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doi:10.1016/j.iji

CorrespondE-mail addrInternational Journal of Impact Engineering 35 (2008) 771783

www.elsevier.com/locate/ijimpengPressureimpulse diagrams for the behavior assessmentof structural components

T. Krauthammera,, S. Astarlioglua, J. Blaskoa, T.B. Sohb, P.H. Ngb

aCenter for Infrastructure Protection and Physical Security, University of Florida, 365 Weil Hall, Gainesville, FL 32611, USAbDefence Science & Technology Agency, 1 Depot Road, #12-05, Singapore 109679, Singapore

Received 26 April 2007; received in revised form 29 November 2007; accepted 6 December 2007

Available online 4 January 2008Abstract

Theoretical and numerical methods for deriving pressureimpulse (PI) diagrams for structural elements subjected to transient loads

are described in this paper. Three different search algorithms for deriving PI diagrams numerically were developed by the authors and

are presented. The PI diagrams of a linear elastic system under rectangular and triangular load pulses are derived using both theoretical

and numerical methods and the results are compared. The application of these approaches to the behavior assessment of tested structural

elements is illustrated.

r 2008 Elsevier Ltd. All rights reserved.

Keywords: PI diagram; Closed-form solution; Numerical methods; SDOF; Blast1. Introduction

For structural dynamic analysis, a designer is frequentlyconcerned with the final states (e.g. maximum displacementand stresses) rather than a detailed knowledge of theresponse histories of the structure. Baker et al. [1]quantified the loading regimes for an undamped, perfectlyelastic system subjected to an exponentially decaying load,where T is the systems natural period and td is thetriangular load pulse duration. One can define threegeneral cases for relative relationships between the loadfunction and the structural response, as illustrated inFig. 1. In the impulsive domain, the load is over before thestructure reaches its maximum response. In the quasi-staticdomain, the structure reaches its maximum deflection wellbefore the load is over. In the dynamic domain, themaximum deflection is reached near the end of the loadfunction.

Plots of a maximum peak response versus the ratio of theload duration or natural period of the system, known asresponse spectra, can be used to simplify the design of aee front matter r 2008 Elsevier Ltd. All rights reserved.

mpeng.2007.12.004

ing author. Tel.: +1352 392 9537; fax: +1 352 392 3394.

ess: tedk@ufl.edu (T. Krauthammer).dynamic system for a given loading. By defining differentsets of axes, the same response spectra for the given dynamicsystem can be represented in different ways. Though thevarious forms of response spectra may look different, theyall describe the relationship between the maximum value ofa response parameter and a characteristic of the dynamicsystem under consideration. A pressureimpulse (PI)diagram is an alternative representation of a responsespectrum, and is widely used for structural componentdamage assessment. Impulse is defined as the area under thepressure vs. time load function.The early application of PI diagrams was based on

empirically derived diagrams for brick houses to determinedamage criteria for other houses, small office buildings,and light-framed industrial buildings [2]. The results ofsuch investigations can be used as the basis for obtainingexplosive safety standoff distances [1,3]. PI diagrams werealso developed to assess human response to blast loadingand to establish damage criteria for specific organs (e.g.eardrum, lungs, etc.) of the human body. This is possible asthe human body responds to blast loading as a complexmechanical system [1]. In protective design, PI diagramshave been extensively used for approximate damageassessments of structural components subjected to blast

www.elsevier.com/locate/ijimpengdx.doi.org/10.1016/j.ijimpeng.2007.12.004mailto:tedk@ufl.edu

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Fig. 1. Typical response domains [1]. (a) Impulse, (b) quasi-static and (c) dynamic.

2.5

2

Quasi-static asymptote Quasi-Staticdomain

P

1.5

0.5

P

Kx m

ax

Impulsive asymptote

Dynamic

t

P

t

1

1

1

1

P0

domain

0

0.5

Impulsivedomain

0 21 3 4 5 6 7 8 9td

10

Fig. 2. Typical response spectrum.

T. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783772loading. For example, the Facility and ComponentEvaluation and Damage Assessment Program or FACE-DAP [4] was developed to predict damage to more than 20structural components based on the assumptions ofselected modes of failures. The PI diagrams in FACE-DAP are mostly based on the analytical work done byBaker et al. [1] that has been adjusted to better fit availableexperimental damage points.

