chapter 4 failure prediction of unflawed...

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CHAPTER 4

FAILURE PREDICTION OF UNFLAWED HIGH STRENGH

STEEL CYLINRICAL PRESSURE VESSELS USING

PROBABILISTIC APPROACH

Cylindrical pressure vessels are used in various technological fields

such as chemical and nuclear industries, rocket motor case manufacturing and

production of many weapon systems. The accurate prediction of bursting

pressure or failure pressure of a cylindrical pressure vessel is very important

in the design and operation of pressure vessels. The design of pressure vessel

under internal pressure requires the study of two modes of failure (Williams et

al 1969; James 1973; Nageswara rao 1992). The first occurs when the

deformation becomes excessive, and there is a possibility of permanent

deformation. The second occurs at a higher pressure and takes the form of

bursting the motor case. Christopher et al (2002 b) made a comparative study

on various failure pressure equations (thirteen methods) of unflawed

cylindrical vessels using literature experimental failure data and concluded

that there is no unique failure theory to predict the failure pressure accurately

for all materials. In choosing design formulae, consideration must be given to

simplicity, availability of data on strength properties and accuracy. This is due

to unavoidable variations in strength properties. Hence, a new method which

duly considers uncertainty in strength properties and geometric parameters are

highly required for accurate prediction of failure of structures.

54

Probabilistic Failure Analysis is becoming increasingly popular for

realistic evaluation of response and reliability of structures. In this study, the

proposed Probabilistic Failure Assessment (PFA) methodology is applied to

predict the safe operating pressure of High Strength Steel cylindrical pressure

vessel for the specified reliability. The uncertainties with respect to strength

properties viz. ultimate tensile strength, yield strength and geometric

parameter thickness are considered. In addition, probabilistic finite element

analysis for the prediction of failure pressure is presented with the use of

finite element tool ANSYS PDS. MCS method is used to perform the

probabilistic failure analysis.

4.1 GOODNESS OF FIT TEST FOR HSS CYLINDRICAL

PRESSURE VESSEL

The uncertainties exist in the input variables are quantified by

performing goodness of fit test. The steps involved in the goodness of fit test

are explained as follows.

4.1.1 Identification of Random Input Variables

In this study, the variations in ultimate strength ( ult), yield strength

ys) and thickness (t) are considered for the prediction of failure pressure of

HSS cylindrical pressure vessels. The literature experimental test failure data

(Benjamin 1965; Margetson 1978) shown in Tables 4.1(a-c) are used to

quantify the uncertainty in strength properties and geometric parameter of the

pressure vessels.

55

Table 4.1(a) Literature Experimental Failure Data of High Strength

D6AC Steel (n=0.05) CPV (Benjamin 1965)

Yieldstrength

ys) MPa

Ultimatestrength

ult) MPa

Innerdiameter(Di) mm

Thickness(t) mm

Literaturetest failure

pressure (pm),MPa

1455 1607 368.3 3.6 35

1420 1551 368.3 3.6 35

1407 1531 368.3 3.6 34

1538 1586 368.3 3.6 35

1448 1620 368.3 3.6 34

1489 1662 368.3 3.6 35

1489 1662 368.3 3.6 35

1538 1613 368.3 3.6 34

1510 1586 368.3 3.6 33

Table 4.1(a) represents the variation in ultimate strength ( ult) and

yield strength ( ys) for D6AC steel CPV sample set. In this study D6AC steel

CPV, inner diameter (Di) and thickness (t) of the vessel are existing as

deterministic. Similarly, Table 4.1(b) and Table 4.1(c) represent the variation

in Air steel X200 and M250 grade Maraging Steel CPV sample set

respectively.

56

Table 4.1(b) Literature Experimental Failure Data of (n=0.0327) High

Strength Air Steel X200 CPV (Benjamin 1965)

Yieldstrength

ys) MPa

Ultimatestrength

ult) MPa


Thickness(t) mm


pressure (pm),MPa

2089 2162 255 1.4 27

2089 2116 255 1.5 27

1884 1884 255 1.7 28

1637 1637 255 1.6 23

1946 1946 255 1.7 26

1961 1961 255 1.6 27

Table 4.1(c) Literature Experimental Failure Data of (n=0.0327) High

Strength M250 Maraging Steel CPV (Margetson 1978)

Yieldstrength

ys) MPa

Ultimatestrength

ult) MPa


Thickness(t) mm


pressure (pm),MPa

2128 2155 90 1.630 86.6

2128 2155 90 1.756 94.5

2128 2155 90 1.793 94.0

2128 2155 90 1.763 94.0

2128 2155 90 1.735 92.3

57

4.1.2 Identification of Suitable Distribution

The first step in the proposed methodology is to find the suitability

of best fit of each random variable. The following four statistical distributions

like Normal, Log-normal, Weibull and Gamma are considered and tested to

quantify the uncertainty in input variables. AD and P-Value are evaluated

parameters are evaluated and their suitability to model the uncertainties is

identified by conducting AD test using the MINITAB statistical software. In

this study, AD test is performed at 5% significance level as detailed in

Section 3. The AD test results for D6AC steel are presented in Table 4.2.

