
assuming that the material is incompressible in the elastic and plastic regions. Doing
this, the equations for the radial and hoop stresses in the elastic region are the same as
equations (75) and (76), and the longitudinal stress and radial displacements are:
Y⋅ c
σz :=
2⋅ b
k⋅ c
u r :=
2
(88)
2
2
(89)
2⋅ G⋅ r
where k is the yield strength in shear and G is the shear modulus.
E
G :=
2⋅ ( 1 + ν )
(90)
Putting equation (89) in another form:
Y⋅ c ⋅ ( 1 + ν )
2
u r :=
2⋅ E⋅ r
(91)
In the plastic region, the Prager and Hodge radial and hoop stresses are the same as those
from Hill, equations (85) and (86), and that the radial displacement uses the same
equation as in the elastic zone, equation (91). The longitudinal stress is given by:
σz :=
Y
2
 c2
⋅
 b2

− 2⋅ ln
c  

 r  
(92)
The Prager and Hodge solution assumes that the material is incompressible in the
elastic region. This does not change the radial or hoop stresses as compared to the Hill
solution. However it does cause a discrepancy between the calculated incompressible
longitudinal stresses and radial displacements and those derived numerically using the
compressible form. Prager and Hodge note this, but suggest multiplying the longitudinal
stress by 2ν and the radial displacement by 2(1-ν). This gives an excellent correlation
between the two approaches. Incorporating these correction factors gives equation (77)
39
for the elastic longitudinal stress, equation (93) for the plastic longitudinal stress, and
equation (94) for the radial displacement (both elastic and plastic):
 c2
 b2

2⋅ E⋅ r
(
2

r 

Y⋅ c ⋅ 1 − ν
u r :=
c  
− 2⋅ ln
σz := ν ⋅ Y⋅ 
2
(93)

)
(94)
E⋅ r
Once again, replacing the yield stress (Y) with equation (83) gives a quasi-VonMises yield solution to this problem. The solutions given by Nadai [10] incorporate this
substitution.
For the strains, the Prager and Hodge solutions were derived using the relationships:
εr :=
d
ur
dr
(95)
u
εθ :=
(96)
r
which gives:
(
2
εr :=
−Y⋅ c ⋅ 1 − ν
)
(97)
2
E⋅ r
2
εθ :=
2
(
Y⋅ c ⋅ 1 − ν
)
2
2
(98)
E⋅ r
These equations hold for both the elastic and plastic regions.
A summary of the Prager & Hodge solution: for the elastic region the stresses are
given by equations (75) - (77). For the plastic region, the stresses are given by equations
(85), (86), and (93). The strains in both regions are given by equations (95) and (97), and
the radial displacement in both regions is given by equation (94). These values are for
the Tresca yield condition. For the Von-Mises yield condition, multiply the yield
strength (Y) by equation (83).
The range of validity for the Prager & Hodge solution is bounded by the pressure
required for initial yielding (c = a) and when the cylinder is fully plastic (c = b). These
pressures can be found using equation (87) to be:
40

p a := k⋅  1 −




2
b 
a
p b := 2⋅ k⋅ ln
2
(99)
b

a
(100)
For the test case cylinder, this equates to a pressure range of 9180 psi to 12840.3 psi
using Tresca yield, and 10600.15 psi to 14826.7 psi using Von-Mises yield. See Figure
19 and Figure 20 on page 46 for plots of the hoop stress and radial displacement and
Appendix B for plots of the other quantities for this test case. For these plots, a pressure
of 13838 psi was added to the internal surface to produce an elastic-plastic boundary, c,
halfway though the cylinder.
3.1.2.2 Plane-Stress End Conditions
Gao [11] developed a solution based on previous work by Nadai [10] for the
stresses, strains, and displacements in an open-ended (plane-stress) cylinder. It is based
upon the Von-Mises yield criterion and the Hencky deformation theory. Although it was
developed for strain-hardening using the elastic power-law plastic material model, the
solution applies to the elastic-perfectly plastic case for n = 0 (the equations shown below
have already made this substitution).
In the elastic zone a modified form of Lame’s solution is presented, where pc is the
pressure at the elastic-plastic interface.
σr :=
σθ :=
p c⋅ c
2

