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13.5 Blast Loading and Blast Effects on Structures

Back

The use of vehicle bombs to attack city centers has been a feature of campaigns by terrorist organizations

around the world. A bomb explosion within or very near a building can have catastrophic effects,

destroying or severely damaging portions of the building’s external and internal structural frames,

collapsing walls, blowing out large expanses of windows, and shutting down critical life-safety systems,

such as fire detection and suppression, ventilation, light, water, sewage, and power systems. Loss of life

and injuries to occupants can result from many causes, including direct blast effects, structural collapse,

debris impact, fire, and smoke. The indirect effects can combine to inhibit or prevent timely evacuation,

thereby contributing to additional casualties. In addition, major catastrophes resulting from gaschemical

explosions or nuclear leakage result in large dynamic loads, greater than the original design

loads, of many structures. Owing to the threat of such extreme loading conditions, efforts have been

made during the past three decades to develop methods of structural analysis and design to resist blast

loads. The analysis and design of structures subjected to blast loads requires a detailed understanding of

blast phenomena and the dynamic response of various structural elements.

13.5.1 Explosions and Blast Phenomenon

An explosion is defined as a large-scale, rapid, and sudden release of energy. Explosions can be

categorized on the basis of their nature as physical, nuclear, or chemical events. In physical explosions,

energy may be released from the catastrophic failure of a cylinder of compressed gas, volcanic eruptions,

or even the mixing of two liquids at different temperatures. In a nuclear explosion, energy is released

from the formation of different atomic nuclei by the redistribution of the protons and neutrons within

the interacting nuclei; whereas the rapid oxidation of fuel elements (carbon and hydrogen atoms) is the

main source of energy in the case of chemical explosions.

Explosive materials can be classified according to their physical state as solids, liquids, or gases. Solid

explosives are mainly high explosives, for which blast effects are best known. They can also be classified

on the basis of their sensitivity to ignition as secondary or primary explosive. The latter is one that can be

easily detonated by simple ignition from a spark, flame, or impact. Materials such as mercury fulminate

and lead azide are primary explosives. Secondary explosives detonate creating blast (shock) waves, which

Trough Wave crest

FIGURE 13.19 A circular cylindrical structure exposed

to ocean waves.

13-34 Vibration and Shock Handbook

result in damage to the surroundings. Examples

include trinitrotoluene (TNT) and ammonium

nitrate and fuel oil (ANFO).

The detonation of a condensed high explosive

generates hot gases under pressure of up to

300 kbar and a temperature of about 3000 to

40008C. The hot gas expands, forcing out the

volume it occupies. As a consequence, a layer of

compressed air (blast wave) forms in front of this

gas volume, containing most of the energy released

by the explosion. The blast wave instantaneously

increases to a value of pressure above the ambient

atmospheric pressure. This is referred to as the

side-on overpressure, and decays as the shock wave

expands outward from the explosion source. After

a short time, the pressure behind the front may drop below the ambient pressure (see Figure 13.20 and

Figure 13.21). During such a negative phase, a partial vacuum is created and air is sucked in. This is also

accompanied by high suction winds that carry the debris for long distances away from the explosion

source.

13.5.2 Explosive Air-Blast Loading

The threat for a conventional bomb is defined by two equally important elements, the bomb size, or

charge weight, W ; and the standoff distance, R; between the blast source and the target (Figure 13.22).

For example, the blast that occurred at the basement of the World Trade Center in 1993 had the charge

weight of 816.5 kg TNT. The Oklahoma City bomb in 1995 had a charge weight of 1814 kg at a stand off

of 4.5 m (Longinow and Mniszewski, 1996). As terrorist attacks may range from a small letter bomb to a

gigantic truck bomb, as experienced in Oklahoma City, the mechanics of a conventional explosion and

their effects on the target must be addressed.

The observed characteristics of air-blast waves are found to be affected by the physical properties of the

explosion source. Figure 13.21 shows a typical blast pressure profile. At an arrival time of tA after the

explosion, pressure at that position suddenly increases to a peak value of overpressure, Pso; over the

ambient pressure, P0: The pressure then decays to the ambient pressure at time td until it reaches a partial

vacuum of peak underpressure P2

so; and eventually returns to the ambient pressure at time td þ t2

d : The

quantity Pso is usually referred to as the peak side-on overpressure, incident peak overpressure, or merely

the peak overpressure (TM 5-1300, 1990).

