Report - PEER - University of California, Berkeley

To determine m and σ , the maximum likelihood theory is used (Shinozuka et al.2000). Considering, for any case with the PHFA a i, the probability of exceeding alimit state is provided by F ( a i) , and for any case in which the limit state is notexceeded, the probability of exceeding that limit state is then provided by ( 1− F(ai)).The likelihood function L ( m,σ ) may then be expressed as:pn−p⎛ ⎞⎛⎞L(m,σ ) = ⎜∏F(a ⎟⎜⎟i)∏ (1 − F(a ))(7)j⎝ i=1 ⎠⎝j=1 ⎠where n = the total number of data points, p = number of cases in which the limit stateis exceeded, therefore, (n-p) = number of cases in which the limit state is notexceeded. To obtain the maximum values of L ( m,σ ) , the following two conditionsmust be satisfied:∂ ln L(m,σ )= 0(8)∂m∂ ln L ( m,σ )= 0(9)∂σSolving the above two-dimensional optimization problem numerically m and σ maybe determined. After obtaining m and σ , the probability of exceeding a limit statefor any PHFA a may be determined using Equation (6).i4.1 Sample Fragility Curves4. RESULTS AND DISCUSSIONFigure 3 provides a sample of the generated fragility curves, where a high and lowbound of µ (= 0.3 and 0.7) andsφ (= 0.5 and 0.9) are selected and damage measuresof (a) DM = 5cm, (b) DM = 10cm, (c) DM = 30 cm/sec and (d) DM = 50 cm/sec areshown. Note that the selection of µ andsφ were based on repeated static pull andinclined base tests performed on a subset of typical bench-mounted laboratoryequipment (Ray Chaudhuri and Hutchinson, 2004a). Selected DM values were basedon review of the general layout of typical laboratories, engineering judgment andobservations during shake table testing of these types of systems. Comparing (a) to(b), as the DM increases, the fragility curve becomes flatter, i.e., both the median andlog-standard deviation increase. It may also be noted that for higher µ values, thesspread between high and low φ (= 0.5 and 0.9) is larger. In Figure 3(b), large changesin PHFA are required to increase the probability of exceedance only moderately, atthe largest resistance parameters ( µ = 0.7 andsφ = 0.9), i.e., the curve is very flat incomparison with other curves shown. Figure 3 implies that when other parameters203