Observed airblast variability and model error from repeatable explosive field trials

Abstract

Explosive field trials have been conducted to measure the peak incident pressure, impulse and time of positive phase duration following the detonation of 15 different masses of the Plastic Explosive No #4. A novel aspect of these field trials was the repeatability of tests. Eight pressure gauges collected data during each blast, and at each scaled distance. In all, 4 blasts were conducted for each scaled distance (i.e. up to 32 measurements recorded for each scaled distance) – 60 blasts were fired in total. Consequently, this repeatability of testing allowed the mean and variance of blast pressure–time histories to be quantified, with a view to better characterise the variability of a blast itself and model error variability. This article describes the explosive field trials, and the statistical analysis of blast load variability and model error for peak incident pressure, impulse and time of positive phase duration. It was found that the mean model error is close to unity with a coefficient of variation of up to 0.15 for pressure and 0.21 for impulse. The lognormal probability distribution best fits the model error data. The probabilistic models derived from these tests can be used for a variety of structural engineering applications, such as calculating reliability-based design load or partial safety factors for explosive blast loading, and estimating the probability of damage and casualties for infrastructure subject to explosive blast loading. This is illustrated for a terrorist explosive scenario involving a spherical free-air burst, where the damage modes of interest are breaching and spalling of a concrete slab. It was found that the variability of charge mass, range and model error have a significant effect on reliability-based design.

Keywords

Reliability airblast explosive blast loading probability risk

Introduction

To properly assess the safety and reliability of buildings, bridges, bunkers and other protective structures subject to accidental or malevolent explosive blasts, there is a need to first understand and quantify the (1) variability of structural materials, dimension and structural configurations that affect structural resistance or capacity and (2) variability of blast loads (e.g. Netherton and Stewart, 2010; Stewart, 2019). The effect of structural resistance variability on the structural reliability of protective structures is relatively well studied (e.g. Hao et al., 2016; Olmati et al., 2014; Shi and Stewart, 2015). However, there is less research that deals with the probabilistic characterisation of blast loads.

The current weaponeering techniques used by the Australian Defence Force (ADF), the US Department of Defense and many other armed forces consider the variability of weapon-platform integration, weapon-launch and a weapon’s delivery to a desired point of impact. No consideration is given to the variability of post-detonation blast loads and/or the variability of damage to people and/or structures given such loads. A similar situation occurs for the safe siting of explosive ordnance (EO). In this case, explosive safety distances proposed by the US Department of Defense Explosives Safety Board (DDESB, 2009) are formulated on the basis of expectancy (mean values), and at the core of their effects analysis they calculate blast load parameters deterministically. On the other hand, the US Department of Defense manual UFC 3-340-01 (2002), which superseded TM5-855-1 (1997), describes the benefits of reliability-based design, which help ‘understand airblast uncertainty, intelligently select design loads, and conduct cost-survivability tradeoff studies’. Hence, UFC 3-340-01 (2002) developed reliability-based load factors to be applied to nominal pressures to give 5th–99th percentiles of loads, for general purpose (GP) bomb detonations. However, the statistics of pressure variability were obtained from only one detonation of a GP 500 lb bomb and three detonations of GP 1000 lb bombs.

Airblast variability is consistently observed from explosive field trials (e.g. Bogosian et al., 2014; Formby and Wharton, 1996; Maserjian and Fisher, 1951; Petes and Tempo, 1984; Rigby and Sielicki, 2014; Rigby et al., 2014a). Twisdale et al. (1994) conducted a statistical analysis of blasts from 325 Mark-83 GP conventional bombs, which found a coefficient of variation (COV) of 0.30 for peak pressure and 0.25 for impulse. Low and Hao (2002) found a similar variability of peak reflected pressure (COV = 0.32) at various scaled distances in their review of available data. Bogosian et al. (2002) reported a COV of 0.23 for peak reflected pressure results of 190 blast tests involving TNT (trinitrotoluene), plastic explosive C-4 and ANFO (ammonium nitrate fuel oil) explosives. However, these statistics are based on data obtained from tests aggregated from various scaled distances. Hence, the observed statistics represent the variability of scaled distance (range, explosive mass) as well as of the blast itself. However, a probabilistic blast load model (P-Blast) developed by Netherton and Stewart (2010) was the first attempt to desegregate the various sources of uncertainty, namely charge mass, net equivalent quantity, range, temperature and pressure, and model error (ME) for free-field detonations. An improved understanding of airblast variability better informs societal appreciation and/or acceptance of risk-based decisions (e.g. Netherton and Stewart, 2016). This is the motivation of this article.

The deterministic models derived by Kingery and Bulmash (1984) are used in ConWep (1991) and UFC-3-340-02 (2008) (which superseded TM5-1300, 1990) to predict blast pressure and impulse for a given charge mass (W), range (R) and other inputs. However, these load models, as well as finite element methods (FEM) or computational fluid dynamics (CFD) such as Air3D (2001), Propagation of Shocks in Air (ProSAir, 2016), LS-DYNA (2016) and AUTODYN (2016), do not take into account the actual variability and uncertainty associated with (Stewart and Melchers, 1997) as follows:

Predictive model – that is, ME defined as the ratio of actual (test) value divided by predicted (model) value;

Input parameters used in the model (i.e. explosive mass, stand-off distance);

Inherent – or aleatory – variability, which relates to the natural (intrinsic, irreducible or fundamental) random uncertainty of a situation.

