Feasibility of a nonlinear spectral approach for peak floor acceleration of multi-story bilinear hysteretic buildings

Abstract

To protect the acceleration-sensitive non-structural components in a multi-story building, a proper seismic design of them against the peak floor acceleration (PFA) is needed. The PFA, therefore, should be estimated in advance. To efficiently estimate the PFA of multi-story buildings at the inelastic stage, a nonlinear modal combination approach is provided in this study. First, for the single-story building (single-degree-of-freedom system), a PFA spectral ratio δ, defined as the ratio between the inelastic PFA and the elastic spectral amplitude, is proposed to measure the nonlinear reductions in PFA. A series of time history analysis (THA) revealed that there is a dependable relationship between the PFA spectral ratio δ, the strength reduction factor R, the post-yield stiffness ratio α, and the period T of the structure. The scatter of δ is notably smaller than the ductility demand, showing the good feasibility of a δ-R-T-α relationship for predicting δ. Second, for the multi-story building (multi-degree-of-freedom system), the floor acceleration is decoupled in the modal space, the contribution of each mode to the elastic PFA is multiplied by δ to account for the nonlinear reduction in PFA at the inelastic stage. Here δ can be determined by an appropriate δ-R-T-α relationship, while R and α are determined via a modal pushover analysis. The modified modal contributions are combined via the complete-quadratic combination (CQC) rule, forming the inelastic CQC approach. The PFAs of some 2-, 5-, and 8-story building structures with different degrees of nonlinearity are computed via the THA and the inelastic CQC approach. Results demonstrate the satisfactory accuracy of the latter. Notably, the effects of nonlinearities associated with the higher-order modes on the nonlinear PFA are well considered in the inelastic CQC, thus the un-acceptable over-estimations of the inelastic PFA, which occurs in conventional methods, are avoided.

Appendix

An empirical δ-R-T-α formula is obtained by fitting the δ-R-T-α curves generated by the earthquake records mentioned in Fig. 7. These 300 randomly selected records are from three different types of sites, yet the computation results show that the δ-R-T-α relationship is basically not influenced by the site class (soil condition). It is worth noting that the record collection does not contain any near-fault excitations. To make it compact, the value of δ is expressed using T and α, while the coefficients in the equation are determined by R, as shown in Eq. (12) and (13).

$$\delta = (f_{1} \cdot \alpha^{3} + f_{2} \cdot \alpha^{2} + f_{3} \cdot \alpha + f_{4} ) \cdot T + f_{5} \cdot \alpha^{3} + f_{6} \cdot \alpha^{2} + f_{7} \cdot \alpha + f_{8}$$

(12)

$$\left\{ \begin{gathered} {}_{{}}f_{1} = {0}{\text{.01664}}R^{{2}} - {0}{\text{.1508}}R + {0}{\text{.1829}} \hfill \\ {}_{{}}f_{2} = - {0}{\text{.01998}}R^{{2}} + {0}{\text{.1901}}R - {0}{\text{.2253}} \hfill \\ {}_{{}}f_{3} = {0}{\text{.4045}}R^{{ - {0}{\text{.23}}}} - {0}{\text{.3967}} \hfill \\ {}_{{}}f_{4} = - {0}{\text{.0359}}R^{{ - {1}{\text{.519}}}} + {0}{\text{.05571}} \hfill \\ {}_{{}}f_{5} = {0}{\text{.008132}}R^{3} - {0}{\text{.1271}}R^{2} + {0}{\text{.6227}}R - {0}{\text{.6894}} \hfill \\ {}_{{}}f_{6} = - {0}{\text{.003757}}R^{3} + {0}{\text{.07128}}R^{2} - {0}{\text{.4004}}R + {0}{\text{.6086}} \hfill \\ {}_{{}}f_{7} = - {1}{\text{.474}}R^{{ - {0}{\text{.3591}}}} + {1}{\text{.409}} \hfill \\ {}_{{}}f_{8} = {1}{\text{.026}}R^{{ - {0}{\text{.9845}}}} - {0}{\text{.005983}} \hfill \\ \end{gathered} \right.$$

(13)

The δ-R-T-α relationship defined by Eq. (12) and (13) exhibits a good accuracy. For demonstration, the mean values of δ from the nonlinear RHA are plotted against those given by Eq. (12) and (13), as shown in Fig. 15. 

Fig. 15

Values of δ from nonlinear RHA and Eq. (12): a T = 0.1 s, b T = 0.2 s, c T = 0.5, d T = 1.0 s, e T = 1.5 s, f T = 2.0 s

The RHA results are marked by black dots, while the simulations are shown using surfaces in blue-brown. For brevity, only the results at T = 0.1 s, 0.2 s, 0.5 s, 1.0 s, 1.5 s, and 2.0 s are given here. From Fig. 15, the simulation well captures the RHA results for T ≥ 0.2 s, while for the very short period (T ≤ 0.1 s), some minor discrepancies can be found. Lastly, it is emphasized that the δ-R-T-α formula given here is an empirical δ-R-T-α relationship established based on limited earthquake excitations. The formula only reveals the general trend of the δ-R-T-α relationship. More specific δ-R-T-α formulas should be developed in the future for different seismic scenarios.