Shock velocity

Pressure

Distance from explosion

FIGURE 13.20 Blast wave propagation.

P(t)

Positive

duration td

Negative

duration td

−

Pso

−

tA tA+td

Positive Specific

Impulse

Negative Specific

Impulse

FIGURE 13.21 Blast wave pressure — time history.

Vibration and Shock Problems of Civil Engineering Structures 13-35

The incident peak overpressure, Pso; is amplified by a reflection factor as the shock wave encounters an

object or structure in its path. Except for the specific focusing of high-intensity shock waves at near 458

incidence, these reflection factors are typically greatest for normal incidence (a surface adjacent and

perpendicular to the source) and diminish with the angle of obliquity or angular position relative to the

source. Reflection factors depend on the intensity of the shock wave. For large explosions at normal

incidence these reflection factors may enhance the incident pressures by as much as an order of

magnitude.

Throughout the pressure – time profile, two main phases can be observed; the portion above ambient

pressure is called positive phase of duration td; while that below ambient is called negative phase of

duration, t2

d : The negative phase is of a longer duration and a lower intensity than the positive duration.

The duration of the positive-phase blast wave increases with range, resulting in a lower amplitude, longer

duration shock pulse the further a target structure is situated from the burst. Charges situated extremely

close to a target structure impose a highly impulsive, high-intensity pressure load over a localized region

of the structure; charges situated further away produce a lower-intensity, longer-duration uniform

pressure distribution over the entire structure. Eventually, the entire structure is engulfed in the shock

wave, with reflection and diffraction effects creating focusing and shadow zones in a complex pattern

around the structure. During the negative phase, the weakened structure may be subjected to impact by

debris that may cause additional damage.

If the exterior building walls are capable of resisting the blast load, the shock front penetrates

through window and door openings, subjecting the floors, ceilings, walls, contents, and people within to

sudden pressures and fragments from shattered windows, doors, and other fixtures. Building

components not capable of resisting the blast wave will fracture and be further fragmented and

moved by the dynamic pressure that immediately follows the shock front. Building contents and people

will be displaced and tumbled in the direction of blast wave propagation. In this manner the blast will

propagate through the building.

13.5.2.1 Blast Wave Scaling Laws

All blast parameters are primarily dependent on the amount of energy released by a detonation in the

form of a blast wave and the distance from the explosion. A universal normalized description of the

blast effects can be given by scaling distance relative to ðE=P0Þ1=3; and pressure relative to P0; where E is

the energy release (kJ) and P0 the ambient pressure (typically 100 kN/m2). For convenience, however,

it is general practice to express the basic explosive input or charge weight W as an equivalent mass of

TNT. Results are then given as a function of the dimensional distance parameter (scaled distance)

Reflected

Pressure

Over-pressure

(side-on)

Blast wave

Stand-off distance

FIGURE 13.22 Blast loads on a building.

13-36 Vibration and Shock Handbook

Z ¼ R=W 1=3; where R is the actual effective distance from the explosion. W is generally expressed in

kilograms. Scaling laws provide parametric correlations between a particular explosion and a standard

charge of the same substance.

13.5.2.2 Prediction of Blast Pressure

Blast wave parameters for conventional high-explosive materials have been the focus of a number of

studies during the 1950s and 1960s. Estimations of peak overpressure due to a spherical blast based on the

scaled distance Z ¼ R=W 1=3 were introduced by Brode (1955) as

Pso ¼

6:7

Z3 þ 1 bar ðPso . 10 barÞ

Pso ¼

0:975

Z þ

1:455

Z2 þ

5:85

Z3 2 0:019 bar ð0:1 bar , Pso , 10 barÞ

ð13:20Þ

Newmark and Hansen (1961) introduced a relationship to calculate the maximum blast overpressure,

Pso; in bars, for a high-explosive charge detonated at the ground surface as

Pso ¼ 6784

R3 þ 93

􀀏 􀀐12

ð13:21Þ

Another expression of the peak overpressure in kPa was introduced by Mills (1987), in which W is

expressed as the equivalent charge weight in kg of TNT, and Z is the scaled distance

