The uncertainty in the probability of collapse is from various sources. The uncertainty in the probability of collapse, calculated using the results of structural response history
Figure 2.23: Collapse fragility curves obtained by fitting lognormal distributions to the data points obtained using the IM-based and EDP-based approaches, from [94]
analyses of numerical models of the building under different ground motion records, have three major sources: (i) the record-to-record (RTR) variability in structural response; (ii) the variability of the building system parameters; and (iii) the uncertainty regarding the numerical modeling decisions and parameters. The RTR variability in structural response is directly included by using the structural response from different ground motion records when conducting an IDA. The contribution of the variability of the building system pa- rameters and the uncertainty regarding the numerical modeling decisions and parameters may be be included using techniques such as Monte Carlo (MC) simulation or first-order second-moment (FOSM) method. The variability of the building system parameters has been included in previous studies. Luco and Cornell [48, 47] study the effect of random connection fractures on the dynamic response of steel moment frames. Foutch et al. [24] considered the effect of hysteretic models for the moment connections.
Ibarra and Krawinkler [35, 34] have studied the effect of the variation of deterioration parameters on the collapse capacity of SDOF oscillators using FOSM method. A peak
Figure 2.24: Backbone curve for hysteresis models, from [35]
oriented deterioration model developed by Ibarra et al. [36] was used for the SDOF os- cillators, as shown in Figure 2.24. The amount of deterioration is defined to be a function of the hysteresis energy dissipated and a selected reference hysteretic energy dissipation capacity of the component, calibrated to experimental results [35]. The ductility capacity (δc/δy), the post-capping stiffness ratio (αc) and the cyclic deterioration rate (γ) are defined as random system parameters.
Ibarra and Krawinkler estimate the variance of collapse the capacity (denoted by Sa,C) using FOSM method as follows [35, 9, 8]:
σln S2 a,C(T OT ) ∼ = n
∑
i=1 n∑
j=1 ∂ g( ¯X) ∂ xi ∂ g( ¯X) ∂ xj ¯ X= ¯µx ρxi,xjσxiσxj+ σ 2 ln Sa,C(RT R) (2.3)where ¯X is the vector of the natural logarithm of the probabilistic system parameters, ¯µxis the vector of the mean values of ¯X, g is the collapse performance function in the log domain, ρxi,xj is the correlation coefficient between the i
th and jth random system parameters, and
σxi is the standard deviation of the i
th random system parameter. The first term on the right hand side of Equation (2.3) is the contribution of the uncertainty in all random system
Figure 2.25: Approximating partial derivative of collapse performance function with re- spect to random system parameter, from [35]
parameters to the variance of the collapse capacity. The second term is the contribution of the RTR variability to the variance of the collapse capacity [35].
Since it is not practical to determine a mathematical expression for the collapse perfor- mance function of structural systems (i.e., g( ¯X)), the partial derivatives of g( ¯X) in the first term of the right hand side of Equation (2.3) are being estimated numerically. The median collapse capacity is calculated for two variations of each system parameter: (i) µx− σx; and (ii) µx+ σx. The variation of the collapse performance function g( ¯X) with respect to a system parameter is ∂ g( ¯X)/∂ xifor the ithrandom system parameter in Equation (2.3). This numerical determination of the partial derivative is schematically shown in Figure 2.25.
Figure 2.26 shows the contribution of different terms of Equation (2.3) to the collapse capacity of SDOF oscillators with different natural periods. The RTR variability in struc- tural response seems to be the greatest contributor and the combined correlation of the system parameters is the second largest contributor. The combined correlation of the sys- tem parameters is the sum of those terms of the double summation of Equation (2.3) when i6= j [35].
Figure 2.26: Relative contribution of different random system parameters to variance of collapse capacity, from [35]
Liel et al. [45] use a combination of the FOSM method and Monte Carlo simulation approach coupled with a response surface methodology to evaluate the collapse uncertainty of concrete MRFs. An example of the response surface developed for the collapse capacity versus the variability in beam and column strength and ductility is shown in Figure 2.27. The assumed plastic hinge backbone curve for beam and columns is similar to the one introduced by Ibarra et al. [35], shown in Figure 2.24. Several parameters of the back- bone curve such as flexural strength, initial stiffness, post-yield stiffness, capping point, post capping deformation capacity, and cyclic deterioration are included as random system parameters for beams and columns. The collapse fragility curve for a 4-story reinforced concrete moment frame developed with and without considering the modeling uncertain- ties is shown in Figure 2.28 [45]. It can be seen that considering the effect of the system uncertainty increases the dispersion in collapse fragility curve.
Figure 2.27: Graphical representation of the polynomial response surface for collapse ca- pacity of 4-story ductile MRF. Each plot represents a slice of a multidimensional surface: (a) effects of column strength and beam strength are shown, while beam ductility and col- umn ductility variables are held constant (at 0, their mean values); (b) effects of varying beam and column ductility, from [45]
Figure 2.29: Sensitivity to variation of ductility capacity of beam hinges: (a) static pushover analysis results; and (b) median IDA curves, from [87]
varying the backbone curve random parameters for the plastic hinges in beams of the MRF. Figure 2.29 shows the sensitivity of the static pushover curve and median IDA curve to variation of the ductility capacity of the beam hinges. The effect of uncertainty in the beam hinge backbone parameters are evaluated using Monte Carlo simulation, an FOSM method, and a point estimation method. The lognormal standard deviation of Sa(T1, 5%), determined at different maximum story drift ratio values, is shown in Figure 2.30, where LHS refers to Monte Carlo simulation results, FOSM refers to the FOSM results, and PEM refers to the point estimate results. The distribution of Sa(T1, 5%) values at four values of maximum story drift ratio due to the variation of MRF plastic hinge ductility is shown in Figure 2.31. The variability in the beam hinge ductility is an important contributor to the overall dispersion in the performance of a steel MRF.