Hence, the P-Blast load model inputs the mean, COV and probability distribution of charge mass, net equivalent quantity, range, temperature and pressure, and ME into a Monte Carlo simulation analysis. P-Blast uses the Kingery and Bulmash (1984) polynomials to then estimate the mean and variability of peak pressure and impulse.

The stochastic characterisations of blast load are only possible from actual explosive field trials where the same blast scenario is repeated many times to measure variability of blast loading. Indeed, while many explosive field trials have been conducted over the decades, the majority of programmes are classified. In the few occasions where field trial data are available (e.g. the trials conducted by Hoffman and Mills, 1956), all of the experiments have been organised to meet an agenda different to that associated with probabilistic modelling. Consequently, the University of Newcastle (UoN) has undertaken explosive field trials with a view to capture blast load data that is better suited in terms of characterising ME and blast load variability. Information from tests such as these is then available for a variety of structural engineering applications, such as calculating reliability-based design load factors for explosive blast loading (Stewart, 2018; Stewart and Netherton, 2015), determining partial factors for the major wavefront parameters of reflected shock wavefronts (Campidelli et al., 2015) or estimating the probability of damage and casualty risks for infrastructure subject to explosive blast loading (e.g. Alterman et al., 2019; Shi and Stewart, 2015; Stewart, 2019).

This article describes the latest series of explosive field trials using Plastic Explosive #4 (PE4). A variety of spherical masses of PE4 were detonated (in free-air) for 15 scaled distances (Z) varying from 0.65 to 3.07 m/kg^1/3. Explosive mass is estimated as W = W_NEQ × W_mass where W_NEQ is the NEQ (net equivalent quantity) factor and W_mass is the measured mass of explosive. The scaled distances is Z = R/W^1/3 where R is stand-off distance in metres and W is explosive mass in kilograms.

A novel aspect of these field trials was the repeatability of tests. Eight pressure gauges collected data during each blast, and at each scaled distance. In all, 4 blasts were conducted for each scaled distance (i.e. up to 32 measurements recorded for each Z value) – 60 blasts were fired in total. Consequently, this repeatability of testing allowed the mean and variance of blast load parameters to be quantified, with a view to better characterise the variability of a blast itself and other parameters, such as ME variability. This article describes the explosive field trials, and the statistical analysis of blast load variability and ME for peak incident pressure and impulse.

P-Blast load model

Like other forms of loading, blast loads are subject to uncertainty and variability. P-Blast has been developed by Netherton and Stewart (2010) that considers variability of the following:

User factor for mass of explosive;

NEQ of an explosive in terms of a mass of TNT;

The range (R) and angle of incidence (AOI);

Air temperature and pressure;

Accuracy (ME) of predictive load models including inherent variability.

A schematic of the output from the P-Blast model is shown in Figure 1. For example, a blast scenario may involve the accidental detonation of a 500 lb (227 kg) Mark 82 GP bomb at an EO storage facility. If the building is 50 m from the EO, then the nominal incident pressure is 13.1 kPa. A comparison with the nominal load shows that it exceeds the Federal Emergency Management Agency (FEMA, 2003) threshold for medium damage (12.4 kPa), indicating that medium damage will occur for this blast scenario for a building 50 m from the EO. However, if variability of NEQ for tritonal is taken as ±5% and variability of ME is ±10%, then the P-Blast model shows that the COV of peak incident pressure is near 0.15 and so there is close to a 10% chance of the threshold for high damage (15.9 kPa) being exceeded. A risk-averse decision-maker may deem a 10% chance to be too high for this blast scenario, and recommend blast mitigation measures. For more details see Stewart and Netherton (2015).

Figure 1.

Schematic of probabilistic blast loads.

Netherton and Stewart (2010) derived probabilistic models for ME and inherent variability obtained from field data of repeatable tests (Hoffman and Mills, 1956). Of note is the age of the data where some of the experiments were conducted more than 60 years ago (e.g. Hoffman and Mills, 1956), and some of these data were lost or not captured properly and so the data record is not complete.

The output from the Netherton and Stewart (2010) P-Blast model is sensitive to changes in ME. The model is also sensitive, but to a lesser degree, to blast wave inherent variability and instrumentation error of the data capture system. The effect of variability in ambient air temperature and pressure is negligible. Consequently, improved characterisation of ME is crucial to improving the accuracy of probabilistic blast load modelling. Thus, it is desirable to conduct explosive field trials with the specific purpose of repeatability and the derivation of variability of ME.

Explosive field trials

The UoN was granted access to an explosives range owned by the ADF, which is a purpose-built open-air explosives bunker at the Royal Australian Air Force (RAAF) Base at Williamtown in New South Wales (see Figure 2).

Figure 2.

The explosives range at RAAF Base Williamtown showing the experimental set-up within the open-air bunker.

