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Article

SimEx: A tool for the rapid evaluation of the effects of

explosions

Juan Sánchez-Monreal †, Alberto Cuadra , César Huete and Marcos Vera*

Citation: Sánchez-Monreal, J.;

Cuadra, A.; Huete, C; Vera, M.

SimEx: A tool for the rapid

evaluation of the effects of

explosions. Appl. Sci. 2022,1, 0.

https://doi.org/

Received:

Accepted:

Published:

Publisher’s Note: MDPI stays neu-

tral with regard to jurisdictional

claims in published maps and insti-

tutional afﬁliations.

Copyright: © 2022 by the authors.

Submitted to Appl. Sci. for possible

open access publication under the

terms and conditions of the Cre-

ative Commons Attribution (CC

BY) license (https://creativecom-

mons.org/licenses/by/ 4.0/).

Departamento de Ingeniería Térmica y de Fluidos, Escuela Politécnica Superior,

Universidad Carlos III de Madrid, 28911 Leganés, Spain

*Correspondence: marcos.vera@uc3m.es; Tel.: +34 91 624 99 87

† Current address: Institute of Engineering Thermodynamics, German Aerospace Center (DLR),

Pfaffenwaldring 38–40, 70569 Stuttgart, Germany

Abstract:

The dynamic response of structural elements subjected to blast loading is a problem

1

of growing interest in the ﬁeld of defense and security. In this work, a novel computational tool

2

for the rapid evaluation of the effects of explosions, hereafter referred to as SimEx, is presented

3

and discussed. The classical correlations for the reference chemical (1 kg of TNT) and nuclear

4

(10

6

kg of TNT) explosions, both spherical and hemispherical, are used together with the blast

5

wave scaling laws and the International Standard Atmosphere (ISA) to compute the dynamic

6

response of Single-Degree-of-Freedom (SDOF) systems subject to blast loading. The underlying

7

simpliﬁcations in the analysis of the structural response follow the directives established by UFC

8

3-340-02 and the Protective Design Center Technical Reports of the US Army Corps of Engineers.

9

This offers useful estimates with a low computational cost that enable in particular the computation

10

of damage diagrams in the Charge Weight-Standoff distance (CW-S) space for the rapid screening

11

of component (or building) damage levels. SimEx is a computer application based on Matlab and

12

developed by the Fluid Mechanics Research Group at University Carlos III of Madrid (UC3M).

13

It has been successfully used for both teaching and research purposes in the Degree in Security

14

Engineering, taught to the future Guardia Civil ofﬁcers at the Spanish University Center of the

15

Civil Guard (CUGC). This dual use has allowed the development of the application well beyond

16

its initial objective, testing on one hand the implemented capacities by undergraduate cadets with

17

end-user proﬁle, and implementing new functionalities and utilities by Masters and PhD students.

18

With this experience, the application has been continuously growing since its initial inception in

19

2014 both at a visual and a functional level, including new effects in the propagation of the blast

20

waves, such as clearing and conﬁnement, and incorporating new calculation assistants, such as

21

those for the thermochemical analysis of explosive mixtures; crater formation; fragment mass

22

distributions, ejection speeds and ballistic trajectories; and the statistical evaluation of damage to

23

people due to overpressure, body projection, and fragment injuries.

24

Keywords:

Effects of Explosions; Blast loading; SDOF systems; Thermochemistry of Explosives;

25

Fragments; Crater formation; Damage to people

26

1. Introduction

27

Unlike the slow energy release exhibited by deﬂagrations, the instantaneous energy

28

deposition associated with the detonation of a high explosive produces an extremely

29

rapid increase in temperature and pressure due to the sudden release of heat, light and

30

gases [

35

]. The gases produced by the explosion, initially at extremely high temperatures

31

and pressures, expand abruptly against the surrounding atmosphere, vigorously pushing

32

away any other object that may be found in their path. This gives rise to the two most

33

notable effects of explosions: the aerial, or blast, wave [

11

], and the projection of shell

34

fragments or other items (i.e., secondary fragments) located in the surroundings of the

35

charge [

23

]. If the explosive device is located at ground level, a fraction of its energy is

36

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Version September 5, 2022 submitted to Appl. Sci. 2 of 23

effectively coupled to the ground, generating seismic waves and a well distinguished

37

surface crater that results from the ejection of the shattered ground materials in direct

38

contact with the charge [

4

]. Quantifying these phenomena and assessing their effect

39

on the environment, including structural elements, vehicles, objects, or people located

40

around the blast site, is a highly complex task that requires a thorough knowledge of the

41

physical-chemistry of explosions [

1

,

14

,

35

,

39

] and their dynamic interactions with nearby

42

structures [45] or the human body [58].

43

As a result of the growing terrorist threat experienced in the last decades [

57

],

44

estimating the effects of explosions has become a critical issue in the design, protection

45

and restoration of buildings and infrastructures, both civil and military [

19

]. However,

46

this task is far from trivial, in that it involves transient compressible ﬂows, non-linear

47

structural response, and highly dynamic ﬂuid-structure interactions. These phenomena

48

can be described with some accuracy using multiphysics computational tools, also

49

known as hydrocodes [

37

], such as Ansys Autodyn, LS-Dyna, or Abaqus, based on the

50

explicit ﬁnite element method [

5

]. In the simulations, all the critical components are

51

modeled, including the detonation of the explosive charge, the resulting blast wave,

52

the induced dynamic loads, and the non-linear structural response. However, the

53

enormous computational effort required to complete detailed computational analyses,

54

which includes not only the calculation time itself, but also complex pre- and post-

55

processing stages, remains a critical issue. For instance, simulating the effect of an

56

explosive charge on a full-scale bridge may require more than 10 million ﬁnite elements

57

[

26

]. For this reason, most engineering analyses still make use of simpliﬁed models

58

for determining the explosive loads and estimating the resulting dynamic structural

59

response in a timely manner. This enables the fast computation of damage diagrams in

60

the Charge Weight-Standoff distance (CW-S) space, of utility to determine the level of

61

protection provided by an input structural component loaded by blast from an input

62

equivalent TNT charge weight and standoff [49].

63

In this regard, the American Uniﬁed Facilities Criteria UFC 3-340-02 [

59

], which

64

supersedes former ARMY TM 5-1300, establishes the requirements imposed by the US

65

Department of Defense in the tasks of planning, design, construction, maintenance,

66

restoration and modernization of those facilities that must be protected against explosive

67

threats. In the absence of similar regulations in other countries, UFC 3-340-02 [

59

] is

68

widely used by engineers and contractors outside the US, as it provides a valuable

69

guide for calculating the effects of blast-induced dynamic loads, including step-by-step

70

procedures for the analysis and design of buildings to resist the effects of explosions.

71

To facilitate the application of the procedures set forth in the UFC 3-340-02 [

59

],

72

as well as other analyses established in classic references of explosives engineering

73

[

1

,

8

,

14

,

23

,

33

,

43

,

58

], fast evaluation software tools have been developed that incorporate

74

the vast amount of data available as tables or graphs in the literature [

39

]. For instance,

75

the United States Army Corps of Engineers (USACE) has developed and provides

76

support for a series of software packages related to the design of explosion-resistant

77

buildings [

47

]. Those tools were developed with public funding, and therefore there

78

are regulations that restrict distributing those products outside of the United States. In

79

addition, given the critical nature of this knowledge, access to these packages is severely

80

limited to US government agencies and their contractors, with use only authorized to

81

US citizens.

82

The inability to access these software packages motivated the authors to develop

83

their own computational toolbox for the rapid evaluation of the effects of explosions.

84

The result was the SimEx platform to be presented in this work. Conceived initially

85

for educational purposes, the main goal was to develop a virtual software platform

86

with an easy and intuitive Graphical User Interface (GUI) to be used in the computer

87

lab sessions of the Explosion Dynamics course of the Degree in Security Engineering,

88

taught at the University Center of the Civil Guard (CUGC) in Aranjuez, Spain. The Civil

89

Guard is the oldest and biggest law enforcement agency in Spain. Of military nature, its

90

Version September 5, 2022 submitted to Appl. Sci. 3 of 23

competencies include delinquency prevention, crime investigation, counter-terrorism

91

operations, coastline and border security, dignitary and infrastructure protection, as

92

well as trafﬁc, environment or weapons and explosives control using the latest research

93

techniques. The paradigm of the Civil Guard’s capacity is its outstanding role in the

94

defeat of the terrorist group ETA, the longest-running terrorist group in Europe and

95

the best technically prepared. In this context, the main target of the Degree in Security

96

Engineering is the training of Guardia Civil cadets (i.e., the Guardia Civil’s future ofﬁcer

97

leadership) in the development, integration and management of last generation civil

98

security systems.

