Characterizing Explosive 

Effects on Underground 

Structures

Office of Nuclear Security and Incident Response

NUREG/CR-7201
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Characterizing Explosive 

Effects on Underground 

Structures

Manuscript Completed: September 2013

Date Published: September 2015

Prepared by

A. H. Chowdhury, CNWRA 

T. E. Wilt, CNWRA

Center for Nuclear Waste Regulatory Analyses 

Southwest Research Institute® 

6220 Culebra Road 

San Antonio, Texas 78238-5166

 

Kris Jamgochian, NRC Project Manager

Peter S. Lee, NRC Technical Monitor

NRC Job Code N4128 

Office of Nuclear Security and Incident Response

NUREG/CR-7201

 

iii

ABSTRACT

A literature review and finite element analyses were conducted to characterize the effects of 

explosions on underground structures for explosive charges located close to and on the ground 

surface. Explosive charge weights were selected to be consistent with Vehicle Borne 

Improvised Explosive Devices (VBIEDs) terrorists make mostly using chemicals and available 

precursor materials. The phenomena associated with air and surface explosions of VBIEDs, 

propagation of their dynamic waves through air, crater formation, and propagation through 

subsurface media are similar to those of conventional high explosive military weapons. 

Empirical equations for predicting these blast-induced effects and designing underground 

structures under explosive loads are available in the literature. These empirical equations, 

however, do have a specific range of applicability. Numerical analysis techniques (e.g., finite 

element) also require specific assumptions and idealizations. A limited finite element parametric 

study was used to investigate the influence of important parameters. These influences include

the explosive charge weight and how the distance of a point in the subsurface soil from the 

location of explosion affects the explosion-generated pressure distribution in the underground 

soil. The blast-induced pressure wave in the soil surrounding an underground structure results 

in time-dependent loading (pressure or impulse force) of the underground structure. The 

current methodology for the designing underground structures subjected to explosive loads 

uses concepts and provisions of other specific codes, standards, and design manuals

commonly used in designing reinforced concrete, prestressed concrete, masonry, and steel 

structures. An underground structure that may be subjected to both explosive and seismic 

loads should consider both types of loads in appropriate combinations.

v

CONTENTS

Section Page

ABSTRACT................................................................................................................................ iii

FIGURES.................................................................................................................................. vii

TABLES.....................................................................................................................................ix

EXECUTIVE SUMMARY ...........................................................................................................xi

ACKNOWLEDGMENTS............................................................................................................xv

ACRONYMS............................................................................................................................xvii

1 INTRODUCTION ..............................................................................................................1-1

1.1 Background .............................................................................................................1-1

1.2 Objectives................................................................................................................1-2

2 EXPLOSIVE EFFECTS.....................................................................................................2-1

2.1 Fundamentals of Explosives ....................................................................................2-1

2.1.1 Vehicle-borne Improvised Explosive Devices ..............................................2-1

2.2 Dynamics of Explosion Propagation in Air and Underground Media.........................2-7

2.3 Blast Consequences..............................................................................................2-12

2.3.1 Effects on the Structure.............................................................................2-12

2.3.2 Effects on Equipment ................................................................................2-13

2.3.3 Effects on Humans....................................................................................2-13

2.4 Surface Blast Effects and Ground Shock ...............................................................2-14

2.4.1 Analytical Methods for Evaluating Blast-Induced Ground Effects ..............2-15

2.4.1.1 Blast Wavefront Parameters .......................................................2-16

2.4.1.2 Blast-Induced Ground Motions....................................................2-19

2.4.2 Calculation of Subsurface Pressure ..........................................................2-22

2.4.3 Calculation of Crater Dimensions ..............................................................2-24

2.5 Underwater Detonations ........................................................................................2-25

3 NUMERICAL EVALUATION OF SURFACE AND SUBSURFACE EFFECTS DUE TO 

AIR AND SURFACE BURSTS..........................................................................................3-1

3.1 Finite Element Lagrangian Model.............................................................................3-1

3.2 Modeling of Soil Behavior ........................................................................................3-1

3.3 Blast Pressure Loading............................................................................................3-2

3.4 Empirical and Numerical Analyses for a 100-kg [220-lb] TNT Explosive 

Charge—Soil Pressures and Crater Formation........................................................3-4

3.4.1 Evaluation of Soil Pressures........................................................................3-4

3.4.2 Evaluation of Predicted Crater Dimensions .................................................3-5

3.5 Finite Element Parametric Study on Different TNT-Equivalent Charge 

Weights and Above-Surface Heights .......................................................................3-9

3.5.1 Predicting Crater Formation Using Element Deletion and 

Continuum-To-Particle Conversion............................................................3-15

3.5.1.1 Element Deletion ........................................................................3-17

3.5.1.2 Continuum-to-Particle Conversion ..............................................3-18

3.6 Eulerian Finite Element Analysis............................................................................3-21

3.7 Alternative Analysis Codes Reviewed....................................................................3-24
vi

CONTENTS (continued)

Section Page

4 CHARACTERIZING STRUCTURE RESPONSE......................................................4-1

4.1 Codes and Standards for Design of Underground Structures...................................4-1

4.1.1 Design and Analysis of Hardened Structures to Conventional 

Weapons Effects and Structures to Resist the Effects of 

Accidental Explosions .................................................................................4-1

4.1.2 Fundamentals of Protective Design for Conventional Weapons ..................4-1

4.2 Analytical Tools for Modeling Underground Structure Response..............................4-1

5 COMPARISON OF EXPLOSIVE LOADS AND SEISMIC LOADS IN 

UNDERGROUND MEDIUM..............................................................................................5-1

6 DESIGN CONSIDERATIONS FOR PROTECTION OF 

UNDERGROUND STRUCTURES ....................................................................................6-1

7 OTHER PUBLICATIONS ..................................................................................................7-1

8 CONCLUSIONS................................................................................................................8-1

9 REFERENCES .................................................................................................................9-1
vii

FIGURES

Figure Page

2-1 U.S. National Counterterrorism Center Bomb Threat Stand-Off Chart ..........................2-6

2-2 Variation of Air Burst Overpressure at a Given Location in Air With Time.....................2-8

2-3 Reflection of Wave at the Earth’s Surface in an Air Blast..............................................2-9

2-4 Air-Burst Generated Crater.........................................................................................2-10

2-5 Engineered Site-Specific Barriers Above an Underground Structure ..........................2-11

2-6 Different Forms of Explosions and Severity of Ground Shock Effects.........................2-15

2-7 Description of Surface Burst.......................................................................................2-23

2-8 Empirical Relationship of Peak Pressure Versus Depth for Different 

Charge Weights..........................................................................................................2-23

3-1 Three-Dimensional Finite Element Model of Soil ..........................................................3-2

3-2 Drucker-Prager Cap Model Yield Surface.....................................................................3-3

3-3 Range of Peak Pressure from Nagy, et al. (2010) ........................................................3-5

3-4 Range of Peak Pressure Scaled for Above Surface Blast.............................................3-6

3-5 Comparison of Empirical Peak Pressure With Finite Element Predictions ....................3-6

3-6 Pressure-Time Histories at Different Depths.................................................................3-7

3-7 Soil Pressure Distribution (Pa) at Time = 0.03 sec........................................................3-8

3-8 Time History of Center Displacement Showing Development of Crater Depth............3-10

3-9 Apparent Crater Diameter and Depth Predicted by Finite Element Analysis ...............3-10

3-10 Plastic Strain Distribution (Dp = Plastic Strain Depth)..................................................3-11

3-11 Plastic Strain Distribution Continued (Dp = Plastic Strain Depth).................................3-12

3-12 Displacement Vectors Indicating Direction of Soil Movement .....................................3-13

3-13 Predicted Crater Sizes for Above-Surface Charge Located at a Height 

of 0.6096 m [24 in]......................................................................................................3-16

3-14 Predicted Crater for 454-kg [1,000-lb]-TNT-Equivalent Charge Located at a

Height of 0.6096 m [24 in]...........................................................................................3-17

3-15 Soil Pressure Distributions for 4,536 and 9,072 kg [10,000 and 20,000 lb] Charges...3-19

3-16 Soil Pressure Distributions for 18,144 and 27,216 kg [40,000 and 

60,000 lb] Charges .....................................................................................................3-20

3-17 ABAQUS/Explicit Smoothed Particle Hydrodynamics Results for a 454-kg

[1,000-lb]-TNT-Equivalent Charge Located at a Height of 0.6096 m [24 in] ................3-21

3-18 Multi-Material Eulerian Analysis Model .......................................................................3-22

3-19 Eulerian-Analysis-Predicted Apparent Crater Size......................................................3-23

4-1 Single Degree of Freedom System...............................................................................4-3

5-1 Seismic Waves.............................................................................................................5-1

5-2 Example of a Seismic Wave’s Travel Time...................................................................5-2

5-3 Comparison of Nuclear Explosion With an Earthquake Generated Wave Signals ........5-3

5-4 Comparison of P- to S-Wave Ratios of a Nuclear Explosion With Those of an

Earthquake at Different Frequencies ............................................................................5-4

5-5 Topographic Map Showing Locations of Nuclear Test Locations, Earthquakes, 

Primary International Monitoring System, and Auxiliary International 

Monitoring System........................................................................................................5-4

ix

TABLES

Table Page

2-1 Homemade Explosives and Ingredients........................................................................2-2

2-2 Characteristics of U.S. Explosives ................................................................................2-3

2-3 Constants for Free Field Peak Pressure.....................................................................2-23

3-1 Drucker-Prager Cap Model Parameters for Silty Clay...................................................3-3

3-2 Apparent Crater Dimensions Using Gould (1981).........................................................3-9

3-3 Comparison of Apparent Crater Dimensions for a 100-kg [220-lb] TNT Charge 

Located at a Height of 0.50 m [20 in] ..........................................................................3-11

3-4 Spherical Charge Dimensions ....................................................................................3-14

3-5 Apparent Crater Dimensions for an Above-Surface Charge Located at a Height 

of 0.5588 m [22 in]......................................................................................................3-14

3-6 Apparent Crater Dimensions for an Above-Surface Charge Located at a Height

of 0.6096 m [24 in]......................................................................................................3-14

3-7 Apparent Crater Dimensions for an Above-Surface Charge Located at a Height

of 0.6604 m [26 in]......................................................................................................3-15

3-8 Apparent Crater Dimensions for an Above-Surface Charge Located at a Height

of 0.7112 m [28 in]......................................................................................................3-15

3-9 Height of Above-Surface Charge................................................................................3-18

3-10 Apparent Crater Dimensions for Above-Surface Charges...........................................3-18

3-11 Ideal Gas Parameters.................................................................................................3-22

3-12 Jones-Wilkens-Lee Equation of State Parameters......................................................3-23

xi

EXECUTIVE SUMMARY

The blast-induced dynamic loads explosions generate above ground and in contact with the 

ground surface can damage underground structures. Underground structures that are designed 

to protect against high explosive charges mostly include sensitive military facilities, civil defense 

facilities, and, most recently, nuclear power plants (e.g., proposed small modular reactors). The 

military facilities include bunkers, missile silos, command and control facilities, and 

communication facilities. The blast-induced shock exerted on an underground structure will 

vary with the type of explosive; the explosive charge weight; whether the explosive charge is in 

direct contact with the ground surface; the distance from the explosion to the underground 

structure (i.e., burial depth); and the characteristics of the underground media, such as the soil 

and/or rock types, presence of different layers of soil, and drainage conditions affecting the 

moisture content and pore water pressure of the underground media. 

The objective of this report is to review and conduct numerical analyses to characterize the 

effects explosions close to the ground surface and in contact with the ground surface have on 

underground structures, with emphasis on the Vehicle Borne Improvised Explosive Devices 

(VBIEDs). The activities include (i) identifying the characteristics and properties of VBIEDs; 

(ii) reviewing the dynamics of explosion propagation through air, underground soil medium, and 

underwater; (iii) identifying the empirical equations for the propagation of the blast-induced 

shock wave through the air, underground soil medium, and underwater; (iv) conducting a limited

finite element parametric study to determine the explosion-generated pressure distribution in the 

underground soil; (v) identifying the empirical equations for evaluating the structural response; 

(vi) identifying structural design standards or guidance for underground structures subjected to 

explosive loads; and (vii) reviewing and identifying the differences between explosive and 

seismic load effects on an underground structure and the differences in the required 

structural design.

Explosives are terrorists’ most popular choice for damaging human lives and property. Although 

military explosives can be illegally obtained and are used by terrorists to make VBIEDs, they are

difficult to obtain, and terrorists often choose to make their own explosives using chemicals and 

precursor materials available to them. This report provides a sample of homemade explosives 

and ingredients used to prepare the VBIEDs. The VBIEDs use a vehicle as the package, 

container, and means of delivery. The types of vehicle used for the detonation of VBIEDs 

determine the aboveground height of detonation. The U.S. National Counterterrorism Center 

has identified the trinitrotoluene (TNT)-equivalent explosive capacities for different vehicle sizes. 

The maximum TNT-equivalent explosives’ holding capacities in U.S. vehicles range from 227 to 

27,215 kg [500 to 60,000 lb].

Because the VBIEDs are conventional weapons, the phenomena of the air and surface 

explosions of the VBIEDs, propagation of their dynamic waves through air, crater formation, and 

propagation through subsurface media are similar to those of conventional military and 

commercial weapons. The magnitude and distribution of the dynamic loads on the underground 

structures created by near-ground surface detonation of VBIEDs are affected by the strength of 

the explosive charge and its distance from the buried structure, the properties of the soil and 

rock between the detonation point and the structure, and the orientation of structural members 

with respect to the direction of propagation of the dynamic waves. This dynamic load on the 

underground structures will also be affected by engineered site-specific conditions, such as the 

presence of an explosion-protected concrete slab, a rock barrier, or a surface building floor slab 

between the point of detonation and the underground structures. 
xii

Empirical equations are available for calculating explosion-induced effects in the air, on the 

ground surface, and under water and are primarily based on the assumption of a 

spherical-shaped explosive charge. The empirical equations are used for calculating important 

parameters that affect the underground structures, such as explosion wave arrival time at a 

given location; duration of the explosion wave; peak free-field pressure; impulse force; and 

particle displacement, velocity, and acceleration. Only a limited amount of test data is available 

to predict the pressures on the underground structures primarily from conventional weapons. 

The empirical and experimental information is aided by various numerical analysis techniques,

with appropriate assumptions, idealizations, and simplifications that have been implemented in 

various restricted-use codes (i.e., defense related and commercially available computer codes,

such as ABAQUS and LS-DYNA).

Empirical equations are also available for the analysis and design of underground concrete and 

steel structures that are subjected to the surface and subsurface explosion-generated pressure 

on the soil that surrounds the underground structures. These empirical equations are based on 

experimental investigations and aided by analytical investigations. These empirical equations 

provide a means to calculate forces imposed on structures, including bending moment and 

shear capacity, beam-column effects, energy absorption capacity under shock loads, cracking, 

and spall of concrete components. 

A finite element parametric study has been conducted on the subsurface effects from a surface 

burst. A Lagrangian approach was used with the blast loading calculated using the 

Conventional Weapons (CONWEP) algorithm available in ABAQUS/Explicit. The size of the 

TNT-equivalent charge was varied, and the resulting soil pressures and apparent crater size 

were predicted. Charge weights varied from 45.3 to 27,216 kg [100 to 60,000 lb]. For the case 

of a 100-kg [220-lb] TNT-equivalent charge, the finite element soil pressures correlated well with 

the pressures calculated from empirical relationships. The finite-element-predicted apparent 

crater dimensions were within the range of the empirically derived dimensions for most of the 

TNT-equivalent charge sizes considered. For the larger charge sizes {4,536; 9,072; 18144; and 

27,216 kg [10,000; 20,000; 40,000; and 60,000 lb]}, the predicted apparent crater radius was 

just below the empirically calculated lower limit. However, for these larger charge sizes,

conventional Lagrangian analysis results in excessive mesh distortion. Thus, for these larger

explosive charges it was necessary to use the element deletion technique with a specified soil 

failure criterion. It was recommended that additional parametric analyses be performed to 

further investigate the choice of the soil failure criterion. The Lagrangian element-to-particle 

conversion technique available in ABAQUS/Explicit was also investigated. However, it was 

determined that element-to-particle techniques in ABAQUS/Explicit are currently not possible 

due to documented code errors. Finally, an Eulerian analysis was performed for a 454-kg 

[1,000-lb] explosive charge, which predicted an apparent crater depth also within the range 

calculated using empirical equations of Gould (1981). However, this was less than the crater 

depth the Lagrangian analysis predicted, which used the CONWEP algorithm to calculate the 

blast pressure. The smaller apparent crater depth may be the result of the Eulerian analysis 

underpredicting the ground surface blast loading pressure. It was recommended that a finer 

mesh be used in modeling the air domain.

The design codes and guidelines the military developed constitute the principal guidance 

documents for the design of underground structures subjected to dynamic forces generated by 

explosions in air, ground surface explosions, and underground detonation of high explosive 

charges. They use concepts and provisions of other specific codes, standards, and design 

manuals commonly used in the design of reinforced concrete, prestressed concrete, masonry, 

and steel structures.
xiii

An underground structure may be subjected to both explosive loads and seismic loads and 

should be designed to perform its intended function considering these loads. The energy 

source for the explosive load is quite different from that of the seismic load, although both of 

them produce P-waves, S-waves, and surface waves. Despite similarities, explosions result in 

higher P/S ratios than similarly located earthquakes, making the energy spectrum very different. 

In addition, there may be a significant difference in the magnitude of spectral quantities. As a 

result, one spectrum will not reasonably envelope another spectrum. However, an approach 

similar to that of combining internal accidental missile load with other loads, as specified in the 

commonly used concrete and steel design codes, will be a reasonable approach for combining 

the external explosive load with other loads for the design of underground structures.

xv

ACKNOWLEDGMENTS

The activities reported here were performed on behalf of the U.S. Nuclear Regulatory 

Commission (NRC), Office of New Reactors. This report is an independent product of the 

Center for Nuclear Waste Regulatory Analyses (CNWRA®) and does not necessarily reflect the 

views or regulatory position of NRC. The authors would like to thank L. Howard for the technical 

review and W. Patrick for the programmatic review. The authors would like to acknowledge 

the input provided by S. Hsiung and A. Ghosh to this report. The authors also appreciate 

L. Naukam for providing word processing support and L. Mulverhill for editorial support in 

preparation of this document. 

