## 4 Dynamic Response Analysis of a Nonlinear Frame Structure  
### Engineering Background  
In order to investigate the dynamic reliability evaluation of a nonlinear frame structure subjected to explosive ground motions, a 10-story frame structure considering material nonlinearity is studied herein as the engineering object. Figure 7 shows the diagrammatic sketch of the nonlinear frame structure, where the height of the first story is \(h_{1}=4\) m, and the height of other stories is \(h=3\) m. The sections of the bottom and rest columns are, respectively, \(750\times 750\) mm\({}^{2}\) and \(600\times 600\) mm\({}^{2}\). It is assumed that the beam stiffness is infinite. The lumped masses from top to bottom are 1.5, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0 and 2.2 (\(\times 10^{5}\) kg), in turn. The initial Young's modulus from top to bottom are, respectively, 2.8, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0 and 3.2 (\(\times 10^{10}\)N/m\({}^{2}\)). The Rayleigh damping is adopted--namely, \(\mathbf{C}=a\mathbf{M}+b\mathbf{K}\), where \(\mathbf{M}\), \(\mathbf{K}\) and \(\mathbf{C}\) indicate the matrix of mass, stiffness and damping, respectively. The parameters of the Rayleigh damping are determined as \(a=0.01\) Hz and \(b=0.005\) s.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Cases**} & \multicolumn{4}{c}{**The Number of the Representative Samples \(\boldsymbol{n_{\text{sel}}}\)**} \\ \cline{3-8}  & & \multicolumn{2}{c}{**144**} & \multicolumn{2}{c}{**377**} & \multicolumn{2}{c}{**610**} \\ \cline{3-8}  & & \(\boldsymbol{\varepsilon_{\text{mean}}(\%)}\) & \(\boldsymbol{\varepsilon_{\text{Std. D}}(\%)}\) & \(\boldsymbol{\varepsilon_{\text{mean}}(\%)}\) & \(\boldsymbol{\varepsilon_{\text{Std. D}}(\%)}\) & \(\boldsymbol{\varepsilon_{\text{mean}}(\%)}\) & \(\boldsymbol{\varepsilon_{\text{Std. D}}(\%)}\) \\ \hline Dimension- & \(\rho=0.02\) & \(1.14\times 10^{-14}\) & 4.62 & \(8.56\times 10^{-15}\) & 4.29 & \(4.27\times 10^{-15}\) & 3.53 \\ reduction & \(\rho=0.03\) & \(1.06\times 10^{-14}\) & 4.38 & \(8.30\times 10^{-15}\) & 3.94 & \(4.19\times 10^{-15}  
For describing the characteristics of structural restoring forces, the extended Bouc-Wen model [43; 44] is adopted in this paper, which can be broken down as the form of the sum of elastic forces and hysteretic forces, followed by:  
\[G(X,Z)=\alpha KX+(1-\alpha)KZ \tag{20a}\] \[\dot{Z}=h(Z)\frac{A\dot{X}-\
u(\beta\Big{|}\dot{X}\Big{|}|Z|^{n-1}Z+\gamma\dot{X }|Z|^{n})}{\eta} \tag{20b}\]  
where \(K\) indicates the initial stiffness, \(\alpha\) indicates the ratio between post-yielding stiffness and initial stiffness, \(X\) denotes the story drift ratio and \(Z\) is hysteretic component with the dimension of displacement, respectively. \(A\), \(\beta\), \(\gamma\) and \(n\) are the parameters controlling the initial stiffness and amplitude of the hysteresis displacement as well as hysteresis shape. \(\
u\) and \(\eta\) are the parameters which, respectively, reflect the degradation of strength and stiffness, of which:  
\[\
u=1+d_{\
u}\varepsilon \tag{21a}\] \[\eta=1+d_{\eta}\varepsilon \tag{21b}\]  
where \(d_{\
u}\) and \(d_{\eta}\) cause the degradation of strength and stiffness, respectively.