elastic plastic response spectra for exponential blast loading

International Journal of Impact Engineering 30 (2004) 323–343

Elastic–plastic response spectra for exponential blast loading

Charis J. Gantes*, Nikos G. Pnevmatikos

Metal Structures Laboratory, School of Civil Engineering, National Technical University of Athens,

9 Heroon Polytechneiou, Zografou GR-15780, Greece

Received 21 June 2001; received in revised form 12 June 2003

Abstract

The design of structures subjected to loads due to explosions is often treated by means of elastic–plasticresponse spectra. Such spectra that are currently available in the literature were computed on the basis oftriangular shape of blast pressure with respect to time. In the present paper, response spectra based on anexponential distribution of blast pressure, which is in better agreement to experimental data, are proposed.To that effect, analytical expressions of the solutions of the pertinent equations of motion have beenobtained via symbolic manipulation software, and have been used to carry out an extensive parametricstudy. A comparison of the spectra obtained by the proposed approach to the existing ones, reveals that thecommonly used assumption of triangular blast load evolution with time can sometimes be slightlyunconservative, particularly for flexible structural systems, but can also be significantly overconservativefor stiffer structures.r 2003 Elsevier Ltd. All rights reserved.

Keywords: Blast loading; Elastic–plastic response spectra; Exponential load distribution; Structural design; Dynamic

analysis

1. Introduction

Structures may experience blast loads due to military actions, accidental explosions or terroristactivities. Such loads may cause severe damage or collapse due to their high intensity, dynamicnature, and usually different direction compared to common design loads. Collapse of onestructural member in the vicinity of the source of explosion, may then create critical stressredistributions and lead to collapse of other members and, eventually, of the whole structure. Arecent example of such a failure was the well-known collapse of the Alfred P. Murrah FederalBuilding in Oklahoma City, Oklahoma following a terrorist attack [1,2].

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*Corresponding author. Metal Structures Laboratory, School of Civil Engineering, National Technical University of

Athens, P.O. Box 31830, Athens 100-35, Greece. Fax: +30-1-7723442.

E-mail address: [email protected] (C.J. Gantes).

0734-743X/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0734-743X(03)00077-0

For some structures blast-resistant design may be required, if their use is such that there is ahigh risk for such a loading incident to be encountered. Examples include government buildings,constructed facilities in petrochemical plants or bunkers in military installations. For suchstructures, it is desirable to establish design procedures and construction techniques necessary toachieve the required strength to resist the applied blast loads [3,4].This problem can be tackled in several different ways. The approach that more accurately

describes the dynamic response of structures to explosive loads is via numerical analysis, usuallyby means of the finite element method. Such analyses can capture the geometry of the structure,the spatial and temporal distribution of the applied blast pressure, as well as the effects of materialand geometric nonlinearity, in a satisfactory manner, and have been performed by severalinvestigators.An example of this type is the work of Louca et al. [5], who presented results of the response of

a typical blast wall and a tee-stiffened panel subjected to hydrocarbon explosions with geometries

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Nomenclature

a ratio of maximum resistance of the SDOF Rm to start peak blast load pressure Psb shape parameterk elastic stiffnessPs start peak blast load pressurePðtÞ blast load pressure P as a function of time t

R distance from the center of a spherical charge in metersRm maximum resistance of the SDOFT period of the systemtd time of reversal of direction of pressuretel time when the system reaches its elastic limittm time when the system reaches its maximum displacementt0m time when maximum displacement in stage (4) occurs.tm2 time when maximum displacement for the second cycle in second stage occurst0m2 time when maximum displacement for the second cycle in fourth stage occurstreb time of transition from stages (3) to (4)vel velocity at the end of stage (1)W charge mass expressed in kilograms of equivalent TNTx displacement response of the systemxel elastic limit of systemxm maximum displacementx0m maximum displacement in stage (4)

xm2 maximum displacement of the second cycle in the second stage.x0m2 maximum displacement of the second cycle in the fourth stage

%xðtÞ ratio of response x to elastic limit xelZ dimensionless scaled distancem ratio of maximum displacement xm to elastic limit xel; or dynamic load factor

