## 2 SDoF Model for Low-Velocity Heavy-Mass Impact
### Loading Stage  
The loading stage is divided into a number of regimes corresponding to the formation of plastic hinges when using the square yield condition Equation (4) relating \(M\) and \(N\) for the plastic flow in a rectangular cross-section. The derivation of Equation (4) can be found in [1]. The details of the response of the elastic-plastic beam during each are given below. \[\frac{M}{M_{p}}+\bigg{(}\frac{N}{N_{p}}\bigg{)}^{2}=1 \tag{4}\]  
where \(M_{p}=\sigma_{0}H^{2}B/4\) and \(N_{p}=\sigma_{0}HB\) are the fully plastic bending moment and fully plastic axial force, respectively, for the beam with a rectangular-shaped cross-section. Figure 2: Force and moment analyses on the fully clamped beam quasi-statically loaded at \(x=0\). \(M_{1}\), \(M_{2}\) and \(M_{0}\): the bending moments at the two supports and the loading point of the beam, respectively. _Regime I_ represents pure elastic deformations developed in the beam with the shape function \(w_{1}(x)\) approximated as  
\[w_{1}(x)=\left\{ \begin{array}{ll}\frac{W}{2}\bigg{[}1+\cos\frac{\pi x}{l_{1}} \bigg{]}&-l_{1}\leq x<0\\ \frac{W}{2}\bigg{[}1+\cos\frac{\pi x}{l_{2}}\bigg{]}&0\leq x\leq l_{2}\end{array} \right. \tag{5}\]  
The explanation to illustrate this approximated shape function will be given later in this section. The axial force following [PERSON]'s law of elasticity in this regime can thus be given by  
\[N=EA\frac{\Delta l}{l}=\frac{EA}{l}\Bigg{[}\frac{1}{2}\int\limits_{l}\left( \frac{dw}{dx}\right)^{2}dx\Bigg{]}=\frac{EA\pi^{2}}{16l_{1}l_{2}}W^{2} \tag{6}\]  
The expressions for the elastic bending moment based on the simple technical theory of bending are used here [18]:  
\[M_{1}=\frac{3 Ell}{l_{1}^{2}l_{2}}W \tag{7}\]  
\[M_{2}=\frac{3 Ell}{l_{1}l_{2}^{2}}W \tag{8}\]  
\[M_{0}=\frac{6 EI}{l_{1}l_{2}}W \tag{9}\]  
where \(I=BH^{3}/12\) is the section moment of inertia and \(W\) is the displacement of the beam at the loading location \(x=0\). Regarding the assumed shape function, it is worth mentioning that in the elastic regime, the maximum deflection does not occur at the loading point, but the maximum deformation point gets closer to the loading point as the beam's deformation increases. In addition, the assumed shape function in this regime is only used to calculate the axial force \(N\), while the elastic bending moment \(M\) is obtained based on the classical bending theory of beams to weaken the influence of the assumed shape function on the establishment of the elastic loading path. The shape function defined in Equation (5), which satisfies the boundary condition, therefore seems reasonable to facilitate a simplified analytical solution that can take into account the influence of the elastic deformations. Substitution of Equations (6)-(9) into Equation (3) gives the following:  
\[F=\frac{EIW}{l_{1}l_{2}}\Bigg{[}3l\Bigg{(}\frac{1}{l_{1}^{2}}+\frac{1}{l_{2}^{ 2}}\Bigg{)}+\bigg{(}6+\frac{3\pi^{2}W^{2}}{4H^{2}}\bigg{)}\bigg{(}\frac{1}{l_{ 1}}+\frac{1}{l_{2}}\bigg{)}\Bigg{]} \tag{10}\]  
The velocity profile of the beam, taking the same form as the shape of the transverse displacement profile (see Equation (5)), is substituted into Equation (2), and then the equivalent mass of the beam can be given as the following:  
\[m_{b}^{e}=\frac{1}{6}\mu(l_{1}+l_{2}) \tag{11}\]  
With the assumption that the striker and the struck region of the beam reach the same velocity immediately after impact and maintain contact during impact [19; 20], the initial velocity of the beam at the impact location just after impact, \(V_{0}^{*}\), can be calculated from the conservation of linear momentum:  
\[V_{0}^{*}=\frac{2m_{0}}{\mu l+2m_{0}}V_{0} \tag{12}\]Thus, substituting Equations (10) and (11) into Equation (1), the dynamic response in this regime can be obtained with the initial conditions for Regime I, namely \(W(0)=0\) and \(\dot{W}(0)=V_{0}^{*}\).