## 2 SDoF Model for Low-Velocity Heavy-Mass Impact
### Loading Stage
In order to ensure continuity of the transverse displacement profile at \(W=W_{3y}\) and to highlight the characteristic that plastic hinges have formed at the supports and the loading point in this regime, the deformation shape function of the beam in this regime is given as  
\[w_{2}(x)=\left\{ \begin{array}{cc}\frac{W_{3y}}{2}\Big{[}1+\cos\frac{\pi x}{ l_{1}}\Big{]}+\big{(}W-W_{3y}\big{)}\Big{(}1+\frac{x}{l_{1}}\Big{)}&-l_{1}\leq x<0 \\ \frac{W_{3y}}{2}\Big{[}1+\cos\frac{\pi x}{l_{2}}\Big{]}+\big{(}W-W_{3y}\big{)} \Big{(}1-\frac{x}{l_{2}}\Big{)}&0\leq x\leq l_{2}\end{array} \right. \tag{16}\]  
Here, based on the normality rule of plasticity, a similar method to that developed by [PERSON] and [PERSON] [23] is adopted for the predictions on the elastic-plastic behavior of the fully clamped beam loaded at any point on the span. The total increment in axial tension is given as  
\[\Delta\epsilon_{ep}=\frac{d\Delta l_{ep}}{dW}\Delta W=\Delta W\frac{d}{dW} \frac{1}{2}\int\bigg{(}\frac{dw}{dx}\bigg{)}^{2}dx=W\bigg{(}\frac{1}{l_{1}}+ \frac{1}{l_{2}}\bigg{)}\Delta W \tag{17}\]  
The total increment in axial extension can be divided into the elastic and plastic components of axial tension, because the elastic extensions occur in the portions of the beam between the plastic hinges. The increment in the elastic extension is expressed as \(\Delta\epsilon_{e}=\Delta Nl/AE\). Thus, the increment in the plastic extension is given as  
\[\Delta\epsilon_{p}=\Delta\epsilon_{ep}-\Delta\epsilon_{e}=W\bigg{(}\frac{1}{l _{1}}+\frac{1}{l_{2}}\bigg{)}\Delta W-\frac{\Delta Nl}{AE}, \tag{18}\]  
which predicts the rate of extension of the beam centre line as  
\[\dot{\epsilon}=\frac{\Delta\epsilon_{p}}{\Delta W}=W\bigg{(}\frac{1}{l_{1}}+ \frac{1}{l_{2}}\bigg{)}-\frac{l}{AE}\frac{\Delta N}{\Delta W} \tag{19}\]  
The rate of rotation across all plastic hinges developing at the supports and the loading point is given as  
\[\dot{\theta}=\frac{\Delta\theta}{\Delta W}=2\bigg{(}\frac{1}{l_{1}}+\frac{1} {l_{2}}\bigg{)} \tag{20}\]  
The normality rule of plasticity requires the following:  
\[\frac{\dot{\epsilon}}{\dot{\theta}}=\frac{2 NM_{p}}{N_{p}^{2}} \tag{21}\]  
Substitution of Equations (19) and (20) into Equation (21) gives  
\[\frac{\Delta N}{\Delta W}=\bigg{(}W-\frac{4 NM_{p}}{N_{p}^{2}}\bigg{)}\frac{ AE}{l}\bigg{(}\frac{1}{l_{1}}+\frac{1}{l_{2}}\bigg{)} \tag{22}\]  
This first-order differential equation involving two variables, \(N\) and \(W\), can be solved with the initial conditions \(N=N_{3y}\) at \(W=W_{3y}\) using Mathematica software version 11.3. The differential equation solution is valid from \(W=W_{3y}\) until the establishment of a membrane state, i.e., \(N=N_{p}\) at \(W=W_{p}\). Thus, the bending moment for a specific displacement \(W\) can be given via Equation (13). Consequently, from Equation (3), the beam's resistance during this regime is determined as  
\[F=2M_{N}\left(\frac{1}{l_{1}}+\frac{1}{l_{2}}\right)+N\left(\frac{1}{l_{1}}+\frac {1}{l_{2}}\right)W \tag{23}\]  
In such a case where \(W\geq W_{3y}\), the transverse velocity field is obtained by taking the derivative of \(w_{2}(x)\) defined in Equation (16) with respect to time:  
\[\dot{w}_{2}(x)=\left\{ \begin{array}{ll}\dot{W}\left(1+\frac{x}{l_{1}}\right) &-l_{1}\leq x<0\\ \dot{W}\left(1-\frac{x}{l_{2}}\right)&0\leq x\leq l_{2}\end{array} \right. \tag{24}\]  
Substituting Equation (24) into Equation (2), the equivalent mass of the beam in this regime is determined as  
\[m_{b}^{\epsilon}=\frac{3}{16}\mu(l_{1}+l_{2}) \tag{25}\]  
The initial conditions are \(W_{3y}\) and \(\dot{W}(t_{3y})\) at \(t=t_{3y}\). The general conditions for Regime IV are \(M_{0}=M_{1}=M_{2}=M_{N}\) and \(N<N_{p}\). The terminating conditions for this regime are \(N=N_{p}\) and \(M=0\) at time \(t=t_{p}\). _Regime V_ represents that full plastic deformation develops along the beam and a membrane state is reached for impact with sufficiently large kinetic energy.