Table 4.2 AD Statistic and P-Value for D6AC Steel CPV

Properties Statistic NormalLog

NormalWeibull Gamma

AD 0.242 0.243 0.552 0.283Yield strength

ys) MPaP-Value 0.681 0.678 > 0.250 > 0.250

AD 0.236 0.236 0.343 0.262Ultimatestrength

ult) MPa P-Value 0.704 0.704 > .250 >0.250closest fit are shown in bold

The probability plot for yield strength ( ys) and ultimate strength

ult) of D6AC steel CPV are analysed for four distributions viz. Normal, Log-

Normal, Weibull and Gamma are shown in Figures 4.1 and Figure 4.2

respectively. The distribution is in closest fit, which is defined by the smallest

observed AD and higher P-Value statistic is chosen as best fit for each

random variable.

58

1600150014001300

99

90

50

10

1

yield strength1600150014001300

99

90

50

10

1

yield strength

1600150014001300

90

50

10

1

yield strength1600150014001300

99

90

50

10

1

yield strength

WeibullAD = 0.312P-Value > 0.250

GammaAD = 0.283P-Value > 0.250

Goodness of Fit Test

NormalAD = 0.242P-Value = 0.681

LognormalAD = 0.243P-Value = 0.678

Probability Plot for yield strengthNormal - 95% CI Lognormal - 95% CI

Weibull - 95% CI Gamma - 95% CI

Figure 4.1 Probability plot of yield strength in AD test for D6AC steel

pressure vessels

Figure 4.2 Probability Plot Of Ultimate Strength in AD test for D6AC

Steel Pressure Vessels

From Table 4.2, Figures 4.1 and 4.2, normal distribution is selected

as best fit since it has smallest AD and higher P-Value for yield strength and

59

ultimate strength which is shown bold in Table 4.2 against each random

variable.

Table 4.3 AD Goodness of Fit Test Results for Different CPV

Best fitting distribution of random input variables

CPV Yieldstrength

ys) MPa

Ultimatestrength

ult) MPa


Thickness(t) mm

D6AC Steel N(1477.1,3.24) N(1602,2.78) 368.3 3.6Air steel X200 N(1934.3,8.25) N(1951,9.19) 255 N(1.58,7.22)Maraging steel 2128 2155 90 N(1.73,3.60)

(Values within parenthesis represents Mean and COV)

4.1.3 Computation of Statistical Parameters of Random Input

Variables

AD goodness of fit test results for different pressure vessels are

shown in Table 4.3. From the test results shown in Table 4.3 it is clear that

thickness (t) follows Normal distribution for Air steel X200 and Maraging

steel and D6AC steel fixed as deterministic. Ultimate strength ( ult) follows

Normal distribution for D6AC steel and Air steel X200 and kept deterministic

for Maraging steel. Yield strength ( ys) follows Normal distribution for both

D6ACsteel and Air steel X200 and kept deterministic for Maraging steel. The

statistical parameters viz. mean and Coefficient of Variation (COV) of

random input variables are shown within parenthesis in Table 4.3.

4.2 PROBABILISTIC FAILURE ANALYSIS

In this study thirteen failure prediction equations are considered to

perform the probabilistic failure analysis. The uncertainties present in the

input variables are modeled in the failure prediction equations by probabilistic

analysis. The steps involved in the second phase of proposed methodology

60

viz. Defining failure criterion, Prediction of failure pressure and Computation

of statistical parameters of failure pressure are explained below.

4.2.1 Defining Failure Prediction Equations

The various methods used to estimate failure pressure of cylindrical

pressure vessel (Christopher et al 2002a) are discussed below. Faupel’s

(1956) expression for failure pressure prediction is given as

ysm ys

ult

2p 2 ln3

(4.1)

where = Ro/Ri, Ri is the inner radius in mm, Ro the outer radius in mm, ys

is yield strength in MPa, and ult is ultimate strength in MPa,

Based on the pressure test results of several steel cylinders,

Wellinger and Uebing (1960) proposed the formula is shown in

Equation (4.2)

um

u

1.1p ln1 2.4

(4.2)

where the true stress u and the true strain u at the ultimate load are,

u ult ult(1 )

u ultln(1 )

ult is the nominal strain at the ultimate load.

