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CHAPTER 9

INFLUENCE OF RESIDUAL STRESSES ON THE FAILURE

PRESSURE OF CYLINDRICAL VESSELS

In the design of pressure vessels, evaluation of failure pressure that

a cylindrical pressure vessel can withstand is an important consideration.

While prediction of failure pressure of pressure vessels it is also necessary to

consider the residual stresses already present in the pressure vessels. The

knowledge of the residual stresses present in the pressure vessel due to

welding is necessary. Of several formulae for calculating bursting pressure of

vessels, the Faupel’s formula is the most popular one. However there is no

unique failure theory applicable to all materials.

9.1 FAILURE PRESSURE ESTIMATES OF UNFLAWED

PRESSURE VESSELS WITHOUT RESIDUAL STRESS

Various methods are being used to estimate the failure pressure

(Faupel 1956, Svensson 1958, Beena et al 1995 and Christopher et al 2002).

Finite element techniques based on the Global Plastic Deformation (GPD) are

also used to evaluate the failure pressure and its results were found to be in

good agreement with test results (Aseer et al 2009). Experimental methods are

useful for verifying the correctness of analytical or computational analysis.

Usually stress cannot be measured directly and hence most of the

experimental methods serve to measure strains by bonding the gauges to the

surface of the structure under test. If nothing is known in advance with the

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strain field, a three element rosette is the best choice for finding the elements

of the small strain tensor. Most of the existing solutions to elasto-plastic

problems are based on Tresca’s yield criterion and in associated flow rule

leading to analytical or quasi analytical solution. Under the application of the

internal pressure, the material deforms both elastically and plastically.

Inelasticity is always present under all types of loading. Hence elasto-plastic

deformations cannot be treated independently. Therefore elasto-plastic

analysis has to be performed while analyzing a pressure vessel. Failure

pressure estimates from FEA based on GPD are found to be in good

agreement with test results of thin as well as thick walled vessels made of

ductile materials (Aseer et al 2009). The pressure corresponding to GPD will

be the failure pressure. Applicability of the present FEA procedure for failure

prediction is examined with the existing test results (Faupel et al 1953).

The geometrical model and FEA model of the ASTM 36 carbon

steel pressure vessel similar to Figures 6.1 and 6.2 is considered for analysis.

Axial displacement is suppressed at both ends of the cylindrical shell to arrest

the axial growth under internal pressure.

= 1 + (9.1)

Equation (9.1) is Inverse Romberg-Osgood relationship (Beena et

al 1995), is a constitutive relationship and gives the stress as an explicit

function of strain. Where = , is the ultimate strength of the material

and ‘ ’ is the parameter defining the shape of the non-linear stress-strain

relationship. = 0.002143, = 1.798 for ASTM 36 carbon steel. The

stress-strain curve (see Figure 9.1) generated using Equation (9.1) is given as

input for FEA to take care of material behavior during the application of

internal pressure. ANSYS has the provision for checking the (GPD).

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It indicates the pressure level to cause complete plastic flow through the

cylinder walls (ie. bursting pressure).

Bursting pressure of pressure vessel (Faupel et al 1953),

= ln (9.2)

Where = Ultimate strength of the material

= Yield strength of the material

= Wall ratio of hollow cylinder (Ro/Ri)

For ASTM 36 steel, = 450 MPa and = 380 MPa

9.1.1 Results and Discussion

Figure 9.2 shows the effective stress at failure of the pressure

vessel. The analysis results indicate the failure pressure as 48 MPa while it is

46 MPa by using Equation (9.2), having good agreement. Table 9.1 gives the

cylinder dimensions, material properties and Table 9.2 gives the material

constants in Equation (9.1) for the stress-strain curve and failure pressure of

cylindrical vessels analyzed in Faupel (1953). Figure 9.3 gives the stress-

strain curve of Cr-Ni-Mo-V steel generated by Equation (9.1). Figure 9.4

gives effective, hoop and meridional stress plot of Cr-Ni-Mo-V steel pressure

vessel No.2 in Table 9.1 with the applied internal pressure up to global plastic

deformation. The failure pressure evaluated through the present analysis

(FEA) is 1358 MPa, where the experimental and calculated values are 1379

MPa and 1325.2 MPa respectively (Faupel et al 1953), and found in good

agreement.

