## Example 1

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

**5. Prepare for Sampling Analysis**

**6. Prepare for Histogram Sampling Analysis**

**7. Prepare for Optimization Analysis**

**0. Example 1**

**This example has a linear limit state function in physical space as Eq. (1) [1–4].**

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

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

**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 name of previous object, which is created in step 2.4.**

**2.6 Generate Solver object: Analysis > Solver**

**In this part, you need to insert 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 show 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 same as FORM Analysis**

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

**4. Prepare for FOTM Analysis**

**4.1 Use Solver object in FORM Analysis. Step 2.5 show 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 show 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 Histogram Sampling Analysis**

**6.1 Generate Limit State Function object: Define > Limit State Function**

**Remove “18” from first limit state function in Eq. (1).**

**6.2 Generate Random Number Generator object: Analysis > Random Number Gen**

**6.3 Generate Accumulator (Histogram) object: Analysis > Accumulator**

**6.4 Generate Output object: Analysis > Output**

**6.5 Generate Histogram Sampling Analysis object: Analysis > Hist Sampling Analysis**

**6.6 Run this Histogram Sampling Analysis object.**

**6.7 Show Plot result:**

**6.8 Show text result:**

**6.9 To determine the probability of failure, using the CDF diagram in the plot, the CDF value is read at 18, which is equal to 0.996972. Given that the negative sign was omitted from the limit state function, the value of the probability of failure is now obtained: using Eq. (3).**

**The reliability index for this failure probability is determined from Eq. (4).**

**7. Prepare for Optimization Analysis**

**7.1 Generate Output object: Analysis > Output**

**7.2 Generate Optimization Analysis object: Analysis > Optimization**

**TRSQP Method**

**SLSQP Method**

**COBYLA Method**

**7.3 Run this Optimization object.**

**7.4 Show Plot and Text result of each method:**

**TRSQP Method**

**SLSQP Method**

**COBYLA Method**

**8. References**

**[1] Ditlevsen O, Madsen HO. Structural Reliability Methods. vol. 178. Wiley New York; 2005.**

**[2] Hong HP, Lind NC. Approximate reliability analysis using normal polynomial and simulation results. Struct Saf 1996;18:329–39. https://doi.org/10.1016/S0167-4730(96)00018-5.**

**[3] Lu ZH, Cai CH, Zhao YG, Leng Y, Dong Y. Normalization of correlated random variables in structural reliability analysis using fourth-moment transformation. Struct Saf 2020;82:101888. https://doi.org/10.1016/j.strusafe.2019.101888.**

**[4] 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.**