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

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. SimEx Capabilities

#### 2.1. Single-Degree-of-Freedom System Analysis

#### 2.1.1. Forcing Term

#### 2.1.2. Resistance Term

#### 2.1.3. Numerical Integration

#### 2.1.4. Post-Processing

#### 2.2. Other Calculation Assistants

#### 2.2.1. Assistant for the Calculation of the Thermodynamic Properties of Explosives

#### 2.2.2. Crater

#### 2.2.3. Primary Fragments

#### 2.2.4. Damage to People

## 3. Example of Application: Façade of a Building under Blast Loading

#### 3.1. Incident Load

#### 3.2. Estimation of the Equivalent SDOF System Response

#### 3.3. SDOF System Integration and CW–S Damage Diagrams

#### 3.4. Crater, Fragments, and Damage to People

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowldgments

## Conflicts of Interest

## SimEx License & Distribution

## Abbreviations

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 |

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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**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.

**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 show 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).

**Figure 3.**Interface of the assistant for the calculation of the theoretical thermodynamic properties of explosives and explosive mixtures.

**Figure 4.**Interface of the assistant for the calculation of craters. HOB denotes the height of burst.

**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.

**Figure 7.**Interface of the primary fragment calculation assistant showing the initial deceleration chart, which provides the fraction of the initial velocity, $u/{u}_{\mathrm{f}}$, achieved at a certain distance (contour lines), given the fragment mass, ${m}_{\mathrm{f}}$, and material (e.g., lead), and the atmospheric conditions (e.g., ISA mean sea level). Fragments are assumed spherical.

**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 in the top-left corner, the blast wave parameters and the statistical damage indicators for the chosen CW–S combination appear in 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 [1].

**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, ${d}_{\mathrm{real},i}$, and the corresponding angles of incidence, ${\delta}_{\mathrm{real},i}$; (

**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.

**Figure 10.**Interface of the Blast Wave calculation assistant for a charge weight of 150 kg of TNT at the 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 ${h}_{0}=5.5$ m above the charge.

**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 × 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.

**Figure 13.**CW–S linear (

**a**) and log-log (

**b**) damage diagrams for reflected blast load on the façade of the first floor (Level 1): Case 1 (◯), Case 2 (✧), Case 3 (△).

**Table 1.**Comparison of the calculated temperature at constant volume, T, detonation pressure, ${p}_{\mathrm{CJ}}$, detonation velocity, ${v}_{\mathrm{CJ}}$, heat release at constant volume, ${Q}_{v}$, and explosive force, ${F}_{e}$, with the results provided by the European Standard EN 13631-15 [43] and by the thermochemical code W-DETCOM [49] for different explosive mixtures using the BKW–S EoS.

Explosive | Source | T [K] | ${\mathit{p}}_{\mathbf{CJ}}$ [GPa] | ${\mathit{v}}_{\mathbf{CJ}}$ [m/s] | ${\mathit{Q}}_{\mathit{v}}$ [kJ/kg] | ${\mathit{F}}_{\mathit{e}}$ [kJ/kg] |
---|---|---|---|---|---|---|

ANFO | CT | 2592 | 7.14 | 5353 | 3845 | 943 |

EN 13631-15 | 2586 | - | - | 3820 | 945 | |

W-DETCOM ${}^{1}$ | 2919 | 6.62 | 5326 | 3849 | - | |

ANFO-Al | CT | 3026 | 7.38 | 5442 | 4666 | 1009 |

EN 13631-15 | 3060 | - | - | 4642 | 1020 | |

W-DETCOM ${}^{1}$ | 3370 | 6.55 | 5215 | 4655 | - | |

Emulsion | CT | 2112 | 15.3 | 6549 | 3263 | 766 |

EN 13631-15 | 2099 | - | - | 3236 | 771 | |

W-DETCOM ${}^{1}$ | 2438 | 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 |

^{1}Calculation performed assuming Chapman-Jouguet detonation.

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/m${}^{3}$] | 850 | 850 | 1300 | 1500 | 1500 |

Oxygen balance [%] | $-1.7$ | $0.08$ | $-5.57$ | $-2.26$ | $0.84$ |

**Table 3.**Probit functions used to estimate the probability of different types of primary and tertiary injuries. Pr is the probit value, ${p}^{\circ}$ [Pa] the peak overpressure, ${p}_{\mathrm{ef}}^{\circ}$ [Pa] the maximum effective overpressure, depending on the relative orientation of the person with respect to the shock wave, ${p}_{1}$ [Pa] the atmospheric pressure, $I/A$ [Pa · s] the impulse per unit area and m [kg] the weight of the person [9].