2. Characteristics of PI diagram

It is well known from structural dynamics that a strongrelationship exists between the structural response and theratio of the load function duration, td, to the structuresnatural period, T [57]. As noted earlier, this relationship isnormally categorized into three regimes: impulsive, quasi-static, and dynamic. Compared to the response spectrum,the PI representation better differentiates the impulsiveand quasi-static regimes, in the form of vertical andhorizontal asymptotes. Fig. 2 shows a typical responsespectrum for an undamped, perfectly elastic SDOF systemunder suddenly applied loads. In this figure, xmax is themaximum dynamic displacement, K is the spring stiffness,P0 is the peak force, M is the lumped mass, td is the loadpulse duration, and T is the natural period of the system.By defining a different set of axes, the same responsespectrum can be transformed into what is known as a PIdiagram (Fig. 3). Humar [7] suggested that for rectangular,triangular, and sinusoidal load pulses with td/T ratio of lessthan 0.25, the dynamic response can be assumed to be inthe impulsive loading regime.

In specific applications for blast-loaded structures, theterm PI diagram is used because the (blast) load istypically defined in terms of a pressure vs. time distribu-tion. For example, various authors consistently use theterm pressureimpulse to describe these diagramsregardless of the nature of the loading [1,3,8,9]. PIdiagrams should be more correctly referred to as loa-dimpulse diagrams, since the ordinate can also bedefined in terms of forces. Here, all loadimpulse diagrams(i.e. PI or forceimpulse diagrams) are collectivelyreferred to as simply PI diagrams. PI diagrams, alsoreferred as Iso-damage curves [3], permit easy structuralresponse assessment to a specified load. With a specificresponse or damage level defined, the points on a PI curveindicate the combinations of load (or pressure) and impulsethat will cause the specified failure (or damage level). Ineffect, such threshold curves divide a PI diagram into twodistinct regions. Combinations of pressure and impulsethat fall to the left of and below the curve will not inducethe specified damage level, while those to the right andabove the graph will produce damage in excess of thespecific limit (e.g., maximum dynamic displacement).

3. Analytical solutions for PI diagrams

Closed-form solutions of PI diagrams can be obtainedfor idealized structures subjected to a specific load pulse.

ARTICLE IN PRESS

3

2.5

Impulsive

1.5

2K

x max

KMxmax

P0

domain

1Impulsiveasymptote

Quasi-Static

Dynamicdomain P

0

0.5

Quasi-staticasymptote

domain

P

t

t

0I

54.543.532.521.510.5

Fig. 3. Typical PI diagram.

T. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783 773This is accomplished by deriving the response spectrumfrom its respective response history functions under bothforced and transient vibration. For example, for anundamped elastic SDOF system subjected to a rectangularload pulse with duration td, the displacement functions areas follows [7]:

xt P0K

1 cos ot 0ptptd,

xt P0K

cos ot td cos otd t4td, (1)

where o is the natural circular frequency. The dimension-less force and impulse terms are defined as

P P0=Kxmax

,

I IffiffiffiffiffiffiffiffiffiKM

pxmax

. (2)

The following expressions for the vertical and horizontalaxis of the PI relationship [7] also represent the responsesin the free and forced domains, respectively:

P sinI

2P 1

21pIp p

2,

P 12

I4p2. (3)Similarly, for a triangular load pulse with zero rise time,the PI curve was defined in [10], as follows:

2I

P2

2 2 2I

P

2 4I

Psin

2I

P

2 cos 2I

P

,

1pIp1:166,2I

P

tan 2I

P

1 1

2P

I41:166. (4)

A more widely used method for obtaining PI diagramsis the energy balance method. The approach, based on theprinciple of conservation of mechanical energy, is con-venient to apply because two distinct energy formulationsalways exist that separate the impulsive loading regimefrom the quasi-static loading regime. To obtain theimpulsive asymptote, it can be assumed that due to inertiaeffects the initial total energy imparted to the system is inthe form of kinetic energy only. Equating this to the totalstrain energy stored in the system at its final state (i.e.maximum response), one obtains an expression for theimpulsive asymptote. For the quasi-static loading regime,the load can be assumed to be constant before themaximum deformation is achieved. By equating the workdone by load to the total strain energy gained by thesystem, the expression for the quasi-static asymptotes isobtained. Expressing these approaches mathematically,

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P New point on branch

I

Predictor stepCorrector steps

Fig. 4. Branch-tracing technique [11].

T. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783774one obtains

K:E: S:E: impulsive asymptote; (5)

W:E: S:E: quasi-static asymptote; (6)

where K.E. is the kinetic energy of the system at time zero,S.E. is the strain energy of the system at maximumdisplacement, and W.E. is the maximum work done by theload to displace the system from rest to the maximumdisplacement. For the case of a perfectly elastic system, theenergy expressions are

K:E: I2

2M, (7)

W:E: P0xmax, (8)

S:E: 12

Kx2max. (9)

Substituting Eqs. (7)(9) into Eqs. (5) and (6), thedimensionless impulsive and quasi-static asymptotes areobtained as 1 and 0.5, respectively. The derivation of theimpulsive and quasi-static asymptotes for several othersimple SDOF systems is available in the literature [10].

Though the energy balance method greatly reducescomputation efforts, its formulation is only applicable tothe impulsive and quasi-static domains of the responsespectrum. The dynamic regime of the PI curve must beapproximated using suitable analytical functions. For thispurpose, Baker et al. [1] recommended the followinghyperbolic tangent squared relationship:

S:E: W:E:tanh2ffiffiffiffiffiffiffiffiffiffiffiK:E:

W:E:

r. (10)

For small values of the above expression, the hyperbolictangent equals its argument, which effectively reduces to Eq.(5). For large values, the hyperbolic function approachesunity and Eq. (6) for the quasi-static asymptote is obtained.Baker et al. [1] reported that less than one percent error isintroduced when Eq. (10) is used to approximate thetransition region for linearly elastic oscillators.

Oswald and Skerhut [4] recommend the simple hyper-bolic function, shown in Eq. (8) to curve-fit the transitionregion, where A and B are the values of the impulsiveasymptote and quasi-static asymptote, respectively. Thisequation is based on limited comparisons to responsecurves developed with dynamic SDOF analyses, whereblast loading has been idealized as a triangular pressurehistory with the same impulse as the positive phase of theblast wave. Modifications of this approach can be obtainedby shifting the curves to fit test data [4]:

P AI B 0:4 A2 B

2

1:5. (11)

All the procedures addressed previously are limited tosimple structural systems, resistance models, and loadfunctions. More involved problems require one to adoptappropriate numerical approaches.

4. Numerical approach to PI diagrams

PI diagrams can be generated numerically by generat-ing a sufficient number of computed points to allow forcurve fitting. Each point represents the result from a singledynamic analysis and indicates that the structure hasreached a specific response-level pressure and impulsecombination. Since running all possible pressure andimpulse combinations is computationally very expensive,a search algorithm must be employed to locate thethreshold points that define the transition from safe todamaged states. Unlike analytical solutions, numericalapproaches allow complex nonlinear resistance functionsand complex loading functions to be used. Furthermore,the numerical approach can describe the behavior of thePI curve in the dynamic response domain accurately.Rhijnsburger et al. [11] presented a procedure to

generate PI diagrams by utilizing multiple analyticaltechniques. The energy balance method estimates theimpulsive and quasi-static asymptotes, while a numericalanalysis procedure generates the dynamic regime using abranch-tracing algorithm. This process, as shown in Fig. 4,extrapolates the slope on the curve from two previouslyknown points and a prediction point is made. A responsecalculation follows with the predicted PI combination,and the ductility of the system is obtained. If the predictedloading point does not agree with the specified ductility,correction steps are then taken to find the next point in thevicinity. The ductility, m, is defined as the ratio of the

ARTICLE IN PRESS

P PSearch interval Search intervalData points

Search direction

Plotlimits

Searchdirection

Searchinterval

I I

Search direction

Estimated locations of asymptotes

Searchinterval

Search direction

Established threshold curve for flexure

Fig. 5. Search algorithm by Soh and Krauthammer [10]. (a) Flexure and (b) direct shear.

Trial 1Trial 2Trial 3

P

SafeDamaged

For agivenpressure

I1/4 I1/2 I1Divided into 20 segments

Check segments in increasing order

I

Fig. 6. Search algorithm by Ng and Krauthammer [12].

T. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783 775maximum displacement, xmax, to the yield displacement,xy. These correction steps are performed until thecalculated ductility is within a certain tolerance of thefailure criterion. As a result, a new point is found on thePI curve. The algorithm assumes that the PI curve issmooth and continuous. However, due to the timediscretization of the numerical method, the PI curvemay violate these assumptions, and abrupt changes in theslope may occur. As a result, the method may becomeunstable, as data points scatter within the overlappedzones [10].

Soh and Krauthammer [10] developed a methodologythat would produce numerically stable PI diagrams. Theprocedure starts by using the energy balance method toestimate the locations of the asymptotes. After deriving theasymptotes, a large number of dynamic analyses areevaluated within specific limits of the asymptote locations.These limits are reduced until the threshold curve isformed. Fig. 5 illustrates the numerical algorithm for thisprocedure. The research in this analysis was based onreinforced concrete beams that were idealized as twoloosely coupled SDOF systems to represent flexure,diagonal shear, and direct shear behaviors.

Ng and Krauthammer [12] generated PI curvesindependently of the asymptotes. In this numericaltechnique, the reinforced concrete slabs were idealized astwo loosely coupled SDOF systems, representing flexuraland direct shear behaviors. The algorithm is based on thedefinition of a threshold curve. Threshold points are foundby keeping the pressure constant and checking whether thePI combination is either safe or damaged. If safe, theimpulse is increased until the point results in damaged.Conversely, reducing the impulse for a damage point willeventually find a safe point. In between these twoboundaries a threshold point is found, as shown in Fig. 6.

The numerical procedures presented by Soh andKrauthammer [10] and Ng and Krauthammer [12]produced reasonably accurate PI diagrams; however,there are a few disadvantages to their algorithms. Bothapproaches are computationally intensive and generate aconsiderable amount of unnecessary data. Due to theseproblems, the computational process is quite lengthy. Sohand Krauthammers [10] numerical analysis was limited toreinforced concrete beams subjected to localized impactloads. While, Ng and Krauthammers [12] study waslimited to reinforced concrete slabs subjected to uniformlydistributed blast loads. These procedures are case specificand do not allow the user to select a different structuralelement or loading scheme.Blasko et al. [13] used a polar coordinate system and the

Bisection method to obtain PI diagrams. The numerical

ARTICLE IN PRESST. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783776procedure starts with normalizing the load function basedon the load assigned to the SDOF system. A pivot point(Ip, Pp), which is located in the fail zone, is set as theorigin of the polar coordinate system. The coordinatesystem is normalized and the radius and angle are mappedto the unit coordinate system (Fig. 7). Iterations using theBisection method are carried out to find the radius Ri to thethreshold point for each angle yi. Since the iterations foreach angle can be carried out independent of each other,PIp , Pp

Ii , Pi

i

I

G(I,P) = 0

I

Fig. 7. Search algorithm by Blasko et al. [13]. (a)

3

2.5

1.5

2

P0

P

1

Kx m

ax

0

0.5

0

K

(*) FACEDAP solution used for e

21.510.5

Fig. 8. Comparison of analytical anthis procedure takes advantage of multi-processor andmulti-core computing capabilities. This method simplifiesthe previous procedures [10,12] by using a single radialsearch direction, instead of two search directions (i.e.horizontal in the impulsive response domain and vertical inthe quasi-static response domain). Although the calcula-tion of the asymptotes is not necessary for the method towork, one can automate the procedure for selecting thelocation of the pivot point by utilizing the asymptotes. The1

i

rmid

rlower

1

F(r,) = 0rupper

1

establist pivot point and (b) data pivot search.

Closed form (*) Numerical

t

P

t P

tt

M xmax

I

xponential pulse

54.543.532.5

d numerical PI solutions [13].

ARTICLE IN PRESST. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783 777vector stemming from the origin and passing through thepoint where the impulsive and quasi-static asymptotesintersect is an ideal direction for locating the pivot point,since the points along this line are expected to be equallydistanced from both of the asymptotes. A randomlyselected point might be close to one asymptote or too farfrom the threshold curve, reducing the resolution of theresults. This approach can be applied effectively to anystructural system for which a resistance function can beobtained.

