Engineering Structures 32 (2010) 1691–1703

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

A new method for progressive collapse analysis of RC frames under blast loading

Yanchao Shi a,b, Zhong-Xian Li a,b,∗, Hong Hao a,b,c

a School of Civil Engineering, Tianjin University, Tianjin 300072, Chinab Key Laboratory of Coastal Civil Engineering Structure and Safety (Tianjin University), Ministry of Education, Tianjin 300072, Chinac School of Civil & Resource Engineering, The University of Western Australia, WA 6009, Australia

a r t i c l e i n f o

Article history:Received 30 June 2008Received in revised form1 February 2010Accepted 4 February 2010Available online 29 March 2010

Keywords:Reinforced concrete (RC) frameBlast loadingProgressive collapseNumerical analysisInitial damageNon-zero initial conditionDirect numerical simulation‘Member-removal’ procedure

a b s t r a c t

The progressive collapse of structures under blast loading has attracted great attention all over theworld. Some guidelines give specific procedures to analyse the progressive collapse of building structures.Numerical analysis and laboratory test results of the progressive collapse of structures have also beenreported in the literature. Because the progressive collapse of a structure induced by blast loadingoccurs only after the blast-loading phase, most of these studies and guideline procedures performprogressive analysis by removing one or a few load-carrying structural members with static and zeroinitial conditions. The damage on adjacent structural members that might be induced by blast loadsand the inevitable non-zero initial conditions when progressive collapse initiates are neglected. Thesesimplifications may lead to inaccurate predictions of the structural collapse process. In this paper, a newmethod for progressive collapse analysis of reinforced concrete (RC) frame structures by considering non-zero initial conditions and initial damage to adjacent structural members under blast loading is proposed.A three-storey two-span RC frame is used as an example to demonstrate the proposedmethod. Numericalresults are compared with those obtained using the alternative load path method, and with those fromcomprehensive numerical simulations by directly applying the blast loads on the frame. It is found thatthe proposed method with a minor and straightforward extension of the simplified ‘member-removal’procedure is efficient and reliable in simulating the progressive collapse process of RC frame structures.It requires substantially less computational effort as compared to direct numerical simulations, and givesmore accurate predictions of the structural progressive collapse process than the ‘member-removal’approach.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Progressive collapse refers to the failure of one or a group ofkey structure load-carrying members that gives rise to a morewidespread failure of the surrounding members and partial orcomplete structure collapse. It is defined as ‘‘the spread of an initiallocal failure from element to element resulting in the collapse of anentire structure or a disproportionately large part of it’’ [1]. Manyaccidental and intentional events, such as false construction order,local failure due to accidental overload, damage of a critical com-ponent by earthquake and explosion, could induce the progressivecollapse of a structure. However, because of the high peak, shortduration and negative phase of the blast load, the progressive col-lapse induced by an explosion is very different from that by earth-quake ground excitations. With the recent progressive collapse of

∗ Corresponding author at: School of Civil Engineering, Tianjin University, Tianjin300072, China. Tel.: +86 22 2740 2397; fax: +86 22 2740 7177.E-mail address: zxli@tju.edu.cn (Z.-X. Li).

0141-0296/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2010.02.017

the Alfred P. Murrah Federal Building and World Trade Centre dueto blast and impact, research is focused more than ever to makebuildings safer from progressive collapse induced by blast and im-pact loading.For an economic and safe design of structures against progres-

sive collapse to blast loads, a reliable progressive collapse analy-sis is essential. Because of the catastrophic nature of progressivecollapse and the potentially high cost of retrofitting buildings toresist it, it is imperative that the progressive analysis methodsbe reliable [2]. Engineers need an accurate and concise method-ology to produce trustworthy and timely results. Thus, many re-searchers have been spending lots of effort in developing reliable,efficient and straightforward progressive collapse analysis meth-ods recently.Krauthammer et al. [3] developed a procedure for studying

progressive collapse both theoretically and numerically, and es-tablished a reliable structural damage assessment procedure topredict a possible future phase of progressive collapse. Luccioniet al. [4] carried out an analysis of the structural collapse of a re-inforced concrete building caused by a blast load. In the analysis,

