s.

Y

e

AAvailable online 24 July 2009

Keywords:Blast effectsFinite difference analysisStrain rate effectRC slabs

rates across a members cross-section is used to derive sectional momentcurvature relationships. Aformula is derived to estimate the distribution of strain rate over the depth of a cross-section alongthe length of the member, and the corresponding strain rate effects are incorporated into the section-based layered analysis model. The Timoshenko beam equations that include both shear deformationsand rotational inertia are solved numerically using an explicit finite difference scheme. The accuracy ofthe proposed finite difference analysis model is part validated using results of blast testing of reinforcedconcrete slabs with combinations of explosive weights and standoff distances. The results are alsocompared with those obtained by conventional single-degree-of-freedom (SDOF) analysis and finiteelement (FE) analysis using solid elements. The finite difference analysis procedure is both fast-runningand accurate and most suitable for design office application, combining the speed of SDOF analysis andthe detail and accuracy of FE analysis.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

Structural components such as beams and columns are typ-ically analyzed for blast loadings using one of two methods ofvery different complexity: (1) equivalent single degree of freedom(SDOF) analysis [1,2]; and (2) finite element analysismethods (e.g.,[39,29]). The SDOFmethod is easy to implement and numericallyefficient and it is widely referenced in current design guidelines[1012] for the blast analysis and design of building components.However, SDOF analysis is incapable of capturing a spatially andtemporally varying distribution of blast loading, cannot allow forvariations of mechanical properties of the cross-section along themember, cannot simultaneously accommodate shear and flexuraldeformations, can only address strain rate effects indirectly, andcan produce very conservative answers. A finite element analysisusing codes such as LS-DYNA [13] and AUTODYN [14] can be ap-plied to analyze the structural response to blast loads [4] but suchan analysis is rarely used for design-office applications because ofits perceived complexity. An accurate, fast-running finite differ-ence analysis method, suitable for design-office applications is de-scribed herein. The proposedmethod captures the key attributes of

Corresponding author.E-mail address: cwu@civeng.adelaide.edu.au (C. Wu).

rigorous finite element analysis but retains much of the simplicityassociated with SDOF analysis.Finite difference (FD) analysis divides a member into discrete

segments of length dx as shown in Fig. 1 to find an approximatesolution to the differential equations of member motion. Thesegments in partial difference analysis are divided by nodes, withthe displacement at each node expressed in terms of the differencein displacement of adjacent nodes. An FD technique is then usedto solve the partial differential beam equations. An FD model canaccommodate any distribution of load along the member, cancapture a variation of mechanical properties of cross-section alongthemember, can incorporate both shear and flexural deformations,and can analyze the deflections along the entire length of themember: all in contrast to a SDOF analysis where none of theseaccommodations is possible. However, only a few studies havebeen conducted to analyze the response of RC members subjectedto blast loads using FD techniques. Krauthammer et al. used anexplicit FD technique to the Timoshenko beam theory to analyzethe response of RC elements under uniformly distributed [15,16]and non-uniformly distributed [17] loads. In both studies, thedynamic increase factor, which is used to account for strain-rateeffects, is estimated by using an average value of the strain rateover the depth of a cross-section; although the strain rate is farfrom constant over the cross-section depth. Recently, Gong andLu [18] used Timoshenko beam theory to analyze the response ofEngineering Structures

Contents lists availa

Engineering

journal homepage: www.el

Finite difference analysis of simply suppoJ. Jones a, C. Wu a,, D.J. Oehlers a, A.S. Whittaker b, Wa School of Civil and Environmental Engineering, The University of Adelaide, SA, Australiab Department of Civil, Structural and Environmental Engineering, State University of Newc Department of Civil Engineering, Huaiyin Institute of Technology, China

a r t i c l e i n f o

Article history:Received 14 August 2008Received in revised form1 July 2009Accepted 8 July 2009

a b s t r a c t

A finite difference procedurepermits variations in cross-sproposed to accurately andmember under blast loads.0141-0296/$ see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.07.01131 (2009) 28252832

ble at ScienceDirect

Structures

evier.com/locate/engstruct

rted RC slabs for blast loadingsSun c, S. Marks a, R. Coppola a

ork at Buffalo, USA

that can account for strain-rate effects, both shear and flexural deformations,ction geometry and strength and loading over the length of a component isefficiently analyze the dynamic response of a simply supported structuralsection-based layered analysis model that accommodates varying strain

ueThe FD procedure involves discretization of a member into aseries of segments joined by nodes to solve the partial differentialequations of motion. It is different from a finite element analysisthat divides a structural member into three dimensional solidblocks to carry out the dynamic response analysis.

