Reliability-based design and robustness for blast-resistant design of RC buildings

Abstract

Explosive blasts from accidental or malevolent sources constitute an extreme event resulting in abnormal loads on buildings and other structures. A reinforced concrete (RC) multistorey building is assumed to be attacked by a terrorist vehicle-borne improvised device. Structural reliabilities are calculated for each RC column in the multistorey building exposed directly to the blast event. The probability of progressive collapse for the building is then estimated using system reliability analysis comprising of ground floor columns exposed to the explosive blast. The RC columns are designed according to United States blast-resistant design standard based on (i) threat dependent and (ii) alternate path design methods. The effects of threat dependent and alternate path design methods on column sizing, column reliability, and building collapse probability are investigated by conservatively assuming that collapse occurs if one or more columns fail. The robustness is also dependent on the location of the explosive. It was also found that a threat-dependent design appears to be more effective than the alternate path method in reducing building collapse risks.

Keywords

Blast robustness reliability-based design load factors terrorism

Introduction

Explosive blast loading may be accidental or malevolent. Either way, there is considerable uncertainty and variability with the likelihood, intensity and damage of explosive blast loading. Airblast variability has been observed from explosive field trials when range, explosive type and mass and other factors are held constant and this variability can exceed 30% in some cases. Such variability is not unexpected, as the sudden release of energy following detonation of an explosive is a complex phenomenon with temporal and spatial variability associated with the emanation of a shock-wave.

Explosive blast loading may arise from a terrorist improvised explosive device (IED), accidental detonation of explosives or munitions, or the deployment of bombs, missiles or other weapons by the military. The variability of explosive mass and range will differ for each blast scenario. For example, the variability of explosive mass (i.e. energetic potential) for a weapon is negligible as they are manufactured to high standards. On the other hand, there is significant uncertainty about (i) the energetic properties of a home-made terrorist IED and (ii) where such a device is located (e.g. Stewart, 2018). Added to this is an understanding of the model error (or model uncertainty) associated with widely used blast load design models, such as ConWep (1991), UFC 3-340-02 (2014), as well as weaponeering software (Driels 2020).

Probabilistic and structural reliability methods underpin the development of modern codes of practice. They can also be applied to explosive blast loading. Hence, the U.S. Department of Defense Design and Analysis of Hardened Structures to Conventional Weapons Effects (UFC 3-340-01 2002) describes the benefits of performance-based or reliability-based design, which helps to ‘understand airblast uncertainty, intelligently select design loads, and conduct cost-survivability tradeoff studies’. A reliability-based design load factor (RBDF) is multiplied by the nominal load to ensure that the actual load is equal to the reliability level. A load factor greater than one increases loads resulting in safer structural designs. A risk-averse decision maker may prefer there to be 95% certainty (0.95 reliability level) that the actual blast is less than the nominal (predicted) value.

RBDFs were first proposed by Twisdale et al., (1994) for general purpose (GP) bombs where the only source of uncertainty was model error. Stewart (2018) developed an analytical model to assess RBDFs that included range, explosive mass and model errors variabilities to derive RBDFs for pressure and impulse for a hemispherical charge detonating against a reflecting surface. This blast scenario provides the basis for the design of many protective structures. The analytical RBDF model developed by Stewart (2018) involved the use of nine design charts due to the large number of combinations of range and explosive mass variabilities where closed-form solutions do not exist. Stewart (2021a) developed a simplified approach where regression equations may be used to predict airblast variability and RBDFs for all combinations of range, explosive mass and model errors for spherical free-air and hemispherical surface bursts. The benefit of this simplified approach is that Monte-Carlo or other probabilistic methods are not needed.

Previous work has focused mainly on how RBDFs affect the reliability-based design of a single reinforced concrete (RC) column exposed to terrorism threats or the accidental detonation of stored explosive ordnance (e.g. Stewart, 2019). This work is somewhat idealised, however, as (i) it assesses damage assuming the location of detonation is directly in front of the building (or column) with an angle of incidence of 0^o, (ii) while a column may be heavily damaged there is no 100% surety that it will collapse, (iii) often more than one column is exposed and vulnerable to blast loading, and (iv) load sharing can occur between damaged columns. Hence, a more realistic assessment is one based on structural reliability that considers the damage and safety of all RC columns exposed to the blast allowing for variability of resistance, dead and live loads, and airblast. Hence, the present paper considers these more realistic features by assessing structural performance as an idealised structural system that considers the effect of load sharing in the estimation of probability of progressive collapse. The paper considers a terrorist VBIED scenario, and compares the effect that reliability-based design and alternate path design methods have on column sizing, column reliability, robustness and building collapse probabilities.

Stochastic approaches to blast-resistant design and damage assessment

Stochastic (probabilistic) methods have a long history for developing disaster risk reduction measures for natural hazards, safety and load rating assessment of bridges, asset management of pipelines, tunnel safety from vehicle fires, safety cases for offshore platforms and chemical process plants, reliability of electricity infrastructure, etc. (Stewart & Melchers, 1997). They also underpin the development of safety factors and design loads for civil engineering design codes and standards; for example, design wind speeds may be set with an annual exceedance probability of 1 in 1000 or 0.001.