⋅ 1 −

b −c 
2
2
p c⋅ c
2

⋅ 1 +

b −c 
2
2
2


r 
b
(101)
2
2


r 
b
2
σz := 0
εr :=
1
2⋅ E
⋅
pc⋅ c
2

⋅  1 − 3⋅

b −c 
2
2
41
(102)
(103)
2


r 
b
2
(104)
εθ :=
1
2⋅ E
⋅
pc⋅ c
2

⋅  1 + 3⋅

b −c 
2
2
2


r 
b
(105)
2
2
−1 p c⋅ c
εz :=
⋅
2
E 2
b −c
u r :=
r
2⋅ E
pc⋅ c
⋅
2
2
b −c

2
(106)
⋅  1 + 3⋅


2

2 
r 
b
(107)
Once the values of c and pc are known, the solution in the elastic range can be
determined.
The key to the solution is to use the Von-Mises yield criteria and the Hencky
deformation theory, along with the substitutions:
−2
σr :=
σθ :=
3
2
3
⋅ σi⋅ cos ( φ)


⋅ σi⋅ sin  φ −
(108)
π

6
(109)
where
2
2
σi := σθ − σr⋅ σθ + σr
(110)
and φ is a auxiliary variable which is a function of r. This is a similar to Nadai’s method,
but the auxiliary variable φ has been slightly modified. For the perfectly-plastic case, σi
is equal to the yield strength is tension (Y).
For the perfectly-plastic case, the final plastic stresses are:
−2
σr :=
σθp :=
σz := 0
⋅ Y⋅ cos ( φ)
(111)
3
2


π
6

(112)
(plane-stress)
(113)
3
⋅ Y⋅ sin  φ −
where φ is found by the relationship:
42


r := a⋅

sin  φ +

sin  φa +
π
3

6
π
2
⋅e
(
)
⋅ φ−φa
(114)

6
and
 3⋅ p 
φa := acos 

 2⋅ Y 
(115)
The location of the elastic-plastic boundary can be found iteratively by using equation
(114):


c := a⋅

sin  φc +

sin  φa +
π
3

6
π
2
⋅e
(
)
⋅ φc− φa
(116)

6
where
 3⋅ b2 + c2 

2
2 
 3⋅ b − c 
φc := atan 

(
)
(117)
Knowing c, pc can be found using:
p c :=
(2
Y⋅ b − c
)
2
4
3⋅ b + c
(118)
4
and then the elastic stresses, strains, and displacements can be determined.
The plastic strains are then given by:
εθ :=
2
Y
2⋅ E
⋅
3⋅ b + c
4
3⋅ b + c
π
(
)
sin ( φ)

3
εz := − ε r + εθ
43
(119)
4


and the plastic radial displacements:
)
3⋅ φc− φ
⋅e
−sin  φ +
εr :=
(
2
⋅ εθ
(120)
(121)
u r :=
(
)
2
2
3⋅ φc− φ
Y⋅ r 3⋅ b + c
⋅
⋅e
2⋅ E
4
4
3⋅ b + c
(122)
See Appendix B for plots of these quantities for the test case with a pressure of
13026 applied to the internal surface. This pressure was selected to create the elasticplastic boundary half-way through the cylinder (c = 8.5).
3.1.3
The Fully-Plastic Cylinder
Timoshenko [4] discusses the stresses and pressures involved when a thick-walled
plane-strain cylinder becomes fully plastic. The pressure involved to make a cylinder
fully plastic is the same as the second part of equation (100) above. When this pressure
is reached, the stresses are (using Tresca yield):
σr := 2⋅ k⋅ ln
r


(123)
b


 b 
σθ := 2⋅ k⋅  1 + ln

r
(124)
At this high strain, the longitudinal stress is equal to the average of the other two
stresses.
σz := 2⋅ k⋅ 
1
2


 b 
+ ln
r
(125)
Also for plane-strain end conditions Nadai [10] calculates the pressures and stresses
in the fully-plastic cylinder. His equations are the same as those for Timoshenko with the
substitution of the Von-Mises yield criterion for Tresca (see equation (83)). Using
Nadai’s approach, a pressure of 14826.7 psi is required to turn the cylinder fully-plastic.
For the plane-stress case. Gao [11] extends his solution and comes up with:
p :=
2
3
( )
⋅ Y⋅ cos φa
(126)
for the pressure required for a fully-plastic state where φa is found by finding the root of:
3
2
⋅ 
b

