Report - PEER - University of California, Berkeley

to allow a quantitative assessment of their relative importance). It can be noted that infact their influence is almost insignificant in this case, which is another way ofconfirming that displacement-related response quantities such as εccare only weaklydependent on strength-related mechanical parameters.For what concerns the shear demand, after yielding of the column extremities, asexpected the curve tends to flatten. Again, the adjacent plots show the sensitivities ofthe shear demand with respect to f cand f . While the first one is on the averageyzero, the second one shows that for a positive variation of the yield stress the increasein shear demand is independent of the PGA value. This is expected, since thevariation in shear force due an increase in the yield stress remains constant along thehardening branch.Finally, it is worth commenting that the sensitivities of the response with respectto the capacity variables εcc, uand ε Vhave not been computed. This fact comesunavoidably from a limitation of the available analytical tools, which do not allow toaccount in the course of the analysis for the modification of the response due to theattainment of the capacity in some of the members. Hence, any perturbation in thecapacity parameters would go undetected during the analysis, yielding identicallyzero derivatives.3.4.1 Fragility CurvesOnce the demand variables are statistically determined, reliability analysis canproceed as indicated in Section 2. The most straightforward and accurate way toevaluate the system probability of failure in Eq. (10) is to resort to Monte Carlosimulation. It is recalled that at this stage no more structural analyses are needed andthat a trial of the MC simulation simply consists of sampling from the jointdistribution of x (in this case f , f ,ε , ε ) and checking the state of the system. Itc y cc, u Vis worth observing that the evaluation of the entire fragility by MC simulation at thislast stage of the procedure usually involves less effort than a single non-lineardynamic analysis.Figure 6 (Left) contains the fragility curves for the structure, evaluated by Eq.(10), as well as by simpler alternative procedures. In particular, the simplest one isthat based on the assumption of independence among the failure modes (Eq. (3)),while in the second alternative the sensitivities are ignored, as suggested by theirmodest influence on the response (see Figure 5).One can note that the independence assumption leads to quite different resultsfrom the other curves, that are considerably more severe in the upper part of thefragility.The closeness between curves (a) and (c), i.e., between considering or ignoringthe dependence of the demands on x through the sensitivities, is a further indicationthat in many cases, such as the present one, the dominant effect is the ground-motioninducedvariability, which tends to overshadow that related to structural randomness.230