117

CHAPTER 7

Thick-Walled Cylinder;

Finite Element Analysis

In this chapter, finite element modeling of the thick-walled cylinder (TWC) under fatigue

loading is presented. As a first step, to provide a base line the stress distribution in TWC

under static internal pressure is estimated using analytical and numerical methods. The

component is then analyzed with internal axial crack under static pressure. Finally, crack

growth analysis is conducted on the component under cyclic pressure applying the

theories of fatigue process. The data obtained from the experimental work is used as the

input for the said analysis.

7.1 Thick-walled cylinder

A thick-walled cylinder or tube is one where the thickness of the wall is greater than one-

tenth of the radius. In the following sections a model of TWC is presented and the stress

distribution under internal/external pressure is discussed.

7.1.1. Model description

Consider a thick walled cylinder with outer diameter, do and inner diameter, di (Fig. 7.1).

The thickness of the cylinder tw is the difference between the inner and outer radius

where the outer radius is always greater than the inner radius. The pressures on the inner

and outer surfaces of the cylinder are pi and po, respectively.

118

Fig. 7.1 Two dimensional section of the TWC showing geometric parameters

In case of a TWC with closed ends, the cylinder experiences three principal stresses

under static internal/external pressure, i.e. tangential (T), radial (R) and axial (A) as

shown in Fig. 7.2. However, in case the cylinder has open ends there will be no axial

component of stress. The exact elastic solution for the cylinder under stress can be

obtained using Lamé’s equations. Among these stresses the tangential or hoop stress is

the maximum.

7.1.2. Model equations

Consider the TWC subjected to an internal pressure above atmospheric pressure. The

resulting stresses and expansion of the cylinder are described by the equations from 7.1 to

pi

do

tw

po di

119

7.5. These equations display how internal/external pressure and the thickness of the

cylinder relate to the stresses. This model shows that the stresses within the thick walled

cylinder depend on the inner and outer pressures and the inner and outer radii.

Fig. 7.2 Schematic of the TWC indicating three principal stresses

Model equations

(7.1)

(7.2)

(7.3)

(7.4)

22

22222)/)((

)(io

oioiooiir

rr

rrrpprprpr

22

22

io

ooiia

rr

rprp

rrr

rrpp

Er

rr

rprp

Eu

io

oioi

io

ooii

r

1)()1()1(22

22

22

22

22

22222)/)((

)(io

oioiooiih

rr

rrrpprprpr

T R

A

Po

Pi

120

(7.5)

where

σh = tangential stress variation within the material of the cylinder

σr = stress variation in the radial direction

σa = longitudinal stress within the material of the cylinder

pi = uniform internal pressure

po = uniform external pressure

ri = inside radius

ro = outside radius

r = radius, ri ≤ r ≤ ro

E = modulus of elasticity of the material

υ = Poisson's ratio of the material

ur = displacement in the radial direction due to pressurization

dua/da = relative increase in length in the axial direction

7.1.3. Parameter description

In the above equations, all the parameters are known except for the position vector ‘r’,

which varies from the inner to the outer radius. If the inner pressure is greater than the

outer pressure, then from the equations the stresses are largest as ‘r’ approaches the inner

radius. However, if the outer pressure is greater than the inner pressure, the stresses will

be largest as ‘r’ approaches the outer radius.

The elements that are located at the same radius but different angle theta will experience

the same tangential and radial stresses; this can be easily inferred from the fact that there

22

222

io

ooiia

rr

rprp

Eda

du

121

is no angular positional variable (i.e. theta) in any of the governing equations. However,

the elements at different radial lengths experience different stresses; this can be observed

from the fact that ‘r’ is a variable in the governing equations.

7.1.4. Stress description

Tangential stress affects an element in a direction tangent to its circumference, i.e.

perpendicular to the radial vector. Radial stress affects the element in a direction that is

parallel to the radial vector. For any pressure-thickness condition the difference between

the tangential and radial stress is a constant for the entire range of ‘r’. That constant can

be arrived by subtracting the radial stress from the tangential stress; the tangential stress

being always greater, the constant will be a positive value.

7.2 Static loading of TWC - without crack

The TWC is analyzed under static loading by classical theory and the results are

compared with the numerical solution. The cylinder was analyzed as an open cylinder

with no axial component of stress. Two types of analyses were conducted; one without

crack and the other with internal axial crack.