99

The purpose of SimEx was initially limited to the blast damage assessment on

100

simple structural elements [

52

], such as beams, columns, pillars, or walls, following the

101

Single-Degree-of-Freedom (SDOF) system analysis established by UFC 3-340-02 [

59

]. The

102

tool has been successfully used since its initial inception in 2014 in both the computer lab

103

sessions of the Explosion Dynamics course, and as a research tool for the development

104

of a number of Bachelor and Master’s thesis on explosion dynamics and blast effects.

105

This double use as end-users and software developers by the Civil Guard cadets and

106

students from other UC3M degrees has enabled the development of the application well

107

beyond the initially planned objectives [

53

]. As a result, the current version of SimEx

108

incorporates advanced topics in blast wave propagation, such as the prediction of cleared

109

blast pressure loads due to the generation of rarefaction waves, as well as conﬁned blast

110

loading in vented structures [

54

]. It also includes several other calculation assistants for

111

the thermochemical analysis of explosive mixtures [

1

,

15

,

39

]; crater formation [

3

,

4

,

14

];

112

fragment mass distributions, ejection speeds and ballistic trajectories [

23

,

40

–

42

]; and

113

the statistical evaluation of damage to people due to overpressure, body projection and

114

fragment injuries [24,51,58].

115

2. SimEx capabilities

116

This section presents the current capabilities of SimEx, starting with the main

117

interface used for computing the dynamic response of SDOF systems subjected to blast

118

loading, and following with the description of the remaining calculation assistants.

119

2.1. Single-Degree-of-Freedom system analysis

120

In many situations of practical interest, the response of structural elements to blast

121

loading can be reduced, in ﬁrst approximation, to that of an equivalent spring-mass

122

SDOF system. As sketched in Figure 1, this system is made up by a concentrated

123

mass subject to external forcing and a non-linear weightless spring representing the

124

resistance of the structure against deformation [

45

]. The mass of the equivalent system is

125

based on the component mass, the dynamic load is imposed by the blast wave, and the

126

spring stiffness and yield strain on the component structural stiffness and load capacity.

127

Generally, a small viscous damping is also included to account for all energy dissipated

128

during the dynamic response that is not accounted by the spring-mass system, such

129

as slip and friction at joints and supports, material cracking, or concrete reinforcement

130

bond slip [48].

131

If the system properties are properly deﬁned, the deﬂection of the spring-mass

132

system,

x(t)

, will reproduce the deﬂection of a characteristic point on the actual system

133

(e.g., the maximum deﬂection). The system properties required for the determination of

134

the maximum deﬂection are the effective mass of the equivalent SDOF system,

Me

, the

135

effective viscous damping,

Ce

, the effective resistance function,

Re(x)

, and the effective

136

load history acting on the system,

Fe(t)

. To systematize the calculations, the effective

137

properties are obtained using dimensionless transformation factors that multiply the

138

actual properties of the blast-loaded component, respectively,

M

,

C

,

R(x)

, and

F(t)

[

10

].

139

These factors are obtained from energy conservation arguments in order to guarantee

140

that the equivalent SDOF system has the same work, kinetic, and strain energies as the

141

Version September 5, 2022 submitted to Appl. Sci. 4 of 23

real component for the same deﬂection when it responds in a given, assumed mode

142

shape, typically the fundamental vibrational mode of the system [48].

143

In the analysis of blast-loaded SDOF systems it is therefore of prime importance to

identify the fundamental vibrational mode of the structural element. This procedure is

not trivial, since obtaining the fundamental mode can entail certain difﬁculties, in which

case its shape must be approximated in some way [

10

]. To determine the equivalent

properties of the SDOF system it is also necessary to determine the type of structure

(beam, pillar, frame, etc.) and how the load is applied (typically, a uniform load is

assumed). The elastic behavior of the material is often modeled as perfect elasto-plastic,

probably the simplest of all nonlinear material models. This assumes that the initial

response follows a linear elastic behaviour described by an apparent elastic constant K,

but once the yield strain is reached,

x≥xu

, the material behaves as plastic, ﬂowing at a

constant stress with an ultimate resistance Ru=Kxu, i.e.,

R(x) = (Kx for |x|<xu

Rufor |x| ≥ xu(1)

Although more complex models could be used, they are not considered here due to the

144

heavy simpliﬁcations introduced in the formulation of the problem.

145

The mass transformation factor,

KM

, is deﬁned as the ratio between the equivalent

146

mass

Me

and the real mass

M

of the blast-loaded component; the load transformation

147

factor

KL

is deﬁned as the ratio between the equivalent load

Fe(t)

and the actual load

148

F(t)

, and usually coincides with the resistance and damping transformation factors; and

149

ﬁnally the load-mass factor

KLM

is deﬁned as the ratio between the mass factor and the

150

load factor

151

KM=Me

M;KL=Fe(t)

F(t)=Re(x)

R(x)=Ce

C;KLM =KM

KL

=Me

M·F(t)

Fe(t)(2)

152

Although all these factors are easy to obtain, even through analytical expressions in

153

some cases, most of them can be found tabulated in the UFC-3-340-02 [59].

154

The linear momentum equation for the equivalent SDOF system then takes the

155

form [10]

156

KLM M¨

x+C˙

x+R(x) = F(t)(3)

157

where, as previously discussed,

C

represents the viscous damping constant of the blast-

158

loaded component. This constant is often speciﬁed as a small percentage,

z

, of the

159

critical viscous damping,

C= (z/

100

)Ccr

, with a damping coefﬁcient

z=

2 being a

160

good value when not otherwise known (for further details see [

48

]). Note, however, that

161

damping has very little effect on the maximum displacement, which typically occurs

162

during the ﬁrst cycle of oscillation, so the actual value of

z

is not of major relevance. The

163

inhomogeneous term,

F(t)

, appearing on the right hand side of Equation

(3)

represents

164

the dynamic load associated with the blast wave, to be discussed in Section 2.1.1 below.

165

SimEx provides an easy and intuitive GUI environment for the study of the dynamic

166

response to blast loadings of a variety of structural elements that can be modeled as

167

SDOF systems. Figure 2shows the main SimEx interface, divided into three calculation

168

assistants for the three basic elements that make up the SDOF system: a module for

169

calculating the properties of the blast wave (forcing term,

F(t)

), a module for calculating

170

the equivalent mechanical properties (resistance term,

R(t)

), and a module for the

171

numerical integration of the problem, which includes the post-processing of the results

172

and their graphic representation in the form of displacements, forces and deformation

173

diagrams (see the bottom plots of Figure 2) and of CW-S damage charts, to be discussed

174

in Section 3.3.

175

As a ﬁnal remark, it is important to note that, following standard practice, the SDOF

176

analysis carried out by SimEx uses the load deﬁned in terms of pressure,

F(t) = p0(t)

177

(Pa), so that both the mass

M

(kg

/

m

2

), the damping coefﬁcient

C

(kg

/(

m

2

s

)

) and the

178

Version September 5, 2022 submitted to Appl. Sci. 5 of 23

Figure 1.

Sketch of the equivalent SDOF system showing the different terms involved in its

mathematical description. Left: forcing term; right: resistance term; center: equivalent spring-

mass SDOF system and its associated differential equation.

ultimate resistance

Ru

(Pa) must all be introduced as distributed values per unit surface

179

(p.u.s.) in the different calculation assistants.

180

2.1.1. Forcing term

181

As previously discussed, the blast wave overpressure deﬁned in Equation (4) below

182

can be used directly in Equation (3) as forcing term,

F(t) = p0(t)

, as long as the analysis

183

is formulated per unit surface and uses distributed masses and forces. In order to

184

determine the blast parameters (arrival time, peak overpressure, positive phase duration,

185

impulse per unit area, waveform parameter, etc.), classical correlations [

8

,

11

,

12

,

33

,

35

,

186

43

,

44

] in terms of scaled distance are used together with the scaling laws for spherical

187

or hemispherical blast waves [

8

,

30

,

35

,

56

], which allow their evaluation for arbitrary

188

CW-S pairs. It is interesting to note that the standoff distance is deﬁned as the minimum

189

distance from the charge to the structural element under study (e.g., a wall). However,

190

the actual distance to a given point of that element, e.g., the centroid (or geometric

191

center), which may be considered the most representative point of the structure, may by

192

slightly different due to the incidence angle being larger than 0 at that point.