C.J. Gantes, N.G. Pnevmatikos / International Journal of Impact Engineering 30 (2004) 323–343324

typical of those used in offshore structures. They performed nonlinear finite element analyses,accounting for the effects of plasticity, strain rate and buckling. They also compared their resultswith experimental data as well as approximate solutions obtained with a single degree-of-freedommodel. Other investigators have performed similar analyses to point out the importance ofdetailing, for example [6], who performed nonlinear dynamic finite element analysis of typicalthree-dimensional reinforced concrete joints in order to investigate the effect of reinforcementdetails.However, such analyses require highly specialized software, elaborate calibration of the models

with experimental [7] or other established results, and extensive computational effort. Therefore,this approach is used, for the time being, primarily for research purposes. Practice-orienteddesign, assessment and protection methods do take advantage of advanced numerical analysis,but focus also on engineering detailing, connections and conceptual design [8,9].An alternative design approach, which involves several approximations, but is rather easily

applicable in routine design, is recommended by the US Department of the Army TM5-1300 [10]and has been adopted by other researchers, for example [11]. It is based upon substituting thestructural element by a stiffness equivalent, single degree-of-freedom structural system, and usingelastic–plastic response spectra to predict the maximum response of this system.The response spectra accompanying the above methodology in the literature have been

computed via numerical integration of the equations of motion, assuming triangularloading evolution with time, and applying the average acceleration method. These spectra wereproduced by Biggs in 1964 [12]. The fact that this design approach is commonly suggested in theliterature is verified by a chapter by Watson in a recent book edited by Kappos [13], where thesame method is proposed. Watson stresses that the response depends on synchronization withthe rebound of the structure, and that a good knowledge of the time and the space variation of theblast load is critical to obtaining the correct response. Nevertheless, he also uses the spectra byBiggs, assuming triangular shape of the loading function and elastoplastic behavior of thematerial.This assumption is acceptable and provides satisfactory results in most cases, considering also

all other uncertainties that are inherent in this design approach, such as material modeling,substitution of the real structure by a single degree-of-freedom (SDOF) model, and explosioncharacteristics. Nevertheless, the fact remains that, according to experimental observations[14–16], the actual time variation of blast pressure can be approximated much better by anexponential function.The objective of this paper is to assess the importance of the load-shape assumption for

practical design, and to provide a more accurate design tool, without sacrificing ease of use for thepracticing engineer. This is achieved by deriving the equations of motion based on a moreaccurate exponential distribution of blast pressure, and accounting for all stages and cycles ofresponse, obtaining analytical expressions of the solutions via symbolic manipulation software,and using these solutions to carry out an extensive parametric study, and draw a new set ofresponse spectra. The intended contribution of this paper is, thus, not in the methodology ofproducing the spectra but in the spectra themselves, which improve the reliability of the designmethod by accounting for a more realistic loading function. To the authors’ knowledge, suchspectra for exponential loading distribution are not available in the literature. A comparison ofthese spectra to the existing ones reveals that the commonly used assumption of triangular blast

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C.J. Gantes, N.G. Pnevmatikos / International Journal of Impact Engineering 30 (2004) 323–343 325

load evolution with time can sometimes be slightly unconservative, particularly for flexiblestructural systems, but can also be significantly overconservative for stiffer structures.

2. Idealization of blast loading

The usual simple way to simulate a blast load for structural analysis and design purposes, iswith a triangular distribution of pressure with respect to time, which has a start peak pressure Psand decreases linearly with time to zero within a time period td; as illustrated by the dotted line inFig. 1, and described by the equation

PðtÞ ¼Ps 1�

t

td

� �; totd;

0; t > td:

8><>: ð1Þ

However, chemical investigation and experimental data have shown that the evolution of blastload pressure P with time t can be simulated more accurately by an exponential distribution[10,11,13], also having a start peak pressure Ps; as described by the following equation [14,15]:

PðtÞ ¼ Ps 1�t

td

� �e�bt=td : ð2Þ

In Eq. (2), plotted by the continuous line of Fig. 1, td is the time of reversal of direction ofpressure, and b is a shape parameter depending on the dimensionless scaled distance Z; given by

Z ¼RffiffiffiffiffiffiW3

p ; ð3Þ

where R is the distance from the center of a spherical charge in meters and W is the charge massexpressed in kilograms of equivalent TNT. Plots of several blast wave parameters, such as peakstatic overpressure, time duration of positive phase and the impulse of the wave with respect to Zare given in a number of references.The parameter b is decisive for the extent of negative phase in the exponential distribution of

Eq. (2). If b is smaller than one, there is an important negative phase, while for b larger than onethe negative phase becomes less significant. Baker et al. [14] provide a graph plotting b with

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0 2 4 6 8 10

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

t/td

Exponential distribution (b=1)

Triangular distribution

P/P

s

Fig. 1. Exponential and triangular distribution of pressure due to blast load.


respect to the scaled distance Z; where it is shown that the parameter b ranges from 0.1 to 10.Fig. 2 illustrates how the shape of the exponential distribution depends on b; and gives thecorresponding values of positive and negative pressure areas. In this work, the values b ¼ 1; 2, and3 are used, with shapes shown in Figs. 2a–c, respectively. The value b ¼ 1 is a reasonable averagevalue, corresponding to the area of negative pressure being equal to the one of positive pressure.The exponential shape as well as the value b ¼ 1 are very well supported by experimental evidenceby Jacinto et al. [16]. The values b ¼ 2 and 3 corresponding to smaller negative pressure have alsobeen treated for comparison purposes. The plot for b ¼ 5; shown in Fig. 2d, is practically identicalto the one for b ¼ 3; thus no larger b-values have been used.