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Figure 9.1 Stress-strain curve of ASTM 36 carbon steel generated by

Equation (9.1)

Figure 9.2 Effective stress plot of ASTM 36 carbon steel pressure vessel

up to global plastic deformation

0

100

200

300

400

500

0 10 20 30 40 50

Eff

ecti

ve

stre

ss,

MP

a

Applied pressure (MPa)

At Inner wall

At Middle

At outer wall

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Figure 9.3 Stress-strain curve of Cr-Ni-Mo-V steel generated by

Equation (9.1)

Figure 9.4 Effective, hoop and meridional stress plot of Cr-Ni-Mo-V

steel pressure vessel (Faupel et al 1953) (Cylinder No.2) with

the applied internal pressure up to global plastic

deformation

0

200

400

600

800

1000

1200

1400

0 5 10 15 20 25 30

Str

ess

-M

Pa

Strain x 103

Cr-Ni-Mo-V steel

155

Table 9.1 Details of cylindrical pressure vessels, materials and

strength properties (Faupel et al 1953)

Cylinder

No

Cylindrical shell

dimensions (mm)

Material

Tensile strength

properties (MPa)

Outer

diameter Thickness

Yield

strength

( )

Ultimate

strength

( )

1 68.00 21.00 Cr-Ni-Mo-V

steel

329.6 641.9

2 68.00 21.00 Cr-Ni-Mo-V

steel

1101.8 1223.8

3 101.60 31.68 SAE 3320 548.1 726.7

4 31.75 9.5 SAE 4340 716.4 855.6

5 31.75 9.5 SAE 4340 596.4 795.0

6 31.75 9.5 SAE 4340 797.0 859.0

7 101.60 31.68 SAE 1045 419.9 701.9

8 101.60 31.68 SAE 1045 562.6 842.5

Table 9.2 Material constants in Equation (9.1) for the stress-strain curve

and failure pressure of cylindrical vessels in Table 9.1

Cylinder

No

Material constants in Eqn (9.1) Failure pressure, (MPa)

Young’s

modulus,

E (GPa)

Observed CalculatedFEA

(Faupel et al 1953)

1 207 0.00310 0.945 689.5 561.9 711.0

2 207 0.00592 3.600 1379.0 1325.2 1358.0

3 207 0.00351 1.710 730.8 770.8 820.0

4 207 0.00414 2.340 917.0 954.9 901.0

5 207 0.00384 1.754 827.4 856.3 836.0

6 207 0.00415 3.512 806.7 892.2 904.0

7 207 0.00339 1.164 737.7 662.6 791.0

8 207 0.00407 1.464 820.5 846.0 950.0

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9.2 INFLUENCE OF RESIDUAL STRESSES ON FAILURE

PRESSURE OF UNFLAWED PRESSURE VESSELS

The ASTM 36 carbon steel pressure vessel as in Figure 6.1 is

considered for failure pressure analysis of butt-welded cylindrical pressure

vessels having residual stresses. Axial displacement is suppressed at both

ends of the cylindrical shell to arrest the axial growth under internal pressure.

Failure pressure of the pressure vessel before welding (ie., without residual

stress) is initially obtained by an elasto-plastic analysis. Then a thermo-

mechanical FEA with the similar procedure and same welding parameters in

the earlier analysis is carried out to assess the weld-induced residual stress

and again failure pressure is obtained by another elasto-plastic analysis for

this pressure vessel having residual stress to assess the effect of residual

stresses in failure pressure. For failure pressure analysis for this pressure

vessel along with the residual stresses present, the analysis is restarted from

the terminating (final) load step of thermal stress analysis along with applied

internal pressure and performed up to GPD. The pressure which is

corresponding to GPD will be the failure pressure.

9.2.1 Results and Discussion

2D Finite element analysis with axisymmetric model has been

carried out using ANSYS software package to access the failure pressure of

cylindrical pressure vessel made of ASTM 36 carbon steel having weld-

induced residual stresses. An elasto-plastic analysis is performed to find out

the failure pressure of the pressure vessel not having residual stresses. The

analysis results indicate the failure pressure as 48 MPa while it is 46 MPa by

using Equation (9.2), having good agreement. Figure 9.5 shows the effective

stress up to failure with and without residual stress. The failure analysis of the

pressure vessel having residual stresses shows that the pressure vessel fails at

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a pressure of 15 MPa, while the failure pressure without residual stresses is 48

MPa.