The principle deliverable for the explosive trial programme was the capture of appropriate pressure–time data to enable the statistical characterisation of peak incident pressure, impulse and time of positive phase duration. This requires repeatable tests where W and R are held constant. Furthermore, the field trials focused on capturing incident pressure values only, and not reflected pressure values. The motivation behind this experimental series was the calculation of a blast wave’s variability, not necessarily capturing what may be the greatest peak load. Consequently, only incident pressures were captured, with a plan to record reflected values in the future. A supplementary consideration for capturing incident pressures was the relatively low cost, ease and simplicity of recording only side-on pressures. Moreover, structural damage criteria are often based on values of incident pressure (e.g. American Society of Civil Engineers (ASCE), 2010; FEMA, 2003). Finally, a key assumption herein is that the variability of an incident pressure wave will describe the variability of that same wave when reflected. Future tests will capture the variability of reflected pressures; however, at present, it is assumed that the variability of incident waves will have a strong positive correlation to their variability post-reflection. It is important to note that the measured (test) variability of reflected pressure is likely to be lower than the measured variability of incident pressures (e.g. Tang et al., 2017). However, once instrumentation variability (see V_instrument in section ‘Probabilistic characterisation of MEs’) is removed from the probabilistic analysis, the variability of ME for reflected and incident pressures may well be similar, however, this is the topic for future research.

Plastic Explosive (PE4)

The preferred explosive for this trial would have been TNT, as this is the universally accepted explosive of reference. However, the licence requirements for this particular range dictated that the explosive PE4 had to be used. PE4 is a conventional plastic explosive used extensively by military forces worldwide. Its principal energetic component is cyclotrimethylenetrinitramine, or, more simply known as RDX.

Of interest is the common use, indeed, interchangeability, of the terms PE4 and C-4. PE4 is considered to be ‘nearly identical’ or ‘very similar’ to the explosive compound C-4 in its explosive properties, albeit with a slight reduction in the percentage of RDX (Bogosian et al., 2016). Experimental studies by Bogosian et al. (2016) and Rigby and Sielicki (2014) conclude that ‘PE4 is in fact the same as C-4 with regard to its pressure and impulse equivalent’. In this case, UFC 3-340-01 (2002) states that the TNT equivalence for C-4 is NEQ = 1.20 for a pressure range of 0.07–1.38 MPa, and NEQ = 1.37 for a pressure range of 1.38–20.7 MPa in terms of pressure. In terms of impulse the NEQ = 1.19 for all pressures. These NEQ values have been used in other studies involving PE4 (e.g. Weckert and Anderson, 2006).

Blast Trial #1 (June 2012)

The first field trial (which is not the subject of this article) was conducted at RAAF Base Williamtown on 4–8 June 2012. This trial is described in full by Netherton et al. (2014), a summary of which is (Stewart and Netherton, 2015) as follows:

Explosive is PE4;

Explosive mass W_mass = 0.250 kg per charge;

Seven blasts at Z ≈ 1.5 m/kg^1/3 and 3.0 m/kg^1/3, and eight blasts at Z ≈ 1.0 m/kg^1/3 and 2.0 m/kg^1/3 – a total of 30 blasts;

Peak observed pressure for all shots was 1.05 MPa, leading to a TNT equivalency for PE4 of NEQ = 1.20 and 1.19 for pressure and impulse, respectively.

Blast Trial #2 (July 2014)

The second experimental blast trial was conducted at RAAF Base Williamtown on 14–31 July 2014. The intent of the second blast trial was that Z values would complement the scaled distances from the earlier 2012 trial. A summary of the second blast trial is as follows:

Explosive is PE4 – manufactured to an ADF Standard, DEF (AUST) 4061, by Thales Australia Limited (Australian Munitions (AM), 2016);

Fifteen explosive masses (W_mass) of PE4 ranging between 0.02 and 1.8 kg;

Four blasts of each mass – 60 blasts were fired in total;

The distance between each explosive’s centre and any blast gauge is R = 0.882 m;

Fifteen different Z values (see Table 1);

Seven Z values in the near-field (Z < 1 m/kg^1/3);

Three Z values between 1 and 2 m/kg^1/3;

Five Z values between 2 and 3 m/kg^1/3;

Average temperature was approximately 15°C and pressure of 1000–1040 hPa;

Peak observed pressure exceeded 1.38 MPa when W_mass was 0.85 kg or higher, leading to a TNT equivalency for PE4 of NEQ = 1.37 and 1.19 for pressure and impulse, respectively;

For W_mass < 0.85 kg, then NEQ = 1.20 and 1.19 for pressure and impulse, respectively.

Table 1.

Nominal explosive mass, range and scaled distance for Blast Trial #2.

Nominal explosive mass (W_mass) of PE4 (g)	Nominal range, R (m)	Nominal scaled distance, Z (m/kg^1/3)
Nominal explosive mass (W_mass) of PE4 (g)	Nominal range, R (m)	Z_p	Z_i
Detonator	0.882	−	−
20	0.882	3.06	3.07
30	0.882	2.67	2.68
40	0.882	2.43	2.43
50	0.882	2.25	2.26
100	0.882	1.79	1.79
300	0.882	1.24	1.24
500	0.882	1.05	1.05
700	0.882	0.94	0.94
850	0.882	0.88	0.88
1000	0.882	0.83	0.83
1200	0.882	0.78	0.78
1400	0.882	0.74	0.74
1600	0.882	0.71	0.71
1800	0.882	0.68	0.68

PE4: Plastic Explosive #4.

Note that one series of shots used the detonator as the explosive source (i.e. no PE4); however, due to the very low and imprecise NEQ of the detonator this series of shots was not included in ME characterisation described in section ‘Probabilistic characterisation of MEs’.