Twisdale et al. (1994) were among the first to estimate airblast variability, in this case, based on a statistical analysis of explosive field trials consisting of GP bombs. The statistical analysis were for GP bombs with known range and explosive mass, so measured blast load variability was categorised as model error (or model uncertainty). Twisdale et al. (1994) found a coefficient of variation (COV) of 0.30 for peak pressure and 0.25 for impulse, and suggested that the high variability was due to casing effects. However, these statistics are only applicable to this weapon type, and ignore the effects of range, explosive mass, temperature, and atmospheric pressure variability. Bogosian et al. (2002) reported a COV of 0.23 for peak reflected pressure results of 190 blast tests involving TNT, C-4 and ANFO explosives for scaled distances of 1.2–15.9 m/kg^1/3 (see also Bogosian et al., 2014). Campidelli et al. (2013) and Rigby and Sielicki (2014) also conducted a statistical analysis of blast test data.

Professor Hong Hao has been an early advocate of probabilistic methods for blast-resistant design and assessment. Low and Hao (2002) found that variability of peak reflected pressure was COV = 0.32 at various scaled distances in their review of available data. The other source of variability is on the resistance side, and Hong Hao and his colleagues have also published extensively in the variability of the resistance, and how blast load and resistance variability affects the reliability of RC slabs and columns (e.g. Low & Hao, 2001; Hao et al., 2010, 2016).

The work of Hong Hao and his colleagues provided motivation for Stewart (2004) being his first publication on this topic. Added to this was airblast reliability and development of RBDFs (e.g. Stewart et al., 2006, 2020; Netherton and Stewart, 2010; Stewart, 2018; 2021a; 2021b), airblast variability in complex urban spaces (Alterman et al., 2019; Marks et al., 2021), damage risks and structural reliability of RC and Ultra High Performance Concrete columns and walls, as well as glazing (e.g. Netherton and Stewart, 2009; Shi and Stewart, 2015a, 2015b; Stewart 2019, 2021b; Stewart and Li, 2021), fragmentation hazards (e.g. Sielicki et al., 2021; Qin and Stewart, 2021), and risk and cost–benefit analysis of blast-resistant design (e.g. Stewart, 2008, 2010, 2011, 2017, 2021c; Stewart and Li, 2021; Thöns and Stewart, 2019, 2020).

There is also much other work related to the use of probabilistic methods to estimate the safety and reliability of structures subject to explosive blast loading, such as Yasseri (2003), Ellingwood (2006, 2007); Olmati et al. (2014, 2017); Le and Xue (2014); Eamon (2007); Razaqpur et al. (2012); Kelliher and Sutton-Swaby (2012); Song (2020); Beck et al. (2020) and Qi et al. (2022).

Airblast variability

There is clear recognition that explosive blast loading is uncertain and variable. An important random variable is model error (or model uncertainty). Model error is defined as actual (test) blast load divided by predicted load. The predicted load is based on the Kingery–Bulmash model that has been incorporated into widely used and well respected blast load design manuals such as ConWep (1991) and UFC 3-340-02 (2014). The ability of the Kingery and Bulmash model to predict blast loads from bare (uncased) spherical explosives in a free-field environment is very good. Explosive field trials show that the mean model error is close to unity for most circumstances, and the COV of airblast is approximately 0.05–0.15 (e.g. Twisdale et al., 1994; Stewart et al., 2020). For example, Stewart et al. (2020) measured the pressure and impulse from 90 free-field explosions of mass of explosive of up to 1.8 kg leading to 593 pressure and impulse observations. When combined with 472 observations from Hoffman and Mills (1956) this leads to model errors as shown in Table 1. These free-field data represent the ‘simplest’ scenario to model and is thus an indication of the minimum variability for airblast parameters. Variability is expected to increase for complex blast environments (such as urban streetscapes), cased munitions, other charge shapes, etc. Note that once instrument error is removed from the analysis, inherent variability is less than 0.01. Hence, this source of variability is included in model error statistics.

Table 1.

Airblast model errors for bare charges (Stewart et al., 2020).

	Mean	COV
Pressure (ME_P)
0.053 m/kg^1/3 ≤ Z < 3.0 m/kg^1/3	1.06–0.005Z	0.1175–0.0125Z
3.0 m/kg^1/3 ≤ Z < 6.0 m/kg^1/3	1.06–0.005Z	0.08
Impulse (ME_I)
0.053 m/kg^1/3 ≤ Z < 3.0 m/kg^1/3	0.99	0.222–0.054Z
3.0 m/kg^1/3 ≤ Z < 6.0 m/kg^1/3	0.99	0.06

Reliability-based design load factors

The magnitude of airblast is often normalised to be a function of scaled distance (Z=R/W^1/3) where R is stand-off distance (m) and W is explosive mass (kg). The mass of explosive, range, type of explosive, angle of incidence (AOI), and air temperature and pressure will influence blast loading. However, airblast variability and RBDFs are independent of AOI, and relatively insensitive to the mean and COV of air temperature and pressure (Stewart 2018). Hence, blast load variability and RBDFs are dependent on:

• nominal scaled distance Z = R_nom/W_nom^1/3,

• variability of range (V_R) and explosive mass (V_W), and

• statistics of airblast model error for pressure (ME_P) and impulse (ME_I).

Note that RBDFs will increase as the variability of mass and range increase. Table 2 shows the RBDFs for a typical scenario as a function of scaled distance. It is observed that, as expected, the RBDF is approximately unity for a 50% reliability level. As the reliability level increases the RBDFs increase, sometimes substantially to more than twice the nominal blast loads. For more details, see Stewart (2018, 2021a).

Table 2.

RBDFs for peak reflected pressure for V_R=0.05 and V_W=0.25.

