Open access peer-reviewed chapter - ONLINE FIRST

Probabilistic Modeling of Failure

By Alireda Aljaroudi

Submitted: March 12th 2018Reviewed: December 11th 2018Published: July 22nd 2019

DOI: 10.5772/intechopen.83461

Downloaded: 57

Abstract

Failure of a system or a component of a system is and has been a major concern to systems’ operators and owners. Failure could be traced back to different causes and may take different forms and shapes. It may result from software malfunction, hardware degraded performance, human error, sabotage, environmental as well as other external factors. There are various techniques found in the literature that can assist in the analysis of failure. These techniques comprise deterministic and probabilistic techniques. Deterministic techniques ignore the variability and uncertainties of the variables in the analysis which may lead to unsatisfactory and inaccurate results. While probabilistic techniques produce accurate and an all-inclusive result because they incorporate the variabilities and uncertainties in the analysis. The focus of this chapter is to present commonly used probabilistic failure analysis techniques and their mathematical derivations. Examples to enhance the understanding of the concept of failure analysis are also presented.

Keywords

  • failure analysis
  • probabilistic methods in engineering
  • reliability engineering

1. Introduction

Traditionally, failure analysis is conducted using deterministic techniques to assess the operability and integrity of industrial systems. These techniques lack the ability to report or predict the probabilistic nature of the systems’ behavior. Moreover, they ignore the probabilistic and random nature of the external factors that have direct impact on the performance of the systems. Implementing these techniques may produce inadequate assessment and eventually, may lead to wrong decisions concerning the integrity and reliability of the evaluated systems. To make an informed and reliable decision about the reliability and operability of such systems, probabilistic failure analysis should be adopted as an alternative analysis technique. This technique should be made as an integral part of the decision process as well as be a part of the overall organization’s risk control.

This chapter presents techniques that assist in the analysis of failure using engineering probabilistic methods. They include simulation as well as analytical methods. Simulation methods can be conducted using Monte Carlo simulation technique. Two different Monte Carlo simulation approaches are presented in this chapter. These are, the counting approach and sample statistics approach. The main drawback with simulation is that it takes great deal of time to perform and may require an extensive processing power. However, it is an essential step in the analysis to validate the results obtained by the analytical methods.

Some of the analytical methods include first order reliability method (FORM) and second order reliability method (SORM). FORM involves two approaches to calculate probability of failure, these are first order and second moment (FOSM) and advanced FOSM.

The focus of this chapter will be only on the FORM with the assumption that all random variables are uncorrelated. Analysis that require the use of FORM for correlated random variables is beyond the scope of this chapter. Likewise, analysis requiring the application of SORM to analyze limit state functions involving second order representation is beyond the scope of this chapter.

2. Failure modeling

Failure can be defined as the inability of industrial systems or subsystems either partially or totally to satisfy operational requirements as set forth by design specifications. Failure of a system could be partial or complete; in either case the consequences of failure may result in adverse consequences. Interruption of services, degraded performance, system shutdown, environmental damage and customers’ dissatisfaction are some of the consequences. Such consequences may lead to financial losses, liabilities and destroyed image of the operating company. As an example, if failure involves leak detection system to detect oil and gas leakage from subsea oil and gas pipelines, consequences could be so severe. Pollution of the ocean, damage to the fishery and tourism industries are some of the major consequences.

The system fails when the imposed demand or load on the system exceeds its capacity or resistance. The strength or the capacity of the system is a design parameter that specifies the maximum load the system can endure or the maximum demand the system can satisfy. The variabilities of the system’s capacity to satisfy the demand or load imposed on it are mainly attributed to the inherent uncertainties of the operation characteristics of the system’s components as well as external environmental factors. Therefore, the capacity of the system is assumed to be probabilistic in nature that varies from time to time due to the reasons mentioned above. Likewise, using same argument the load or demand imposed on the system are considered probabilistic in nature due to the effect of the varying environmental conditions.

Considering the above, the performance function of the system or sometime is called limit state function can be formulated as the difference between the system’s capacity and the load or demand imposed on it. The same argument can be used for production facility, the performance is the difference between supply and demand, supply being the capacity, or the strength and demand is the load. If the two parameters are the same then it can be said that the system is at a limit state, if the system cannot meet the demand then the system is at a failure state, and if the system capacity exceeds the load imposed on it, the system is at a satisfactory state.