Pso ¼

1772

Z3 2

114

Z2 þ

108

Z ð13:22Þ

As the blast wave propagates through the atmosphere, the air behind the shock front is moving outward

at a lower velocity. The velocity of the air particles, and hence the wind pressure, depends on the peak

overpressure of the blast wave. This later velocity of the air is associated with the dynamic pressure, qðtÞ:

The maximum value, qs; say, is given by

qs ¼ 5P2

so=2ðPso þ 7P0Þ ð13:23Þ

If the blast wave encounters an obstacle perpendicular to the direction of propagation, reflection

increases the overpressure to a maximum reflected pressure Pr as

Pr ¼ 2Pso

7P0 þ 4Pso

7P0 þ Pso

􀀘 􀀙

ð13:24Þ

A full discussion and extensive charts for predicting blast pressures and blast durations are given by TM

5-1300 (1990) and Mays and Smith (1995). Some representative numerical values of peak-reflected

overpressure are given in Table 13.7.

TABLE 13.7 Peak-Reflected Overpressures Pr (in MPa) with Different W – R Combinations

R (m) W

100 kg TNT 500 kg TNT 1000 kg TNT 2000 kg TNT

1 165.8 354.5 464.5 602.9

2.5 34.2 89.4 130.8 188.4

5 6.65 24.8 39.5 60.19

10 0.85 4.25 8.15 14.7

15 0.27 1.25 2.53 5.01

20 0.14 0.54 1.06 2.13

25 0.09 0.29 0.55 1.08

30 0.06 0.19 0.33 0.63

Vibration and Shock Problems of Civil Engineering Structures 13-37

For design purposes, reflected overpressure can be idealized by an equivalent triangular pulse of

maximum peak pressure Pr and time duration td; which yields the reflected impulse ir:

ir ¼

Prtd ð13:25Þ

The reflection effect dissipates as the perturbation propagates to the edges of the obstacle at a velocity

related to the speed of sound ðUsÞ in the compressed and heated air behind the wave front. Denoting the

maximum distance from an edge as S (for example, the lesser of the height or half the width of a

conventional building), the additional pressure due to reflection is considered to reduce from Pr 2 Pso to

0 in time 3S=Us: Conservatively, Us can be taken as the normal speed of sound, about 340 m/sec, and the

additional impulse to the structure evaluated on the assumption of a linear decay.

After the blast wave has passed the rear corner of a prismatic obstacle, the pressure similarly propagates

on to the rear face; linear build-up over duration 5S=Us has been suggested. For skeletal structures, the

effective duration of the net overpressure load is thus small, and the drag loading based on the dynamic

pressure is then likely to be dominant. Conventional wind-loading pressure coefficients may be used,

with the conservative assumption of instantaneous build-up when the wave passes the plane of the

relevant face of the building, the loads on the front and rear faces being numerically cumulative for the

overall load effect on the structure. Various formulations have been put forward for the rate of decay of

the dynamic pressure loading; a parabolic decay (i.e., corresponding to a linear decay of equivalent wind

velocity) over a time equal to the total duration of positive overpressure is a practical approximation.

13.5.3 Gas Explosion Loading and Effect of Internal Explosions

In the circumstances of a progressive build-up of fuel in a low-turbulence environment, typical of

domestic gas explosions, flame propagation on ignition is slow and the resulting pressure pulse is

correspondingly extended. The specific energy of combustion of a hydrocarbon fuel is very high

(46,000 kJ/kg for propane, compared with 4520 kJ/kg for TNT) but widely differing effects are possible

according to the conditions at ignition.

Internal explosions often produce complex pressure loading profiles as a consequence of having two

loading phases. The first results from the blast overpressure reflection and, due to the confinement

provided by the structure, re-reflection will occur. Depending on the degree of confinement of the

structure, the confined effects of the resulting pressures may cause different degrees of damage to the

structure. On the basis of the confinement effect, target structures can be described as either vented or

unvented. The latter must be stronger to resist a specific explosion yield than a vented structure where

some of the explosion energy would be dissipated by the breaking of window glass or fragile partitions.

