reliability analysis of direct shear and flexural failure modes of rc slabs under explosive...

Engineering Structures 24 (2002) 189198www.elsevier.com/locate/engstruct

Reliability analysis of direct shear and flexural failure modes ofRC slabs under explosive loading

Hsin Yu Low, Hong Hao *School of Civil and Structural Engineering, Nanyang Technological University, Nanyang Ave., Singapore 639798

Received 12 February 2001; received in revised form 3 September 2001; accepted 3 September 2001

Abstract

Two loosely coupled SDOF systems are used to model the flexural and direct shear responses of one-way reinforced concreteslabs subjected to explosive loading. Incorporating the effects of random variations of the structural and blast loading properties,as well as the strain rate effect caused by rapid load application, failure probabilities of the two failure modes are analyzed. Themodel is capable of predicting the failure probability of the slab with random material and geometrical parameters and subjectedto random blast loading. Considering the random variations of structural properties and blast loading, the failure probabilities ofone-way RC slab designed according to BS 8110 (Stuctural use of concrete, parts 1 and 2 (1985)) are calculated. The effect ofspan length of the slab on its failure probability to blast loading is also investigated. Based on numerical results, a semi analyticalboundary that separates the slabs flexural and shear failure modes is derived as a function of peak reflected pressure and durationof blasting wave. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Reliability; Flexural; Direct shear; Blast loading; SDOF

1. Introduction

Inherent in every structure are the uncertainties in itsmaterial and dimensions. As parameters related to struc-tural materials and geometry, as well as loading on astructure, cannot be completely certain, it is an over-simplification to perform only deterministic structuralanalysis although the deterministic limit state designimplicitly includes the uncertainty effects by applyingsafety factors to both structural material and appliedload. A challenging task in non-deterministic structuralanalysis is to more accurately account for the ran-domness in a given problem with the use of provennumerical algorithm. The result from such an analysiswould be in the form of statistical quantities describingthe response. Many efforts have been focused on thedevelopment of reliability methods and algorithms inrecent years. With these algorithms, reliability problemsare solved more readily. Many researchers have alsostudied the random variations of RC structural material

* Corresponding author. Tel.: +65-791-5278; fax: +65-791-0676.E-mail address: [email protected] (H. Hao).

0141-0296/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0141- 02 96 (01)00 08 7- 6

and dimensional properties. These will be brieflyreviewed and used in the present study.

The analysis of the dynamic response of reinforcedconcrete slabs subjected to blast loading is complicatedbecause the impulsive load caused by the explosion ishighly nonlinear and occurs in an extremely short dur-ation. An experimental phenomenon observed by manyresearchers indicated that reinforced concrete structuressubjected to distributed load of short duration may notbehave plastically at mid-span and fail there. Some ofthe beams might fail at positions very close to the sup-port owing to direct shear failure, i.e., failure is notnecessarily caused by its flexural mode [25]. Althoughit is generally agreed that a large loading with short dur-ation is more likely to cause a slab to fail by its shearfailure mode while a relatively small amplitude load withlonger duration will result in flexural failure, thisphenomenon is not well understood, and a lot of modelshave been proposed to account for this problem. Therehave been studies of structural element response inassessing the dynamic elastoplastic deformation ofplates both analytically as presented, for example, byYankelevsky [6] and experimentally by, for example,Gurke and Bucking [7]. A Timoshenko beam model was

190 H.Y. Low, H. Hao / Engineering Structures 24 (2002) 189198

also used to model the direct shear failure by Ross [8],Karthaus and Leussink [2] and by Krauthammer et al.[9]. However, the simplest approximation is by a SDOFsimplification. As commented in conwep [10], problemsthat involve non-oscillatory loads such as blast load,when only the peak response is required, many structuralsystems may be sufficiently analyzed using only the firstmode. Works in this area include the references [4,5,1115]. The behavioral prediction by the more advancedTimoshenko beam theory has been shown by Krau-thammer et al. [16] to be exactly the same as thosederived by the SDOF approach. In all these studies, thetwo failure modes of a slab to blast load, namely thedirect shear failure and flexural failure, are modelledindependently. This is usually acceptable because thetwo failure modes normally do not occur at the sametime. A slab will enter the flexural response mode onlyif it has survived the direct shear force. The decouplingof the flexural and direct shear responses in SDOF analy-sis was also justified in Krauthammer et al. [16].