3. Reliability analysis methods

Knowing in advance when the system is going to fail or degrade in performance is an essential step in the failure analysis. Under this step, the probability of failure is calculated in terms of the random variables affecting the performance of the system. There are several approaches found in the literature that can be used to evaluate the probability of failure either analytically or by simulation. Analytical methods approximate the probability of failure by using first order reliability method (FORM) or second order reliability method (SORM). The FORM uses two approximation techniques that evaluates the probability of failure, these are the first order and second moment (FOSM) and advanced first order second moment (AFOSM) techniques.

Calculating the probability of failure based on the methods mentioned above can be used to predict the ability of the system to satisfy operational as well as safety requirements during its life cycle. Combining this analysis with risk analysis, the consequences of failure can be easily determined. First order reliability method consists of two techniques namely:

3.1 Analytical methods

3.1.1 First order second moment (FOSM)

FOSM makes use of second moment statistics (mean and variance) and ignores higher moments (skewness and kurtosis) of the random variables. It evaluates the performance function by using the first order Taylor series expansion of the limit state function (LSF) at the mean value. This method is used when the performance function is linear having statistically independent, normally distributed and noncorrelated random variables Xis.

Performance function can be defined as [3, 5, 10]:

Z=CDE1

where C is the capacity and D is the demand are statistically independent random variables and are assumed to be normally distributed. Failure occurs when:

Z<0E2

Then the probability of failure (Pf) can be computed as:

Pf=(PC<DE3

Or

Pf=PZ<0E4

Figure 1 shows the probability density function (PDF) of the random variable Z, as it can be depicted from the figure that the probability of failure comprises the shaded area where Z < 0.

Figure 1.

Probability density function of the performance function.

The probability of failure is expressed as [3, 5, 10]:

Pf=PZ<0=0fzZdzE5

Alternatively, the performance function Zcan be formulated in terms of many random variables designated by a vector X as [3, 5, 10]:

ZX=ZX1X2,.XnE6

X1,.Xnare the random variables in the performance function.

The integration of the performance function as indicated in Eq. (5) is performed for the region where Z<0, this type of integration is difficult to solve, alternatively Taylor series expansion is used. The first order Taylor series approximation about the mean of the random variables is shown in Eq. (7). The expansion is truncated at the linear terms to obtain the first order approximation of the performance function [7, 8, 9].

Z=ZμX+i=1nZXiXiμXi+12!i=1n j=1n2ZXiXjXiμXiXjμXjE7
Z=ZμX+i=1nZXiXiμXiE7.1

Then the mean and variance are given as:

μZZμX1,μX2..μXnE8
VarZ=VarZμX+i=1nZXiXiμXiE9
VarZ=VarZμX+Vari=1nZXiXiVari=1nZXiμXiXiE10
VarZμX=0E10.1
Vari=1nZXiμXiXi=0E10.2
VarZ=i=1nZXi2VarXiE10.3
σZ=i=1nZXi2VarXiE10.4

The reliability index (β) is taken as ratio of the mean to the standard deviation of the performance function.

β=μZσZE11

The reliability index is computed for every failure mode, where the probability of failure is expressed as:

Pf=ϕβ=1ϕβE12

This method is simple to use and assumes that the random variables are normally distributed. All is needed for the calculation is the knowledge of the mean and the standard deviation and it is not necessary to know the distribution of the random variables. The downside of this method is that it can cause error in the final results if the function is nonlinear or if the tail of the distribution cannot be approximated by normal distribution. Moreover, if the function is nonlinear it will be provided different answer than that if it is linear. Advanced FOSM is used to deal with the limitations of the FOSM mentioned above.

3.1.2 Advanced first order second moment method (AFOSM)

AFOSM provides solution for linear and nonlinear performance function by determining the shortest distance from the origin to the failure surface. This method is also called Hasofer-Lind method. It evaluates the probability of failure for the limit state function or the performance function by determining the most probable failure point instead of the mean. Hasofer and Lind developed this advanced method in 1974 which is called Hasofer-Lind method and is abbreviated as H-L method. As stated above, the main objective of this method is to estimate the failure point which is the shortest distance from the origin to the failure surface that separates the failure region from the safe region. This can be clearly shown in Figure 2. The failure point is sometimes called in the literature design point or check point, but in this chapter, it will be referred to as the most probable point of failure (MPPF). Let us consider a limit state function/performance function with normally distributed and independent random variables X as:

Figure 2.

