Example 3

I'm excited to share the recent project I worked on that involved developing software called BI (Beta Index). BI is a powerful tool designed to solve complex structural reliability problems and help engineers make more informed decisions. In this post, I'd like to highlight a specific analytical example of how BI can be implemented to estimate the failure probability.

0. Example 3

This example has a linear limit state function in physical space as Eq. (1) [1,2].?

Table 1 shows the correlation matrix, the number, and the statistical moment of random variables for this example.

The probability density function for random variables is in accordance with Eq. (2) and Eq. (3).

1. Steps to prepare this example in BI (before analysis):

1.1 Generate Random Variable object: Define > Variable > RandomVariable

1.2 Generate Correlation Coefficient object: Define > Correlation

1.3 Generate Limit State Function object: Define > Limit State Function

2. Prepare for FORM Analysis

2.1 Generate Nataf Transformation object: Analysis > Transformer

2.2 Generate Convergence Checker object: Analysis > Convergence Checker

2.3 Generate Step Direction Searcher object: Analysis > Step Direction Searcher

2.4 Generate Merit Checker object: Analysis > Merit Checker

2.5 Generate Step Size Searcher object: Analysis > Step Size Searcher

In this part, you need to insert the name of the previous object, which is created in step 2.4.

2.6 Generate Solver object: Analysis > Solver

In this part, you need to insert the names of previous objects, which are created in steps 2.1, 2.2, 2.3, and 2.5.

2.7 Generate Output object: Analysis > Output

2.8 Generate FORM Analysis object: Analysis > FORM Analysis

2.9 Run this FORM Analysis object.

2.10 Show Plot result:

2.11 Show text result:

3. Prepare for SORM Analysis

3.1 Use Solver object in FORM Analysis. Step 2.5 shows this object.

3.2 Generate Output object: Analysis > Output

3.3 Generate SORM Analysis object: Analysis > SORM Analysis

3.4 Run this SORM Analysis object.

3.5 Plot result is the same as FORM Analysis

3.6 Show text result: (After FORM Analysis results, there is the SORM Analysis result.)

4. Prepare for FOTM Analysis

4.1 Use Solver object in FORM Analysis. Step 2.5 shows this object.

4.2 Generate Output object: Analysis > Output

4.3 Generate FOTM Analysis object: Analysis > FOTM Analysis

4.4 Run this FOTM Analysis object.

4.5 Show Plot result:

4.6 Show text result:

5. Prepare for Sampling Analysis

5.1 Use Nataf Transformer object in FORM Analysis. Step 2.1 shows this object.

5.2 Generate Random Number Generator object: Analysis > Random Number Gen

5.3 Generate Accumulator (Failure Probability) object: Analysis > Accumulator

5.4 Generate Output object: Analysis > Output

5.5 Generate Sampling Analysis object: Analysis > Sampling Analysis

5.6 First Run Form Analysis, then run this Sampling Analysis object. (For fast convergence)

5.7 Show Plot result:

5.8 Show text result:

6. Prepare for Optimization Analysis

6.1 Generate Output object: Analysis > Output

6.2 Generate Optimization Analysis object: Analysis > Optimization

TRSQP Method

SLSQP Method

COBYLA Method

6.3 Run this Optimization object.

6.4 Show the Plot and Text result of each method:

TRSQP Method

SLSQP Method

COBYLA Method

7. References

[1] Lu Z-H, Cai C-H, Zhao Y-G. Structural Reliability Analysis Including Correlated Random Variables Based on Third-Moment Transformation. J Struct Eng 2017;143:04017067.?https://doi.org/10.1061/(asce)st.1943-541x.0001801.

[2] Der Kiureghian A, Liu P. Structural Reliability under Incomplete Probability Information. J Eng Mech 1986;112:85–104.?https://doi.org/10.1061/(asce)0733-9399(1986)112:1(85).