Experimental set-up

The experiment used eight sensors, arranged so they were all in-plane with the explosive’s centre of mass and all equidistant from the point of detonation. Each explosive sphere was placed inside a stocking, which was then hung from overhead steel cables and securely located (underneath) via a set of cross-strings. The intent was that the centre of the explosive was always at the same point in three-dimensional (3D) space for each detonation (see Figures 3 to 5). Hence, the centre-of-mass of the explosive was the same distance – or range (R) – from the eight ‘in-plane’ blast gauges that surrounded the charge. It is important to reiterate that all eight gauges were set up to record incident, and not reflected pressure values.

Figure 3.

Experimental set-up showing the location of the explosive (in the black-sock) relative to the eight equidistant blast gauges (recording incident pressure).

Figure 4.

Experimental set-up (in plan view), showing the location of the explosive relative to all eight ‘in-plane’ gauges, thus capturing incident pressure values only.

Figure 5.

Experimental set-up showing a stocking holding a 1.2 kg sphere of PE4, where the charge is suspended, centred and placed relative to detonation-location cross-strings.

A key aspect of the field trials was the repeatability of tests. Care was taken with respect to the repeatability of explosive mass, detonator location, and the distance between the explosive and instruments. That said, there will, of course, be some variability of these parameters. So, while the nominal mass values (as listed in Table 1) were desired, the actual mass values used were within a tolerance of ±0.6%. Full details of each explosive mass, as assembled and detonated, are described by Netherton et al. (2016).

Each explosive blast used a spherical mass of PE4, with each sphere handshaped from portions of standard-military-cartridges of PE4 using moulded-rubber forms (see Figure 6). An electric detonator (Detonator Electric Demolition F2) was inserted into the top of each sphere such that the tip of the detonator was located in the middle of each sphere. The intent was that each explosive was centre-detonated. Once the detonator is electrically initiated, a high-order detonation of the PE4 sphere ensues, as shown in Figure 7.

Figure 6.

A sphere of PE4 (0.25 kg) hand-moulded via a rubber mould.

Figure 7.

Detonation of a 0.050 kg sphere of PE4 (still image taken from high-speed video).

Data capture

In consultation with the Australian Defence Science and Technology (DST) Group, the UoN developed a bespoke blast data recording system. This included the following:

A sensor sub-system: piezoelectric gauges (PCB: Model 113B) held within gauge support discs (DST Group design) and instrumentation support frames (see Figure 8);

A data collection sub-system: an integrated electronic piezoelectric excitation power supply unit and a 2-MHz, 24-channel, signal acquisition and data storage unit.

Figure 8.

Blast gauge in the centre of the much larger support disc.

The data captured consisted of voltage–time histories from each of the eight blast gauges, and from each of the 60 blasts fired, for a total of 480 individual data records. Relevant information for each record is as follows:

The voltage–time history was converted into a pressure–time history using the voltage/pressure calibration data appropriate for each particular gauge.

The raw pressure-time data was ‘smoothed’ using 100,000 curve-fit points within the programme KaleidaGraph. An example of raw and smoothed plots is shown in Figure 9.

The incident pressure value (P_i) was taken from the raw pressure–time history.

The time of arrival of the blast wave (t_a) was determined from the smoothed pressure–time history, and was taken as that time when pressure values first passed through the x-axis, and continued trending upward towards the maximum value.

The time at the end of the blast wave’s first positive phase (t_e) was taken as that time when the smoothed curve first passed below the x-axis.

The time of duration of the blast wave’s first positive phase is t_d = t_e − t_a.

The blast wave’s incident impulse (I_i) was calculated via trapezoidal integration between all data points of the raw pressure–time history, between values of t_a and t_e.

Blast wave values P_i, I_i and t_d for all 480 data records are provided in Netherton et al. (2016).

The summary of blast load variability is shown in Table 2.

Figure 9.

The pressure–time history from a blast of 0.1 kg PE4, recorded at a distance of 0.882 m. The black line is the raw data, while the red line shows the smoothed data line.

Table 2.

Statistical summary of blast-wave incident pressure, impulse and duration.

