Article

SimEx: A Tool for the Rapid Evaluation of the Effects of Explosions

Abstract and Figures

The dynamic response of structural elements subjected to blast loading is a problem of growing interest in the field of defense and security. In this work, a novel computational tool for the rapid evaluation of the effects of explosions, hereafter referred \reved{to} as SimEx, is presented and discussed. The classical correlations for the reference chemical (1 kg of TNT) and nuclear (10^6 kg of TNT) explosions, both spherical and hemispherical, are used together with the blast wave scaling laws and the International Standard Atmosphere (ISA) to compute the dynamic response of Single-Degree-of-Freedom (SDOF) systems subject to blast loading. The underlying simplifications in the analysis of the structural response follow the directives established by UFC 3-340-02 and the Protective Design Center Technical Reports of the US Army Corps of Engineers. This offers useful estimates with a low computational cost \reved{that enable} in particular the computation of damage diagrams in the Charge Weight-Standoff distance (CW-S) space for the rapid screening of component (or building) damage levels. SimEx is a computer application based on Matlab and developed by the Fluid Mechanics Research Group at University Carlos III of Madrid (UC3M). It has been successfully used for both teaching and research purposes in the Degree in Security Engineering, taught to the future Guardia Civil officers at the Spanish University Center of the Civil Guard (CUGC). This dual use has allowed the development of the application well beyond its initial objective, testing on one hand the implemented capacities by undergraduate cadets with end-user profile, and implementing new functionalities and utilities by Masters and PhD students. With this experience, the application has been continuously growing since its initial inception in 2014 both at a visual and a functional level, including new effects in the propagation of the blast waves, such as clearing and confinement, and incorporating new calculation assistants, such as those for the thermochemical analysis of explosive mixtures; crater formation; fragment mass distributions, ejection speeds and ballistic trajectories; and the statistical evaluation of damage to people due to overpressure, body projection, and fragment injuries.
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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,
xxu
, 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=p1t0
tdexpαt0
td(4)
212
where
p=p2p1
represents the peak overpressure measured from the undisturbed
213
atmospheric pressure
p1=pa
, with
p2
denoting the peak post-shock pressure,
t0=tta
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
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=ptd1/α(1eα)/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
πKLMM/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 57. 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 =55.74 ln 4.2
p
ef/p1+1.3
i/(p1/2
1m1/3)!
Tertiary injuries
Death due to displacement
and whole-body impact Pr =52.44 ln7380
p+1.3 ×109
pi
Death due to displacement
and skull impact Pr =58.49 ln2430
p+4×108
pi
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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