**7.1 Example 1: a nonlinear limit state function**

A nonlinear limit state function was studied in literature, as [21, 49, 50]:

$$\mathbf{g(x)} = \exp\left(\mathbf{0.2x\_1} + \mathbf{6.2}\right) - \exp\left(\mathbf{0.47x\_2} + \mathbf{5.0}\right) \tag{25}$$

where *x*<sup>1</sup> and *x*<sup>2</sup> are independent random variables following standard normal distributions (mean = 0; standard deviation = 1). The failure probability *PF* = 0.009372 was obtained based on Direct MCS and used to compare with other solutions. The RBF surrogate models were constructed using the two variables sampled in the range of �3.0 to 3.0. All three surrogate models started with 10 sample points in the first iteration. With 10 sample points, the error of the estimated failure probability was 7.0, 3.0, and 1.8% for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP, respectively. In each subsequent iteration 10 more sample points were added. At convergence, the accuracy of SRBF models was improved; the error was reduced to 0.9, 0.8, and 1.3% for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP, respectively. Adequate accuracy of reliability analysis was achieved for all three SRBF surrogate models. The failure probability values obtained based on three surrogate models and the associated errors as compared to the solution obtained using Direct MCS are listed in **Table 2**. It took 4, 3, and 2 iterations for SRBF-MQ-LP, SRBF-CS20-, and SRBF-CS30-LP methods to converge, corresponding to 40, 30, and 20 sample points, respectively. A total of 40, 30, and 20 function evaluations (original limit state function) were required for the three SRBF-based MCS, respectively.


#### **Table 2.**

6.Calculate failure probability *PF* for iteration *k* using MCS.

*Flowchart of reliability analysis using a SRBF surrogate technique.*

*Reliability and Maintenance - An Overview of Cases*

*k* ¼ *k* þ 1.

**78**

**Figure 1.**

Step 8, then go to Step 4.

7. Check the convergence criterion. If the convergence criterion is satisfied, stop; otherwise go to Step 8. In this study the convergence criterion is that the relative error of the failure probability *PF* between two successive iterations is less than the tolerance. A tolerance value of 1% was applied in this study. For practical applications, another convergence criterion may be defined, e.g., the maximum number of response simulations has been reached. This will help control the total number of iterations performed in the reliability analysis.

8.Generate additional sample set with *m* sample points; set the iteration number

9.Evaluate limit state function *g*ð Þ x for the additional sample set *m* generated in

*Example 1: numerical results.*