Nominal explosive mass (kg)	Range (m)	Blast-wave parameter	Mean	Standard deviation	COV
Detonator only	0.882	P_i (kPa)	21.64	3.27	0.15
		I_i (Pa s)	2.81	0.55	0.20
		t_d (ms)	0.40	0.04	0.10
0.020	0.882	P_i (kPa)	89.92	10.25	0.11
		I_i (Pa s)	18.36	1.63	0.09
		t_d (ms)	0.59	0.03	0.05
0.030	0.882	P_i (kPa)	113.40	13.30	0.12
		I_i (Pa s)	23.45	2.45	0.10
		t_d (ms)	0.64	0.06	0.09
0.040	0.882	P_i (kPa)	139.98	16.42	0.12
		I_i (Pa s)	27.08	2.54	0.09
		t_d (ms)	0.62	0.06	0.09
0.050	0.882	P_i (kPa)	158.14	19.65	0.12
		I_i (Pa s)	30.91	3.02	0.10
		t_d (ms)	0.62	0.07	0.12
0.100	0.882	P_i (kPa)	258.34	27.68	0.11
		I_i (Pa s)	47.10	4.95	0.11
		t_d (ms)	0.66	0.08	0.12
0.300	0.882	P_i (kPa)	622.67	91.96	0.15
		I_i (Pa s)	90.14	19.01	0.21
		t_d (ms)	0.69	0.15	0.21
0.500	0.882	P_i (kPa)	954.02	127.65	0.13
		I_i (Pa s)	127.87	18.01	0.14
		t_d (ms)	0.72	0.10	0.14
0.700	0.882	P_i (kPa)	1262.22	134.28	0.11
		I_i (Pa s)	166.65	28.23	0.17
		t_d (ms)	0.77	0.11	0.15
0.850	0.882	P_i (kPa)	1514.26	175.42	0.12
		I_i (Pa s)	207.93	30.92	0.15
		t_d (ms)	1.03	0.29	0.28
1.000	0.882	P_i (kPa)	1674.68	206.75	0.12
		I_i (Pa s)	200.40	27.70	0.14
		t_d (ms)	0.79	0.18	0.23
1.200	0.882	P_i (kPa)	1861.20	253.46	0.14
		I_i (Pa s)	218.87	29.15	0.13
		t_d (ms)	0.78	0.21	0.26
1.400	0.882	P_i (kPa)	2187.98	325.97	0.15
		I_i (Pa s)	243.25	32.32	0.13
		t_d (ms)	0.83	0.32	0.38
1.600	0.882	P_i (kPa)	2319.56	258.04	0.11
		I_i (Pa s)	255.62	31.06	0.12
		t_d (ms)	0.95	0.35	0.37
1.800	0.882	P_i (kPa)	2430.40	265.33	0.11
		I_i (Pa s)	269.77	39.56	0.15
		t_d (ms)	0.78	0.32	0.41

COV: coefficient of variation.

Note that data records were included only if the blast waves were unaffected, in that the wave had to pass a visual inspection where rate of decay of the first positive wave was examined. If a wave had (clearly) been affected by something (i.e. was not smooth), then the data associated with that particular wave, from that particular gauge, were not included in the analysis. Once the series of tests involving the detonator alone were removed from the analysis, this situation occurred 78 times out of the possible 480 data records.

The maximum incident pressure values (P_i) taken from the raw pressure–time history are taken prima-facie. This does not presume that sensor-ringing or other electrical noise – as described by Rigby et al. (2014b) – is not present, although it is not obvious from the raw data. Indeed, such matters may affect the maximum pressure value. Rather, in this article, these matters are taken to be implicitly characterised (later, via equation (3)) as part of the variability associated with instrument error. Future testing will be conducted with a specific agenda to better characterise sensor-ringing and other electrical noise, and what it may mean in terms of the actual observed variability.

Table 2 shows that variability is the highest for duration t_d. This is not unexpected as the determination of t_d was often not easily identified from the pressure–time plots. Figure 9 shows a ‘smoothed’ pressure–time history, thus permitting a reasonable estimate of t_d. However, in some instances, the wave’s profile seems to take a relatively long time to pass from positive pressure to negative, as shown in Figure 10. In this instance, the (smoothed) pressure–time history does not pass below the x-axis when expected, at approximately 1.8 ms; rather, the pressure seems to ‘hang-on’ until approximately 2.2 ms. This 0.4 ms, or 22%, increase in t_d will cause significant differences in the estimation of t_d. There are a number of possible reasons for these situations, such as reflection of the wave from the ground, following which the incident and reflected waves coalesce; or, there could be a general increase in gas pressure around the base of the bunker, such that rapid clearing of the blast wave does not occur. Notwithstanding, it is not the scope of this article to desegregate and/or identify the various reasons for a wave’s disruption. For a fuller discussion see Netherton et al. (2016).

Figure 10.

The raw pressure–time history from 0.85 kg of PE4, range = 0.882 m.

Mass variability of PE4 cartridges

Of supplementary interest was the measured mass of the as-supplied cartridges of PE4. Each individual cartridge was weighed, thus permitting the characterisation of the mass variability of an industrially produced explosive compound. Full details of the mass of each PE4 cartridge are provided in Netherton et al. (2016), with the statistical characterisation of explosive mass summarised in Table 3. As expected, mean is very close to the nominal mass and variability is very low.

Table 3.

Statistical characterisation of explosive mass of the PE4 cartridges.

Parameter	Value
Nominal mass of each cartridge	230.000 g
Number of cartridges weighed	139
Minimum and maximum weights	213.225–243.164 g
Mean	229.437 g
Standard deviation	4.802 g
Coefficient of variation (COV)	0.021

Probabilistic characterisation of MEs

ME or model uncertainty is determined from a comparison of pressure and impulse data observed during tests with predicted results as

Blast load model error {(ME}_{T / M}) = \frac{Observed (test) value}{Model prediction}

(1)

In the present case, the model prediction is based on the widely used Kingery and Bulmash (1984) polynomials for AOI = 0°, and standard temperature and pressure of 15°C and 1013.25 hPa. Equation (1) calculates the ratios of test-to-model values (ME_T/M) for P_i and I_i using the following:

Hoffman and Mills (1956) test data (n = 472) used for derivation of ME statistics described by Netherton and Stewart (2010);

Blast Trial #1 test data (n = 223) used for derivation of updated ME statistics described by Stewart and Netherton (2015);

Blast Trial #2 test data (n = 370).