Faupel et al (1953) have analyzed eight number of pressure vessels

shown in Tables 9.1 and 9.2, in which three cylinders of each material were

heat treated under identical conditions. One cylinder was then used for the

residual stress test, the second for the static internal pressure test, and the third

one was used to determine the mechanical properties resulting from the heat

treatment. They have concluded that the residual stresses in pressure vessels

due to heat treatment do not appear to influence overstrain or bursting

pressure. It should be noted that the material properties were measured after

the heat treatment process. Now while calculating the failure pressure

analytically, in which if these material properties were used then the effect of

residual stresses on failure pressure will not be felt. If the properties of the material

before the heat treatment were used, then the effect would have been felt.

Figure 9.5 Effective stress plot of ASTM 36 carbon steel pressure vessel

with and without residual stress up to global plastic

deformation

0

100

200

300

400

500

0 10 20 30 40 50

Eff

ecti

ve

stre

ss,

(MP

a)

Applied pressure (MPa)

Inner without RS

Middle without RS

Outer without RS

Inner with RS

Outer with RS

GPD

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9.3 INFLUENCE OF RESIDUAL STRESSES ON FAILURE

PRESSURE OF FLAWED PRESSURE VESSELS

Christopher et al (2002) applied the three parameter fracture

criterion to correlate the fracture data on aluminium, titanium and steel

materials from test results on flawed cylindrical tanks/pressure vessels

without considering residual stresses. In this work to quantify the influence of

residual stresses on failure pressure of flawed pressure vessels, the point stress

criterion with two parameter fracture criterion is used in AA2014-T6. Figure

9.6 shows the cracked configuration of specimen and cylindrical pressure

vessel. First a failure assessment diagram is drawn for tensile specimens

having through thickness cracks with the data given in Table 9.3 used in

Christopher et al (2002). Figure 9.7 shows the failure assessment diagram for

AA2014-T6 specimen. For cylindrical pressure vessels with axial surface

crack (see Figure 9.6) the value of is calculated using Equations (9.3 to

9.12) from the stress intensity factor expressions (Newman 1985).

= ( ) / (9.3)

= (9.4)

= (9.5)

(9.6)

= 1+ 1.464 .

for a c, (9.7)

= 1 + 1.464.

for a > c,

= + ( - ) ( ) , (9.8)

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= 2 + 8( ) (9.9)

= (1 + 0.52 + 1.29 - 0.074 ) for 0 10 (9.10)

= , q = 2+ 8 (9.11)

= ln 1 + (9.12)

where = crack depth, c = half the crack length, , : outer and inner

diameter of cylindrical vessel, = stress intensity factor at failure, =

failure pressure of unflawed cylindrical vessel, = failure pressure of

flawed pressure vessel, = internal pressure, = inner radius of cylinder,

= thickness of cylinder, = failure stress, = nominal stress required to

produce a fully plastic region on the net section, = ultimate tensile

strength and = yield strength or 0.2 % proof stress. Table 9.4 shows the

failure pressure of pressure vessels with different crack sizes without

considering residual stresses. Figure 9.8 shows the failure assessment diagram

for AA2014-T6 center crack tension specimen in which design points of

specimen and cylindrical pressure vessel were plotted. Figure 9.9 shows the

FAD for AA2014-T6 center crack tensile specimen and design points for four

cylindrical pressure vessels of dimensions and crack sizes given in Table 9.4.