61

Section VIII of the ASME Boiler code (1962) gives the formula for

allowable maximum pressure for unfired pressure vessels is shown in

Equation (4.3)

m ult1p

0.6 0.4 For 1.5

2

ult 2

11

For 1.5 (4.3)

Soderberg (1941) proposed the formula that is shown in

Equation (4.4) based upon assumption of uniform stress distribution

throughout the wall by considering the average stress value and failure as a

function of the significant stress or octahedral shear stress.

m ult4 1p

13 (4.4)

Failure pressure prediction based on Maximum stress criterion is

presented in Equation (4.5)

m ultp ( 1) (4.5)

Failure pressure prediction based on Maximum shear stress criterion

is shown in Equation (4.6)

m ult1p 21

(4.6)

The Equations (4.4), (4.5) and (4.6) can be considered as

approximate estimate due to the assumption of uniform stress distribution

62

throughout the wall. In order to provide the actual stress distribution, the

plastic range must be considered.

Turner (1910) has developed a formula, considering the material to

be perfectly plastic with no strain-hardening. Assuming small strains and

maximum shear stress theory of failure, the maximum pressure is expressed in

Equation (4.7)

m ultp ln (4.7)

Assuming perfectly plastic material with octahedral shear stress in

place of the maximum shear stress, Nadai (1931) obtained the relation shown

in Equation (4.8) for failure pressure prediction

m ult2p ln3

(4.8)

Using the torsional stress-strain n-power relation ( no ), where

o and n are material constants) in place of the tensile stress-strain relation,

as suggested by Nadai, Bailey (1930) determined a theoretical equation for

the maximum pressure, based upon the maximum shear stress plasticity

relation as shown in Equation (4.9)

ultm 2n

1p 12n

(4.9)

Equation (4.9) has been referred to as Bailey-Nadai equation.

Based upon the tensile stress-strain relation, Nadai (1950) obtained

the maximum pressure relation shown in Equation (4.10).

63

ultm 2n

1p 13n

(4.10)

Marin and Rimrott (1958) developed an approximate expression

for maximum pressure as shown in Equation (4.11).

ultm

ult

2p ln(1 )3

(4.11)

Svensson (1958) independently derived a simplified equation for

fully plastic yielding of a rigid plastic material by applying a factor to

Turner’s equation for the effect of strain hardening. The pressure expansion

relation suggested by Svensson is in the form of series, while Marin and

Rimrott found that because of rapid convergence of the series, the first two

terms give sufficiently accurate results. The simplified formula proposed by

Svensson is shown in Equation (4.12)

u

m ultu u

0.25 ep ln0.227

(4.12)

The pressure expansion relation for an isotropic cylindrical vessel

under internal pressure is expressed as shown in Equation (4.13).

o

i

n

i o 3p d

1 e (4.13)

where n0 , is stress, 0 and n are Material constant in stress-strain

equation. The effective strains at the inner surface i(r R ) and at the outer

surface 0(r R ) are shown in Equations (4.14) and (4.15)

64

i i2 ln x3

(4.14)

i3

0 2

1 ln 13

1-e (4.15)

At the point of instability, pm is maximum as presented in

Equation (4.16)

i

ii

dp 0*d

(4.16)

After simplification, one gets a non-linear equation

i o

n3 o

i

e 1 0 (4.17)

Since, Equation (4.17) is a non-linear equation in terms of i, it is

solved by using Newton-Raphson method. Substituting the values of strains in

Equation (4.13) and integrating, gives the maximum pressure or failure

pressure. A Ten point gauss rule method is adopted for evaluating the integral

in Equation (4.13).

4.2.2 Prediction of Failure Pressure

The uncertainty in ultimate strength ( ult), yield strength ( ys) and

thickness (t) are quantified and are incorporated in the failure prediction

equation. Diameter of the pressure vessel is existed as deterministic. Due to

the uncertainties in strength properties and geometric parameters, the failure

pressure prediction through Equations (4.1-4.13) mentioned is also uncertain.

From Table 4.3, the mean, COV and probability distributions of all random

65

input variables for D6AC Steel, Air steel X200 and Maraging steel pressure

vessels are used to perform the analysis.

4.2.2.1 Random number generation

In this work, MCS method is used to generate random variate using

distribution with the specified mean and standard deviation for each random

input variable. Box and Muller (Law and Kelton 2004) method is used to

generate normal random variates for simulation study. As the number of

sample increases the prediction is accurate. The detailed procedure is

explained below. With the use of statistical parameters of random input

variables in Table 4.3, probabilistic failure analysis is performed as explained

in Section 3.3 using various failure pressure prediction equations. 1000

samples of each random variable are generated as per the following

procedure.