193

The local atmospheric pressure,

pa

, and temperature,

Ta

, are determined using

194

the International Civil Aviation Organization (ICAO) Standard Atmosphere (ISA) [

32

]

195

with a temperature offset (ISA

±∆T

). The user must specify the geopotential height,

196

in meters, and the non-standard offset temperature

±∆T

, although arbitrary ambient

197

temperature and pressure can also be introduced directly [

22

]. TNT is used as reference

198

explosive, although the results can be extrapolated to other compositions using either the

199

equivalence tables included in SimEx for selected explosives [

31

], or the thermochemical

200

calculation assistant, to be presented in section 2.2.1, for less conventional formulations

201

or explosive mixtures.

202

To estimate the dynamic load exerted by the blast wave, the angle of incidence of

203

the incoming shock wave must be considered, the worst-case conditions being usually

204

those of normal incidence. UFC 3-340-02 [

59

] contains scaled magnitude data for both

205

spherical and hemispherical blast waves. It also provides methods to calculate the

206

properties of the blast wave with different incidence angles, including both ordinary and

207

Mach reﬂections for oblique shocks. The time evolution of the blast wave overpressure

208

p0(t0)

at a ﬁxed distance,

d

, sufﬁciently far from the charge (at least, larger than the

209

ﬁreball scaled distance) is approximated using the modiﬁed Friedlander’s equation,

210

which captures also the negative overpressure phase [8,21,35]

211

p0(t0) = p(t0)−p1=p◦1−t0

tdexp−αt0

td(4)

212

where

p◦=p2−p1

represents the peak overpressure measured from the undisturbed

213

atmospheric pressure

p1=pa

, with

p2

denoting the peak post-shock pressure,

t0=t−ta

214

Version September 5, 2022 submitted to Appl. Sci. 6 of 23

File

Help

Blast wave

Explosive

Charge weigth (kg)

150

Distance (m)

20.74

Incident angle (deg)

15.38

Explosive

TNT

Atmosphere (ISA +/- DT)

pa (kPa)

101.325

Ta (ºC)

15

DT (ISA +/- DT)

0

Altura (m - ISA)

0

UFC 3-340-02 Hemi Friendlander

Compute

Reset

Clearing effect

S_c (m)

10

t_c (ms)

6.52

OnOff

Wave parameters

pº (kPa)

169.58

t_d (ms)

17.96

I/A (kPa ms)

688.20

alpha (-)

3.04

Confinement

V (m³)

200

A_f (m²)

50

W_f (kg/m²)

0

P_g (kPa)

0

i_g (kPa ms)

0

t_g (ms)

0

OnOff

Integration

dt_max (ms)

0.01

x_0 (mm)

0

t_f (ms)

50

v_0 (m/s)

0

t_0 (ms)

0

Integrate

Reset

Average acceleration

Results

mu (-)

1.64

x_max (mm)

3.879

theta_max (deg)

0.1482

Damage analysis. CW-S diagram

Standoff distance interval (m)

10

50

Charge weight interval (kg)

1

100

Type

mu

theta

B1

B2

B3

B4

mu

1

0

mu-theta

3

3

mu-theta

12

10

mu-theta

25

10

Number of points

8

Diagram

Damage

to people

Fragments

Fragment

trajectories

Blast wave

wizard

Crater

Thermochemistry

of explosives

Resistance

M (kg/m²)

226.8

z (% C_cr)

2

L (m)

3

T_n (ms)

9.394

C_cr (kg/(m² s))

2.003e+05

x_u (mm)

2.365

Equivalent mechanical properties

K (kPa/mm)

66.98

R_u (kPa)

158.4

K_LM (-)

0.66

Compute

Reset

Metal beams

Concrete beams

Fluid Mechanics

Figure 2.

Main interface of SimEx showing the “Blast wave”, “Resistance” and “Integration” assistants for the computation

of the structural response of perfect elasto-plastic SDOF systems under blast loading. The access buttons to the other

calculation assistants are seen under the top toolbar. The bottom plots shows the post-processing pop-up window that

displays the results of the numerical integration in terms of displacements, forces and deformation diagrams (for a detailed

discussion of these diagrams see Section 2.1.4).

is time measured from the blast arrival time,

td

is the positive phase duration, and

α

215

is the waveform parameter, closely related to the impulse per unit area of the positive

216

phase

I/A=Rtd

0p0(t0)

d

t0

(area under the positive phase of the overpressure-time curve)

217

according to

I/A=p◦td1/α−(1−e−α)/α2

. SimEx performs by default the complete

218

integration of the Friedlander waveform, but the equivalent triangular pressure pulse

219

can also be used without signiﬁcant errors [

10

]. This simpliﬁed waveform has the same

220

maximum peak overpressure,

p◦

, but a ﬁctitious positive duration computed in terms of

221

the total positive impulse and the peak over pressure, td=2(I/A)/p◦.

222

The “Blast wave” calculation assistant allows the activation of the effects of clearing

223

and conﬁned explosions, which increases the computational capabilities to more realistic

224

situations. The clearing effect takes into account the time required for reﬂected pressures

225

to clear a solid wall that has received the impact of a blast wave as a result of the

226

propagation of rarefaction waves from the edges of the wall. In the case of conﬁned

227

Version September 5, 2022 submitted to Appl. Sci. 7 of 23

explosions, SimEx implements the procedure outlined in UFC 3-340-02 [

59

] to estimate

228

the gas phase peak overpressure and duration of the equivalent triangular pressure

229

pulse in terms of the chamber’s total vent area and free volume. These effects can be

230

activated on the lower part of the “Blast wave” calculation assistant.

231

2.1.2. Resistance term

232

The “Resistance” calculation assistant provides a means to deﬁne the equivalent

233

mechanical properties (i.e., structural mass, damping coefﬁcient, and structural strength)

234

of the SDOF system under study modeled as a perfectly elasto-plastic system with elastic

235

stiffness

K

until the yield strain, as given in Equation

(1)

. The characteristic length,

236

L

, of the structural element must also be provided, as it is required to determine the

237

maximum rotation angle at its boundaries, often referred to as support rotation,

θ

. For

238

the equivalent SDOF system, the assistant computes the fundamental natural period,

239

Tn=

2

π√KLMM/K

, the critical damping,

Ccr =

2

√KLMK M

, and the deﬂection at which

240

plastic deformation initiates in the system,

xu

. Direct access to calculation assistants that

241

compute the equivalent properties (

M

,

K

,

KLM

,

Ru

) required for the calculations are also

242

provided for various types of systems. Currently, standard European wide ﬂange “metal

243

beams” [

17

] and reinforced “concrete beams” are included (see Section 3.2), although it

244

could be possible to incorporate additional assistants for other elements, such as metal

245

panels/plates, open-web steel joists, reinforced concrete slabs, reinforced/unreinforced

246

masonry, or wood panels/beams. The metal beams assistant also provides the possibility

247

of studying custom (i.e., non-normalized) proﬁles and materials in order to widen the

248

computation capabilities.

249

2.1.3. Numerical integration

250

Once the characteristics of the equivalent SDOF system have been deﬁned, the

251

resulting ordinary differential equation that models the transient non-linear response

252

of the equivalent structural system

(3)

must be integrated numerically. The integration

253

module implements the two numerical methods recommended by UFC-3-340-02 [

59

],

254

namely the “Acceleration-Impulse-Extrapolation Method” and the “Average Accelera-

255

tion Method” [

59

], which can be selected from a drop-down menu. Text boxes are also

256

included to set the initial conditions (displacement and initial speed, which are zero by

257

default) as well as the ﬁnal integration time. Since both numerical methods use constant

258

time steps, a sufﬁciently short time increment, typically of the order of a few percentage

259

of the natural period or the positive phase duration (usually, fractions of a millisecond),

260

should be used in order to ensure the numerical convergence of the integration.

261

2.1.4. Post-processing

262

After integration, three plots appear in a pop-up window and a summary table

263

is provided at the bottom left corner of the main window. The left plot shows the

264

instantaneous displacement (solid line) and the permanent displacement, or deformation

265

(dashed line). The central plot shows the temporal variation of the forcing term (i.e.,

266

the blast pressure wave, solid line) together with the resistance strength of the SDOF

267

system (dashed line). The right plot shows the displacement-resistance graph, in which

268

it is possible to determine more clearly whether permanent deformations occur or not.