It is important to note that equation (1) includes inherent variability and variability of instrument error and test procedures, so it represents more than just the accuracy of the model itself. Hence the statistical parameters of ME are

mean (ME) = mean ({ME}_{T / M})

(2)

V_{M E} = \sqrt{V_{T / M}^{2} - V_{i n s t r u m e n t}^{2} - V_{t e s t}^{2}}

(3)

where V_T/M is the COV obtained directly from a comparison of the measured and model (predicted) blast loads, V_instrument represents the COV in the measured loads due to the accuracy of instrumentation, and V_test represents uncertainties due to differences in measured range, mass, spherical charge and so on. Equations (2) and (3) assume that there is no systematic bias in variability, instrument or set-up errors, and there is statistical independence between variables. Note that inherent variability is included in the ME statistics.

Military Applications of Blast Simulators (MABS, 1995) provides little information on calibration accuracies (real or perceived) for the data acquisition systems they describe; nevertheless, some of the example blast pressure gauges discussed are given an accuracy of ±10%. Smith (Engineering Systems Department, Defence College of Management and Technology, Cranfield University, 23 June 2009, Private email communication) states that it was often necessary to conduct multiple experiments to have confidence in the measurements, and reported that their instrumentation gave output within ±10% of the actual values. For these reasons, Netherton and Stewart (2010) recommend that V_instrument is likely to be 0.05 – that is, instrumentation accuracy is ±10% and that such errors are normally distributed and that they represent at least 95% of all possible values. Clearly, the upper bound for instrument error is the variability of observed blast response, which according to Table 2 is up to 0.21 for impulse.

The blast trials went to considerable effort to accurately measure range, mass and ensuring that the charge was spherical and located accurately as part of the test set-up. Nonetheless, there still remains some variability in the test set-up. For example, if the actual range deviates from the measured range by ±10 mm (or ±1%), then pressure and impulse vary by ±1%–2%. Hence, if test set-up accuracy is ±2% and such errors are normally distributed and they represent at least 95% of all possible values, then V_test = 0.01.

ME statistics are not inferred for time of positive phase duration (t_d) due to less confidence in measuring that variable (see section ‘Data capture’). This is confirmed by Kingery (1966), who noted that positive duration is very difficult to measure with consistency and repeatability. Moreover, t_d can be inferred directly if P_i and I_i are known.

The ratio of observed test to model data (ME_T/M) for P_i and I_i are plotted against scaled distance in Figures 11 and 12. These figures show that ME_T/M for pressure and impulse have a mean ME close to unity and can be approximated by the best-fit lines described in Table 4. That ME close to unity is not unexpected as the Kingery and Bulmash polynomials are derived from a best-fit regression analysis to extensive field data.

Figure 11.

Observed model errors for pressure.

Figure 12.

Observed model errors for impulse.

Table 4.

Statistical parameters for blast loading model errors.

	Mean	COV	Distribution
Pressure
0.053 m/kg^1/3 ⩽ Z ⩽ 3.0 m/kg^1/3	1.06–0.005Z	0.1175–0.0125Z	Lognormal
3.0 m/kg^1/3 < Z ⩽ 6.0 m/kg^1/3	1.06–0.005Z	0.08	Lognormal
Impulse
0.053 m/kg^1/3 ⩽ Z ⩽ 3.0 m/kg^1/3	0.99	0.222–0.054Z	Lognormal
3.0 m/kg^1/3 ⩽ Z ⩽ 6 m/kg^1/3	0.99	0.06	Lognormal

COV: coefficient of variation.

The ME_T/M versus Z plots in Figures 11 and 12 contain data for 0.58 m/kg^1/3 < Z < 6.0 m/kg^1/3, which are the limits of the scaled distances of Hoffman and Mills (1956) and Blast Trials #1 and #2. It is desirable to have an ME methodology for the full range of Z values for the blast loading polynomials of Kingery and Bulmash (1984) – that is, 0.0531 m/kg^1/3 < Z < 40.0 m/kg^1/3. Given the choice that a horizontal line best describes the ME_T/M values for pressure and impulse, then such a line is readily extrapolated to Z values below 0.58 and above 6.0 m/kg^1/3. However, the quantification of ME_T/M within the 0.0531 m/kg^1/3 < Z < 0.58 m/kg^1/3 and 6 m/kg^1/3 < Z < 40 m/kg^1/3 domains are a topic of future field trials and research. For more details see Baldacchino (2017). These statistics are not overly sensitive to instrument or test variability. For example, COV of ME is not affected if test set-up variability (V_test) is increased from 0.01% to 0.025% (±5%).

It is observed from Figures 11 and 12 that the variability of pressure and impulse appears to be a function of scaled distance, with seemingly greater variability at lower Z values. The COV of ME (V_ME) is obtained from equation (3) when V_instrument = 0.05 and V_test = 0.01. The COV of ME is shown in Figures 13 and 14 for pressure and impulse, respectively. These figures show that the variability of ME tends to reduce with scaled distance. The lines of best fit are shown and are also described in Table 4.

Figure 13.

COV of model error for pressure as function of scaled distance.

Figure 14.

COV of model error for impulse as function of scaled distance.

