## 3 Impact of Pressure Wave on a Ship Structure  
The formulas described in Table 1 make it possible to determine the pressure value on the shock wave front in water as a function of distance from the detonation epicenter, charge mass and time. Taking into account the angle of incidence of the shock wave on the structure, it is possible to calculate the load on the structure at discrete time moments. This approach can be directly implemented in the CAE program. Empirical formulas shows that the explosion is a rapid process, therefore, in numerical calculations of a discrete nature, it is necessary to select an appropriately \"small\" time step, which will allow to reflect the entire history of the structure load. The time appearing in the formulas is counted from the moment of the wave pressure occurrence at a given point in space, it does not take into account the time of wave transition from the epicenter. After taking into account the speed of propagation of the pressure wave in water, for a unit of significant length, a different load history is obtained.  
Determining the load for a given structure element, the time appearing in the formulas given in Table 1 should be shifted by the value of \(t_{r}\) which is:  
\[t_{r}=\frac{r}{c_{o}} \tag{6}\]  
where:  
* \(t_{r}\)--time when the pressure wave reaches the given element of the structure, s
* \(r\)--distance to given place of the structure, m
* \(c_{o}\)--sound speed in surrounding medium, m/s  
For example, if 250 kg TNT is detonated, at a depth of 15 m from a distance of 20 m in front of the mine destroyer's bow (206 FM project), she will be loaded with a pressure wave of 11 MPa, which decreases along the ship to a value of 3 MPa (Figure 7). The total load duration on the ship's structure is 0.0376 s [34].  
Figure 7: Propagation of pressure wave along the ship, detonation of 250 kg of TNT at a distance of 20 m in front of the ship, at a depth of 15 m [34]. A significant drop in the pressure wave value between the bow and the stern is noticeable.  
For the same load, detonated at a depth of 15 m from a distance of 20 m on a traverse, the maximum shock wave pressure reaches 11 MPa amidships and decreases along the ship to 5.9 MPa at the bow and stern. The total load duration on the ship's structure is more than half shorter and equals 0.0124 s.  
Using finite element method requires the examined structure division into mesh elements. Each element is loaded with a different pressure value at different angles. To be able to take into account the angle of incidence of a shock wave on a given FEM element (Figure 8), it is necessary to determine two vectors: a normal vector for element n and a pressure wave vector R, which begins at the epicenter of the explosion \(K\) and ends in the middle of the structure element \(S\).  
The pressure value changes depending on the angle of incidence on the object (Figure 8), therefore:  
\[cos\alpha=\frac{R}{R+c_{o}\Delta t} \tag{7}\]  
hence:  
\[c_{o}\Delta t=\frac{R-Rcos\alpha}{cos\alpha} \tag{8}\]  
The time difference between the first contact of the pulse with the surface and the subsequent contact with the structure depending on the incidence angle is equal:  
\[\Delta t=\frac{R-Rcos\alpha}{c_{o}cos\alpha} \tag{9}\]  
The pressure drop depending on the angle of incidence from [PERSON]'s formula equals:  
\[p(\alpha)=52,3\left(\frac{\sqrt[3]{m}}{R+c_{o}\Delta t}\right)^{1,13} \tag{10}\]  
If the examined object is relatively small in relation to the distance from the epicenter, it can be assumed that the radius of curvature is large enough to model the wave as flat.