Elastic-Plastic Behavior of an 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

2

© Copyright 2008

by

Peter P. Poworoznek

All Rights Reserved

3

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

4

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

5

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

6

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

7

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.

8

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 finite-

element 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, thin-

walled cylinders are addressed for the linear elastic/pressure case as is the boundary

between what constitutes thin and thick walls. Then the effects of a 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.

9

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 plane-

strain 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.

10

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−:=

The hoop stress can be readily found by examining the free body diagram of a half-

cylinder and is given by the equation [1]:

σθp r⋅t

:= (2)

(1)

11

Figure 1 – Typical Cylinder

From Hooke’s law, the strains are calculated using:

εr1

Eσr ν σθ σz+( )⋅− ⋅:=

εθ1

Eσθ ν σr σz+( )⋅− ⋅:=

εz1

Eσz ν σr σθ+( )⋅− ⋅:=

In this case, the longitudinal (σz) stress is zero, therefore:

εrp−

E1

ν r⋅t

+

⋅:=

εθp

E

r

tν+

⋅:=

εzν p⋅E

1r

t−

⋅:=

(3)

(4)

(5)

(6)

(7)

(8)

12

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:

ur r εθ⋅:=

urp r⋅E

r

tν+

⋅:=

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:

εz1

Eσz ν σr σθ+( )⋅− ⋅ 0:=

σz ν σr σθ+( )⋅:=

The radial stress is constant (-p), therefore:

σz ν− p⋅ 1r

t−

⋅:=

The hoop and radial strains, using the same equations as in plane stress are:

εrp−

E1 ν

2−

ν r⋅t

1 ν+( )⋅+

⋅:=

εθp

Eν 1 ν+( )⋅

r

t1 ν

2−( )⋅+

⋅:=

And the change in radius is:

urp r⋅E

ν 1 ν+( )⋅r

t1 ν

2−( )⋅+

⋅:=

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.

(9)

(11)

(12)

(13)

(14)

(15)

(16)

(10)

13

Wall Plane Stress Thick. r/t psi σr σθ σz εr εθ εz ur 2.000 4.0 7019 -7019 28706 0 -0.00051 0.00100 -0.00021 0.00804 1.500 5.7 5323 -5323 30164 0 -0.00048 0.00106 -0.00025 0.009 1.000 9.0 3577 -3577 32193 0 -0.00044 0.00111 -0.00029 0.00998 0.750 12.3 2690 -2690 33177 0 -0.00042 0.00113 -0.0003 0.01048 0.500 19.0 1797 -1797 34143 0 -0.0004 0.00116 -0.00032 0.01098 0.250 39.0 900 -900 35100 0 -0.00038 0.00118 -0.00034 0.0115 0.125 79.0 450 -450 35550 0 -0.00037 0.00119 -0.00035 0.01175

Table 1 – Thin-Walled Cylinder Plane Stress Results

Wall Plane Strain Thick. r/t psi σr σθ σz εr εθ εz ur 2.000 4.0 7482 -7482 29928 6734 -0.00062 0.00101 0 0.00804 1.500 5.7 5768 -5768 32685 8075 -0.0006 0.00107 0 0.00906 1.000 9.0 3949 -3949 35541 9478 -0.00058 0.00113 0 0.01016 0.750 12.3 3001 -3001 37012 10203 -0.00057 0.00116 0 0.01075 0.500 19.0 2026 -2026 38494 10940 -0.00056 0.00119 0 0.01134 0.250 39.0 1026 -1026 40014 11696 -0.00055 0.00123 0 0.01196 0.125 79.0 516 -516 40764 12074 -0.00055 0.00124 0 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:

σr

a2pi b

2po⋅−

b2

a2−

pi po−( ) a2⋅ b

2⋅

r2

b2

a2−( )⋅

−:=

σθa2pi b

2po⋅−

b2

a2−

pi po−( ) a2⋅ b

2⋅

r2

b2

a2−( )⋅

+:=

(17)

(18)

14

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:

σrp a

2⋅

b2

a2−

1b

2

r2

−

⋅:=

σθp a

2⋅

b2

a2−

1b

2

r2

+

⋅:=

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

a2

b2+

2 a2⋅

:=σθ_max

σθ_min

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:

εrp a

2⋅

E b2

a2−( )⋅

1 ν−b

2

r2

1 ν+( )⋅−

⋅:=

εθp a

2⋅

E b2

a2−( )⋅

1 ν−b

2

r2

1 ν+( )⋅+

⋅:=

εz2− ν⋅ p⋅ a

2⋅

E b2

a2−( )⋅

:=

And the change in radius (rεθ) is:

urp a

2⋅

E b2

a2−( )⋅

1 ν−( ) b2

r2

1 ν+( )⋅+

⋅ r⋅:=

(19)

(20)

(21)

(22)

(23)

(24)

(25)

15

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:

σz2 ν⋅ p⋅ a

2⋅

b2

a2−

:=

εrp a

2⋅

E b2

a2−( )⋅

1 ν−b

2

r2

1 ν+( )⋅− 2 ν2

⋅−

⋅:=

εθp a

2⋅

E b2

a2−( )⋅

1 ν−b

2

r2

1 ν+( )⋅+ 2 ν2

⋅−

⋅:=

ur1 ν+

E

p a2⋅

b2

a2−

⋅ 1 2 ν⋅−b

2

r2

+

⋅ r⋅:=

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.