2.1. Timoshenko beam theory

and curvature, a simplified bilinear stress strain relationship forconcrete material is adopted and the maximum compressiveconcrete strain is assumed to be 0.0035, the neutral axis depthis computed by horizontal force equilibrium on the cross section,accounting for the effect of strain rate, and the strain profile isassumed to be linear. The flexural resistance of the cross-sectionis calculated by moment equilibrium about the neutral axis.

2.2.1. Strain rate profile and dynamic increase factors (DIF)2826 J. Jones et al. / Engineering Str

i =0 i =1 i =2

dx

= 0

M0

V0

= 0

0 = 1

= 2V1 V2

Fig. 1. Beam discretization for simpl

Fig. 2. Equilibrium of a discrete segment [20].

a beamcolumn frame to blast-induced ground shock, but did notinclude the effects of the strain rate. Hereinwe extend and validatethe FD model proposed by Krauthammer et al. [1517] and Gongand Lu [18] to incorporate strain rate effects to solve the partialdifferential equations of motion of a structural member subjectedto blast loads. We acknowledge that similar results would likelybe obtained using FE analysis and Timoshenko beam elements butemphasize that FE analysis of blast effects has typically involvedsolid finite elements to date.In this paper, the Duhamel integral [19] is used to derive the

formula to calculate the strain rate profile along the depth of across-section of a simple supported structural member. The de-rived strain rate profile is incorporated into a section-based layeredanalysis model for calculating the momentcurvature function ofthe member under blast loads. The derived momentcurvaturefunction of the structural member is then incorporated into theTimoshenko beam theory [20] and the FD approach is then usedto solve the Timoshenko beam equations. The FD procedure pro-posed herein can accommodate changes in the mechanical prop-erties of a members cross section along its length and through itsdepth in each time step, making it possible to directly incorporateboth strain rate effects (whichwill vary along the length and depthof a member) and non-uniform member loading to solve the par-tial differential equation of motion of the member. The accuracyof the proposed FD method is validated using data from field blasttesting [21]. The results of the FD analysis are then compared withthose from both a SDOF analysis and a detailed finite element anal-ysis. The FD procedure is implemented easily and enables accuratepredictions of member response.

2. Finite difference analysis of blast effectsThe Timoshenko beam equations that were derived fromconsiderations of both vertical force equilibrium and momentctures 31 (2009) 28252832

nn - 1n -2

= 0

Vn = 2Vn-1 Vn-2

Mn = 0

n = n-1

supports in finite difference model.

Fig. 3. Layered analysis of a cross-section.

equilibrium of a beam segment of length dx as shown in Fig. 2were used in this study to model the member responses underblast loads as they take into account both shear deformationsand rotational inertia. These effects can be significant in blastloading due to the fast propagation of stresses in the member.The Timoshenko beam equations [20,22] to describe the membermotion are as follows

Mx V = mI

2

t2(1)

Vx+ p = mA

2v

t2(2)

whereM and V are bendingmoment and shear force, respectively;I is moment of inertia; m is material density; A is the cross-sectional area; p is the any distributed dynamic load transverse tobeam length; is the rotation of the cross section due to bending;and v is the transverse displacement of themid-plane of the beam.

2.2. Moment curvature relationship

In Eqs. (1) and (2), nonlinear flexure can occur anywhere alongthe length of the member. The resistance function for flexuralnonlinearity is represented by the moment versus curvaturerelationship of the member that can be determined using thesection-based layered analysis model of Fig. 3. In the layeredanalysis model, the cross-section is sliced into a large number oflayers; stress and strain are assumed to be constant in a given layer.The resistance of the ith slice is equal to Fi = i(i)DIFi(i)wbs,where i(i) is the stress, DIFi(i) is the dynamic increase factor inthe slice that changes in every step, and wbi andsi are the widthand thickness respectively. The momentcurvature relationshipis computed as per [23] but considering the effect of strainrate as noted above. For the calculation of the ultimate momentFor the SDOF model of Fig. 4 subjected to a triangular forcefunction with a peak force of P and duration td, the following

uJ. Jones et al. / Engineering Str

P(t)

y(t)K

M

P(t)

P

td

T

(a) Single degree of freedomsystem.