a
2


:= sin  φa +
π
 ⋅e
6
 π −φ 
a
2 
3⋅ 
For this test case, the pressure using the Gao solution is approximately 13815 psi.
44
(127)
3.2 Finite-Element Model
For the partially-plastic cylinder, the ABAQUS models used are the same as in the
previous elastic cylinder under internal pressure analysis, except that a yield stress of 36
ksi is assigned and added to the material card. As no strain hardening is added, the
program treats the material as perfectly-plastic. A pressure of 13838.64 psi is then added
to the inside surface of the plane-strain model and a pressure of 13026 psi is added to the
inside surface of the plane-stress model. These pressures are calculated using the above
equations to produce elastic-plastic boundaries halfway through the cylinders (c = 8.5
in.). See Appendix A for a listing of the ABAQUS input files. See Appendix B for a full
set of plots showing the stresses, strains, and displacements, and the next section for
plots of the hoop stresses and radial displacements.
For the yield pressure and fully-plastic case, the same model is used with an applied
internal pressure of 15000 psi. However, a RIKs [5] arc-length solution method is
employed to capture the non-linear behavior. Appendix A contains a snippet of the
solution step for this analysis.
3.3 Comparison of Results
Partially-Plastic Cylinder
Figure 19 and Figure 20 show the exact and ABAQUS hoop stresses and radial
displacements for identically-loaded elastic-plastic plane-strain thick-walled cylinders
(see Appendix B for plots of the other quantities). In general, the exact solutions and
finite-element models showed excellent correlation for the stresses. The plastic portion
of the longitudinal stress in the plane-strain case was the one exception, but that was
most likely due to some of the approximations made in order to get a simpler solution.
The strains did not exhibit the same degree of correlation though. While they generally
followed the exact curves fairly closely, there was still significant error in some areas.
Whether this is a fault with some of the assumptions in the exact solutions or a limitation
to the elements chosen is unclear. The displacements however were fairly accurate, with
a maximum error of 12% near the inside radius in the plane-strain solution and a
maximum error of 6% in the plane-stress solution.
45
Elastic-Plastic Hoop Stress - Plane Strain
5.00E+04
4.50E+04
4.00E+04
Hoop Stress (psi)
3.50E+04
3.00E+04
Exact-Elastic
2.50E+04
Exact-Plastic
ABAQUS
2.00E+04
1.50E+04
1.00E+04
5.00E+03
0.00E+00
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 19 – Exact vs. ABAQUS Hoop Stresses (Plastic - Pressure Load)
Elastic-Plastic Radial Displacement - Plane Strain
0.014
0.012
Radial Displacement (in)
0.01
0.008
Exact
ABAQUS
0.006
0.004
0.002
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 20 – Exact vs. ABAQUS Radial Displ. (Plastic - Pressure Load)
Fully-Plastic Cylinder
Figure 21 shows the pressure histories for a fully-plastic plane-strain and planestress cylinder. Steadily-increasing pressures were applied to the internal surfaces until
the solutions became unstable. The curved lines represent the applied pressure over
“time” (i.e. the solution increments). As the cylinders became more plastic, the pressures
required to increase the plasticity decreased (for perfectly-plastic materials). The straight
46
line is a continuation of the linear parts of the curves added for comparison. Inspection
of the plot reveals that for the plane-strain cylinder, it begins to yield at about 10700 psi
(the two curves begin to separate) and is fully-plastic at about 14720 psi (the curve levels
out). The difference between the calculated yield pressure and the ABAQUS pressure is
1.6%, and -0.7% for the fully-plastic pressure. For the plane-stress cylinder, it begins to
yield at about 10400 psi and is fully-plastic at about 13770 psi. The difference between
the calculated yield pressure and the ABAQUS pressure is 1.96%, and -0.33% for the
fully-plastic pressure.
Yield Pressure History
20000
18000
16000
Pressure (psi)
14000
12000
Plane-Strain
Plane-Stress
10000
Linear Response
8000
6000
4000
2000
0
0
0.5
1
1.5
2
Solution Time Increment
Figure 21 – ABAQUS Yield Pressure vs. Time Increment – (Fully Plastic)
47