QUALITY OF DATA, ANALYSES, AND CODE DEVELOPMENT

DATA: All CNWRA-generated original data contained in this report meet the quality assurance 

requirements described in the Quality Assurance Manual. Data used in this report are primarily 

obtained from other sources. Each data source is cited in this report and should be consulted 

for determining the level of quality for those cited data. The work presented in this report is 

documented in Scientific Notebook 1160E (Wilt, 2013).

ANALYSES AND CODES: CNWRA conducted finite element analyses using the commercial 

computer code ABAQUS/Explicit Version 6.12 (Dassault Systèmes Simulia Corp., 2012b). 

ABAQUS/Explicit Version 6.12 (Dassault Systèmes Simulia Corp., 2012b) is controlled under 

the software quality assurance procedure Technical Operating Procedure–018, Control, 

Development, and Modification of Scientific and Engineering Software. Spreadsheet 

calculations were accomplished using Microsoft® Excel® (Microsoft Corporation, 2012).

References

Dassault Systèmes Simulia Corp. “ABAQUS Analysis User’s Manual, Version 6.12.”

Providence, Rhode Island: Dassault Systèmes Simulia Corp. 2012.

Microsoft Corporation. “Microsoft Excel 2007.” Redmond, Washington: 

Microsoft Corporation. 2007.

Wilt, T.E. “Characterizing Explosive Effects on Underground Structures.” Electronic Scientific 

Notebook 1160E. San Antonio, Texas: Center for Nuclear Waste Regulatory Analyses. 2013
.
xvii

ACRONYMS

ACI American Concrete Institute

AISC American Institute of Steel Construction

ANSYS

ASCE American Society of Civil Engineers

CONWEP Conventional Weapons

DOD U.S. Department of Defense

DOE U.S. Department of Energy

EFG element free Galerkin

EOS equations-of-state

EPIC Elastic-Plastic Impact Computations

ERDA Energy Research and Development Administration

GM General Motors

HOB height-of-burst

IED Improvised Explosive Device

JWL Jones-Wilkens-Lee

LSTC Livermore Software Technology Corporation

MDOF multiple-degree-of-freedom

PETN pentaerythritol tetranitrate

PP peak pressure

PTX Picatinny Ternary Explosive

RDX Cyclotrimethlenetrinitramine

SDOF single-degree-of-freedom

SPH smoothed particle hydrodynamics

SSI soil-structure interaction

TNT trinitrotoluene

UFC Unified Facilities Criteria

VBIED Vehicle Borne Improvised Explosive Device

1-1

1. INTRODUCTION

1.1 Background 

The dynamic waves generated in the subsurface media by surface explosions and underground 

explosions may exert dynamic loads on the underground structures and severely damage them. 

An explosion above but near the ground surface produces overpressure in the air that is 

transmitted underground through refraction at the ground surface. Explosions at the ground 

surface and underground near the ground surface couple more directly, and may produce a 

crater and then propagate dynamic waves through the underground media. Baker (1973),

American Society of Civil Engineers (ASCE, 1985), and Cooper (1996) detail propagation of 

waves due to these explosions.

Explosives are the most popular choice of terrorists to harm or kill people and damage property 

(Rostberg, 2005). The characteristics and properties of military and commercial explosives are 

well documented (Department of Army, 2008; Department of Navy, 1947; Cooper, 1996). As 

military explosives become more difficult to obtain, terrorists may choose to make their own 

explosives using chemicals and precursor materials available to them. The conventional 

homemade explosive devices that are used in terrorist attacks are called Improvised Explosive 

Devices (IEDs). The IEDS fall into three categories: (i) Vehicle Borne Improvised Explosive 

Device (VBIED), (ii) Package-Type IED, and (iii) Suicide Bomb IED (U.S. National 

Counterterrorism Center, 2006). VBIEDs use a vehicle as the package, container, and means 

of delivery.

In general, VBIEDs are detonated near the ground surface. The types of vehicles used to 

convey VBIEDs to the target location determine the aboveground height of detonation. Thus,

the damage delivered by a VBIED explosion on an underground structure depends on the 

resulting air blast of the VBIED, propagation of blast waves through air, crater formation, and 

propagation of waves through subsurface soil and rock. 

The magnitude of the dynamic load on the underground structures created by near-surface 

(above ground) detonation of VBIEDs is affected by the strength of the explosive charge and its 

distance from the underground structure; the properties of the soil and rock between the 

detonation point and the structure; and the orientation of structural members with respect to the 

direction of propagation of the dynamic waves. This dynamic load on the underground 

structures also are affected by engineered site-specific conditions, such as the presence of an 

explosion protection concrete slab, a rock barrier, or a surface building floor slab between the 

point of detonation and the underground structures (DOD, 2008, 1986).

The VBIEDs are conventional weapons. The phenomena of the air and surface explosions of 

the VBIEDs, propagation of their dynamic waves through air, crater formation, and propagation 

through subsurface media are similar to those of conventional military weapons and commercial 

explosives (Baker, 1973; Cooper, 1996). The explosive loads generated from the detonation of 

nuclear weapons (ASCE, 1985) also are discussed herein to the extent applicable for VBIEDs.

Various investigators have developed empirical equations for calculating explosion-induced 

effects in the air, on the ground surface, in the subsurface, and under water (ASCE, 1985; 

Baker, 1973; Brode, 1954; Cooper, 1996; Larcher, 2007; Kinney and Graham, 1985;

Krauthammer and Altenberg, 2000; Smith and Hetherington, 1994). These empirical equations 

are based primarily on the assumption of a spherically shaped explosive charge. However, 

limited empirical equations also are available for non-spherical explosive charges. The 
1-2

empirical equations are for calculating important parameters that affect the underground 

structures, such as explosion wave arrival time at a given location; duration of the explosion 

wave; peak free field pressure; impulse force; and particle displacement, velocity, and 

acceleration. Only a limited amount of test data is available to predict the pressures on the 

underground structures primarily from conventional weapons (DOD, 2008). Empirical and 

experimental information is supported by various numerical analysis techniques, with 

appropriate assumptions, idealizations, and simplifications implemented in Department of 

Defense (DOD, 2008) and commercial computer codes (Dassault Systèmes Simulia Corp., 

2012a; LSTC, 2013).

Empirical equations are available for the analysis and design of underground concrete and steel 

structures that are subjected to the surface and subsurface explosion-generated pressure on 

the soil that surrounds the underground structures (DOD, 2008, 2002, 1986; Smith and 

Hetherington, 1994). These empirical equations are based on experimental investigations aided 

by analytical investigations. These empirical equations provide a means to calculate factors 

such as bending moment and shear capacity of various structural components, beam-column 

effects, energy absorption capacity under shock loads, and cracking and spall of concrete 

components. Various computer codes are also available to model underground structures 

subjected to explosive loads (e.g., Dassault Systèmes Simulia Corp., 2012a; LSTC, 2013). 

In addition to possible explosive loads, underground structures may be subjected to dynamic 

waves generated by natural earthquakes. In broad terms, earthquakes occur from the sudden 

release of strain energy caused by ruptures at plate boundaries or at faults. Many investigators 

and authors (e.g., Das, 1993; Kramer, 1996; Gere and Shah, 1984; Newmark and Rosenblueth, 

1971; Kana, et al., 1991) have discussed the propagation of seismic waves through 

underground media. There are important differences between the characteristics of 

explosion induced and earthquake-induced responses of the geologic media, which influence 

the characteristics of the input loads on the underground structures. These result from 

differences in parameters such as source dimension, source time function, source mechanism,

and focal depth (Dahy and Hassib, 2009). Explosive loads to underground structures are 

P-wave dominated short duration loads, whereas earthquakes loads are S-wave dominated 

long duration loads. For underground structures that may be subjected to both explosive 

loads and seismic loads, consideration of the differences between explosion-induced and 

seismic-induced loads on underground structures is important in the design of underground 

structures as discussed in Chapter 6 of this report.

1.2 Objectives

The overall objective of this report is to identify and evaluate potential effects on underground 

structures of a VBIED explosion that is assumed to occur in air near or on the ground surface. 

This report describes the characteristics of VBIEDs; dynamics of explosion propagation through 

air, underground soil medium, and underwater; computational methods to quantify blast-induced 

pressures; empirical relationships and numerical analyses to calculate soil pressure distribution 

and estimate crater formation; computational methods to evaluate underground structural 

response; and relevant design standards and guidance applicable to underground structures. 

This report also provided numerical determination of the “zones of influence” in the underground 

medium due to air and surface explosions of different magnitudes. The explosive and seismic 

loads to which an underground structure may be subjected are compared. The report includes 

a qualitative assessment of the design of an underground structure that will need to be done for 

(i) two individual design spectra (one for explosive load and the other for seismic load) or 

(ii) only one of these two design spectra that may envelope the other spectrum.
2-1

2. EXPLOSIVE EFFECTS

The physical phenomena and resulting effects of detonation of high explosive charges in air 

near the ground surface and at the ground surface are reviewed and summarized to provide a 

basis for evaluating potential threats to shallow buried underground structures. The discussion 

emphasizes Vehicle Borne Improvised Explosive Devices (VBIEDs). The mechanism of 

propagation of blast waves caused by conventional weapons (i.e., high explosives) is examined 

using available information from the literature. Because of the limited amount of information on 

this topic, publicly available information related to nuclear explosions also is considered (ASCE, 

1985). Nuclear-explosion information is included because the shock-wave propagation 

mechanism is similar to that for conventional weapons (Baker, 1973).

For purposes of this discussion, a surface burst is associated with detonation of an explosive 

charge that is in contact with the ground. An air burst is a detonation in which the explosive

charge is not in contact with the ground surface. The discussion includes transmission of 

explosion-generated pressure waves through air and underground media, as well as the 

analytical tools available for modeling explosive yields and explosive effects in air, underground, 

and in water. The available information summarized here is used to characterize the physical 

phenomena of surface and air burst effects on underground structures.

2.1 Fundamentals of Explosives 

Explosives are the most popular choice of terrorists for inflicting damage to infrastructure and 

taking lives (Rostberg, 2005). Explosive materials may be solids, liquids, or gases (DOD, 

2008). Detonation occurs when the rate of reaction exceeds the speed of sound in the 

explosive material, creating a shock wave and rapid release of energy (Baker, 1973), which 

proceeds through the explosive material at a supersonic speed, called the detonation velocity. 

The blast pressure produced by an explosion depends on the physical and chemical 

characteristics of the explosive materials; the methods and procedures used in manufacturing, 

storing, and handling of the explosive; and the properties of the medium in which it explodes 

and/or propagates.

2.1.1 Vehicle-borne Improvised Explosive Devices

Conventional explosive devices used in terrorist attacks are called Improvised Explosive 

Devices (IEDs). An IED is a homemade device that is usually unique in nature because its 

builder has had to improvise by creating it with the materials at hand. These materials could be 

explosives alone or used in combination with toxic chemicals, or biological or radiological 

materials. IEDs can vary in size and have different functioning methods, containers, and 

delivery methods. IEDs fall into three categories: (i) VBIED, (ii) Package-Type IED, and 

(iii) Suicide Bomb IED (U.S. National Counterterrorism Center, 2006). In general, IEDs become 

more difficult to detect and protect against as they become more sophisticated.

The most common materials used for making IEDs, particularly in battlefield situations or where 

access otherwise is available, are military munitions. Thefts from U.S. military bases and 

depots are potential sources, as are illegal cross-border transfers. As military explosives 

become more difficult to obtain, terrorists make their own explosives from available chemicals 

and precursor materials. Commonly available agricultural, mining, and civilian munitions 

manufacturing are readily available sources. Chemical precursors to explosives have wide 

variety of uses and include such chemicals as acetone, ammonia, glycol, nitric acid, peroxide, 
2-2

butane, benzene, ether, glycerin, iodine, methane, sulfuric acid, urea and certain highly reactive 

metals (Rostberg, 2005). Explosives derived from these common chemicals are relatively easy 

to manufacture. Table 2-1 provides a sample of homemade explosives and ingredients (Nardin, 

2005). Table 2-2 contains information on characteristics of U.S. military and commercial 

explosives (Cooper, 1996; Department of Army, 2008; Department of Navy, 1947). 

Table 2-1 Homemade Explosives and Ingredients*

Explosive Synonyms Raw Ingredients

Nitroglycerin Nitroglycerine; trinitroglycerin; 

glyceryl trinitrate. 

Product may be mixed with ethylene 

glycol dinitrate and stored chilled 

until ready to use. 

Glycerin is slowly added to a 

50:50 mix of concentrated nitric and 

sulfuric acid under controlled 

temperature conditions; wash with 

sodium bicarbonate solution. 

Potassium nitrate may be used 

instead of nitric acid. 

Acetone peroxide Tricycloacetone peroxide 30% (preferred) or 3% hydrogen 

peroxide; acetone; sulfuric or 

hydrochloric acid.

Methyl ethyl ketone 

peroxide 

MEKAP Methylethylketone, 30% hydrogen 

peroxide, sulfuric or hydrochloric acid.

DDNP 4,6-Dinitrobenzene-2-diazo-1-oxide; 

diazodinitrophenol 

Picric acid, potassium or sodium 

nitrite; more than 85% sulfuric acid, 

sodium hydroxide, sulfur.

Silver acetylide Double salts 

Ag2C2

.

Ag2NO3

Silver metal; acetylene or calcium 

carbide+water; 70% nitric acid; 

alcohol.

Copper acetylide Copper sulfate; sodium hydroxide; 

acetylene or calcium carbide + water.

Trimercury chlorate 

acetylide 

Chloate-trimercury acetylide Mercuric nitrate; sodium chlorate; 

acetylene or calcium carbide + water.

HMTD hexamethylenetriperoxidediamine Hydrogen peroxide; hexamine; citric 

acid. Alternative procedure uses 37% 

formaldehyde solution, 3% hydrogen

peroxide, and ammonium sulfate.

Lead azide Sodium azide (also an explosive); 

dextrin; sodium hydroxide; 

lead nitrate.

Sodium azide 85% hydrazine hydrate; butyl nitrite or 

isopropyl nitrite; ethyl alcohol; sodium 

hydroxide.

TACC Tetraaminecopper chlorate Ammonium hydroxide, copper sulfate, 

sodium chlorate, alcohol.

Lead picrate Picric acid, lead monoxide, methanol 

Mercury fulminate Mercury metal, 70% nitric acid, 

ethanol.

Nitrated milk 

powder 

Milk booster; casein nitrate Milk; 70% nitric acid; conc. Sulfuric 

acid; vinegar; baking soda.

Nitromannite Mannitol hexanitrate Mannose; conc. nitric acid; conc. 

sulfuric acid; ethanol; sodium chloride; 

sodium bicarbonate (baking soda).
2-3

Table 2-1 Homemade Explosives and Ingredients* (continued)

Explosive Explosive Explosive

Armstrong’s 

explosive 

Chlorate impact explosive Mix almost any chlorate salt 

(e.g., potassium chlorate) with sulfur 

and then add red phosphorous while 

very wet; mixture when dry explodes 

upon touch.

Diaminesilver 

chlorate 

Silver nitrate; sodium chlorate; 

25% ammonium hydroxide; produces 

shock-sensitive dark crystals 

Potassium chlorate 

primer 

Friction primer Mix potassium chlorate with antimony 

sulfate, wet, in 5% gum Arabic 

solution; some preparations add sulfur 

and ground glass, sometimes also 

calcium carbonate and/or 

meal powder.

Nitrogen trichloride Chloride of azode Ammonium chloride or ammonia and 

hydrogen chloride or mix ammonium 

hydroxide with hydrochloric acid; 

carbon or lead rods from battery; 

electrolyze solution using battery 

charger; nitrogen trichloride explodes 

above 60 oC or on shock or in contact 

with dust or organic material.

Nitrogen triiodide Iodine crystals; conc. ammonium 

hydroxide; nitrogen triiodide crystals 

will settle in mixture. Very unstable 

explosive.

Silver fulminate 70% nitric acid, silver metal, ethanol.

Mixture of 

potassium nitrate, 

potassium 

carbonate 

Yellow powder If mixture is heated resulting in melting 

especially in a metal container or with 

trace metal salts it may detonate.

Lead nitroanilate Lead nitrate; sodium nitrite; salicylic 

acid; hydrochloric acid; potassium 

chlorate; ethanol. Easily detonated.

Nitrogen sulfide Nitrogen sulfide may be mixed with 

dry potassium chlorate 

Sulfur; chlorine (generated from 

bleach or swimming pool chemicals); 

hydrochloric acid; oil; sodium chloride; 

manganese dioxide; benzene; 

anhydrous ammonia. 

*Nardin, J. “Homemade Explosives.” Laramie, Wyoming: Arista Tek, Inc. 2005.