7.2.1. Analytical solution

For a TWC with the following parameters the principal stresses calculated by the model

equations are given in Tables 7.1 and 7.2. The sketch of the cylinder half section is

shown in Fig. 7.3.

pi = 5 – 100 MPa po = 0 MPa

di = 100 mm do = 150 mm tw = 25 mm

122

Fig. 7.3 Half model of the cylinder section subjected to internal pressure pi

Table 7.1 The variation in stresses and displacements with internal pressure calculated by

the model equations at inner radius, ri

pi, MPa

Principal stress, MPa Radial

displacement,

mm Tangential Radial

5 13 -5 0.0103169

10 26 -10 0.0206338

15 39 -15 0.0309507

20 52 -20 0.04126761

25 65 -25 0.05158451

30 78 -30 0.06190141

35 91 -35 0.07221831

40 104 -40 0.08253521

45 117 -45 0.09285211

50 130 -50 0.10316901

55 143 -55 0.11348592

60 156 -60 0.12380282

65 169 -65 0.13411972

70 182 -70 0.14443662

75 195 -75 0.15475352

80 208 -80 0.16507042

85 221 -85 0.17538732

90 234 -90 0.18570423

95 247 -95 0.19602113

100 260 -100 0.20633803

ri = 50 mm

pi = 5–100 MPa ro = 75 mm

tw = 25 mm

123

Table 7.2 The variation in stresses and displacements along the wall thickness calculated

by the model equations at pi = 50 MPa, po = 0 MPa

r, mm

Principal stress, MPa Radial

displacement,

mm Tangential Radial

50 130.00 -50.00 0.103169

51 126.51 -46.51 0.101894

52 123.21 -43.21 0.100682

53 120.10 -40.10 0.099530

54 117.16 -37.16 0.098435

55 114.38 -34.38 0.097393

56 111.75 -31.75 0.096402

57 109.25 -29.25 0.095459

58 106.88 -26.88 0.094562

59 104.64 -24.64 0.093708

60 102.50 -22.50 0.092894

61 100.47 -20.47 0.092120

62 98.53 -18.53 0.091383

63 96.69 -16.69 0.090682

64 94.93 -14.93 0.090014

65 93.25 -13.25 0.089378

66 91.65 -11.65 0.088773

67 90.12 -10.12 0.088197

68 88.66 -8.66 0.087650

69 87.26 -7.26 0.087129

70 85.92 -5.92 0.086634

71 84.63 -4.63 0.086163

72 83.40 -3.40 0.085716

73 82.22 -2.22 0.085292

74 81.09 -1.09 0.084889

75 80.00 0.00 0.084507

124

7.2.2. Finite element modeling

The TWC as shown in Fig. 7.3 was numerically analyzed by finite element method and

the results were compared with the analytical solution. The commercially available

ANSYS 9.0 finite element software was used for this purpose. Two dimensional finite

element analysis (FEA) was conducted using 4-noded quadrilateral elements under plane-

strain conditions.

7.2.2.1 Model Geometry

Fig. 7.4 shows the two dimensional model geometry of the cylinder used for FEA. The

symmetry of the cylinder was taken advantage of and a solid model for a half section of

the cylinder was created in the ANYSYS pre-processor. The same symmetry conditions

can also be used in the presence of axial crack. The outer diameter, do of the cylinder is

150 mm while the inner diameter, di is 100 mm. The wall thickness, tw of the cylinder is

25 mm.

Fig. 7.4 TWC Model used for FEA

125

7.2.2.2 Material properties

During FEA, an isotropic material with modulus of elasticity E = 71 GPa and Poisson’s

ratio, = 0.33 was used [147].

7.2.2.3 Element selection and meshing

The TWC was meshed using two dimensional 4-noded, PLANE42 solid elements. The

element geometry is shown in Fig. 6.2. The parametric study was conducted to see the

effects of element size on the results. Meshed model is shown in Fig. 7.5.

(a)

(b)

Fig. 7.5. a) Meshed model using PLANE42 element b) magnified view of boxed area;

element size is 0.5 mm

126

7.2.2.4 Boundary conditions and solution

The boundary conditions (BCs) applied on the TWC are shown in Fig. 7.6. The half

section of the cylinder was constrained applying symmetry boundary conditions along the

wall thickness on both edges. The model was loaded by applying pressure on the inner

wall of the cylinder, simulating internal pressure. The pressure was varied from 5 to 100

MPa. There was no outer pressure applied. Solutions were obtained at different internal

pressures and the results were compared with the analytical one.

The von Mises stress distribution obtained after solution is shown in Fig. 7.7. This value

is normally used in both fatigue and static load design of such cylinders. The parametric

study conducted to see the effect of element size reveals that the results obtained using

element size of 1 mm and less are in good agreement with the analytical results.