269

Finally, the table of results shows the maximum displacement obtained,

xmax

, along with

270

two damage indicators: the ductility ratio,

µ=xmax/xu

, deﬁned as the ratio of the peak

271

deﬂection to the ultimate elastic deﬂection, and the maximum support rotation,

θ

, whose

272

calculation depends on the type of structure under study.

273

By integrating different combinations of charge weights and standoff distances

274

for the same structural element, damage level diagrams can be rapidly obtained in

275

the CW-S distance space. SimEx has a function for it located in the central part of the

276

integrator module. One can select the range of charge weights and standoff distances,

277

the number of intermediate values and the type of damage in terms of the quantitative

278

Version September 5, 2022 submitted to Appl. Sci. 8 of 23

indicators

µ

and

θ

[

49

]. From the two quantitative indicators, the structural damage level

279

can be classiﬁed qualitatively into: superﬁcial, moderate, heavy, hazardous failure, and

280

blowout, with response limit boundaries between these levels denoted respectively by B1

281

(superﬁcial to moderate), B2 (moderate to heavy), B3 (heavy to hazardous failure), and

282

B4 (hazardous failure to blowout). Convenient limits for the boundaries of component

283

damage levels for common structural components in terms of

µ

and

θ

are provided in

284

[

49

]. An example of a damage level diagram for the façade of a conventional building

285

subject to blast loading computed with SimEx will be presented in Section 3.3 .

286

2.2. Other calculation assistants

287

The main SimEx interface gives access to several other calculation assistants. These

288

include: a module for the calculation of the theoretical (i.e., thermochemical) properties

289

of explosives and explosive mixtures; a module for estimating the initial velocity, mass

290

distribution and ballistic trajectories of primary fragments; a crater formation calculator;

291

and a module for estimating damage to people, including both primary and tertiary

292

injuries. The fragment assistant also provides estimations of the secondary injuries due

293

to the impact of primary fragments on people. In this section we shall brieﬂy present

294

and discuss the above-mentioned assistants.

295

2.2.1. Assistant for the calculation of the thermodynamic properties of explosives

296

For the calculation of the theoretical thermodynamic properties of explosives and

297

explosive mixtures, SimEx includes an extensive database of pure CHNO propellants

298

and explosives extracted from Kinney & Graham [

35

], updated with data from Meyer

299

[

39

] and Akhavan [

1

] for more recent explosives. From the properties of pure explosives,

300

the thermochemical assistant estimates the properties of explosive mixtures formed by

301

two or more components by specifying the mass fractions and the density of the mixture.

302

First, it computes the apparent chemical formula of the explosive mixture along

303

with its molecular weight and maximum density. For the calculation of the decom-

304

position reaction in nominal products, which provides the heat of explosion and the

305

volume of gases generated, one can choose different calculation hypotheses: Kamlet-

306

Jacobs (KJ), Kistiakowsky-Wilson (KW), Modiﬁed Kistiakowsky-Wilson (modiﬁed KW),

307

Springall-Roberts (SR), or chemical equilibrium [

1

]. In the latter case, SimEx determines

308

the composition of the product mixture following the chemical equilibrium approach

309

considering a constant-volume explosion transformation that uses the ideal gas Equation

310

Entropy [kJ/(kg K)]

8.81

gamma = cp/cv [-]

1.19

Volume gases [m3/kg]

0.8924

Internal energy [kJ/kg]

-3840

-3840

Sound speed [m/s]

1093

Mean Molecular Weight [g/mol]

95.83

25.12

cp [kJ/(kg K)]

2.074

Enthalpy [kJ/kg]

-3964

-2961

Density [kg/m3]

1100

1100

Pressure [bar]

1.164e+05

Products

Reactants

Temperature [K]

298.1

1

3030

Heat release [kJ/kg]

4003

Detonation speed [m/s]

6097

Gurney constant [m/s]

2830

Explosive force [kJ/kg]

1003

Parameters

Composition

Components

Mass fraction

NG

0.0350

EGDN

0.0350

N2O3H4

0.7200

TNT

0.1400

C6H10O5

0.0500

CaCO3

0.0100

TALC

0.0100

Density [kg/m3]

1100

Oxygen Balance [%]

-1.756

Charge weight [kg]

1

C 7.1885 CA 0.0999 H 43.9106 MG 0.0791

N 20.7720 O 35.6249 SI 0.1055

Reactants

UNE 31-002-94

Reset

Compute

Equation of State

Ideal

Figure 3.

Interface of the assistant for the calculation of the theoretical thermodynamic properties

of explosives and explosive mixtures.

Version September 5, 2022 submitted to Appl. Sci. 9 of 23

Table 1.

Composition [mass %], density, and oxygen balance of different explosive mixtures tested.

Component ANFO ANFO-Al Emulsion Dinamite I Dinamite II

Aluminium — 5 — — —

Ammonium nitrate 94 91 80 — 49

Cellulose — — — — 3

2,4-Dinitrotoluene — — — — 4

Nitrocellulose 12% — — 10 — 4

Nitroglycerin — — — 45 20

Nitroglycol — — — 45 20

Fuel oil 6 4 7 — —

Sodium nitrate — — 5 — —

Water — — 8 — —

Density [kg/m3]850 850 1300 1500 1500

Oxygen balance [%] −1.7 0.08 −5.57 −2.26 0.84

Table 2.

Comparison of the calculated temperature at constant volume,

T

, detonation pressure,

pCJ

, detonation velocity,

vCJ

, heat release at constant volume,

Qv

, and explosive force,

Fe

, with the

results provided by the European Standard EN 13631-15 [

18

] and by the thermochemical code

W-DETCOM [55] for different explosive mixtures using the BKW-S EoS.

Explosive Source T[K] pCJ [GPa] vCJ [m/s] Qv[kJ/kg] Fe[kJ/kg]

ANFO

CT 2592 7.14 5353 3845 943

EN 13631-15 2586 — — 3820 945

W-DETCOM12919 6.62 5326 3849 —

ANFO-Al

CT 3026 7.38 5442 4666 1009

EN 13631-15 3060 — — 4642 1020

W-DETCOM13370 6.55 5215 4655 —

Emulsion

CT 2112 15.3 6549 3263 766

EN 13631-15 2099 — — 3236 771

W-DETCOM12438 13.9 6758 3214 —

Dinamite I CT 4173 25.03 7960 6452 1147

EN 13631-15 4130 — — 6338 1138

Dinamite II CT 3165 23.58 7729 5049 987

EN 13631-15 3151 — — 4989 984

1Calculation performed assuming Chapman-Jouguet detonation.

of State (EoS) for the products according to the norm UNE 31-002-94 [46], as illustrated

311

in Figure 3.

312

More complex computations based on the European Standard EN 13631-15 [

18

],

313

which use the semi-empirical Becker-Kistiakowsky-Wilson (BKW) EoS [

6

,

29

,

38

] or the

314

Heuzé (H9) EoS [

28

] for the products, are also supported in the last version of SimEx. As

315

sample results of these computations, Table 2shows the detonation properties obtained

316

by SimEx for different explosive mixtures (see Table 1for its composition) compared

317

with the results reported in the European Standard EN 13631-15 [

18

], and obtained with

318

the W-DETCOM code [36,55], which computes directly the Chapman-Jouguet state.

319

Version September 5, 2022 submitted to Appl. Sci. 10 of 23

The equilibrium calculations are carried out using Combustion Toolbox (CT), an in-

320

house thermochemical equilibrium package developed at UC3M [

15

,

16

]. CT determines

321

the equilibrium composition of the product mixture through the Gibbs free energy

322

minimization method by using Lagrange multipliers combined with a multidimensional

323

Newton-Raphson method. The thermodynamic properties (speciﬁc heat, enthalpy, and

324

entropy) are computed as a function of temperature derived from NASA’s 9-coefﬁcient

325

polynomial ﬁts for combustion of ideal and non-ideal gases and condensed phases.

326

From the resulting composition of the product mixture at equilibrium, the assistant

327

computes the volume of gases generated, the heat of explosion, the Gurney constant, the

328

detonation pressure, the detonation velocity, and the explosive force (or power index).