5. Influence of load and structural propertieson PI diagrams

Fig. 8 shows the PI diagrams of an undamped simpleperfectly elastic system subjected to three different loadpulses that are shown in Fig. 9. For the rectangular and1

P(t)

1

P(t)

10 0

t

Fig. 9. Characteristic load functions for simple load pulses

3

2.5

1.5

2

Kx m

ax

P0

1tr = 0.5td

tr = 0.1

0

0.5

1

K

Fig. 10. Influence of rise timtriangular pulses, both the closed-form [10] and thenumerical [13] solutions yield similar results. For the morecomplex exponential load pulse, the results obtained usingthe numerical procedure are reasonably close to thoseobtained using FACEDAP [4]. For all three load pulses theimpulsive and quasi-static asymptotes were correctlycomputed by the numerical procedure as 1.0 and 0.5,respectively. The insignificant scatter in numerical datapoints about the analytical solution reflects truncation andround-off errors inherent in the numerical method, and canbe reduced by using a smaller time step. In these load cases,the resulting PI curves follow the shape of a hyperbolicfunction, and the impulsive and quasi-static asymptotes areunaffected by the different pulse shapes. This is due to thefact that all three loading cases have instantaneous risetimes. The shift from the impulsive domain to the quasi-static domain becomes softer and the dynamic response1

P(t)

(1t)e8t

1 10

tt

. (a) Rectangular, (b) triangular and (c) experimental.

P0

tr tdt

tdtr = 0.01td

tr = 0

10010

M xmax

I

e on the PI curve [13].

ARTICLE IN PRESS

3

2.5P

t

1.5

2

P0

: Damping ratio (%)

1

Kx m

ax

0.5

0100101

KM xmax

I

= 0 = 5%

= 10%

= 20%

Fig. 11. Influence of damping on the PI curve [13].

3

2.5

P

t

1.5

2

P0

1

Kx m

ax

0

0.5

= 1

= 2

= 5

= 20

= 50

100101

KM xmax

I

Fig. 12. Influence of ductility on the PI curve [10].

T. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783778

ARTICLE IN PRESS

180

140

160

100

120

60

80Load

(kN

)

0

20

40

0Displacement (mm)

25020015010050

Fig. 14. Flexural resistance function for beam 1c [13].

T. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783 779domain becomes wider as the order of the load pulseincreases from constant to linear and then to exponential.

For load pulses with a finite rise time, the PI curve maydepart from the usual shape [10,13]. Fig. 10 shows theeffect of the rise time on the same perfectly elastic systemconsidered before. While the impulsive response remainsunaffected, the quasi-static response fluctuates greatly,depending on the rise time to load duration ratio, a [1]. Asa increases, the quasi-static asymptote moves from 0.5 to1.0. This occurs because the load is applied relativelyslowly to the structure, and as a result there is no dynamicamplification. The quasi-static asymptote when equal to 1is equivalent to a static loading.

The damping ratio assumed for the structure may alsohave a significant effect on the PI diagram. Typicaldamping ratios of reinforced concrete structures arebetween 2% and 7% under usual conditions. For casesinvolving significant external damping, such as soilinteraction for buried structures, the damping ratio mightbe well above 10% [14].

Fig. 11 shows the influence of the damping ratio on thePI curve for a perfectly elastic system under triangularload. Increasing the damping from 0% to 20% increasesthe magnitude of the impulsive and quasi-static asymptotesby 29% and 18%, respectively.

The ductility ratio, m, also has a pronounced effect on thePI curve. Fig. 12 shows the PI curves of an elasticper-fectly plastic system under triangular load with varyingdegrees of ductility [10]. For more complicated resistancefunctions, it may be more appropriate to use materialevents, such as yielding, or fracture of reinforcing steel, orcrushing of concrete to define damage instead of ductility.

6. Complicated resistance functions and multiplebehavior modes

Typically, the resistance function of the SDOF systemused for plotting the PI curve is in the form of a simplifiedelasticperfectly plastic relationship. For these types ofAStub

Load

A

2.7 m

Fig. 13. Detail of bsystems, it is natural to plot the PI diagram in terms ofdimensionless P and I quantities in Eq. (2) as opposed tothe non-scaled pressure and impulse values. However, formore complex resistance functions, which are eitherobtained through nonlinear incremental load-deflectionanalysis or through more advanced analytical solutionsthan the usual plastic theory, finding out the necessaryparameters to scale the PI diagram may not be possible.In this case, the PI curve will be specific to the particularstructural component under consideration.Consider the simply supported reinforced concrete beam

(beam 1c) shown in Fig. 13, which was tested by Feldmanand Siess [15]. The resistance and the equivalent masscharacteristics of the beam were determined through anincremental nonlinear analysis procedure described inSection A-A

25.5 cm

3.8 cm

30.5 cm

15.2 cm

Top: 2 #6Bottom: 2 #7Transverse: #3@178mm

fy = 318 MPafc = 40.2 MPa

eam 1c [15].