Uniform random variates are randomly generated 1, 2 for each

variable (t, ys, and ult).

Then normal random variates are generated using the following

equations

12

1 21

1u 2ln cos 2

12

1 21

1u 2ln sin 2

Normal random variable, Y u * (4.18)

Log-Normal random variable, YX e (4.19)

66

where , are the mean and standard deviation of random

variable. The generated random input variables are used to execute the

deterministic failure analysis for thirteen failure prediction equations defined

in Section 4.2.1. The corresponding failure pressure for D6AC steel, Air steel

X200 and Maraging steel pressure vessels are calculated and studied. The

sample simulated data of Air steel X200 CPV are mentioned in Table 4.4.

Table 4.4 Simulated Sample Data of Air Steel X200 CPV using MCS

Method

Thickness,t (mm)

Yield strength, ys(MPa)

1.0e+003

Ultimate strength, ult(MPa)

1.0e+003

Predicted failurepressure,pm (MPa)

1.62211.48781.56991.73941.72851.37221.53001.45951.44651.47211.75681.49581.44831.80771.33661.59281.60791.57851.59101.61511.73901.52881.65331.55371.65661.60201.49611.66621.57341.51991.4905

2.00251.83942.06022.15212.06691.75771.74111.77312.09231.76902.00651.85731.96402.01321.72251.95792.11502.14831.91742.16971.93982.02281.76301.78012.08031.86691.79331.66942.21671.89011.8197

1.91291.96481.95661.90841.91682.05951.96322.26622.24651.64731.80871.84911.99681.90092.18941.81841.75241.80181.97411.92822.15961.53881.81661.75571.66672.07211.97691.82271.83882.10492.0272

29.174424.527029.040233.240331.969021.085023.592321.500827.107223.338231.398125.013425.608432.625719.219227.925129.705629.729727.431131.146929.954126.246326.205124.891729.784226.591623.902924.816230.482125.528824.1029

67

Table 4.4 (Continued)

Thickness(t), mm

Yield strengthys), MPa

1.0e+003 *

Ultimate strengthult), MPa

1.0e+003 *

Predicted Failurepressure (pm),

MPa

1.57601.54501.69361.57681.63821.44841.35081.48461.66961.79031.79151.58661.61291.83881.39701.47191.39691.51111.6872

1.83312.10791.59571.78742.18412.28171.72382.13371.96201.83721.97891.63291.86472.02561.70792.09011.74932.09322.1626

1.85481.89281.86391.91532.38341.58832.05641.91141.89041.71161.90441.82802.07961.83251.54161.96171.86361.97671.8473

25.999229.011523.627225.237931.929827.012420.195728.213629.435929.443731.838022.984726.708433.186821.287427.598021.916428.390032.1306

Simulation is done for 1000 samples, here a sample data of 50 values are

shown

4.2.3 Computation of Statistical Parameters of Failure Pressure

The statistical parameters of predicted probabilistic failure pressure

are computed for three different pressure vessels viz. mean, standard

deviation and COV and are summarized in Table 4.5 (a-c).

68

Table 4.5(a) Statistical Parameters of Predicted Probabilistic Failure

Pressure of D6AC steel CPV

Predicted failure pressure, pm

(MPa)Formulae

MeanStandarddeviation

COV

Faupel Equation (4.1)

Wellinger & Uebing

Equation (4.2)

ASME Boiler Code

Equation (4.3)

Soderberg Equation (4.4)

Maximum stress Equation (4.5)

Max. shear stress Equation (4.6)

Turner Equation (4.7)

Nadai Equation (4.8)

Baily-Nadai Equation (4.9)


Marin & Rimrott Equation (4.11)

Svensson Equation (4.12)


35.559

33.143

30.968

35.873

31.363

30.989

31.028

35.772

31.055

35.811

35.044

34.205

34.910

0.880

0.943

0.864

0.975

0.867

0.882

0.866

1.021

0.844

1.003

0.992

0.942

0.991

2.47

2.84

2.79

2.71

2.76

2.84

2.79

2.85

2.71

2.80

2.83

2.75

2.84

(Literature Test Failure Pressure, Mean=34.44 MPa; COV= 11%)

69

Table 4.5(b) Statistical Parameters of Predicted Probabilistic Failure

Pressure of Air Steel X200 CPV

Predicted Failure Pressure, pm

(MPa)Formulae

MeanStandarddeviation

COV


Wellinger & Uebing Equation (4.2)

ASME Boiler Code Equation (4.3)