3. Static and dynamic elastic–plastic behavior for SDOF systems

An elastic–perfectly plastic SDOF system without damping has the static resistancefunction R; shown in Fig. 3, where resistance is plotted in the vertical axis with respect to the

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0 2 4 6 8 10

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

P/P

s

t/td

0 2 4 6 8 10

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

P/P

s

t/td

(a) b=1

Epos=0.3678, Eneg=-0.3678

(b) b=2

Epos=0.2838, Eneg=-0.0338

0 2 4 6 8 10

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

P/P

s

t/td

0 2 4 6 8 10

-0.2

0.0

0.2

0.4

0.6

0.8

1.0P

/Ps

t/td

(c) b=3

Epos=0.2277, Eneg=-0.0055

(d) b=5

Epos=0.1602, Eneg=-0.0002

Fig. 2. Exponential distribution of pressure due to blast load for different values of the b-parameter and corresponding

areas of positive and negative pressure.


degree-of-freedom x; plotted in the horizontal axis. The response of the system is divided in fourstages, as described for example by Bangash [15] and Chopra [17]:

(1) Response up to the elastic limit xel; corresponding to a maximum resistance Rm; characterizedby the elastic stiffness k of the system, defined as the ratio of Rm to xel:

(2) Plastic response with constant resistance Rm from the elastic limit xel up to a maximumdisplacement xm:

(3) Rebounding, where the response starts to decrease with the same absolute stiffness as in stage(1) up to a maximum (in absolute terms) negative resistance �Rm:

(4) Plastic response during rebounding, corresponding to a resistance �Rm:

The analytical expressions describing the response in each stage are

R ¼

kx; 0oxoxel ðstage 1Þ;

Rm; xeloxoxm ðstage 2Þ;

Rm � kðxm � xÞ; xm � 2xeloxoxm ðstage 3Þ;

�Rm; xoxm � 2xel ðstage 4Þ:

8>>><>>>:

ð4Þ

The response of such a system without damping subjected to dynamic excitation PðtÞ is describedby the following equations of motion:

3.1. Stage (1)

In this stage the equation describing the response of the system is

m .xðtÞ þ kxðtÞ ¼ PðtÞ: ð5Þ

According to Watson [13], the influence of damping can be neglected because the peak response ofthe system occurs in the first few cycles. The solution xðtÞ of Eq. (5) depends on the load functionPðtÞ: The initial conditions are usually zero. The typical profile of the displacement in the first

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(2)

(4)

(1)

(3)

xel xm x

R

Rm

-Rm

Fig. 3. Elastic–perfectly plastic behavior for SDOF system.


stage can be seen in Figs. 4a and 6a, corresponding to the cases when the system reaches plasticity(stage 2) before or after elastic rebounding, respectively. If tel is the time at the end of the elasticregion, where the response is xel; then we have the following displacement and velocity at the endof stage (1):

xðtelÞ ¼ xel;

vel ¼ ’xðtelÞ: ð6Þ

The time tel is obtained by solving the first of Eqs. (6), as we know the displacement xel from thestatic resistance function of the SDOF system, given by Fig. 3 and Eq. (4). Then, vel is calculated

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x

t

xel

tel

(a) First stage of first cycle (1-1)

x

ttm

xm

xel

tel

(b) Second stage of first cycle (1-2)

x

treb t

xm-2xel

tm

xm

(c) Third stage of first cycle (1-3)

x

treb t

xm-2xel

(d) Fourth stage of first cycle (1-4)

x

t

(e) First stage of the second cycle(2-1)

x

t

(f) Second stage of the second cycle (2-2)

x

t

(g) Third stage of the second cycle (2-3)

x

t

(h) Fourth stage of the second cycle (2-4)

-x'mt'm

-x'm

x'm-2xel

t'm

x'm-2xel

xm2

tm2

-x'm2 t'm2

xm2-2xel

treb2

xm2-2xel

treb2

tel2 tel2

xm2

tm2

Fig. 4. Stages of response in the first and second cycle, when plasticity occurs before elastic rebounding.


from the second of Eqs. (6). The displacement xel and the velocity vel at the end of this stage arethe initial conditions for the second stage.

