## 2 Non-Linear Finite Element Analysis
### Finite Element Modeling
#### 2.1.5 Loadings
\[F_{D}=\frac{1}{2}\rho v_{gas}^{2}C_{D}A \tag{6}\]  
where \(C_{D}\) is the drag coefficient of the object which is dependent on the shape of the structure (projected area), \(\rho\) is density of gas (kg/m\({}^{3}\)), \(v_{gas}\) is velocity of the unburned gas mixture (m/s) and \(A\) is structures projected area (m\({}^{2}\)). Since the prediction of gas density and velocity in an explosion event are very difficult, [PERSON] et al. Reference [30] reported that the dynamic pressure (drag load) calculation in Equation (6) can be simplified to the following empirical expression given by Equation (6). \[q_{D}(t)=C_{D}\times p(t)\times OD \tag{7}\]  
where \(q_{D}(t)\) is the line load on a pipe function with respect to time, \(p(t)\) is maximum overpressure time-history and \(OD\) is outer pipe diameter. According to Equation (7), the dynamic pressure can be calculated by using the maximum overpressure values. The maximum dynamic pressure \(P_{max}\) of 2.5 bar was selected based on [PERSON] et al. [30] whereas the peak negative pressure \(P_{min}\) of \(-0.83\) bar was taken as one-third of the maximum dynamic pressure as suggested by in the Chevron Engineering Standard and [PERSON] et al. [34]. The pressure duration of 0.136 s for pressures \(P_{max}\) of 2.5 bar and the duration of 0.24 s for \(P_{min}\) of \(-0.83\) bar were determined using the curve in Figure 9.  
The dynamic pressure wave profile can be developed using the pressures and durations information. The idealized triangular waveform shown in Figure 10 was considered to describe the dynamic pressure history that was applied in the FE models. Figure 9: Pressure durations relationship based on API RP 2 FB [24]. In the analyses, a uniform distributed pressure of 2.5 bar was applied on the column flanges. To simplify the problem, the drag loads on the unmodeled pipes and non-structural items on the frame were ignored. Since the blast direction and the beam longitudinal direction are parallel, the beams are to be minimally affected by the lateral pressures, and thus no blast loads were applied to the beams. In a blast event, the structures might buckle and undergo large geometry changes, hence the necessity to also consider geometric non-linearity [20]. In the initial step, initial conditions, boundary conditions, and predefined fields that are applicable with the analysis can be specified. In addition to the initial step, three steps were specified to simulate the blast analysis, namely; self-weight (quasi-static analysis with structural weight and dead load), blast (dynamic analysis using dynamic pressure history with self-weight propagation) and post-blast (only dead load is set active, no propagation of blast load from previous analysis step). In each analysis step, the incrementation settings were set to automatic incrementation so that ABAQUS automatically adjusts the time increment size depending on the numerical stability limit [20]. Before applying dynamic pressure, the initial condition of the structure was in a static condition under self-weight consisting of structural dead load and piping masses. Unfortunately, ABAQUS cannot directly combine static analysis with dynamic explicit analysis in the same model. When dynamic explicit analysis is being used, all the following analysis in the next steps must also be in the dynamic explicit mode. Therefore, the self-weight member forces also need to be analysed using dynamic explicit analysis. When the self-weight is applied instantaneously in the dynamic analysis, oscillatory vertical reaction force occur as shown in Figure 11. This oscillation does not represent the correct static condition in which the structure should be before the blast is initiated. This oscillation will create incorrect stress variations during and after the actual blast dynamic analysis. To overcome this problem, the ramp loading was implemented in the self-weight dynamic analysis whereby the load was applied slowly to avoid the dynamic effect.