27.368

25.722

23.981

27.761

24.337

24.083

24.052

27.766

24.266

27.612

27.120

26.535

27.021

3.281

3.068

2.788

3.293

2.854

2.812

2.809

3.185

2.812

3.319

3.240

3.040

3.108

11.98

11.92

11.62

11.86

11.73

11.67

11.67

11.47

11.59

12.02

11.94

11.45

11.50

(Literature Test Failure Pressure, Mean=26.33 MPa; COV=6.65 %)

70

Table 4.5(c) Statistical Parameters of Predicted Probabilistic Failure

Pressure of Maraging Steel Cylindrical Pressure Vessel

Predicted Failure Pressure, pm

(MPa)Formulae

Mean(µ)

Standarddeviation ( )

COV


Wellinger & Uebing Equation (4.2)

ASME Boiler Code Equation (4.3)











93.942

86.841

80.896

93.843

82.771

81.332

81.269

93.937

81.261

93.612

91.939

89.791

91.086

3.229

3.035

3.008

3.308

2.924

2.951

2.828

3.345

2.752

3.439

3.410

3.179

3.314

3.43

3.49

3.71

3.52

3.53

3.62

3.48

3.56

3.38

3.67

3.70

3.54

3.63

(Literature Test Failure Pressure, Mean=92.28MPa; COV=3.56 %)

71

4.3 COMPUTATION OF SAFETY FACTOR AND RELIABILITY

Due to incomplete information, uncertainty always exists in the

system, which leads to uncertainty in the performance prediction. A

probabilistic approach provides a rational and consistent framework for

treating uncertainties and plausible reasoning. The reliability of a system has

been the center of concern when engineering systems are analyzed in the

presence of uncertainties.

4.3.1 Identification of Suitable Method

In the present study, Stress-Strength interference theory (Kapur and

Lamberson 1977) is used to find a suitable failure pressure prediction

equation for identified three CPV. The standard normal variate (z0) is

computed using Equation (3.23) for all the methods by considering the

literature experimental failure pressure as strength (S) variable and the

predicted failure pressures as stress (L) variable. If the S and L are

independent, then the interference area between the probability density

functions of S and L gives a measure of the probability of failure ( Rao 1992).

The mean and standard deviation of strength and stress random variables are

utilised from Tables 4.5(a-c). The computed standard normal variate (z0) for

the three CPV materials using thirteen failure pressure equations is tabulated

in Table 4.6 (a-c). It is observed from Tables 4.6 (a-c) that the ASME Boiler

code Equation (4.3) yields the highest standard normal variate (z0) and is

identified as the more suitable failure pressure equation for all the three CPV

considered for study among the analysed thirteen failure pressure prediction

equations and are shown in bold in Tables 4.6(a-c).

72

Table 4.6(a) Standard Normal Variate of D6AC Steel CPV for Various

Failure Pressure Prediction Equations using Stress-Strength

Interference Theory

Statistical parameters of failure pressure

Literature experimental(S) MPa

Predicted (L) MPaFailure pressureequations

Mean ( S)Standarddeviation

S)Mean ( L)

Standarddeviation

L)

Standardnormal

variate (z0)

Faupel Equation(4.1)

34.44 0.726 35.559 0.880 -0.98

Wellinger & UebingEquation (4.2)

34.44 0.726 33.143 0.943 1.05

ASME Boiler CodeEquation (4.3)

34.44 0.726 30.968 0.864 3.07

Soderberg Equation(4.4)

34.44 0.726 35.873 0.975 -1.18

Maximum stressEquation (4.5)

34.44 0.726 31.363 0.867 2.72

Max. shear stressEquation (4.6)

34.44 0.726 30.989 0.882 3.02

Turner Equation(4.7)

34.44 0.726 31.028 0.866 3.02

Nadai Equation (4.8) 34.44 0.726 35.772 1.021 -1.06

Baily-NadaiEquation (4.9)

34.44 0.726 31.055 0.844 3.04

Nadai Equation(4.10)

34.44 0.726 35.811 1.003 -1.11

Marin & RimrottEquation (4.11)

34.44 0.726 35.044 0.992 -0.50

Svensson Equation(4.12)

34.44 0.726 34.205 0.942 0.20


34.44 0.726 34.910 0.991 -0.33

73

Table 4.6(b) Standard Normal Variate of Air Steel CPV for Various

Failure Pressure Prediction Equations using Stress-Strength

Interference Theory


Literatureexperimental

(S) MPaPredicted (L) MPaFailure pressure

equations


S)Mean ( L)

Standarddeviation

L)

Standardnormalvariate

(z0)


26.33 1.751 27.368 3.281 -0.278

Wellinger &Uebing Equation

26.33 1.751 25.722 3.068 0.173

ASME BoilerCode Equation(4.3)