Generally, venting following the failure of windows (typically at 7 kN/m2) greatly reduces the peak

values of internal pressures. A study of this problem at the Building Research Establishment (Ellis and

Crowhurst, 1991) showed that an explosion fuelled by a 200 ml aerosol canister in a typical domestic

room produced a peak pressure of 9 kN/m2 with a pulse duration over 0.1 sec. This is long by comparison

with the natural frequency of wall panels in conventional building construction, and a quasi-static design

pressure is commonly advocated. Much higher pressures with a shorter timescale are generated in

turbulent conditions. Suitable conditions arise in buildings in multiroom explosions on the passage of

the blast through doorways, but can also be created by obstacles closer to the release of the gas. These may

be presumed to occur on a release of gas due to a failure of industrial pressure vessels or pipelines.

13.5.4 Structural Response to Blast Loading

Complexity in analyzing the dynamic response of blast-loaded structures involves the effect of high

strain-rates, nonlinear inelastic material behavior, uncertainties of blast-load calculations, and timedependent

deformations. Therefore, to simplify the analysis, a number of assumptions related to the

response of structures and loads have been proposed and widely accepted. To establish the principles of

13-38 Vibration and Shock Handbook

this analysis, the structure is idealized as an single-DoF system and the link between the positive duration

of the blast load and the natural period of vibration of the structure is established. This leads to blast-load

idealization and simplifies the classification of the blast-loading regimes.

13.5.4.1 Elastic Single-Degree-of-Freedom Systems

The simplest discretization of transient problems is by means of the single-DoF approach. The actual

structure can be replaced by an equivalent system of one concentrated mass and one weightless spring,

representing the resistance of the structure against deformation. Such an idealized system is illustrated in

Figure 13.23. The structural mass, M; is under the effect of an external force, FðtÞ; and the structural

resistance, R; is expressed in terms of the vertical displacement, y; and the spring constant, K:

The blast load can also be idealized as a triangular pulse having a peak force Fm and positive-phase

duration td (see Figure 13.23). The forcing function is given as

FðtÞ ¼ Fm 1 2

􀀏 􀀐

ð13:26Þ

The blast impulse is approximated as the area under the force – time curve, and is given by

I ¼

Fmtd ð13:27Þ

The equation of motion of the undamped elastic single-DoF system for a time ranging from 0 to the

positive-phase duration, td; is given by Biggs (1964) as

My€ þ Ky ¼ Fm 1 2

􀀏 􀀐

ð13:28Þ

The general solution can be expressed as

Displacement yðtÞ ¼

K ð1 2 cos vtÞ þ

Ktd

sin vt

2 t

􀀏 􀀐

Velocity y_ðtÞ ¼

dt ¼

v sin vt þ

td ðcos vt 2 1Þ

􀀒 􀀓 ð13:29Þ

in which v is the natural circular frequency of vibration of the structure and T is the natural period of

vibration of the structure given as

v ¼

T ¼

ffiffiffiffiffi

ð13:30Þ

The maximum response is defined by the maximum dynamic deflection ym; which occurs at time tm: The

maximum dynamic deflection ym can be evaluated by setting dy=dt in Equation 13.29 equal to zero, that

is, when the structural velocity is zero. The dynamic load factor (DLF) is defined as the ratio of the

maximum dynamic deflection ym to the static deflection yst which would have resulted from the static

Stiffness, K Displacement

y(t)

Force

F(t)

Time

F (t)

(a) (b)

FIGURE 13.23 (a) Single-DoF system and (b) blast loading.

Vibration and Shock Problems of Civil Engineering Structures 13-39

application of the peak load Fm; as follows:

DLF ¼

ymax

yst ¼

ymax

Fm=K ¼ cðvtdÞ ¼ C

􀀏 􀀐

ð13:31Þ

The structural response to blast loading is significantly influenced by the ratio td=T or vtd ðtd=T ¼

vtd=2pÞ: Three loading regimes are categorized as follows:

1. vtd , 0:4: impulsive loading regime

2. vtd , 0:4: quasi-static loading regime

3. 0:4 , vtd , 40: dynamic loading regime

13.5.4.2 Elasto-Plastic Single-Degree-of-Freedom Systems

Structural elements are expected to undergo

large inelastic deformation under a blast load or

high-velocity impact. The exact analysis of

dynamic response is then only possible by a

step-by-step numerical solution requiring nonlinear

dynamic finite-element software. However,

the degree of uncertainty in both the determination

of the loading, and the interpretation of

acceptability of the resulting deformation, is

such that the solution of a postulated equivalent

ideal elasto-plastic single-DoF system (Biggs, 1964) is commonly used. Interpretation is based on

the required ductility factor m ¼ ym=ye (Figure 13.24).