3.2. Stage (2)

In this stage the system has reached plasticity, and the equation that describes its response is

m .xðtÞ7Rm ¼ PðtÞ: ð7Þ

The sign of Rm in Eq. (7) depends on whether the system comes to plasticity (stage 2) before elasticrebounding (Fig. 4), in which case ðþÞ is used, or after elastic rebounding (Fig. 6), when ð�Þ isused. The solution xðtÞ of Eq. (7) depends again on the load function PðtÞ and the initialconditions xel and vel: The displacement in the second stage can be seen in Figs. 4b and 6b,respectively, between the time tel; corresponding to the first occurrence of plasticity, and the timetm of maximum response.By setting the velocity in the second stage, given by the derivative of the solution of Eq. (7),

equal to zero, the time of maximum response tm is obtained,

’xðtmÞ ¼ 0: ð8Þ

Then, the maximum response xm can be obtained replacing the time t with tm in the solution xðtÞof Eq. (7),

xm ¼ xðtmÞ: ð9Þ

The new displacement xm; and the corresponding velocity vm ¼ 0 are the initial conditions forstage (3).

3.3. Stages (3) and (4)

A similar procedure is followed to obtain the response in stages (3) and (4). Graphically, we cansee these stages in Figs. 4c and d for the case when the system reaches plasticity (stage 2) beforeelastic rebounding. The time of transition from stage (3) to stage (4) is denoted by treb; and themaximum displacement in stage (4) is named x0

m and the corresponding time t0m:When the fourth stage finishes, then the second cycle starts and four new stages are possible to

occur. The response of the second cycle can be treated with the same procedure. In Figs. 4e–h,these four stages of the new cycle can be seen, where the maximum displacement and thecorresponding time in the second and fourth stage are named xm2; x0

m2 and tm2; t0m2; respectively.In Fig. 5 the total response of system in the two cycles is shown. The absolute maximumdisplacement may occur either in the second or in the fourth stage of each cycle, in other wordsthe maximum displacement of the system could be anyone of xm; x0

m; xm2 and x0m2: The number

of cycles depends on the ratio of maximum system resistance to external load Rm=P: As this ratioincreases, the cycles decrease until the ratio becomes equal to 2, in which case there is no cycle andthe system behaves elastically.When plasticity occurs after elastic rebounding, the system usually works elastically after the

third stage of the first cycle and does not enter the plasticity region again (Fig. 6c). As can beobserved from Fig. 7, where the total response is presented for that case, the maximum

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displacement could be either xm or x0m; corresponding to first plasticity and to the elastic region

which follows, respectively.Because of the shape of the loading function PðtÞ in the case of explosions, the maximum

response of the system is obtained in most cases from the first cycle and only rarely from thesecond or third cycle, as could be the case if we had a harmonic loading function PðtÞ: This will beillustrated better in the following sections.

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x

tel

t

-xel

xel

(a) First stage (1-1)

x

tel

t

xel

tm

xm

(b) Second stage (1-2)

x

ttm

xm

(c) Third stage (1-3)

Fig. 6. Stages of response when plasticity occurs after elastic rebounding.

x

t

(1-1)

(2-1)

(2-2)

(1-4)

(1-3)

(1-2)

(2-4)

(2-3)

-x'm

t'm

xm

tm

-x'm2

t'm2

xm2

tm2

First cycle Second cycle

Fig. 5. Combined response in the first and second cycle when plasticity occurs before elastic rebounding.


4. Solutions of equations of motion of SDOF systems under explosive loads

The solutions of dynamic equations of motion (5) and (7) of an SDOF system subjected toexplosive loads have been obtained analytically, using the symbolic manipulation softwareMathematica [18]. Both cases of exponential and triangular evolution of the loading function withtime have been considered, as described by Eqs. (1) and (3), respectively. The resulting solutionsfor the different stages of response are the following.

4.1. Triangular loading

Stage (1):

%xðtÞ ¼2pðtd � tÞ � 2ptd cosð2pt=TÞ þ T sinð2pt=TÞ

2aptd; ð10Þ

where

%xðtÞ ¼xðtÞxel

; a ¼Rm

Ps: ð11Þ

Stage (2):Case I: telotd: (a) For telototd; the equation of motion is m .xðtÞ7Rm ¼ PðtÞ; and its solution is

given by

%xðtÞ ¼1

6apT2td�2pð2p2t3 þ 3tT2 � 6p2t2td þ 6ap2t2td � 3T2td þ 12tp2tdtel � 6tp2t2el

� 6p2tdt2el þ 6ap2tdt

2el þ 4p

2t3elÞ þ 6pT2ðt � td � telÞ cos2ptel

T

þ 3TðT2 þ 4tp2td � 4p2tdtelÞ sin2ptel

T

: ð12Þ

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x

ttel

xel

tm

xm

Fig. 7. Combined response when plasticity occurs after elastic rebounding.