Reliability level	Z = 1 m/kg^1/3	Z = 2 m/kg^1/3	Z = 5 m/kg^1/3	Z = 10 m/kg^1/3	Z = 20 m/kg^1/3
0.05	0.66	0.63	0.73	0.79	0.81
0.10	0.73	0.71	0.79	0.84	0.85
0.50	1.03	1.04	1.04	1.04	1.03
0.90	1.46	1.54	1.37	1.28	1.25
0.95	1.61	1.72	1.48	1.36	1.32
0.99	1.93	2.11	1.71	1.52	1.47

It is important to note that airblast variability is mostly dominated by variability of range and model error. Variability of explosive mass has a relatively minor effect on airblast variability and RBDFs.

Damage probabilities for RC columns

In the present paper, building damage is predicted from Pressure-Impulse (P-I) curves developed by Mutalib and Hao (2011) for far-field detonations. Figure 1 shows a typical P-I diagram for a RC column. The damage index is defined as

DI = \frac{P_{residual}}{P_{design}}

(1)

where P_design is the axial load-carrying capacity of the undamaged column, and P_residual is the axial load-carrying capacity of the column due to blast damage. The damage index DI is correlated to different damage degrees as:

• DI = (0–0.2) low damage

• DI= (0.2–0.5) medium damage

• DI = (0.5–0.8) high damage

• DI = (0.8–1) collapse

Figure 1.

P-I diagrams for a RC column.

The design axial capacity for a RC column is

P_{design} = 0.85 f_{c}^{'} (A_{c} - A_{st}) + f_{y} A_{st}

(2)

where

f_{c}^{'}

is the unconfined concrete compressive strength, A_c is concrete cross-sectional area, f_y is the reinforcing yield strength and A_st is cross-sectional area longitudinal reinforcing steel. The P-I curve can be expressed as

(P_{r} - P_{o}) (I_{r} - I_{o}) = 12 {(\frac{P_{o}}{2} + \frac{I_{o}}{2})}^{1.5}

(3)

where P_o and I_o are the reflected pressure and reflected impulse asymptotes, respectively. The P-I diagrams are the best fit to results from numerical (FEA) modelling that considered shear and flexural damage in the estimation of P_residual for a range of parameter values.

The reflected pressure and reflected impulse asymptotes for threshold of RC column collapse (DI = 0.8) are (Mutalib & Hao, 2011)

P_{o} (0.8) = {ME}_{Po} [11 f_{c}^{'} + 3.456 d - 0.268 H - 1.552 b + 14753.3 ρ (\frac{f_{y}}{550}) + 8924.068 ρ_{s} (\frac{f_{y}}{550}) + 851.90]

(4)

I_{o} (0.8) = {ME}_{Io} [59 f_{c}^{'} + 13.16 d - 0.43 H - 0.26 b + 1091.78 ρ (\frac{f_{y}}{550}) + 489.97 ρ_{s} (\frac{f_{y}}{550}) - 3302.33]

where column width (b), column depth (h) and column height (H) are in mm, f^′_c and f_y are in MPa, ρ is the longitudinal reinforcement ratio, ρ_s is the transverse reinforcement ratio, ME_Po and ME_Io are model errors (Stewart 2019), and P_o is in kPa, and I_o is in kPa.msec. If these parameters are treated as random variables then P_o and I_o are dependent variables. Refer to Mutalib and Hao (2011) for equations of similar form for DI = 0.2 and DI = 0.5.

The probability of damage conditional on a hazard H (e.g. explosive blast loading) is (Hao et al., 2016)

Pr (damage = DI|H) = P r [G (X) \leq 0] = P r (D - D^{*} \leq 0)

(5)

where G(X) is the limit state function where damage occurs if G(X) ≤ 0, and

D = \sqrt{P_{l}^{2} + I_{l}^{2}} and D^{*} = \sqrt{P_{r}^{2} + I_{r}^{2}}

(6)

where P_r and I_r are random variables for peak reflected pressure and reflected impulse, respectively, and

P_{l} = \frac{P_{r}}{I_{r}} I_{l} I_{l} = \frac{(I_{o} + \frac{I_{r}}{P_{r}}) P_{o} + \sqrt{{(I_{o} - \frac{I_{r}}{P_{r}} P_{o})}^{2} + 48 \frac{I_{r}}{P_{r}} {(\frac{I_{o}}{2} + \frac{P_{o}}{2})}^{1.5}}}{2}

(7)

Parameter D* is the distance to the loading point (P_r,I_r) and D is the distance to the intersection point for each P-I curve.

Case study – terrorist VBIED

VBIED terrorist scenario

The terrorist blast scenario involves a vehicle-borne improvised explosive device (VBIED) containing 500 kg of home-made grade ANFO (ammonium nitrate fuel oil). If commercial explosives are unavailable to terrorists, they will likely attempt to make home-made explosives, which will most likely exhibit considerably higher variabilities than military or commercial explosives. Terrorists in Western countries are mostly lone-wolves or self-starters (Mueller and Stewart 2016). Hence, they often lack knowledge and training, lack access to bomb making equipment, and lack the opportunity to test their product. Thus, it may be hypothesised that a terrorist produced home-made ANFO explosive will have low net equivalent quantity (NEQ) of 0.6 with lower and upper bounds of approximately 0.35 and 0.9. Mass variability might be quite high as well. If it is assumed that there is ±20% accuracy in measuring, mixing and storing the quantity of the home-made explosive. If these limits represent 95% percentile bounds of a lognormal distribution the variability of home-made ANFO may be considerable with V_W = 0.25. If the VBIED is located within a car spacing of ±2.5 m tolerance then COV of R is approximately V_R = 0.12 for a range of R = 10 m. The quantification of the variability of R and W is heuristic and will highlight the type of information that may be required for probabilistic analysis, for more details see Stewart (2019, 2019).

