## 2 Simulation of Blast Events
### Determination of Shock Wave Parameter  
The intensity of the shock wave and thus the peak overpressure, but also the positive and negative phase duration, depend on the distance to the position of the explosive charge. The farther a target is from the source, the smaller the peak overpressure, but the longer the positive and negative phase duration. Various blast load equations that enable the calculation of the parameter of the shock pressure time history curve are available in the literature. The Kingery-Bulmansh [35] equation is the most widely used. It has been adopted by UFC 3-340-02 [32] and is referred to, for example, in [32]. It provides rules to determine the blast parameter depending on the scaled distance \(Z\) to the explosive charge:  
\[Z=R/W^{1/3}, \tag{1}\]  
where R is the distance from the explosive charge and \(W\) is the equivalent TNT weight of the charge. Blast parameters considered in Kingery-Bulmansh [35] and in comparable research to determine the pressure time history are the maximal incident and reflected blast overpressures, the corresponding impulses, the arrival time of the shock wave, the duration of positive (and negative) pressure phase, and the velocity of the shock wave. To sum up, the blast load application should consider the following aspects:  
* The velocity of propagation of the shock wave is considered; elements located farther from the explosive charge are loaded later than elements located at a shorter distance; the velocity and the arrival time of the shock wave are the relevant parameters. * The peak pressure decreasing with the distance from the detonation source is considered; elements located farther from the explosive charge have a lower peak pressure than elements at a shorter distance; the determination of the maximal incident and reflected blast overpressures and their corresponding impulses is relevant. * The loading duration of the applied shock wave is considered; elements located farther away from the explosive charge are loaded for a longer time than elements located at a shorter distance; the determination of the positive (and negative) pressure phase duration is relevant. In the simplest way, the time-dependent decay behavior of the incident and reflected blast overpressures of free field detonations can be described by [PERSON]'s equation [36]:  
\[P(t)=P\cdot e^{(-b\frac{t}{t_{0}})}\left(1\ -\frac{t}{t_{a}}\right), \tag{2}\]  
where P is the pressure, \(t_{d}\) is the positive phase duration and \(b\) describes the decay behavior of the curve (see also Figure 1a). The blast load applied to a structure is defined as the  
Figure 1: Explosion pressure propagation in free air: (**a**) characteristic shape of a shock wave; (**b**) characteristic shape of a pressure wave. effective pressure. When the blast wave crosses a target surface at an oblique angle of incidence \(\theta\), the effective pressure \(\mathrm{P_{eff}}\) is given using the following equation:  
\[\mathrm{P_{eff}}=\mathrm{P_{ref}}\cdot\mathrm{cos^{2}}\theta+\mathrm{P_{in}} \cdot(1+\mathrm{cos}\theta-2\ \mathrm{cos^{2}}\theta), \tag{3}\]  
with the angle of incidence \(\theta\) as depicted in Figure 2. This approach, described in [37], for example, can be summarized as follows:  
* Structural parts that are oriented perpendicular to the blast wave direction receive the maximal reflected pressure: \(\theta=0^{\circ}\): \(\mathrm{P_{eff}}=\mathrm{P_{ref}}\). * Structural parts that face away from the charge receive only the incident pressure: \(90^{\circ}\leq\theta\leq 180^{\circ}\): \(\mathrm{P_{eff}}=\mathrm{P_{inc}}\). Further explanations and definitions of the blast load and its wave propagation are given in the literature [23; 30; 32]. The described phenomena of shock wave propagation and their effect on structures can be modelled in LS-DYNA (smp d R13) [18] software by means of several techniques.