(b) For telotdot; the equation of motion is m .xðtÞ7Rm ¼ 0; and the response is obtained as

%xðtÞ ¼1

6apT2td�2pð3tT2 þ 6ap2t2td � 3T2td � 6tp2t2d þ 2p

2t3d þ 12tp2tdtel � 12atp2tdtel

� 6tp2t2el � 6p2tdt

2el þ 6ap

2tdt2elÞ þ 6pT2ðt � td � telÞ cos

2ptel

T

þ 3TðT2 þ 4tp2td � 4p2tdtelÞ sin2ptel

T

: ð13Þ

Case II: tdotel: (a) For tdototel; the equation of motion is m .xðtÞ þ kxðtÞ ¼ 0; and the solu-tion is

%xðtÞ ¼ �2ptd cosð2pt=TÞ � T sinð2pt=TÞ þ T sinð2pðt � tdÞ=TÞ

2aptd: ð14Þ

(b) For tdotelot; the equation of motion is m .xðtÞ7Rm ¼ 0; and its solution is given by

%xðtÞ ¼1

2apT2td�4ap3t2td þ 8atp3tdtel � 4ap3tdt2el � 2pT2ðt � telÞ cos

2pðtd � telÞT

þ 2pT2ðt � td � telÞ cos2ptel

Tþ T3 sin

2pðtd � telÞT

þ T3 sin2ptel

T

þ 4p2tTtd sin2ptel

T� 4p2Ttdtel sin

2ptel

T

: ð15Þ

The analytical solutions for stages (3) and (4) are obtained in a similar manner.

4.2. Exponential loading

Stage (1):

%xðtÞ ¼1

aðb2T2 þ 4p2t2dÞ22ptd �2e�bt=tdpðb2ðt � tdÞT2 þ 2btdT

2 þ 4p2ðt � tdÞt2dÞ�

� 2ptdððb � 2ÞbT2 þ 4p2t2dÞ cos2pt

Tþ Tððb � 1Þb2T2 þ 4ðb þ 1Þp2t2dÞ sin

2pt

T

�: ð16Þ

Stage (2): For b ¼ 1

%xðtÞ ¼ 2p2

t2el � t2

T2þ2ða � e�tel=tdÞtelt � 2tde�t=tdðt þ tdÞ þ 2tde�tel=tdðtel þ tdÞ þ 2ðe�tel=td � aÞt2el

aT2

þð2tdð4pTt2d sinð2ptel=TÞ þ ðT2 � 4pt2dÞtd cosð2ptel=TÞ � e�tel=tdð�4p2t3d þ 4p

2telt2d þ T2td þ T2telÞÞÞ

aðT2 þ 4p2t2dÞ2

:

ð17Þ

The analytical solution for other values of b is not given here for the sake of brevity, but can beobtained in a similar manner, as well as the solutions for stages (3) and (4).

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5. Elastic response of SDOF systems to explosive loads

As long as the system remains elastic, the main response parameter is the ratio of thecharacteristic blast loading duration td to the system’s fundamental period T : A typical elasticresponse is shown in Fig. 8, for the case of exponential blast loading with b ¼ 1; if T=td ¼ 0:1: Theinfluence of T=td is illustrated very well in the 3D plot of Fig. 9, showing elastic response toexponential blast load, for varying values of the T=td ratio. Fig. 10 shows an elastic responsespectrum, containing the maximum response for different values of T=td:

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Fig. 8. Elastic response ratio %xðtÞ for exponential blast loading for b ¼ 1 and T=td ¼ 0:1:

Fig. 9. Response ratio %xðtÞ for exponential blast loading and elastic behavior, for td ¼ 0:1 s:


6. Elastic–plastic response spectra

The elastic analysis described briefly in Section 5, is not suitable for the design of structureswhich are in the vicinity of an explosion source, as it is highly unlikely that such structures willrespond elastically. Therefore, elastic–plastic design should be applied. Given the dynamic natureof the response, but also the uncertainties in the characteristics of the blast loading, a responsespectrum analysis seems to be appropriate.To that effect, the elastic–plastic analysis described in Section 3 has been performed for a

characteristic blast loading duration of td ¼ 0:1 s; and a wide range of fundamental periods of theSDOF system and range of ratios Rm=P: The maximum values of displacement ratios m ¼xmax=xel ðxmax ¼ maxfxm; xm2; x0

m; x0m2gÞ have been plotted as response spectra for both

triangular and exponential blast loading, and are shown in Figs. 11 and 12, respectively. In

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Fig. 10. Response spectrum D ¼ maxf %xðtÞg for exponential blast loading with b ¼ 1 and elastic behavior.