It is assumed that R, W and Z are lognormally distributed. Note that blast load variability (and hence, load factors) is expected to increase if a more realistic assessment of charge shape and orientation, casing effects, etc. are considered.

Structural configuration

The RC building is assumed to be 10 stories high. The RC columns of interest are the exterior columns – that is, those closest to the external blast loading (see Figure 2). The results to follow assume that the explosive detonates on or very near to the ground. It is thus considered a hemispherical charge detonating against a reflecting surface. The blast load is from a single uninterrupted emanation of the shock-wave, and that reflections from other structures or surfaces are not considered. All detonations are assumed to occur within air that is at 15^o C and 1013.25 hPa. These assumptions provide the basis for the design of many protective structures. It is also assumed that only the outer face of the RC is exposed to the blast, as the facade will shield blast loading in the y-direction. Blast loading is calculated from the Kingery–Bulmash model that also accounts for AOI.

Figure 2.

Building layout showing 5 columns and VBIED.

Limit states design is predicated on the concept that structural reliabilities are consistent for different structural elements and loadings, and that the reliability should exceed a target value (e.g. Stewart and Melchers 1997). Hence, a conventional RC column not designed to be blast-resistant is sized to ensure that its annual reliability does not exceed the target reliability index defined by ISO 2394 (2015). For the present design situation the failure class is large. The Joint Committee on Structural Safety Probabilistic Model Code (JCSS 2001) recommends that the relative cost of safety is medium. Hence, the annual target reliability for economic optimisation is β_T = 4.4 or p_f = 5.4 × 10⁻⁶. For a 50-year lifetime, this is equivalent to p_f =2.7 × 10⁻⁴.

In the present study it is assumed that the design live load Q = 3 kPa, design dead load is G = 3.75 kPa leading to Q/G = 0.8, and tributary area is 50 m². Failure occurs if load exceeds resistance, hence, the lifetime (50-year) probability of column collapse is

Pr (failure) = Pr {ME [0.85 f_{c}^{'} (A_{c} - A_{st}) + f_{y} A_{st}] - (DL + L_{50}) \leq 0}

(8)

where ME is the model error for axial capacity, DL is the actual dead load effect, and L₅₀ is the lifetime (50-year) maximum live load effect. The statistical parameters for the RC column and applied dead and live loads are given in Table 3.

Table 3.

Statistical parameters for structural reliability analysis.

	Mean	COV	Distribution	Reference
Column width and breadth	1.005	0.04	Normal	Hao et al. (2016)
Column height	1.00	0.03	Normal	Hao et al. (2016)
Longitudinal reinforcement	1.02	0.05	Modified lognormal	Mirza and MacGregor (1979)
Transverse reinforcement	1.03	0.06	Modified lognormal	Mirza and MacGregor (1979)
Concrete compressive strength	1.06	0.15	Normal	Foster et al. (2016)
Yield stress	1.15	0.05	Normal	Wiśniewski et al. (2012)
Model error for axial capacity (ME)	1.01	0.08	Normal	Stewart et al. (2016)
Model error for P-I curves (ME_Po, ME_Io)	1.00	0.05	Normal	Stewart (2019)
Dead load – DL	1.05G	0.10	Lognormal	Pham (1985)
Arbitrary point-in-time live load – L_apt	0.27Q	0.72	Weibull	Pham (1985)
Peak (50-year) live load – L₅₀	0.74Q	0.25	Gumbel	Pham (1985)
Airblast model error (ME_P, ME_I)	Table 1		Lognormal	Stewart et al. (2020)

The RC column is sized to ensure that its reliability calculated from equation (8) meets the target reliability of p_f = 2.7 × 10⁻⁴. The size of a RC column designed for a 10 storey building is shown in Table 4.

Table 4.

RBDFs, RC column sizes, and system probabilities of collapse conditional on the explosive hazard.

	Not	Threat-dependent
	Blast-resistant	No load factors	0.95 Rel
RBDF – pressure	—	1.0	2.26
RBDF – impulse	—	1.0	1.57
Column size (mm)	425	625	850
Column height (mm)	4600	4600	4600
Longitudinal reinf. ratio	0.01	0.014	0.02
Transverse reinf. ratio	0.005	0.006	0.01
Concrete strength (MPa)	32	42	40
Yield strength (MPa)	500	500	500
Probability of collapse: R = 5 m	1.0	2.7×10⁻²	8.6×10⁻⁷
R = 10 m (design threat)	0.73	1.4×10⁻³	6×10⁻⁹
R = 15 m	7.3×10⁻²	2.4×10⁻⁶	1×10⁻¹³
R = 20 m	4.1×10⁻³	9×10⁻¹⁰	7×10⁻²³
Probability of collapse: W = 250 kg	0.14	6.1×10⁻⁶	6×10⁻¹⁵
W = 500 kg (design threat)	0.73	1.4×10⁻³	6×10⁻⁹
W = 1000 kg	0.99	2.0×10⁻²	3.0×10⁻⁷
W = 2000 kg	1.0	5.4×10⁻²	2.2×10⁻⁶

Blast damage and collapse probabilities

In reality, the VBIED location will be variable in x and y directions. In this case, column spacing and the overall size of the building are important. The office building is of RC construction with the perimeter of the building comprising of five RC columns at 6 m spacings (see Figure 2). The variability of range (V_R) is measured along the x-axis. However, there is also uncertainty of where along the building the VBIED will be placed and detonated. A reasonable assumption is it will be desired to have it centrally placed (i.e. directly in front of the middle column) with a 95% tolerance of ±5 m leading to a standard deviation of σ_Y=2.5 m assumed normally distributed. There is also a need to more accurately predict how column damage affects its load-carrying capacity and the probability of failure (collapse).