For example, a uniform simply supported beam has first mode shape fðxÞ ¼ sin px=L and the

equivalent mass is M ¼ ð1=2ÞmL; where L is the span of the beam and m is mass per unit length.

The equivalent force corresponding to a uniformly distributed load of intensity p is F ¼ ð2=pÞpL: The

response of the ideal bilinear elasto-plastic system can be evaluated in closed form for the triangular load

pulse comprising rapid rise and linear decay, with maximum value Fm and duration td. The result for the

maximum displacement is generally presented in chart form (TM 5-1300) as a family of curves for

Deflection

Resistance

FIGURE 13.24 Simplified resistance function of an

elasto-plastic single-DoF system.

0.1 0.5 1 5 10 20

0.1

0.5

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

td/T

ym /ye

Numbers next to curves are Ru/Fm

0.9

1.0

1.2

1.5

2.0

FIGURE 13.25 Maximum response of an elasto-plastic single-DoF system to a triangular load (TM 5-1300).

13-40 Vibration and Shock Handbook

selected values of Ru =Fm showing the required ductility m as a function of td=T; in which Ru is the

structural resistance of the beam and T is the natural period (Figure 13.25).

13.5.5 Material Behaviors at High Strain Rate

Blast loads typically produce very high strain-rates in the range of 100 to 10,000 sec21. This high

straining (loading) rate would alter the dynamic mechanical properties of target structures and,

accordingly, the expected damage mechanisms for various structural elements. For reinforced concrete

structures subjected to blast effects, the strength of concrete and steel reinforcing bars can increase

significantly due to the strain-rate effect. Figure 13.26 shows approximate ranges of the expected strain

rates for different loading conditions. It can be seen that the ordinary static strain rate is located in the

range of 1026 to 1025 sec21, while blast pressures normally yield loads associated with strain rates in the

range of 100 to 10,000 sec.

13.5.5.1 Dynamic Properties of Concrete under High Strain Rates

The mechanical properties of concrete under dynamic loading conditions can be quite different from that

under static loads. While the dynamic stiffness does not change very much compared with the static

stiffness, the stresses that are sustained for a certain period under dynamic conditions may gain values

that are remarkably higher than the static compressive strength (Figure 13.27). Strength magnification

factors as high as four in compression and up to six in tension for strain rates in the range of 100 to

1000 sec21 have been reported (Grote et al., 2001).

10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 102 103 104

Quasi-static Earthquake Impact Blast

Strain rate (s−1)

FIGURE 13.26 Strain rates associated with different types of loading.

100

150

200

250

0.002 0.004 0.006 0.008 0.01

Strain

Stress (MPa)

Static

e. = 49

e. = 97

e. = 233

e. = 264

FIGURE 13.27 Stress – strain curves of concrete at different strain rates. (Source: Data from Ngo,T. et al., Proc. 18th

Australasian Conf. on Mechanics of Structures and Materials, Perth, Australia. With permission.)

Vibration and Shock Problems of Civil Engineering Structures 13-41

For the increase in peak compressive stress ðf 0cÞ;

a dynamic increase factor (DIF) is introduced in

the Comite´ Euro-International du Be´ton (CEBFIP;

1990) model (Figure 13.28) for strain-rate

enhancement of concrete as follows:

DIF ¼

1_s

􀀏 􀀐1:026a

for 1_ # 30 sec21 ð13:32Þ

DIF ¼ g

1_s

􀀏 􀀐1=3

for 1_ . 30 sec21 ð13:33Þ

where

1_ ¼ strain rate

1_s ¼ 30 £ 1026 sec21 (quasi-static strain rate)

log g ¼ 6:156a 2 2

a ¼ 1=ð5 þ 9f 0c=fcoÞ

fco ¼ 10 MPa ¼ 1450 psi

13.5.5.2 Dynamic Properties of Reinforcing Steel under High Strain Rates

Owing to the isotropic properties of metallic materials, their elastic and inelastic response to dynamic

loading can easily be monitored and assessed. Norris et al. (1959) tested steel with two different static

yield strengths (330 and 278 MPa) under tension at strain rates ranging from 1025 to 0.1 sec21. Strength

increases of 9 – 21% and 10 – 23% were observed for the two steel types, respectively. Dowling and