(26)

(27)

(28)

(29)

16

Hoop Stress - Plane-Stress

15000.00

17000.00

19000.00

21000.00

23000.00

25000.00

27000.00

29000.00

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

Exact

Figure 2 – Exact Hoop Stress (Pressure Load)

Radial Displacement - Plane-Stress

0.00640000

0.00660000

0.00680000

0.00700000

0.00720000

0.00740000

0.00760000

0.00780000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

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

17

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+( )⋅:= S11

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.

(30)

18

σθ_a ur_b Nodes

Radially Exact FEA % Err Exact FEA % Err

3 2573.59 -11.91 0.000640518 -7.81E-04

5 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 2876.00 -1.56 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

2921.569

2897.92 -0.81

0.000640523

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 (thirty-

five 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.

19

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

15000

17000

19000

21000

23000

25000

27000

29000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

ABAQUS

Figure 5 – ABAQUS Hoop Stress (Pressure Load)

20

Radial Displacement - Plane-Stress

0.0064

0.0066

0.0068

0.007

0.0072

0.0074

0.0076

0.0078

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

ABAQUS

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.

21

Hoop Stress - Plane-Stress

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure 7 – ABAQUS vs. Exact Hoop Stresses as a Function of r/t Ratios

Radial Displacement - Plane-Stress

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Dis

plac

emen

t

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

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

22

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

0.95

0.96

0.97

0.98

0.99

1

1.01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure 9 – ABAQUS vs. Exact Hoop Stresses as a Function of r/t Ratios

23

Radial Displacement - Plane-Stress

0.99

0.995

1

1.005

1.01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Dis

plac

emen

t

r/t = 4.0

r/t = 2.3

r/t = 1.5

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.

24

For the plane stress case, the radial stress is given by:

σr α E⋅1−

r2 a

r

rT r⋅⌠⌡

d⋅r2

a2−

r2

b2

a2−( )⋅ a

b

rT r⋅⌠⌡

d⋅+

⋅:= α

and the hoop stress can be found by the relationship:

σθ σr rrσr

d

d

⋅+:= σr

which in turn gives:

σθ α E⋅1

r2 a

r

rT r⋅⌠⌡

d⋅r2

a2−

r2

b2

a2−( )⋅ a

b

rT r⋅⌠⌡

d⋅+ T−

⋅:= α

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:

TTi

lnb

a

lnb

r

⋅:=Ti

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

E α⋅ Ti⋅

2 lnb

a

⋅ln

b

r

−a2

b2

a2−

1b

2

r2

−

⋅ lnb

a

⋅−

⋅:=α

σθE α⋅ Ti⋅

2 lnb

a

⋅1 ln

b

r

−a2

b2

a2−

1b

2

r2

+

⋅ lnb

a

⋅−

⋅:=α

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

νE

σθ σz+( )⋅− α T⋅+:=σr

(31)

(32)

(33)

(34)

(35)

(36)

(37)

25

εθσθE

νE

σr σz+( )⋅− α T⋅+:=σθ

εz

σz

E

νE

σr σθ+( )⋅− α T⋅+:= σr

The resulting strains are:

εr

α Ti⋅

2 lnb

a

⋅ν− 1 ν+( ) ln

b

r

⋅+a2

b2

a2−

1 ν−( ) 1 ν+( ) b2

r2

⋅−

⋅ lnb

a

⋅−

⋅:=α

εθα Ti⋅

2 lnb

a

⋅1 1 ν+( ) ln

b

r

⋅+a2

b2

a2−

1 ν−( ) 1 ν+( ) b2

r2

⋅+

⋅ lnb

a

⋅−

⋅:=α

εz

α Ti⋅

2 lnb

a

⋅ν− 2 1 ν+( )⋅ ln

b

r

⋅+2 ν⋅ a

2⋅

b2

a2−

lnb

a

⋅+

⋅:=α

Of interest is that unlike the pressure-only case where the longitudinal strain is

constant, under a thermal load the longitudinal strain is a function of the radius. The

radial displacement is calculated by:

ur r εθ⋅:= εθ

ur

α Ti⋅

2 lnb

a

⋅1 1 ν+( ) ln

b

r

⋅+a2

b2

a2−

1 ν−( ) 1 ν+( ) b2

r2

⋅+

⋅ lnb

a

⋅−

⋅ r⋅:=α

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):

σr

E α⋅ Ti⋅

2 1 ν−( )⋅ lnb

a

⋅ln

b

r

−a2

b2

a2−

1b

2

r2

−

⋅ lnb

a

⋅−

⋅:=α

σθE α⋅ Ti⋅

2 lnb

a

⋅1 ln

b

r

−a2

b2

a2−

1b

2

r2

+

⋅ lnb

a

⋅−

⋅:=α

(40)

(41)

(42)

(43)

(44)

(38)

(39)

(45)

(46)

26

The longitudinal stress is found using the equation:

σz ν σr σθ+( )⋅ α E⋅ T⋅−:= σr

which results in:

σz

E α⋅ Ti⋅

2 1 ν−( )⋅ lnb

a

⋅ν 2 ln

b

r

⋅−2 ν⋅ a

2⋅

b2

a2−

lnb

a

⋅−

⋅:=α

The radial and hoop strains become:

εr

α Ti⋅

2 1 ν−( )⋅ lnb

a

⋅ν− ν

2− 1 ν+( ) ln

b

r

⋅+a2

b2

a2−

1 ν−( ) 1 ν+( ) b2

r2

⋅− 2 ν2

⋅−

⋅ lnb

a

⋅−

⋅:=α

εθα Ti⋅

2 1 ν−( )⋅ lnb

a

⋅1 ν

2− 1 ν+( ) ln

b

r

⋅+a2

b2

a2−

1 ν−( ) 1 ν+( ) b2

r2

⋅+ 2 ν2

⋅−

⋅ lnb

a

⋅−

⋅:=α

And the radial displacement:

ur

α Ti⋅

2 1 ν−( )⋅ lnb

a

⋅1 ν

2− 1 ν+( ) ln

b

r

⋅+a2

b2

a2−

1 ν−( ) 1 ν+( ) b2

r2

⋅+ 2 ν2

⋅−

⋅ lnb

a

⋅−

⋅ r⋅:=α

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.

(48)

(49)

(50)

(51)

(47)

27

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 6.944E-

04 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

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

160.00

180.00

200.00

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)

Tem

pera

ture

(°F

)

Exact

ABAQUS

Figure 11 – Exact vs. ABAQUS Temperature Distribution (Thermal Load)

28

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

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

10000

15000

20000

25000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

Exact

ABAQUS

Figure 12 – Exact vs. ABAQUS Hoop Stress (Thermal Load)

Radial Displacement - Plane-Stress

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

ABAQUS

Figure 13 – Exact vs. ABAQUS Radial Displacement (Thermal Load)

29

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_pressure

σrp a

2⋅

b2

a2−

1b

2

r2

−

⋅E α⋅ Ti⋅

2 lnb

a

⋅ln

b

r

−a2

b2

a2−

1b

2

r2

−

⋅ lnb

a

⋅−

⋅+:=α

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.

(52)

(53)

30

Hoop Stress - Plane-Stress

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

Exact

ABAQUS

Figure 14 – Exact vs. ABAQUS Hoop Stress (Combined Load)

Radial Displacement - Plane-Stress

0.012

0.0122

0.0124

0.0126

0.0128

0.013

0.0132

0.0134

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

ABAQUS

Figure 15 – Exact vs. ABAQUS Radial Displacement (Combined Load)

31

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 Curvefor ASTM-A36 Steel

0

5000

10000

15000

20000

25000

30000

35000

40000

0 0.002 0.004 0.006 0.008 0.01 0.012

ASTM-A36 Steel

Figure 16 – Stress-Strain Curve for an Elastic-Perfectly Plastic Material

32

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:

kY

2:=

Y

or

Y 2 k⋅:= 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−

2k:=

σ1 σ3−

2 or

(55)

(56)

(54)

(57)

33

σ1 σ3− Y:=σ1 σ3−

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−( )2 σ2 σ3−( )2+ σ1 σ3−( )2+ 2 Y

2⋅:=σ1 σ2−( )2 σ2 σ3−( )2+ σ1 σ3−( )2+

or

σ1 σ2−( )2 σ2 σ3−( )2+ σ1 σ3−( )2+ 6 k

2⋅:=σ1 σ2−( )2 σ2 σ3−( )2+ σ1 σ3−( )2+

With the subsequent relationship that with Von-Mises yielding:

kY

3:=

or

Y 3 k⋅:=

From this it can be seen that the yield shear strength in the Von-Mises criterion is:

kVM2

3kT:= kT

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:

σθ σr−Y

2:=σθ σr−

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.

(58)

(60)

(62)

(63)

(59)

(61)

34

σθ σr−

2 p⋅ b2⋅

r2

b2

a2

1−

:=σθ σr−

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:

pY

21

a2

b2

−

⋅:=

For the test case shown in Table 3, this gives a yield pressure of 9180 psi.

For the Von-Mises criterion:

σr σθ−( )2 σθ σz−( )2+ σz σr−( )2+ 2 Y2⋅:=σr σθ−( )2 σθ σz−( )2+ σz σr−( )2+

or

σr σθ−( )2 σθ σz−( )2+ σz σr−( )2

+6 b

4⋅ p2⋅

b2

a2

1−

2

r4⋅

2p

b2

a2

1−

σz−

2

⋅+:=σr σθ−( )2 σθ σz−( )2+ σz σr−( )2

+

Once again, this is the greatest when r = a. Using this and setting equation (67)

equal to equation (66):

3 b4⋅ p

2⋅

b2

a2

1−

2

r4⋅

p

b2

a2

1−

σz−

2

+ Y2:=

3 b4⋅ p

2⋅

b2

a2

1−

2

r4⋅

p

b2

a2

1−

σz−

2

+

For the plane-stress condition, σz is zero, and for plane-strain it is given by equation

(26). Solving for p gives:

py

Y

31

a2

b2

−

⋅

1a4

3 b4⋅

+

:=

(plane-stress)

(64)

(65)

(66)

(67)

(68)

(69)

35

py

Y

31

a2

b2

−

⋅

1 1 2ν⋅−( )2 a4

3 b4⋅

⋅+

:=

(plane-strain)