Fig. 11. Response spectra for triangular blast loading and elastoplastic behavior.


addition, the time tmax of occurrence of maximum displacement has been plotted for both loadingassumptions, and is illustrated in Figs. 13 and 14.The objectives of this analysis were the following:

(i) To compare the response spectra for triangular loading to those found in the literature (USDepartment of the Army Technical Manual), and thus verify the proposed solutionapproach.

(ii) To compare the response spectra for triangular loading to those for exponential one, in orderto assess the influence of the assumption of triangular distribution on practical designproblems.

(iii) To draw qualitative conclusions regarding the response of structures subjected to explosiveloadings.

(iv) To provide designers of structures subjected to explosions with a more accurate design toolthan the one currently in use.

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Fig. 12. Response spectra for exponential blast loading and elastoplastic behavior.


First, it should be noted that, in spite of the nonlinearity of the system, it is possible to usenondimensional parameters in order to describe the response. It is the ratios Rm=P and td=T thatdetermine the response, and not the actual values of these four parameters.

In order to meet the first objective, characteristic values of maximum response are listed inTable 1 for different combinations of ratios Rm=P and td=T : The first observation is that there aresome small differences between our results for triangular loading and those given in the USDepartment of the Army Technical Manual TM5-1300 [10]. This is attributed to the fact that theproposed analysis considers analytical expressions for several cycles of response and four stages ineach cycle, while the analysis in TM5-1300 employs the average acceleration and the acceleration

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Fig. 13. Time of occurrence of maximum response for triangular blast loading and elastoplastic behavior.

Fig. 14. Time of occurrence of maximum response for exponential blast loading and elastoplastic behavior ðb ¼ 1Þ:


impulse extrapolation methods to numerically integrate the equation of motion of the elasticsystem, and the predictor–corrector method to accommodate the nonlinear term R (restoringforce), and does not specify in detail how the subsequent stages and cycles are accounted for. Thesmaller the ratio Rm=P; the larger difference is observed in m ¼ xmax=xel:Comparing also the results for the basic exponential loading (b ¼ 1; shown in Fig. 12a) with

those for triangular one, it is noted that differences do occur, which are significant in some cases.When the ratio Rm=P is large, ranging between 1 and 2, the response for exponential loading issmaller compared to the one for triangular loading. This can be explained qualitativelyconsidering the shape of loading. The exponential loading decreases faster than the triangular one,and this has more influence in elastic–plastic situations than in purely elastic ones.Some exceptions to this were encountered in case of low Rm=P ratios, for example for Rm=P ¼

0:1 and td=T ¼ 0:4; where the exponential loading gives higher response than the triangular one.This happens in cases when the maximum response is obtained at times larger than td; and theexponential loading has changed sign and may be in phase with the system motion, while thetriangular loading at this time is zero.

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Table 1

Comparison of maximum response for triangular and exponential loading

Rm=P TM5-1300 Proposed procedure Proposed procedure

(triangular loading) triangular loading exponential loading

b ¼ 1 b ¼ 2 b ¼ 3

0.1 td=T m ¼ xmax=xel m ¼ xmax=xel m ¼ xmax=xel m ¼ xmax=xel m ¼ xmax=xel0.1 5.2 5.32 3.97 1.88 1.48