In this study, parametric reliability analysis is carriedout to estimate the failure probabilities of RC slabs sub-jected to blast loading. Two loosely coupled SDOF sys-tems are used to model the direct shear and flexural fail-ure modes. The random variations of the RC materialproperties and structural dimensions suggested by otherresearchers as well as the random variations of blastingload proposed by the authors [17], based on a literaturesurvey of the various available empirical formulae, areused in the analysis. Failure probabilities of slabs withdifferent dimensions subjected to blast pressures of vari-ous amplitudes and duration are estimated by the First-order reliability method (FORM). Based on the numeri-cal results obtained, a boundary, which is a function ofblast pressure amplitude, duration and the aspect ratioof slab, is derived to separate the direct shear and flex-ural failure modes. This boundary can be easily used todetermine the failure mode of a given slab under aknown blast pressure history.

2. Random variables

To account for the random variation of basic para-meters in structural analysis, statistical descriptions ofthe variability of loads and material properties ofreinforced concrete (RC) members are required. Theflexural and direct shear strengths and stiffness of RCmembers may vary from the expected values due to vari-ations in material properties and dimensions of the mem-ber, as well as uncertainties inherent in the models usedto compute them. The explosive loading parameters likepressure, rise time and duration might also vary fromtheir expected values owing to its highly nonlinear nat-ure and short duration. In fact, it is well known that suchparameters are very difficult to measure.

2.1. Concrete strength

Random variations of concrete properties have beenstudied by many researchers. It was found that the con-crete cube strength is normally distributed with coef-ficient of variation (COV) of 0.07 if construction qualityis well controlled [18]. Analyzing data obtained from anumber of published sources, the same reference pro-posed the following equation for the COV of concretesin-situ strengthCOV(insitu strength)[COV(cylinder strength)2 (1)0.0084]1/2

Assuming the COV of cylinder strength is the sameas that of cube strength, the COV of in-situ concretestrength is estimated to be approximately 0.11 from theabove equation.

The mean strength of concrete is often related to itscharacteristic strength. Specifically, BS 8110 [1] definesthe characteristic strength of concrete as that value ofthe cube strength below which not more than 5% of thetest results may be expected to fall. Hence, the relation-ship between the characteristic strength (fcu)k and themean strength (fcu)m can be written as(fcu)k(fcu)m1.64sfcu MPa (2)where sfcu is the standard deviation, and subscripts k andm denote characteristic and mean values. Using theabove relation, the mean strength of grade 30 concrete((fcu)k=30 MPa) is estimated to be 36.56 MPa and stan-dard deviation s=COV(fcu)m=0.1136.56=4 MPa.

Many codes and researchers suggest slightly differentapproaches to estimate the Youngs modulus of concreteaccording to the weight, compressive strength and den-sity [19,20]. However, BS 8110s recommendation isadopted here for purpose of consistency, which treats theYoungs modulus as normally distributed with a COVof 0.1. The mean static value of Youngs modulus forGrade 30 concrete is 26 GPa.

2.2. Reinforcement strength

A few statistical distribution types for yieldingstrength of reinforcement have been proposed such asnormal, log-normal and beta distributions [21], or nor-mal distribution [22]. In the latter study, it was proventhat normal distribution is more appropriate forreinforcement yield strength at the 95% confidence level.In the present study, normal distribution is adopted.

The mean yield strength of high-strength deformedbar is related to characteristic yield strength through theequation [23] below

fym fyk11.64COV and COV0.08 (3)

191H.Y. Low, H. Hao / Engineering Structures 24 (2002) 189198

where fyk is the characteristic yield strength and COV isthe coefficient of variation of the yield strength. Usingthis relation, it can be estimated that the mean yieldstrength of reinforcement steel is 530 MPa if its charac-teristic yield strength is 460 MPa. Many studies indi-cated that the variation of Youngs modulus of reinforce-ment steel is minimal [24]. Hence, it is taken asdeterministic with a value of 200 GPa in this study.