Fig. 7.6 Static loading - Boundary conditions applied for analysis

Symmetry BCs

Pressure

127

(a)

(b)

Fig. 7.7 Static loading – Nodal solution showing von Mises stress distribution at internal

pressure of a) 5 MPa b) 100 MPa

7.2.3. Comparison of the analytical and numerical results

The results of the stress distribution obtained from analytical (thick-walled cylinder

theory, Lamé’s equations) and numerical techniques were compared to see the validity of

the model. Figs. 7.8a and 7.8b show the graphical presentation of the analytical and the

128

FEA results of stress versus internal pressure at inner radius. The stress variation along

the wall thickness of the cylinder obtained from the two methods is shown in Fig. 7.9.

(a)

(b)

Fig. 7.8 Stress versus internal pressure - comparison of the two results at inner radius a)

tangential b) radial

0

75

150

225

300

0 20 40 60 80 100 pi, MPa

Tan

gen

tial

str

ess,

MP

a .

. FEA

TWC theory

-100

-75

-50

-25

0

0 20 40 60 80 100 pi, MPa

Rad

ial

stre

ss, M

Pa

MP

a

FEA

TWC theory

129

(a)

(b)

Fig. 7.9 Stress variation along the wall thickness of the cylinder obtained from the two

methods at an internal pressure of 100 MPa a) tangential b) radial

150

175

200

225

250

275

50 55 60 65 70 75

r, mm

Tan

gen

tial

str

ess,

MP

a

MP

a

Analytical

FEA

-100

-75

-50

-25

0

25

50 55 60 65 70 75 r, mm

Rad

ial

stre

ss, M

Pa

MP

a

Analytical

FEA

130

In Fig. 7.8, both the tangential and radial stresses obtained from the analytical and FEA

methods change linearly with the applied internal pressure. It can be seen that the results

obtained from the two techniques are in good agreement. Fig. 7.9a shows a gradual

decrease in the tangential stress from inner to outer radius. The highest tangential (hoop)

stress is found at the inner radius i.e. at the inner wall of the cylinder. In Fig. 7.9b the

change in radial stress along the wall thickness of the cylinder is presented. A

compressive stress is found which varies from 100 MPa at the inner radius to a value of 0

MPa at the outer radius. Again the results obtained from the two techniques and

presented in Figs. 7.9a and 7.9b are in fairly good agreement. This concludes that the half

model used for the stress analysis is providing satisfactory results and can be used for the

analysis of the cylinder with internal crack.

7.3 Static loading of TWC - with internal axial crack

The TWC with internal axial crack and under static loading was analyzed at different

internal pressures. Analysis was conducted to determine the stress intensity factor (KI) at

the crack tip; (KI) is used to estimate the crack growth rate under cyclic loading. The two

dimensional model of the TWC was analyzed analytically and the values for KI obtained.

These results were used for finite element analysis of the cylinder under fatigue loading.

7.3.1. Geometry of the model

Fig. 7.10 shows the modified two dimensional model geometry of the cylinder, with

internal axial crack, used for FEA. The crack was modeled on the inner bore of the

cylinder in axial direction (perpendicular to the plane of the paper). The crack depth is ‘a’

mm.

131

Fig. 7.10 Schematic of two dimensional half cylinder model with internal axial crack

7.3.2. Material properties, element type and meshing

The cracked model was analyzed using the same material properties and employing the

same element type as was used for un-cracked model. The half model with an initial

crack length of 3 mm, a/tw = 0.12 was used in FEA due to the geometrical symmetry of

the cylinder.

7.3.3. Boundary conditions and solution

The boundary conditions applied on the TWC are shown in Fig. 7.11. The half section of

the cylinder was constrained applying symmetry boundary conditions along the wall

thickness on both sides. A 3 mm long crack was modeled by applying no constraints from

ri to 3 mm along the x direction at the right wall, thus providing the crack tip node at 3

mm from the inner wall. The model was loaded by applying tractions at the inner wall of

the cylinder, simulating internal pressure. After loading the model and obtaining the

solution, KI was obtained at the crack tip by defining the path and using KCALC

command. Solutions were obtained at internal pressures varying from 5 to 100 MPa.

ri = 50 mm

pi = 5 –100 MPa ro = 75 mm

tw = 25 mm

Crack a

132

Fig. 7.11 Static loading of TWC with crack - Boundary conditions applied for analysis

The nodal solution showing von Mises stress distribution at internal pressure of 5 and 100

MPa is shown in Fig. 7.12. The maximum stress is at the crack tip node which can be

seen more clearly in the Fig. 7.13.

