## 3 Simulation Model for Numerical Analyses
### Calculation Approach  
A multi-step simulation method with explicit time integration is selected. In the first step, the static loads (dead load and internal design pressure of 50 mbar-g) are applied, linearly increasing within a time range of 300 ms. The control of the energy balance shows that with this slow load increase, no undesired dynamic effects are inserted into the system. After that, the structure remains in the loading condition for 200 ms to further control the quasi-static state. Afterwards, the blast load is applied as a pressure time history. At time 608 ms, the blast load hits the first element of the model; at time 719 ms, the last element starts being loaded. After 1022 ms, the pressure wave leaves the last elements of the tank model. The calculation continues until a total duration of 1.1 s. According to the recommendation outlined in [23] (paragraph 6.4.1), elastic material damping is conservatively ignored in the blast analyses. This is justified by the short time in which the structure reaches its maximum response. Thus, damping effects have little influence on peak displacements. A comprehensive numerical dynamic study that reflects the behavior of the tank at different levels of computational acuity is performed. This may provide guidance for future studies to choose the optimal and most efficient calculation approach. The underlying basics of the different calculations are explained in detail below. Two calculation runs are performed to comprehensively capture the roof stress and overall uplift loading (see Section 2.5):  
Figure 10: Assumption of blast load source in simulation model. Different colors of the tank model indicate different model parts with different sections (red: tank wall, green: tank roof panel, blue: compression ring of wall and roof, grey: base plate, black: beam elements of ring stiffeners, rafter, and anchorage). Figure 9: Transfer of pressure wave to shock wave via impulse equality. * Roof compression load;
* Roof suction load. Additionally, two calculations each are performed considering different load applications with the idealized pressure wave according to Figure 8:  
* A uniform peak pressure amplitude of 30 kPa to one half of the tank model. In this easily applicable approach, pressure wave propagation effects related to the decrease in the peak pressure explained in Section 2.2 are neglected. The peak pressure of 30 kPa is applied to each shell element of the blast-loaded side of the simulation model according to the proposal of [31] depicted in Figure 3. The rear side remains unloaded. * Consideration of varying pressure amplitude depending on scaled distance and angle of incidence according to the procedure is described in Section 3.4. In this more accurate but also more elaborate approach, the peak reflected pressure, peak incident pressure, and positive phase duration are determined for each shell element of the simulation model. The effective pressure according to Equation (3) is applied to each shell element. Next, two different material approaches will be investigated (see Section 3.4):  
* A simplified bilinear approach (BL);
* A Johnson-Cook material formulation (JC) considering strain rate effects. Furthermore, an analysis is also carried out using a shock wave approach. The blast scenario with an equivalent TNT mass of W = 826.44 to and a distance R = 785 m is applied using LBE in LS-Dyna. A spherical air burst without amplification of the initial shock wave due to interaction with the ground surface and without negative phase duration is considered. The results of this impulse-congruent shock wave load are compared with the results of the pressure wave load to quantify differences in the structural response. Since, in this approach, the roof loading follows the approach depicted in Figure 2, compression of the roof load will result. Finally, a static implicit analysis is performed with the same numerical model according to the proposal of [47].