Effect | Probit Function |
---|---|

Primary injuries | |

Eardrum rupture | $\mathrm{Pr}=-12.6+1.52\mathrm{ln}{p}^{\circ}$ |

Death due to lung damage | $\mathrm{Pr}=5-5.74\mathrm{ln}\left(\right)open="("\; close=")">\frac{4.2}{{p}_{\mathrm{ef}}^{\circ}/{p}_{1}}+\frac{1.3}{i/\left({p}_{1}^{1/2}{m}^{1/3}\right)}$ |

Tertiary injuries | |

Death due to displacement and whole-body impact | $\mathrm{Pr}=5-2.44\mathrm{ln}\left(\right)open="("\; close=")">\frac{7380}{{p}^{\circ}}+\frac{1.3\times {10}^{9}}{{p}^{\circ}i}$ |

Death due to displacement and skull impact | $\mathrm{Pr}=5-8.49\mathrm{ln}\left(\right)open="("\; close=")">\frac{2430}{{p}^{\circ}}+\frac{4\times {10}^{8}}{{p}^{\circ}i}$ |

**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/kg${}^{1/3}$) | 3.86 | 3.76 | 3.73 |

**Table 5.**Incident load parameters and component damage indicators per floor. According to the PDC-TR-06-08 [15], the response limits for hot rolled structural steel can be defined in terms of the ductility ratio, $\mu $, and support rotation angle, $\theta $, as follows: B1—superficial $\{\mu ,\theta \}=\{1,-\}$; B2—moderate $\{\mu ,\theta \}=\{3,{3}^{\circ}\}$; B3—heavy $\{\mu ,\theta \}=\{12,{10}^{\circ}\}$; B4—hazardous $\{\mu ,\theta \}=\{25,{20}^{\circ}\}$. 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 | $\Delta p$ (kPa) | 168.30 | 182.50 | 186.80 |

$I/A$ ($\mathrm{kPa}\xb7\mathrm{ms}$) | 406.70 | 724.40 | 922.20 | ||

${d}_{\mathrm{real}}$ (m) | 12.17 | 20.10 | 25.08 | ||

${\delta}_{\mathrm{real}}$ (deg) | 9.46 | 5.71 | 4.57 | ||

Damage level indicators | $\mu $ (-) | 1.60 | 3.26 | 4.40 | |

$\theta $ (deg) | 0.19 | 0.39 | 0.53 | ||

1 | Incident load parameters | $\Delta p$ (kPa) | 139.40 | 169.50 | 178.00 |

$I/A$ ($\mathrm{kPa}\xb7\mathrm{ms}$) | 349.90 | 688.10 | 893.00 | ||

${d}_{\mathrm{real}}$ (m) | 13.20 | 20.74 | 25.60 | ||

${\delta}_{\mathrm{real}}$ (deg) | 24.62 | 15.38 | 12.41 | ||

Damage level indicators | $\mu $ (-) | 0.90 | 1.64 | 2.10 | |

$\theta $ (deg) | 0.08 | 0.15 | 0.19 | ||

2 | Incident load parameters | $\Delta p$ (kPa) | 110.20 | 152.00 | 165.40 |

$I/A$ ($\mathrm{kPa}\xb7\mathrm{ms}$) | 293.00 | 630.50 | 845.50 | ||

${d}_{\mathrm{real}}$ (m) | 14.71 | 21.73 | 26.41 | ||

${\delta}_{\mathrm{real}}$ (deg) | 35.31 | 23.03 | 18.78 | ||

Damage level indicators | $\mu $ (-) | 1.67 | 5.85 | 9.26 | |

$\theta $ (deg) | 0.87 | 3.05 | 4.83 |

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**MDPI and ACS Style**

Sánchez-Monreal, J.; Cuadra, A.; Huete, C.; Vera, M.
SimEx: A Tool for the Rapid Evaluation of the Effects of Explosions. *Appl. Sci.* **2022**, *12*, 9101.
https://doi.org/10.3390/app12189101

**AMA Style**

Sánchez-Monreal J, Cuadra A, Huete C, Vera M.
SimEx: A Tool for the Rapid Evaluation of the Effects of Explosions. *Applied Sciences*. 2022; 12(18):9101.
https://doi.org/10.3390/app12189101

**Chicago/Turabian Style**

Sánchez-Monreal, Juan, Alberto Cuadra, César Huete, and Marcos Vera.
2022. "SimEx: A Tool for the Rapid Evaluation of the Effects of Explosions" *Applied Sciences* 12, no. 18: 9101.
https://doi.org/10.3390/app12189101