To assess the reliability of a RC column subject to damage from explosive loading, the actual resistance for column i is

R_{i} = ME \times (1 - DI) [0.85 f_{c}^{'} (A_{c} - A_{st}) + f_{y} A_{st}]

(9)

The probability of failure given occurrence of damage is

Pr (failure|damage = DI) = Pr [R_{i} - (DL + L_{apt}) \leq 0]

(10)

where L_apt is the arbitrary point-in-time live load effect. The model errors and load effects are statistically independent for each column.

Given occurrence of an explosive blast event (the hazard H), the probability of failure for each column i is thus

p_{f | H, i} = \int_{0}^{1} Pr (damage = DI|H) \times Pr (failure|damage = DI) dDI

(11)

The damage index used to estimate resistance R_i given in equation (9) is linearly interpolated between the P-I curves for DI = 0.2, 0.5 and 0.8.

The COV for reflected pressure and impulse are 0.48 and 0.29 for the middle column. These high COVs are due to the variability of range, explosive mass, AOI, and model error. The mean-to-nominal resistance (mean (R_i)/P_design) for an undamaged RC column is 1.14 with a COV of 0.15.

Mitigation of progressive collapse

Progressive collapse can be minimised by two structural design approaches applicable to new or leased United States federal government (civilian and defence) buildings:

• Threat Dependent – ‘a reduction of progressive collapse potential can be achieved either by precluding failure of load-carrying elements or by bridging over their loss. The first approach reduces the risk of progressive collapse for a defined threat by directly limiting the initial damage through hardening of structural elements’ (GSA 2016). In other words, each column is designed to be blast-resistant, for example, by ensuring compliance to US Department of Defense Structures to Resist the Effects of Accidental Explosions (UFC 3-340-02 2014).

• Alternate Path – ‘reduces the risk of progressive collapse by limiting the propagation of the initial damage, without explicit consideration for the cause of the initial event, through implementation of GSA Guidelines’ (GSA 2016). This is generally achieved by ensuring that the building is stable when one exterior supporting column is removed from the structure (UFC 4-023-03 2016; GSA 2016).

Threat-dependent approach

The US Department of Defense Structures to Resist the Effects of Accidental Explosions (UFC 3-340-02 2014) is typically used for the design of RC columns subject to explosive blast loading. The RC column is not permitted to attain large plastic deformation; hence, design is based on elastic or slightly plastic action. This leads to a design damage index of DI = 0.2 (Stewart 2019). In the following examples it is assumed that blast-resistant RC columns will be designed using the Mutalib and Hao (2011) P-I curves for a damage index of 0.2.

The RC columns are designed assuming:

1. Not blast-resistant (conventional design)

2. Blast-resistant design with no load or safety factors, and

3. Reliability-based design load factors (RBDF) for 0.95 reliability estimated from the procedure described by Stewart (2018, 2021a).

It is assumed that progressive collapse occurs if at least one column fails. This is conservative. If the building is a special moment resisting frame or designed against progressive collapse it is likely that loads from the most heavily damaged column will be redistributed to adjacent columns. This will reduce collapse probabilities. However, there is no surety about the ability of a building to redistribute loads so this consideration is ignored leading to slightly conservative collapse probabilities.

Table 4 shows the RC column sizes for a conventional design, or when designed to be blast-resistant considering no load factors, or load factors associated with the 0.95 Reliability RBDFs. This assumes that the VBIED will attempt to be centred directly in front of the middle column (e.g. σ_Y = 2.5 m). The probability of at least one column failing is modelled as a series system of five columns estimated using Monte-Carlo simulation. As expected, the collapse probabilities are very high at 73% for the conventional columns exposed to the design threat (R = 10 m, W = 500 kg - i.e. a far-field detonation). For a blast-resistant design with no load factors the collapse probability reduces to 1.4 × 10⁻³ for the design threat. This is to be expected, as a blast-resistant design should provide a reasonable margin of safety against major structural damage. Designing RC columns for a 0.95 reliability level leads to load factors of 2.26 and 1.57 for pressure and impulse, respectively. In this case, the columns are much larger leading to significantly reduced collapse probabilities. These larger columns reduce collapse risks by a further five orders of magnitude for the design threat.

Table 4 also shows that as the stand-off increases or the charge mass reduces these probabilities reduce. If the actual attack is worse than anticipated, such as a 50% reduction in stand-off, or a doubling of charge mass, the probability of building collapse can increase by at least ten-fold. As expected, collapse risks are more sensitive to reductions in stand-off than to increases in charge mass.

Table 5 shows Pr (damage = DI|H) and the corresponding probability of failure for each column given by equation (11) for the design threat. The damage and failure probabilities are highest for the central column (the one closest to the VBIED – that is, i = 3); however, damage is also very high for the adjacent columns. The probability that at least one column collapses is also shown in Table 5.

Table 5.