Harding (1967) conducted tensile experiments using the tensile version of the Split Hopkinton’s Pressure

Bar (SHPB) on mild steel using strain rates varying between 1023 and 2000 sec21. It was concluded from

this test series that materials of body-centered cubic (BCC) structure (such as mild steel) showed the

greatest strain rate sensitivity, the lower yield tensile strength of mild steel was almost doubled, the

ultimate tensile stress was increased by about 50%, the upper yield tensile strength considerably

increased, and the ultimate tensile strain decreased by different percentages, depending on the strain rate.

Malvar (1998) also studied the strength enhancement of steel reinforcing bars under the effect of high

strain rates. This was described in terms of the DIF, which can be evaluated for different steel grades and

for yield stresses, fy ; ranging from 290 to 710 MPa as

DIF ¼

1024

􀀏 􀀐a

ð13:34Þ

where for calculating yield stress a ¼ afy ;

afy ¼ 0:074 2 0:04 ð fy =414Þ ð13:35Þ

and for ultimate stress calculation a ¼ afu

afu ¼ 0:019 2 0:009 ð fy =414Þ ð13:36Þ

13.5.6 Failure Modes of Blast-Loaded Structures

Blast-loading effects on structural members may produce both local and global responses associated

with different failure modes. The type of structural response depends mainly on the loading rate, the

orientation of the target with respect to the direction of the blast wave propagation, and boundary

conditions. The general failure modes associated with blast loading can be flexure, direct shear, or

punching shear. Local responses are characterized by localized breaching and spalling, and generally

1.E−04 1.E−02 1.E+00 1.E+02 1.E+04

Strain rate (s−1)

Dynamic factor

FIGURE 13.28 Dynamic increase factor for peak stress

of concrete (CEB-FIP model).

13-42 Vibration and Shock Handbook

result from the close-in effects of explosions, while global responses are typically manifested as

flexural failure.

13.5.6.1 Global Structural Behavior

The global response of structural elements is generally a consequence of transverse (out of plane) loads

with long exposure time (quasi-static loading), and is usually associated with global membrane

(bending) and shear responses. Therefore, the global response of above-ground reinforced concrete

structures subjected to blast loading is referred to as membrane/bending failure.

The second global failure mode to be considered is shear failure. It has been found that under the effect

of both static and dynamic loads, four types of shear failure can be identified: diagonal tension, diagonal

compression, punching shear, and direct (dynamic) shear (Woodson, 1993). The first two types are

common in reinforced concrete elements under static loads, while punching shear is associated with local

shear failure; for example, the familiar case is column punching through flat slabs. These shear response

mechanisms have relatively minor structural effect in case of blast loading and can be neglected. The

fourth type of shear failure is direct (dynamic) shear. This failure mode is primarily associated with

transient short duration dynamic loads that result from blast effects, and it depends mainly on the

intensity of the pressure waves. The associated shear force is many times higher than the shear force

associated with flexural failure modes. The high shear stresses may lead to a direct global shear failure and

it may occur very early (within a few milliseconds of shock wave arrival to the facing structure’s surface)

even prior to any significant bending deformations.

13.5.6.2 Localized Structural Behavior

The close-in effect of an explosion may cause localized shear or flexural failure in the closest

structural elements. This depends mainly on the distance between the explosion center and the target,

and the relative strength/ductility of the structural elements. The localized shear failure takes the form

of localized punching and spalling, which produces low and high-speed fragments. The punching

effect is frequently referred to as breaching, which is well known in high-velocity impact applications

and in the case of explosions close to the surface of structural members. Breaching failures are

typically accompanied by spalling and scabbing of concrete covers, as well as fragments and debris

(Figure 13.29).

FIGURE 13.29 Breaching failure due to a close-in explosion of 6000 kg TNT equivalent (photograph by

Tuan Ngo).