329

To estimate the detonation pressure and velocity, the approximate expressions of Kamlet

330

& Jacobs [

34

,

50

] are used, whereas the explosive force is estimated using the well-known

331

Berthelot approximation [

35

]. These data are subsequently used to calculate the TNT

332

equivalent of the explosive composition under study.

333

2.2.2. Crater

334

SimEx also has an assistant for the direct and inverse calculation of craters based on

335

the classical correlations for craters reviewed by Cooper [

14

] (see also Refs. [

3

] and [

4

]),

336

whose interface is shown in Figure 4. With this assistant, one can calculate the radius

337

of the crater generated by the detonation of a certain amount of a given explosive at a

338

certain height above the ground, considering different types of soil. It is also possible to

339

calculate the explosive charge required to produce a crater of a certain size, which may

340

be useful for the forensic analysis of explosions [

7

]. Buried craters are not yet included in

341

the assistant, but could be incorporated in future versions following the work of Westine

342

[60], as reviewed by Baker et al. [9].

343

Figure 4.

Interface of the assistant for the calculation of craters. HOB denotes the height of burst.

2.2.3. Primary fragments

344

SimEx incorporates assistants for calculating the mass distribution, ejection velocity

345

and ballistic trajectory of primary fragments. The corresponding interfaces are shown

346

in Figures 5–7. The fragment size distribution is estimated using Mott’s statistical

347

theory for fragmentation of steel cylindrical shells [

23

,

40

–

42

], as suggested by UFC-

348

3-340-03 [

59

]. As shown in Figure 5, this model determines the average number of

349

fragments and their average weight. It also provides the size of the largest fragment

350

corresponding to a given Conﬁdence Level (CL). SimEx also includes a ballistic trajectory

351

assistant for primary fragments that, in addition to the ﬂight path, provides the ﬂight

352

time, velocity and maximum distance, as illustrated in Figure 6. The initial velocity of

353

primary fragments is computed using Gurney’s analysis [

25

] for cylindrical, spherical,

354

and symmetrical/asymmetrical sandwich charges. Although this analysis assumes that

355

Version September 5, 2022 submitted to Appl. Sci. 11 of 23

all fragments have the same the initial velocity, given the different fragment sizes both

356

their initial kinetic energy and their subsequent aerodynamic deceleration are different.

357

The assistant thus includes an initial aerodynamic deceleration chart, shown in Figure 7,

358

that provides the fraction of the initial velocity achieved at a certain distance, given the

359

fragment mass and material, and the local air density, speciﬁed through the ISA

±∆T

360

model. The aerodynamic assistants assume spherical fragments with a variable drag

361

coefﬁcient for all Mach numbers [

13

], although the model could be extended to account

362

for more realistic (i.e., irregular) fragment shapes in future versions [

27

]. The results of

363

these models are also used to estimate the lethality risk by impact of primary fragment

364

in the event of a strike on a person, which is found to depend on the speed and the mass

365

of the fragment, as illustrated by Figure 5.

366

Charge-shell configuration

Charge weight (kg)

150

Explosive

TNT

Shape

Cilinder

Shell weight (kg)

20

Diameter (cm)

50

Thickness (mm)

2

Maximum distance (m)

500

150

Compute

Reset

TNT eq (kg)

Cte Gurney (m/s)

2438

Fragments statistics and secondary injuries

Confidence level (%)

99

Average weight (g)

0.59

CL weight (g)

12.44

Average fragment (m)

15.13

CL fragment (m)

378.72

16.13

17.13

400.48

422.23

Number

34088

Number

51

Velocity vs distance

Number of fragments

99 %

50 %

1 %

Lethality

Fluid Mechanics

Figure 5. Interface of the primary fragment mass distribution and lethality assistant.

Fragment configuration

Weight (g)

1

Material

Lead

Compute

Reset

Density (kg/m3)

11340.00

Shooting conditions

Height (ISA) (m)

0

Initial velocity (m/s)

2330

DT (ISA +/- DT)

0

Diameter (m)

5.52e-04

Shooting angle

min alpha (deg)

1

max alpha (deg)

90

step alpha (deg)

10

Fluid Mechanics

Ballistic fragment trajectories

Initial deceleration chart

Figure 6.

Interface of the primary fragment calculation assistant showing the ballistic fragment

trajectory, ﬂight time, velocity and maximum distance charts. Fragments are assumed spherical.

Version September 5, 2022 submitted to Appl. Sci. 12 of 23

Fragment configuration

Weight (g)

1

Material

Lead

Compute

Reset

Density (kg/m3)

11340.00

Shooting conditions

Height (ISA) (m)

0

Initial velocity (m/s)

2330

DT (ISA +/- DT)

0

Diameter (m)

5.52e-04

Shooting angle

min alpha (deg)

1

max alpha (deg)

90

step alpha (deg)

10

Fluid Mechanics

Ballistic fragment trajectories

Initial deceleration chart

Figure 7.

Interface of the primary fragment calculation assistant showing the initial deceleration

chart, which provides the fraction of the initial velocity,

u/uf

, achieved at a certain distance

(contour lines), given the fragment mass,

mf

, and material (e.g., lead), and the atmospheric

conditions (e.g., ISA mean sea level). Fragments are assumed spherical.

Non-tabulated explosives or explosive mixtures can also be considered, with the

367

Gurney constant being computed by the thermochemical assistant presented in Section

368

2.2.1. In this case, the user must select a “custom” explosive, and the thermochemical

369

assistant will open to specify the desired explosive composition. Once the wizard is

370

closed, the Gurney constant is automatically exported to the fragment wizard.

371

2.2.4. Damage to people

372

SimEx includes an assistant for estimating damage to people using the widely

373

accepted probit (probability unit) functions [

20

,

27

] provided by the TNO’s Green Book [

58

]

374

and summarized in Table 3. For each type of injury or cause of death (eardrum rupture,

375

lung injury, etc.), a probit function is deﬁned that depends on the blast parameters: side-

376

on, dynamic or reﬂected peak overpressure (depending on the body position), impulse

377

per unit area, etc. For primary injuries, lethality due to lung damage is evaluated together

378

with the probability of eardrum rupture. For tertiary injuries, lethality is evaluated for

379

shock-induced body displacement and subsequent direct impact, either with the head or

380

the whole body [51].

381

The appearance of the interface is shown in Figure 8. All necessary parameters

382

can be selected on the left: size, type and geometry of the explosive charge, as well as

383

the body position relative to the incoming pressure wave, which determines whether

384

side-on, dynamic or reﬂected pressure is used to compute the peak overpressure and

385

impulse. The rest of the window presents the results both numerically and graphically,

386

using overpressure-impulse diagrams on the left and CW-S diagrams on the right,

387

with primary injuries shown above and tertiary injuries below. Overpressure-impulse

388

diagrams display the characteristic overpressure–impulse–distance curve for the selected

389

charge weight to facilitate the interpretation of results [

2

], while CW-S diagrams include

390

a diagonal dashed line indicating the approximated position of the ﬁreball radius,

391

corresponding roughly to an scaled distance

Z=d/W1/3 =

1 m. Above this line

392

the Freidlander waveform is not valid and the blast wave parameters are increasingly

393

imprecise [35].

394

Version September 5, 2022 submitted to Appl. Sci. 13 of 23

Figure 8.

Interface of the assistant for estimating blast-induced damage to people. The CW-S and atmospheric data,

along withe the body position relative to the incoming pressure wave, are introduced on the top-left corner, the blast

wave parameters and the statistical damage indicators for the chosen CW-S combination appear on the bottom left corner.

The right plots represent graphically the statistical damage indicators in the form of overpressure-impulse and CW-S

diagrams. Both show the conditions corresponding to the speciﬁed CW-S combination with a solid red dot, while the CW-S

diagrams include also a diagonal dashed line indicating the approximated position of the ﬁreball radius. Above this line the

Freidlander waveform is not valid and the blast wave parameters are increasingly imprecise [35].

3. Example of application: façade of a building under blast loading

395

To illustrate the capabilities of SimEx, this section presents a preliminary study

396

to asses the ability of a conventional three-story steel frame building, such as the one

397

shown in Figure 9, to resist three different combinations of charge weight,

W

, and

398

standoff distance,

d

, preserving a similar scaled distance,

Z=d/W1/3

. The three CW-S

399

combinations are summarized in Table 4. For simplicity, we assume mean sea level ISA

400

conditions for all the calculatons. For illustrative purposes, the ﬁgures quoted below

401

show results corresponding to the ﬁrst ﬂoor of the building (hereafter referred to as

402

Level 1) and Case 2 conditions. That is, we shall consider as reference conditions a

403

ground explosion of 150 kg of TNT at 20 m standoff distance from the front façade of the

404

building, as depicted in Figure 9a.