For the test case, the yield pressure for plane-stress is 10199.84 psi and for plane-

strain 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

0

5000

10000

15000

20000

25000

0 1 2 3 4 5 6 7 8 9 10

b/a (Outside Radius/Inside Radius)

Yie

ld P

ress

ure

(psi

)

Tresca

Von-Mises (Plane-Stress)

Von-Mises (Plane-Strain)

Figure 17 – Yield Pressure vs. Ratio of b/a

(70)

36

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:

σr C−b

2

r2

1−

⋅:=r

σθ C−b

2

r2

1+

⋅:=r

(71)

(72)

(73)

37

σz E εz⋅ 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 work-

hardening, 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:

σθ σr− Y:=σθ σr−

holds everywhere in the plastic region and with the longitudinal strain being zero for

plane-strain, the stresses in the elastic region are:

σrY−2

c2

b2

⋅b

2

r2

1−

⋅:=r

σθY

2

c2

b2

⋅b

2

r2

1+

⋅:=r b

σz ν Y⋅c2

b2

⋅:=

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.

εrY c

2⋅

2 E⋅ b2⋅

1 ν−b

2

r2

1 ν+( )⋅− 2 ν2

⋅−

⋅:=r

εθY c

2⋅

2 E⋅ b2⋅

1 ν−b

2

r2

1 ν+( )⋅+ 2 ν2

⋅−

⋅:=r

εz 0:= (plane-strain)

And the radial displacement:

ur r εθ⋅:= εθ

(74)

(75)

(76)

(77)

(78)

(79)

(80)

(81)

38

urY c

2⋅ r⋅

2 E⋅ b2⋅

1 ν−b

2

r2

1 ν+( )⋅+ 2 ν2

⋅−

⋅:=r

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:

Y2

3Y:=

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:

rσr

d

d

σθ σr−

r

Y

r:=

rσr

d

d

This gives the solutions for the radial and hoop stresses:

σrY−2

1 2 lnc

r

⋅+c2

b2

−

⋅:=r

σθY

21 2 ln

c

r

⋅−c2

b2

+

⋅:=r

And by applying continuity of the radial stress at r = c, the internal pressure required to

produce this yield is:

pY

21 2 ln

c

a

⋅+c2

b2

−

⋅:=c

a

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

(82)

(83)

(84)

(85)

(86)

(87)

39

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:

σzY c

2⋅

2 b2⋅

:=

urk c

2⋅2 G⋅ r⋅

:=r

where k is the yield strength in shear and G is the shear modulus.

GE

2 1 ν+( )⋅:=

Putting equation (89) in another form:

urY c

2⋅ 1 ν+( )⋅2 E⋅ r⋅

:=r

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:

σzY

2

c2

b2

2 lnc

r

⋅−

⋅:=r

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)

(89)

(90)

(91)

(92)

(88)

40

for the elastic longitudinal stress, equation (93) for the plastic longitudinal stress, and

equation (94) for the radial displacement (both elastic and plastic):

σz ν Y⋅c2

b2

2 lnc

r

⋅−

⋅:=r

2 E⋅ r⋅

urY c

2⋅ 1 ν2

−( )⋅E r⋅

:=r

Once again, replacing the yield stress (Y) with equation (83) gives a quasi-Von-

Mises 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 rur

d

d:=

r

εθu

r:=

r

which gives:

εrY− c

2⋅ 1 ν2

−( )⋅

E r2⋅

:=r

εθY c

2⋅ 1 ν2

−( )⋅

E r2⋅

:=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:

(93)

(95)

(97)

(94)

(96)

(98)

41

pa k 1a2

b2

−

⋅:=

pb 2 k⋅ lnb

a

⋅:=

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

pc c2⋅

b2

c2−

1b

2

r2

−

⋅:=pc

σθpc c

2⋅

b2

c2−

1b

2

r2

+

⋅:=pc

σz 0:=

εr1

2 E⋅

pc c2⋅

b2

c2−

⋅ 1 3b

2

r2

⋅−

⋅:=pc

(100)

(101)

(104)

(99)

(103)

(102)

42

εθ1

2 E⋅

pc c2⋅

b2

c2−

⋅ 1 3b

2

r2

⋅+

⋅:=pc

εz1−

E

pc c2⋅

b2

c2−

⋅:=pc

urr

2 E⋅

pc c2⋅

b2

c2−

⋅ 1 3b

2

r2

⋅+

⋅:=r

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:

σr2−

3σi⋅ cos φ( )⋅:= φ

σθ2

3σi⋅ sin φ

π6

−

⋅:= φ

where

σi σθ2

σr σθ⋅− σr2

+:= σθ

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:

σr2−

3Y⋅ cos φ( )⋅:= φ

σθp2

3Y⋅ sin φ

π6

−

⋅:= φ

σz 0:=(plane-stress)

where φ is found by the relationship:

(107)

(108)

(110)

(111)

(112)

(113)

(105)

(106)

(109)

43

r a

sin φaπ6

+

sin φπ6

+

⋅ e

3

2φ φa−( )⋅

⋅:=φ

and

φa acos3 p⋅

2 Y⋅

:=

The location of the elastic-plastic boundary can be found iteratively by using equation

(114):

c a

sin φaπ6

+

sin φcπ6

+

⋅ e

3

2φc φa−( )⋅

⋅:=φc

where

φc atan3 b

2⋅ c2+

3 b2

c2−( )⋅

:=c

Knowing c, pc can be found using:

pcY b

2c2−( )⋅

3 b4⋅ c

4+:=

c4

and then the elastic stresses, strains, and displacements can be determined.