Table 2-2 Characteristics of U.S. Explosives*

Name Applications

Detonation Velocity

RE Factor

Heat of Detonation

m/sec ft/sec MJ/kg ft-lb/lb

Ammonium 

nitrate

Earthmoving 2,700 8,900 0.42 — —

Nitroguanidine Gun propellants 7,437 24,400 — — —

PETN Detonating cord, 

blasting caps, 

demolition charges

8,300 27,200 1.66 6.90 2.31 × 106
2-4

Table 2-2 Characteristics of U.S. Explosives* (continued)

Name Applications

Detonation Velocity

RE Factor

Heat of Detonation

m/sec m/sec MJ/kg ft-lb/lb

RDX Blasting caps, 

composition 

explosive

8,350 27,400 1.60 6.78 2.27 × 106

Torpex Torpedo war 

heads, depth 

bombs, mines

7,315 24,000 — — —

TNT Demolition charge 

composition 

explosive

6,900 22,600 1.00 5.90 1.97 × 106

Tetryl Booster charge 

composition 

explosive

7,100 23,300 1.25 6.32 2.11 × 106

Haleite EDNATOL 

constituent

7,620 25,000 — — —

Nitroglycerin Commercial 

dynamite

7,700 25,200 1.50 — —

Black powder Time fuse 400 1,300 0.55 — —

Amatol 80/20 Bursting charge 4,900 16,000 1.17 — —

Composition A3 Booster charge 

Bursting charge

8,100 26,500 — — —

Composition B Bursting charge 7,800 25,600 1.35 6.44 2.15 × 106

Composition 

C4 (M112)

Cutting charge 

Breaching charge

8,040 26,400 1.34 6.65 2.22 × 106

Mercury 

fulminate

Blasting cap, 

Primer

5,029 16,500 — — —

Composition H6 Cratering charge 7,190 23,600 1.33 — —

Ammonium 

nitrate

Cratering charge 2,700 8,900 0.42 — —

Tetrytol 75/25 Demolition charge 7,000 23,000 1.20 — —

Pentolite 50/50 Booster charge 

Bursting charge

7,450 24,400 — 6.40 2.14 × 106

M1 dynamite Demolition charge 6,100 20,000 0.92 — —

Detonating cord Priming, 

demolition charge

6,100 to 

7,300

20,000 to 

24,000

1.66 — —

Sheet explosive Cutting charge 7,300 24,000 1.14 — —

Bangalore 

torpedo, M1A2

Demolition charge 7,800 25,600 1.17 — —

Picratol Bomb M103 6,972 22,875 — — —

Shaped charges 

M2A3, M2A4, 

and M3A1

Cutting charge 7,800 25,600 1.17 — —

PTX-1 Shaped charges, 

bombs, demolition 

blocks, grenades, 

mines

7,376 24,200 — — —
2-5

VBIEDs are a class of IED that uses a vehicle as the package, container, and means of delivery

for the explosive. The VBIEDs come in all shapes and sizes, depending upon the type of 

vehicle used (e.g., compact sedans to semi-trailers) (U.S. National Counterterrorism 

Center, 2006). 

For explosives, it is common to express the explosive energy as an equivalent weight of 

trinitrotoluene (TNT) for the detonating materials by relating the explosive energy of the 

“effective charge weight” of those materials to that of an equivalent weight of TNT (DOD, 2008). 

In addition to the energy output, other factors may affect the equivalency of material, such as 

shape, the number of explosive items, explosive confinement, and the pressure range being 

considered. The energy-equivalency-based equation could be obtained in various ways. A 

commonly used equation for unconfined detonation is given by (Cooper, 1996; DOD, 

2008, 2002).

WE = 

HHEEEEEE

HHTTTTTT

WEXP (2-1)

where

WE = effective charge weight

WEXP = weight of the explosive in question

HEXP = heat of detonation of explosive in question

HTNT = heat of detonation of TNT

The energy output per unit weight of an explosive depends on the relative amount of oxygen 

present in that explosive (Cooper, 1996, Section 2.3). An explosive produces the maximum 

energy output per unit weight if it is exactly oxygen balanced in stoichiometric terms, neither rich 

nor lean. For example, the chemical composition of TNT is CH3N3O6 and its overall reaction 

formula is C7 H5N3O6, giving the TNT a negative oxygen balance of about 74 percent. This is

Table 2-2 Characteristics of U.S. Explosives* (continued)

Name Applications

Detonation Velocity

RE Factor

Heat of Detonation

m/sec m/sec MJ/kg ft-lb/lb

PTX-2 Shaped charge, 

main charge for 

fragmentation 

ammunition, 

booster

7,986 26,200 — — —

Binary mix, 

sodium 

perchlorate, and 

aluminum 

powder

FPE main charges 4,000 13,100 1.60 — —

PETN–pentaerythritol tetranitrate

PTX–Picatinny Ternary eXplosive

RDX–Research Department eXplosive

TNT–trinitrotoluene

*Cooper, P.W. Explosives Engineering. Hoboken, New Jersey: John Wiley & Sons, Inc. 1996.

Department of Army. “Explosives and Demolitions.” FM 3–34.214 (FM 5–250). Washington, DC: Department 

of Army. 2008. <http://info.publicintelligence.net/USArmy-Explosives.pdf> (April 16, 2013).

Department of Navy. “U. S. Explosive Ordnance.” OP 1664. Washington, DC: Bureau of Ordnance Publication. 

May 28, 1947.
2-6

compared to cyclotrimethlenetrinitramine (RDX), which has a chemical composition of C3H6N6O3

with an overall reaction formula of C3H6N6O6, giving it a negative oxygen balance of about 22 

percent (Cooper, 1996, Section 2.4). Because RDX has less negative oxygen balance than 

TNT, it has an effective charge weight, wE, of about 1.60, (i.e., energy output per unit weight of 

RDX is 60 percent more than that of TNT). Table 2-2 provides detonation velocity and effective 

charge weight of a sample of U.S. explosives (Department of Army, 2008; Department of Navy, 

1947; Cooper, 1996).

The U.S. National Counterterrorism Center has identified the TNT-equivalent explosive 

capacities for different vehicle sizes. The U.S. National Counterterrorism Center has a Bomb 

Threat Stand-Off Chart (U.S. National Counterterrorism Center, 2006), which is provided in 

Figure 2-1. This chart presents TNT-equivalent capacities based on the maximum weight of 

explosive material that could reasonably fit in a similarly sized container. Some of the 

containers identified in the table include a typical pipe bomb, briefcase/suitcase bomb, and six 

different vehicle types.

For the vehicles listed in Figure 2-1, the maximum TNT-equivalent explosives’ holding 

capacities range from 227 to 27,215 kg [500 to 60,000 lb]. 

Figure 2-1 U.S. National Counterterrorism Center Bomb Threat Stand-Off Chart 

(U.S. National Counterterrorism Center, 2006)
2-7

The type of vehicle used for VBIED detonation determines the height at which explosives 

detonate above the ground, making it an important parameter for explosive effects on 

underground and surface structures. For compact cars, the aboveground heights of vehicles 

vary from 56.5 to 58.1 cm [22.2 to 22.9 in] depending on the vehicle make and model. The 

vehicle height for sedans varies from 55.9 to 58.9 cm [22.0 to 23.2 in], and passenger vans may 

range from 67.4 to 96.1 cm [26.5 to 37.8 in]. The floor heights of vehicles are not readily 

available; however, they can be easily measured. With the vehicle height and floor height 

known, aboveground blast height may be approximated.

2.2 Dynamics of Explosion Propagation in Air and Underground Media

VBIED detonation in air near the ground or at the ground surface, and transmission of explosion 

waves through air and underground media are evaluated to estimate the pressures generated. 

This evaluation includes the phenomena of blast overpressure in air and the hemispherical 

shock front that reflects from the ground and refracts into the ground.

Two hypothetical case examples are considered here to the dynamics of a VBIED explosion. 

The first case is where the explosive is surrounded by open air. The second case is when the 

explosive is encased in a structure, providing a measure of confinement. 

In the first case, if the detonating explosive is surrounded by open air, the hydrodynamic 

expansion of the explosive device generates debris and heated air produces a strong shock 

wave, expanding the surrounding air. The detonation wave propagates out onto the 

surrounding air as an intensive shock or blast wave and is driven by the expanding hot gases, 

which were the explosive material. As the blast wave expands, it decays in strength, lengthens 

in duration, and slows down, both because of spherical divergence and because the chemical 

reaction is over, except for afterburning, as the hot explosion products mix with the 

surrounding air.

If the explosive is encased and the explosive energy is greater than the mechanical integrity of 

the casing material, it will drive the casing material outward at high velocity until the casing falls 

into fragments. The high pressure gases then vent out past the casing fragments and again 

drive a strong blast wave into the surrounding atmosphere.

The single most important parameter for determining air blast wave characteristics of explosives 

is the total heat of detonation, which is directly proportional to the total weight or mass of the 

explosive. Each explosive has a specific heat of detonation per unit weight or mass.

An air blast produces overpressure in the air as shown in Figure 2-2. An overpressure is 

defined as the difference between the transient pressure and the ambient pressure (surrounding 

atmospheric). The maximum pressure that occurs at a location is defined as the peak 

overpressure and is one of the values used to evaluate the response of a structure to an 

explosive event (Baker, 1973). As the blast wave arrives at a given point, the overpressure 

rapidly increases from zero to the peak overpressure. The main characteristics of the pressure 

wave or pulse in Figure 2-2 are

• Arrival time ttaa: time taken by the pulse to reach the location under consideration

• Peak pressure ppmm mm: maximum pressure of the pulse which is reached with extremely 

fast rise time and then decreases until it reaches the atmospheric pressure, p0 (also 

called side-on pressure)
2-8

Figure 2-2 Variation of Air Burst Overpressure at a Given Location in Air With Time

• Positive phase duration ttdd

+: time when the pressure is above atmospheric pressure 

• Negative phase duration ttdd

−: time when the pressure is below the atmospheric pressure

• Impulse ii: time integral of pressure, normally limited to the positive phase of 

the pressure

As an air burst progresses, the surrounding air is compressed and accelerated away from the 

point of detonation. The time of arrival of the blast front at a given location in air depends 

primarily upon the explosive yield and the distance from the point of burst (ASCE, 1985). The 

duration of the blast wave at that location in the air is characterized by two distinct phases: the 

positive and the negative phases, as shown qualitatively in Figure 2-2. During the duration of 

the positive phase, the blast wave overpressure rises very rapidly from ambient to peak value, 

subsides more slowly to ambient pressure, and further subsides to the negative phase.

During the negative phase, a partial vacuum is created, sucking the air toward the initial source 

of the explosion [i.e., the flow is directed toward the point of detonation (ASCE, 1985)]. As in 

the case of peak overpressure, underpressure peak values also decrease with distance from 

the explosion, but at a slower rate. The peak values of underpressure are usually much smaller 

than during the positive phase, but may be important for the design of some structural 

components. A more detailed quantitative discussion of blast wave propagation in air appears 

in Section 2.4.

Although it is important to understand the phenomenology of explosive effects propagating 

through air, their progression into the soil surrounding an underground structure is more 

complex and more important as a potential threat. When an incident air blast wave encounters 

denser medium, it is reflected partially back into the air, as shown in Figure 2-3 (Glasstone and 

Pmax

, P0

ta
2-9

Figure 2-3 Reflection of Wave at the Earth’s Surface in an Air Blast; t1 to t4 Represent

Successive Times

Dolan, 1977), and refracted partially into the ground. Figure 2-3 shows four stages of the 

outward motion of the spherical blast wave originating from an air burst. In the first and second 

stages, the wave front has not reached the ground, and in the third and fourth stages, the 

pressure wave has interacted with the ground, producing a reflected wave, indicated by the 

dashed lines. The reflected blast wave would produce an instantaneous overpressure at the 

surface more than twice the peak overpressure of the incident wave for an ideal surface. This 

reflected overpressure decreases rapidly with time, as do the associated blast effects and 

ground responses.

The portion of the wave refracted into the ground produces Body and Raleigh waves that travel 

into the medium. Depending on overpressure conditions, proximity to the surface, and 

properties of the medium, the refracted wave could also create a crater. The mechanism of 

crater formation and potential effects on the underground medium are briefly reviewed as a 

foundation for identifying important parameters affecting the distribution of underground 

stresses, which in turn affect the potential for damage to an underground structure. 

When an explosion occurs at or near the ground surface, the ground shock is transmitted 

through the Earth downward and outward. Depending on material properties and strength of 

the explosion, this may create a crater (ASCE, 1985). For bursts near (above) the surface, 

ground shock waves can be produced in two primary ways. The first mechanism is by direct 

coupling of explosive energy to the ground in the vicinity of the crater, causing shock waves to 

contribute to the crater formation and the plastic zones immediately around the crater. The 

second mechanism is by pressure of the air blast wave as it radiates outward along the ground 

surface. For air bursts (where the explosive is not embedded or emplaced in the underlying 

medium), the air blast pressure is the source of most of the stress on the underground medium 

beyond the crater area (DOD and ERDA, 1977).

Radial propagating 

blast waves 1 to 4

Reflected waves 3’ and 

4’ from ground surface

1 2 3 4

3’ 4’
2-10

Formation of a depression crater is a result of overpressure forces (Figure 2-4). Surface 

material can be removed by being pushed, thrown, and scoured out by the pressure wave 

developed by the air. Some of this material may fall back into the crater, but most of it is 

deposited around the edges to form the lip of the crater or is scattered beyond the crater. The 

primary variables that affect the size and shape of the crater include the yield of the explosive, 

the height or depth of burst, and the underlying medium and its properties (ASCE, 1985).

The shock wave generated by a near-surface blast produces body waves traveling into the 

medium, assuming an elastic half-space, with hemispherical wave fronts and Raleigh waves 

propagating radially outward. These body and surface waves are attenuated by two basic 

mechanisms. The geometric effect (Smith and Hetherington, 1994) occurs because as the 

waves propagate away from the source, the energy is distributed over a large area, reducing its 

effect. The hysteresis effect (Smith and Hetherington, 1994) results from the energy being 

dissipated as a result of plastic deformation of the geologic media. The underground wave 

propagation is discussed quantitatively in Section 2.4.

U.S. Department of Defense (DOD, 1986) notes that the propagation of ground shock in a soil 

medium is a complex function of the dynamic constitutive properties of the soil, the detonation 

products, and the geometry of the explosion. No single soil index, or combination of indices, 

can adequately describe the process in a simple way for all cases (DOD, 1986). For example, 

the water content of the soil significantly influences the propagation of shock waves in cohesive 

soils, particularly if the degree of soil saturation is 95 percent or more. This is because water 

that is typically bound within the skeletal structures of cohesive soils provides a significant 

contribution to the overall stiffness and strength of the soil structure. Wang, et al. (2004) found 

that, with 4 percent air content by volume, the peak pressure (PP) in the soil could be more than 

one order of magnitude smaller than that in water-saturated soils and the peak particle velocity

Figure 2-4 Air-Burst Generated Crater 
2-11

reduces by two to six times. They also found that at relatively high air content in the soil, the 

attenuation relations show a noticeable nonlinear trend for soils. The variation in PPs indicates 

the need to understand soil properties in order to characterize an explosive event. DOD (1986) 

also stated that as soil saturation approached 100 percent, peak stresses and accelerations 

increased significantly in wet clays, clay shales, and sandy clays. Seismic surveys generally 

show a sharp jump in the soil seismic velocity, to more than 1,524 mps [5,000 fps], at the depth 

of a saturated zone.

DOD (1986) pointed out that granular soils with high relative density are generally not as 

strongly influenced by water saturation as are the cohesive soils discussed earlier. This is 

because the stiffness in granular soil is provided by the grain-to-grain contacts in the skeleton 

with only a small contribution by the free water. This has been demonstrated by controlled 

laboratory and field experiments in dense, nearly saturated sands that showed no large 

influence of the pore water on shock wave propagation in granular soils. In contrast, the effects 

of water in sands with low relative densities can produce effects similar to those seen in 

cohesive soils. In sands with low relative densities, the soil skeleton can collapse, and the loss 

of grain-to-grain contacts results in high pore pressures as the sand liquefies.

The shock load on an underground structure caused by detonation of high explosives also is

affected by engineered site-specific conditions, such as the presence of a surface building floor 

slab, a rock barrier, or an explosive protection slab some distance above the underground 

structure (Figure 2-5). The protection slab, building floor slab, or rock barrier will reflect the 

explosion wave propagating through the air, thereby reducing effects on the underground 

structures. These barriers also could prevent an aboveground launched weapon planned for 

underground detonation from penetrating the soil and detonating adjacent to the structure. To 

be effective, barriers should extend beyond the edge of an underground structure to form at 

least a 45 degree angle with the bottom edge of the underground structure (DOD, 2008). To 

mitigate the effects of an underground detonation adjacent to a buried structure due to a 

weapon penetrating the soil at an angle less than 45 degrees (Figure 2-5), the protection slab or 

rock barrier may have to be extended further.

Figure 2-5 Engineered Site-Specific Barriers Above an Underground Structure
2-12

In summary, the attenuation of ground shock in soil is controlled by the irreversible crushing of 

the void volume within a soil matrix by the passage of a stress wave. The volume of the 

air-filled voids is considered as the index for the attenuation of ground motions in cohesive soils, 

whereas in granular soils, relative density is considered as the index for attenuation. Based on 

the previous discussions, soils with high relative density or a low volume of air voids will 

attenuate the ground shock less than low relative density or high-air-void soils.

2.3 Blast Consequences 

This section briefly discusses how a surface blast may affect the structure, personnel, and 

equipment within an underground structure. The amount of damage to the structure will be a 

function of the depth of the structure and the explosive charge size. Because the structure is 

buried, injury to humans due to direct exposure to the air blast pressure wave would occur if 

there is perforation of the structure; however, blast-induced fragmentation (spallation) is also a 

potential hazard to humans. For an underground structure that is directly connected to the 

surface by a tunnel, shaft, stack, or other conduit, some of the air blast pressure and shock 

wave may propagate into the structure. Furthermore, equipment contained within the 

underground structure and attached to either walls or the floor could be affected by ground 

shock depending on the magnitude of the ground shock and the resulting structural vibration. 

2.3.1 Effects on the Structure

From the standpoint of the structure, the most important quantity is the magnitude of the 

stresses that are generated by the ground shock wave (Slawson, et al., 1986). The factors that 

determine the strength of the ground shock are discussed in Section 2.4. Blast-induced effects 

that are important include fire, temperature increase, fragmentation due to spallation, and dust 

(fine particulate blast debris) pose a hazard to personnel.

Ground shock is particularly important for close-in blasts in which an explosive charge is located 

on the surface directly above a buried structure. In this case, the pressure loading is distributed 

over a localized area. The degree of localization depends on the depth of the structure. As will 

be shown in Section 3 of this report, depending on the size of the explosive charge, substantial 

cratering can develop, which could partially exhume or expose the underground structure. The 

localized dynamic pressure loading can cause different concrete failure modes: (i) scabbing, 

which is loss of material from the back face of the wall; (ii) spalling, which is the loss of material 

from the wall front face directly exposed to the blast wave; (iii) penetration, which is the 

formation of a crater in wall front face; and (iv) perforation, which is complete penetration 

through the wall forming a hole. Attempting to quantify or predict these modes of damage by 

analytical methods is extremely difficult (Bangash, 2001). McVay (1988) compiled an extensive 

theoretical and experimental study on spall damage to concrete structures subjected to air blast 

from bare and cased explosive charges. The empirical and experimental data were used to 

estimate whether local damage would occur. McVay (1988) found, however, that the smallscale test damage results were difficult to scale up to actual damage states observed in fullscale tests. In addition, damage predictions based on empirical equations did not always 

predict small-scale test observations well (McVay, 1988). Therefore, from a structural analysis 

perspective, close-in blasts with localized loading are the most challenging to solve. 