(b) Simplified blast loads.

Fig. 4. Single degree of freedom analysis for blast loads.

displacement response is given by Clough and Penzien [19]

y(t) = yst(1 cost + 1

tdsint t

td

)0 t td (3)

y (t td) = y(td)sin (t td)+ y(td) cos(t td) td < t (4)

where is the natural circular frequency of the SDOF system,yst = P/K and K is the stiffness of the SDOF system.When a simple supported member under blast loading is in its

elastic range of response, its shape function can be assumed to besinusoidal,

y(x, t) = y(t)L/2 sinpi xL (5)where x is a distance from the left support of themember; y(t)L/2 isthe displacement response at mid-span of the member, which canbe determined using SDOF using Eqs. (3) and (4). If plane sectionsremain plane after bending, the curvature of a cross-section asshown in Fig. 3 at a distance of x is the second derivative of thedeflection along the member,

(x, t) = d2y(x, t)dx2

. (6)

Substituting Eq. (5) into Eq. (6) yields

(x, t) = y(t)L/2pi2

L2sinpi

xL. (7)

From Fig. 3, the strain at a height of h above the neutral axis ofa cross section is (x, t) = (x, t)h. The strain distribution at adistance x for the tensile rebar, s, and for the concrete, c , in thecompressive zone are

s(x, t) = y(t)L/2pi2

L2(h0 Kud) sinpi xL (8)

c(x, t) = y(t)L/2pi2

L2h sinpi

xL

(9)

and the corresponding strain rates for the tensile rebar s andconcrete c are

s(x, t) = y(t)L/2pi2

L2(h0 Kud) sinpi xL (10)

c(x, t) = y(t)L/2pi2

L2h sinpi

xL

(11)

where Kud is the depth to the neutral axis from the extreme com-pression fiber; h0 is the distance from the extreme compressionfiber to the centroid of tensile reinforcement; and y(t)L/2 is the

velocity response of the member at the mid-span of the member.Substituting Eq. (3) into Eqs. (10) and (11), it yieldsctures 31 (2009) 28252832 2827

a b c d

Fig. 5. Layered analysis of cross section at tensile yielding.

s(x, t) = pi2

L2(h0 Kud)yst

( sint + 1

tdcost 1

td

)sinpi

xL

0 t td (12)c(x, t) = pi

2

L2hyst

( sint + 1

tdcost 1

td

)sinpi

xL

0 t td. (13)The strain rates for both rebar and concrete after td < t can alsobe derived by substituting Eq. (4) into Eqs. (10) and (11). Usingthe above strain rate profile, DIFi(i) for concrete can be calculatedusing CEB code [24],

DIF = fcdfcs=(

s

)1.026sfor 30 s1 (14)

DIF = (

s

)1/3for > 30 s1 (15)

where fcs and fcd are the unconfined uniaxial compressivestrength in quasi-static and dynamic loading, respectively. s =10(6.156s2.0), s = 1/(5 + 9fcs/10) and is the static strain rate(30 106 s1). The DIF formula for the rebar is [25]

DIF =(

104

). (16)

where = 0.074 0.040(fy/414) for yield stress and =0.0190.009(fy/414) for ultimate stress, where fyis the steel yieldstrength (MPa). After DIFi(i) for the concrete and the rebar havebeen determined, the corresponding stress profile and flexuralresistance can be established by cross-section analysis as shownin Fig. 3.We note that the assumed displacement shape function is

strictly applicable only to simply supported beams respondingin the elastic range. For incremental inelastic response involvinghinge formation in the middle of the span, the shape function isbilinear, with incremental deformation associated only with hingerotation in the middle of the span. However, as seen in the tablesin Biggs [1], the differences in the shape functions are modest andso the sinusoidal (elastic) shape function is adopted herein.

