### 9.3.2. Theory

The following Analytical Blast topics are discussed in this Theory section:

#### 9.3.2.1. Pressure Load Evaluation Without Clearing

When clearing is switched off, the impulse on a face which faces the blast is determined without account for pressure relief from structure edges. The applied pressure load on a face depends on the distance from the blast, the angle of incidence of the blast wave, and the orientation of the face with respect to the blast. The angle of incidence, θ, is computed by finding the angle between the face normal and the vector joining the face center and the blast center. The distance from the blast, d, is computed as the straight-line distance between the blast center and the face center. The scaled distance, ,is computed from the blast weight, W, and is used to compute the blast parameters including the incident overpressure, POI, positive phase duration, t0, arrival time, tarr, and incident impulse, II, using the formulae reported in 1984 report ARBRL-TR-02555 by Kingery and Bulmash [2].

If the face does not face the blast, the applied pressure load is given by the Friedlander equation [3] for side on blast loads:

 (9–1)

where the decay parameter, λI, is calculated such that . If t is less than tarr then no pressure is applied.

However, if the face is determined to face the blast, the normal reflected impulse (for 0° angle of incidence), IR, is also computed using the formulae reported in Kingery and Bulmash [2] and the peak reflected over pressure, POR, is computed from the angle of incidence and POI using Figure 3-3 in TM 5-855-1 [1]. The reflected impulse is corrected for non-normal angles of incidence using

 (9–2)

and the applied pressure load is given by the Friedlander equation

 (9–3)

where λR is calculated such that .

#### 9.3.2.2. Clearing ‘On’

When clearing is switched on, the impulse on a face which faces the blast is corrected to account for pressure relief from structure edges as described in TM 5-855-1 [1].

The time for the reflected pressure wave to be cleared, tc, is computed as tc =3l/u where l is the distance to the nearest edge from which pressure relief can occur, and u is the blast wave speed (calculated using the equations in Kingery and Bulmash [2]). A fictitious pressure-time curve as shown in Figure 9.1: Overpressure-Time Curve is constructed which is used to compute the new impulse, Ic (area shaded in Figure 9.1: Overpressure-Time Curve). This approach can overestimate the impulse (typically when the pressure loads are large), so the impulse is capped by the reflected impulse as given in Equation 9–2.

Overpressure-time curve used to calculate the reduced impulse on a structure due to clearing. q is the dynamic pressure which is computed from Figure 3-8 in TM 5-855-1 [1] using POI and θ. CD is the drag coefficient which is assumed to be 1 [1] and .

The pressure-load applied to a face is a Friedlander equation

 (9–4)

where the decay parameter, λc, is taken such that . Figure 9.2: Effect of the Clearing Option on Pressure Load shows the difference in pressure load applied to two shell parts both with analytical blast boundary conditions applied, but one with clearing on and the other with clearing off.

The effect of the clearing option on pressure load applied due to an analytical blast boundary condition. Both shells have analytical blast boundary conditions applied with the center of the charge in-line with the center of the shell and positioned 500 mm away (charge weight is 1 kg). The shell on the left has clearing turned on and the shell on the right has clearing turned off. The snapshot has been taken approximately 6 μs after the blast arrival at the shells.

The load applied to a face due to an analytical blast boundary condition can be visualised both using contour plots and history plots at gauges using the variable BLAST.LOAD. This variable is only available for contour plots when an analytical blast boundary condition has been applied to some parts. The pressure load applied to a face due to an analytical blast boundary condition is plotted in the element to which the face belongs. Note that if two faces of an element both have analytical blast boundary conditions applied (e.g. at corners), from which face the value of BLAST.LOAD is taken is arbitrary. In order to plot the BLAST.LOAD variable in a history plot, it needs activating in the Select Gauge Variables (under History in the Outputs tab) menu prior to solving.

#### 9.3.2.4. Limitations

The formulae in ARBRL-TR-02555 for the blast parameters are only valid for scaled distances (ratio of distance to blast and the cube root of the blast weight) between 0.147 and 40.0 m/kg1/3, therefore, it is necessary to ensure that all faces with an analytical blast boundary condition applied satisfy this condition (if the scaled distance between the face and blast is not within this range the pressure load is set to zero).

The blast parameters are computed only once for each face during the model initialization, which therefore assumes negligible movement of the faces to which the analytical blast boundary condition are applied during the duration of the blast loading.

This implementation does not take into account shadowing, diffraction, multiple reflections or channelling by structures in the model as the distance of a face from the blast is computed as the straight line distance between the blast and the face. This is also true for faces not facing the blast which can lead to overestimation of pressure loads and impulses on these faces. It is assumed that the face has a direct line of sight to the blast if the face is determined to face the blast origin, irrespective of any structured obstructing the direct line of sight. Furthermore, since reflections from other structures are not considered, the implementation is not applicable for internal blasts or for buried charges.