(a) Nonlinear limit state function—original coordinates. (b) Nonlinear limit state function—transformed coordinates.

ZX=Zx1x2xn=0E13

This method transforms the random variables into reduced form as:

ui=xiμxiσxi,i=1,2,.,E14
xi=uiσxi+μxi,i=1,2,.,E15

The performance function is then formulated in terms of the reduced random variables as:

ZU=Zu1σx1+μx1u2σx2+μx2.unσxn+μxn=0E16

Figure 2 shows the plot of the limit state function in the original as well as the transformed coordinates. It shows that the MPPF is the tangent point on the curve ZX=0and the reliability index βas the shortest distance from the origin to the limit surface.

To find the MPPF xion the limit surface under the condition that ZX=0, Taylor series expansion is used around the MPPF, considering the first order terms only, this gives:

ZU=ZU+i=1nZUUiuiuiE17
ZU=Zu1σx1+μx1)u2σx2+μx2.unσxn+μxn=0E18

Using chain rule for derivative and considering that relationship between U and X as:

Zfunction ofxZ=gx
Xfunction ofuX=fu

and using Eqs. (14) and (15) the partial derivative gUuibecomes:

ZUi=ZXiXUi=ZXiXuiσxi+μxiUi=ZXiσxiE19

Substituting Eq. (19) into Eq. (17) gives:

ZU=ZU+i=1nZXxiσxiuiuiE20

The mean of ZUis:

uz=EZU+Ei=1nZXxiσxiuiEi=1nZXxiσxiuiE21
EZU=ZUE22
Ei=1nZXxiσxiui=Ei=1nZXxixiuxi=Exii=1nZXxiEμxii=1nZXxiE23
μxii=1nZXxiμxii=1nZXxi=0E24
Ei=1nZXxiσxiui=i=1nZXxiσxiuiE25
uz=ZUi=1nZXxiσxiuiE26

The variance is expressed as:

σZ2=Var(ZU=VarZU+Vari=1nZXxiσxiuiuiE27

It must be noted that constants have no variance, their variances equal to zero. The first term of the Taylor expansion in Eq. (27) is constant; therefore, its variance equals to zero. Similarly, the variance at the mean value is zero.

VarZU=0E28
Var(ZU=Vari=1nZXxiσxiuiVari=1nZXxiσxiuiE29
Vari=1nZXxiσxiui=0E30
Var(ZU=Vari=1nZXxiσxiui=Vari=1nZXxixiuxiE31
Vari=1nZXxixiuxi=Vari=1nZXxixiVari=1nZXxiuxiE31.1
Vari=1nZXxiuxi=0E32
Var(ZU=i=1nZXxi2VarxiE33
σz=i=1nZXxi2σxi2E34

Alternatively, Eq. (31) can be calculated as [6]:

Vari=1nZXxixiuxi=Ei=1nZXxixiuxi2=i=1nZXxi2Exiuxi2=i=1nZXxi2VarxiE34.1

The reliability index is calculated as:

β=μZσZ=ZXi=1nZXxiσxiuii=1nZXxiσxi2E35

The directional cosine αialong the coordinate axes is computed as:

αi=ZUuii=1nZUui2=ZXxiσxii=1nZXxiσxi2E36

It can be shown from Figure 2 that:

ui=βCosθiorui=αxiβE37

Using Eqs. (14), (15) and (36) we can determine the design point in the original coordinates as:

xi=βαxiσxi+μxi=uiσxi+μxii=1,2,,nE38

The probability of failure, Pfcan be computed as:

Pf=Φβ=1ΦβE39

The steps to estimate the reliability index are:

  1. Formulate the performance function in terms of the original random variables,xi:

    Zx1x2xn

  • Assume the initial design points as the given mean of each variables.

  • Compute the initial reliability index β in terms of the mean values of the random variables usingEq. (11).

  • Compute the partial derivatives of the limit state function (LSF)/performance function in terms of mean value of the random variables.

  • Compute the directional cosinesαiusingEq. (36).

  • Compute the new design point usingEq. (38).

  • Compute the LSF in terms of new design points.

  • Compute the partial derivatives at the new design points.

  • Compute the new reliability index β usingEq. (35).

    1. Another alternative is to compute β from the limit state functionZX=Zβαxiσxi+μxi=0using the newly determined design points in step 7 and solve for β.

  • Compute the directional cosine.

  • Repeat steps six through nine until β converges to a pre-established tolerance level.

  • UseEq. (39)to calculate the probability of failure.