2.2.2. Yield and ultimate moment capacityTo develop a simplified momentcurvature relationship of a

reinforced concrete member, the yield and ultimate points aredetermined and the relationship between the origin, yield andultimate points is assumed to be linear. For analysis at the yieldpoint, the strain in the tensile steel is assumed to be equal to theyield strain and an estimate of the neutral axis depthKud ismade asshown in Fig. 3. From the strain diagramwith DIF values calculatedfrom the strain-rate formula in Eqs. (12) and (13), a stress profilecan be derived as in Fig. 5(c) from which the longitudinal forcesacting are calculated as per Fig. 5(d). The curvature is then varied

until equilibrium of the cross-section in axial direction is reachedas indicated in Fig. 5.

u2828 J. Jones et al. / Engineering Str

Fig. 6. Layered analysis of cross-section at concrete crushing.

The point of failure by concrete crushing is analyzed by a similarprocedure. For this point, the strain in the outermost compressionfibre of the concrete is held constant at the assumed failure strainof 0.0035 and the curvature is varied as indicated in Fig. 6 untilhorizontal equilibriumon the cross-section is achieved, the neutralaxis depth is identified and the flexural resistance corresponding toconcrete crushing is computed.The yield and ultimate moment capacities of the cross section

and their corresponding curvatures as well as the momentcurvature diagram in the entire domain are then calculated fromthe section-based layered analysis model.

2.3. Shear behavior

In the Timoshenko beam equations, the shear force V at a cross-section is calculated asV = KAxz = KAGxz (17)whereG is shear stiffness;K is the correction factorwhich is used totake into account the assumption that the shear stress is constantacross the cross-section. For rectangular cross-sections, the valuewas given by Krauthammer et al. [15] to be K = pi2/12.2.4. Numerical technique

A first-order Taylor series expansion for Mx and

Qx and a

second-order Taylor series expansion 2t2and

2t2are used to

approximately solve the Timoshenko beam equations (Eqs. (1)and (2)) in this paper. The magnitude of the error resulting fromtruncating the Taylor expansion to the above terms in the finitedifference scheme is shown in Fig. 7. Small increments in timeand space must be used to limit the error as discussed later. Acomplete description of the numerical procedure using the finitedifference method for the analysis of a beam member under blastloads is provided by Krauthammer et al. [15]. A similar explicitfinite difference schemewas implemented in this study as follows.

2.4.1. Calculating displacement and rotationEq. (18) describes how the rotation of node i at time t + 1 is

calculated from values at adjacent nodes at previous time steps.Eq. (19) describes how the displacement of node i at time t + 1 iscalculated.

t+1i = 2 ti t1i dt2

mIi

[M ti+1 M ti1

2dx V (i)

](18)

t+1i = 2ti t1i +dt2

mAi

[V ti+1 V ti12dx

+ p(i)]

(19)

where all terms have been defined previously.

2.4.2. Calculation of shear force from shear strainThe calculations of rotations and displacements in Eqs. (18) and

(19) require the shear force at each node at the previous time stepsto be known. The shear at a given node is calculated as a function ofthe shear strain. The shear strain is calculated from Eq. (20), withthe partial derivative converted into a finite difference term [15]. txzi =

x =

ti+1 ti12dx

ti . (20)ctures 31 (2009) 28252832

Fig. 7. The approximate slope of a continuous function f (x) at point i.

2.4.3. Moment curvature modelEq. (18) requires the moment at each node to be known.

The moment at a given node is calculated as a function of thecurvature. Since the momentcurvature relationship is nonlinearand the internal stresses in the member are cyclic, it is computednumerically. In the finite difference code developed in this study,the curvature at each node is input to a subroutine that obtainsthe moment at each node. The curvature is given by Krauthammeret al. [15]

ti =

x=

ti+1 ti12dx

. (21)

2.4.4. Member boundary conditionsThe member in the program is divided into n segments as

shown in Fig. 1, with the left support corresponding to i = 0and the right support corresponding to i = n. If one considerscalculating Eqs. (18) and (19) at i = 0 or i = n, it can be seen thatthere are no nodes at i1 or i+1 to complete the calculation. Thisis dealt with by introducing boundary conditions at the supports,and performing the analysis from the node i = 1 to i = n 1. Theboundary conditions at the simple supports can be seen in Fig. 1and the moment and transverse displacement at the two supportsare zero. The shear and rotation at the supports are approximatedby linear interpolation. This is a reasonable assumption becausethemoment, and therefore the curvature, is zero at the support. Asthe rotation is the integral of the curvature, a very small curvaturewill result in only a minor change in rotation at the support.