0.2 17 19.08 18.53 6.01 4.33

0.3 40 42.21 44.61 12.59 8.87

0.4 70 72.28 79.03 21.61 15.08

0.5 0.1 0.6 0.62 0.6 0.34 0.28

0.3 1.95 2 2.19 1.08 0.82

0.9 9 9.42 6.38 3.23 2.36

2 35 35 16.78 9.76 6.62

3.3 85 87.51 39.37 21.43 15.09

1 0.1 0.32 0.31 0.28 0.17 0.14

0.2 0.6 0.60 0.65 0.36 0.28

0.5 1.1 1.05 1.00 0.76 0.62

1 1.8 1.91 1.37 1.1228 0.9725

3 2.8 2.91 2.02 2.01 1.76

10 7 6.93 4.89 3.96 3.40

2 0.1 0.17 0.16 0.15 0.08 0.07

0.2 0.3 0.30 0.32 0.18 0.14

0.5 0.6 0.598 0.52 0.38 0.31

1 0.8 0.79 0.69 0.55 0.48

3 0.9 0.92 0.85 0.79 0.74

10 1 0.99 0.95 0.93 0.90


The response spectra for larger b-values are shown in Figs. 12b ðb ¼ 2Þ and c ðb ¼ 3Þ: It isobserved that the response is reduced for larger values of b: This is more clearly illustrated inFig. 15, showing a case for which significant differences between response to triangular andexponential loading were observed even for b ¼ 1; particularly for rather large values of td=T : Forb ¼ 2 and 3 even larger differences are observed between response to triangular and exponentialloading for all ranges of td=T : Considering that the duration td usually varies within a small rangeof values, large values of td=T correspond to small values of the fundamental period T ; therefore,to stiff systems.An explanation of why the exponential loading for b ¼ 1 leads to a larger displacement than

that produced by the triangular one in the region of td=T between 0.1 and 0.4 is provided inFig. 16. In this figure, the response of a system with ratio td=T ¼ 0:2 and Rm=P ¼ 0:5 is shown forboth exponential and triangular load. For triangular loading it is observed that the maximumresponse xm=xel is 1.22 and occurs at first time when the system comes to plasticity right (beforeelastic rebounding) of the equilibrium position. The plasticity phase (stage (2)) starts at timet ¼ 0:22 s:For exponential loading with b ¼ 1 the maximum response xm=xel is 1.37 and occurs for the

first time when the system comes to plasticity left (after elastic rebounding) of the equilibriumposition. The plasticity phase (stage (2)) starts at time t ¼ 0:62 s: At this time, as can be seen fromthe scaled loads in Fig. 16a, the external loading is zero for triangular loading but nonzero andacting in opposite direction for the exponential loading. This nonzero load and the change ofthe direction of the exponential load, combined with the fact that the plasticity occurs left of theequilibrium position contributes to increasing the system’s response. In other words, theexponential loading and the corresponding displacement have the same direction duringthe plasticity phase, compared to the response of the system under triangular loading wherethe loading is zero during the plasticity phase. Then, the system moves elastically around thenew equilibrium position for both cases of loading.

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0.1 1 100.1

1

10

100

TM5-1300

td/T

x max

/xel

b=1

b=2b=3

Triangular

Fig. 15. Comparison of response spectra for Rm=P ¼ 0:5:


One other example, in the region of td=T above 0.6, is a system with ratio td=T ¼ 1:2 andRm=P ¼ 0:5 for both exponential ðb ¼ 1Þ and triangular load (Fig. 17a). The time response is seenin Fig. 17b. The maximum response xm=xel is 14.8 for triangular and 10.58 for exponentialloading. The system with the triangular loading comes to plasticity for the first time and thenmoves elastically around the new equilibrium position. The system with the exponential loading

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(Triangular load) (Exponential load) (a) Pressure vs. time

(b) Displacement vs. time

(c) Response vs. displacement

xmax/xel=-1.37

x/x e

l

x/x e

l

t

P/P

s

P/P

s

t t

Plastic (stage2)Elastic (stage1)

Elastic (stage3)


Elastic (stage3)

x/xel

R

Rm

-Rm

1 1.22


Elastic (stage3)

x/xel

R

Rm

-Rm

1.37 1


Elastic (stage3)

t

xmax/xel=1.22

Fig. 16. Comparison of response for triangular and exponential loading for Rm=P ¼ 0:5; td ¼ 0:2; T ¼ 1:


comes to plasticity four times making two cycles and then moves elastically around the newequilibrium position. The above behavior is illustrated in Fig. 17c.Summarizing the results presented in Table 1 and Fig. 15, it can be said, that the commonly

used assumption of triangular blast load evolution with time can sometimes be slightlyunconservative, particularly for flexible structural systems, but can also be significantly

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(Triangular load) (Exponential load)

(a) Pressure vs. time

(b) Displacement vs. time

(c) Response vs. displacement

x/x e

l

x/x e

l

t

t t

P/P s

P/P s

t

c

x/xel

R

Rm

-Rm

14.801


Elastic (stage3)

x/xel

R

Rm

-Rm

10.581

Plastic (stages 2, 4) ), 2 cycles Elastic (stage1), 2 cycles

Elastic (stage3) ), 2 cycles

xmax/xel=10.5xmax/xel=14.8

Fig. 17. Comparison of response for triangular and exponential loading for Rm=P ¼ 0:5; td ¼ 0:2; T ¼ 0:166:


overconservative for stiffer structures, even for b ¼ 1: If the rebound of the system occurs in aperiod of negative phase of the blast load, then this may lead to maximum displacement of thesystem due to the negative phase. Thus, the negative plastic deformations due to the negativeplastic forces may exceed in magnitude the positive plastic deformations caused by the largerforces of the triangular pulse. This is a matter of dynamic ‘‘tuning’’ or resonance, and it occurs foronly a small range of ratios of the system’s natural period to the pulse duration.A common observation for both exponential and triangular loading is that when the ratio

