# SimEx: A Tool for the Rapid Evaluation of the Effects of Explosions
## 2 SimEx Capabilities
### Single-Degree-of-Freedom System Analysis  
In many situations of practical interest, the response of structural elements to blast loading can be reduced, in first approximation, to that of an equivalent spring-mass SDOF system. As sketched in Figure 1, this system is made up of a concentrated mass subject to external forcing and a nonlinear weightless spring representing the resistance of the structure against deformation [8]. The mass of the equivalent system is based on the component mass, the dynamic load is imposed by the blast wave, and the spring stiffness and yield strain on the component structural stiffness and load capacity. Generally, a small viscous damping is also included to account for all energy dissipated during the dynamic response that is not accounted by the spring-mass system, such as slip and friction at joints and supports, material cracking, or concrete reinforcement bond slip [31]. Figure 1: Sketch of the equivalent SDOF system showing the different terms involved in its mathematical description. **Left**: forcing term; **right**: resistance term; **center**: equivalent spring-mass SDOF system and its associated differential equation. If the system properties are properly defined, the deflection of the spring-mass system, \(x(t)\), will reproduce the deflection of a characteristic point on the actual system (e.g., the maximum deflection). The system properties required for the determination of the maximum deflection are the effective mass of the equivalent SDOF system, \(M_{e}\), the effective viscous damping, \(C_{e}\), the effective resistance function, \(R_{e}(x)\), and the effective load history acting on the system, \(F_{e}(t)\). To systematize the calculations, the effective properties are obtained using dimensionless transformation factors that multiply the actual properties of the blast-loaded component, respectively, \(M\), \(C\), \(R(x)\), and \(F(t)\)[32]. These factors are obtained from energy conservation arguments in order to guarantee that the equivalent SDOF system has the same work, kinetic, and strain energies as the real component for the same deflection when it responds in a given, assumed mode shape, typically the fundamental vibrational mode of the system [31]. In the analysis of blast-loaded SDOF systems, it is therefore of prime importance to identify the fundamental vibrational mode of the structural element. This procedure is not trivial, since obtaining the fundamental mode can entail certain difficulties, in which case its shape must be approximated in some way [32]. To determine the equivalent properties of the SDOF system, it is also necessary to determine the type of structure (beam, pillar, frame, etc.) and how the load is applied (typically, a uniform load is assumed). The elastic behavior of the material is often modeled as perfect elasto-plastic, probably the simplest of all nonlinear material models. This assumes that the initial response follows a linear elastic behavior described by an apparent elastic constant \(K\), but once the yield strain is reached, \(x\geq x_{\rm u}\), the material behaves as plastic, flowing at a constant stress with an ultimate resistance \(R_{\rm u}=Kx_{\rm u}\), i.e.,  
\[R(x)= \begin{cases}Kx&\text{for}\quad|x|<x_{\rm u}\\ R_{\rm u}&\text{for}\quad|x|\geq x_{\rm u}\end{cases}  \tag{1}\]  
Although more complex models could be used, they are not considered here due to the heavy simplifications introduced in the formulation of the problem.