## 2 SDoF Model for Low-Velocity Heavy-Mass Impact
### Loading Stage
The general conditions for Regime I are as follows: \(M_{0}<M_{N}\), \(M_{1}<M_{N}\) and \(M_{2}<M_{N}\), where \(M_{N}\) is the bending moment at a plastic hinge related to axial force \(N\):  
\[M_{N}(W)=\left[1-\left(\frac{N(W)}{N_{p}}\right)^{2}\right]M_{p} \tag{13}\]  
It is evident from Equations (7)-(9) that \(M_{1}>M_{0}>M_{2}\) for \(l_{1}<l_{2}\), which suggests that the initiation of plastic yielding occurs at the left support \(x=-l_{1}\). Thus, the terminating condition for Regime I is \(M_{1}=M_{T}\) at the inception of yield \(t=t_{1y}\). At \(t=t_{1y}\), the critical displacement \(W_{1y}\), obtained by substituting Equations (6) and (7) into Equation (4), and the instantaneous velocity \(\dot{W}\big{(}t_{1y}\big{)}\) are used as the initial condition for Regime II. _Regime II_ represents the elastic-plastic deformations developed in the beam when the plastic hinge has formed at the left support \(x=-l_{1}\). To simplify the theoretical analysis of the unloading path, the deformation shape function is assumed to remain the same as that defined in Equation (5) until the plastic hinges are formed at the two supports and the impact point. As a result, the equivalent mass of the beam in regimes II and III can be estimated by Equation (11). In addition, owing to the fact that the bending moment dominates the beam's behavior under a relatively smaller deformation, it is reasonable to assume that prior to the plastic hinge formation at the impact point, the elastic behavior in terms of the axial force persists [20]. With these assumptions, in this regime, the axial force can be calculated via Equation (6), and the bending moments at \(x=-l_{1}\), \(l_{2}\) and \(0\) can be calculated via Equations (13), (8) and (9), respectively. Then, when substituting the corresponding values of \(M\) and \(N\) into Equation (3), the beam's resistance during this regime can be obtained as  
\[F=\left[\frac{M_{N}+M_{0}}{l_{1}}+\frac{M_{2}+M_{0}}{l_{2}}\right]+N\left( \frac{1}{l_{1}}+\frac{1}{l_{2}}\right)W \tag{14}\]  
The general conditions for Regime II are: \(M_{0}<M_{N}\), \(M_{1}=M_{N}\) and \(M_{2}<M_{N}\). This regime continues until a critical displacement \(W_{2y}\) is attained to define the plasticity hinge formation at \(x=0\) for \(M_{0}>M_{2}\). Thus, the terminating conditions for this regime are \(M_{0}=M_{N}\) at \(t=t_{2y}\). When the terminating conditions for Regime II have been reached, the response of the beam is about to enter the next regime, with the initial conditions, namely \(W=W_{2y}\) and \(\dot{W}=\dot{W}\big{(}t_{2y}\big{)}\). _Regime III_ represents the elastic-plastic deformations developed in the beam when the plastic hinges have formed at the left support and the impact point, and plastic yielding is about to occur at the right point. In this regime, the axial force \(N\) and the bending moment \(M_{2}\) are obtained via Equations (6) and (9), respectively, while the bending moments \(M_{0}\) and \(M_{1}\) are obtained by Equation (13). Hence, the beam's resistance during this regime can be formulated as  
\[F=\left[M_{N}\left(\frac{2}{l_{1}}+\frac{1}{l_{2}}\right)+\frac{M_{2}}{l_{2}} \right]+N\left(\frac{1}{l_{1}}+\frac{1}{l_{2}}\right)W \tag{15}\]  
The general conditions for Regime III are \(M_{0}=M_{N}\), \(M_{1}=M_{N}\) and \(M_{2}<M_{N}\). This regime continues until the plasticity hinge forms at the support \(x=l_{2}\), implying that the terminating condition for this regime is \(M_{2}=M_{N}\) at time \(t=t_{3y}\). The critical displacement \(W_{3y}\) when reaching the terminating condition of Regime III is attained from Equation (4) with the substitution of Equations (6) and (9). The axial force \(N=N_{3y}\) at \(W=W_{3y}\) is obtained by Equation (6), which is used as the initial value to determine the dependence of the axial force on the displacement of the beam in Regime IV. _Regime IV_ represents the elastic-plastic deformations developed in the beam when the plastic hinges have formed at the supports and the loading point.