The plastic strains are then given by:

εθY

2 E⋅3 b

2⋅ c2+

3 b4⋅ c

4+⋅ e

3 φc φ−( )⋅⋅:=

φ

εr

sin φπ3

+

−

sin φ( ) εθ⋅:=φ

εz εr εθ+( )−:= εr

and the plastic radial displacements:

(114)

(115)

(116)

(117)

(118)

(119)

(121)

(120)

44

urY r⋅2 E⋅

3 b2⋅ c

2+

3 b4⋅ c

4+⋅ e

3 φc φ−( )⋅⋅:=

r

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 elastic-

plastic 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⋅ lnr

b

⋅:=r

σθ 2 k⋅ 1 lnr

b

+

⋅:=r

At this high strain, the longitudinal stress is equal to the average of the other two

stresses.

σz 2 k⋅1

2ln

r

b

+

⋅:=r

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:

p2

3Y⋅ cos φa( )⋅:=

for the pressure required for a fully-plastic state where φa is found by finding the root of:

3

2

b

a

2

⋅ sin φaπ6

+

e

3π

2φa−

⋅⋅:=

3

2

b

a

2

⋅

For this test case, the pressure using the Gao solution is approximately 13815 psi.

(126)

(127)

(123)

(124)

(125)

(122)

45

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.

46

Elastic-Plastic Hoop Stress - Plane Strain

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

3.00E+04

3.50E+04

4.00E+04

4.50E+04

5.00E+04

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tress

(ps

i)

Exact-Elastic

Exact-Plastic

ABAQUS

Figure 19 – Exact vs. ABAQUS Hoop Stresses (Plastic - Pressure Load)

Elastic-Plastic Radial Displacement - Plane Strain

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

ABAQUS

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 plane-

stress 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

47

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

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0 0.5 1 1.5 2

Solution Time Increment

Pre

ssur

e (p

si)

Plane-Strain

Plane-Stress

Linear Response

Figure 21 – ABAQUS Yield Pressure vs. Time Increment – (Fully Plastic)

48

4. BIBLIOGRAPHY

[1] Young, W.C., 1989, Roark’s Formulas for Stress & Strain, 6th Edition, McGraw-Hill, 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.

49

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-st ress # node_circum = nodes around 1/16 circumference # node_radial = nodes through the thickness (minim um 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 **

50

*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, transfo rmation, and 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 t ype ** *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>,<nod e_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 thi ckness assumed ** *solid section, elset=cylinder, material=steel 1.0, ** ** define material

51

** *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 (minim um 2) # node_circum = 9 node_radial = 35 ## ## dependent parameters (do not modify)

52

## 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, transfo rmation, and 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 t ype ** *element,type=<elem> 1,1,2,<node_circum2>,<node_circum1> ** ** generate remainder of elements

53

** *elgen,elset=cylinder 1,<node_circum0>,1,1,<node_int>,<node_circum4>,<nod e_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 thi ckness 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 #

54

# 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-st ress # node_circum = nodes around 1/16 circumference # node_radial = nodes through the thickness (minim um 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

55

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, transfo rmation, and 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 t ype ** *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>,<nod e_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 thi ckenss assumed ** *solid section, elset=cylinder, material=steel 1.0, ** ** define material ** *material,name=steel *elastic <young>,<poisson>

56

*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-st ress # node_circum = nodes around 1/16 circumference # node_radial = nodes through the thickness (minim um 2) # elem_type = 'PE' node_circum = 9 node_radial = 35 ## ## dependent parameters (do not modify) ##

57

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, transfo rmation, and 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 t ype ** *element,type=<elem> 1,1,2,<node_circum2>,<node_circum1> ** ** generate remainder of elements

58

** *elgen,elset=cylinder 1,<node_circum0>,1,1,<node_int>,<node_circum4>,<nod e_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 thi ckness 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

59

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

60

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

-11000.00

-10000.00

-9000.00

-8000.00

-7000.00

-6000.00

-5000.00

-4000.00

-3000.00

-2000.00

-1000.00

0.00

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial S

tres

s (p

si)

Exact

Figure A1– Exact Radial Stress (Pressure Load)

Hoop Stress - Plane-Stress

15000.00

17000.00

19000.00

21000.00

23000.00

25000.00

27000.00

29000.00

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

Exact

Figure A2 – Exact Hoop Stress (Pressure Load)

61

Radial Strain - Plane-Stress

-0.00070000

-0.00060000

-0.00050000

-0.00040000

-0.00030000

-0.00020000

-0.00010000

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)

Rad

ial S

trai

n

Exact

Figure A3 – Exact Radial Strain (Pressure Load)

Hoop Strain - Plane-Stress

0.00000000

0.00020000

0.00040000

0.00060000

0.00080000

0.00100000

0.00120000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

trai

n

Exact

Figure A4 – Exact Hoop Strain (Pressure Load)

62

Longitudinal Strain - Plane-Stress

-0.00025000

-0.00020000

-0.00015000

-0.00010000

-0.00005000

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)

Long

itudi

nal S

trai

n

Exact

Figure A5 – Exact Longitudinal Strain (Pressure Load)

Radial Displacement - Plane-Stress

0.00640000

0.00660000

0.00680000

0.00700000

0.00720000

0.00740000

0.00760000

0.00780000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

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.