2.3. Structural dimensions

Dimension is another parameter that will affect thestrength and stiffness of a structure. Geometric imperfec-tion in RC elements is caused by deviations from thespecific values of the cross-sectional shape and dimen-sions, the position of reinforcing bars, ties and stirrups,the horizontality and verticality of the concrete lines, andthe alignment of columns and beams. Most researchersrecommended the use of normal distribution to modelthe statistical variation of dimensions of structural mem-bers [25,26]. Mirza and MacGregor [26] also analyzedthe variations of dimensions of cast in situ slabs. Basedon the above information on variability in structuresdimensions, a COV of 0.03 is adopted for all dimensions(slab height, h; effective depth, d; length, L; breadth, b),and the designed dimensions are taken as the meanvalues and their variations are assumed following nor-mal distributions.

3. Strain rate effect on strengths

As mentioned in Bischoff and Perry [27], propertiesof the materials used in reinforced concrete structuresare almost all strain rate dependent. It is also pointed outin the same reference that the expected magnitude ofstrain rate for blast loading ranges from 100 to 1000/s.However, owing to the difficulty in carrying out concretematerial tests at high loading rate, knowledge on con-crete material property enhancement is limited to strainrates of 100/s. Moreover, during the blast loading pro-cess, a slab will experience varying strain rates. For thesereasons, in the present study, a constant strain rate of100/s is used in the analysis. This is acceptable as shownby Krauthammer and his co-authors [28] that a constantstrain rate with a reasonable order of magnitude is suf-ficient to yield good results.

3.1. Concrete

The strain rate enhancement formulae for concretesuniaxial compressive strength (s), Youngs modulus (E)and critical axial strain (e0) from the CEB [29] rec-ommendation shown below are adopted in the presentstudy.

sd/ss=(ed/es)1.026a ed30s1sd/ss=g(ed)1/3 ed30s1

(4)

Ed/Es(ed/es)0.026 (5)e0,d/e0,s(ed/es)0.020 (6)where es=30106s1 and a=(5+3fcu/4)1,logg=6.156a0.492, and subscripts d and s rep-resent dynamic and static conditions respectively.

3.2. Steel

The steels dynamic enhancement is based on themodel proposed by Liu and Owen [30], which suggestsfydfysl log10 ees1 (7)in which the parameter l is 0.03 and es for steel isapproximately 102/s.

Coupled with the strain rate effect, the statistical vari-ations of the basic random variables considered are tabu-lated in Table 1. From these basic random variables ofmaterial strengths, the flexural and direct shear strengthscan be obtained from the following models.

4. Structural resistances

4.1. Flexural resistance

The flexural resistance of the structure analyzed isobtained first by computing the momentcurvaturerelation of the section. Then, by considering the supportconditions, the deflection of the structure under uni-formly distributed loading is calculated from the onsetof loading to failure. The incremental procedure forobtaining the static load-deflection relationship of thestructural element is shown in Fig. 1.

In the analysis, it was assumed that strains had a lineardistribution over the beam cross section and that tensilebehavior of concrete located below the neutral axis isneglected. An iterative process is utilized to satisfy theforce equilibrium condition of the section. The entireprocedure is repeated for incremental values of strain intensile reinforcement until failure of the cross section isreached. Here, since ultimate failure is of concern, itimplies a severe deformation of the steel bars (whenreinforcement strain reached esu where esu=10 esy) orcrushing of the concrete (when concrete strain reachesecu). The section analyzed is designed according to BS8110 [1] without considering the severe blast load. Theelastic limit of the momentcurvature relation thus corre-sponds to the point of the first yielding of the reinforce-ment.

