Results for elastocplastic systems
By using the first set of records and carrying out nonlinear dynamic analysis of elastic – perfectly plastic single-degree-of-freedoom (SDOF) system, the obtained mean of the ductility demand is presented in Figures 1a to 1c for several values of the natural vibration period and $ = 2%, 5%, and 10%.
The results shown in the figure suggest that one may consider that the logarithmic of the expected ductility demand, mp, is a power function of 1п(Ф). This leads to
mM= exp((-«1ln^)), (1)
where a and P are parameters to be determined. The parameters for the models given in Eq. (1) may depend on the natural vibration period Tn and the damping ratio $. By minimizing the error є defined by,
є=Z (- mp(ф))2, (2) the estimates of a and P can be obtained. In Eq. (2), mis the mean of p obtained from the samples such as those shown in Figure 4, mp(0) represents mp predicted using Eq. (1) for each given set of values of Tn and $.
If one is interested in obtaining simple empirical equations for predicting a and P, the following fitted equations may be employed,
a = ajexp(a2/ Ta), (3)
where values of the parameters a, and bb i = 1,2,3, are given in Table 1. An example of the predicted mp obtained by using the model given in Eq. (1) with a and P calculated from Eqs. (3) and (4) is illustrated in Figure 2 for $ equal to 5%. Comparison of the results shown in this figure and those presented in Figure 1b suggests that the empirical predicting model provides a good approximation to those given in Figure 1b.
Table 1. Parameters for Eqs. (3) and (4)
The obtained cov of the ductility demand is illustrated in Figure 3 for $ equal to 0.05. The results shown in the figure suggest that for the mean ductility demand less than about 10 (see Figure 1) the cov of p increases as ф decreases. The cov of p for relatively rigid structures is larger than that for the flexible structures, and decreases as the damping ratio increases. In almost all cases with a mean ductility demand less than 10, the cov of p can be considered to be less than
1. 0. Similar trends of the cov values were observed for the results obtained for £ equal to 0.02 and 0.10.
Figure 2. Predicted expected ductility demand using the model given in Eq. (1) for £ = 5%.
Figure 3. Coefficient of variation of the ductility demand for elastic-perfectly-plastic SDOF system with £ = 5%.
Now if the second set of records mentioned in the previous section is employed, the obtained mean and cov of the ductility demand for £ = 0.05 are shown in Figures 4a and 4b. Comparison of the results shown in Figure 1b and Figure 4a and the results shown in Figure 3 and Figure 4b suggest that:
1) the difference between the predicted ductility demand obtained by using the first set of records and the second set of records is not very significant; and
2) The values of the cov of the ductility demand depend somewhat on the set of records used; however, the conclusion, that the cov of q is less than about 1.0 for the mean of q less than 10, is still adequate.
Note that no detailed analysis of the cov of ц was presented. This is because that the uncertainty in peak elastic displacement rather than that in the ductility demand is likely to play a dominant role in characterizing the uncertainty in the peak inelastic displacement since the cov of the peal elastic displacement usually ranges from 0.8 to higher than 10 for different sites in Canada.
Figure 4. Statistics of ductility demand obtained using the second set of records for elastic-perfectly-plastic SDOF system with £ = 5%.
To investigate possible probabilistic distribution models for the ductility demand, we plot the samples of the ductility demand in the lognormal probability paper for a given value of ф. It was found that the samples slightly curved, therefore, the assumption that the ductility demand is lognormally distributed can be very convenient but may not be very adequate. However, if the samples are presented in the Frechet probability paper as illustrated in Figure 5, the ductility demand samples could be approximate by straight liners for each given values of ф. Therefore,
the lognormal variate could be adopted to model the ductility demand but the use of Frechet distribution is preferred.
Figure 5. Frechet probability paper plot for the ductility demand samples Results for Bilinear systems
To investigate the effect of the strain-hardening on the statistics of the ductility demand, bilinear hysteretic systems are considered in this section. Let у denote the ratio of the post yield stiffness to the initial stiffness.
The analysis carried out for the elasto-perfectly-plastic SDOF systems is repeated for the bilinear system for the combinations of у (= 0.01, 0.05 and 0.1) and £ (= 0.01, 0.05 and 0.1) values. The obtained results are employed to find the values of a1 and в for the model given in Eq. (1) as was done for elasto-perfectly-plastic SDOF systems. Eqs. (3) and (4) are then employed to fit these values leading to the parameters presented in Table 1.