Damage Likelihoods and Failure Probabilities for Each Column Not Designed to be Blast-Resistant for Design Threat.

	Columns not designed to be blast-resistant
Damage state	i = 1	i = 2	i = 3	i = 4	i = 5
Low damage – Pr (DI < 0.2)	59.1%	7.1%	0.5%	7.1%	59.1%
Medium damage – Pr (0.2 ≤ DI < 0.5)	39.0%	55.2%	24.2%	55.2%	39.0%
High damage – Pr (0.5 ≤ DI < 0.8)	1.0%	13.0%	16.0%	13.0%	1.0%
Collapse – Pr (DI > 0.8)	0.9%	24.7%	59.3%	24.7%	0.9%
pf for column – pf\|H,i	0.017	0.32	0.68	0.32	0.017
System probability of failure			0.73 or 73%

Tables 4 and 5 assume that the VBIED will attempt to be centred directly in front of the middle column. However, in reality, this may be difficult to achieve due to placement of vehicle barriers, presence of parked vehicles, etc. Hence, the placement of the VBIED may be randomly placed anywhere in front of the target building. In this case, the location of the VBIED in the y-direction may be uniformly distributed from −12 m to + 12 m. The probabilities of building collapse are given in Table 6 for the design threat. It is observed that this increased uncertainty of VBIED location reduces the collapse probabilities as there is now an increased chance that the VBIED will not be directly in front of any column – that is, increased likelihood the VBIED will be placed between columns hence reducing the stand-off to the nearest column.

Table 6.

Probabilities of Collapse Conditional on the Explosive Hazard when VBIED is Uniformly Distributed Anywhere in Front of the Building.

	Centrally located VBIED	Uniformly distributed VBIED
Not blast-resistant	0.73	0.69
Threat-dependent
No load factors	1.4 × 10⁻³	1.2 × 10⁻³
0.95 Rel	6 × 10⁻⁹	1 × 10⁻⁹
Alternate load path	6.8 × 10⁻²	8.1 × 10⁻²

Alternate path approach

It is assumed that loads from the most heavily damaged column will be redistributed to the adjacent columns. In this case, the Alternate Path approach would require that adjacent columns have their structural resistance to conventional loads (not designed to be blast resistant) increased by 50% to ensure safe load transfer to adjacent columns in the event of column removal. Since the structure is designed to ensure that there is no progressive collapse following the removal of one column, collapse is now defined to occur if more than one column is removed – that is, the probability that at least two columns fail. To be sure, this is a simplification of the Alternate Path method; however, it will suffice for a preliminary comparison of collapse probabilities. The probability of at least two columns failing is estimated using Monte-Carlo simulation.

Table 7 shows that the collapse probabilities are reduced quite markedly compared to columns not designed to be blast resistant (see Table 4). The collapse risk for the design threat is relatively high at 6.8%, but is 10 times safer than if the RC columns are not designed to be blast-resistant. However, the collapse probabilities for the alternate path method are about 50 times higher than columns designed to be blast-resistant for the design threat. For a higher threat level (R = 5 m or W = 1000 kg) the blast-resistant design is about 20–25 times safer than the alternate path method.

Table 7.

Probabilities of failure conditional on the explosive hazard, for alternate path method.

	Alternate path method	Threat-dependent (no load factors)
R=5 m	0.47	2.7 × 10⁻²
R = 10 m (design threat)	6.8 × 10⁻²	1.4 × 10⁻³
R = 15 m	6.2 × 10⁻⁴	2.4 × 10⁻⁶
R = 20 m	1.2 × 10⁻⁶	9 × 10⁻¹⁰
W = 250 kg	5.5 × 10⁻⁴	6.1 × 10⁻⁶
W = 500 kg (design threat)	6.8 × 10⁻²	1.4 × 10⁻³
W = 1000 kg	0.53	2.0 × 10⁻²
W = 2000 kg	0.85	5.4 × 10⁻²

It might be that to protect against progressive collapse column strength may need to be increased by more than 50% for adjacent columns to safely redistribute loads. If a 75% increase in column capacity is assumed, the collapse probability for the design threat reduces from 6.8% to 2.0%.

Hence, it appears that a threat-dependent design is more effective than the alternate path method in reducing collapse risks. To be sure, this is a preliminary finding, and needs to be validated with more detailed structural design and reliability analyses.

If the placement of the VBIED is now uniformly distributed anywhere in front of the target building, Table 6 shows that the collapse probabilities increase slightly from the case where the desired placement is more central (e.g. Table 7). As there is now an increased chance that the VBIED will not be directly in front of the central column this reduces the probability of failure of the central column. However, the probability of failure of the adjacent columns increase due to higher likelihood that the VBIED will be located further from the central column. Since at least two column failures are needed to cause progressive collapse, the probability of collapse increases.

Conclusions

The paper described how RBDFs and structural reliability methods can be applied to assessing failure probabilities for RC columns subject to explosive blast loading. Hence, the present paper predicts damage and collapse risks to a reinforced concrete office building from a large terrorist VBIED. It is conservatively assumed that progressive collapse occurs if at least one column fails. It was found that RC columns designed to be blast resistant has a building collapse probability of about 1 in 1000. This is to be expected, as a blast-resistant design should provide a reasonable margin of safety against major structural damage. However, the building collapse probability reaches close to 100% for the design threat if the building is not designed to be blast-resistant. It was also found that a threat-dependent design appears to be more effective than the alternate path method in reducing collapse risks. However, a more detailed structural design and reliability analysis is needed to validate these preliminary findings.

