# SimEx: A Tool for the Rapid Evaluation of the Effects of Explosions
## 2 SimEx Capabilities
### Single-Degree-of-Freedom System Analysis
#### 2.1.1 Forcing Term  
As previously discussed, the blast wave overpressure defined in Equation (4) below can be used directly in Equation (3) as forcing term, \(F(t)=p^{\prime}(t)\), as long as the analysis is formulated per unit surface and uses distributed masses and forces. In order to determine the blast parameters (arrival time, peak overpressure, positive phase duration, impulse per unit area, waveform parameter, etc. ), classical correlations [1; 2; 17; 18; 19; 33; 34] in terms of scaled distance are used together with the scaling laws for spherical or hemispherical blast waves [1; 17; 35; 36], which allow their evaluation for arbitrary CW-S pairs. It is interesting to note that the standoff distance is defined as the minimum distance from the charge to the structural element under study (e.g., a wall). However, the actual distance to a given point of that element, e.g., the centroid (or geometric center), which may be considered the most representative point of the structure, may be slightly different due to the incidence angle being larger than 0 at that point. The local atmospheric pressure, \(p_{a}\), and temperature, \(T_{a}\), are determined using the International Civil Aviation Organization (ICAO) Standard Atmosphere (ISA) [37] with a temperature offset (ISA \(\pm\Delta T\)). The user must specify the geopotential height, in meters, and the non-standard offset temperature \(\pm\Delta T\), although arbitrary ambient temperature and pressure can also be introduced directly [38]. TNT is used as reference explosive, although the results can be extrapolated to other compositions using either the equivalence tables included in SimEx for selected explosives [39], or the thermochemical calculation assistant, to be presented in Section 2.2.1, for less conventional formulations or explosive mixtures. To estimate the dynamic load exerted by the blast wave, the angle of incidence of the incoming shock wave must be considered, the worst-case conditions being usually those of normal incidence. UFC 3-340-02 [16] contains scaled magnitude data for both spherical and hemispherical blast waves. It also provides methods to calculate the properties of the blast wave with different incidence angles, including both ordinary and Mach reflections for oblique shocks. The time evolution of the blast wave overpressure \(p^{\prime}(t^{\prime})\) at a fixed distance, \(d\), sufficiently far from the charge (at least, larger than the fireball scaled distance) is approximated using the modified Friedlander's equation, which captures also the negative overpressure phase [1; 17; 40]  
\[p^{\prime}(t^{\prime})=p(t^{\prime})-p_{1}=p^{\circ}\left(1-\frac{t^{\prime}}{ t_{d}}\right)\exp\left(-\alpha\,\frac{t^{\prime}}{t_{d}}\right) \tag{4}\]  
where \(p^{\circ}=p_{2}-p_{1}\) represents the peak overpressure measured from the undisturbed atmospheric pressure \(p_{1}=p_{a}\), with \(p_{2}\) denoting the peak post-shock pressure, \(t^{\prime}=t-t_{a}\) is time measured from the blast arrival time, \(t_{d}\) is the positive phase duration, and \(\alpha\) is the waveform parameter, closely related to the impulse per unit area of the positive phase \(I/A=\int_{0}^{t_{d}}p^{\prime}(t^{\prime})\ \mathrm{d}t^{\prime}\) (area under the positive phase of the overpressure-time curve) according to \(I/A=p^{\circ}\,t_{d}\big{[}1/\alpha-(1-e^{-\alpha})/\alpha^{2}\big{]}\). SimEx performs by default the complete integration of the Friedlander waveform, but the equivalent triangular pressure pulse can also be used without significant errors [32]. This simplified waveform has the same maximum peak overpressure, \(p^{\circ}\), but a fictitious positive duration computed in terms of the total positive impulse and the peak over pressure, \(t_{d}=2(I/A)/p^{\circ}\). The \"Blast wave\" calculation assistant allows the activation of the effects of clearing and confined explosions, which increases the computational capabilities to more realistic situations.