## 2 Theoretical Concepts
### Energy Approach to Structural Response  
To analyze the insights of the inelastic behavior, energy formulation is used in this paper. Although the energy balance has little practical significance to most of the practitioners in the field of civil engineering, energy is a fundamental parameter for all hydrocodes used in the analysis of special problems like impact analysis, fluid-structure interaction, blast analysis, and in general in continuum mechanics. Thus, the energy balance will give an insight into the energy dissipation capacity of a civil engineering structure. The energy balance is obtained by a simple integration of the equation of motion (Equation (1)), over the path of the total displacement vector **u(t)**. The terms given in Equation (18) are rearranged to illustrate the kinetic energy \(KE_{k}\), the energy dissipated by viscous damping \(DE_{k}\), and the input energy \(IE_{k}\). The third term in the left hand side of the equation of motion can be separated in accordance with the total displacement decomposition given in Equation (2) for \(\mathbf{du}=\mathbf{d\ddot{u}}+\mathbf{du}^{{}^{\prime\prime}}\), to obtain Equation (23). The significance of the terms given in Equation (23) is the strain energy \(SE_{k}\) and hysteresis energy, as they are detailed in Equations (24) and (25).  
\[\int_{t_{0}}^{t_{k}}(\mathbf{M}\cdot\mathbf{\ddot{u}}+\mathbf{C}\cdot\mathbf{ \dot{u}}+\mathbf{K}_{0}\cdot\mathbf{\ddot{u}})\mathbf{du}=-\int_{t_{0}}^{t_{k }}(\mathbf{M}\cdot\mathbf{h}\cdot a_{g})\mathbf{du} \tag{18}\]  
Figure 1: Capping bending value to capacity.  
\[\begin{split}&\int_{t_{0}}^{t_{k}}(\tilde{\mathbf{u}}^{T}\cdot \mathbf{M})\mathbf{du}+\int_{t_{0}}^{t_{k}}(\tilde{\mathbf{u}}^{T}\cdot\mathbf{ C})\mathbf{du}+\\ &+\int_{t_{0}}^{t_{k}}(\tilde{\mathbf{u}}^{T}\cdot\mathbf{K}_{0}) \mathbf{du}=-\int_{t_{0}}^{t_{k}}[(\mathbf{h}\cdot a_{g})^{T}\mathbf{M}] \mathbf{du}\\ & KE_{k}=\int_{t_{0}}^{t_{k}}(\tilde{\mathbf{u}}^{T}\cdot\mathbf{ M})\mathbf{du}=\int_{t_{0}}^{t_{k}}(\tilde{\mathbf{u}}^{T}\cdot\mathbf{M}) \mathbf{d}\tilde{\mathbf{u}}=\\ &=\frac{1}{2}\mathbf{u}_{\mathbf{k}}^{T}\cdot\mathbf{M}\cdot \tilde{\mathbf{u}}_{k}\end{split} \tag{19}\]  
\[\begin{split} HE_{k}=\int_{t_{0}}^{t_{k}}(\tilde{\mathbf{u}}^{T} \cdot\mathbf{C})\mathbf{du}=\mathbf{DE}_{\mathbf{k}-\mathbf{1}}+\tilde{ \mathbf{u}}_{\mathbf{k}}^{T}\cdot\mathbf{C}\cdot(\mathbf{u}_{\mathbf{k}}- \mathbf{u}_{\mathbf{k}-\mathbf{1}})\end{split} \tag{20}\]  
\[\begin{split} IE_{k}&=-\int_{t_{0}}^{t_{k}}[(\mathbf{h }\cdot a_{g})^{T}\mathbf{M}]\mathbf{du}=\\ &=IE_{k-1}-a_{g}\cdot\mathbf{h}^{T}\cdot\mathbf{M}\cdot(\mathbf{ u}_{\mathbf{k}}-\mathbf{u}_{\mathbf{k}-\mathbf{1}})\end{split} \tag{22}\]  
\[\begin{split}\int_{t_{0}}^{t_{k}}(\tilde{\mathbf{u}}^{T}\cdot \mathbf{K}_{0})\mathbf{du}=\int_{t_{0}}^{t_{k}}(\tilde{\mathbf{u}}^{T}\cdot \mathbf{K}_{0})\mathbf{d}\tilde{\mathbf{u}}+\int_{t_{0}}^{t_{k}}(\tilde{ \mathbf{u}}^{T}\cdot\mathbf{K}_{0})\mathbf{du}^{{}^{\prime\prime}}\end{split} \tag{23}\]  
\[\begin{split} SE_{k}&=\int_{t_{0}}^{t_{k}}(\tilde{ \mathbf{u}}^{T}\cdot\mathbf{K}_{0})\mathbf{d}\tilde{\mathbf{u}}=\frac{1}{2} \cdot\tilde{\mathbf{u}}_{\mathbf{k}}^{T}\cdot\mathbf{K}_{0}\cdot\tilde{ \mathbf{u}}_{\mathbf{k}}=\\ &=\frac{1}{2}\cdot(\mathbf{u}_{\mathbf{k}}-\mathbf{u}_{\mathbf{k }}^{{}^{\prime\prime}})\cdot\mathbf{K}_{0}\cdot(\mathbf{u}_{\mathbf{k}}- \mathbf{u}_{\mathbf{k}}^{{}^{\prime\prime}})\end{split} \tag{24}\]  
\[\begin{split} HE_{k}&=\int_{t_{0}}^{t_{k}}(\tilde{ \mathbf{u}}^{T}\cdot\mathbf{K}_{0})\mathbf{du}^{{}^{\prime\prime}}=\int_{t_{0 }}^{t_{k}}(\tilde{\mathbf{u}}^{T}\cdot\mathbf{K}_{P})\mathbf{d}\theta^{{}^{ \prime\prime}}=\\ &=HE_{k-1}+(\mathbf{u}_{\mathbf{k}}-\mathbf{u}_{\mathbf{k}}^{{}^{ \prime\prime}})^{T}\cdot\mathbf{K}_{P}\cdot(\theta_{\mathbf{k}}^{{}^{\prime \prime}}-\theta_{\mathbf{k}-\mathbf{1}}^{{}^{\prime\prime}})\end{split} \tag{25}\]