PSI - Issue 48

Behrooz Keshtegar et al. / Procedia Structural Integrity 48 (2023) 348–355 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

354

7

compared to PSO and DSTM. Generally, the PSO is an applicable non-gradient method for reliability problems compared to gradient-based methods (see Example 1). The performance of FORM using DSTM formula enhanced its performance compared to HL-RF, but this approach provides unstable results for some gray performance functions, whereas the non-gradient methods-based refined FORM are stable approaches. The refined FORM using PSO can produce more accurate results than the FORM using DSTM (see examples 2 and 3) but its efficiency depends on the nonlinearity of the performance functions and the PSO formulation.

Fig. 3. The steel frame structure

Table. 3. Random variables for steel frame structure

Random variable

Standard deviation 9.1×10 -4 8.09×10 -6 5.38×10 -4 8.36×10 -6 10.6×10 -4

Random variable 1 E (kg/m 2 E ( kg/m

Standard deviation 1.9×10 9 1.74×10 9

Mean

Distribution

Mean

Distribution

1 A (m 1 I (m 2 A (m 2 I (m 3 A (m 3 I (m

2 )

2 )

91×10 -4

Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal

2.38×10 10 2.17×10 10

Normal Normal Normal Normal Gumbel Gumbel

4 )

2 )

80.9×10 -6 53.8×10 -4 83.6×10 -6 106×10 -4

2 ) 4 ) 2 ) 4 )

L (m) H (m)

6

0.48 0.36 630 350

4.5

W (kg/m)

4500 2000

P (kg)

112.6×10 -6 11.26×10 -6

Table 4. Reliability results for steel frame example Method Iteration Call function

Reliability index

Error (%)

P f

MCS

--

12×10 5

3.07

0.00107

-- --

HL-RF DSTM

Not-converged

Chaos

--

38

2466 4500 2250

3.17 3.38 3.08

0.00076 0.00036 0.00104

3.6

HS

900 450

10.1

PSO

0.3

5. Conclusions The robustness and accuracy of two non-gradient methods using MPP search based on refined first order reliability method have been investigated. The non-gradient methods are harmony search optimization (HS) and particle swarm optimization (PSO). Two gradient approaches of HL-RF and directional stability transformation method (DSTM) are applied to illustrate the convergence performances of traditional FORM-based iterative formulations. Three nonlinear/complex problems are used to show the robustness and accuracy of these reliability methods. The non gradient methods are more robust approaches than the gradient approaches. The PSO is more robust and accurate compared to traditional FORM, and more accurate than HS. The performances of the PSO can improve the efficiency and accuracy using refined FORM optimization model in future.