**7.2 Example 2: a cantilever beam**

The reliability analysis of a cantilever beam with a concentrated load is conducted in this example [50]. The beam has a rectangular cross section. The performance requirement is the displacement at tip should be <0.15 in. Therefore, the limit state function is.


#### **Table 3.**

*Example 2: random variables [50].*


#### **Table 4.**

*Example 2: numerical results.*

$$\log(l, b, h) = 0.15 - \frac{4Pl^3}{Ebh^3} \tag{26}$$

*g*ð Þ¼ x *x*1*x*2*x*<sup>3</sup> � *x*<sup>4</sup>

*Reliability Analysis Based on Surrogate Modeling Methods*

*DOI: http://dx.doi.org/10.5772/intechopen.84640*

lists the six input random variables and their statistical properties.

tions, representing the associated computational effort. Compared with

and SRBF-CS30-LP were 0.8, 1.1, and 0.9%, respectively.

**Table 5.**

**Table 6.**

**81**

*Example 3: random variables [70].*

*Example 3: numerical results.*

Eq. (27) included six independent random variables: *x*<sup>1</sup> is the total crosssectional area of rebars, *x*<sup>2</sup> is the yield strength of rebars, *x*<sup>3</sup> is the effective depth of section, *x*<sup>4</sup> is a dimensionless factor related to concrete stress-strain curve, *x*<sup>5</sup> is the compressive strength of concrete, and *x*<sup>6</sup> is the width of the concrete section. The limit state was for the ultimate bending moment strength of the section, and a bending moment limit *Mn* <sup>¼</sup> <sup>211</sup>*:*<sup>20</sup> � <sup>10</sup><sup>6</sup> N-mm was adopted in this study. **Table 5**

To start the reliability analysis, 30 sample points were used in the first iteration of all three SRBF surrogate models, and 10 additional samples were included in each subsequent iteration. **Table 6** lists the failure probability *PF* values obtained using different methods, in addition to the required number of original function evalua-

*PF* = 0.01102 obtained by Direct MCS, the errors of SRBF-MQ-LP, SRBF-CS20-LP

**Figure 2** is the plot showing failure probability estimation versus sample size. All three SRBF models worked well and smooth convergence histories can be observed. The three SRBF models produced similar failure probabilities. The results by SRBF-CS20-LP and SRBF-CS30-LP were shown to be better than that using SRBF-MQ-LP when the sample size was small. Among the three SRBF models, SRBF-CS30-LP generated the most accurate approximation with the same sample size. As expected, more sample points resulted in reduced SRBF approximation errors. With the increase of the number of sample points or function evaluations (i.e., computational effort), a reduction in estimation error of the failure probability using the proposed SRBF models was observed. For example, the estimation error of *PF* was reduced from 10.7 to 0.8% for SRBF-MQ-LP, 4.9–1.1% for SRBF-CS20-LP, and 4.1–0.9% for

*x*2 1*x*2 2 *x*5*x*<sup>6</sup>

� *Mn* (27)

where *P* is the concentrated load, *l* is the beam length, *b* and *h* are the width and depth of the beam cross-section, and *E* = 10<sup>7</sup> psi is the Young's modulus. In this example *P* = 80 lb. was considered. **Table 3** lists the three random variables in this problem, i.e., *l*, *b*, and *h*.

All three SRBF surrogate models started with 20 sample points in the first iteration, with 10 more samples generated in each following iteration. The reliability analysis results and the corresponding sample sizes required for SRBF surrogate models were examined, as listed in **Table 4**. The failure probability estimated based on Direct MCS using Eq. (26) was 0.02823, which was regarded as the actual solution. It took 4, 7, and 5 iterations for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP to converge, respectively. With the initial 20 samples, the error of the estimated failure probability was 35.9, 19.4, and 9.7% for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP, respectively. With 50, 80, and 60 sample points, the error was reduced to 9.7% for SRBF-MQ-LP, 0.3% for SRBF-CS20-LP, and 1.7% for SRBF-CS30-LP. The errors in estimating the failure probability by SRBF surrogate models decreased as the sample size increased. The SRBF-MQ-LP model did not produce as accurate estimation of *PF* as SRBF-CS20-LP and SRBF-CS30-LP, when the same sample size was used. In all three SRBF surrogate models, SRBF-CS20-LP provided the most accurate estimate of *PF*, and the surrogate model SRBF-MQ-LP did not converge close to the actual solution. In this example, 60–80 sample points were required for SRBF-CS20-LP and SRBF-CS30-LP to achieve reasonably accurate surrogate models and estimates of the failure probability.

