EXAMPLE
313
–3
×10
4. 5
Failure probability
4 3. 5 3 2. 5 2 1. 5 1 0. 5 0 –1 –0.8 –0.6 –0.4 –0.2
0
0.2 0.4
0.6
0.8
1
Correlation coefficient Figure 17.11 Effect of correlation of variables on failure probability of the steel beam
17.6.2 A Steel Portal Frame In this example a steel portal frame, as shown in Figure 17.12, is investigated. Uncertainties of the frame are represented by the limit yielding moments Mc and Mb respectively for the columns and the beam employed in the portal frame. The statistics of Mc and Mb and those of the load, P, are given in Table 17.9, with assumed normal distribution. The limit state function of the portal frame can be expressed as G ¼ R S ¼ l S S;
ð17:70Þ
where R ¼ l S is the structural resistance (i.e. limit load-bearing capacity), S is the load effect including the actions of the two concentrated loads, P and 2P, and l is the load factor. Two different models for structural analysis are used to determine the structural resistance. One is the second-order inelastic structural analysis and the other is the rigid-plastic structural analysis. Failure probabilities of this frame with DMCS are Pf1 ¼ 2:548 10 3 (by second-order inelastic analysis model, 2P
P
Mc
Mc
l=5m
h = 5m
Mh
l = 5m
Figure 17.12 A simple steel portal frame
Table 17.9 Statistics of the variable in the portal frame Variables Mean value Coefficient of variation
Mc
Mb
P
75 kN m 0.05
150 kN m 0.05
20 kN 0.3