  • The steps mentioned above are used with the assumption that the random variables are normally distributed. For non-normally distributed variables additional steps are needed to determine the mean and standard deviation of the equivalent normal distribution as listed below. These steps should be carried out after step number two to determine mean and standard deviation of the equivalent normal distribution. Assuming the random variables are statistically independent and non-normally distributed:

    1. Determine the distribution parameters.

    2. Compute the cumulative distribution function cdf, Fxi, the probability density function pdf, fxiand the inverse cdf, Φ1Fxiof the original non-normal random variables at the initial design point.

    3. Compute the values of the standard deviation, σxiand the mean, μxiof the equivalent normal distribution as:

      σxi=ϕΦ1FxifxiE40

    It must be noted that fxirefers to the pdf of the original non-normal random variable and ϕrefers to the pdf of the equivalent standard normal random variable.

    μxi=xiΦ1FxiσxiE41

  • Compute the standard normal variable (ui)of xiby:

  • ui=xiμxiσxi,i=1,2,,nE42

    For a log normally distributed random variable, distribution parameters μlnx, σlnxare defined using the following equations:

    σlnx2=lnσxμx2+1,σlnx=σlnx2E43
    μlnx=lnμx0.5σlnx2E44

    The pdf and cdf are defined as:

    fx=12πxσlnxexp0.5lnxμlnxσlnx2E45
    Fx=ΦlnxμlnxσlnxE46

    For other distribution types the readers are referred to Refs. [4, 7, 8].

    3.2 Simulation methods

    Alternatively, the probability of failure is computed using Monte Carlo simulation method. Two methods are considered in this chapter, the counting and sample statistics methods. The simulation is conducted using computer programs such MATLAB, C++ or MINITAB or any other simulation programming packages.

    3.2.1 Monte Carlo counting method

    The counting method is formulated by dividing the number of simulation cycles at the events when the Z function becomes less than 0 (Nf<0)by the total number of simulation cycles (N).

    Pf=NfNE47

    The steps for Monte Carlo simulation counting method are listed below:

    Counting method

    1. Formulate the performance function in terms of the original random variables,xi:

      Zx1x2xn
      .

  • Determine the distribution and its parameters for each random variable.

  • Assign N to be the number of simulation cycles.

  • Assign M to be the number of calculation times.

  • Assign Nf to be the number of simulation cycles when the Z function becomes less than 0.

  • Initialize:

    1. Set M to zero

    2. Set Nf to zero

  • Generate random values from the given distribution. With the determined distribution parameters for each variable.

    1. As an example, if the random variable is normally distributed use the following Matlab function to generate the random values:

      x=normrndmuvarn1E48

    N is number of simulation cycles, mu and var. are the distribution parameters (the mean and standard deviation of the random variable).

  • If the random variable is log normally distributed with a mean,μxiand standard deviation,σxi:

    1. Determine the distribution parameters;Eqs. (43)and(44)can be used use to calculate these two parameters.

    2. Use the following Matlab function to generate the random values:

      x=lognrndμlnxσlnxN1E49

  • Calculate Pf:

    1. Is Z < 0?

      1. If yes:

        • Nf = Nf +1,

        • M = M + 1,

      2. If no:

        • Is M = N?

          1. If no—go to step 7

          2. If yes—calculatePf=NfN

  • Alternative approach to random number generation is to use the following steps:

    • Generate uniformly distributed random numbersuibetween 0 and 1; this can be accomplished by using software packages such as MATLAB, excel or C++ and other software packages.

    • Equate the inverse cumulative distribution function cdf of the random variable to the generated random numbers,uiby using the following equation:

    ui=FxxiE49.1
    xi=Fx1uiE49.2

    As an example, ifxifollows normal distribution with a meanμxi, and standard deviationσxithen its random number becomes:

    xi=μxi+σxiFx1uiE49.3

    Fx1uican be determined using the cdf of the standard normal distribution, these tables are included in most probability and statistics books.

    The random numbers can be generated N times using the following MATALB function:

    ui=randN1E49.4

    3.2.2 Monte Carlo sample statistics method

    The Monte Carlo sample statistics methods considers the meanμzand the standard deviationσzin computing the reliability index.