Of interest is the observation that as Z decreases the variability of the fireball’s perimeter (and thus, clearly, its effects) can be quite varied, as per Figure 7. An understanding of why this perimeter variation occurs is important, particularly in terms of blast-wave pressure and impulse variability. Rigby et al. (2019) suggest that, for near-field explosive blasts, their observed variability is evidence of the growth of various instabilities on the surface of the expanding detonation product cloud. While there is clearly a limit to fireball expansion, the closely related – and inflating – blast wave ‘sphere’ proceeds well beyond this threshold. In these circumstances, it is regularly observed that the blast-front (almost immediately) becomes uniform, and a ‘smooth’ spherical emanation ensues, and out to distances well beyond the fireball. However, even with this apparently uniform wave, recorded pressure values (at the same radius, but different locations) are clearly variable, as evidenced by the blast load statistics in Table 2. It is assessed that the increases in the standard deviation of pressure and impulse (when closer to the detonation source) are related to the physical perturbations of the fireball; however, these assertions are yet to be tested, and will be the subject of further explosive trials.

The most appropriate probability distribution is also needed to fully characterise ME. Figure 15 shows the ME data histogram and five fitted probability distributions (normal, lognormal, Weibull, gamma and Gumbel). The Kolmogorov–Smirnov test found that only the lognormal probability model was not rejected at the 5% significance level for pressure MEs. Figure 16 shows a range of probability distributions fitted to the MEs using an inverse a cumulative distribution function (CDF⁻¹) plot. When the CDF⁻¹ of a particular probabilistic model sits on the 1:1 line, then this indicates that the probabilistic model is a good fit for the data. Of most interest to the safe performance of EO storage and maintenance and protective and civilian structures is the upper tail of ME – when the actual blast load is higher than that predicted by existing design models. In this case, Figure 15 shows that the normal distribution underestimates high MEs. It is observed from Figure 16 that the lognormal distribution slightly overestimates the upper tail of ME. Therefore, to be slightly conservative, the lognormal distribution is selected as the best fit to observed ME data for pressure.

Figure 15.

Histogram of observed model errors compared with five probability distribution models, for pressure.

Figure 16.

Distribution fitting to model error data, for pressure.

The Kolmogorov–Smirnov test found that only the Weibull probability model is rejected at the 5% significant level for impulse. The lognormal distribution best fits the observed MEs as it is slightly above the 1:1 line (Figure 17), indicating that the lognormal distribution slightly overestimates the upper tail of ME (see also Figure 18). Therefore, the lognormal distribution is selected as the best fit to observed ME data for impulse.

Figure 17.

Distribution fitting to model error data, for impulse.

Figure 18.

Histogram of observed model errors compared with five probability distribution models, for impulse.

Finally, the estimation of correlation coefficients shows little or no correlation between pressure and impulse. Hence, there is reasonable statistical independence between each ME data set, as such, and blast load MEs for P_i and I_i may be assumed as statistically independent.

Illustrative example: person-borne improvised explosive device damage to reinforced concrete slabs

Person-borne improvised explosive device threat

A terrorism blast scenario is used to illustrate the effect of airblast variability, and ME in particular, on decision-making. In this case, a person-borne improvised explosive device (PBIED) may contain 10 kg of homemade explosive (HME) that is detonated 0.6 m above a reinforced concrete (RC) floor slab. The damage modes of interest are breaching and spalling. The following example is a hypothetical scenario for a specific threat, and the variability of W and R is heuristic and will highlight the type of information that is required for risk-based assessment of blast damage.

The HME triacetone triperoxide (TATP) was used in the London underground bombing (2005), Paris attacks (2015), Brussels attacks (2016) and the Manchester concert bombing (2017). TATP is a very volatile and unstable explosive and is easily detonated accidently. Landucci et al. (2015) estimates that TATP has an NEQ of 0.5–0.6 if manufactured in a controlled laboratory environment. Hargather and Settles (2007) found from experimental testing that the NEQ for TATP is 0.1–0.45, and is a function of distance from the charge. Pachman et al. (2014) find from experimental testing that the average NEQ is 0.7 for pressure and 0.55 for impulse. On the other hand, Price and Ghee (2009) used a thermo-chemical theoretical model to predict that the NEQ for TATP is a high 0.92. In all experimental studies, TATP was manufactured in controlled laboratory environments. If TATP is stabilised with carbonaceous liquids and waxes its NEQ is even lower. The NEQ of TATP is also highly dependent on the quality of the final product, and is very sensitive to the temperature during its synthesis (Landucci et al., 2015). Even when made under ideal conditions the NEQ of TATP is highly variable. It is assumed herein that terrorist-made TATP will have a mean NEQ of 0.6, with lower and upper bounds of 0.35 and 0.85, then if these limits represent the 95% percentile bounds of a lognormal distribution, the COV of NEQ is 0.23. If mass tolerance for a PBIED is ±20%, then COV of explosive mass (W) is 0.25 for a terrorist PBIED (see Stewart, 2019 for more details on improvised explosive device (IED) variability).

The precise height of burst of an PBIED is also uncertain. In this example the mean value is taken as 0.6 m. However, the placement of a PBIED may vary by, say ±0.2 m. In this case, the COV of range is approximately 0.15 assumed lognormally distributed.