7.3.4. Determination of the stress intensity factor (KI)

After obtaining the solution, the stress intensity factor was determined by defining the

path and using KCALC command. In order to see the effect on KI, element size was

varied from 2 to 0.25 mm; solutions were obtained at internal pressures varying from 5 to

100 MPa. Plot in Fig. 7.14 shows the KI versus internal pressure at a crack length of 3

mm. KI increases linearly with the pressure and the effect of element size was found

negligible. Fig. 7.15 shows the KI versus internal pressure at crack length from 3 to 10

mm. Again KI increases linearly with internal pressure for all the crack sizes analyzed.

Pressure

Symmetry BCs

crack

133

(a)

(b)

Fig. 7.12 Static loading of cylinder with crack – Nodal solution showing von Mises stress

distribution at internal pressure of a) 5 MPa b) 100 MPa

crack

134

Fig. 7.13 Magnified view of the crack region shown in Fig. 7.12a with BCs

Fig. 7.14 Plot of KI versus internal pressure at a crack length of 3 mm

a = 3 mm

0

10

20

30

0 20 40 60 80 100 pi, MPa

E2 E1

E0.5

KI, M

Pa.

sqrt

(m)

MP

a.sq

rt(m

)

135

Fig. 7.15 Plot showing KI versus internal pressure at crack length of 3, 5, 7 and 10 mm

Fig. 7.16 shows the variation of KI with the increase in crack length along the wall

thickness of the cylinder at different internal pressures. The data was obtained using an

element size of 0.5 mm. The curves obtained from the data show polynomial fits which

are used for the fatigue calculations.

7.4 FEA of fatigue crack growth in TWC

The fatigue crack growth analysis of TWC was performed based on linear elastic fracture

mechanics and using Paris law. The FEA results obtained in the previous section were

used for this purpose. The relations between the stress intensity factor KI and the crack

size were used and the fatigue calculations were performed employing the same approach

as was applied in the case of M(T) samples. The analysis was conducted using the

0

20

40

60

80

0 20 40 60 80 100 pi, MPa

KI , M

Pa.

sqrt

(m)

MP

a.sq

rt(m

)

a-3

a-5

a-7

a-10

136

experimental data obtained in the CR direction which corresponds to the hoop stress in

the cylinder. The cyclic pressure was applied with R ratio equal to 0.1 and the crack was

advanced in steps of 0.05 mm. The fatigue crack growth life (Ng) of the cylinder was

determined at different internal pressures. The fatigue crack growth life was the total

applied cycles from the initial crack length to the final fracture [24].

0

50

100

150

0 5 10 15 20a, mm

KI, M

Pa.

sqrt

(m)

P10

P20

P30

P40

P50

P60

P75

P100

Fig. 7.16 Variation of KI with the increase of crack length at different internal pressures

7.4.1. Crack propagation in TWC

Fig. 7.17 shows the plots of the applied pressure cycles versus crack length of the

simulated TWC model with an initial crack size of 3 mm.

The analysis of the results showed that the crack grows faster at higher pressures and vice

versa as was observed in the case of M(T) samples. It is also clear from the plot that the

fatigue crack growth life decreases with an increase in the internal pressure.

137

1.0

10.0

100.0

10 100 1000 10000 100000 1000000ln N, cycles

ln a

, m

m.

P20

P25

P30

P40

P50

P60

Fig. 7.17 Applied cycles versus crack length of the simulated TWC model with an initial

crack length of 3 mm

7.4.2. Predicted FCG rate – Experimental vs FEA

The variation of fatigue crack growth rate with K obtained experimentally in CR

samples and from the FEA of the cylinder at different applied pressures is shown in Fig.

7.18. The smooth crack growth rate achieved using the FE analysis is based on the

calculations using Paris equation. It can be seen that the fatigue crack growth rate

obtained by the FEA lied within the upper and lower bounds of the crack growth rate

achieved from the experimental data.

7.4.3. Fatigue crack growth life prediction of the cylinder

The fatigue crack growth life of the cylinder was predicted from the FEA and is

presented in Fig. 7.19. The plot provides variation in the internal pressure versus the total

applied cycles, starting from the initial crack length to the final fracture. The curve fitting

138

of the data provides the best fit with power relation between the two values and is given

in the figure. As expected, the fatigue crack growth life of the cylinder obtained from

FEA shows that the fatigue lifetime increases as the applied pressure decreases.

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1 10 100

K, MPa.sqrt(m)

da/

dN

, m

/cycl

e

EXP

FEA-P20

Fig. 7.18 The variation of fatigue crack growth rate with K – Experimental vs FEA

Fig. 7.19 Predicted fatigue crack growth life of the thick-walled cylinder at different

internal pressures

y = 360.19 x -0.23

R 2 = 0.9927

0

20

40

60

80

100 1000 10000 100000 1000000 10000000

ln Ng, cycles

pi,

MP

a

MP

a