Footnotes

Acknowledgements

The support of the Australian Research Council Discovery Project DP210101487 is gratefully acknowledged.

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

References

Alterman

Stewart

Netherton

(2019) Probabilistic assessment of airblast variability and fatality risk estimation for explosive blasts in confined building spaces. International Journal of Protective Structures 10(3): 306–329.

Beck

Ribeiro

Valdebenito

(2020) Risk-based cost-benefit analysis of frame structures considering progressive collapse under column removal scenarios. Engineering Structures 225: 111295.

Bogosian

Ferritto

Shi

(2002) Measuring uncertainty and conservatism in simplified blast models. 30th Explosives Safety Seminar. Atlanta, GA.

Bogosian

Dailey

, et al. (2014) Statistical Variation in Reflected Airblast Parameters from Bare Charges, 23rd International Symposium on Military Aspects of Blast and Shock. Oxford, UK: MABS.

Campidelli

Razaqpur

Foo

(2013) Reliability-based load factors for blast design. Canadian Journal of Civil Engineering 40(5): 461–474.

ConWep (1991) Conventional Weapons Effects Program 1991 Prepared by D.W. Hyde. Vicksburg: US Waterways Experimental Station.

Driels

(2020) Weaponeering: Conventional Weapon System Effectiveness. 3rd edition. Reston, VA: American Institute of Aeronautics and Astronautics Education Series.

Eamon

(2007) Reliability of concrete masonry unit walls subjected to explosive loads. Journal of Structural Engineering 133(7): 935–944.

Ellingwood

(2006) Mitigating risk from abnormal loads and progressive collapse. Journal of Performance of Constructed Facilities 20(4): 315–323.

10.

Ellingwood

(2007) Strategies for mitigating risk to buildings from abnormal load events. International Journal of Risk Assessment and Management 7(6–7): 828–845.

11.

Foster

Stewart

Loo

, et al. (2016) Calibration of Australian standard AS3600 concrete structures: part I statistical analysis of material properties and model error. Australian Journal of Structural Engineering 17(4): 242–253.

12.

Hao

Shi

(2016) Reliability analysis of RC columns and frame with FRP strengthening subjected to explosive loads. Journal of Performance of Constructed Facilities 30(2): 04015017.

13.

Hoffman

Mills

(1956) Air Blast Measurements about Explosive Charges at Side-On and Normal Incidence, BRL Report No 988. MD, US: US Army Armament Research and Development Centre.

14.

GSA (2016) General Services Administration Alternate Path Analysis & Design Guidelines for Progressive Collapse Resistance. Washington, DC: General Services Administration. January 28.

15.

Hao

Stewart

, et al. (2010) RC column failure probabilities to blast loads. International Journal of Protective Structures 1: 571–591.

16.

Hao

Shi

(2016) Reliability analysis of RC columns and frame with FRP strengthening subjected to explosive loads. Journal of Performance of Constructed Facilities 30: 04015017.

17.

ISO (2015) General Principles on Reliability for Structures. Geneva: International Organization for Standardization, p. 2394.

18.

JCSS (2001) Probabilistic Model Code: Part I –Basis of Design. [Online] Joint Committee On Structural Safety. Available From: http://www.Jcss.Byg.Dtu.Dk/Publications/Probabilistic_Model_Code.Aspx (Accessed 29 April 2018).

19.

Kelliher

Sutton-Swaby

(2012) Stochastic representation of blast load damage in a reinforced concrete building. Structural Safety 34: 407–417.

20.

Xue

(2014) Probabilistic analysis of reinforced concrete frame structures against progressive collapse. Engineering Structures 76: 313–323.

21.

Low

Hao

(2001) Reliability analysis of reinforced concrete slabs under explosive loading. Structural Safety 23: 157–178.

22.

Low

Hao

(2002) Reliability analysis of direct shear and flexural failure modes of RC slabs under explosive loading. Engineering Structures 24(2): 189–198.

23.

Marks

Stewart

Netherton

, et al. (2021) Airblast variability and fatality risks from a VBIED in a complex urban environment. Reliability Engineering and System Safety 209: 107459.

24.

Mirza

MacGregor

(1979) Variations in dimensions of reinforced concrete members. Journal of the Structural Division, ASCE 105(ST4): 751–766.

25.

Mueller

Stewart

(2016) Chasing Ghosts: The Policing of Terrorism. Oxford and New York: Oxford University Press.

26.

Mutalib

Hao

(2011) Development of P-I diagrams for FRP strengthened RC columns. International Journal of Impact Engineering 38: 290–304.

27.

Netherton

Stewart

(2009) The effects of explosive blast load variability on safety hazard and damage risks for monolithic window glazing. International Journal of Impact Engineering 36(12): 1346–1354.

28.

Netherton

Stewart

(2010) Blast load variability and accuracy of blast load prediction models. International Journal of Protective Structures 1(4): 543–570.

29.

Olmati

Petrini

Gkoumas

(2014) Fragility analysis for the performance-based design of cladding wall panels subjected to blast load. Engineering Structures 78: 112–120.

30.

Olmati

Vamvatsikos

Stewart

(2017) Safety factor for structural elements subjected to impulsive blast loads. International Journal of Impact Engineering 106(8): 249–258.

31.

Pham

(1985) Load combinations and probabilistic load models for limit states codes. Civil Engineering Transactions, IEAust CE27(1): 62–67.

32.

