## 4 Dynamic Response Analysis of a Nonlinear Frame Structure
### Engineering Background
\(\varepsilon\) indicates the hysteretic energy, represented as:  
\[\varepsilon(t)=\int_{0}^{t}Z\dot{X}\ \ \ \mathrm{d}t \tag{22}\]  
in which \(h(Z)\) is the function which reflects the pinch effect, defined as the following:  
\[h(Z)=1-\zeta_{1}\exp(-\frac{\left[Z\mathrm{sgn}(\dot{X})-qZ_{u}\right]^{2}}{ \zeta_{2}^{2}  
denote the \(n_{\rm d}\times 1\) vectors of structural displacement, velocity and acceleration relative to ground, respectively. \(\mathbf{F}(\cdot)\) indicates the nonlinear restoring force vector.  
It can be acknowledged from that the randomness of the frame structure is caused by the elementary random variable \(\Theta\). To this goal, the following is the generalized probability density evolution equation (PDEE) of a stochastic dynamic system, where \(Q\) is some physical-response quantity of interest (it might be displacement, stress or anything else) [24]:  
\[\frac{\partial p_{\rm QQ}(q,\theta,t)}{\partial t}+\tau(\theta,t)\frac{ \partial p_{\rm QQ}(q,\theta,t)}{\partial q}=0 \tag{26}\]  
where \(p_{\rm QQ}(q,\theta,t)\) indicates the joint probability density function (PDF) of the stochastic process \(Q(\Theta,t)\). Further, the velocity of \(Q\) can be defined as \(\dot{Q}=\tau(\Theta,t)\). Finally, the PDF of interested physical quantity \(Q(t)\) can be obtained through the following integral equation:  
\[p_{\rm Q}(q,t)=\int_{\Omega}p_{\rm QQ}(q,\theta,t)\ \ d\theta \tag{27}\]  
where \(\Omega\) denotes the distribution domain of the elementary random variable \(\Theta\). Figure 8a shows the mean and Std. D of the dynamic response of story drift ratio under the explosive ground motion representative time-histories with \(\rho=0.04\) of the fourth and eighth stories. Figure 8b shows the restoring force vs. the story drift ratio under a typical explosive ground motion representative time-history of the fourth and eighth stories. It can be seen from Figure 8a that the dynamic response of higher stories is greater than that of lower stories, and the time instance of the greatest dynamic response is around 1 s, which corresponds to the arrival time of the PGA of the simulated explosive ground motions as shown in Figure 4. It can be acknowledged from Figure 8b that the strong nonlinearity involved in this frame structure model and the dynamic response of higher stories are greater than that of lower stories. Figure 9 demonstrates how the PDEM can be used to accurately capture the probability information of the dynamic response of the 10-story frame structure. In specific, the PDFs of the story drift ratio of the eighth story at the three given time instances, i.e., \(t=2\) s, 3 s and 4 s, are presented in Figure 9a, from which it can be clearly acknowledged that the PDFs of dynamic response at different time instances are quite different. According to the simulated representative samples of the explosive ground motions, the intensity approaches the peak value around 1 s and then gradually decays over the duration, indicating that the maximum story drift ratio of the nonlinear frame structure is most likely to occur at about 1 s. In fact,  
Figure 8: The statistical characteristics of dynamic response and restoring force: (**a**) mean and Std. D of dynamic response; (**b**) restoring force. the story drift ratio reaches nearly 0.003 at 2 s, and then gradually decreases as time goes by. Figure 9b depicts the probability density evolution information for the eighth story's drift ratio in the time interval from \(t=2\) s to \(t=4\) s. It depicts the PDF's contour and resembles a streaming liquid, describing the probability density's changes over time. Apparently, the PDFs of story drift ratio from 2 s to 4 s gradually decreases from 0.0025. It has thus been proven without any reasonable doubt that the proposed dimension-reduction method for modeling explosive ground vibrations is successful and can be used in conjunction with the PDEM to execute the refined dynamic response.