#### **7.3 Example 3: a reinforced concrete beam section**

This example is the reliability analysis of a singly-reinforced concrete beam section [51, 70]. Based on static equilibrium, the following nonlinear limit state function can be developed, as:

*Reliability Analysis Based on Surrogate Modeling Methods DOI: http://dx.doi.org/10.5772/intechopen.84640*

$$g(\mathbf{x}) = \mathbf{x}\_1 \mathbf{x}\_2 \mathbf{x}\_3 - \mathbf{x}\_4 \frac{\mathbf{x}\_1^2 \mathbf{x}\_2^2}{\mathbf{x}\_5 \mathbf{x}\_6} - M\_n \tag{27}$$

Eq. (27) included six independent random variables: *x*<sup>1</sup> is the total crosssectional area of rebars, *x*<sup>2</sup> is the yield strength of rebars, *x*<sup>3</sup> is the effective depth of section, *x*<sup>4</sup> is a dimensionless factor related to concrete stress-strain curve, *x*<sup>5</sup> is the compressive strength of concrete, and *x*<sup>6</sup> is the width of the concrete section. The limit state was for the ultimate bending moment strength of the section, and a bending moment limit *Mn* <sup>¼</sup> <sup>211</sup>*:*<sup>20</sup> � <sup>10</sup><sup>6</sup> N-mm was adopted in this study. **Table 5** lists the six input random variables and their statistical properties.

To start the reliability analysis, 30 sample points were used in the first iteration of all three SRBF surrogate models, and 10 additional samples were included in each subsequent iteration. **Table 6** lists the failure probability *PF* values obtained using different methods, in addition to the required number of original function evaluations, representing the associated computational effort. Compared with *PF* = 0.01102 obtained by Direct MCS, the errors of SRBF-MQ-LP, SRBF-CS20-LP and SRBF-CS30-LP were 0.8, 1.1, and 0.9%, respectively.

**Figure 2** is the plot showing failure probability estimation versus sample size. All three SRBF models worked well and smooth convergence histories can be observed. The three SRBF models produced similar failure probabilities. The results by SRBF-CS20-LP and SRBF-CS30-LP were shown to be better than that using SRBF-MQ-LP when the sample size was small. Among the three SRBF models, SRBF-CS30-LP generated the most accurate approximation with the same sample size. As expected, more sample points resulted in reduced SRBF approximation errors. With the increase of the number of sample points or function evaluations (i.e., computational effort), a reduction in estimation error of the failure probability using the proposed SRBF models was observed. For example, the estimation error of *PF* was reduced from 10.7 to 0.8% for SRBF-MQ-LP, 4.9–1.1% for SRBF-CS20-LP, and 4.1–0.9% for


#### **Table 5.**

*g l*ð Þ¼ *; <sup>b</sup>; <sup>h</sup>* <sup>0</sup>*:*<sup>15</sup> � <sup>4</sup>*Pl*<sup>3</sup>

All three SRBF surrogate models started with 20 sample points in the first iteration, with 10 more samples generated in each following iteration. The reliability analysis results and the corresponding sample sizes required for SRBF surrogate models were examined, as listed in **Table 4**. The failure probability estimated based on Direct MCS using Eq. (26) was 0.02823, which was regarded as the actual solution. It took 4, 7, and 5 iterations for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP to converge, respectively. With the initial 20 samples, the error of the estimated failure probability was 35.9, 19.4, and 9.7% for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP, respectively. With 50, 80, and 60 sample points, the error was reduced to 9.7% for SRBF-MQ-LP, 0.3% for SRBF-CS20-LP, and 1.7% for SRBF-CS30-LP. The errors in estimating the failure probability by SRBF surrogate models decreased as the sample size increased. The SRBF-MQ-LP model did not produce as accurate estimation of *PF* as SRBF-CS20-LP and SRBF-CS30-LP, when the same sample size was used. In all three SRBF surrogate models, SRBF-CS20-LP provided the most accurate estimate of *PF*, and the surrogate model SRBF-MQ-LP did not converge close to the actual solution. In this example, 60–80 sample points were required for SRBF-CS20-LP and SRBF-CS30-LP to achieve reasonably accu-

rate surrogate models and estimates of the failure probability.

This example is the reliability analysis of a singly-reinforced concrete beam section [51, 70]. Based on static equilibrium, the following nonlinear limit state

**7.3 Example 3: a reinforced concrete beam section**

function can be developed, as:

**80**

problem, i.e., *l*, *b*, and *h*.

*Example 2: numerical results.*

**Table 3.**

**Table 4.**

*Example 2: random variables [50].*

*Reliability and Maintenance - An Overview of Cases*

where *P* is the concentrated load, *l* is the beam length, *b* and *h* are the width and depth of the beam cross-section, and *E* = 10<sup>7</sup> psi is the Young's modulus. In this example *P* = 80 lb. was considered. **Table 3** lists the three random variables in this

*Ebh*<sup>3</sup> (26)

*Example 3: random variables [70].*


**Table 6.** *Example 3: numerical results.*

**Figure 2.** *Example 3: failure probability iterations.*

SRBF-CS30-LP, respectively. SRBF-CS20-LP and SRBF-CS30-LP created with 40 samples and SRBF-MQ-LP created with 50 samples could provide fairly accurate reliability analysis results (<2% error of *PF*).