Appropriate analytical techniques for evaluating the structural response of a buried structure are 

further discussed in Section 4.2 of this report.
2-13

The shock wave also results in an impulse force, which may have a large magnitude over a very 

short duration. This impulse force, coupled with soil–structure interaction effects, can cause 

significant vibration in the structure.

2.3.2 Effects on Equipment

Blast-induced pressures, motion (vibration), and shock may affect internal equipment. The 

equipment can potentially be subjected to blast pressures due leakage into the structure 

through openings. If the opening is sufficiently small, “jetting” may occur, which results in an 

increased blast pressure and may cause the equipment to fall or tip over (DOD, 2008). 

However, if the equipment is sufficiently attached to the structure, it is typically not affected by 

the increased pressure. Equipment damage can be classified as either temporary or permanent 

failure. Temporary failure typically results only in the disruption of operation of the affected 

equipment for a period of time. Permanent failure is either the actual destruction of equipment 

or a failure that prevents the equipment from performing its intended function over an 

unacceptably long period of time (DOD, 2008). The degree of damage a piece of equipment 

can withstand is typically referred to as its fragility level, which is the amount of acceleration 

(shock) that the equipment can withstand and still perform its intended function. The fragility 

level depends on the piece of equipment and how it is attached to the structure. In addition to 

acceleration, equipment vibration is important especially if the vibration frequency results in 

resonance. However, whether continued vibration of the structure occurs depends on the 

amount of structural damping due to soil-structure-interaction effects. Therefore, to determine 

the equipment fragility level, the natural frequency and damping characteristics of the equipment 

supports/mounting need to be evaluated also (DOD, 2008). The equipment fragility level is 

determined by testing. Testing has found that most commercial mechanical equipment 

can withstand an acceleration level of 3 g {1 g is the gravitational acceleration constant of 

9.80665 m/s2 [32.174 ft/s2

]}. Electronic components are considerably more fragile and can 

withstand acceleration levels of approximately 1.5 g. DOD (2008, Table 1-4) provides a list of 

peak acceleration levels for typical equipment (e.g., light fixtures and pumps). The peak 

acceleration levels range from 10 to 70 g (DOD, 2008). Note that for equipment to withstand 

these acceleration levels, it must be sufficiently attached to the structure and should employ 

dampers at the supports to provide some form of shock isolation to protect the equipment 

(DOD, 2008).

2.3.3 Effects on Humans

For personnel not directly exposed to an unabated air blast shock wave, human tolerance of 

blast effects can be considered relatively high. Air blast effects on humans can be classified as 

primary, secondary, and tertiary (Richmond and White, 1966). Primary effects are from direct 

exposure to the blast-induced pressure wave. Secondary effects are from being struck by 

debris generated from the blast. Tertiary effects occur when the body is thrown by the blast 

wave and the body subsequently impacts other objects (e.g., floor or wall) (Richmond and 

White, 1966). The extent of injury is dependent on the weight and position of the person relative 

to the blast wave and the orientation of the person (i.e., standing, sitting, or prone) (DOD, 2008). 

Other effects are fire and the inhalation of high concentrations of dust (White, 1961).

A critical factor is the duration of the blast pressure increase (i.e., fast rise with short duration, 

“fast-fill” rooms versus slow rise with long duration “slow-fill” rooms) (Richmond and White, 

1966). Humans can tolerate higher pressures for short durations, but injury can occur at much 

lower pressures for long duration pressure increases (DOD, 2008).
2-14

Hirsch (1966) stated that when comparing different types of blast-induced trauma, injury to ears 

is of secondary importance when compared to the potential trauma that can occur in the lungs 

and other organs which have air emboli (gas bubbles) in their vascular elements. This air 

embolism occurs when a damaged lung allows the air bubbles to pass into the circulatory 

system and then pass to the brain or heart, resulting in death (Richmond and White, 1966). 

However, for lung damage to occur, the air blast must strike the chest directly, and the damage 

is caused by the pressure acting on the chest wall and not the pressure entering the lungs 

through the respiratory passage (Richmond and White, 1966). Data presented in DOD (2008, 

Table 1-1) show that lung damage can occur for short duration (3 to 5 ms) pressures between 

207 and 552 kPa [30 and 80 psi]. Death can occur at a threshold pressure of 689 kPa [100 psi], 

and certain death occurs for pressures of 1,379 kPa [200 psi]. By comparison, for long duration 

blast loads, petechial hemorrhaging, which is a mild form of hemorrhaging due to rupture of 

blood vessels in the skin resulting in red or purple spots can occur at pressures of as low as 

69 to 103 kPa [10 to 15 psi]. Thus for long duration loads injury is due to a prolonged 

“squeezing” effect.

Damage to hearing can be quantified in terms of eardrum rupture and temporary hearing loss. 

As shown in DOD (2008, Table 1-1), the threshold for eardrum rupture to occur is 34 kPa [5 psi]; 

at a pressure of 103 kPa [15 psi], eardrum rupture will occur for 50 percent of people exposed to 

this pressure. It is also noted that temporary hearing loss can occur at pressure levels lower 

than 34 kPa [5 psi] depending on whether the blast wave occurs in a normal direction to the 

eardrum (DOD, 2008; DOE, 1981).

Being struck by flying debris is considered a secondary blast effect. The debris is typically 

either fragments of the explosive charge casing or fragments of the structure. For an 

underground structure subjected only to a surface charge, there is a low likelihood that any 

blast-induced fragments could injure personnel. However, if the explosion results in perforation 

of a structural component [e.g., ceiling slab or substantial damage that results in spalling 

(fragmentation) of the concrete], flying debris could be generated.

As mentioned previously, a tertiary effect of blast is the body being thrown about the structure. 

DOD (2008) states that an acceleration of 0.5 g is tolerable for personnel who are standing,

sitting, or prone. If the body is thrown, an impact velocity of 3 m/s [10 ft/s] is considered 

tolerable (DOD, 2008).

2.4 Surface Blast Effects and Ground Shock 

Air burst and surface explosions are dynamic events which result in the generation of ground 

shock, which travels on the surface and through the subsurface. The severity of the ground 

shock highly depends on the amount of coupling between the explosive charge and the ground 

(Smith and Hetherington, 1994). Figure 2-6 shows the different forms of explosions and the 

resulting severity of ground shock. Air bursts, with the exception of nuclear explosions, 

generate moderate to low amounts of ground shock because of limited coupling with the 

ground. Surface bursts have a high severity of ground shock due to the potential for a 

significant amount of coupling between the explosive charge and ground. Buried charges can 

generate the most severe amounts of ground shock. Charges that are “tightly” buried, (i.e., fully 

confined with no surrounding void space), are the most severe, while charges surrounded by 

void, (e.g., in a tunnel), generate a lesser amount of ground shock. Subsurface effects from 

buried charges are not within the scope of this report.
2-15

Figure 2-6 Different Forms of Explosions and Severity of Ground Shock Effects

The primary interest in ground shock is to determine its effects on a buried structure in the form 

of structural loads. Ground shock loading on a buried structure is a function of the following 

variables: (i) explosive charge size, (ii) degree of coupling between the charge and the ground,

(iii) properties of the geologic media, and (iv) the distance from the explosive charge to the 

structure (Smith and Hetherington, 1994).

For an air burst, the amount of energy transmitted as ground shock highly depends on the 

degree of coupling between the explosive charge and the ground and is determined by the 

location of the charge relative to the soil surface; this is referred to as height-of-burst (HOB). In 

empirical expressions, HOB is usually expressed as a function of the radius of an assumed 

spherically-shaped charge. 

Finally, it needs to be noted that a high water content, or a high water table, can potentially lead 

to liquefaction in low-cohesion and cohesionless soils when subject to blast-induced ground

shock (Bretz, 1990). If liquefaction develops, there is the potential for structural instability to 

occur. Numerical studies have investigated instability of surface structures due to blast-induced 

liquefaction using finite element analysis in Wang, et al. (2008). It is well established that 

liquefaction can cause buried structures to displace upward towards the surface.

2.4.1 Analytical Methods for Evaluating Blast-Induced Ground Effects

Analytical methods are available for calculating blast-induced effects in the air, on the ground 

surface, below the ground surface, and underwater. These analytical methods provide a means 

to calculate important quantities affecting underground structures, such as blast wave arrival 

time, time duration of the blast wave, peak free-field pressure, and particle (e.g., soil, 

displacement, velocity, acceleration, and specific impulse force). Most empirical equations are 

based on the assumption of a spherically shaped explosive charge. However, limited empirical 

equations are also available for non-spherical explosive charges (e.g., cylinders, cubes, and

cones). If an explosive charge with a nonspherical shape is detonated, the shock wave 

resulting from the entire charge surface will not enter the surrounding air simultaneously; 

Air Burst

Surface Burst

Buried Charge

Nuclear Explosion – High

Chemical Explosion – Moderate to Low

High

Tightly Buried – Very High

Loose within a Void – Moderate to Low
2-16

consequently, the blast waves will not be spherical. The shape and magnitude of the shock 

wave will be functions of the charge geometry and relative location of initiation of the detonation 

process (DOE, 1981). Therefore, the blast parameters will depend on azimuth and, possibly,

the elevation, as well as the radial distance. Experimental observations from detonation of 

nonspherical but regular-shaped (e.g., cylinders, cubes, cones) explosive charges show that the 

largest blast pressure wave was generated from the charge face which had the largest surface 

area (DOE, 1981). Furthermore, multiple peaks occur in the positive overpressure phase (DOE, 

1981). However, U.S. Department of Energy (DOE, 1981) notes that as the stand-off distance 

increases, the blast wave becomes more spherical.

The following discussion focuses on spherical charges because the numerical studies 

presented in Section 3 use pressure-time histories based on this geometry. Empirical 

relationships presented in this section are compared with the numerical results in Section 3.

2.4.1.1 Blast Wavefront Parameters

Three characteristic parameters of the blast wavefront are the blast wavefront velocity, Us; the 

air density, ρs, behind the wavefront; and the dynamic pressure, qs. These three parameters 

can be calculated from the following expressions given in Smith and Hetherington (1994)

Us= �6Ps + 7P0

7P0

a0 (2-2)

ρs = 6Ps + 7P0

Ps + 7P0

ρ0 (2-3)

qs = 5Ps

2

2(Ps+7P0) (2-4)

where P0 is the ambient air pressure, Ps is the peak side-on overpressure, ρ0 is the air density 

at ambient pressure, and a0 is the speed of sound at ambient air pressure.

It is not feasible to categorize all of the different geologic media and know the in-situ state 

of the geologic media without site-specific assessment. Therefore, dimensional analysis is 

commonly used to obtain a first-order quantification of the effects of explosives (Cooper, 1996). 

Dimensional analysis typically takes the form of so-called “cube-root scaling.” The empirical 

relationships presented in this section for PP, particle velocity, and acceleration are typically 

scaled by W1/3 such that the scaled distance, Z, becomes

Z = R

W1/3 (2-5)

where R is the distance from the charge center and W is the TNT-equivalent charge weight. 

The use of scaling allows test data using different explosive types, geologic media, and in-situ

states to be combined. However, Cooper (1996) states that even with cube-root scaling, a 
2-17

significant amount of scatter is observed in empirically derived constants, which can lead to 

significant uncertainty in results calculated from empirical relationships.

Kinney and Graham (1985) give the side-on overpressure distance relation as 

Ps

P0

=

808 �1+ � Z

4.5�

2

�

�1+ � Z

0.048�

2

�1+ � Z

0.32�

2

�1+ � Z

1.35�

2 (2-6)

Alternatively, Brode (1954) developed overpressure relationships and given by Smith and 

Hetherington (1994)

Ps = 

6.7

Z3 + 1 (for Ps > 0.1 MPa)

Ps = 0.975

Z

+

1.455

Z2 +

5.85

Z3 - 0.019 (for 0.1 < Ps < 1 MPa)

(2-7)

In these expressions, the peak side-on overpressure is considered in the near field when 

greater than 1 MPa [145 psi] and when less than 1 MPa [145 psi], the peak static overpressure 

is considered in the medium and far field.

The complete air blast pressure-time history curve, Figure 2-2, can be determined using the 

modified Friedlander equation as given by Baker (1973)

P(t) = P0 + Ps �1-

t

td

+� e

-

b t

td

+ (2-8)

where the parameter, b [sometimes referred to as the “waveform parameter” (Smith and 

Hetherington, 1994)], relates to pressure decay and is a function of the peak side-on

overpressure Ps (Smith and Hetherington, 1994), and td

+

 is the time duration of the positive 

overpressure phase. The specific impulse, is, is calculated by integrating the area under the 

positive part of the air blast pressure-time history curve

is = � P(t) dt

ta+ td

+

ta

(2-9)

Kinney and Graham (1985) and Larcher (2007) give the following equation to estimate the 

specific impulse (Pa·s)
2-18

is = 100 ∙ 

0.067�1+ � Z

0.23�

4

Z2 ∙ �1+ � Z

1.55�

3 3

∙ √W3(2-10)

Positive Air Blast Pressure Duration td

+

Duration of the air blast pressure wave is one important parameter associated with the damage 

of a structure. Because the positive pressure phase is generally measured more precisely and 

is typically the more damaging phase, the positive air blast pressure phase td

+ is taken as the 

index of the air blast duration (Kinney and Graham, 1985; Larcher, 2007). Kinney and Graham 

(1985) give the following equation for estimating the duration, td

+

, for chemical explosions

td

+

W1⁄3 =

980 �1+ � Z

0.54�

10�

�1+ � Z

0.02�

3

��1+ � Z

0.74�

6

� �1+ � Z

6.9�

2 (2-11)

where td

+ W1/3 ⁄ is the duration of the positive phase in milliseconds for 1 kg [2.2 lb] of TNT 

explosion and Z is the scaled distance in meters scaled to 1 kg [2.2 lb] of TNT.

Negative Air Blast Pressure Pmin and Duration td

˗

The air blast pressure in the negative phase drops below the atmospheric pressure. Although 

for most structural evaluations the positive phase of the air blast pressure is the most important, 

for structural panels (e.g., glass and composite), negative pressures can lead to additional 

failure modes (Krauthammer and Altenberg, 2000). Thus, depending on the structure subjected 

to the air blast, the negative phase may have considerable influence on the structural response 

(Krauthammer and Altenberg, 2000). For example, at scaled distances Z larger than 50, 

Krauthammer and Altenberg (2000) showed that the positive and negative air blast pressures 

can be similar in magnitude. Thus, the negative air blast pressure cannot always be neglected.

The negative air blast overpressure Pmin and its duration td

- are given by Larcher (2007)

 Pmin = �

3.5 × 104

Z for Z > 3.5

104

 for Z < 3.5

(2-12)

and

td

- = �

0.0101∙ W1/3 for Z < 0.3

(0.003125 ∙ log Z + 0.01201) ∙ W1/3 for 0.3 ≥ Z ≤ 1.9 

0.0139 ∙ W1/3 for Z > 1.9

(2-13)
2-19

2.4.1.2 Blast-Induced Ground Motions

Murphy (1981) investigated the near-field ground motion from surface explosions and 

determined that the low-frequency components of the ground motions are Rayleigh waves 

induced by the air blast acting on the ground surface. The amplitude of the Rayleigh wave is 

directly proportional to the cube-root of yield of the explosion and is independent of surface 

geology. In the following discussion, expressions are given to describe the motion of a particle 

both on the ground surface and below the surface.

Surface Ground Motion

Energy from an explosion that occurs near or on the ground surface can result in two forms 

of ground shock effects: air-induced ground motion and direct-induced ground motion 

(DOD, 2008). Air-induced ground shock is caused when the shock wave developed from the 

detonation compresses the ground, generating a stress pulse propagating downward. The 

maximum amplitude of this pulse occurs at the ground surface and attenuates with depth 

unless a strata interface, water table at shallow depth, or other discontinuities are present

(DOD, 2008). Direct-induced ground shock is caused by the explosive energy that is directly 

transmitted through the ground. Direct-induced ground motion has a longer duration than

air-induced ground motion. Expressions to estimate the surface ground motion are 

presented next.

Air-induced Ground Motion

DOD (2008) describes the motion of the ground surface in terms of one-dimensional wave 

propagation theory. Here, the peak vertical velocity, VV, on the ground surface is given by 

VV = Pmax

ρ Cp

(2-14)

where ρ is the mass density of the soil, Cp is the soil compressive wave propagation velocity in 

ft/s, and Pmax is the peak positive air blast pressure in psi. The peak vertical ground surface 

displacement, DV, is given by

DV = 

is

+

1000 ρ Cp

(2-15)

where iis

+ is specific impulse calculated from the positive pressure phase of the air blast 

pressure-time history. Assuming a linear velocity increase during the pulse rise time (taken as 

1 millisecond), the following formula to estimate the peak vertical ground surface acceleration, 

AV, is given by

AV = 100 Pmax

ρ Cp g (2-16)

where g is the gravitational constant. In Eqs. (2-14) to (2-16), the moisture content of the soil is 

accounted for via the soil density, ρ. However, DOD (2008) notes that Eq. (2-15) is only directly 

applicable to dry soils; for saturated soils and rock, it is recommended that the calculated 

acceleration value from Eq. (2-16) be doubled (DOD, 2008).
2-20

The peak horizontal ground surface motions can be estimated using

DH = DV tan �sin-1 �

Cp

12000 U � �� (2-17)

VH = VV tan �sin-1 �

Cp

12000 U � �� (2-18)

AH = AV tan �sin-1 �

Cp

12000 U � �� (2-19)

where U is the velocity of propagation of the shock front (ft/s).