The stressstrain relationship employed for the con-


Table 1Basic structural variables adopted (dynamically enhanced)

Variable Mean (strain rate enhanced) Enhancement factor COV Distribution

fcu 84.088 MPa 2.3 0.11 NormalEc 38.5 GPa 1.48 0.1 Normalfy 594 MPa 1.12 0.08 NormalEs 200 GPa 1 Deterministic ecu 0.0054 1.35 Deterministic Dimensions as designed 0.03 Normalmass 2446.5 kg/m3*Volume 0.05 NormalPrmax As recommended 0.3227 Normala As recommended 0.13 Normal

Fig. 1. Flow chart of flexural resistance computation.

crete material is the idealized stressstrain curve of con-crete under uniaxial compression proposed by Hognestad[31]. The problems of concrete confinement and possiblebond slip between concrete and reinforcement are notconsidered. The support conditions are taken as ideal andsymmetrical, where differential settlement and secondorder effect (P- effect) are not considered. The strain

rate effect on flexural resistance is accounted for by theuse of dynamically enhanced material strengths as dis-cussed above.

4.2. Direct shear resistance

The direct shear resistance function of RC structuresis not well developed and thus is more empirical. Themodel used, as shown in Fig. 2, is based originally onKrauthammer et als work [5]. It consists of five straightline segments, namely the elastic response segment OA,hardening segment AB, plastic flow segment BC, soften-ing segment CD and final yielding segment DE. Themodel was developed by modifying a few existing shearstressslip models. The elastic resistance (segment OA)is modelled by equation te given in the figure, for whichthe shear slip is up to 0.1 mm. The segment AB startsfrom a shear slip of 0.1 mm to 0.3 mm and its resistanceis represented by tm as indicated in the figure. In thethird segment BC, the shear strength remains a constant,and the point C corresponds to a shear slip of 0.6 mm.In actual application, Krauthammer et al. [9] did not con-sider the softening region, and the model used was a tri-linear one. In the present study, the tri-linear model isfurther simplified to a bi-linear one for use in the

Fig. 2. Direct shear resistance model.


dynamic analysis (Fig. 2). The simplification is based onthe justification that as long as the area under the stressdisplacement curve remains constant, the energyabsorbed by the system would be the same, and thus thedisplacement calculated would be the same as well [32].The yielding and the maximum allowable shear slip aretaken as 0.1 mm and 0.6 mm, respectively as shown inFig. 2. An enhancement factor of 2 is multiplied to thedirect shear strength, as done in Krauthammer et al. [5]to account for the rapid loading rate. This factor isindeed obtained from tests by Chung [33]. In a series ofpush-off tests for investigating the shear strengthenhancement under dynamic load, Chung [33] reporteddynamic increase factors for shear strength from 1.80 to2.02, under a stressing rate of 10,000 to 12,800 N/mm2/s.

5. Blast loading

The blast load pressure time history on a structure isusually simplified to an exponential or triangular shapeas

P(t)Prmaxeat or Prmax1 tto (8)where Prmax is the peak reflected pressure of the blastwave, a is the decaying rate for exponential represen-tation and to the duration for triangular loading simplifi-cation. The peak reflected pressure Prmax can be easilyestimated by Mills [34] in the following equation.

Prmax2Pmax(710+4Pmax)

710+PmaxKPa (9)

in which Pmax is the peak pressure of the blast wave infree air.

Many empirical formulae and charts are available toestimate Pmax and duration to. From a statistical analysisof eight available publications carried out by the author[17], it was found that the mean value of the peakreflected pressure at various scaled distances Z=R/w1/3(R and w are the stand-off distance in metres and equiv-alent TNT charge weight in kg) can be estimated by

Prmax154.67

Z

617.19Z2

3069.3

Z31.2024 kPa (10)

and the average COV at each Z is 0.3227. The statisticalanalysis also revealed that the average COV of loadingduration is about 0.13.

From the principle of conservation of impulse, thedecaying rate a is found to be related to duration tothrough the relation a=2/to. Since the decay rate andloading duration are inversely related, it is reasonable toassume that they share the same COV.

6. Equivalent SDOF systems

Two coupled SDOF systems with bilinear resistancefunctions are used to represent the shear and flexuralresponse mode of the slab. The first system is used formodelling the flexural response at the point of maximumdisplacement along the span of the structure (ie. at mid-span as the loading and structure are symmetrical), andthe second is for monitoring the direct shear response atthe support.