ARTICLE IN PRESS

500

150

350

400

450

0

50

0

Load

(kN

)

Time (sec)

250

300Steel fracture

Steel strain hardening

150

200Pea

k Lo

ad (k

N)

Test

0

50

100

Steel yield

1001010.1Impulse (kN-sec)

0.10.05

Fig. 15. Loadimpulse diagram for beam 1c [13].

Fig. 16. Details of slab DS1-1 [17].

T. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783780

ARTICLE IN PRESS

2

1.6

1.8

1

1.2

1.4

0.6

0.8Pre

ssuu

re (M

Pa)

0

0.2

0.4

0Deflection (mm)

40035030025020015010050

Fig. 17. Flexural resistance function for slab DS1-1 [13].

2500

2000

1500

1000She

ar F

orce

(kN

)

0

500

0Shear Slip (mm)

87654321

Fig. 18. Direct shear resistance function for slab DS1-1 [13].

T. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783 781

ARTICLE IN PRESS

25

20Test

P

t

15

10Pre

ssuu

re (M

Pa)

Direct

0

5

Flexural

shear

00.001

Impulse (MPa-sec)1010.10.01

Fig. 19. PI diagram for slab DS1-1 [13].

T. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783782Krauthammer et al. [16], and the flexural resistancefunction is shown in Fig. 14. Using the numerical approachdescribed in Blasko et al. [13], one can plot theloadimpulse curve for the beam without having toapproximate the resistance function or the response inthe dynamic range. Furthermore, if the resistance functioncontains the displacements corresponding to specificmaterial events, the threshold curves for these events canalso be included in the PI diagram as shown in Fig. 15. Inthe test, the beam suffered damage, its displacement wasabout 79mm, and its residual displacement was about56mm. The damage was limited to cracking at variouslocations along the beam and noticeable concrete crushingin the compression zone at the midspan. The PI pointcorresponding to the test is also shown to indicate that theobserved response was confirmed by the analysis.

While most components, such as the beam describedabove, are prone to failure in a flexural response mode, forcomponents such as reinforced concrete slabs, the failuremight be initiated in the direct shear mode for some PIcombinations. Fig. 16 shows the construction details andthe resistance function for slab DS1-1 tested by Slawson[17]. The flexural resistance function, shown in Fig. 17, wasobtained using the empirical model proposed byKrauthammer et al. [14] and includes the effects ofcompression and tension membrane behavior [18]. Thedirect shear resistance function, shown in Fig. 18, wasobtained using the Hawkins direct shear model [19]. Fig. 19shows the composite PI diagram for flexural and directshear modes of failure. From this diagram, it is apparentthat these types of components are very prone to directshear-type failure under impulsive loading conditions,which agrees well with test observations [17].

7. Conclusions

Various possibilities need to be considered in allvulnerability assessments of an engineered system subjectedto severe dynamic loads. In the case of building structuresunder the threat of blast, each structural component atdifferent scaled distances from the explosive source issubjected to different transient loadings. Traditionally,pressureimpulse (PI) diagrams are plotted for theassessment and design of structures under severe transientloads, to indicate combinations of pressure and impulsethat will cause a specific damage level. Both closed-formsolutions and energy balance methods are easy to use, butthey are quite restrictive in the types of the structures andloads that can be analyzed. Closed-form solution becomevery involved, if at all possible, for cases where theresistance function is no longer linear elastic, or elasticplastic, or when multiple behavior modes are present. Forcases where the resistance function is nonlinear and/or theload pulse is irregular, numerical solutions are the onlyreasonable means for deriving PI curves, as described,herein.

ARTICLE IN PRESST. Krauthammer et al. / International Journal of Impact Engineering 35 (2008) 771783 783References

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Pressure-impulse diagrams for the behavior assessment of structural componentsIntroductionCharacteristics of P-I diagramAnalytical solutions for P-I diagramsNumerical approach to P-I diagramsInfluence of load and structural properties on P-I diagramsComplicated resistance functions and multiple behavior modesConclusionsReferences