4. 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.
[7] Mase, G.E, 1970, Schaum’s Outlines – Continuum Mechanics, McGraw-Hill, New
York, NY.
[8] Prager, W. & Hodge, P.G, 1951, Theory of Perfectly Plastic Solids, John Wiley &
Sons, New York, NY.
[9] Hill, R., 1950, The Mathematical Theory of Plasticity, Oxford University Press, New
York, NY.
[10] Nadai, A., 1931, Plasticity, McGraw-Hill, New York, NY.
[11] Gao, X., 1992, “An Exact Elasto-Plastic Solution for an Open-Ended Thick-Walled
Cylinder of a Strain-Hardening Material,” International Journal of Pressure
Vessels and Piping 52 (1992) 129-144.
48
APPENDIX A – SAMPLE ABAQUS FILES
1) Sample ABAQUS input file (.inp) for the linear elastic cylinder under pressure.
*heading
10-Inch OD, 3.0-Inch Wall Thickness, Plane-Strain, 10199 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 = 3.000
pressure = 10199
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 = 'PS'
node_circum = 9
node_radial = 35
##
## 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
**
49
*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 thickness assumed
**
*solid section, elset=cylinder, material=steel
1.0,
**
** define material
50
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 = 35
##
## dependent parameters (do not modify)
51
##
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
52
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 thickness 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
#
53
# 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 = 35
##
## 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
54
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>
55
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
4) Sample ABAQUS input file (.inp) for the elastic-plastic analysis of a perfectly-plastic
cylinder under internal pressure.
*heading
10" OD, 3" WT, Plane-Strain, 13838.64 (c=8.5) psi int. 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 = 3.000
pressure = 13838.64315
press_type = 'int'
#
# elastic/plastic material properties
#
young = 30e+06
poisson = 0.3
sigma_y = 36000
#
# 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 = 35
##
## dependent parameters (do not modify)
##
56
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
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
57
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 thickness assumed
**
*solid section, elset=cylinder, material=steel
1.0,
**
** define material
**
*material,name=steel
*elastic
<young>,<poisson>
*plastic
<sigma_y>,
**
** define boundary conditions
**
*boundary
ends,2,2
**
** define pressure load step
**
*step, name=Pressure_Load, nlgeom, inc=50
*static
*dsload
<load_surf>, P, <pressure>
**
** Output variable requests
**
*output, field
*node output
CF, RF, U
*element output, directions=yes
EE, IE, LE, P, PE, PEEQ, PEMAG, PS, S
*output,field, variable=preselect
*output, history, variable=preselect
*end step
5) Sample ABAQUS input file (.inp) snippet of the solution step for the fully-plastic
model of a perfectly-plastic cylinder under internal pressure.
*step,nlgeom, amplitude=ramp
Fully-plastic analysis using Riks
*static,riks
58
0.01,1.0,,0.02,,1093,1,
*monitor,node=1093,dof=1
*dsload
<load_surf>, P, <pressure>
*node file,freq=20,nset=inside
u,rf
*output,field,freq=20
*node output
u,rf
*output,history,freq=1
*node output,nset=inside
u,rf
*output,field, variable=preselect
*output, history, variable=preselect
*end step
59
APPENDIX B – ADDITIONAL PLOTS
Thick-Walled Cylinder Under Internal Pressure (Exact Solution)
The following plots show the exact solution for the typical thick-walled cylinder test
case shown in Table 3, with 10199 psi internal pressure and plane-stress conditions.
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
10
Radius (in)
Figure A2 – Exact Hoop Stress (Pressure Load)
60
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
10
Radius (in)
Figure A4 – Exact Hoop Strain (Pressure Load)
61
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)
The following plots show the ABAQUS solution for the typical thick-walled
cylinder discussed above.
62
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)
63
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)
64
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 solutions and