Huang

Zhi

, et al. (2022) External blast load factors for dome structures based on reliability. Defence Technology 18(2): 170–182.

33.

Qin

Stewart

(2021) Casualty risks induced by primary fragmentation hazards from high-explosive munitions. Reliability Engineering and System Safety 215: 107874.

34.

Razaqpur

Campidelli

Foo

(2012). In: Hao

(eds), Experimental versus Analytical Response of Structures to Blast Loads, Recent Research Advances on Protective Structures. The Netherlands: CRC Press, pp. 163–194.

35.

Rigby

Sielicki

(2014) An investigation of TNT equivalence of hemispherical PE4 charges. Engineering Transactions 62(4): 423–435.

36.

Shi

Stewart

(2015a) Spatial reliability analysis of explosive blast load damage to reinforced concrete columns. Structural Safety 53(March): 13–25.

37.

Shi

Stewart

(2015b) Damage and risk assessment for reinforced concrete wall panels subjected to explosive blast loading. International Journal of Impact Engineering 85: 5–19.

38.

Sielicki

Stewart

Gajewski

, et al. (2021) Field test and probabilistic analysis of irregular steel debris casualty risks from a person-borne improvised explosive device. Defence Technology 17(6): 1862–1863.

39.

Song

(2020) Parameterized fragility analysis of steel frame structure subjected to blast loads using bayesian logistic regression method. Structural Safety 87: 102000.

40.

Stewart

Melchers

(1997) Probabilistic Risk Assessment of Engineering Systems. London: Chapman & Hall.

41.

Stewart

(2004) Risk Assessment as a Decision-Making Tool to Mitigate Blast Damage to Built Infrastructure. IEAust: Australian Journal of Multi-disciplinary Engineering: Engineering a Secure Australia, pp. 1–12.

42.

Stewart

Netherton

Rosowsky

(2006) Terrorism risks and blast damage to built infrastructure. Natural Hazards Review 7(3): 114–122.

43.

Stewart

(2008) Cost effectiveness of risk mitigation strategies for protection of buildings against terrorist attack. Journal of Performance of Constructed Facilities 22(2): 115–120.

44.

Stewart

(2010) Risk-informed decision support for assessing the costs and benefits of counter-terrorism protective measures for infrastructure. International Journal of Critical Infrastructure Protection 3(1): 29–40.

45.

Stewart

(2011) Life-safety risks and optimisation of protective measures against terrorist threats to infrastructure. Structure and Infrastructure Engineering 7(6): 431–440.

46.

Stewart

Foster

Ahammed

, et al. (2016) Calibration of Australian standard AS3600 concrete structures part II: reliability indices and changes to capacity reduction factors. Australian Journal of Structural Engineering 17(4): 254–266.

47.

Stewart

(2017) Risk of progressive collapse of buildings from terrorist attacks: are the benefits of protection worth the cost? Journal of Performance of Constructed Facilities 31(2): 04016093.

48.

Stewart

(2018) Reliability-based load factor design model for explosive blast loading. Structural Safety 71: 13–23.

49.

Stewart

(2019) Reliability-based load factors for airblast and structural reliability of reinforced concrete columns for protective structures. Structure and Infrastructure Engineering 15(5): 634–646.

50.

Stewart

Netherton

Baldacchino

(2020) Observed airblast variability and model error from repeatable explosive field trials. International Journal of Protective Structures 11(2): 235–257.

51.

Stewart

(2021a) Simplified calculation of airblast variability and reliability-based design load factors for spherical air burst and hemispherical surface burst explosions. International Journal of Protective Structures. (in press).

52.

Stewart

(2021b) Simplified reliability-based load design factors for explosive blast loading, weapons effects, and its application to collateral damage estimation. Journal of Defense Modeling and Simulation. (in press).

53.

Stewart

(2021c) Terrorism risks and economic assessment of infrastructure protection against progressive collapse. Journal of Structural Engineering 147(10): 04021165.

54.

Stewart

(2021) Risk-based assessment of blast-resistant design of ultra-high performance concrete columns. Structural Safety 88: 102030.

55.

Thöns

Stewart

(2019) On decision optimality of terrorism risk mitigation measures for iconic bridges. Reliability Engineering and System Safety 188: 574–583.

56.

Thöns

Stewart

(2020) On the cost-efficiency, significance and effectiveness of terrorism risk reduction strategies for buildings. Structural Safety 85: 101957.

57.

Twisdale

Sues

Lavelle

(1994) Reliability-based design methods for protective structures. Structural Safety 15(1–2): 17–33.

58.

UFC 3-340-01 (2002) Design and Analysis of Hardened Structures to Conventional Weapons Effects, Unified Facilities Criteria. Washington D.C: Department of Defense.

59.

UFC 3-340-02 (2014) Structures to Resist the Effects of Accidental Explosions, Unified Facilities Criteria, UFC 3-340-02. Washington D.C: Department of Defense.

60.

UFC 4-023-03 (2016) Design Of Buildings To Resist Progressive Collapse . Washington, DC: United Facilities Criteria, Dept. of Defense (1 November 2016).

61.

Wiśniewski

Cruz

PJS

Henriques

AAR

, et al. (2012) Probabilistic models for mechanical properties of concrete, reinforcing steel and pre-stressing steel. Structure and Infrastructure Engineering 8(2): 111–123.

62.

Yasseri

(2003) Probabilistic cost-benefit analysis for blast resistant design. FABIG Newsletter, Fire and Blast Information Group 34: 9–13.