The residual stress values are calculated through SINTAP procedure used in

Chapter-8 are:

= = 3 , = 12 ,

= = 560 MPa, = 680 MPa,

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Welding speed = 6 mm/sec and plate thickness =1.52 mm

R = 71.1 mm, a = t = 1.52 mm, = 554 Mpa, = 687 MPa

9.3.1 Results and Discussion

In SINTAP procedure the welding residual stresses values are given

as trapezoidal profile, in which the residual stress values are available up to

yield zone only ( ). Beyond yield zone the residual stress values are taken as

zero. Hence from Table 9.4 the crack sizes less than yield zone is considered

for analysis. Figure 9.9 also shows that three of the four pressure vessels are

critical with the available residual stresses. Therefore it is necessary for the

pressure vessel to undergo stress relieving process. Figure 9.10 shows the

failure assessment diagram for AA2014-T6 cylindrical pressure vessel with

70% residual stress relieved. Figure 9.10 and Table 9.5 show that the point D

is unsafe with the existing residual stress and an applied internal pressure of

3.25 MPa. When the applied internal pressure is increased to 3.75 MPa the

point C also becomes unsafe. When the applied internal pressure is increased

to 5 MPa the point B also becomes unsafe. When the applied internal pressure

is increased to 7.5 MPa the point A also becomes unsafe. This analysis brings

out the reduced internal pressure withstanding capacity in the presence of

residual stress as indicated in Table 9.6.

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Table 9.3 Comparison of analytical and test results of center crack

tension specimens made of AA2014-T6 ( = MPa,

= MPa, KF = 44.38 MPa m, m = 0.3022)

Specimen dimensions (mm) Fracture strength,

(MPa) Relative

error (%) Width, W Thickness, t

Crack

length, 2c

Test

(MCIC 1975) Analysis

76.2 1.56 7.04 408.9 410.97 -0.51

76.2 1.58 7.06 405.4 410.62 -1.29

76.2 1.57 14.15 320.6 320.07 0.16

76.2 1.52 19.68 288.9 277.28 4.02

76.2 1.56 21.49 272.4 265.75 2.44

76.2 1.55 26.21 235.8 239.44 -1.54

76.2 1.53 30.15 222.0 220.31 0.76

76.2 1.53 30.58 217.2 218.37 -0.54

76.2 1.54 35.53 195.8 196.94 -0.58

76.2 1.55 36.37 189.6 193.51 -2.06

Standard error (SE) = 0.018

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Table 9.4 Comparison of analytical and test results of cylindrical

pressure vessels made of AA2014-T6 ( = MPa,

= , = . , = . )

Points in

FAD

Crack

length,

2c(mm)

Failure

pressure,

(MPa)-Test

(MCIC 1975)

,

( )

A 2.64 12.15 38.14 0.83

B 6.35 9.37 49.45 0.64

C 12.7 5.85 51.58 0.40

D 19.05 4.73 60.53 0.32

E 25.4 3.1 53.59 0.21

F 31.75 2.93 65.19 0.20

G 44.45 1.95 65.04 0.13

H 50.8 1.76 69.33 0.12

Table 9.5 Fracture parameters of AA2014-T6 pressure vessels with

70% residual stress relieved

Points

in

FAD

Residual

stress

For 30%

RS only

30% RS+

3.25 MPa

30% RS+

3.75 MPa

30% RS+

5 MPa

30% RS+

7.5 MPa

100% 30%

A 560 168 0.25 11.3 0.45 20.7 0.48 22.3 0.55 25.4 0.72 33.3

B 540 162 0.24 18.3 0.44 34.1 0.47 36.8 0.54 42.0 0.71 55.2

C 340 102 0.15 19.2 0.35 45.7 0.39 50.1 0.45 58.9 0.63 80.9

D 160 48 0.07 13.1 0.27 51.5 0.31 57.9 0.38 70.7 0.55 102.7

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Table 9.6 Failure pressure of AA2014-T6 pressure vessels with and

without residual stress

Points

in FAD

c

(mm)

(MPa)

without RS

(MPa)

with 30% RS

A 1.32 12.15 7.50

B 3.18 9.37 5.00

C 6.35 5.85 3.75

D 9.53 4.73 3.25

Figure 9.6 Cracked configurations

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Figure 9.7 Failure assessment diagram for AA2014-T6 center crack

tension specimen

Figure 9.8 Failure assessment diagram for AA2014-T6 center crack

tension specimen and cylindrical pressure vessel

165

Figure 9.9 Failure assessment diagram for AA2014-T6 showing

residual stress levels without any internal pressure

Figure 9.10 Failure assessment diagram for AA2014-T6 cylindrical

pressure vessel with 70% residual stress relieved