405

3.1. Incident load

406

As previously discussed, SimEx allows the user to enter directly the desired CW-S

407

combination to deﬁne the incident blast load. Figure 2shows the results corresponding

408

to the reference conditions (Level 1, Case 2). For a more detailed analysis of the load

409

induced by the blast wave, the “Blast Wave” calculation assistant shown in Figure 10

410

allows a fast evaluation of all blast parameters as a function of the standoff distance. To

411

Version September 5, 2022 submitted to Appl. Sci. 14 of 23

Table 3.

Probit functions used to estimate the probability of different types of primary and tertiary

injuries. Pr is the probit value,

p◦

[Pa] the peak overpressure,

p◦

ef

[Pa] the maximum effective

overpressure, depending on the relative orientation of the person with respect to the shock wave,

p1

[Pa] the atmospheric pressure,

I/A

[Pa

·

s] the impulse per unit area and

m

[kg] the weight of

the person [58].

Effect Probit function

Primary injuries

Eardrum rupture Pr =−12.6 +1.52 ln p◦

Death due to lung

damage Pr =5−5.74 ln 4.2

p◦

ef/p1+1.3

i/(p1/2

1m1/3)!

Tertiary injuries

Death due to displacement

and whole-body impact Pr =5−2.44 ln7380

p◦+1.3 ×109

p◦i

Death due to displacement

and skull impact Pr =5−8.49 ln2430

p◦+4×108

p◦i

Table 4.

Standoff distance,

d

, explosive charge,

W

, and scaled distance,

Z

, of the different case

studies. The reference case is shown in blue.

Variables Case 1 Case 2 Case 3

d(m) 12 20 25

W(kg) 30 150 300

Z(m/kg1/3)3.86 3.76 3.73

this end, the user must provide the following input data: the ground distance from the

412

explosion to the point of calculation,

d

, the elevation of the explosive charge,

hc

, and the

413

elevation of the calculation point, h0, both measured from the ground.

414

For

hc=

0, an hemispherical surface burst computed from Kingery & Bulmash

415

parameters for TNT [

33

] is considered, although other correlations for hemispherical

416

explosions [

35

] could also be selected. For

hc>

0, hemispherical or spherical blasts are

417

both available, letting the user decide what is the best option based on the height of

418

burst. The code does not include correlations for more complex conﬁgurations, such as

419

air bursts producing regions of regular and Mach reﬂections that eventually modify the

420

incident shock wave. The user must also introduce the angle formed by the normal to

421

the structural element at the point of calculation with the horizontal projection of the

422

line joining the center of the explosion with that point,

δ

, which is identically zero in

423

our case studies if we assume a symmetric conﬁguration with a pillar in the center of

424

the front façade. These distances and angles are employed for simplicity in obtaining

425

in-ﬁeld measurements.

426

With these data, the wizard is able to compute the real distance and incidence

427

angle, thereby providing the peak overpressure,

p◦

, the impulse per unit area,

I/A

,

428

the duration of the positive phase,

td

, the blast arrival time,

ta

, the average speed

429

of the pressure front,

σ=dreal/ta

, the positive phase length,

Lw

, and the waveform

430

parameter,

α

. The results are presented in a table for several standoff distances,

d

, which

431

gives also the real distances,

dreal

, and angles of incidence,

δreal

. The maximum and

432

minimum distances that appear in the table can be easily modiﬁed by the user, who

433

can select any intermediate value using a slider bar to compute the blast parameters

434

at a ﬁxed speciﬁed distance. A button has also been included to graphically represent

435

the variation of any of the blast parameters as a function of the distance to the center of

436

Version September 5, 2022 submitted to Appl. Sci. 15 of 23

Figure 9.

Schematic diagram of the three-story building under study, composed of equally spaced pillars and an outer

enclosure wall, including: (a) the distances and angles used for the different ﬂoor levels (

i=

0, 1, and 2), including the

standoff distance,

d

, the real distance to the midpoint of the different levels,

dreal,i

, and the corresponding angles of incidence,

δreal,i

; (b) schematic of the façade constructive details and dimensions; and (c) diagram of the equivalent façade element

used in the SDOF analysis. L

i

denotes the height of Level

i

, representing the length of the pillars, and

S

is the spacing

between pillars, representing the tributary loaded width.

the explosion. The results are also exportable as a "comma-separated-value" format for

437

further postprocessing.

438

For more qualitative information, two exportable graphs are presented in the lower

439

part. The graph on the left displays the time evolution of the overpressure at a ﬁxed

440

horizontal distance. The user can change this distance easily with the slider bar. All

441

the characteristics of the blast wave are shown for the particular distance chosen by the

442

user. The graph on the right represents the maximum overpressure and the impulse per

443

unit area as a function of the horizontal distance. As previously indicated, the range of

444

distances is also adjustable by the user. Using the "Export and exit" button, the module

445

is closed and the weight and type of explosive, the distance to the charge, and the real

446

angle of incidence to be used in the integration of the SDOF system are exported to the

447

main SimEx module. Figure 10 shows the calculation of the blast parameters for an

448

explosive charge of 150 kg of TNT on a point at a height of 5.5 m above the horizontal,

449

i.e., the geometric center of the façade of the ﬁrst ﬂoor, corresponding to the reference

450

case (Level 1, Case 2). Other distances are also included in the top table, showing how

451

the angle of incidence tends to become normal as the charge moves away from the target.

452

3.2. Estimation of the equivalent SDOF system response

453

To study the structural response to an explosive charge it is necessary to know in

454

detail the type of construction. However, when using a simpliﬁed SDOF model, the

455

study can be simpliﬁed and generalized for many different cases. In the present example

456

we will analyze a façade structure like the one in Figure 9b, composed of equally spaced

457

pillars and an outer enclosure wall.

458

The ﬁrst element that receives the blast wave is the enclosure of the façade. This,

459

in turn, transmits the load to the rest of the structure. In most constructions the façade

460

is only an enclosure without structural function (glass façades, brick, etc.). In ﬁrst

461

approximation, it can be considered that the exterior enclosure transmits the full load

462

received directly to the pillars. The pillars are structural elements whose integrity is

463

considered critical. It will therefore be the ﬁrst element to be studied since the protection

464

of the supporting structure is pivotal to avoid the potential collapse of the building. The

465

enclosure can be considered as a secondary element in most constructions and therefore

466

a signiﬁcantly higher level of damage than in primary elements can be allowed.

467

Version September 5, 2022 submitted to Appl. Sci. 16 of 23

Explosive

Charge weight (kg)

150

Explosive

TNT

W TNT eq Dp (kg)

150

W TNT eq i (kg)

150

Standoff distance

h_c (m)

0

h_o (m)

5.5

delta (deg)

0

Atmosphere

pa (kPa)

101.325

Ta (ºC)

15

DT (ISA +/- DT)

0

Height (m - ISA)

0

Blast wave type

UFC 3-340-02 Hemispheric

Export and exit

Export *.CSV

Fluid Mechanics

Standoff d (m)

d_real (m)

delta_real (deg)

pº (kPa)

I/A (kPa·ms)

t_d (ms)

t_a (ms)

sigma (m/s)

L_w (m)

alpha (-)

0.5000

5.5227

84.8056

1.4217e+03

1.0928e+03

10.0400

2.6651

1.2402e+04

12.2614

11.9697

1

5.5902

79.6952

1.5770e+03

1.2045e+03

10.2585

2.7243

1.2254e+04

12.3923

12.3428

1.5000

5.7009

74.7449

1.6696e+03

1.3203e+03

10.5761

2.8240

1.2022e+04

12.6144

12.2856

2

5.8523

70.0169

1.8977e+03

1.4295e+03

10.8971

2.9630

1.1715e+04

12.9469

13.3858

2.5000

6.0415

65.5560

2.0390e+03

1.5254e+03

11.1878

3.1412

1.1353e+04

13.4052

13.8768

5

7.4330

47.7263

2.1914e+03

1.6975e+03

11.7122

4.6148

9.2655e+03

17.9179

14.0430

10

11.4127

28.8108

776.5040

1.2304e+03

11.0972

10.2888

6.3779e+03

32.3697

5.7995

15

15.9765

20.1363

318.7900

890.3983

15.0684

18.9096

5.1534e+03

44.1625

4.1013

20

20.7425

15.3763

169.5334

688.1037

17.9609

29.5218

4.6027e+03

53.3798

3.0386

25

25.5979

12.4074

106.1733

544.2806

19.8142

41.3644

4.3200e+03

60.6029

2.4015

30

30.5000

10.3889

75.0865

463.3434

21.1770

53.9345

4.1568e+03

66.3909

1.8908

35

35 4295

8 9306

58 0521

393 3842

22 2938

66 9363

4 0543 03

71 1384

1 7193

Compute

Standoff distance

d min (m)

d max (m)

0.5

20 m

50

Blast wave

pº (kPa)

169.5

I/A (kPa ms)

688.1

t_d (ms)

17.96

t_a (ms)

29.52

sigma (m/s)

4603

L_w (m)

53.38

alpha (-)

3.039

delta_real (deg)

15.38

d_real (m)

20.74

Plot

Plot

Plot

Plot

Plot

Plot

Plot

Plot

Plot

Help

Figure 10.

