## 1 Introduction  
Impact on flexible structures is common in a wide range of engineering fields, such as marine, automobile and civil engineering. In order to provide useful information on the preliminary designs of structures and systems subjected to impact, the demand for deep insight into the impact performance and response mechanism of main structural members, such as restrained beams, has rapidly increased. In order to predict the nonlinear response of dynamically loaded structures with high efficiency, some efforts have been made to develop simplified analytical approaches. Most theoretical analyses adopt the rigid-plastic approximations, reviewed and discussed in detail by [PERSON] [1]. Nevertheless, given that material elasticity is found to play a significant role in accurate predictions of important response characteristics, such as the spring-back and deformation energy dissipation [2; 3; 4; 5; 6; 7; 8], the incorporation of material elasticity is essential in developing a more accurate analytical model for structures under impact. The equivalent single-degree-of-freedom (SDoF) mass-spring model is an important tool that has been widely used to predict the dynamic behavior of elastic-plastic structures due to its simplicity and accuracy [9; 10]. In fact, owing to the high complexity of the material and geometric nonlinear behavior of the actual structures, the equivalent resistance functions in the SDoF model are often determined by experimental and/or numerical methods [11; 12; 13]. Hence, some simplifications are employed so as to develop an analytical solution of the SDoF model. Several representative research studies on predicting the impact response of beams using the SDoF model are reviewed here. For example, on the basis of the traditional equivalent SDoF method considering only flexural behavior, while the influence of the axial force on the bending moment is ignored [14], [PERSON] and [PERSON] [15] examined the dynamic responses of reinforced concrete beams under blast and impact loads. In order to simulate the behavior of structures that undergo large displacements more accurately, some improved SDoF models are proposed [16]. [PERSON] et al. [17] numerically and theoretically investigated the dynamic response of axially loaded CFST memberswith the fixed-sliding boundary conditions subjected to lateral impact at its mid-span. A constant axial force was applied to the sliding support. The effect of the axial force was incorporated into the classical SDoF model to calculate the deflection, and reasonable accuracy was obtained when compared with the numerical simulation. [PERSON] et al. [18] presented a non-linear elastic-plastic SDoF model for a steel beam-column subjected to impact at any point on its span. Although this model assumes a linear yield surface to account for the interaction between axial force and bending moment, the changes of axial forces along the beam are ignored until the formation of the plastic hinges at the three critical locations, i.e., the two fixed ends and the impact point. Noticeably, although the influence of the material elasticity on the loading behavior of the structures is considered in the previously mentioned SDoF methods, the unloading behavior is still neglected. One such improvement in the SDoF model is made by [PERSON] et al. [19] to predict the elastic-plastic behavior of the clamped-clamped beam under the impact of a heavy mass with a low velocity at its mid-span. The explicit expressions of the resistance functions of the beam are proposed during the loading/unloading/elastic vibration stage. This model adopts a non-linear relation between the axial force and the bending moment to account for the membrane effect. In their recent study [20], the strain hardening effect on the structural resistance of an elastic-linear strain hardening beam is estimated for the problem of repeated impacts.