26.33 1.751 23.981 2.788 0.714

SoderbergEquation (4.4)

26.33 1.751 27.761 3.293 -0.383


26.33 1.751 24.337 2.854 0.596


26.33 1.751 24.083 2.812 0.679


26.33 1.751 24.052 2.809 0.689

Nadai Equation(4.8)

26.33 1.751 27.766 3.185 -0.394


26.33 1.751 24.266 2.812 0.624


26.33 1.751 27.612 3.319 -0.341


26.33 1.751 27.120 3.240 -0.214

SvenssonEquation (4.12)

26.33 1.751 26.535 3.040 -0.058

SvenssonEquation (4.13)

26.33 1.751 27.021 3.108 -0.193

74

Table 4.6(c) Standard Normal Variate of Maraging Steel CPV forVarious Failure Pressure Prediction Equations usingStress-Strength Interference Theory


Literatureexperimental

(S) MPaPredicted (L) MPaFailure pressure

equations


S)Mean ( L)

Standarddeviation

L)

Standardnormalvariate

(z0)


92.28 3.28 93.942 3.229 -0.36

Wellinger &Uebing Equation(4.2)

92.28 3.28 86.841 3.035 1.22

ASME BoilerCode Equation(4.3)

92.28 3.28 80.896 3.008 2.56

SoderbergEquation (4.4)

92.28 3.28 93.843 3.308 -0.34


92.28 3.28 82.771 2.924 2.16


92.28 3.28 81.332 2.951 2.48


92.28 3.28 81.269 2.828 2.53

Nadai Equation(4.8)

92.28 3.28 93.937 3.345 -0.35


92.28 3.28 81.261 2.752 2.54


92.28 3.28 93.612 3.439 -0.28


92.28 3.28 91.939 3.410 0.07


92.28 3.28 89.791 3.179 0.54


92.28 3.28 91.086 3.314 0.26

Stress-strength interference graph is obtained by drawingdistribution plot for predicted and literature test failure pressures. In order toassure that the maximum standard normal variate yields higher reliability, thestress-strength interference graph for the equation having maximum andminimum z0 value for D6AC steel has been drawn and is shown in Figure 4.3(a-b).

75

37363534333231302928

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Failure pressure, MPa

34.44 0.72630.968 0.864

Mean StDev

NormalDistribution Plot - D6AC steel pressure vessel for z = 3.07

Figure 4.3(a) Interference graph For Maximum Standard Normal

Variate z0=3.07 for D6AC Steel Vessels

37363534333231

0.6

0.5

0.4

0.3

0.2

0.1

0.0


34.44 0.72634.205 0.942

Mean StDev

Norm alDistribution Plot - D6AC steel for minimum z = 0.2

Figure 4.3(b) Interference Graph for Minimum Standard Normal

Variate z0=0.2 for D6AC Steel Vessels

Test failurepressure

Predicted failurepressure

Test failure

pressure

Predicted failure

pressure

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It is clear from the Figures 4.3(a) and (b) that the method having

highest z0 has less interference area indicating a minimum probability of

failure and the method having lowest z0 has more interference area indicating

a maximum probability of failure. The method which is having less

interference area and the predicted failure pressure values falling well below

the experimental failure pressure value is identified as a suitable method.

Table 4.7 shows the predicted probabilistic failure pressure for the three CPV

considered for study, in that ASME boiler code equation is identified as

suitable failure prediction equation.

Table 4.7 Recommended Failure Pressure Prediction Equations

Predicted probabilistic failurepressure, Pm (MPa)Cylindrical

pressure vesselsMean (µL)

Standarddeviation ( L)

Suitablemethod

D6AC steel 30.968 0.864Air steel X200 23.980 2.788Maraging steel 80.896 3.008

ASME BoilerCode Equation

4.3.2 Determination of Safety Factor

To predict the operating pressure of pressure vessel for the specified

reliability, a safety factor is determined from the probabilistic analysis results

obtained in the previous section. Using the Equation (3.25), the safety factor

is computed for the specified reliability levels which are presented in

Table 4.8.

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Table 4.8 Predicted Safety Factor for the Specified Reliability

Reliability(%)

Standardnormal

variate (z0)D6AC steel Air steel

Maragingsteel

90 1.281 1.046 1.180 1.068

95 1.645 1.060 1.234 1.088

99 2.326 1.084 1.341 1.128

99.99 3.719 1.137 1.584 1.212

4.3.3 Determination of Operating Pressure

Since the reliability based safety factor considers the uncertainty in

the design variables in terms of COV, the operating pressure (po) of the

pressure vessel for the specified reliability is determined using equation given

below from the computed safety factor. Table 4.9 presents the operating

pressure of vessels for the specified reliability levels.