The SDOF system for modelling the flexural responseis based on Biggs [32] where the deflected shape of thestructure is assumed to be the same as that resulting fromthe static application of the dynamic load. The equival-ent mass and stiffness parameters are derived based onthe mass density, Youngs modulus, moment inertia,span length and boundary condition of the slab, as wellas the deflection shape of the slab under distributed staticload. More details can be found in Biggs [32]. The accu-racy of using a SDOF system to model the slab responseto blast loads was proven by Krauthammer and his co-authors [5]. This simplification for flexural response hasbeen widely accepted and recommended, for instance,by the US Air Force Manual AFM 8822 [35] and USDepartment of Army [36].

The second SDOF system is used to model the directshear response of the slab. Since the direct shear modeis expected to occur within a very short duration afterthe initiation of the explosive loading, the structurewould not have any significant deformation at that time,and because the failure plane occurs very near to thesupport, the phenomenon is very much like a suddencollapse of the entire beam. This implies that the shapefunction of the structure can be taken as unity with negli-gible deflection. The transformation factors for the shearmass, direct shear stiffness as well as the loading aretaken as unity too [11].

7. Performance function formulation

The exponentially decaying function in Eq. (8) isselected to represent the explosive pressures time his-tory. Upon transforming the structural slab into its equiv-alent flexural SDOF system, the post-yielding responseof the equivalent system is given by

zw2yzAeatzy(KyKn)

my(11)

in which z=mid-span displacement; wy=post-yieldingfrequency; A=equivalent explosive force per unit mass;zy=yield displacement; Kn, Ky=pre- and post-yieldingflexural stiffness, and my=post-yielding mass.

The motion equation of the equivalent direct shearSDOF system is


yw21y1/ms[0.5 PrmaxbLeta0.5KLmsz(t)] (12)

where y=shear slip at support; w1=pre-yielding directshear stiffness; L,b=length and breadth of slab; KL=loadfactor, and ms=shear mass. On the right hand side is theapplied shear force at the support, where the first termis contributed by the pressure loading and the secondterm from the inertia force of the structure. The twoSDOF systems are loosely coupled through the acceler-ation of the flexural response that appears in the directshear equation as inertia force.

Their maximum responses can be derived as follows.

z(tmax,f)C1sinwytmax,fC2coswytmax,f (13)

A

a2+w2yetmax,fa

zy(KyKn)w2ymy

y(tmax,ds)H1sinw2tmax,dsH2cosw2tmax,ds (14)

0.5 PrmaxbLms(a2+w22) e

tmax,dsa0.5KLz(tmax,ds)

w22

y1(K2K1)w22ms

where z(tmax,f) and y(tmax,ds) are the maximum responsesof the flexural and direct shear modes respectively wherew2=post-yielding direct shear frequency, C1, C2, H1 andH2 are constants dependent on initial conditions.

The performance functions of both the flexural anddirect shear modes relate the maximum responses to theultimate limits. This is done by dividing the maximumdisplacement reached under the blast loading by the ulti-mate displacement; if the value is greater than one, fail-ure is initiated. Therefore, the performance functions ofthe two modes are shown as follows.

gf1zmaxzu

(15)

gds1ymaxy2

(16)

where zu and y2 are the ultimate displacement of thestructure at the mid-span and the ultimate direct shearslip at the support corresponding to the flexural anddirect shear responses respectively. zu needs be calcu-lated iteratively as indicated in the flowchart given inFig. 1. The description of the procedure for calculatingzu is given in Section 4.1. The ultimate shear slip, y2, ischosen as 0.6 mm as discussed in Section 4.2.

With the performance functions of both the flexuraland direct shear modes defined, the corresponding failureprobabilities of them can be computed by the first-orderreliability method (FORM). The standard reliabilitymethod is used in the present study [37]. The numericalcalculation is performed by using a computer softwarecalrel [38].