2.4.5. Stability and convergenceAn explicit time integration method is used to solve Eqs. (18)

and (19). In an explicit scheme, the values of rotations anddisplacements at time step i + 1 are calculated from values attime steps i and i 1 and the time step must be small toguarantee stability of the solution process as described by theCourantFriedrichsLewy condition.The number of segments of equal length dx along the length

of the member, or the number of equally spaced nodes along themember length must be chosen carefully because it influencessignificantly the runtime and accuracy of the FD analysis. For themomentcurvature relationship of Fig. 8 and simply supported endconditions, once the moment at the central node d has reachedthe ultimate moment capacity of the cross-section, an increase incurvature and decrease in bending moment at the center of thespan will accompany an increase in displacement. To maintainequilibrium, the neighboring non-yielded nodes c and e will beunloaded with decreasing curvature. A plastic hinge of length 2dx forms with much of the displacement being associated withconcentrated rotation in the hinge region. Convergence analysiswas conducted by decreasing the size of the segment by half untilthe difference in the displacement response for two consecutivesegment sizes was less than 5% and it was found that a lengthof approximately 0.75 times the depth of the slab (i.e. a plastic

hinge length suggested byWarner et al. [26]) satisfied the requiredaccuracy.

uMajor Bending

Plane

2000

1000

100

Minor Bending Plane

Fig. 9. RC test specimens.

2.4.6. Blast load historiesThe blast loading in the finite difference program is applied on

the member as a function of space and time. This allows an arbi-trary load distribution to be applied to the member. The shape ofthe distributed load across themember is dependant on the chargeweight, shape and standoff distance and the pressure history canbe predicted by current code such as TM5 [10] or using the mea-sured pressure history directly.

3. Validation of models using experimental data

To test the utility of the FD analysismodel, the predictionswerecompared with maximum displacement data from blast testingresults on 2m long, simply supported RC slabs that can bemodeled

diametermesh,with a 10mmconcrete cover (see Fig. 9 for details).The mesh bars were spaced at 100 mm centres ( = 1.34%) in themajor bending plane and 200mm in theminor plane ( = 0.74%).The concrete had a cylinder compressive strength of 39.5 MPa, atensile capacity of 8.2 MPa and a Youngs modulus of 28.3 GPaat the time of testing. The reinforcing bar had a yield strength of600 MPa and a Youngs modulus of 200 GPa.A linear variable displacement transducer (LVDT), accelerome-

ters, and pressure transducers were used to record the response ofthe specimen under blast loads. Wu et al. [21] presents completedetails. Fig. 10 identifies the locations of the instruments. The pres-sure transducers were installed at the center of the specimen (PT1)and near the supports (PT2) tomeasure the distribution of pressureover the face of the specimen.(c) Change in curvature and moment at central and adjacentnodes after maximummoment attained.

Fig. 8. Momentcurvature relationships along the span.

1000

A10

SectionA-AAJ. Jones et al. / Engineering Str

(a) Curvature distribution before yielding.as simply supported beams [21]. The RC specimens were designedwith both tension and compression reinforcement using a 12 mmctures 31 (2009) 28252832 2829

(b) Curvature distribution after yielding.Two specimens NRC (normal reinforced concrete) A and NRCB were subjected to different explosive loads at different standoff

(a) Pentagonal loading at time zero. (b) Variation of pressure with time.LVDT Pressure Transducer (PT)

Support

LVDT

50 mm

PT2

20 mm

Fig. 10. Specimen Instrumentation.