    β=μzσzE50
    Pf=Φβ=1ΦβE51

    The steps for Monte Carlo simulation sample statistics method are listed below:

    Sampling method

    1. Formulate the performance function in terms of the original random variables,xi:Zx1x2··xn.

    2. Determine the distribution and its parameters for each random variable.

    3. Assign N to be the number of simulation times.

    4. Assign M to be the number of calculation times.

    5. Initialize:

      1. Set M to zero

      2. Set Nf to zero

    6. Generate random values from the given distribution with the determined distribution parameters for each variable.

    7. Calculate Z function.

    8. Calculate Pf:

      c. Calculateβ=μzσz

      d. CalculatePf=1Φβ

      e. Is M = N?

      i.If no—go to step 6

      ii.If yes—stop

    It must be noted that Monte Carlo sample statistics method can be used only for linear functions having uncorrelated normal random variables.

    Example 1

    The performance function for a system has been formulated as: z=2.5x1x2

    where

    x1N20.2,x2N3.10.32

    1. Determine the probability of failure using simulation methods: Monte Carlo (MCS) sample statistics and Monte Carlo counting simulation methods.

    2. Determine the probability of failure using the analytical methods: FOSM and AFOSM (Hasofer-Lind) methods.

    Solution:

    1. Monte Carlo simulation was conducted for both methods, the sample statistics and counting methods. The number of simulation cycles used in the analysis are 2e5 and 1e6 cycles (Table 1).

    2. See Table 2.

    MCS—sample statistics methodMCS—counting method
    Number of simulation cyclesReliability index—βPfReliability index—βPf
    2e53.19930.0006893.19670.000695
    1e63.20300.000683.20050.000686

    Table 1.

    Simulation methods results—Example 1.

    FOSMAFOSM
    Reliability index—βPfReliability index—βPf
    3.20060.0006863.20060.000686

    Table 2.

    Analytical methods results—Example 1.

    This example adopted from [9].


    Example 2

    The performance function for a leak detection system has been formulated as [2]:

    z=2.8252x1x2,x1N1.10.02,x2N2.370.04

    1. Determine the probability of failure using Monte Carlo counting simulation method.

    2. Determine the probability of failure using AFOSM method.

    This example is adopted from reference [1, 2], with some modifications.

    Solution

    Part a

    Monte Carlo simulation was conducted for both methods, the sample statistics and counting methods as indicated in the table.

    Table 3 indicates that the probability of failure converges to 0.000476.

    MCS—counting method
    Number of simulation cyclesReliability indexβPf
    1e53.29620.000490
    1e63.30430.000476

    Table 3.

    Simulation methods results—Example 2.

    Part b

    First the Z function is computed in terms of the mean values of the random variables:

    x1=μx1=1.1
    x2=μx2=2.37
    u1=x1μx1σx1=1.11.10.02=0
    u2=x2μx2σx2=2.372.370.04=0
    ZX=Z1.12.37=0.1984

    The partial derivatives:

    zx1=2.8252x12=2.3349
    zx2=1
    σz=i=1nZXxiσxi2=2.33490.022+10.042=0.061487
    α1=ZXx1σx1i=1nZXxiσxi2=2.33490.020.061487=0.75947
    α2=ZXx2σx2i=1ngXxiσxi2=10.040.061487=0.65054
    β=μzσz=0.198360.061487=3.2261
    Pf=Φβ=1Φβ=1Φ3.2261=0.000627

    Iteration 1

    u1=βαx1=3.22610.75947=2.45013
    u2=βαx2=3.22610.65054=2.09872
    x1=βα1σx1+μx1=u1σx1+μx1=2.450130.02+1.1=1.149003
    x2=βα2σx2+μx2=u2σx2+μx2=2.098720.04+2.37=2.453949
    ZX=Z1.1490032.453949=0.004880
    Zx1=2.8252x12=2.82521.1490032=2.13997
    Zx2=1
    α1=ZXx1σx1i=1nZXxiσxi2=2.139970.022.139970.022+10.042=0.0427990.05858=0.7306
    α2=ZXx2σx2i=1nZXxiσxi2=10.042.139970.022+10.042=0.6828
    βHL=ZXi=1nZXxiσxiuii=1nZxiσxi2=0.004880(2.139970.022.45013+10.042.098720.05858=0.193690.05858=3.3064
    Pf=Φβ=1Φβ=1Φ3.3064=0.0004726

    The probability of failure obtained by counting method is very close to that obtained by AFOSM (Table 4).