Interface of the Blast Wave calculation assistant for a charge weight of 150 kg of TNT at ISA mean sea level,

showing the variation of the blast parameters with the standoff distance from the front façade (top table). The lower part of

the assistant shows the blast parameters calculated at a point located at

d=

20 m standoff distance and

h0=

5.5 m above

the charge.

Figure 9c shows the simplest element in which the façade is to be divided. Each

468

pillar receives loads from a part of the façade corresponding to the distance between

469

pillars and the height between ﬂoors. The load generated by the explosion is applied to

470

the pillars crosswise, so they behave in ﬁrst approximation as bending elements. For

471

the calculation of the equivalent properties, the beam assistants available in SimEx are

472

employed. Either for metal or concrete beams, the length corresponds to the height

473

between ﬂoors, while the span is the spacing between pillars. In the case of pillars, the

474

boundary condition between ﬂoors is that of embedment on both sides, whereas a free

475

conditions is preferred at the roof. As a result, we use ﬁxed-ﬁxed conditions for Levels

476

0 and 1 and cantilever (or ﬁxed-free) for Level 2. The presence of a roof diaphragm

477

element may require additional considerations regarding the boundary condition at the

478

roof top, but we prefer to use a ﬁxed-free boundary condition for the second ﬂoor both

479

for simplicity and for illustrating the effect of considering different boundary conditions

480

on different ﬂoors.

481

In the case of metal beams, it is only necessary to indicate the standard shape of the

482

proﬁle and the size. SimEx uses European cross-section proﬁles HEB, IPE, and IPN in

483

accordance with Euronorm 53–62 (DIN 1025) [

17

]. Figure 11 shows the result for a HEB

484

340 proﬁle with a length of 3 m and a separation between pillars of 5 m. The assistant

485

uses standardized proﬁles, so if a non-existent measure is introduced, it corrects down

486

to the nearest lower normalized proﬁle. However, it is also possible to select custom

487

proﬁles and materials. In this case, the area, ﬁrst moment of area about the bending axis,

488

moment of inertia about the bending axis, density, Young’s modulus, and resistance

489

must be provided by the user. Once the structural properties have been introduced,

490

closing the assistant incorporates the computed data into the main SimEx interface.

491

Figure 2shows the result for the case under study. It should be noted that the additional

492

enclosure mass supported by the pillar when ﬂexed must also be included in the mass

493

of the equivalent SDOF system in the main interface.

494

If a rectangular reinforced concrete pillar is considered, SimEx requires that the

495

external measurements

b

and

h

(perpendicular and parallel to the direction of application

496

Version September 5, 2022 submitted to Appl. Sci. 17 of 23

Shape

Section

350

Tipo de perfil

HEB

Area (cm2)

170.9

1st moment of area (cm3)

1200

Moment of inertia (cm4)

3.666e+04

Reinforcement

Reinforcement

A-36

Density (kg/m3)

7850

Young's modulus (kPa)

2e+08

Resistance f_y (kPa)

2.482e+05

Beam geometry

Length (m)

3

Span (m)

5

Type of edge

Fixed-Fixed

Equivalent properties

K_LM

0.66

K (kPa/mm)

66.98

R_u (kPa)

158.4

Total mass (kg)

402.5

Mass p.u.s. (kg/m2)

26.83

Used section

340

Compute

Reset

Fluid Mechanics

Figure 11.

Metal beam calculation assistant showing results for a HEB 340 pillar with a length of 3

m and a spacing between pillars of 5 m. Note that even though a HEB 350 is requested, which is

not included in the norm, the assistant corrects down to the nearest normalized value, HEB 340.

Figure 12.

Reinforced concrete beam calculation assistant showing results for a pillar of 45 x 45

cm

2

with a length of 3 m and a spacing between pillars of 5 m. The pillar is reinforced using 5 A36

steel reinforcement bars of 22.5 mm of diameter per side spaced apart 37 cm.

of the load, respectively) be introduced. In addition, the properties of the reinforcement

497

should be indicated in a simpliﬁed manner, that is, interior spacing,

dc

, and reinforcement

498

area,

As=nπd2

bar/

4, where

n

represents the number of steel reinforced bars per side.

499

Figure 12 shows results for a pillar of 45 x 45 cm

2

with 5 A36 steel reinforcement bars

500

of # 7 (approximately 22.5 mm in diameter) per side, for a length of 3 m and a spacing

501

between pillars of 5 m. The distance

dc

must be estimated according to the constructive

502

detail. In this particular case, it is assumed that the reinforcement centers are located at

503

4 cm from the edge, resulting in an interior reinforcement spacing of dc=37 cm.

504

It is worth noting that neglecting axial load can be considered a conservative

505

approach, particularly in the case of columns or pillars. These elements are initially

506

subjected to a signiﬁcant compression load due to the weight of the supported structure,

507

which reduces the tensile stresses caused by bending. This simpliﬁcation constitutes a

508

ﬁrst approximation in the study of the structural response. For a more detailed analysis,

509

the wall should be the next element to be analyze in order to assure that it is able to fully

510

transmit the blast load to the load-bearing element. If the wall was made of concrete, this

511

could be done using the concrete beam assistant with

b=S

. In this case, the mass of the

512

Version September 5, 2022 submitted to Appl. Sci. 18 of 23

element under study would be the total mass of the equivalent SDOF system. However,

513

in the case considered here of load-bearing elements (beams or columns/pillars) the

514

total mass can be signiﬁcantly larger than the mass of the element.

515

3.3. SDOF system integration and CW-S damage diagrams

516

Once the user sets the explosive charge and the properties of the equivalent SDOF

517

system, SimEx is ready to integrate the resulting mathematical problem. Figure 2shows

518

the results for the case of a HEB 340 proﬁle with a 5 m span between pillars. The

519

main results are the maximum deﬂection,

xmax

, the ductility ratio,

µ

, and the maximum

520

rotation angle,

θ

. The two latter parameters are used as indicators to quantify the

521

component damage levels [

49

]. Assuming that the Level of Protection (LOP) required is

522

very low, in case of a hot rolled compact steel shape for the columns, according to [

49

],

523

the allowable component damage is heavy (response between B2-B3).

524

For ﬁxed values of the structural parameters, a parametric sweep can be carried

525

out in CW-S space to obtain damage diagrams such as the ones shown in Figure 13. To

526

this end, it is enough to indicate in the assistant the charge weight and standoff distance

527

ranges to be analyzed and the number of intervals to be used for each parameter. In

528

addition, the desired damage level criteria must be indicated to separate the zones.

529

Figure 2shows characteristic values of

µ

and

θ

for metallic elements, although other

530

values could be selected from [

49

] for other structural elements and materials. Note that

531

CW-S damage diagrams are presented both in linear and log-log scales.

532

As can be seen, the CW-S damage diagrams shown in Figure 13 include three points

533

corresponding to the three cases considered in Table 4. As the three scaled distances are

534

almost equal then the damage levels are also very similar, although differences in real

535

distances an incidence angles make them grow from superﬁcial-moderate (B1) to (almost)

536

moderate-heavy (B2) for increasing charge weights and standoff distances. According

537

to the PDC-TR 06-08 [

49

], a superﬁcial damage level implies “no visible permanent

538

damage”, whereas a moderate damage level implies “some permanent deﬂection” that

539

generally can be repaired. By way of contrast, a heavy damage is associated with

540

“signiﬁcant permanent deﬂections” that cause the component to be unrepairable.