Operating pressure Lop

N(4.20)

Where L is mean of predicted probabilistic failure pressure (MPa) and N is

safety factor for the specified reliability

Table 4.9 Predicted Operating Pressure for the Specified Reliability

Operating pressure, po (MPa)Reliability (%)Motor case

90 95 99 99.99D6AC steel 29.61 29.22 28.52 27.24Air steel 20.32 19.43 17.88 15.14Maragingsteel

75.72 74.35 71.72 66.77

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4.4 PROBABILISTIC FAILURE PRESSURE PREDICTION

USING FINITE ELEMENT ANALYSIS

The failure pressure prediction is verified by probabilistic finite

element analysis using ANSYS PDS. In this study, an attempt is made to

predict the failure pressure probabilistically using ANSYS PDS for the three

sample sets of pressure vessel. ANSYS PDS executes or "loops through"

multiple times the deterministic analysis file during the probabilistic analysis.

The following steps are executed to perform probabilistic finite element

analysis.

4.4.1 Deterministic Finite Element Analysis

Nonlinear finite element analysis is carried out for the pressure

vessels mentioned in Tables 4.1(a-c). The material curve is modeled using

inverse Ramberg Osgood relation shown in Equation (4.21).1

n n

1 (4.21)

where is stress,(MPa), E is young’s modulus of material (MPa),

is engineering strain, ulto

; ult is ultimate strength (MPa) of the material

and n is parameter defining the shape of the stress-strain-hardening relations.

Table 4.10 shows the young’s modulus value and poisons ratio for the three

CPV considered for study to perform the non-linear finite element analysis.

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Table 4.10 Young’s Modulus Value and Poissons Ratio of CPVs

Materials Young’sModulus, E (MPa) Poisson’s ratio

D6AC Steel 204000 0.3

Air Steel 204000 0.3

Maraging Steel 186300 0.3

The cylindrical vessel is modeled using an axi-symmetric four node

quadrilateral finite element (Element type: 2D PLANE 42) available in

ANSYS software. The Model geometry and the finite element mesh are as

shown in Figure 4.4. The axial displacement is suppressed at both ends of the

vessel to have no-axial growth under internal pressure.

Figure 4.4 A Typical Axi-Symmetric Finite Element Model of A

Cylindrical Vessel

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The failure pressure is predicted deterministically(Aseer Brabin et al

2009) with the inbuilt facility in ANSYS for checking the global plastic

deformation (GPD). It indicates the pressure level to cause complete plastic

flow through the cylinder wall i.e., bursting pressure.

4.4.2 Probabilistic Finite Element Analysis

The thickness (t) is defined as random input parameter and substep

number in non-linear analysis is defined as random output parameter. The

statistical parameters and type of distribution of thickness for Air steel CPV

are obtained from Table 4.3. The uncertainties in yield strength and ultimate

strength are modeled through material curves generated using inverse

Ramberg-Osgood relation (4.21). Totally, five different material curves are

generated corresponding to each set of material property. The material curves

generated for the Air steel CPV corresponding to each set of strength

properties are shown in the Figure 4.5. From the material curves drawn, the

upper, lower and the middle curves are used in the analysis to model the

uncertainties in the strength properties. The probabilistic finite element

analysis is performed using MCS technique. In MCS method, Latin hyper

cube sampling is used. To perform the PFEA 1000 samples are generated.

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Figure 4.5 Material Curves for Air Steel CPV

The cumulative distribution function of thickness variable is shown

in Figure 4.6. Table 4.11 shows the Statistical parameters of thickness for

Maraging steel CPV.

Figure 4.6 CDF for Maraging Steel CPV

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Table 4.11 Statistics of Thickness for Maraging Steel CPV

Name MeanStandarddeviation

minimum maximum

Thickness(t) mm

1.735 6.2506E-02

1.569 1.913

The statistical characteristic of the random output parameter sub-

step number is obtained from the PFEA results. Then the failure pressure of

the CPV is computed from the results of PFEA and its statistics is

summarized in Table 4.12. Figure 4.7 shows the cumulative distribution

function of probabilistic failure pressure.

105100959085

100

80

60

40

20

0


Mean 93.43Standard deviation 3.416

Normal Cumulative Distribution Function of failure pressure

Figure 4.7 CDF of Failure Pressure for Maraging Steel CPV

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The statistical parameters of predicted probabilistic failure

pressure are computed using ANSYS PDS for three CPVs and summarized in

Table 4.12.

