## 2 Equation of Motion and Its Solution Considering Structural Nonlinearity  
The equation of motion of an \(N\) degree-of-freedom damped linear structural system subjected to external excitation can be represented as  
\[\mathbf{M\ddot{x}}+\mathbf{C\dot{x}}+\mathbf{Kx}=\mathbf{L}\mathbf{F} \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrices of the structural system, respectively. \(\mathbf{F}\) is the vector of external excitation forces on the structure and \(\mathbf{L}\) is the mapping matrix for the input forces. \(\mathbf{\ddot{x}}\), \(\mathbf{\dot{x}}\), and \(\mathbf{x}\) are vectors of acceleration, velocity, and displacement of the structural system, respectively. The Rayleigh damping model was used in this study and assumed with Equation (2)  
\[\mathbf{C}=a_{1}\cdot\mathbf{M}+a_{2}\cdot\mathbf{K} \tag{2}\]  
where \(a_{1}\) and \(a_{2}\) are the Rayleigh damping coefficients. When the structural system is nonlinear, the discrete equation of motion can be represented with  
\[\mathbf{M\ddot{x}}_{k}+\mathbf{C\dot{x}}_{k}+\mathbf{R}_{k}(\mathbf{x})= \mathbf{L}\mathbf{F}_{k} \tag{3}\]  
where \(\mathbf{R}_{k}(\mathbf{x})\) is the restoring force of the structure which is a nonlinear function of the displacement and \(k\) represents the integration time step. When the structure is subjected to seismic excitation, the equation of motion for nonlinear structure becomes  
\[\mathbf{M\ddot{x}}_{k}+\mathbf{C\dot{x}}_{k}+\mathbf{R}_{k}(\mathbf{x})=- \mathbf{M\ddot{x}}_{g,k} \tag{4}\]  
where \(\mathbf{\ddot{x}}_{g}\) is the acceleration at ground level. The Newmark-\(\beta\) method can be used to solve the response of linear system shown in Equation (1) with the following assumption:  
\[ \begin{array}{l}\mathbf{\dot{x}}_{k+1}=\frac{2}{\Delta t}(\mathbf{x}_{k+1}- \mathbf{x}_{k})-\mathbf{\dot{x}}_{k}\\ \mathbf{\ddot{x}}_{k+1}=\frac{2}{\Delta t}\left(\mathbf{\dot{x}}_{k+1}-\mathbf{ \dot{x}}_{k}\right)-\mathbf{\ddot{x}}_{k}\end{array}  \tag{5}\]  
where \(\Delta t\) represents the time interval in the integration. The Newmark-\(\beta\) method for a linear structural system can be extended to solve Equations (3) and (4) representing a nonlinear system. The time-step solution with the Newmark-\(\beta\) method was commonly used, however, it was found that the Newmark-\(\beta\) method cannot guarantee stability in the case of geometric nonlinearity [26; 27]. To improve the accuracy and stability of the dynamic analysis, a new equation of motion in discrete form is provided in the middle of each time step instead of Equation (3) or Equation (4) with the following assumption:  
\[\dot{\mathbf{x}}_{k+1}=\dot{\mathbf{x}}_{k}+\Delta t\mathbf{M}^{-1}[\mathbf{LF}_ {\text{m}}-\mathbf{Cv}_{\text{m}}-\mathbf{R}_{\text{m}}(\mathbf{x})] \tag{6}\]  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}+\Delta t\frac{\dot{\mathbf{x}}_{k+1}+\dot{ \mathbf{x}}_{k}}{2} \tag{7}\]  
where \(\mathbf{F}_{\text{m}},\mathbf{v}_{\text{m}}\), and \(\mathbf{R}_{\text{m}}\) are the average external force, velocity, and resisting force between the \(k\) and \(k+1\) time step with the definition as follows  
\[ \begin{array}{l}\ddot{\mathbf{x}}_{\text{m}}=\frac{\ddot{\mathbf{x}}_{k+1}+ \ddot{\mathbf{x}}_{k}}{2}\\ \dot{\mathbf{x}}_{\text{m}}=\frac{\ddot{\mathbf{x}}_{k+1}+\dot{\mathbf{x}}_{k }}{2}\\ \mathbf{x}_{\text{m}}=\frac{\mathbf{x}_{k+1}+\mathbf{x}_{k}}{2}\\ \mathbf{F}_{\text{m}}=\frac{\mathbf{F}_{k+1}+\mathbf{F}_{k}}{2}\end{array}