63

Radial Stress - Plane-Stress

-11000

-9000

-7000

-5000

-3000

-10007 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial S

tres

s (p

si)

ABAQUS

Figure A7 – ABAQUS Radial Stress (Pressure Load)

Hoop Stress - Plane-Stress

15000

17000

19000

21000

23000

25000

27000

29000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

ABAQUS

Figure A8 – ABAQUS Hoop Stress (Pressure Load)

64

Radial Strain - Plane-Stress

-0.00070000

-0.00060000

-0.00050000

-0.00040000

-0.00030000

-0.00020000

-0.00010000

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)

Rad

ial S

trai

n

ABAQUS

Figure A9 – ABAQUS Radial Strain (Pressure Load)

Hoop Strain - Plane-Stress

0.00000000

0.00020000

0.00040000

0.00060000

0.00080000

0.00100000

0.00120000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

trai

n

ABAQUS

Figure A10 – ABAQUS Hoop Strain (Pressure Load)

65

Longitudinal Strain - Plane-Stress

-0.00025000

-0.00020000

-0.00015000

-0.00010000

-0.00005000

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)

Long

itudi

nal S

trai

n

ABAQUS

Figure A11 – ABAQUS Longitudinal Strain (Pressure Load)

Radial Displacement - Plane-Stress

0.0064

0.0066

0.0068

0.007

0.0072

0.0074

0.0076

0.0078

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

ABAQUS

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-wall-

thickness (r/t) ratio is increased.

66

Radial Stress - Plane-Stress

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A13 – Radial Stress vs. r/t Ratios for Plane-Stress

Radial Stress - Plane-Strain

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A14 – Radial Stress vs. r/t Ratios for Plane-Strain

67

Hoop Stress - Plane-Stress

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A15 – Hoop Stress vs. r/t Ratios for Plane-Stress

Hoop Stress - Plane-Strain

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A16 – Hoop Stress vs. r/t Ratios for Plane-Strain

68

Longitudinal Strain - Plane-Stress

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A17 – Longitudinal Strain vs. r/t Ratios for Plane-Stress

Longitudinal Stress - Plane-Strain

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A18 – Longitudinal Stress vs. r/t Ratios for Plane-Strain

69

Radial Strain - Plane-Stress

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A19 – Radial Strain vs. r/t Ratios for Plane-Stress

Radial Strain - Plane Strain

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A20 – Radial Strain vs. r/t Ratios for Plane-Strain

70

Hoop Strain - Plane-Stress

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A21 – Hoop Strain vs. r/t Ratios for Plane-Stress

Hoop Strain - Plane-Strain

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A22 – Hoop Strain vs. r/t Ratios for Plane-Strain

71

Radial Displacement - Plane-Stress

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Dis

plac

emen

t

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

Figure A23 – Radial Displacement vs. r/t Ratios for Plane-Stress

Radial Displacement - Plane-Strain

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Dis

plac

emen

t

r/t = 4.0

r/t = 5.7

r/t = 9.0

r/t = 12.3

r/t = 19.0

r/t = 39.0

r/t = 79.0

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-wall-

thickness (r/t) ratio is decreased.

72

Radial Stress - Plane-Stress

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A25 – Radial Stress vs. r/t Ratios for Plane-Stress

Radial Stress - Plane-Strain

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A26 – Radial Stress vs. r/t Ratios for Plane-Strain

73

Hoop Stress - Plane-Stress

0.95

0.96

0.97

0.98

0.99

1

1.01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A27 – Hoop Stress vs. r/t Ratios for Plane-Stress

Hoop Stress - Plane-Strain

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A28 – Hoop Stress vs. r/t Ratios for Plane-Strain

74

Longitudinal Strain - Plane-Stress

0.999

0.9992

0.9994

0.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A29 – Longitudinal Strain vs. r/t Ratios for Plane-Stress

Longitudinal Stress - Plane-Strain

0.999

0.9992

0.9994

0.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ess

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A30 – Longitudinal Stress vs. r/t Ratios for Plane-Strain

75

Radial Strain - Plane-Stress

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A31 – Radial Strain vs. r/t Ratios for Plane-Stress

Radial Strain - Plane Strain

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A32 – Radial Strain vs. r/t Ratios for Plane-Strain

76

Hoop Strain - Plane-Stress

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A33 – Hoop Strain vs. r/t Ratios for Plane-Stress

Hoop Strain - Plane-Strain

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Str

ain

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A34 – Hoop Strain vs. r/t Ratios for Plane-Strain

77

Radial Displacement - Plane-Stress

0.99

0.995

1

1.005

1.01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Dis

plac

emen

t

r/t = 4.0

r/t = 2.3

r/t = 1.5

Figure A35 – Radial Displacement vs. r/t Ratios for Plane-Stress

Radial Displacement - Plane-Strain

0.99

0.995

1

1.005

1.01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Distance Through Thickness

Nor

mal

ized

Dis

plac

emen

t

r/t = 4.0

r/t = 2.3

r/t = 1.5

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.