Rm=P is larger than 1.5 the system behaves elastically and no significant difference in the responseis encountered. A point made in the literature [17] was verified by the present analysis, namely thatin elastic response spectrum calculations, if the time duration td=T is longer than 0.5, then themaximum deformation occurs during the pulse and the shape of loading is of great significance. Ifon the other hand, the time duration td=T is less than 0.5, then the maximum deformation occursduring the free vibration and is mainly controlled by the time integral of the pulse, independentlyof the shape of loading. For elastoplastic behavior and when the ratio Rm=P is smaller than 1.5,the ratio td=T is also a critical parameter for which the shape of blast loading (exponential ortriangular) plays a significant role in the response.

7. Design implications

According to the design procedure suggested by Mays and Smith [11] for a structural memberin flexure, the following steps must be followed:

1. Carry out a preliminary design assuming an equivalent static ultimate resistance andappropriate deformation limits for the desired protection category.

2. Calculate the natural period of the element.3. Refer to the appropriate SDOF response spectrum for an elastic–plastic system to obtain themaximum response and compare this response to the desired deformation limits.

It is apparent that the third step is directly dependent upon the response spectrum to be used.Hence, in cases of significant discrepancies between existing and proposed response spectra, theproposed approach will have important design implications. For example, with the use of thespectrum of exponential loading for b ¼ 1 in the region of td=T between 0.1 and 0.4 heaviersections will be needed, while for td=T larger than 0.4 the design procedure will lead to moreeconomical choices.Another example is the criterion for steel beams and plates that the ratio m ¼ xmax=xel should be

less than 10 for protection category 1. According to the triangular spectra (Fig. 15), for values oftd=T between 1 and 1.5 the element should be redesigned. If, however, the proposed spectra forexponential loading are used, then m is inside the desired limits and the criterion is satisfied.

8. Summary and conclusions

The response of structures subjected to loads due to explosions has been investigated. Thepreliminary and potentially also the final design of such structures, as well as the assessment of the

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bearing capacity of existing structures, is often treated by means of elastic–plastic responsespectra. Such spectra that are currently available in the literature are based on triangular timeevolution of the blast pressure. In the present paper, response spectra based on a more accurateexponential distribution of blast pressure, and accounting for all stages and cycles of response,have been computed. This has been achieved by deriving the pertinent equations of motion, andobtaining analytical expressions of their solutions via symbolic manipulation software. Acomparison of the spectra obtained by the proposed approach to the ones for triangular pressurefunction, led to the conclusion that design based on the commonly used assumption of triangularblast load evolution can sometimes be slightly unconservative, particularly for flexible structuralsystems, but can also be significantly overconservative for stiffer structures. Regarding the valueof the exponential load b-parameter to be used for design purposes, the conservative choice ofb ¼ 1 is recommended, which is still, in most cases, less adverse than using a triangulardistribution.

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the murrah building. ASCE J Performance Constr Facilities 1998;12(3):120–36.

[3] Rittenhouse T. Designing terrorist-resistant buildings. Fire Engineering 1995; Nov.

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[6] Otani RK, Krauthammer T. Assessment of reinforcing details for blast containment structures. ACI Struct J

1997;94(2):124–32.

[7] Krauthammer T, Flathau WJ, Smith JL, Betz JF. Lessons from explosive test on RC buried arches. ASCE J Struct

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[11] Mays GC, Smith PD. Blast effects on buildings. London: Thomas Telford; 1995.

[12] Biggs JM. Introduction to structural dynamics. New York: McGraw-Hill; 1964.

[13] Watson AJ. Loading from explosions and impact. In: Kappos AJ, editor. Dynamic loading and design of

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[15] Bangash MYH. Impact and explosion, analysis and design. Oxford: Blackwell Scientific Publications; 1993.

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[17] Chopra AK. Dynamics of structures: theory and applications to earthquake engineering. Upper Saddle River, NJ:

Prentice-Hall; 2001.

[18] Wolfram S. The mathematica book, reference manual. Cambridge: Wolfram Media, Cambridge University Press;

1996.

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http://www/wai.com/AppliedScience/index-blast.html

elastic plastic response spectra for exponential blast loading

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