Direct-Induced Ground Motion

The direct-induced ground surface motion also can be estimated using empirical relationships

given in DOD (2008). The peak vertical ground surface displacement, DV, and peak horizontal 

ground surface displacement, DH, for rock medium is

DV = 0.025 ∙ R1/3 ∙ W1/3

Z1/3 (2-20)

DH = 0.5 DV (2-21)

where R is the distance on the ground from the charge center, W is the TNT-equivalent charge 

weight, and Z is the cube-root scaling factor, Z = R/W1/3. For soils, both dry and saturated, DV

and DH are estimated using

DV = 0.17 ∙ R1/3 ∙ W1/3

Z2.3 (2-22)

DH = DV (2-23)

The peak vertical ground surface velocity, VV, and horizontal ground surface velocity, VH for all 

types of soil and rock are 

VV = 150

Z1.5 (2-24)

VH = VV (2-25)

Similarly, the peak vertical ground surface acceleration, AV, for all cases is

AV = 10000

W1/3 ∙ Z2 (2-26)

The peak horizontal ground surface acceleration, AH, is
2-21

AH = 0.5 AV for rock (2-27)

AH = AV for soil (2-28)

Subsurface Ground Motion

As discussed in DOD (1986), the time history for shock stress and particle velocity are typically 

expressed in an exponential form. That is, the magnitude of the free-field shock stress, P(t), at 

a given time t, is 

P(t) = P0e

-α � t

ta

� (2-29)

and the particle velocity, V(t), at a given time t, is 

V(t) = V0[1 - β(t/ta)]e

-β� t

ta

� (2-30)

In Eqs. (2-29) and (2-30), α and β are constants that typically depend on the specific site 

(geologic media); however, DOD (1986) provides values of 1.0 and 0.4 for α and β, respectively. 

Also in Eqs. (2-29) and (2-30), the quantity ta is the arrival time of the shock wave at a specified 

location. This arrival time can be calculated based on the seismic velocity, c, of the geologic 

media and the distance, R, from the explosion source

ta = R

c

(2-31)

Finally, the quantities P0 and V0 are the peak free-field shock stress and velocity, respectively. 

Empirical expressions for both these quantities as well as other ground motion quantities are 

given by DOD (1986) as follows

peak free-field pressure (psi)

P0 = f ∙ (ρc) ∙ 160 ∙ � R

W1/3�

-n

(2-32)

peak particle velocity (ft/s)

V0 = f ∙ 160 ∙ � R

W1/3�

-n

(2-33)

peak acceleration (gs)

a0 W1/3 = f ∙ 50 ∙ c ∙ � R

W1/3�

(-n-1)

(2-34)

peak displacement (ft)
2-22

d0

W1/3 = f ∙ 500 ∙ �

1

c

� � R

W1/3�

(-n+1)

(2-35)

impulse (lb-s2

/in2)

i0

W1/3 = f ∙ ρ0 ∙ 1.1 ∙ � R

W1/3�

(-n+1)

(2-36)

In Eqs. (2-32) to (2-36), f is a coupling factor, (ρc) is the acoustic impedance [psi/(ft/s)], n is an 

attenuation coefficient, c is the seismic velocity (ft/s), and ρ0 is mass density (lb-s2

/ft4). The 

acoustic impedance, seismic velocity, and mass density are for a specific geologic media. 

Typical values are given in DOD (1986, Tables 5-1 and 5-2). The coupling factor is used to 

account for the location of the explosive charge in relation to the ground surface; higher 

amounts of coupling occur for charges that are in contact or buried below the ground surface. 

For example, the coupling factor for an air burst has a constant value of 0.14, while a surface 

charge has a coupling factor of 0.4 for soil (DOD, 1986, Figure 5-3); the coupling factor 

increases to a limiting value of 1.0 as the charge depth increases.

With the parameters defined in Eqs. (2-32) to (2-36), the response of a buried structure can then 

be determined using idealized single-degree-of-freedom (SDOF) or multiple-degree-of-freedom 

(MDOF) analyses, as well as more general dynamic finite element analyses. The theoretical 

basis and numerical algorithms for performing SDOF and MDOF analyses for structural 

response analyses are detailed in Section 4.

2.4.2 Calculation of Subsurface Pressure

Nagy, et al. (2010) provide an expression based on the empirical relations for ground shock, as 

presented in DOD (1986), for the free-field PP

PP = c � R

W1/3�

-n

(2-37)

where R is the distance from the charge center (Figure 2-7), W is the TNT-equivalent charge 

weight, and c and n are empirical constants that are dependent on the soil type. Because of the 

inherent uncertainty in soil parameters, both upper and lower limits are given for c and n 

(Table 2-3) and will be used in the following calculations.

Figure 2-8 shows the calculated PP as a function of distance using Eq. (2-35) from the surface 

explosion for explosive charge weights from 45.3 to 4,530 kg [100 to 10,000 lb]. Figure 2-8

includes the peak pressure upper and lower limits corresponding to the values of c and n given 

in Table 2-3. Upper and lower limits are given because of the inherent uncertainty in the soil 

parameters, making bounding calculations the most appropriate approach. For a 45.3-kg 

[100 lb] charge, the pressure is calculated to dissipate to zero at a depth of approximately 20 m 

[66 ft], and for the 4,530-kg [10,000-lb] charge, the pressure is calculated to dissipate to zero at 

a depth of approximately 90 m [295 ft].
2-23

Figure 2-7 Description of Surface Burst

Table 2-3 Constants for Free Field Peak Pressure*

c n

Upper Empirical Limit 1.12 2.75

Lower Empirical Limit 0.65 2.5

*Values taken from Nagy, N., M. Mohamed, and J.C. Boot. “Nonlinear Numerical Modelling for the Effects of 

Surface Explosions on Buried Reinforced Concrete Structures.” Geomechanics and Engineering. Vol. 2.

pp. 1–18. 2010.

Figure 2-8 Empirical Relationship of Peak Pressure Versus Depth for Different Charge Weights. 

 Explosive Charge Located on the Soil Surface.

Surface Burst

R

Soil

Air

0.0

0.1

1.0

10.0

100.0

1,000.0

10,000.0

0.10 1.00 10.00 100.00

Peak Pressure (MPa)

Distance (depth) (m)

45.3 kg upper

45.3 kg lower

227 kg upper

227 kg lower

453 kg upper

453 kg lower

2268 kg upper

2268 kg lower

4530 kg upper

4530 kg lower
2-24

2.4.3 Calculation of Crater Dimensions

As in ground shock, crater size is primarily influenced by the explosive charge weight, height of 

burst, the type of geologic media (e.g., soil or rock), and the water content in the geologic media 

(Gould, 1981). As a blast wave contacts the ground surface, the ground surface is scoured and 

detonation gases penetrate the surface. The expanding gases cause the formation of ejecta,

which is thrown into the air. Further propagation of the blast wave causes compaction and 

plastic flow in the soil. A rarefaction wave with a reversed particle velocity develops and

produces spalling and more ejecta (Cooper, 1996). The pressure wave in the ground results in 

the development of a plastic slip zone around the crater causing an upheaval of the geologic 

media and formation of a “lip” around the crater.

Gould (1981) notes that crater size can vary widely from test to test even when the previously 

mentioned four variables appear identical in each test. The deformation response of different 

types of geologic media and the potential uncertainty in the in-situ state make it difficult to 

formulate empirical models for predicting crater formation. Similarly, determining the 

appropriate method to account for this variability in numerical models makes numerical 

prediction of crater size quite challenging.

The apparent crater radius, Ra, in feet, can be approximated using the empirical relationships 

given by Cooper (1996). For an explosive charge located on the ground surface

Ra = (0.46 + 0.027 PCJ)(2ECRW)1/3 (2-38)

where PCJ is the Chapman-Jouguet (CJ) pressure in GPa, ECR is the cratering efficiency, and W 

is the weight of the explosive charge in pounds. The cratering efficiency for a sandy clay soil is 

given as 0.475 (Cooper, 1996, Table 29.1). The sandy clay soil listed in Cooper (1996) is 

assumed to be the closest match to the soil used in the finite element model in Section 3.2.

The Chapman-Jouguet pressure, PCJ, is calculated from Cooper (1996)

PCJ = ρD2

4

(2-39)

where ρ is the density of the unreacted explosive in g/cm3 and D is the detonation velocity 

in km/s.

The apparent crater radius for an explosive charge located above the ground surface can be 

approximated by the empirical relation given by Cooper (1996)

Ra=(0.46 + 0.027 PCJ)(2 ECRW e-1.457∙ HOB)

1/3 (2-40)

where HOB is the ratio of the charge height above the surface to the charge radius. Cooper 

(1996) does not provide any relationships to calculate apparent crater volume or apparent 

crater depth.
2-25

The apparent crater dimensions can also be calculated using the empirical relationship given by 

Gould (1981). Gould (1981) gives an expression for the apparent crater volume, Va,

Va= V0W e(-5.2 H (V0W)-1/3) (2-41)

where V0 is the cratering efficiency of the explosive in ft3/ton when the height of the burst is 

zero, W is the TNT-equivalent charge weight in tons, and H is the height of the explosive above 

the surface in feet. Gould (1981) also gives empirical expressions for the apparent crater radius 

in feet, Ra, and apparent crater depth in feet, Da, as

Ra = 1.2 Va

1/3 (2-42)

Da = 0.5 Va

1/3 (2-43)

Because of the inherent uncertainty in the soil parameters associated with each empirical 

relationship, it cannot be determined which empirical relationship is most accurate. The 

empirical relationships of Cooper (1996) and Gould (1981) will be used to predict crater 

dimensions in Sections 3.4 and 3.5.

2.5 Underwater Detonations 

An underwater explosion will develop two phases of behavior: a shock wave and a bubble 

pulse. These two phases of behavior result in a partition of energy. According to Smith and 

Hetherington (1994), approximately 53 percent of the energy is partitioned in the shock wave 

and 47 percent of the energy is partitioned in the bubble pulse.

The transient shock wave significantly increases fluid velocity. The PP of the shock wave is 

very high, compared to a shock wave generated by a surface explosion, and results in a large 

impulse loading; the duration of the shock wave, however, is very short. Smith and 

Hetherington (1994) provide approximate expressions for the maximum pressure at the 

shock front.

(2-44)

The pressure, Pm, in Eq. (2-44) is calculated in bar (1 bar = 1 × 105 Pa), and Z is in m/kg1/3. 

Analysis and calculation of the shock wave pressure-time history follow a similar procedure as 

for the air blast, although there are physical and quantitative differences in the results due to the 

difference in the surrounding medium (i.e., water versus air).

The bubble pulse is the result of the expansion of the detonation products (i.e., gases) (Smith 

and Hetherington, 1994), with the size and shape of the bubble oscillating over time. At first, the 

bubble radius is larger than what would correspond to hydrostatic equilibrium due to the initial 

expansion and inertia of the hot compressed gases. Subsequently, the bubble pressure 

decreases and the bubble collapses in response to the hydrostatic pressure. As the collapse
2-26

progresses, it recompresses the detonation gas, which eventually leads to a reexpansion of the 

bubble. A series of expansions and contractions continues until eventually the energy is 

dissipated and the bubble rises to the surface or contacts an underwater object. During the 

oscillatory phase, the bubble can take a mushroom-like shape; during contraction, or in the 

extreme case, the bubble has a torus shape with a powerful water jet passing through the

central hole in the torus (Geersb and Hunter, 2002). 

Bubble size depends on weight and composition of the explosive, as well as the depth of 

explosion below the water surface. Smith and Hetherington (1994) give an approximate 

expression for the initial maximum bubble radius, amax

amax = JexW1/3

(H + H0)1/3 (2-45)

where Jex is an empirical constant that depends on the explosive type. For example, for TNT 

Jex is 3.5 m4/3/kg1/3 [13.1 ft

4/3/lb1/3]. H is the charge depth in meters, H0 is atmospheric head in 

meters, and W is charge weight in kg. The duration of the bubble pulse is at least two orders of 

magnitude longer than the shock wave (Geersb and Hunter, 2002); thus, damage to an

underwater structure from the longer duration bubble pulse can be significant (Smith and 

Hetherington, 1994).

The most important aspect of an underwater detonation is its effect on a surface or underwater 

structure. In general, underwater explosions have three potential damaging mechanisms: high 

pressure, a so-called “whipping” effect, and water jet impact (Klaseboer, et al., 2005). High 

pressure is produced by the transient shock wave and can cause damage when it strikes the 

structure. The whipping effect can occur if the frequency of the bubble pulse (expansion and 

contraction) matches the eigen-frequency of the structure that the bubble contacts (Klaseboer,

et al., 2005). Finally, the third damaging mechanism occurs when the bubble moves toward the 

structure and a high-speed water jet forms. If the water jet is in the direction of the structure, 

damage to the structure can occur. As Riley (2010a,b) discussed, damage severity is a function 

of the stand-off distance at which the explosion occurs relative to the structure. For example, if 

the bubble collapses after impinging on a structure, at close-in distances, it can impart a load

more severe than the shock wave (Riley, 2010a,b). This severity may be caused by bubbles

repeatedly reforming and pulsating against the structure causing several loading cycles (Riley, 

2010a,b). Formation of the water jet at a close-in distance would also result in potentially 

severe loading.
3-1

3. NUMERICAL EVALUATION OF SURFACE AND SUBSURFACE

EFFECTS DUE TO AIR AND SURFACE BURSTS

Numerical analyses using the finite element method were conducted assuming a 

spherical-shaped explosive charge and associated pressure-time histories, consistent with the 

empirical formulae presented in Section 2.4. This approach permits direct comparisons 

between the empirical and numerical results and also provides confidence that the numerical 

results are consistent with similar previous research in this topical area.

3.1 Finite Element Lagrangian Model

The finite element analyses used a Lagrangian representation of a soil block, which is loaded by 

a time-dependent surface pressure. Because the analyses focus on soil behavior, particularly 

soil stresses and crater formation, including air in the computational domain was considered 

unnecessary. However, this approach does have a limitation with respect to crater formation. 

Dynamic air blast pressure results in the production of soil ejecta during crater formation. Using 

the Lagrangian approach does not permit capturing this particular soil deformation mechanism. 

As will be shown in Sections 3.4 and 3.5, because the soil remains as a continuum, the ejecta is 

included as part of the “lip” that forms around the crater. Although a Lagrangian approach such 

as the mesh-free smoothed particle hydrodynamics (SPH) could be used to account for ejecta 

by modeling the soil as discrete particles (Liu and Liu, 2003), doing so was beyond the scope of 

this report. However, future analyses could utilize this approach to further refine the crater 

formation predictions using Lagrangian finite elements.

The finite element model in Figure 3-1 shows a block of soil modeled using Lagrangian

three-dimensional solid continuum elements. The soil block represents a

100 × 100 × 50-m [328 × 328 × 164-ft] domain. The dimensions were selected based on 

previous experience [e.g., Wilt, et al. (2012)] so that the soil domain was sufficiently large, 

thereby preventing artificial pressure wave reflection from the domain boundaries.

A sufficiently fine mesh was constructed to allow a reasonable fidelity of the soil pressure 

distribution and crater formation without excessive computation time. The mesh shown in 

Figure 3-1 contains approximately a half-million elements. To capture complex near-field 

phenomena, the mesh was constructed with the finest discretization directly beneath the 

explosive charge.

3.2 Modeling of Soil Behavior

The deformation behavior of the soil was modeled using the Drucker-Prager Cap model 

available in ABAQUS/Explicit (Dassault Systèmes Simulia Corp., 2012a). The Drucker-Prager 

model is appropriate for soil behavior because it accounts for stress path dependence, accounts 

for volume dilatancy, and provides both hardening and softening behavior (Huang and 

Chen, 1990). 

Figure 3-2 shows the yield surface for the Drucker-Prager Cap model. The cap surface controls 

both hardening and softening behavior and is a function of volumetric plastic strain. The cap 

surface limits the yield surface under hydrostatic compression by inelastic hardening, which 

represents plastic compaction of the soil. The cap surface also limits volume dilatancy by 

introducing softening when the soil yields on the shear failure surface. Two key material model 

parameters are β, which is the friction angle, and d, which measures cohesion; both define 
3-2

Figure 3-1 Three-Dimensional Finite Element Model of Soil

hydrostatic pressure-dependent shear failure. Huang and Chen (1990) offer additional 

discussion on determining the Drucker-Prager Cap model material parameters.

A literature search identified suitable material parameters to use in this particular 

Drucker-Prager Cap model. The material parameters for a silty clay soil were identified in 

Nagy, et al. (2010) and are provided in Table 3-1. This particular set of material parameters 

was chosen from Nagy, et al. (2010) because it is consistent with the empirical expressions also 

given in Nagy, et al. (2010) and will allow direct comparison between the empirical and finite 

element results.

3.3 Blast Pressure Loading

The dynamic, time-dependent air-blast loading was calculated using the CONWEP algorithm 

available in ABAQUS/Explicit (Dassault Systèmes Simulia Corp., 2012b). The CONWEP 

algorithm implemented in ABAQUS/Explicit is based on a program the U.S. Army developed to 

calculate conventional weapons effects. The equations used in the CONWEP algorithm are 

based on the Friedlander equation (see Section 2.4). Because CONWEP is an empirical 

method, there is a specific range of validity. The minimum valid distance is equal to the 

explosive charge radius. Therefore, for the parametric study presented in Section 3.5, some 

limitations on the height of burst as a function of the explosive charge weight are imposed and 

will be further discussed in that section.

Location of 

100 m Spherical Charge

50 m
3-3

Figure 3-2 Drucker-Prager Cap Model Yield Surface

Table 3-1 Drucker-Prager Cap Model Parameters for Silty Clay*

Parameter Value

Young’s modulus (E) 51.7 MPa [7.5 ksi]

Poisson’s ratio (ν) 0.45

Density (ρ) 1920 kg/m3 [120 lb/ft3

]

Material cohesion (d) 0.036 MPa [5 × 10−3 ksi]

Material angle of friction (β) 24 degrees

Cap eccentricity (R) 0.3

Initial cap yield surface position (εv) 0.02

Transition surface radius (α) 0.0

Cap hardening behavior

(stress vs. plastic strain)

2.75 MPa [0.4 ksi], 0.00

4.83 MPa [0.7 ksi], 0.02

5.15 MPa [0.75 ksi], 0.04

6.20 MPa [0.9 ksi], 0.08

* Values taken from Nagy, N., M. Mohamed, and J.C. Boot. “Nonlinear Numerical Modeling for the Effects of 

Surface Explosions on Buried Reinforced Concrete Structures.” Geomechanics and Engineering. Vol. 2. 

pp. 1–88. 2010.

