**7.4 Example 4: burst margin of a rotating disk**

This example is the reliability analysis of a disk with an angular velocity of *ω*, as shown in **Figure 3** [50, 51]. The inner and outer radii of disk are *Ri* and *Ro*, respectively. The burst margin, *Mb,* of the disk refers to the safety margin before overstressing the disk, which is expressed as:

$$M\_b(a\_m, \mathbf{S}\_u, \rho, \boldsymbol{\alpha}, \mathbf{R}\_o, \mathbf{R}\_i) = \sqrt{\frac{a\_m \mathbf{S}\_u}{\left(\frac{\rho \left(\frac{2w}{60}\right)^2 \left(\boldsymbol{R}\_o^3 - \boldsymbol{R}\_i^3\right)}\right)}}\tag{28}$$

If a lower bound value of 0.37473 is used, the limit state function of *Mb* can be written as:

$$\mathbf{g}(\mathbf{x}) = \mathbf{M}\_b(a\_m, \mathbf{S}\_u, \rho, a, R\_o, R\_i) - \mathbf{0}.\mathbf{37473} \tag{29}$$

estimated failure probability *PF* in this study based on different SRBF surrogate models and the associated errors as compared to the solution obtained using Direct MCS. The augmented SRBF-based methods required 60–70 original function evaluations to converge. **Figure 4** illustrates the variation of the failure probability *PF* versus number of sample points. In general with the increase of the sample size, a reduction was observed in the estimation errors of the failure probability *PF*, from 67.1, 6.6, and 12.8% when 30 sample points were used, to 5.6, 0.8, and 0.5% at convergence for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP, respectively. The reliability analysis results based on surrogate models SRBF-CS20-LP and SRBF-CS30-LP were shown to be better that using SRBF-MQ-LP. It showed that with around 50 sample points very accurate SRBF-CS20-LP and SRBF-CS30-LP surro-

gate models could be created for reliability analysis.

**Table 7.**

**Table 8.**

**Figure 4.**

**83**

*Example 4: failure probability iterations.*

*Example 4: numerical results.*

*Example 4: random variables [50, 51].*

*Reliability Analysis Based on Surrogate Modeling Methods*

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

where *Su* is the ultimate material strength, *α<sup>m</sup>* is a dimensionless material utilization factor, and *ρ* is the mass density of material. **Table 7** lists the six random variables used in the example.

Similar as Example 3, all three surrogate models started with 30 sample points. In each subsequent iteration, 10 sample points were added. **Table 8** lists the

**Figure 3.** *Example 4: a rotating disk.*

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


#### **Table 7.**

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

This example is the reliability analysis of a disk with an angular velocity of *ω*, as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *αmSu*

(28)

*<sup>ρ</sup>* <sup>2</sup>*ωπ* ð Þ <sup>60</sup> <sup>2</sup> *<sup>R</sup>*<sup>3</sup> *<sup>o</sup>*�*R*<sup>3</sup> ð Þ*<sup>i</sup>* 3 385 ð Þ *:*82 ð Þ *Ro*�*Ri* � �

*g*ð Þ¼ x *Mb*ð*αm; Su; ρ;ω; Ro; Ri*Þ � 0*:*37473 (29)

vuuut

If a lower bound value of 0.37473 is used, the limit state function of *Mb* can be

where *Su* is the ultimate material strength, *α<sup>m</sup>* is a dimensionless material utilization factor, and *ρ* is the mass density of material. **Table 7** lists the six random

Similar as Example 3, all three surrogate models started with 30 sample points.

In each subsequent iteration, 10 sample points were added. **Table 8** lists the

shown in **Figure 3** [50, 51]. The inner and outer radii of disk are *Ri* and *Ro*, respectively. The burst margin, *Mb,* of the disk refers to the safety margin before

*Mb*ð*αm; Su; ρ;ω; Ro; Ri*Þ ¼

reliability analysis results (<2% error of *PF*).

*Reliability and Maintenance - An Overview of Cases*

*Example 3: failure probability iterations.*

overstressing the disk, which is expressed as:

written as:

**Figure 3.**

**82**

*Example 4: a rotating disk.*

**Figure 2.**

variables used in the example.

**7.4 Example 4: burst margin of a rotating disk**

*Example 4: random variables [50, 51].*


#### **Table 8.** *Example 4: numerical results.*

#### **Figure 4.**

*Example 4: failure probability iterations.*

estimated failure probability *PF* in this study based on different SRBF surrogate models and the associated errors as compared to the solution obtained using Direct MCS. The augmented SRBF-based methods required 60–70 original function evaluations to converge. **Figure 4** illustrates the variation of the failure probability *PF* versus number of sample points. In general with the increase of the sample size, a reduction was observed in the estimation errors of the failure probability *PF*, from 67.1, 6.6, and 12.8% when 30 sample points were used, to 5.6, 0.8, and 0.5% at convergence for SRBF-MQ-LP, SRBF-CS20-LP, and SRBF-CS30-LP, respectively. The reliability analysis results based on surrogate models SRBF-CS20-LP and SRBF-CS30-LP were shown to be better that using SRBF-MQ-LP. It showed that with around 50 sample points very accurate SRBF-CS20-LP and SRBF-CS30-LP surrogate models could be created for reliability analysis.