    Iterationx1x2u1u2Z(X)∂z/∂x1∂z/∂x2α1α2βpf
    11.1000002.3700000.0000000.0000000.198364−2.334876−10.7594690.6505433.2261040.000627
    21.1490032.4539492.4501262.0987210.004880−2.139968−10.7305960.6828103.3063750.000473
    31.1483122.4603052.4156242.2576260.000001−2.142540−10.7310050.6823723.3063890.000473
    41.1483402.4602482.4169872.2561880.000000−2.142439−10.7309890.6823893.3063890.000473

    Table 4.

    Summary of the analytical methods results—Example 2.

    Example 3

    A pipeline segment is suffering corrosion that grows annually at steady rate. The extent of the initial growth has been estimated to be 4.7 mm and it is assumed that it follows log normal distribution with standard deviation of 1.1.

    The corrosion annual growth follows log normal distribution with a mean and standard deviation values of 0.2 and 0.01. The pipeline wall thickness follows normal distribution with a mean and standard deviation values of 14 mm and 4.7 respectively. The critical pipeline wall thickness has been determined to be 80% of the wall thickness [3]. A summary of the relevant information pertaining to the pipeline corrosion is presented in the net table, Table 5.

    Wall thicknessInitial corrosionCorrosion growth
    Distribution typeNormal distributionLog normal distributionLog normal distribution
    Mean14 mm4.7 mm0.2 mm
    Standard deviation0.071.10.01

    Table 5.

    Pipeline corrosion data—Example 3.

    The owner of the pipeline decides not to repair the corrosion and wants to know if the pipeline can survive for the next 14 years without causing a leakage. It has been decided that in order to be in the safe side the maximum acceptable probability of failure has been set to 1e4 [3].

    Solve this problem using analytical method as well as Monte Carlo (MCS) simulation method.

    Solution

    Monte Carlo simulation

    x1: Wall thickness,x1N1.10.02

    x2: Extent of the initial corrosion,x2LGN4.71.1

    x3: Corrosion annual growth,x3LGN0.20.01

    The limit state function can be formulated as indicated below:

    The capacity portion is formulated as:0.8x1

    The demand portion is formulated as:x2+14x3

    ZX=0.8x1x2+14x3

    Pipeline wall thickness:

    μx1=14,σx1=0.07
    x1=normrndμx1σx1N1

    Initial corrosion:

    μx2=4.7,σx2=1.1
    σlnx22=lnσx2μx22+1=0.0533
    σlnx2=σlnx22=0.2309
    μlnx2=lnμx20.5σlnx22=1.5209
    x2=lognrndμlnx2σlnx2N1

    Corrosion growth rate:

    μx3=0.2,σx3=0.01
    σlnx32=lnσx3μx32+1=0.002497
    σlnx3=σlnx32=0.04997
    μln3=lnμx30.5σlnx32=1.61069
    x3=lognrndμlnx3σlnx3N1

    Monte Carlo simulation produced the following results as shown in Table 6. The probability of failure and beta converge to the following values:

    MCS—counting method
    Number of simulation cyclesReliability index—βPf
    1e62.62490.004334
    1e72.62060.004389

    Table 6.

    Simulation methods results—Example 3.

    Pf=NfN=43961e6=0.004389
    Beta=Φ1Pf=2.6206

    Here only the counting method is used because the sampling method produces different results. The sampling method produces accurate results for linear and normal limit state/performance function only.

    Analytical Solution

    Assume the initial value for each random variable to be its mean.

    ZX=Z14,4.7,0.2=3.7
    Zx1=0.8
    Zx2=1
    Zx3=14

    For non-normal variables, the standard deviation and mean values of the equivalent normal variables are calculated using Eqs. (40) and (41).

    Pipeline wall thickness,x1:

    μx1=14,σx1=0.07

    Initial corrosion,x2:

    μx2=4.7,σx2=1.1
    σlnx22=lnσx2μx22+1=0.0533
    σlnx2=σlnx22=0.2309
    μlnx2=lnμx20.5σlnx22=1.5209

    Compute the pdf of the original non-normal variable (log normal distribution) using Eq. (45):

    fx=12πxσlnxexp0.5lnxμlnxσlnx2=12π4.70.2309exp0.5ln4.71.52090.23092=0.36512

    Compute the cdf of the original non-normal variable (log normal distribution) using Eq. (46):

    Fx=Φlnx2μlnxσlnx=Φln4.71.52090.2309
    Φ1Fxi=ln4.71.52090.2309=0.115466

    Compute the mean and standard deviation of the equivalent normal variable at the design point using Eqs. (40) and (41):