Table 4.12 Predicted Probabilistic Failure Pressure using PFEA

Failure Pressure, pm (MPa)Cylindricalvessel Mean (µ) Standard

deviation ( )COV

D6AC Steel 33.33 2.31 6.93

Air Steel 26.76 3.65 13.65

Maraging Steel 93.43 3.416 3.65

Using the stress-strength interference theory, safety factor was

estimated by considering test failure pressure as strength variable (S) and the

predicted probabilistic failure pressure (using ANSYS PDS) as stress

variables (L) using Equation (3.25) for the specified reliability. The predicted

safety factor for all CPVs is summarized. The cell value represents the safety

factor (N) shown in Table 4.13.

Table 4.13 Predicted Safety Factor using PFEA

Reliability (%)CPV

90 95 99 99.99

D6AC Steel 1.094 1.120 1.171 1.277

Air Steel 1.096 1.126 1.188 1.336

Maraging Steel 1.068 1.088 1.126 1.210•Cell value represents the safety factor (N)

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The operating pressure of the CPVs was computed by substituting the safety

factor and the mean of the predicted probabilistic failure pressure using

ANSYS PDS in Equation (4.20). The cell values represent operating pressure

(p0) shown in Table 4.14.

Table 4.14 Predicted Operating Pressure using PFEA

Reliability (%)CPV

90 95 99 99.99

D6AC Steel 30.48 29.75 28.46 26.11

Air Steel 24.42 23.76 22.53 20.03

Maraging Steel 87.48 85.87 82.98 77.21*Cell value represents operating pressure (p0) in MPa

4.5 RESULTS AND DISCUSSION

In this study, probabilistic failure assessment methodology was

proposed to predict the operating pressure for the specified reliability. In this

methodology uncertainties in the strength and geometric parameters were

given due consideration. The results obtained are summarized as below:

(i) Probabilistic failure pressure for all the cylindrical pressure

vessels follows normal distribution and its statistical

measures are presented in Table 4.5 (a-c)

(ii) Thirteen failure pressure equations are considered. in which

ASME boiler code Equation was recommended as a suitable

failure pressure prediction equation, using stress-strength

interference theory for all the CPV under study.

(iii) Using the recommended failure pressure prediction equation,

a reliability-based safety factor was estimated. The

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computed reliability-based safety factor is for specified

reliability is more reliable than the deterministic approach.

(iv) It is clear from Figure 4.8, as the reliability increases, safety

factor also increases which leads to reasonable conservatism.

Figure 4.8 Relationships between Safety Factor and Reliability

(v) Figure 4.9 shows the relationship between the operating

pressure and the reliability. The relationship drawn helps the

design engineer to select an appropriate operating pressure

for the required reliability.

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Figure 4.9 Relationships between Operating Pressure and

Reliability

(vi) The PFEA was also carried out using the numerical

simulation in ANSYS PDS. The prediction using the

proposed PFA methodology was verified by PFEA. From

the probabilistic finite element results, the safety factor was

computed and was summarized in Table 4.13.

(vii) A comparative study was made between proposed

Probabilistic approach and probabilistic finite element

approach for three CPVs is shown in Figure 4.10. It is clear

from the graph that both approaches ensure the safe design

for the specified reliability when all the uncertainties are

properly quantified.

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Figure 4.10 Comparison of Probabilistic Approach and Probabilistic

Finite Element Approach

4.6 SUMMARY

Probabilistic design is a methodology aimed at producing robust and

reliable products. A common slip-up made by design engineers is that they do

not consider variations in the input variables that affect the reliability or

performance of the structure due to usage of nominal or mean of input

variables. When the distribution of material properties and geometric

parameters are known, it is possible to design a structure for a specified

reliability. In this present study, different failure pressure prediction equations

have been used to model the uncertainties in strength properties and

geometric parameter of the pressure vessel. The proposed research

methodology was successfully implemented in the prediction of failure of

pressure vessel with the use of literature experimental test data. Further, the

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probabilistic failure prediction methodology adopted for the pressure vessel

was verified through Probabilistic Finite element analysis using ANSYS PDS.

Both the analytical and finite element approaches ensure the safe

design of pressure vessel for the specified reliability when all the uncertainties

are exactly quantified. The choice of reliability level depends upon the

statistical scatter in the strength and geometric parameters of the pressure

vessels. The effect of variation in operating pressure on safety and reliability

was studied. Further, this methodology can be extended for different materials

used in various applications.

The safety assessment of structures without a fracture mechanics

analysis is insufficient, because, structural components generally contain

crack like defects, which are either inherent in the material or introduced

during a fabrication process. To quantify the effects of the presence of cracks

on material strength, a simple elastic-plastic fracture mechanics procedure for

the development of fracture criteria in the design of pressure vessels will be

described in the next chapter.

chapter 4 failure prediction of unflawed...

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