Because the analyses in this report focus on the soil behavior (i.e., soil pressures and crater 

formation), the CONWEP algorithm was chosen to generate the air blast, which eliminated the 

need to model the explosive charge and surrounding air environment. This approach 

substantially reduces the computational cost of the analyses.

β

1

Shear Failure 

Surface

d

Hardening 

Cap Surface

Deviatoric 

Stress

Pressure Stress
3-4

The input required for the ABAQUS/Explicit CONWEP algorithm is the trinitrotoluene

(TNT)-equivalent explosive charge weight, the location of the charge, and the target surface 

area to which the air-blast load will be applied.

3.4 Empirical and Numerical Analyses for a 100-kg [220-lb] TNT 

Explosive Charge—Soil Pressures and Crater Formation

A case was considered with a 100-kg [220-lb] TNT explosive charge located 0.5 m [20 in] above 

the soil surface. Soil pressure calculations using empirical expressions and the finite element 

results are compared. Predicted crater dimensions using empirical expressions and the crater 

produced in the finite element analysis are also discussed. 

Comparing the finite element results to the empirical results also provides a method to establish 

the validity and accuracy of the finite element modeling. This same finite element model is used 

later in a parametric study of charge height, height of burst, and charge weight.

3.4.1 Evaluation of Soil Pressures

In this problem, the soil peak pressure (PP) was calculated using Eq. (2-37) and the data given 

previously in Table 2-3. The PP as function of soil depth is plotted in Figure 3-3. Note that the 

peak soil pressure is calculated to dissipate to zero at an approximate depth of 25 m [984 ft].

In the analyses reported by Nagy, et al. (2010), the center of the 100-kg [220-lb] TNT spherical 

charge was in direct contact with the soil surface; thus the bottom hemisphere of the spherical 

charge was buried in the soil. Embedded in the constant c is a coupling factor which adjusts the 

amount of energy that is transferred from the explosive charge to the soil. This coupling factor 

is discussed in DOD (1986). For the case of Nagy, et al. (2010), the coupling factor would be 

0.40. However, in the present analysis, in order to use the CONWEP option, the center of the 

charge must be located at a distance above the soil surface such that no portion of the charge is 

located below the surface. Therefore, the coupling factor should be reduced to 0.14 as given in 

DOD (1986). Figure 3-4 shows values of PP now scaled by the factor (0.14/0.40) to account for 

the center of the explosive charge being located above the surface. Thus, because the 

explosive charge is now elevated above the surface, the peak soil pressure is predicted to 

dissipate to zero at a depth of approximately 15 m [590 ft].

An ABAQUS/Explicit analysis was run using the finite element model discussed in Section 3.1 to

determine how the PP in the soil at different locations compared with the empirical predictions 

given in Figure 3-4. The comparison of the empirical and finite element results is shown in 

Figure 3-5. Two finite element analyses were performed: an elastic–plastic analysis (FE 

Plastic) using the Drucker-Prager Cap model, discussed in Section 3.2, and a simple elastic

analysis (FE Elastic). Interestingly, the elastic analysis peak soil pressures match more closely 

with the empirically calculated upper limit. As compared to the elastic–plastic analysis, the 

elastic analysis produces a higher peak soil pressure at depths less than 2 m [6.6 ft] and lower 

soil pressure at depths greater than 3 m [9.8 ft]. At shallower depths, the formation of plasticity 

(plastic dissipation) in the soil does not permit the soil pressure to reach the magnitude 

predicted by the elastic analysis. At depths greater than 3 m [9.8 ft], however, the peak soil 

pressure from the elastic–plastic analysis is higher than the elastic analysis. This may be due to 

the permanent plastic deformation (crater), which constrains the soil displacement and does not 

allow the soil pressure (stress) to redistribute and relax at these depths.
3-5

Figure 3-3 Range of Peak Pressure from Nagy, et al. (2010)

{100-kg [220-lb]-TNT Charge}

Figure 3-6 shows the pressure-time history in the soil at selected depth locations of 1.115, 

2.175, 3.132, and 5.105 m [3.6, 7.1, 10.3, and 16.7 ft]. These depths correspond to locations of 

element centroids, where pressure values can be obtained directly without interpolation. The 

ABAQUS/Explicit analysis terminated at a time of 0.03 seconds. Figure 3-6 shows that at 

0.03 seconds, the soil pressure at all locations has decreased below 0.5 MPa [72.5 psi], 

indicating that significant reduction in the soil pressure occurs rapidly.

Finally, Figure 3-7(a and b) show the pressure distribution in the soil (units are in Pa). For 

clarity, a half cross section of the soil block is shown. Figure 3-7(a) shows the extent of the 

pressure propagation throughout the soil block and that the pressure is localized around 

the crater. At no time did the pressure interact with the soil block boundary resulting in 

artificial reflection. 

3.4.2 Evaluation of Predicted Crater Dimensions

Empirical and numerical crater dimension predictions are compared next. The empirical 

methods of Cooper (1996) and Gould (1981) are used to calculate apparent crater dimensions.
3-6

Figure 3-4 Range of Peak Pressure Scaled for Above Surface Blast

{100-kg [220-lb]-TNT Charge}

Figure 3-5 Comparison of Empirical Peak Pressure With Finite Element Predictions

{100-kg [220-lb]-TNT Charge}
3-7

Figure 3-6 Pressure-Time Histories at Different Depths {100-kg [220-lb]-TNT Charge}

The empirical method of Cooper (1996), Eqs. (2-38) to (2-39), requires the detonation velocity, 

D; the Chapman-Jouguet pressure, PCJ; the cratering efficiency, ECR; and the explosive charge 

weight, W. From Cooper (1996), D for TNT is approximately 6.97 km/s [15,591 mi/hr] (Cooper, 

1996, Table 5.1) with a density of 1.630 g/cm3 [0.0598 lb/in3

]. For the 100-kg [220-lb] TNT 

charge, PCJ is calculated to be 20.1 GPa [2,915 ksi]. Using these values for PCJ, ECR, and W, 

the apparent crater radius, Ra, is approximately 1.8 m [5.9 ft].

The empirical method of Gould (1981) requires the cratering efficiency of the explosive, V0; the 

charge weight, W; and the height of the explosive above the surface, H. From Gould (1981), 

values of V0 = 312 m3

/tonne [10,000 ft3/ton] and V0 = 47 m3/tonne [1,500 ft3/ton] for wet and dry 

clay, respectively, were chosen because the precise moisture content of the soil represented in 

the finite element model is not known. Recall that the soil of Nagy, et al. (2010) was described

as “silty clay”; therefore, the value of V0 = 34 m3

/tonne [1,100 ft3/ton] corresponding to dry 

alluvium was also selected as a lower bound. Gould (1981) does not give data for a 

wet alluvium.

For the 100-kg [0.11-ton]-TNT-equivalent charge at 0.5 m [1.7 ft] above the surface, using 

Eqs. (2-41) to (2-43), the apparent crater volume, radius, and depth are given in Table 3-2. 

Thus, the apparent crater radius ranges from 2.85 to 1.01 m [9.35 to 3.31 ft] and the apparent

crater depth ranges from 1.19 to 0.42 m [3.9 to 1.38 ft].
3-8

Figure 3-7 Soil Pressure Distribution (Pa) at Time = 0.03 sec 

{100-kg [220-lb]-TNT Charge}

a)

b)
3-9

Table 3-2 Apparent Crater Dimensions using Gould (1981)*

Soil Type Apparent Crater Radius (m) Apparent Crater Depth (m)

Wet Clay 2.85 1.19

Dry Clay 1.18 0.49

Dry Alluvium 1.01 0.42

*Based on Gould, K.E. “High-Explosive Field Tests: Explosion Phenomena and Environmental Impacts.” 

DNA 6187F. Washington, DC: Defense Nuclear Agency. October 1981.

In the finite element analysis, the crater dimensions were measured at 0.03 seconds after 

detonation because it was determined that the center node displacement had reached 

steady-state at that time (Figure 3-8). Figure 3-9 shows the apparent crater radius and depth, 

which is measured relative to the original surface shown by the dashed line.

The finite element analysis predicts an apparent crater radius, Ra, of 1.37 m [4.49 ft] and an 

apparent crater depth, Da, of 1.2 m [3.9 ft]. A comparison of the apparent crater dimensions 

calculated from Cooper (1996), Gould (1981), and finite element results are given in Table 3-3. 

Both of these predicted crater dimensions lie within the range of the empirically calculated 

values. This provides a degree of confidence that the current Lagrangian finite element model 

produces a reasonably consistent prediction of crater formation compared to the empirical data.

Figures 3-10 and 3-11 show the progression of the plastic strain distribution in the soil around 

the crater as a function of time. The quantity Dp has been introduced here, which quantifies the 

depth of the plastic strain. Note the nonspherical plastic strain zone that forms in the soil 

around the crater boundary.

Figure 3-12 shows plots of the soil displacement vectors at selected times in the analysis. The 

red vectors signify the largest displacements. Initially at time 0.0006 seconds, the primary 

movement of the soil is downward; however, as time proceeds, the soil begins an upward 

movement (upheaval) forming the “lip” around the crater. This upward movement also indicates 

the development of ejecta. However, as discussed in Section 3.1, actual formation of ejecta is 

not possible using the current Lagrangian continuum approach. A Lagrangian finite element 

analysis incorporating mesh-free SPH in conjunction with an appropriate failure criterion would 

be required to attempt to simulate the formation of ejecta.

3.5 Finite Element Parametric Study on Different TNT-Equivalent Charge 

Weights and Above-Surface Heights

Based on the discussion in Section 2.1, a small moving van is chosen as the representative 

vehicle. The maximum payload is approximately 4,536 kg [10,000 lb] (GM, 2012a). For a 2008 

full-size cutaway van produced by General Motors (GM), the dimension from the ground to the 

top of rear load floor ranges from 57 to 64 cm [22.3 to 25.1 in] (GM, 2012b). The explosive 

charge sizes used in the analyses are 45.3, 227, 453, 2,268, and 4,536 kg [100, 500, 1,000, 

5,000, and 10,000 lb].

The finite element results are presented in the form of soil pressure distributions throughout the 

soil. Plots of pressure stress contours in the soil are used to determine the location of shock 

wave attenuation. Finite element results showing the predicted crater in the soil surface are 

also shown.
3-10

Figure 3-8 Time History of Center Displacement Showing Development of Crater Depth 

{100-kg [220-lb]-TNT Charge}

Figure 3-9 Apparent Crater Diameter and Depth Predicted by Finite Element Analysis 

{100-kg [220-lb]-TNT Charge}

Ra

Da

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Displacement (m)

Time (sec)
3-11

Table 3-3 Comparison of Apparent Crater Dimensions for a 100-kg [220-lb] TNT

 Charge Located at a Height of 0.50 m [20 in]

Results Apparent Crater Radius m [ft] Apparent Crater Depth m [ft]

Cooper* 1.8 [5.9] -

Gould† 2.85–1.01 [9.35–3.31] 1.19–0.42 [3.90–1.38]

Finite Element 1.37 [4.49] 1.2 [3.9]

* Cooper, P.W. Explosives Engineering. Hoboken, New Jersey: John Wiley & Sons, Inc. 1996.

†Gould, K.E. “High-Explosive Field Tests: Explosion Phenomena and Environmental Impacts.” DNA 6187F. 

Washington, DC: Defense Nuclear Agency. October 1981.

Figure 3-10 Plastic Strain Distribution (Dp = Plastic Strain Depth) 

{100-kg [220-lb]-TNT Charge}

Time = 0.0006 sec

Dp = 0.6 m

Time = 0.0048 sec

Dp = 1.9 m
3-12

Figure 3-11 Plastic Strain Distribution Continued (Dp = Plastic Strain Depth)

{100-kg [220-lb]-TNT Charge}

Time = 0.01 sec

Dp = 3.3 m

Time = 0.02 sec

Dp = 5.7 m

Time = 0.03 sec

Dp = 7.7 m
3-13

Figure 3-12 Displacement Vectors Indicating Direction of Soil Movement

{100-kg [220-lb]-TNT Charge}

Time = 0.0006 sec

Time = 0.0048 sec

Time = 0.02 sec
3-14

For the weight range between 2,268 and 4,535.9 kg [5,000 and 10,000 lb] TNT, the CONWEP 

algorithm cannot be used for charge heights less than the charge radius. This is because the 

corresponding dimensions would cause part of the spherical charge to be buried in the soil, a 

geometry for which the CONWEP algorithm is not valid. Table 3-4 shows the spherical charge 

radii for each charge weight. Thus, ABAQUS/Explicit analyses were not performed for those 

cases where the spherical charge would have been partially buried in the soil.

Tables 3-5 to 3-8 show the empirically calculated and finite element predicted apparent crater 

dimensions for different charge weights located at different heights. The empirical values were 

calculated using Eqs. (2-37) to (2-39) (Gould, 1981). Figure 3-13 shows the craters predicted 

by the finite element analyses for charge weights of 45.4, 226.8, and 453.6 kg [100, 500, 

1,000 lb].

Table 3-4 Spherical Charge Dimensions

Charge Weight WTNT kg [lb] Charge Volume m3 [ft3] Charge Radius m [ft]

45.4 [100] 0.028 [0.989] 0.19 [0.62]

226.8 [50] 0.139 [4.91] 0.32 [1.05]

453.6 [1,000] 0.278 [9.82] 0.41 [1.34]

2268.0 [5,000] 1.391 [49.12] 0.69 [2.26]

4535.9 [10,000] 2.783 [98.28] 0.87 [2.85]

For TNT: density = 1,630 kg/m3 

volume = weight/density = 4

3 π r

3 where r is radius

Table 3-5 Apparent Crater Dimensions* for an Above-Surface Charge Located at a Height of

 0.5588 m [22 in]

Charge Weight 

WTNT kg [lb]

Empirical Finite Element

Ra m [ft] Da m [ft] Ra m [ft] Da m [ft]

45.4 [100] 1.94–0.60 [6.36–1.97] 0.81–0.25 [2.66–0.82] 0.78 [2.56] 0.18 [0.59]

226.8 [500] 3.92–1.46 [12.9–4.79] 1.63–0.61 [5.35–2.00] 1.26 [4.13] 1.06 [3.48]

453.6 [1,000] 5.18–2.03 [17.0–6.66] 2.16–0.85 [7.09–2.79] 1.63 [5.35] 1.41 [4.62]

2268.0 [5,000] 9.56–4.07 [31.4–13.3] 3.98–1.70 [13.1–5.58] – –

4535.9 [10,000] 12.31–5.38 [40.39–17.65] 5.13–2.24 [16.83–7.35] – –

*Based on Gould, K.E. “High-Explosive Field Tests: Explosion Phenomena and Environmental Impacts.” DNA 

6187F. Washington, DC: Defense Nuclear Agency. October 1981. 

– Indicates that charge height is less than charge radius, making the CONWEP algorithm invalid.

Table 3-6 Apparent Crater Dimensions* for an Above-Surface Charge Located at a Height of

 0.6096 m [24 in]

Charge Weight 

WTNT kg [lb]

Empirical Finite Element

Ra m [ft] Da m [ft] Ra m [ft] Da m [ft]

45.4 [100] 1.87–0.56 [6.14–1.84] 0.78–0.23 [2.56–0.75] 0.52 [1.71] 0.15 [0.49]

226.8 [500] 3.83–1.39 [12.56–4.56] 1.60–0.58 [5.25–1.90] 1.21 [3.97] 1.04 [3.41]

453.6 [1,000] 5.09–1.96 [16.70–6.43] 2.12–0.82 [6.96–2.69] 1.57 [5.15] 1.38 [4.53]

2268.0 [5,000] 9.46–3.99 [31.03–13.09] 3.94–1.66 [12.93–5.45] – –

4535.9 [10,000] 12.22–5.29 [40.09–7.36] 5.09–2.20 [16.70–7.22] – –

*Based on Gould, K.E. “High-Explosive Field Tests: Explosion Phenomena and Environmental Impacts.” 

DNA 6187F. Washington, DC: Defense Nuclear Agency. October 1981.

– Indicates that charge height is less than charge radius, making the CONWEP algorithm invalid.
3-15

Table 3-7 Apparent Crater Dimensions* for an Above-Surface Charge Located at a

 Height of 0.6604 m [26 in]

Charge Weight 

WTNT kg [lb]

Empirical Finite Element

Ra m [ft] Da m [ft] Ra m [ft] Da m [ft]

45.4 [100] 1.80–0.52 [5.91–1.71] 0.75–0.22 [.46–0.72] 0.67 [2.20] 0.12 [0.39]

226.8 [500] 3.75–1.33 [12.30–4.36] 1.56–0.55 [5.12–1.80] 1.09 [3.58] 0.96 [3.15]

453.6 [1,000) 5.00–1.89 [16.4–6.20] 2.09–0.79 [6.86–2.59] 1.49 [4.89] 1.37 [4.49]

2268.0 [5,000] 9.37–3.91 [30.75–12.83] 3.90–1.63 [12.80–5.35] – –

4535.9 [10,000] 12.12–5.20 [39.76–17.06] 5.05–2.17 [16.57–7.12] – –

*Based on Gould, K.E. “High-Explosive Field Tests: Explosion Phenomena and Environmental Impacts.” DNA 

6187F. Washington, DC: Defense Nuclear Agency. October 1981.

– Indicates that charge height is less than charge radius, making the CONWEP algorithm invalid.