finite-element models for a series of thick-to-thin-walled cylinders as the radius-to-wallthickness (r/t) ratio is increased.
65
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
66
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
67
Longitudinal Strain - Plane-Stress
1.2
1.18
1.16
Normalized Strain
1.14
1.12
1.1
r/t = 4.0
r/t = 5.7
1.08
r/t = 9.0
1.06
r/t = 12.3
r/t = 19.0
1.04
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 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
68
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
69
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
70
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 solutions and
finite-element models for a series of thick-walled cylinders as the radius-to-wallthickness (r/t) ratio is decreased.
71
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
72
Hoop Stress - Plane-Stress
1.01
Normalized Stress
1
0.99
r/t = 4.0
0.98
r/t = 2.3
0.97
r/t = 1.5
0.96
0.95
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
73
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
74
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
75
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
76
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 (Comparison)
The following plots show the exact solution vs. the ABAQUS model for the typical
thick-walled cylinder test case shown in Table 3, with a 200°F temperature at the inner
surface, and a 0°F temperature at the outer surface with plane-stress conditions.
77
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 A37 – 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 A38 – Exact vs. ABAQUS Radial Stress – Plane-Strain (Thermal Load)
78
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 A39 – 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 A40 – Exact vs. ABAQUS Hoop Stress – Plane-Strain (Thermal Load)
79
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 A41 – Exact vs. ABAQUS Long. 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 A42 – Exact vs. ABAQUS Long. Stress – Plane-Strain (Thermal Load)
80
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 A43 – 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 A44 – Exact vs. ABAQUS Radial Strain – Plane-Strain (Thermal Load)
81
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 A45 – 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 A46 – Exact vs. ABAQUS Hoop Strain – Plane-Strain (Thermal Load)
82
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 A47 – 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 A48 – 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 model for the typical
thick-walled cylinder test case shown in Table 3, with a 200°F temperature at the inner
surface, and a 0°F temperature at the outer surface with either a 10199 psi internal
pressure (plane-stress) or a 10600 psi internal pressure (plane strain).
83
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 A49 – 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 A50 – Exact vs. ABAQUS Radial Stress – Plane-Strain (Combined Load)
84
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 A51 – 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 A52 – Exact vs. ABAQUS Hoop Stress – Plane-Strain (Combined Load)
85
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 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 A54 – Exact vs. ABAQUS Long. Stress – Plane-Strain (Combined Load)
86
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 A55 – 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 A56 – Exact vs. ABAQUS Radial Strain – Plane-Strain (Combined Load)
87
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 A57 – 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 A58 – Exact vs. ABAQUS Hoop Strain – Plane-Strain (Combined Load)
88
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 A59 – 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 A60 – Exact vs. ABAQUS Radial Displ. – Plane-Strain (Combined Load)
Elastic-Plastic Thick-Walled Cylinder Under Pressure Load (Comparison)
The following plots show the exact solution vs. the ABAQUS model for the typical
thick-walled cylinder test case shown in Table 3, with either a 13868 psi internal
pressure (plane-strain) or a 13026 psi internal pressure (plane-stress).
89
