# SimEx: A Tool for the Rapid Evaluation of the Effects of Explosions
## 2 SimEx Capabilities
### Single-Degree-of-Freedom System Analysis
The mass transformation factor, \(K_{M}\), is defined as the ratio between the equivalent mass \(M_{e}\) and the real mass \(M\) of the blast-loaded component; the load transformation factor \(K_{L}\) is defined as the ratio between the equivalent load \(F_{e}(t)\) and the actual load \(F(t)\), and usually coincides with the resistance and damping transformation factors; and finally the load-mass factor \(K_{LM}\) is defined as the ratio between the mass factor and the load factor  
\[K_{M}=\frac{M_{e}}{M}\ ;\quad K_{L}=\frac{F_{e}(t)}{F(t)}=\frac{R_{e}(x)}{R(x)}= \frac{C_{e}}{C}\ ;\quad K_{LM}=\frac{K_{M}}{K_{L}}=\frac{M_{e}}{M}\cdot\frac{F(t)}{F_{e}(t)} \tag{2}\]  
Although all these factors are easy to obtain, even through analytical expressions in some cases, most of them can be found tabulated in the UFC-3-340-02 [16]. The linear momentum equation for the equivalent SDOF system then takes the form [32]  
\[K_{LM}M\ddot{x}+C\dot{x}+R(x)=F(t) \tag{3}\]  
where, as previously discussed, \(C\) represents the viscous damping constant of the blast-loaded component. This constant is often specified as a small percentage, \(z\), of the critical viscous damping, \(C=(z/100)C_{\rm cr}\), with a damping coefficient \(z=2\) being a good value when not otherwise known (for further details see [31]). Note, however, that damping has very little effect on the maximum displacement, which typically occurs during the first cycle of oscillation, so the actual value of \(z\) is not of major relevance. The inhomogeneous term, \(F(t)\), appearing on the right-hand side of Equation (3) represents the dynamic load associated with the blast wave, to be discussed in Section 2.1.1 below. SimEx provides an easy and intuitive GUI environment for the study of the dynamic response to blast loadings of a variety of structural elements that can be modeled as SDOF systems. Figure 2 shows the main SimEx interface, divided into three calculation assistants for the three basic elements that make up the SDOF system: a module for calculating the properties of the blast wave (forcing term, \(F(t)\)), a module for calculating the equivalent mechanical properties (resistance term, \(R(t)\)), and a module for the numerical integration of the problem, which includes the post-processing of the results and their graphic representation in the form of displacements, forces, and deformation diagrams (see the bottom plots of Figure 2) and of CW-S damage charts, to be discussed in Section 3.3. As a final remark, it is important to note that, following standard practice, the SDOF analysis carried out by SimEx uses the load defined in terms of pressure, \(F(t)=p^{\prime}(t)\) (Pa), so that both the mass \(M\) (kg/m\({}^{2}\)), the damping coefficient \(C\) (kg/(m\({}^{2}\) s)) and the ultimate resistance \(R_{\text{u}}\) (Pa) must all be introduced as distributed values per unit surface (p.u.s.) in the different calculation assistants. Figure 2: Main interface of SimEx showing the “Blast wave”, “Resistance”, and “Integration” assistants for the computation of the structural response of perfect elasto-plastic SDOF systems under blast loading. The access buttons to the other calculation assistants are seen under the top toolbar. The bottom plots show the post-processing pop-up window that displays the results of the numerical integration in terms of displacements, forces, and deformation diagrams (for a detailed discussion of these diagrams, see Section 2.1.4).