Table 3-8 Apparent Crater Dimensions* for an Above-Surface Charge Located at a

 Height of 0.7112 m [28 in]

Charge Weight 

WTNT kg [lb]

Empirical Finite Element

Ra m [ft] Da m [ft] Ra m [ft] Da m [ft]

45.4 [100] 1.74–0.48 [5.71–1.57] 0.72–0.20 [2.36–0.66] 0.70 [2.30] 0.10 [0.33]

226.8 [500] 3.67–1.27 [12.04–4.17] 1.53–0.53 [5.02–1.74] 1.10 [3.61] 0.86 [2.82]

453.6 [1,000] 4.92–1.82 [16.14–5.97] 2.05–0.76 [6.73–2.49] 1.43 [4.69] 1.21 [3.97]

2268.0 [5,000] 9.28–3.83 [30.45–12.57] 3.87–1.59 [12.70–5.22] 2.72 [8.92] 2.50 [8.20]

4535.9 [10,000] 12.03–5.12 [39.47–16.80] 5.01–2.13 [16.44–6.99] – –

*Based on Gould, K.E. “High-Explosive Field Tests: Explosion Phenomena and Environmental Impacts.” DNA 

6187F. Washington, DC: Defense Nuclear Agency. October 1981.

– Indicates that charge height is less than charge radius, making the CONWEP algorithm invalid.

Tables 3-5 to 3-8 show that the crater radius (depth) from the finite element analyses is 

consistently smaller than the empirical calculations. One possible explanation is that there is 

uncertainty in whether the soil parameters used in the empirical relationships are consistent with 

the soil parameters used in the Drucker-Prager Cap model. This demonstrates the difficulty in 

using empirical constants obtained from tables (from a variety of sources) because there is 

typically wide variability in data and the data are not always complete. However, considering 

that in Section 3.4.1 the finite element analysis soil pressures were comparable to those Nagy, 

et al. (2010) calculated, the differences in crater radius is most likely due to discrepancies 

between the empirical and Drucker-Prager Cap model soil parameters.

3.5.1 Predicting Crater Formation Using Element Deletion and 

Continuum-To-Particle Conversion

The formation of a crater in soil also was predicted using the element deletion approach

available in ABAQUS/Explicit. A soil failure criterion was used to determine when an element 

would be deleted from the finite element mesh. Calculating failure utilizes effective strain, ε�, 

which is based on a von Mises form

ε� = �(ε1 – ε2)2+(ε2 – ε3)2+(ε3 – ε1)2�

1/2

(3-1)
3-16

Figure 3-13 Predicted Crater Sizes for Above-Surface Charge Located at a Height of

 0.6096 m [24 in]

Charge Weight = 45.4 kg 

[100 lb]

Charge Weight = 226.8 kg

[500 lb]

Charge Weight = 453.6 kg

[1,000 lb]
3-17

where the quantities ε1, ε2, ε3 are principal strains. This expression was implemented using the 

ABAQUS/Explicit user utility subroutine VUSDFLD (Dassault Systèmes Simulia Corp., 2012a). 

The failure criterion used principal plastic strains in the effective strain calculation [Eq. (3-1)]. 

For the analyses in this report, a failure criterion of 15 percent effective plastic strain was used.

3.5.1.1 Element Deletion

A 454-kg [1,000-lb] TNT-equivalent charge located at a height of 0.6096 m [24 in] was analyzed. 

Figure 3-14 shows the predicted crater size using the element deletion technique. The 

predicted apparent crater radius, Ra, is approximately 2.41 m [7.91 ft] which is within the range 

of 5.09 to 1.96 m [16.7 to 6.4 ft] as previously given in Table 3-6. The apparent crater depth, 

Da, is approximately 1.89 m [6.2 ft] which is within the range of 2.12 to 0.82 m [6.96 to 2.69 ft], 

also previously given in Table 3-6.

Additional analyses were performed for other TNT-equivalent charge sizes of 4,536; 9,072; 

18,144; and 27,216 kg [10,000; 20,000; 40,000; and 60,000 lb]. Table 3-9 shows the location of 

the center of the spherical charge, which is equal to the radius of the spherical charge. These 

charge heights result in the charge just touching the ground surface. Table 3-10 shows both the 

empirical and finite element predictions of the apparent crater dimensions for each charge size. 

For each charge size, the finite element predicted apparent crater depth falls within the range 

given by the empirical expressions of Gould (1981), but the predicted apparent crater radius is

Figure 3-14 Predicted Crater for 454-kg [1,000-lb]-TNT-Equivalent Charge Located at a 

 Height of 0.6096 m [24 in]
3-18

Table 3-9 Height of Above-Surface Charge 

Charge Weight 

WTNT kg [lb]

Radius of Spherical Charge Height of Charge Center Above Surface

m [ft] m [ft]

4,536 [10,000] 0.87 [2.85] 0.9 [2.95]

9,072 [20,000] 1.10 [3.61] 1.1 [3.64]

18,144 [40,000] 1.39 [4.56] 1.4 [4.59]

27,216 [60,000] 1.59 [5.22] 1.6 [5.25]

Table 3-10 Apparent Crater Dimensions* for Above-Surface Charges

Charge Weight 

WTNT kg [lb]

Empirical Finite Element

Ra m [ft] Da m [ft] Ra m [ft] Da m [ft]

4,536 [10,000] 11.68–4.82 [38.32–15.82] 4.87–2.01 [15.98–6.59] 3.46 [11.35] 2.79 [9.15]

9,072 [20,000] 14.76–6.10 [48.42–20.01] 6.15–2.54 [20.18–8.33] 5.99 [19.65] 3.88 [12.73]

18,144 [40,000] 18.59–7.68 [60.99–25.20] 7.74–3.20 [25.39–10.50] 7.50 [24.61] 5.91 [19.39]

27,216 [60,000] 21.27–8.80 [69.78–28.87] 8.86–3.67 [29.07–12.04] 8.67 [28.44] 6.87 [22.54]

* Based on Gould, K.E. “High-Explosive Field Tests: Explosion Phenomena and Environmental Impacts.” 

DNA 6187F. Washington, DC: Defense Nuclear Agency. October 1981.

– Indicates that charge height is less than charge radius, making the CONWEP algorithm invalid.

just below the lower limit. For the 454-kg [1,000-lb] charge, the predicted apparent crater radius 

was also within the limits of Gould (1981). At this point, there is not a clear explanation other 

than the selected failure criterion for the discrepancy at the larger charge weights (Table 3-10). 

In this study, a soil failure criterion of 15 percent equivalent plastic strain was chosen based on 

a range of values found in the literature. However, because of the variability in soil parameter 

values reported in the literature it is recommended that additional failure criterion values be 

investigated to determine their effect on apparent crater dimensions. In addition, alternative 

failure criteria, such as equivalent total strain and maximum principal tensile strain, also have 

been used. Further investigation of these alternative failure criteria may prove useful.

Figures 3-15 and 3-16 show the finite element predicted soil pressure distributions. The times 

given correspond to when the finite element model maximum energy occurred and when the 

kinetic energy reached steady state.

3.5.1.2 Continuum-to-Particle Conversion

A third approach for predicting crater formation was investigated using the 

“continuum-to-particle” conversion option in which the solid continuum elements are 

converted to SPH particles once a specified failure criterion is satisfied. As in the element 

deletion approach, an effective plastic strain failure criterion of 15 percent was used. 

Figure 3-17 shows results where the failed continuum elements have been converted to 

SPH particles.

Figure 3-17 shows that SPH particles penetrate the contact surface that represents the crater 

wall; this is not physically realistic. Dassault Systèmes Simulia Corp. user support was 

contacted regarding this observation and it was confirmed that ABAQUS/Explicit has a “bug” 

that allows SPH particles to penetrate contact surfaces. Dassault Systèmes Simulia Corp. 

stated their intent to correct this error, possibly in the release of ABAQUS/Explicit Version 6.14

in May 2014. The continuum-to-particle conversion technique in ABAQUS/Explicit cannot be 

used until this error is corrected by the software vendor.
3-19

Figure 3-15 Soil Pressure Distributions for 4,536 and 9,072 kg [10,000 and 20,000 lb]

 Charges. Pressure Is in Pa.

4,536 kg [10,000 lb] Charge at Time = 2.4 ms

9,072 kg [20,000 lb] Charge at Time = 5.3 ms
3-20

 

18,144 kg [40,000 lb] Charge at Time = 6.6 ms

27,216 kg [60,000 lb] Charge at Time = 7.8 ms
3-21

Figure 3-16 Soil Pressure Distributions for 18,144 and 27,216 kg [40,000 and 60,000 lb]

 Charges. Pressure Is in Pa.

Figure 3-17 ABAQUS/Explicit Smoothed Particle Hydrodynamics Results for a 454-kg 

 [1,000-lb]-TNT-Equivalent Charge Located at a Height of 0.6096 m [24 in]

3.6 Eulerian Finite Element Analysis

As an alternative to the Lagrangian approach with CONWEP generated blast pressures 

(Sections 3.4 and 3.5), a multi-material Eulerian approach was investigated. This approach 

involved constructing an Eulerian mesh containing all the materials in the solution domain: air, 

explosive charge, and soil, Figure 3-18. Although Figure 3-18 does not show the mesh, 

approximately 1.5 million elements were used to provide sufficient mesh density. Air was 

modeled using the ideal gas equation-of-state (EOS) (Dassault Systèmes Simulia Corp., 2012b) 

Particles penetrating 

contact surface (crater wall)

Particles penetrating 

contact surface (crater wall)
3-22

and the necessary parameters are given in Table 3-11. A spherically shaped explosive charge 

having a weight of 454 kg [1,000 lb] was modeled and its detonation behavior modeled using 

the Jones-Wilkens-Lee (JWL) EOS (Dassault Systèmes Simulia Corp., 2012b). The JWL EOS 

parameters are given in Table 3-12 and are the same as those given in Nagy, et al. (2010). The 

soil was represented by the same Drucker-Prager Cap model and soil parameters as described 

in Section 3.2 of this report.

Figure 3-18 Multi-Material Eulerian Analysis Model 

(One Half of the Solution Domain Is Shown)

Table 3-11 Ideal Gas Parameters

Parameter Value

Reference Density (ρa) 1.225 kg/m3 [0.0765 lb/ft3

]

Gas constant for air (Rair) 286.9 J/kg∙K [1716 ft∙lb/slug∙R]

Specific heat capacity (cv) 717.3 J/kg∙K [0.1714 BTU/lbm∙F]

Reference temperature (T0) 288.2° K [59.4° F]

Figure 3-19 shows results of the Eulerian analysis and the predicted shape of the crater at 

6.24 ms. The yellow line denotes the original ground surface. The apparent crater depth is 

0.9 m [2.95 ft]. The Lagrangian analysis predicted an apparent crater depth of 1.37 m [4.49 ft],

Explosive 

Charge

Soil

Air

Plane of Symmetry
3-23

a difference of approximately 34 percent. Interestingly, the value of 0.9 m [2.95 ft] falls within 

the empirically predicted range of 2.09–0.79 m [6.86–2.59 ft] given in Table 3-7. There are a 

number of possible reasons for the difference between the Eulerian and Lagrangian results. 

The primary reason may be insufficient mesh refinement of the air domain surrounding the 

explosive charge. If the elements are too large, inaccuracy in the propagation of the air blast 

shock wave may result, which would in turn under-predict the ground surface blast pressure 

loading compared to the loading calculated by the CONWEP algorithm used in the Lagrangian

Table 3-12 Jones-Wilkens-Lee Equation of State Parameters

Parameter Value

Detonation wave speed (Cd) 6,930 m/s [22,736 ft/s]*

A 373.8 GPa [54,215 ksi]*

B 3.747 GPa [543 ksi]*

R1 4.15*

R2 0.9*

ω 0.35*

Density of explosive (ρ0) 1,630 kg/m3 (102 lb/ft3)*

Initial specific energy (Em0) 4.56 MJ/kg (1961.8 Btu/lb)†

*Values taken from Nagy, N., M. Mohamed, and J.C. Boot. “Nonlinear Numerical Modelling for the Effects of 

Surface Explosions on Buried Reinforced Concrete Structures.” Geomechanics and Engineering. Vol. 2. pp. 1–

18. 2010.

†Value taken from Smith, P.D. and J.G. Hetherington. Blast and Ballistic Loading of Structures. Oxford, 

England: Butterworth-Heinemann. 1994.

Figure 3-19 Eulerian-Analysis-Predicted Apparent Crater Size

analyses. It is recommended that a finer mesh be used in modeling the air domain to 

investigate the accuracy of the ground surface blast loading pressure.

Ground Surface

Air

Soil

Ejecta
3-24

3.7 Alternative Analysis Codes Reviewed

This section briefly discusses alternative codes that can be used to model blast-induced shock 

waves and their effects on underground structures. Open literature sources provide examples 

of both commercially available and restricted use codes that are commonly used for blast 

analyses will be discussed. The rationale for using ABAQUS/Explicit for the analyses in this 

report was to utilize commercially available software that does not have access restrictions. 

Analyses conducted by nuclear power plant licensees may not qualify for access to specialized 

codes, which typically have use restrictions. Nevertheless, it is recognized that these highly 

specialized codes are useful when performing blast analyses.

Commercial Codes

Both Autodyn (ANSYS, 2013) and LS-DYNA (LSTC, 2013) have wide acceptance for 

performing explicit dynamic analyses. Both of these codes are capable of multi-material

Eulerian, Lagrangian, Arbitrary Lagrangian-Eulerian analysis, and SPH modeling capabilities. 

LS-DYNA also has an element free Galerkin (EFG) analysis capability which is useful for crack 

propagation analysis. Both codes have a large number of material models for applicable to large 

deformation, high strain rate behavior and EOS for solids, fluids, and gases. There are no 

restrictions on the use of either Autodyn or LS-DYNA.

Restricted Codes

The shock physics code CTH (Sandia, 2013) is well-known and widely used to calculate 

blast-induced pressures produced by explosive charges. CTH is an Eulerian code capable of 

modeling shock waves and large deformations of materials. The JWL EOS used in the 

ABAQUS/Explicit analyses is also available in CTH for modeling the explosive charge. A 

common approach is to use CTH to calculate the blast-induced pressure distributions, which are 

then input as time-dependent pressure loads in a separate Lagrangian finite element structural 

analysis. Access to CTH is restricted to use by the U.S. government or its contractors working 

for the U.S. government.

Another software tool is the code EPIC (Elastic-Plastic Impact Computations) (Gerlach and 

Johnson, 2009). This code was developed for the purpose of understanding the effects of 

Improvised Explosive Devices (IEDs) buried in soil. EPIC is a Lagrangian code that uses a 

so-called hybrid particle-finite element approach involving both solid continuum elements and 

hybrid particles. Distribution of EPIC is limited to the U.S. Department of Defense and 

its contractors.
4-1

4. CHARACTERIZING STRUCTURE RESPONSE 

4.1 Codes and Standards for Design of Underground Structures

This section briefly describes codes, specifications, and standards that could be used to design 

underground structures subjected to dynamic loads generated by detonation of high explosives

in air, at the ground surface, and underground.

4.1.1 Design and Analysis of Hardened Structures to Conventional Weapons 

Effects and Structures to Resist the Effects of Accidental Explosions

The Unified Facilities Criteria (UFC) Joint Services Manuals (DOD, 2008, 2002) are regarded as 

the principal guidance documents for design of structures subjected to explosive loads. 

Although the design approach presented in these manuals is directed primarily toward 

structures subjected to the effects of a high explosive detonation, it is also applicable to the 

design of structures exposed to other explosive detonations. The design methods account for 

the close-in effects of an explosion on structures. These design methods also are applicable for

intermediate and far-range effects on structures located away from the explosion. These 

guidance documents use concepts and provisions from other specific codes, standards, and 

design manuals commonly used in the design of reinforced concrete, prestressed concrete,

masonry, and steel structures (e.g., ACI, 2011; AISC, 1994; PCI, 1978; J. Healey, et al., 1975). 

The design procedure is applicable to both surface and underground structures subjected to 

explosive loads.

4.1.2 Fundamentals of Protective Design for Conventional Weapons

This U.S. Army Manual (DOD, 1986) was originally published in 1949 as the first standard to 

provide design guidelines for structures subjected to explosive loads in a single document. 

This document used extensive structural damage data produced during World War II and was 

revised in 1986. This document provides guidance on the design of reinforced concrete 

structures under explosive loads. It uses concepts and provisions of the American Concrete 

Institute code to design reinforced concrete structures, with special consideration for the rate of 

loading effects on the strength properties of concrete and reinforcing steel. This manual is 

referenced in U.S. Department of Defense (DOD, 2008, 2002) and provides more detailed 

design guidelines for reinforced concrete underground structures.

4.2 Analytical Tools for Modeling Underground Structure Response 

The effects of surface blast-induced ground shock on an underground structure are different 

than the effects of an air blast on a surface structure. The structural response of a freestanding 

surface structure is primarily from the air blast, with ground shock effects being secondary. For 

an underground structure, this is not the case, because the structure is surrounded by a 

medium (e.g., soil) that is closer in density to the material of the underground structure than to 

air, which facilitates stronger energy coupling. In the region where the crater forms, there is 

typically significant structural damage, but the depth at which the structure will remain 

undamaged is difficult to predict because it depends on the soil characteristics, which control 

pressure wave attenuation and ground motion. Therefore, the responses of the ground and the 

structure are closely related. The motion of the underground structure (i.e., displacement, 

velocity, and acceleration) is closely coupled with the motion and constraining effect of the 

ground (Glasstone and Dolan, 1977). Because of this coupling, commonly referred to as 
4-2

soil-structure interaction (SSI), developing purely analytical methods to analyze underground 

structures is difficult.

Baylot (1992) investigated various parameters that affect the loading on buried structures when 

a conventional weapon (explosive) detonates near a buried structure. As Baylot (1992) states,

the most difficult part of designing an underground structure is to accurately determine the 

blast-induced loading. The typical approach is to calculate the free-field stresses at a specific 

location that would be produced without the presence of the structure. These stresses are then 

modified to approximate the interface stresses that would occur on the structure (Baylot, 1992).