Vibration and Shock Problems of Civil Engineering Structures 13-43

13.5.6.3 Pressure – Impulse (P – I ) Diagrams

The P– I diagram is an easy way to mathematically relate a specific damage level to a combination of

blast pressures and the corresponding impulses for a particular structural element. An example P– I

diagram is given in Figure 13.30. This figure shows the levels of damage of a structural member, in

which region (I) corresponds to severe structural damage and region (II) refers to no or minor damage.

There are other P– I diagrams that are concerned with human responses to blasts, in which three

categories of blast-induced injury are identified as primary, secondary, and tertiary injury (Baker et al.,

1983).

13.5.7 Blast Wave – Structure Interaction

The structural behavior of an object or structure exposed to such a wave may be analyzed by dealing with

two main issues. Firstly, blast-loading effects, that is, forces that result from the action of the blast

pressure; secondly, the structural response, or the expected damage criteria associated with such loading

effects. It is important to consider the interaction of the blast waves with target structures. This might be

quite complicated in the case of complex structural configurations. However, it is possible to consider

some equivalent simplified geometry. Accordingly, in analyzing the dynamic response to blast loading,

two types of target structures can be considered: diffraction-type and drag-type structures. As these

names imply, the former would be affected mainly by diffraction (engulfing) loading and the latter by

drag loading. It should be emphasized that actual buildings will respond to both types of loading and the

distinction is made primarily to simplify the analysis. The structural response will depend upon the size,

shape, and weight of the target, how firmly it is attached to the ground, and also on the existence of

openings in each face of the structure.

13.5.8 Effect of Ground Shocks

Above ground or shallow-buried structures can be subjected to ground shock resulting from the

detonation of explosive charges that are on, or close to, the ground surface. The energy imparted to the

Pressure Ps (kPa)

100 101 102 103

101

100

10−1

10−2

(I) − Severe damage

(II) − No damage / minor damage

Impulse is (kPa.sec)

FIGURE 13.30 Typical pressure – impulse (P– I) diagram.

13-44 Vibration and Shock Handbook

ground by the explosion is the main source of ground shock. A part of this energy is directly transmitted

through the ground as direct-induced ground shock, while part is transmitted through the air as airinduced

ground shock. Air-induced ground shock results when the air-blast wave compresses the ground

surface and sends a stress pulse into the ground underlayers. Generally, motion due to air-induced

ground shock is maximum at the ground surface and attenuates with depth (TM 5-1300, 1990).

The direct-induced shock results from the direct transmission of explosive energy through the ground.

For a point of interest on the ground surface, the net experienced ground shock results from a

combination of both the air-induced and direct-induced shocks.

13.5.8.1 Loads from Air-Induced Ground Shock

To overcome complications of predicting actual ground motion, one-dimensional wave propagation

theory has been employed to quantify the maximum displacement, velocity, and acceleration in terms of

the already known blast wave parameters (TM 5-1300). The maximum vertical velocity at the ground

surface, Vv ; is expressed in terms of the peak incident overpressure, Pso; as

Vv ¼

Pso

rCp ð13:37Þ

where r and Cp are, respectively, the mass density and the wave seismic velocity in the soil.

By integrating the vertical velocity in Equation 13.37 with time, the maximum vertical displacement at

the ground surface, Dv ; can be obtained as

Dv ¼

1000rCp ð13:38Þ

Accounting for the depth of soil layers, an empirical formula is given by TM 5-1300 to estimate the

vertical displacement in meters so that

Dv ¼ 0:09W 1=6ðH=50Þ0:6ðPsoÞ2=3 ð13:39Þ

where W is the explosion yield in 109 kg and H is the depth of the soil layer in meters.

13.5.8.2 Loads from Direct Ground Shock

As a result of the direct transmission of the explosion energy, the ground surface experiences

vertical and horizontal motions. Some empirical equations were derived (TM 5-1300) to predict

the direct-induced ground motions in three different ground media; dry soil, saturated soil, and rock

media. The peak vertical displacement in m/sec at the ground surface for rock, DVrock and dry soil, DVsoil

are given as

DVrock ¼

0:25R1=3W 1=3

Z1=3 ð13:40Þ

DVsoil ¼

0:17R1=3W 1=3

Z2:3 ð13:41Þ

The maximum vertical acceleration, Av ; in m/sec2 for all ground media is given by

Av ¼

1000

W 1=8Z2 ð13:42Þ

13.5.9 Technical Design Manuals for Blast-Resistant Design

This section summarizes applicable military design manuals and computational approaches to predicting

blast loads and the responses of structural systems. Although the majority of these design guidelines were

focused on military applications, this knowledge is relevant for civil design practice.