    σx2=ϕΦ1Fx2fx2=ϕ0.1154660.36512,

    ϕrefers to the equivalent standard normal random variable and is calculated as:

    ϕ0.115466=12πexp0.5lnx2μlnxσlnx2=12πexp0.50.1154662=0.3963
    σx2=0.39630.36512=1.0854
    μx2=x2Φ1Fx2σx2=4.70.1154661.0854=4.5747

    Corrosion growth,x3

    The same procedures outline for initial corrosion extent, x2: to produces the following results:

    fx3=39.907,Φ1Fx3=0.02526,ϕΦ1Fx3=0.398815,
    σx3=0.0099936andμx3=0.19975
    σz=i=1nZXxiσxi2=1.0958
    β=μzσz=3.71.0958=3.3766
    Pf=Φβ=0.00037
    α1=∂ZXx1σx1i=1n∂ZXx1σx12=0.0511
    α2=ZXx2σx2i=1nZXx2σx22=0.9905
    α3=ZXx3σx3i=1nZXx3σx32=0.1277

    Iteration 1

    u1=βαx1=0.1013
    u2=βαx2=3.3445
    u3=βαx3=0.4311
    x1=βα1σx1+μx1=u1σx1+μx1=13.9879
    x2=βα2σx2+μx2=u2σx2+μx2=8.2047
    x3=βα3σx3+μx3=u2σx3+μx3=0.20406
    gx1=0.8,gx2=1,gx3=14

    Mean and standard deviation for normal variable x1remain the same:

    μx1=14,σx1=0.07

    Mean and standard deviation for the equivalent normal distribution for the non-normal variables, x2andx3(Table 7):

    Iteration123456
    x114.000013.987913.992913.995313.994713.9947
    x24.70008.20479.90918.56418.45958.3734
    x30.20000.20410.20230.20150.20170.2016
    u10.0000−0.1726−0.1013−0.0675−0.0753−0.0750
    u20.00003.34453.42762.75762.66092.6164
    u30.00000.43110.25820.17060.18960.1889
    mu114.000014.000014.000014.000014.000014.0000
    mu24.57473.41472.25383.19723.26203.3145
    mu30.19980.19970.19970.19970.19970.1997
    σ10.07000.07000.07000.07000.07000.0700
    σ21.08541.89472.28831.97771.95361.9337
    σ30.01000.01020.01010.01010.01010.0101
    Z(x)3.70000.1288−1.5475−0.1882−0.0869−0.0006
    ∂z/∂x10.80000.80000.80000.80000.80000.8000
    ∂z/∂x2−1.0000−1.0000−1.0000−1.0000−1.0000−1.0000
    ∂z/∂x3−14.0000−14.0000−14.0000−14.0000−14.0000−14.0000
    σ1.09581.90092.29331.98351.95941.9396
    α1−0.0511−0.0295−0.0244−0.0282−0.0286−0.0289
    α20.99050.99670.99780.99710.99700.9969
    α30.12770.07510.06170.07110.07200.0727
    β3.37663.43882.76372.66872.62432.6240
    Pf0.00040.00030.00290.00380.00430.0043

    Table 7.

    Analytical methods results—Example 3.

    μx2=3.4147,σx2=1.8947,μx3=0.1997,σx3=0.010197
    α1=0.0295,α2=0.9967,α3=0.0751

    σz=0.80.072+11.89472+140.0101972=1.9009

    ZX=Z13.98798.20470.20406=0.1288
    βHL=ZXi=1nZXxiσxiuii=1nZxiσxi2=0.12886.40801.9009=3.4388
    Pf=Φβ=0.00037

    Table 8 shows that the obtained values for beta and the probability of failure are so close to each other.

    Analytical methodMCS—counting method
    Reliability index—βPfReliability index—βPf
    2.62400.0043452.62060.004389

    Table 8.

    Comparison of results—Example 3.

    The calculated probability of failure exceeds the target probability of failure 104.

    Example 4

    Two systems, system A and system B, each system has three main components and each component with it; the probability distribution type and its parameters are shown in Table 9. It is required to determine the probability of failure of each component using FOSM, AFOSM and Monte Carlo simulation [9].