An approach provided in DOD (1986) uses semi-empirical methods to predict the free-field 

stresses, displacements, velocities, accelerations, and impulse forces. These expressions were 

presented in Section 2.4.1.2, Eqs. (2-32) to (2-36). To calculate the shock of the buried 

structure, the parameters R1 and R2 are the distance from the explosive charge to the front face 

and back face of the structure, respectively (Dove, 1992). The original expressions given in 

Eqs. (2-32) to (2-36) are then integrated over the range R1 to R2 to give uniform displacements, 

velocities, and accelerations (Dove, 1992)

davg

W1/3 = 

500 f W

(n-1)

3 (R1

-n+2- R2-n+2)

c (n - 2) (R2 - R1)

(4-1)

Vavg = 

160 f W

n

3 (R1

-n+1- R2-n+1)

(n - 1) (R2 - R1)

(4-2)

aavgW1/3=

50 f c W

n+1

3 (R1

-n - R2-n

)

n (R2 - R1)

(4-3)

Equations (4-1) to (4-3) give the average free-field motions, and the parameters f, c, n, and W 

are the same as described in Section 2.4.1.2. Finally, a reduction factor is applied, which is 

based on the geometry of the structure. Dove (1992) states that although this approach 

conservatively estimates structural response it has limited applicability because it uses a

box-like structure. Baylot (1992) also states that this method significantly overpredicts the 

loading on the structure, which results in an overprediction of the structural response.

A second approach is the single-degree-of-freedom (SDOF) analysis, which reduces the 

problem to a simple spring, mass, and damper system, as shown in Figure 4-1.

Here, the spring represents the stiffness (K) of a structural component (e.g., wall, slab), and the 

mass (M) and damper (C) are calculated to give the same frequency and damping 

characteristics as the actual structure (Dove, 1992). The term F(t) is the time-dependent 

loading. Because the SDOF approach is very common, it is not discussed in this report. Instead 

the reader is referred to Paz and Leigh (2004) and Weidlinger and Hinman (1988), who present 

a thorough discussion on SDOF analysis, as well as multiple-degree-of-freedom (MDOF) 
4-3

Figure 4-1 Single Degree of Freedom System

analysis. Dove (1992) states that SDOF analyses tend to be inconsistent because of the 

difficulty calculating the structural loading.

To accurately calculate loading of the structure, it is necessary to know the deformability of the 

structure relative to the surrounding soil (Glasstone and Dolan, 1977). Glasstone and Dolan 

(1977) state that if the structure is at least as deformable as the surrounding soil, then the 

free-field pressure can be applied to the structure and considered as an upper limit. If the 

structure is more deformable than the surrounding soil, then the pressure loading can be lower 

(Glasstone and Dolan, 1977). In this case the structure may deflect away from the soil, which 

creates the so-called “soil arching” effect. Soil arching will result in part of the pressure being 

transmitted around the structure. Chen, et al. (1990) state that for burial depths equal to or 

greater than approximately 20 percent of the span of the structure, accounting for soil arching is 

important. Dove (1992) and Baylot (1992) conclude that the complicating factor in the methods 

discussed previously is the inability to accurately account for SSI. Therefore, the overall 

conclusion is that to accurately analyze an underground structure subjected to a surface 

explosion, numerical methods such as the finite element method should be used. Despite the 

shortcomings of the empirical methods, the current design methodology for underground 

structures (DOD, 2008, 2002, 1986) makes significant use of the conservative empirical 

approach discussed in Chapter 6 of this report.

M

C

K

F(t) u

5-1

5. COMPARISON OF EXPLOSIVE LOADS AND SEISMIC LOADS IN

UNDERGROUND MEDIUM

Three sources of dynamic waves that may propagate through an underground medium. These 

are (i) natural earthquakes, (ii) surface explosions, and (iii) underground explosions. Although 

this discussion focuses on conventional explosive loads and seismic loads, explosive loads from

nuclear weapons are discussed herein to the extent applicable (ASCE, 1985). Differences 

between the characteristics of explosion-induced and earthquake-induced responses of the 

geological media are examined here to support analyses on the effects of these sources in 

Chapter 6 of this report.

Natural earthquakes primarily occur due to the sudden release of strain energy when the rock 

ruptures at plate boundaries or at faults. As the earthquake occurs at depth, dynamic waves 

radiate away from the source and travel rapidly through Earth’s crust, as shown in Figure 5-1

(Gere and Shah, 1984). These waves are either body waves or surface waves (Kramer, 1996).

When an explosive is detonated in air near the surface, it produces overpressure in the air 

(Section 2.2, Figure 2-2), which interacts with the ground surface and, under certain 

circumstances may create a crater (Section 2.2, Figure 2-4, and Section 3.4.2). When an

Figure 5-1 Seismic Waves (After Gere and Shah, 1984)
5-2

explosion occurs at the surface, it is more likely to produce a crater, because of the intense 

interaction associated with more direct coupling of the energy into the ground (Sections 2.2 and 

3.4.2). The shock wave produces body waves and surface waves, as discussed in Sections 2.2

and 2.4.

Body waves are created by the energy released at relatively great depth by an earthquake and 

at shallow depths in the vicinity of a surface or near-surface explosion. Two types of body 

waves travel through the underground media: P-waves and S-waves. P-waves, also known as 

primary, compressional, or longitudinal waves, involve successive compression and rarefaction 

of the underground media through which they pass. The P-waves can travel through both solid 

and fluid media. Underground media are stiffest in compression, so P-waves travel faster than 

other seismic waves and are therefore the first to arrive at a particular site. S-waves, also 

known as secondary, shear, or transverse waves, produce shear deformations as they travel 

through a material. S-waves can travel only through solid media. S-waves are divided into two 

components: SV (vertical plane movement) and SH (horizontal plane movement). The speed 

at which body waves travel varies with the stiffness of the underground media. Figure 5-2

shows travel time of a seismic wave. 

When body waves interact with either the surface or surficial layers of the Earth, surface waves 

result. The waves that travel along the Earth’s surface have amplitudes that decrease roughly 

exponentially with depth. Surface waves are more prominent at lateral distances farther from

Figure 5-2 Example of a Seismic Wave’s Travel Time

0

2

4

6

8

10

12

14

0 1,000 2,000 3,000 4,000

Time, min

Distance, km

P Wave

S Wave

Love Wave

Rayleigh Wave
5-3

the source of an earthquake. This is because of the nature of the interactions that take place 

between the body waves and the surface and surficial layers of the Earth that produce them. 

For engineering purposes, Rayleigh waves and Love waves are the two most important surface 

waves. Raleigh waves have both vertical and horizontal particle motion, whereas Love waves 

have only horizontal particle motion.

There are differences between the waves generated by earthquakes and those of explosions.

These may be due to differences in various parameters, such as source dimension, source time 

function, source mechanism and focal depth, or a combination of these parameters (Dahy and 

Hassib, 2009). Because explosions are primarily spherically symmetric disturbances, the 

explosions radiate p-waves efficiently. In contrast, earthquakes that result primarily from sliding 

or rupture along a buried fault surface, over a relatively large area for a longer time, excite the 

transverse motions of S-waves efficiently. Berry (1967) found that for earthquakes and 

underground nuclear tests with body waves of similar magnitude, the surface waves for the

underground explosions are usually much smaller in amplitude than those for the earthquakes. 

Thus, all explosions will show stronger P-waves and weaker S-waves than observed for similar 

magnitude earthquakes (Figure 5-3) (University of California at Berkeley, 2009; Vortman, 1982,

1981; Walter, 2013; Walter, et al., 1998). Another reason for the relatively high amplitude

P-waves and relatively lower amplitude surface waves for explosions is that these types of 

events release energy rapidly from a “point” source (Berry, 1967). However, despite these 

differences, seismic velocity is often used as a crude index of soil properties for ground shock 

prediction purposes.

To further distinguish the explosions from earthquakes, seismic waves for two cases have been

studied in detail by Walter (2013). One way to quantify the difference between explosions and 

earthquakes is by determining the ratio of P-wave to S-wave (P/S) energy measured for both 

explosions and earthquakes. Explosions should result in higher P/S ratios than similarly located

earthquakes. However, the frequency at which the best separation of explosion P/S ratio and 

earthquake P/S ratio occurs varies by conditions of geology of the region where explosions and 

earthquakes take place. Figure 5-4 shows the P/S ratio from a May 11, 1998, nuclear explosion 

test in India and for earthquakes in Asia (Figure 5-5). The measurements in Figure 5-4 were 

made at four different frequencies. The India test has a higher P/S ratio than the earthquakes, 

as expected. The other approach is to analyze the moment tensor of seismic data collected 

from various monitoring stations. The moment tensor solution should yield information on the 

source mechanism that triggers the ground shaking (Ford, et al., 2007; University of California 

at Berkeley, 2009).

Figure 5-3 Comparison of Nuclear Explosion With an Earthquake Generated Wave 

 Signals (After Walter, 2013)
5-4

Figure 5-4 Comparison of P- to S-Wave Ratios of a Nuclear Explosion With Those of 

an

 Earthquake at Different Frequencies (After Walter, 2013)

Figure 5-5 Topographic Map Showing Locations of Nuclear Test Locations 

(Diamonds), 

 Earthquakes (Circles), Primary International Monitoring System (Stars), and 

 Auxiliary International Monitoring System (Triangles) (After Walter, 2013) 

The underground structures may be subjected to explosive and/or seismic loads, which have 

different characteristics, as shown in Figures 5-3 and 5-4. The differences in effects of these 

loads on underground structures are discussed qualitatively in Chapter 6 of this report.
6-1

6. DESIGN CONSIDERATIONS FOR PROTECTION OF

UNDERGROUND STRUCTURES

The stress transients generated by high explosive detonations in air, near the surface, and 

underground can cause severe damage to underground structures. Underground structures 

that are designed to withstand detonation of high explosive charges mostly include sensitive 

military facilities, civil defense facilities, and, most recently, nuclear power plants [e.g., proposed 

small modular reactors (Braverman, et al., 1997)]. Military facilities include bunkers, missile 

silos, command and control facilities, and communication facilities. The shock exerted on an 

underground structure by detonation of high explosives will vary with the intensity of detonation 

force (Table 2-2), the distance of detonation from the underground structure (Figure 2-6), any

barrier or protective system between the detonation point and the structure,(Figure 2-5), soil and 

rock types, and various drainage conditions of the underground media.

The pressure–time relationships of different components of underground structures, such as 

roof panels and exterior walls, depend not only on their distances from the location of detonation 

but also their geometric orientation with respect to detonation location. For example, an 

overhead burst produces the most critical loading for a roof panel, while a side burst produces 

the most critical loading for an exterior wall. Furthermore, the magnitude and distribution of the 

load acting on an underground structure component is greatly influenced by the deformability of 

the underground structure because of “soil arching” phenomena (Kiger, et al., 1984).

The effect of the soil is to modify the shock exerted on an underground structure as discussed in 

Section 2.2 of this report. The available capacity of the underground structure to resist blast 

load is reduced by the dead (or gravitational) load of the soil. At the same time, a portion of the 

soil surrounding the underground structure acts with the structural elements to increase the 

mass without proportional increase of stiffness, thereby increasing the natural period of vibration 

of the underground structure. A common approach is to treat the increase of mass 

approximately by assuming that the (i) mass of 0.61 m [2 ft] of soil acts with the mass of wall 

and (ii) entire mass of the soil supported by the roof or a depth of soil equal to one-quarter of the 

roof span (short span for a two-way panel), whichever is smaller, acts with the mass of the roof 

(DOD, 2008).

Underground structures should be designed so the dynamic response is limited to 

comparatively small deformations to prevent structural collapse due to Earth loads. Concrete 

underground structures should be designed so that the failure is caused by yielding of the 

reinforcing steel with sufficient ductility (DOD, 2008, 2002, 1986). The design of steel structures 

should use appropriate U.S. Department of Defense steel design codes and standards (DOD, 

2008, 2002, 1986). To ensure any failure of underground structures is in the ductile regime, 

DOD (2008, 1986) suggests various ductility factors for different components of structures 

subjected to explosive loads.

An underground structure that may be subjected to both explosive and seismic loads should be 

designed to perform its intended function under both these loads. Although both of these 

dynamic loads require high energy-absorbing capacity of the underground structures, their 

energy spectra are very different. There are differences both in the spectral shape and in the 

magnitude of the spectral quantities. As a result, one spectrum will not reasonably envelope 

another spectrum. Furthermore, explosive and seismic loads can be assumed to occur 

independently of each other.
6-2

The combination of seismic load with other loads is well established in the design codes and 

standards (e.g., ACI, 2011, 2006; AISC, 2007, 1994). Although the design load combinations 

for reinforced concrete and steel nuclear structures in American Concrete Institute (2006) and 

American Institute of Steel Construction (2007), respectively, include missile impact load as an 

abnormal load, that impact load is generated internally by an operational accident, such as pipe 

whipping generated by or during the postulated accident. This is not relevant to the external 

explosive loads on underground structures addressed in this report. However, a similar 

approach for combining external explosive load with other loads will be reasonable for the 

design of underground structures.
7-1

7. OTHER PUBLICATIONS

The listed references are provided in this section for information only. They are not an NRC 

endorsement or acceptance of material content, methods, or approaches. Exceptions are 

Regulatory Guides, which provide methods or approaches acceptable for meeting NRC 

regulatory requirements.

Bailey, A. and S. Murray. Explosives, Propellants, and Pyrotechnics. London, United Kingdom: 

Brassey’s. 1989.

Bowen, I.G., E.R. Fletcher, and D.R. Richmond. “Estimate of Man’s Tolerance to the Direct 

Effects of Air Blast.” Technical Report to Defense Atomic Support Agency, DASA 2113. 

Albuquerque, New Mexico: Lovelace Foundation for Medical Education and Research, 

AD 693105. October 1968.

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8-1

8. CONCLUSIONS

An underground structure may located with its upper elements at the ground surface or be deep 

underground. The physical positions of underground structures make them potentially 

vulnerable to terrorist attacks, though being located underground can provide a measure of 

protection. Explosive devices used in terrorist attacks are typically homemade, chemical-based 

explosive devices that could be fabricated from either military or commercial grade explosives, 

through theft or purchase, or from explosives that could be manufactured by combining publicly 

available common chemicals. An Improvised Explosive Device (IED) is the most prevalent form 

of explosive device that is used by terrorists. A Vehicle Borne Improvised Explosive Device 

(VBIED) is an IED in which a vehicle is used as a package and delivery means to conduct an 

attack. The energy released by an explosion is normally expressed as an equivalent weight of 

trinitrotoluene (TNT). The maximum TNT-equivalent explosive holding capacities of vehicles in 

the US range from about 227 to 27,215 kg [500 to 60,000 lb]. The type of vehicle used for a 

VBIED determines the height at which the VBIED detonates above the ground, making it an 

important parameter for explosion effects on surface and underground structures.

A detonation above the ground surface (often referred to as an air blast) produces a blast wave 

characterized by two distinct phases: the positive phase and the negative phase. During the 

positive phase, the blast wave overpressure rises very rapidly from zero (ambient) to peak 

value, subsides more slowly to zero (ambient), and further subsides to the negative phase. The 

peak value of negative pressure, which is called peak underpressure, is usually much smaller 

than the peak overpressure. The portion of the air-blast wave refracted into the ground and the 

underground wave generated by the blast at the ground surface are propagated through the 

underground media radially outward, potentially creating a crater and generating underground 

soil pressures that attenuate because of geometric and hysteresis effects. The underground 

soil pressure at a given location depends on the strength of the explosive, the properties of the 

media through which the blast waves propagate, and the distance from the explosion. This 

underground soil pressure affects the response of underground structures. Empirical equations

are available for calculating important parameters that affect underground structures, such as 

explosion wave arrival time at the location of the structure; duration of the explosive wave; peak 

free-field pressure; impulse force; and particle displacement, velocity, and acceleration.

This report presents a parametric study on the subsurface effects caused by a surface burst. 

The size of the TNT-equivalent charge was varied and the resulting soil pressures and surface 

crater size were predicted. Charge weights varied from 45.3 kg [100 lb] to 4,536 kg [10,000 lb]. 

The finite element predicted crater dimensions were within the range of the empirically derived 

dimensions for all the TNT-equivalent charge sizes considered. For the case of a 100-kg 

[220-lb]-TNT-equivalent charge, the finite element soil pressures correlated well with the 

pressures calculated from empirical relationships. Additional analyses were conducted for 

TNT-equivalent charge sizes of 4,536; 9,072; 18144; and 27,216 kg [10,000; 20,000; 40,000; 

and 60,000 lb]. For the larger explosive charges, it was necessary to use the element deletion 

technique to eliminate excessive mesh distortion. Using this element deletion technique 

resulted in predicted apparent crater depths that were within the empirically calculated range; 

however, the predicted crater radii were slightly less than the empirically calculated lower limit. 

It is recommended that the equivalent plastic strain soil failure criterion value be further 

investigated, as well as alternative failure criteria, such as equivalent total strain and maximum 

principal tensile strain. Element-to-particle conversion techniques were considered; however, it 

was determined that element-to-particle techniques in ABAQUS/Explicit are currently not 

possible because of documented code errors that remain unresolved by the code vendor. An 

Eulerian analysis was performed for a 454-kg [1,000-lb] explosive charge, which predicted an 
8-2

apparent crater depth within the range calculated using empirical equations of Gould (1981). 

Further investigation is recommended to evaluate the adequacy of mesh refinement in the 

air domain.

Design considerations are needed to protect underground structures subjected to dynamic 

forces generated by explosions in air, near the surface, and underground. These include

special features associated with the behavior of underground structures (e.g., closely coupled 

motion of underground structures and surrounding soil and larger secondary effects due to 

lateral displacement of underground structures) and their analysis and design using specific 

design methodology.

In general, underground structures should be designed to withstand both explosive loads and 

seismic loads. Although explosive and seismic loads are similar in that they both produce 

P-waves, S-waves, and surface waves, explosions result in higher P/S ratios than similarly 

located earthquakes, making the energy spectra very different. In addition, there may be 

significant differences in the magnitude of spectral quantities. As a result, one spectrum will not 

reasonably envelope another spectrum. Nevertheless, an approach similar to that in American 

Concrete Institute (2011, 2006) and American Institute of Steel Construction (2007, 1994) of 

combining internal accidental missile load with other loads may be a reasonable approach for 

combining the external explosive load with other loads for the design of underground structures. 
9-1

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NUREG/CR-7201 Characterizing Explosive Effects on Underground Structures