Elastic-Plastic Mises Stress - Plane Strain
6.00E+04
5.00E+04
Mises Stress (psi)
4.00E+04
Exact
3.00E+04
ABAQUS
2.00E+04
1.00E+04
0.00E+00
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 A61 – Exact vs. ABAQUS Mises Stress Plane-Strain (Plastic – Pressure)
Elastic-Plastic Mises Stress - Plane Stress
6.00E+04
5.00E+04
Mises Stress (psi)
4.00E+04
Exact
3.00E+04
ABAQUS
2.00E+04
1.00E+04
0.00E+00
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 A62 – Exact vs. ABAQUS Mises Stress Plane-Stress (Plastic – Pressure)
90
Elastic-Plastic Radial Stress - Plane Strain
2.00E+03
0.00E+00
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-2.00E+03
Radial Stress (psi)
-4.00E+03
-6.00E+03
Exact-Elastic
-8.00E+03
Exact-Plastic
ABAQUS
-1.00E+04
-1.20E+04
-1.40E+04
-1.60E+04
-1.80E+04
Radius (in)
Figure A63 – Exact vs. ABAQUS Radial Stress Plane-Strain (Plastic – Pressure)
Elastic-Plastic Radial Stress - Plane Stress
2.00E+03
0.00E+00
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-2.00E+03
Radial Stress (psi)
-4.00E+03
-6.00E+03
Exact-Elastic
Exact-Plastic
ABAQUS
-8.00E+03
-1.00E+04
-1.20E+04
-1.40E+04
-1.60E+04
Radius (in)
Figure A64 – Exact vs. ABAQUS Radial Stress Plane-Stress (Plastic – Pressure)
91
Elastic-Plastic Hoop Stress - Plane Strain
5.00E+04
4.50E+04
4.00E+04
Hoop Stress (psi)
3.50E+04
3.00E+04
Exact-Elastic
2.50E+04
Exact-Plastic
ABAQUS
2.00E+04
1.50E+04
1.00E+04
5.00E+03
0.00E+00
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 A65 – Exact vs. ABAQUS Hoop Stress Plane-Strain (Plastic – Pressure)
Elastic-Plastic Hoop Stress - Plane Stress
4.50E+04
4.00E+04
3.50E+04
Hoop Stress (psi)
3.00E+04
2.50E+04
Exact-Elastic
2.00E+04
ABAQUS
Exact-Plastic
1.50E+04
1.00E+04
5.00E+03
0.00E+00
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 A66 – Exact vs. ABAQUS Hoop Stress Plane-Stress (Plastic – Pressure)
92
Elastic-Plastic Longitudinal Stress - Plane Strain
1.40E+04
1.20E+04
Longitudinal Stress (psi)
1.00E+04
8.00E+03
Exact-Elastic
Exact-Plastic
ABAQUS
6.00E+03
4.00E+03
2.00E+03
0.00E+00
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 A67 – Exact vs. ABAQUS Long. Stress Plane-Strain (Plastic – Pressure)
Elastic-Plastic Longitudinal Strain - Plane Stress
0.00E+00
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-1.00E-04
Longitudinal Strain
-2.00E-04
-3.00E-04
Exact-Elastic
Exact-Plastic
ABAQUS
-4.00E-04
-5.00E-04
-6.00E-04
-7.00E-04
Radius (in)
Figure A68 – Exact vs. ABAQUS Long. Strain Plane-Stress (Plastic – Pressure)
93
Elastic-Plastic Radial Strain - Plane Strain
0.00E+00
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-2.00E-04
-4.00E-04
-6.00E-04
Radial Strain
-8.00E-04
Exact
-1.00E-03
ABAQUS
-1.20E-03
-1.40E-03
-1.60E-03
-1.80E-03
-2.00E-03
Radius (in)
Figure A69 – Exact vs. ABAQUS Radial Strain Plane-Strain (Plastic – Pressure)
Elastic-Plastic Radial Strain - Plane Stress
0.00E+00
7
7.25
7.5
7.75
8
8.25
8.5
8.75
9
9.25
9.5
9.75
10
-2.00E-04
Radial Strain
-4.00E-04
-6.00E-04
Exact-Elastic
ABAQUS
Exact-Plastic
-8.00E-04
-1.00E-03
-1.20E-03
-1.40E-03
Radius (in)
Figure A70 – Exact vs. ABAQUS Radial Strain Plane-Stress (Plastic – Pressure)
94
Elastic-Plastic Hoop Strain - Plane Strain
2.00E-03
1.80E-03
1.60E-03
1.40E-03
Hoop Strain
1.20E-03
Exact
1.00E-03
ABAQUS
8.00E-04
6.00E-04
4.00E-04
2.00E-04
0.00E+00
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 Hoop Strain Plane-Strain (Plastic – Pressure)
Elastic-Plastic Hoop Strain - Plane Stress
1.80E-03
1.60E-03
1.40E-03
Hoop Strain
1.20E-03
1.00E-03
Exact-Elastic
ABAQUS
Exact-Plastic
8.00E-04
6.00E-04
4.00E-04
2.00E-04
0.00E+00
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 A72 – Exact vs. ABAQUS Hoop Strain Plane-Stress (Plastic – Pressure)
95
Elastic-Plastic Radial Displacement - Plane Strain
0.014
0.012
Radial Displacement (in)
0.01
0.008
Exact
ABAQUS
0.006
0.004
0.002
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 A73 – Exact vs. ABAQUS Radial Displ. Plane-Strain (Plastic – Pressure)
Elastic-Plastic Radial Displacement - Plane Strain
0.014
0.012
Radial Displacement (in)
0.01
0.008
Exact-Elastic
ABAQUS
Exact-Plastic
0.006
0.004
0.002
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 A74 – Exact vs. ABAQUS Radial Displ. Plane-Stress (Plastic – Pressure)
96
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