541

To summarize the results obtained in the different case studies, Table 5reports the

542

incident blast load parameters and the corresponding component damage indicators

543

per ﬂoor for Cases 1, 2, and 3. The reference case (Level 1, Case 2) and the worst-case

544

scenario (Level 2, Case 3) are both highlighted for clarity. As can be seen, damage levels

545

are signiﬁcantly higher in the upper ﬂoor (Level 2) as a result of the lowest rigidity

546

imposed by the cantilever boundary condition at the roof top, resulting in heavy damage

547

levels for cases 2 and 3.

548

Figure 13.

CW-S linear (left) and log-log (right) damage diagrams for reﬂected blast load on the

façade of the ﬁrst ﬂoor (Level 1): Case 1 (), Case 2 (♦), Case 3 (4).

Version September 5, 2022 submitted to Appl. Sci. 19 of 23

Table 5.

Incident load parameters and component damage indicators per ﬂoor. According to the

PDC-TR-06-08 [

49

], the response limits for hot rolled structural steel can be deﬁned in terms of the

ductility ratio,

µ

, and support rotation angle,

θ

, as follows: B1 - superﬁcial

{µ

,

θ}={

1,

−}

; B2 -

moderate

{µ

,

θ}={

3, 3

◦}

; B3 - heavy

{µ

,

θ}={

12, 10

◦}

; B4 - hazardous

{µ

,

θ}={

25, 20

◦}

. The

reference case and worst-case scenario are indicated in blue and gray, respectively.

Level Type Variables Case 1 Case 2 Case 3

0

Incident load parameters

∆p(kPa) 168.30 182.50 186.80

I/A(kPa ·ms) 406.70 724.40 922.20

dreal (m) 12.17 20.10 25.08

δreal (deg) 9.46 5.71 4.57

Damage level indicators µ(-) 1.60 3.26 4.40

θ(deg) 0.19 0.39 0.53

1

Incident load parameters

∆p(kPa) 139.40 169.50 178.00

I/A(kPa ·ms) 349.90 688.10 893.00

dreal (m) 13.20 20.74 25.60

δreal (deg) 24.62 15.38 12.41

Damage level indicators µ(-) 0.90 1.64 2.10

θ(deg) 0.08 0.15 0.19

2

Incident load parameters

∆p(kPa) 110.20 152.00 165.40

I/A(kPa ·ms) 293.00 630.50 845.50

dreal (m) 14.71 21.73 26.41

δreal (deg) 35.31 23.03 18.78

Damage level indicators µ(-) 1.67 5.85 9.26

θ(deg) 0.87 3.05 4.83

3.4. Crater, fragments and damage to people

549

Figure 4presents an estimation of the crater generated in the reference case on a

550

sandstone soil, with an approximated radius of 1.6 m. For surface bursts, HOB

=

0

551

m, as the one considered here, the equivalent charge radius is irrelevant, as it is only

552

used to determine the dimensionless height of burst which is identically zero in our

553

example. The ﬁgure also shows that for above-surface bursts, HOB

>

0 m, the crater

554

radius is signiﬁcantly smaller for the same amount of explosive due to the air cushion

555

that exists between the load and the ground, which reduces to a great extent the pressure

556

that reaches the ground surface [14].

557

Figure 5shows the interface of the fragment assistant using the input data of the

558

reference case. For the application of Mott’s statistical theory for fragmentation of

559

steel cylindrical shells [

23

,

40

–

42

], the explosive charge is approximated to a cylinder of

560

approximately 50 cm diameter surrounded by a steel fragmentation shell with a mass of

561

the order of about 13% of the charge and a thickness of 2 mm.

562

Finally, Figure 8shows the calculating assistant for estimating damage to people in

563

the reference case. As an illustrative example, the ﬁgure presents the results of lethality

564

due to different types of injuries at a distance of 20 m from the origin of the explosion,

565

assuming the worst-case scenario of an average person located close to the façade of the

566

building being attacked. In the pressure-impulse graphs, representative distances are

567

indicated using red dots plotted along the characteristic overpressure–impulse–distance

568

curve [

2

]. As can be seen, at 20 m standoff distance lethality due to lung damage or

569

whole-body projection is negligible, but large primary fragments (e.g., CL 99%) may still

570

produce secondary injuries with fatal results, as indicated by Figure 5.

571

Version September 5, 2022 submitted to Appl. Sci. 20 of 23

4. Conclusions

572

SimEx is a computational tool that allows a rapid and easy estimation of the effects

573

of explosions on structural elements and their damage to people. It has been developed

574

in accordance with the speciﬁcations of American standard UFC-3-340-02 and other

575

widely accepted directives published in the open literature. It provides assistants for the

576

calculation of the blast-wave load; SDOF dynamic response, including the calculation

577

of the equivalent structural properties of standardized metal and reinforzed concrete

578

beams; thermodynamic properties of explosive mixtures; crater formation; projection of

579

primary fragments; and damage to people.

580

After presenting the main calculating assistants, a preliminary study has been

581

presented to illustrate the full capabilities of SimEx in the assessment of the ability of a

582

building to resist a given explosive charge. The analysis enables the determination of

583

component damage levels for the main structural components, and a further study of

584

the reference case has led to the computation of CW-S damage diagrams for a pillar of

585

the ﬁrst ﬂoor. These diagrams are very useful to provide design guidelines for those

586

facilities that must be protected against explosive threats.

587

Although still under development, SimEx is being successfully used for research

588

and teaching activities at the Spanish University Center of the Civil Guard. Due to its

589

advanced stage of maturation, it could also be used in other areas within the Army and

590

Law enforcement Agencies involved in the ﬁght against terrorism and the design of

591

blast resistant buildings and structures.

592

SimEx License & Distribution:

SimEx is a closed-source proprietary software that may be licensed

593

by the copyright holders, UC3M & Guardia Civil, under speciﬁc conditions. Please contact the

594

corresponding author for further information.

595

Author Contributions:

Conceptualization, J.S.-M. and M.V.; methodology, J.S.-M. and M.V.; soft-

596

ware, J.S.-M. and A.C.; validation, all authors; formal analysis, J.S.-M. and M.V.; investigation,

597

all authors; resources, C.H. and M.V.; data curation, J.S.-M. and A.C.; writing—original draft

598

preparation, J.S.-M., A.C. and M.V; writing—review and editing, C.H and M.V.; visualization,

599

J.S.-M and A.C.; supervision, C.H. and M.V.; project administration, M.V.; funding acquisition,

600

C.H. and M.V. All authors have read and agreed to the published version of the manuscript.

601

Funding:

This research was partially funded by UE (H2020-SEC-2016-2017-1) grant number # SEC-

602

08/11/12-FCT-2016 and by proyect H2SAFE-CM-UC3M awarded by the Spanish Comunidad de

603

Madrid. The authors would like to thank Dr. Henar Miguelez from University Carlos III of Madrid

604

and Col. Fernando Moure from the University Center of the Civil Guard for their continuous

605

support, as well as Com. Miguel Ángel Albeniz from the SEDEX-NRBQ (EOD-CBRN) service of

606

the Civil Guard for many enlightening discussions. Fruitful discussions with Dr. Lina López from

607

the School of Mines at Technical University of Madrid are also gratefully acknowledged.

608

Institutional Review Board Statement: Not applicable

609

Informed Consent Statement: Not applicable

610

Conﬂicts of Interest:

The authors declare no conﬂict of interest. The funders had no role in the

611

design of the study; in the collection, analyses, or interpretation of data; in the writing of the

612

manuscript, or in the decision to publish the results.

613

Abbreviations

614

The following abbreviations are used in this manuscript:

615

BKW Becker-Kistiakowsky-Wilson EoS

CL Conﬁdence Level

CT Combustion Toolbox

CUGC Centro Universitario de la Guardia Civil

CW-S Charge Weight-Standoff

EoS Equation of State

GUI Graphical User Interface

Version September 5, 2022 submitted to Appl. Sci. 21 of 23

H9 Heuzé EoS

HOB Height of Burst

IED Improvised Explosive Device

ISA International Standard Atmosphere

LOP Level of Protection

PDC Protective Design Center

SDOF Single Degree of Freedom

SEDEX-NRBQ Explosive Ordnance Disposal (EOD) and CBRN Defense Service

UC3M University Carlos III of Madrid

UFC Uniﬁed Facilities Criteria

US United States

USACE United States Army Corps of Engineers

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