## Example 5

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

**4. Prepare for Sampling Analysis**

**5. Prepare for Optimization Analysis**

**0. Example 5**

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

**Table 1 shows 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 FOTM 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 FOTM Analysis object: Analysis > FOTM Analysis**

**3.4 Run this FOTM Analysis object.**

**3.5 Show Plot result:**

**3.6 Show text result:**

**4. Prepare for Sampling Analysis**

**4.1 Use Nataf Transformer object in FORM Analysis. Step 2.1 show this object.**

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

**4.3 Generate Accumulator (Failure Probability) object: Analysis > Accumulator**

**4.4 Generate Output object: Analysis > Output**

**4.5 Generate Sampling Analysis object: Analysis > Sampling Analysis**

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

**4.7 Show Plot result:**

**4.8 Show text result:**

**5. Prepare for Optimization Analysis**

**5.1 Generate Output object: Analysis > Output**

**5.2 Generate Optimization Analysis object: Analysis > Optimization**

**TRSQP Method**

**SLSQP Method**

**COBYLA Method**

**5.3 Run this Optimization object.**

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

**TRSQP Method**

**SLSQP Method**

**COBYLA Method**

**6. 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.**