distances. Specimen NRC A was subjected to two shots, one smalland one large. The experimental test program is summarized inTable 1. The explosive charge was suspended above the center ofthe slab as described in [21].The experimental pressure readings demonstrate that the

pentagonal distributed load as shown in Fig. 11(a) is a betterapproximation of the actual blast load; an expected result giventhat the standoff distance and angle of incidence change as afunction of location on the panel. Variables Pr maxC and Pr maxare the peak pressures at the center (PT1) and the edge (PT2),respectively. The pressure time histories at the center and edgeare simplified as triangular blast loads as shown in Fig. 11(b). Theduration of the positive pressure wave td was back-calculated as

td = 2irprmax(22)

where ir is the measured positive impulse at the center and theedge. For analysis, the reported values of the peak load wereassumed to apply over the full width of the slab. Since thenegative pressure phase does not affect significantly themaximumtransient displacement of the panel and should have only asmall effect on the residual displacement, we chose to ignore thenegative phase for our computations.The test specimens were analyzed using the finite difference

analysis model with 23 nodes along the length. Fig. 12 enablesFig. 11. Simplified pr1

00 0.004 0.008 0.012

Time (s)

Fig. 12. Predicted and measured displacement histories for NRC 1A.

Table 2Comparison of maximum deflections from predictions and tests.

Test Max deflection (mm)Experimental Predicted

NRC-1 1.8 1.9NRC-2 7.9 9.6NRC-3 14.0 13.3

a comparison of the predicted and experimental responses forthe NRC 1A. The finite difference model considering strain rateeffect predicted the measured displacement response very well.Ignoring strain rate effects in this instance had little effect onthe maximum displacement but overestimated the displacementsduring the rebound phase of the response history. Table 2compares the predicted responses from the finite differencemodelwith the experimental responses [21]. The finite difference modelaccurately predicted maximum displacement responses in allthree tests.

4. Comparison of different analysis methods

To illustrate issues associated with analysis of structural com-ponents subjected to blast loads, the displacement response ofspecimens 1A and 1B of Table 1 were estimated by SDOF, finiteelement and the finite difference analyses. The SDOF analysis wasperformed using the industry-standard approach first proposed by2830 J. Jones et al. / Engineering Structures 31 (2009) 28252832

Table 1Experimental air blast program.

Blast Slab name Dimension (mm) Reinforcement ratio (%) Standoff distance (m) Scaled distance (m/kg1/3) Explosive used (g)

NRC-1 1A 2000 1000 100 1.34 3 3.0 1007NRC-2 1A 2000 1000 100 1.34 3 1.5 8139NRC-3 1B 2000 1000 100 1.34 1.4 0.93 3440

AccelerometerSlab

Support

PT1

3

2essure distributions.

uJ. Jones et al. / Engineering Str

Table 3SDOF transformation factors.

Elastic region Plastic region

Mass factor, KM 0.50 0.33Load factor, KL 0.64 0.50Loadmass factor, KLM 0.78 0.66

K

Fig. 13. Bilinear resistance-deflection curve.

Biggs [1] and later adapted in US Army Technical Manuals suchas TM5-1300 [10] and ASCE standards and guidelines [11,12]. TheSDOF analysis is based on the transformation of distributed com-ponent mass, stiffness, resistance and loads to equivalent SDOFvalues. The derivation of the transformation factors is reportedelsewhere [1] and not repeated here. Table 3 lists the transforma-tion factors for a simply supported structure under a uniformly dis-tributed load. In the SDOF model, a bilinear resistance-deflectioncurve as shown in Fig. 13 was used in the analysis. For a SDOF sys-tem undergoing elastic deformation, the yield resistance capacityRyield of the cross-section is given by ASCE [11] as

Ryield = 8Myield/L (23)where Myield is the yield moment and L is the span. The deflectionat which the yield resistance is reached can be calculated from

yyield = Ryield/K (24)where

K = 384EI5L3

(25)

and E is Youngs modulus and I is the average moment of inertiaof the cross-section and equal to (Ig + Ic)/2, where Ig is the mo-ment of inertia of the gross concrete cross-section and Ic is themo-ment of inertia of the cracked concrete cross-section that can bedetermined from layered analysis. The ultimate resistance (Rult) ofthe section is back-calculated from the ultimate moment capacity(Mult) that is computed from the layered analysis:

Rult = 8Mult/L. (26)The ultimate deflection (yult) in Fig. 12 is given by

yult = L2 (27)where is the rotation which is derived by assuming that all of therotation in the member takes place over the plastic hinge length[27,28].The computer code LSDYNA [13] with the Mat_PSEUDO_

TENSOR solid element model was used to perform the finite

element analysis. In the Mat_PSEUDO_TENSOR model, two yieldversus pressure curves, as illustrated in Fig. 14, with the means ofctures 31 (2009) 28252832 2831

Fig. 14. Two-curve concrete model with damage and failure.