78

Radial Stress - Plane-Stress

-2500

-2000

-1500

-1000

-500

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)

Rad

ial S

tres

s (p

si)

Exact

ABAQUS

Figure A37 – Exact vs. ABAQUS Radial Stress – Plane-Stress (Thermal Load)

Radial Stress - Plane-Strain

-3000

-2500

-2000

-1500

-1000

-500

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)

Rad

ial S

tres

s (p

si)

Exact

ABAQUS

Figure A38 – Exact vs. ABAQUS Radial Stress – Plane-Strain (Thermal Load)

79

Hoop Stress - Plane-Stress

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

10000

15000

20000

25000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

Exact

ABAQUS

Figure A39 – Exact vs. ABAQUS Hoop Stress – Plane-Stress (Thermal Load)

Hoop Stress - Plane-Strain

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

Exact

ABAQUS

Figure A40 – Exact vs. ABAQUS Hoop Stress – Plane-Strain (Thermal Load)

80

Longitudinal Strain - Plane-Stress

-0.0005

0

0.0005

0.001

0.0015

0.002

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Long

itudi

nal S

trai

n

Exact

ABAQUS

Figure A41 – Exact vs. ABAQUS Long. Strain – Plane-Stress (Thermal Load)

Longitudinal Stress - Plane-Strain

-60000

-50000

-40000

-30000

-20000

-10000

0

10000

20000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Long

itudi

nal S

tres

s (p

si)

Exact

ABAQUS

Figure A42 – Exact vs. ABAQUS Long. Stress – Plane-Strain (Thermal Load)

81

Radial Strain - Plane-Stress

-0.0005

0

0.0005

0.001

0.0015

0.002

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial S

trai

n

Exact

ABAQUS

Figure A43 – Exact vs. ABAQUS Radial Strain – Plane-Stress (Thermal Load)

Radial Strain - Plane Strain

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial S

trai

n

Exact

ABAQUS

Figure A44 – Exact vs. ABAQUS Radial Strain – Plane-Strain (Thermal Load)

82

Hoop Strain - Plane-Stress

0.00063

0.00064

0.00065

0.00066

0.00067

0.00068

0.00069

0.0007

0.00071

0.00072

0.00073

0.00074

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

trai

n

Exact

ABAQUS

Figure A45 – Exact vs. ABAQUS Hoop Strain – Plane-Stress (Thermal Load)

Hoop Strain - Plane-Strain

0.00082

0.00084

0.00086

0.00088

0.0009

0.00092

0.00094

0.00096

0.00098

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

trai

n

Exact

ABAQUS

Figure A46 – Exact vs. ABAQUS Hoop Strain – Plane-Strain (Thermal Load)

83

Radial Displacement - Plane-Stress

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

ABAQUS

Figure A47 – Exact vs. ABAQUS Radial Displ. – Plane-Stress (Thermal Load)

Radial Displacement - Plane-Strain

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

ABAQUS

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).

84

Radial Displacement - Plane-Stress

0.012

0.0122

0.0124

0.0126

0.0128

0.013

0.0132

0.0134

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

ABAQUS

Figure A49 – Exact vs. ABAQUS Radial Stress – Plane-Stress (Combined Load)

Radial Stress - Plane-Strain

-12000

-10000

-8000

-6000

-4000

-2000

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)

Rad

ial S

tres

s (p

si)

Exact

ABAQUS

Figure A50 – Exact vs. ABAQUS Radial Stress – Plane-Strain (Combined Load)

85

Hoop Stress - Plane-Stress

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

Exact

ABAQUS

Figure A51 – Exact vs. ABAQUS Hoop Stress – Plane-Stress (Combined Load)

Hoop Stress - Plane-Strain

-10000

0

10000

20000

30000

40000

50000

60000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tres

s (p

si)

Exact

ABAQUS

Figure A52 – Exact vs. ABAQUS Hoop Stress – Plane-Strain (Combined Load)

86

Longitudinal Strain - Plane-Stress

-0.0005

0

0.0005

0.001

0.0015

0.002

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Long

itudi

nal S

trai

n

Exact

ABAQUS

Figure A53 – Exact vs. ABAQUS Long. Strain – Plane-Stress (Combined Load)

Longitudinal Stress - Plane-Strain

-60000

-50000

-40000

-30000

-20000

-10000

0

10000

20000

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Long

itudi

nal S

tres

s (p

si)

Exact

ABAQUS

Figure A54 – Exact vs. ABAQUS Long. Stress – Plane-Strain (Combined Load)

87

Radial Strain - Plane-Stress

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial S

trai

n

Exact

ABAQUS

Figure A55 – Exact vs. ABAQUS Radial Strain – Plane-Stress (Combined Load)

Radial Strain - Plane Strain

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial S

trai

n

Exact

ABAQUS

Figure A56 – Exact vs. ABAQUS Radial Strain – Plane-Strain (Combined Load)

88

Hoop Strain - Plane-Stress

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

trai

n

Exact

ABAQUS

Figure A57 – Exact vs. ABAQUS Hoop Strain – Plane-Stress (Combined Load)

Hoop Strain - Plane-Strain

0

0.0005

0.001

0.0015

0.002

0.0025

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

trai

n

Exact

ABAQUS

Figure A58 – Exact vs. ABAQUS Hoop Strain – Plane-Strain (Combined Load)

89

Radial Displacement - Plane-Stress

0.012

0.0122

0.0124

0.0126

0.0128

0.013

0.0132

0.0134

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

ABAQUS

Figure A59 – Exact vs. ABAQUS Radial Displ. – Plane-Stress (Combined Load)

Radial Displacement - Plane-Strain

0.0132

0.0134

0.0136

0.0138

0.014

0.0142

0.0144

0.0146

0.0148

0.015

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

ABAQUS

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).