8. Assessment of the two failure modes

To the authors knowledge, there is no definition yetin the available literature to define the conditions underwhich the flexural or direct shear failure occurs. Ross[8] has come up with failure curves that divide the peakpressurerise time diagram into two parts; for loadingconditions above the failure curve, direct shear failure isexpected, and for the rest of the region, flexural failureor no failure can be anticipated. He compared thedemand and supply of flexural and direct shear strengthsof the slab in each time increment. The mode in whichthe strength demand reaches its capacity first is the fail-ure mode. However, no information is given on the prob-ability of such damage to the structure.

Krauthammer et al. [5,9,16] used SDOF and Timosh-enko beam theory to model the structural behavior ofslabs under blast. Although their numerical results com-pare well with the test data, no prediction is given as towhen one failure mode happens instead of the other.

Fig. 3(a)(c) show the failure probabilities of the twofailure modes of a one-way simply supported RC slab,of dimensions 1*3*0.17 m3 with 1% tension reinforce-ment and nominal top reinforcement. Detailed infor-mation of the slab used is given in Table 2. The peakreflected pressure is 1.5 and 2 MPa, with varying loadingduration on the horizontal axis. The results are calculatedby using calrel [38] with the above two performancefunctions and the statistical properties of structural para-meters defined in Table 1.

From the figure, a few important characteristics canbe observed:

1. When the peak pressure increases, failure prob-abilities of both modes increase at the same loadingduration, as expected.

2. For a low enough peak reflected pressure (Fig. 3(a)),flexural failure mode dominates the whole range ofloading duration considered.

3. When the peak reflected pressure increases, the inter-mediate behavior occurs, where both modes areactive within the probability of failure range of zeroand one. In this case, as demonstrated in Fig. 3(b),the direct shear mode dominates in the shorter loadingduration, and the flexural mode is important for longerloading duration.

4. If the peak reflected pressure further increases, failureis dominated solely by the direct shear mode in theloading duration range considered, as shown in Fig.3(c).

If the dominating failure probabilities for some particularvalues are of concern, say 0.1, 0.5 and 0.8, then the peakreflected pressure and the corresponding loading dur-ation that give these failure probabilities can beextracted, as shown in Fig. 4. In this figure, each contour


Fig. 3. Failure probabilities of two modes.

Table 2Standard values adopted in analysis

Slab parameters Mean value COV

Slab dimensions L 3 m 0.03B 1 m 0.03h 0.17 m 0.03d 0.145 m 0.03m 1.24*103 kg 0.05r 1% Deterministic

Slabs strengths Kn 1.11*107 N/m 0.13Ky 1.07*106 N/m 0.13K1 2.05*1010 N/m 0.1K2 2.10*109 N/m 0.22zy 0.027 m 0.11zu 0.1713 m 0.13y1 0.1 mm Deterministicy2 0.6 mm Deterministic

Fig. 4. Failure contours of 0.1, 0.5 and 0.8.

indicates the same failure probability and here, failureprobability refers to the one with the higher value of thetwo failure modes under that particular loading. The twomodes are differentiated by their symbols in the figure.

It can be seen that generally each failure contour inFig. 4 is an exponential curve with the peak reflectedpressure Prmax causing failure decaying with increase ofloading duration to. This indicates that the higher thepeak reflected pressure, the shorter the loading durationthat a structure can sustain. Also, with higher peakreflected pressure and shorter loading duration, whichmeans higher loading intensity, the direct shear mode isthe dominating failure mode. The flexural mode domi-nates in the region with relatively lower peak reflectedpressure and longer loading duration. In order to obtainthe transition between the two modes in each failure con-tour, more data points are computed to ensure that thedifference in peak reflected pressure between two con-secutive points that fail by different failure modes do notexceed 0.05 MPa. A straight line can be fitted throughthese transition points in the peak reflected pressure loading duration diagram as shown in Fig. 5. The dia-gram can be used to predict the failure probability and


Fig. 5. Failure contours with failure mode transition.

failure mode for the slab with a given peak reflectedpressure and loading duration.