Fig. 15. Finite element model of the specimen.

migrating from one curve to the other, are adopted for reinforcedconcrete. The two yield versus pressure curves take the form

= ao + p/(a1 + a2p) (28)where ao and aof are the cohesions for both undamaged andfailed materials, respectively; a1 and a2 are pressure hardeningcoefficients, a1f is the pressure hardening coefficient for thefailed material and p is the hydrostatic pressure. In Fig. 14, maxis the maximum yield strength curve and failed is the failedmaterial curve. The strain-rate-effect relationships for the rebarand concrete from the CEB code [24] are also input in theMat_PSEUDO_TENSOR model.Simple tensile failure in Mat_PSEUDO_TENSOR model, namely,

the yield strength, is taken from the maximum yield curve untilthe maximum principal stress (1) in the element exceeds thetensile-cut off cut . For every time step 1 > cut , the yieldstrength is scaled back by a fraction of the distance betweenthe two curves. The set of parameters required in this functioncan be computed through the following formulas where f c is theunconfined concrete compressive strength [13]: ao = f c /4; a1 =1/3; a2 = 1/3f c ; aof = 0; a1f = 0.385; and b1 = 0. Thespecimen was modeled using solid brick elements as shown inFig. 15. Convergence tests were conducted to investigate howmany elements were needed to achieve a reliable estimation.This was realized by decreasing the size of the element by halfwhile keeping loads on the specimen until the difference in theresults between two consecutive element sizes was less than 5%.The convergence tests resulted in the selection of 400,000 solidelements employed in the simulation.The predictedmaximumdeflections of the tests using the SDOF,

finite element and FD model are summarized in Table 4. Thefinite element and FD predictions of the maximum deflection aremuch closer to the measured maximum deflections than the SDOFpredictions. The use of the SDOF model gave a very conservativeprediction (the smallest error is 38%), in part because Pr maxC wasused in the calculation of the equivalent SDOF load. Although the

LSDYNAmodel also predicted themaximum deflection well, muchtime was required to prepare the FE model and reduce the data.

u2832 J. Jones et al. / Engineering Str

Table 4Predictions of maximum deflections.

Test Max deflection (mm)Experimental FDA SDOF

NRC-1 1.8 1.9 2.9NRC-2 7.9 9.6 12.8NRC-3 14.0 13.3 19.3

5. Conclusion

A finite difference analysis model is proposed in this paper forthe analysis and design of simply supported structural memberssubjected to blast loads. A theoretical formula for the strain rateprofile along the depth of a cross-section varying with time andspan is derived for concrete and steel. The stain rate effects are in-corporated into the section-based layered capacity model for cal-culation of the momentcurvature relationship of the member.Using the Timoshenko theory, variation of blast loads, distributedstiffness and mass, as well as mechanical properties are codedinto the finite difference analysis model to solve partial differen-tial equations ofmotion of themember. A comparison between themeasured and analytical responses was made and the largest dif-ference was only 21%, indicating that the finite difference analysismodel can accurately predict the response of a simply supportedstructural member to blast loads. However, unlike the finite ele-ment analysis that divides a member into three dimensional solidelements, the finite difference method discretizes a member into anumber of segments joined at nodes. Far fewer segments are usedin the finite difference model than elements in the finite elementmodel, leading to a substantial reduction in the computationaleffort. SDOF analysis is straightforward and suitable for use in a de-sign office but the results can be substantially conservative. The fi-nite difference analysis can capturemany of the important featuresof a finite element analysis, provides accurate results, is computa-tionally efficient and is ideally suited for routine use in a designoffice.

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Finite difference analysis of simply supported RC slabs for blast loadingsIntroductionFinite difference analysis of blast effectsTimoshenko beam theoryMoment curvature relationshipStrain rate profile and dynamic increase factors (DIF)Yield and ultimate moment capacity

Shear behaviorNumerical techniqueCalculating displacement and rotationCalculation of shear force from shear strainMoment curvature modelMember boundary conditionsStability and convergenceBlast load histories

Validation of models using experimental dataComparison of different analysis methodsConclusionReferences