90

Elastic-Plastic Mises Stress - Plane Strain

0.00E+00

1.00E+04

2.00E+04

3.00E+04

4.00E+04

5.00E+04

6.00E+04

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Mis

es S

tres

s (p

si)

Exact

ABAQUS

Figure A61 – Exact vs. ABAQUS Mises Stress Plane-Strain (Plastic – Pressure)

Elastic-Plastic Mises Stress - Plane Stress

0.00E+00

1.00E+04

2.00E+04

3.00E+04

4.00E+04

5.00E+04

6.00E+04

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Mis

es S

tress

(ps

i)

Exact

ABAQUS

Figure A62 – Exact vs. ABAQUS Mises Stress Plane-Stress (Plastic – Pressure)

91

Elastic-Plastic Radial Stress - Plane Strain

-1.80E+04

-1.60E+04

-1.40E+04

-1.20E+04

-1.00E+04

-8.00E+03

-6.00E+03

-4.00E+03

-2.00E+03

0.00E+00

2.00E+03

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial S

tres

s (p

si)

Exact-Elastic

Exact-Plastic

ABAQUS

Figure A63 – Exact vs. ABAQUS Radial Stress Plane-Strain (Plastic – Pressure)

Elastic-Plastic Radial Stress - Plane Stress

-1.60E+04

-1.40E+04

-1.20E+04

-1.00E+04

-8.00E+03

-6.00E+03

-4.00E+03

-2.00E+03

0.00E+00

2.00E+03

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial S

tres

s (p

si)

Exact-Elastic

Exact-Plastic

ABAQUS

Figure A64 – Exact vs. ABAQUS Radial Stress Plane-Stress (Plastic – Pressure)

92

Elastic-Plastic Hoop Stress - Plane Strain

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

3.00E+04

3.50E+04

4.00E+04

4.50E+04

5.00E+04

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tress

(ps

i)

Exact-Elastic

Exact-Plastic

ABAQUS

Figure A65 – Exact vs. ABAQUS Hoop Stress Plane-Strain (Plastic – Pressure)

Elastic-Plastic Hoop Stress - Plane Stress

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

3.00E+04

3.50E+04

4.00E+04

4.50E+04

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

tress

(ps

i)

Exact-Elastic

Exact-Plastic

ABAQUS

Figure A66 – Exact vs. ABAQUS Hoop Stress Plane-Stress (Plastic – Pressure)

93

Elastic-Plastic Longitudinal Stress - Plane Strain

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

1.40E+04

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Long

itudi

nal S

tres

s (p

si)

Exact-Elastic

Exact-Plastic

ABAQUS

Figure A67 – Exact vs. ABAQUS Long. Stress Plane-Strain (Plastic – Pressure)

Elastic-Plastic Longitudinal Strain - Plane Stress

-7.00E-04

-6.00E-04

-5.00E-04

-4.00E-04

-3.00E-04

-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)

Long

itudi

nal S

trai

n

Exact-Elastic

Exact-Plastic

ABAQUS

Figure A68 – Exact vs. ABAQUS Long. Strain Plane-Stress (Plastic – Pressure)

94

Elastic-Plastic Radial Strain - Plane Strain

-2.00E-03

-1.80E-03

-1.60E-03

-1.40E-03

-1.20E-03

-1.00E-03

-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)

Rad

ial S

trai

n

Exact

ABAQUS

Figure A69 – Exact vs. ABAQUS Radial Strain Plane-Strain (Plastic – Pressure)

Elastic-Plastic Radial Strain - Plane Stress

-1.40E-03

-1.20E-03

-1.00E-03

-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)

Rad

ial S

trai

n

Exact-Elastic

ABAQUS

Exact-Plastic

Figure A70 – Exact vs. ABAQUS Radial Strain Plane-Stress (Plastic – Pressure)

95

Elastic-Plastic Hoop Strain - Plane Strain

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

1.20E-03

1.40E-03

1.60E-03

1.80E-03

2.00E-03

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

train

Exact

ABAQUS

Figure A71 – Exact vs. ABAQUS Hoop Strain Plane-Strain (Plastic – Pressure)

Elastic-Plastic Hoop Strain - Plane Stress

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

1.20E-03

1.40E-03

1.60E-03

1.80E-03

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Hoo

p S

trai

n Exact-Elastic

ABAQUS

Exact-Plastic

Figure A72 – Exact vs. ABAQUS Hoop Strain Plane-Stress (Plastic – Pressure)

96

Elastic-Plastic Radial Displacement - Plane Strain

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact

ABAQUS

Figure A73 – Exact vs. ABAQUS Radial Displ. Plane-Strain (Plastic – Pressure)

Elastic-Plastic Radial Displacement - Plane Strain

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10

Radius (in)

Rad

ial D

ispl

acem

ent (

in)

Exact-Elastic

ABAQUS

Exact-Plastic

Figure A74 – Exact vs. ABAQUS Radial Displ. Plane-Stress (Plastic – Pressure)