8.1. Effect of span length

A series of simulations is run for slabs with differentspan lengths. It should be noted that in all the cases, thetwo shorter sides of the slabs are simply supported, andthe other two sides are free. To ensure that the total forceacting on the slabs under the same pressure loading isapproximately the same, the area of the slab is main-tained at around 10 m2. Hence, the slab width b alsovaries with its span length L. The actual dimensions ofeach slab used in the simulation are given in Table 3.Other parameters of the slab such as the thickness, effec-tive depth and reinforcement ratio, remain the same asin Table 2. It should be noted that although the L/b ratioin some cases is less than 2.0, the slabs are designed asone-way slabs and analyzed by the beam theorydescribed above. Since the flexural rigidity of a beamdepends on I/L3, in which I is the moment inertia of thecross section, increasing L and reducing b results in asmaller flexural rigidity and a larger allowable deflec-tion. On the other hand, it reduces its strength capacityas the cross sectional area decreases. Hence, for slabswith the same surface area, the larger the span length is,the better is its capacity to resist blast loading in flexuralmode, but lower is its capacity in direct shear failuremode.

Table 3Slabs of different span lengths

L (m) B (m) A (m2)

5.4 1.8 9.725 2 104.4 2.2 9.683.9 2.6 10.143.2 3.2 10.24

This is demonstrated in the contour of failure prob-ability of 0.1, corresponding to different span lengths inFig. 6 with the failure modes differentiated by symbol.To present a clearer picture, only three span lengths stud-ied are shown in this figure. It shows that the load resist-ance of the slab in the direct shear mode is higher asthe span length L decreases. However, when the peakpressure is small and the dominating failure mode isflexural, the reverse is true. This can be seen from thepoints corresponding to the peak reflected pressures of0.5 and 0.7 MPa in the figure. It is clear that the orderswapped as the failure mode changed from direct shearto flexural. The above observations indicate that as theslab approaches a square shape, it tends to fail in flexuralmode. In other words, if a structure does not fail in directshear mode during the earlier stage when subject to anexplosion, a rectangular slab stands a better chance ofsurviving the flexural failure mode than a square onewith the same surface area.

Fig. 7 demonstrates the failure mode transitions forslabs with different L, with their equations shown againstthe transitions. As loading points above a transition ismore likely to initiate a direct shear failure and the onesbelow in flexural failure, a slab with a smaller spanlength has less chance of failing in direct shear mode.

Fig. 6. Failure contours of 0.1 for slabs with different span lengths.


Fig. 7. Failure transitions of slabs with different span lengths.

It is more likely to fail in flexural mode. The oppositeis true for the slab with a larger span length. If the sameinformation is fitted in a three-dimensional space, asshown in Fig. 8, a function is found relating the peakreflected pressure, loading duration and span length L asgiven below.

PrmaxL2A16.3847107.1571to (17)where A is the surface are of the slab. Again, any pointof combinations of Prmax, to and span length L at a givensurface area A that is lying above the plane is expectedto fail in direct shear mode, while those below in flex-ural mode.

9. Conclusion

Using two loosely coupled SDOF systems to representthe flexural and direct shear failure modes, the failureprobabilities of RC slabs designed to BS 8110 underblast loading have been computed by the first-orderreliability method. Statistical variations of materialstrengths, structural dimensions and parameters of theexplosive loading were considered. Rapid strain rate

Fig. 8. Failure contours of slabs with different span lengths.

effects on material strengths were also accounted inthe analysis.

The numerical results of parametric calculation indi-cate that a slab tends to fail in a direct shear mode ifthe blast load amplitude is high but of short duration. Ittends to fail in flexural failure mode if load amplitudeis relatively low and duration is relatively long. Inaddition, the span length effect of the RC slab with thesame surface area has also been analyzed. Results indi-cate that a slab tends to fail in direct shear mode whenit is relatively stiffer with a smaller span length. Whenit is relatively flexible with a larger span length, itschance of survival of the direct shear failure modeincreases. A semi-analytical boundary has been derivedto predict the failure mode of a slab in terms of its spanlength L, surface area A, and the peak reflected pressureand pressure loading duration.

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