Vibration and Shock Problems of Civil Engineering Structures 13-45

Structures to Resist the Effects of Accidental Explosions, TM 5-1300 (U.S. Departments of the Army, Navy,

and Air Force, 1990): This manual appears to be the most widely used publication by both military and

civilian organizations for designing structures to prevent the propagation of explosion, and to provide

protection for personnel and valuable equipment. It includes step-by-step analysis and design

procedures, including information on such items as (1) blast, fragment, and shock-loading; (2) principles

of dynamic analysis; (3) reinforced and structural steel design; and (4) a number of special design

considerations, including information on tolerances and fragility, as well as shock isolation. Guidance is

provided for the selection and design of security windows, doors, utility openings, and other components

that must resist blast and forced-entry effects.

A Manual for the Prediction of Blast and Fragment Loadings on Structures, DOE/TIC-11268 (U.S.

Department of Energy, 1992): This manual provides guidance to the designers of facilities subject to

accidental explosions and aids in the assessment of the explosion-resistant capabilities of existing

buildings.

Protective Construction Design Manual, ESL-TR-87-57 (Air Force Engineering and Services Center,

1989): This manual provides procedures for the analysis and design of protective structures exposed to

the effects of conventional (nonnuclear) weapons, and is intended for use by engineers with a basic

knowledge of weapons effects, structural dynamics, and hardened protective structures.

Fundamentals of Protective Design for Conventional Weapons, TM 5-855-1 (U.S. Department of the

Army, 1986): This manual provides procedures for the design and analysis of protective structures

subjected to the effects of conventional weapons. It is intended for use by engineers involved in designing

hardened facilities.

The Design and Analysis of Hardened Structures to Conventional Weapons Effects (DAHS CWE, 1998):

This new joint services manual, written by a team of more than 200 experts in conventional weapons and

protective structures engineering, supersedes U.S. Department of the Army TM 5-855-1, Fundamentals

of Protective Design for Conventional Weapons (1986), and Air Force Engineering and Services Centre

ESL-TR-87-57, Protective Construction Design Manual (1989).

Structural Design for Physical Security — State of the Practice Report (Conrath et al., 1995): This report

is intended to be a comprehensive guide for civilian designers and planners who wish to incorporate

physical security considerations into their designs or building retrofit efforts.

13.5.10 Computer Programs for Blast and Shock Effects

Computational methods in the area of blast effects mitigation are generally divided into those used for

the prediction of blast loads on the structure and those for the calculation of structural responses to the

loads. Computational programs for blast prediction and structural response use both first-principle and

semiempirical methods. Programs using the first-principle method can be categorized into uncouple and

couple analyses. The uncouple analysis calculates blast loads as if the structure (and its components) were

rigid, and then applies these loads to a responding model of the structure. The shortcoming of this

procedure is that, when the blast field is obtained with a rigid model of the structure, the loads on the

structure are often overpredicted, particularly if significant motion or the failure of the structure occurs

during the loading period.

For a coupled analysis, the blast simulation module is linked with the structural response module. In

this type of analysis, the computational fluid mechanics (CFD) model for blast-load prediction is solved

simultaneously with the computational solid mechanics (CSM) model for structural response. By

accounting for the motion of the structure while the blast calculation proceeds, the pressures that arise

due to the motion and failure of the structure can be predicted more accurately. Examples of this type of

computer software are AUTODYN, DYNA3D, LS-DYNA, and ABAQUS. Table 13.8 provides a listing of

computer programs that are currently being used to model blast effects on structures.

Prediction of the blast-induced pressure field on a structure and its response involves highly nonlinear

behavior. Computational methods for blast-response prediction must therefore be validated by

comparing calculations to experiments. Considerable skill is required to evaluate the output of the

13-46 Vibration and Shock Handbook

computer software, both as to its correctness and its appropriateness to the situation modeled; without

such judgment, it is possible through a combination of modeling errors and poor interpretation to

obtain erroneous or meaningless results. Therefore, successful computational modeling of specific blast

scenarios by engineers unfamiliar with these programs is difficult, if not impossible.

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