Probabilistic Modeling of Failure

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 Xi′s.

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

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:

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

Or

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.

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

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

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].

Then the mean and variance are given as:

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

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

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:

This method transforms the random variables into reduced form as:

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

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 xi′on 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:

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

and using Eqs. (14) and (15) the partial derivative ∂gU∗∂uibecomes:

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

The mean of ZUis:

The variance is expressed as:

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.

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

The reliability index is calculated as:

The directional cosine αialong the coordinate axes is computed as:

It can be shown from Figure 2 that:

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

The probability of failure, Pfcan be computed as:

The steps to estimate the reliability index are:

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

   Zx1x2⋯⋯⋯⋯⋯⋯xn

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, Φ−1Fxiof 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=ϕΦ−1FxifxiE40

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.

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

The pdf and cdf are defined as:

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

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).

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:

   Zx1x2⋯⋯⋯⋯⋯⋯xn
   .

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

• Generate uniformly distributed random numbers uibetween 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:

As an example, if xifollows normal distribution with a mean μxi, and standard deviation σxithen its random number becomes:

Fx−1uican 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:

3.2.2 Monte Carlo sample statistics method

The Monte Carlo sample statistics methods considers the mean μzand the standard deviation σzin computing the reliability index.

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. Calculate β=μzσz

   d. Calculate Pf=1−Φβ

   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=2.5x1−x2

where

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.

Example 2

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

z=2.8252x−1−x2, x1∼N1.10.02,x2∼N2.370.04

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.

Part b

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

The partial derivatives:

Iteration 1

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

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.

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, x1∼N1.10.02

x2: Extent of the initial corrosion, x2∼LGN4.71.1

x3: Corrosion annual growth,x3∼LGN0.20.01

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

Pipeline wall thickness:

Initial corrosion:

Corrosion growth rate:

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

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.

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:

Initial corrosion, x2:

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

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

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

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

Corrosion growth, x3

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

Iteration 1

Mean and standard deviation for normal variable x1remain the same:

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

σz=0.8∗0.072+−1∗1.89472+−14∗0.0101972=1.9009

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

The calculated probability of failure exceeds the target probability of failure 10−4.

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].

Solution

Formulate a general LSF for each component as:

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

Analytical Solution

Analytical solution see (Table 11).