    System ADistribution type and its parameters
    Electric motorCapacityDemand
    System energyN∼(0.52, 0.05)N∼(0.46, 0.08)
    Number of partsN∼(2, 0.5)N∼(1, 0.3)
    WeightN∼(0.42 0.05)N∼(0.400, 0.06)
    System BDistribution type and its parameters
    Pneumatic cylinderCapacityDemand
    System energyN∼(0.52, 0.05)N∼(0.4, 0.04)
    Number of partsN∼(2, 0.5)N∼(1, 0.3)
    WeightN∼(0.420, 0.05)N∼(0.37, 0.04)

    Table 9.

    Capacity and demand variables—Example 4.

    Solution

    Formulate a general LSF for each component as:

    ZX=x1x2

    Formulate a general LSF for each component see the (Table 10).

    System AMCS—sample statistics methodMCS—counting method
    Number of simulation cyclesReliability index—βPfReliability index—βPf
    Electric motor—system energy
    1e50.63340.26320.63260.2635
    1e60.63630.26230.63630.2623
    Electric motor—number of parts
    1e51.72070.04271.71970.0427
    1e61.71680.04301.71830.0429
    Electric motor—weight
    1e50.25560.39910.25720.3985
    1e60.25580.39910.25640.3988
    System BMCS—sample statistics methodMCS—counting method
    Number of simulation cyclesReliability index—βPfReliability index—βPf
    Pneumatic cylinder—system energy
    1e51.88080.03001.89270.0292
    1e61.87480.03041.87700.0303
    Pneumatic cylinder—number of parts
    1e51.70880.04371.70360.0442
    1e61.71500.04321.71560.0431
    Pneumatic cylinder—weight
    1e50.78080.21750.78010.2177
    1e60.7814434670.21727090.781276310.21732

    Table 10.

    Simulation methods results systems A and B—Example 4.

    Analytical Solution

    Analytical solution see (Table 11).

    Electric motor—system energy
    Iterationx1x2u1u2σ1σ2Z(X)∂z/∂x1∂z/∂x2α1α2σ_zβpf
    10.52000.46000.00000.00000.05000.08000.06001.0000−1.0000−0.53000.84800.09430.63600.2624
    20.50310.5031−0.33710.53930.05000.08000.00001.0000−1.0000−0.53000.84800.09430.63600.2624
    Electric motor—number of parts
    Iterationx1x2u1u2σ1σ2Z(X)∂z/∂x1∂z/∂x2α1α2σ_zβpf
    12.00001.00000.00000.00000.50000.30001.00001.0000−1.0000−0.85750.51450.58311.71500.0432
    21.26471.2647−1.47060.88240.50000.30000.00001.0000−1.0000−0.85750.51450.58311.71500.0432
    Electric motor—weight
    Iterationx1x2u1u2σ1σ2Z(X)∂z/∂x1∂z/∂x2α1α2σ_zβpf
    10.42000.40000.00000.00000.05000.06000.02001.0000−1.0000−0.64020.76820.07810.25610.3989
    20.41180.4118−0.16390.19670.05000.06000.00001.0000−1.0000−0.64020.76820.07810.25610.3989
    Pneumatic cylinder—system energy
    Iterationx1x2u1u2σ1σ2Z(X)∂z/∂x1∂z/∂x2α1α2σ_zβpf
    10.52000.40000.00000.00000.05000.04000.12001.0000−1.0000−0.78090.62470.06401.87410.0305
    20.44680.4468−1.46341.17070.05000.04000.00001.0000−1.0000−0.78090.62470.06401.87410.0305
    Pneumatic cylinder—number of parts
    Iterationx1x2u1u2σ1σ2Z(X)∂z/∂x1∂z/∂x2α1α2σ_zβpf
    12.00001.00000.00000.00000.50000.30001.00001.0000−1.0000−0.85750.51450.58311.71500.0432
    21.26471.2647−1.47060.88240.50000.30000.00001.0000−1.0000−0.85750.51450.58311.71500.0432
    Pneumatic cylinder—weight
    Iterationx1x2u1u2σ1σ2Z(X)∂z/∂x1∂z/∂x2α1α2σ_zβpf
    10.42000.37000.00000.00000.05000.04000.05001.0000−1.0000−0.78090.62470.06400.78090.2174
    20.38950.3895−0.60980.48780.05000.04000.00001.0000−1.0000−0.78090.62470.06400.78090.2174

    Table 11.

    Analytical method results—Example 4.

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    Alireda Aljaroudi (July 22nd 2019). Probabilistic Modeling of Failure [Online First], IntechOpen, DOI: 10.5772/intechopen.83461. Available from:

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