The clearing option is intended only for very simple geometries and is designed only to work if each analytical blast boundary condition is applied only to one part. ‘Edges’ from which clearing can occur for a particular blast boundary condition on a particular part are determined by finding the blast load faces that face the blast which share a common edge only with faces that either have no analytical blast boundary condition applied, or have blast boundary conditions applied but do not face the blast. The clearing distance from a face is then calculated by finding the distance to the nearest ‘edge’ face. The distance is taken as the straight-line distance between the center of the two faces.

Joins are not accounted for when determining the ‘edge’ faces, therefore clearing could potentially come from the edge of joined faces.

The solver has no concept of the plane in which the floor lies in for surface bursts. Therefore clearing could potentially come from what the user intends to be the floor. As clearing cannot come from symmetry planes, a symmetry plane could be used to represent the floor.

#### 9.3.2.5. Example – Small Scale Urban Blast

The results of an Autodyn analytical blast simulation were compared to the results of a 1/50th scale experimental study of a blast in an urban environment as described by Feng [4]. The experiments consisted of an 8 g TNT charge positioned 30 mm above a horizontal steel plate. Structures representing buildings on the street were rested on the plate with the charge at the street center as depicted in Figure 9.3: Small Scale Blast Experiment. Two experiments were conducted: one with and one without the concrete blocks shown in Figure 9.3: Small Scale Blast Experiment.

Plan of the small-scale blast experiments as described in [4]. The red circle represents the charge and the green circles represent gauge locations.

The Autodyn analytical blast simulation was set up to mimic the experimental setup with a Lagrangian volume representing the building with an analytical blast boundary condition applied to the external faces. Gauges were positioned on the volume in the same locations as in the experimental set-up to monitor the applied pressure load using the variable BLAST.LOAD. Additionally, Autodyn Numerical Blast (using the Euler Ideal Gas solver) simulations of the blast were performed, and the pressure load histories (using theory from TM 5-855-1) were computed.

Figure 9.4: Side-On Overpressure and Figure 9.5: Reflected Overpressure compare the side-on overpressure time history and the reflected overpressure time history from the gauge on the front of the building for the Autodyn Numerical Blast model, TM 5-855-1, the Autodyn analytical blast mode,l and the experimental results when the concrete blocks were not present. The peak overpressure and positive phase impulse for these results are summarized in Table 9.1: Peak incident overpressure and positive phase impulses at the gauge on the front of the building for a spherical explosion without the concrete blocks and Table 9.2: Peak reflected overpressure and positive phase impulses at the gauge on the front of the building without the concrete blocks. Note that the Autodyn Numerical Blast results agree well with the experimental results and that the analytical blast Autodyn results agree well with the TM 5-855-1 results. The peak incident and reflected overpressures computed by the Autodyn analytical blast model are in reasonable agreement with experiment, although the incident and reflected positive phase impulses are slightly overestimated.

Figure 9.6: Channelled Street Blast Results shows the results from the Autodyn Numerical Blast model, TM 5-855-1, the Autodyn analytical blast model and the experimental results when the concrete blocks were present (a channelled blast). Table 9.3: Peak overpressures and positive phase impulses at both the front and side gauges when the concrete blocks are present summarises the peak overpressure and positive phase impulses for these results. The disagreement between the Autodyn analytical blast results and those from experiment are because the analytical blast formulation does not account for multiple reflections or shadowing by structures.

Table 9.1:  Peak incident overpressure and positive phase impulses at the gauge on the front of the building for a spherical explosion without the concrete blocks

Peak Incident Overpressure (kPa)Positive Phase Impulse (kPa.ms)
Autodyn Numerical Blast35.179.11
TM 5-855-136.4911.10
Autodyn Analytical Blast36.6011.12

Table 9.2:  Peak reflected overpressure and positive phase impulses at the gauge on the front of the building without the concrete blocks

Peak Incident Overpressure (kPa)Positive Phase Impulse (kPa.ms)
Autodyn Numerical Blast71.7380.17
TM 5-855-183.6523.20
Autodyn Analytical Blast80.1723.17
Experimental74.7218.28

Table 9.3:  Peak overpressures and positive phase impulses at both the front and side gauges when the concrete blocks are present

Peak Incident Overpressure (kPa)Positive Phase Impulse (kPa.ms)
FrontSideFrontSide
Autodyn Numerical Blast242.927.9863.6211.68
TM 5-855-183.65-23.20-
Autodyn Analytical Blast80.1729.3323.179.88
Experimental295.824.7434.498.46

Side-on overpressure experienced 1150 mm away from a 8g TNT charge as predicted by TM 5-855-1, an Autodyn Numerical Blast model analysis, and an Autodyn analytical blast analysis.

Reflected overpressure at the gauge on the front of the building shown in Figure 9.3: Small Scale Blast Experiment. The results from an Autodyn analytical blast analysis are compared to experiment, an Autodyn Numerical Blast model analysis (shown as AUTODYN-3D in the figure), and to the results predicted by TM 5-855-1.

Channelled street blast results A) overpressure at front gauge B) impulse at front gauge C) overpressure at side gauge D) impulse at side gauge. The results are for the Autodyn Numerical Blast solution (blue), Autodyn Analytical Blast solution (red dashed), TM 5-855-1 (black) and the experimental results (green).