Spalling and breaching of RC slab

The US Department of Defense design manual UFC 3-340-02 (2008) provides formulae to predict the minimum thicknesses to prevent (1) spalling of the soffit of an RC slab, and (2) breaching of an RC slab. The explosive is a spherical free-air burst which is the same explosive scenario used to derive MEs in section ‘Probabilistic characterisation of MEs’. The work of Marchand et al. (1994) provides the basis for the design formulae in UFC 3-340-02 (2008), where spalling and breaching predictions are inferred from peak reflected pressure (P_r), reflected impulse (I_r) and concrete compressive strength $(f_{c}^{'})$ .

The pressure and impulse are estimated for a spherical free-air burst with standard temperature and atmospheric pressure. Table 5 shows the minimum slab thickness to prevent spalling and breaching for the nominal (deterministic) case when W = W_NEQ × W_mass = 0.6 × 10 kg = 6 kg TNT equivalency. Range is R = 0.6 m and concrete compressive strength is taken as $f_{c}^{'}$ = 40 MPa. Nominal scaled distance is Z = 0.33 m/kg^1/3.

Table 5.

Probabilistic blast loads and their effect on structural performance (values rounded to nearest 10 mm).

Blast scenario	PressureP_r (kPa)	ImpulseI (kPa ms)	Minimum thickness to prevent breaching (mm)	Minimum thickness to prevent spalling (mm)
Nominal (deterministic)	70,342	4943	100	180
Probabilistic analysis
(1) Exclude model errors
Rel = 50%	70,553	4904	100	180
Rel = 90%	101,996	7365	130	250
Rel = 95%	111,844	8305	140	280
(2) Include model errors
Rel = 50%	73,859	4795	100	180
Rel = 90%	110,437	7712	140	280
Rel = 95%	122,396	8849	150	310

Rel: reliability.

It is now of interest to consider the probabilistic analysis using P-Blast. The COV of W and R are 0.25 and 0.15, respectively. Table 5 also shows the pressure and impulse for reliability (Rel) levels of 50%, 90% and 95% (i.e. probability of load exceedance is 50%, 10% and 5%, respectively). It is initially assumed that blast loading MEs are excluded from the probabilistic analysis (i.e. mean(ME) = 1.0, V_ME = 0.0). As expected, the blast loads with a 50% reliability (median) are close to the nominal loads, providing similar estimates of slab performance. However, if blast loads are based on a 10% chance of exceedance (reliability = 90%), then pressure and impulse increase by 45% and 49%, respectively, leading to higher slab thicknesses to prevent breaching and spalling. In other words, the reliability-based design load factors (RBDF) are 1.45 and 1.49 – for more details see Stewart (2018). Blast loads increase for high levels of reliability, leading to thicker slabs. For example, a risk-averse decision-maker may want to ensure that the design thickness only has a 5% chance of being breached by a PBIED, hence the minimum slab thickness would be increased from 100 to 140 mm.

Table 5 also shows results if MEs given in Table 4 are now included in the probabilistic analysis. The effect is to increase high reliability (Rel ⩾ 90%) blast loads by up to 25% when compared to the loads without ME. The effect on minimum thicknesses is not overly significant. However, if for example, the variability of W or R was lower, then the effect of ME variability would increase in relative terms. To be sure, other blast scenarios will reveal different sensitivities to variability of W, R and ME. However, in all cases the effect of variability of W, R and ME will be important for protective design where the decision-maker may wish to make a decision on the basis of reliability and risk.

Future work

The current explosive field trials and associated MEs considered a free-field airburst detonation. This the ‘simplest’ blast scenario to simulate, and it forms the basis for much research on protective design and assessment. Airblast uncertainty and variability is likely to increase for more complex blast environments (e.g. urban environments), for other charge shapes and orientation, angles of incidence, detonator locations, height of detonation, topography of ground surface, casing effects and so on. These are all topics for further study, but are challenging as explosive field testing programmes need to isolate each of these influencing factors in turn, which will result in many firings and care in desegregating each influencing factor. Future work will also focus on measuring variability of reflective pressures and comparing these to incident pressures.

Conclusion

To characterise the variability of explosive blast waves, there is a need for repeatable blast testing. Hence, explosive field trials were conducted to measure the peak incident pressure, impulse and time of positive phase duration following the detonation of 15 different masses of spherical masses of PE4 for 15 scaled distances varying from 0.68 to 3.07 m/kg^1/3. The test set-up allowed for a free-air detonation with eight pressure gauges to collect data, each for the same range. Four blasts were conducted for each scaled distance providing up to 32 measurements for each Z value. The repeatability of testing allowed the mean and variance of blast load parameters to be quantified, which was then used to probabilistically characterise the variability of a blast itself and ME variability. It was found that the mean ME is close to unity with a COV of up to 0.15 for pressure and 0.21 for impulse. The lognormal probability distribution best fits the ME data. It was found that the variability of charge mass, range and ME are important for reliability-based protective design. This was illustrated for a terrorist explosive scenario involving a spherical free-air burst, where the damage modes of interest were breaching and spalling of an RC slab.

Footnotes

Acknowledgements

The support of the following people and organisations are gratefully acknowledged: Australian Research Council Discovery Project DP160100855; Air Vice-Marshall G.N. Davies, Deputy Chief of Air Force, Mr Ross Gibson, Mr Ian Jeans and Mr Michael Goodwin from the Civil Engineering Laboratory; and final year project students Samuel Buttershaw, Kaitlin Reidy and Bryn Rodger.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Mark G Stewart

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