1692 Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703

the building was modelled using 3D solid elements, including thereinforced concrete columns, beams and masonry walls. The vol-ume of air in which the structure was immersed was also mod-elled. The comparison of numerical resultswith photographs of thecollapsed structure by blast load showed that the numerical anal-ysis reproduced the collapse of the building under the blast load.Marjanishvili [5] summarized the progressive collapse proceduresdefined in the US General Service Administration (GSA) [6] and USDepartment of Defence (DoD) [7] guidelines and discussed theiradvantages and disadvantages. Kaewkulchai and Williamson [8]proposed a framework for computing the dynamic response offrame structures during a progressive collapse event to overcomethe limitations of the Alternative Load Path method, i.e., the GSAmethod and DoD method. Sasani [9] evaluated the response of asix-storey reinforced concrete infilled-frame structure followingthe simultaneous removal of two adjacent exterior columns usingthe finite element method and the applied element method. Theyfound that the analytical results show good agreement with ex-perimental data. Tsai and Lin [10] carried out nonlinear static andnonlinear dynamic analyses to estimate the progressive collapseresistance of a building subjected to column failure. The resultsshowed that different assessed results are obtained by the linearstatic method and the nonlinear acceptance criterion suggested bythe GSA guidelines. Mohamed [11] investigated the implementa-tion of UFC 4-023-23 to protect against the progressive collapseof corner floor panels when their dimensions exceed the damagelimits through analysing the progressive collapse potential of areinforced concrete building using the alternative path method.Kwasniewski [12] carried out the progressive collapse analysis ofan existing eight-storey steel framed structure built for fire testsusing nonlinear dynamic finite element simulations following theGSA guidelines. A detailed 3D model with a large number of fi-nite elements was developed for the entire structure, and themainmodelling parameters affecting the numerical results were identi-fied. Hao et al. [2] found that both the GSA and DoD methods maynot give reliable predictions of structural progressive collapse andusually underestimate the stress and strain response at the sup-porting joint of adjacent columns. The authors [13] also found thateven the nonlinear dynamic alternative load Path method wouldunderestimate the collapse potentials of RC frames and its dy-namic responses. Starossek [14] developed a typology and classifi-cation of the progressive collapse of structures based on a study ofthe various underlying mechanisms of collapse. Six different typesand four classes of progressive collapse are discerned; the char-acteristic features of each category are described and compared.They are pancake-type collapse, zipper-type collapse, domino-type collapse, section-type collapse, instability-type collapse andmixed-type collapse. Vlassis et al. [15] proposed a novel simplifiedframework for the progressive collapse assessment ofmulti-storeybuildings, considering the sudden column loss as a design scenario.Using the proposedprocedure, they conducted a case study to learnthe progressive collapse process of a typical steel-framed compos-ite building. The result demonstrated that steel-framed compos-ite buildings with typical structural configurations could be proneto progressive collapse initiated by local failure of a vertical sup-porting member. Vlassis et al. [16] also proposed a new design-oriented methodology for the progressive collapse assessment offloor systemswithinmulti-storey buildings subject to impact froman above failed floor. The proposed method was applied to anal-yse the progressive collapse of a typical multi-storey steel-framedcomposite buildingwith the impact of a floor plate. Saffen [17] car-ried out a simple analysis of the progressive collapse of the WorldTrade Center. In the analysis, a simplified variable-mass collapsemodel was used. By solving the governing equation of motion, in-formation about the overall collapse conditions was obtained.As reviewed above, the current methods of analysing structural

progressive collapse consist of two major approaches, namely the

direct simulation of blast-loading effects on structural damage andcollapse, and uncoupled alternative load path analysis of struc-tural progressive collapse without considering the blast-loadingeffects. The direct simulationmethod can yield reliable predictionsof structural collapse to blast loads [2,4], but it is extremely timeconsuming, and requires a profound knowledge of structuraldynamics, damage mechanics, dynamic material properties andcomputational skills. It is therefore not practical for common en-gineering applications. The uncoupled alternative load path anal-ysis as specified in the GSA [6] and DoD [7] guidelines is easy toimplement, but does not necessarily yield reliable predictions ofstructural progressive collapse induced by blast loading. The pri-mary drawback of the alternative load path approaches is that theyneglect the ‘initial damage’, or damage in adjacent structure mem-bers caused by the blast load, and the non-zero initial condition [2].Obviously, if the blast load is big enough to knock off one or a fewstructural columns, a certain degree of damage in adjacent struc-turalmembers is inevitable, whichwill definitely reduce themem-bers’ stiffness and strength. Moreover, the structure will not havezero initial conditionswhen progressive collapse initiates althoughstructural progressive collapse usually occurs after the action ofblast loads.The objective of this paper is to develop a new method for the

progressive collapse analysis of RC frames with consideration ofboth the non-zero initial condition and existing damage in struc-tural members. The method consists of three steps: (1) determi-nation of critical blast scenarios of the RC frame for progressivecollapse analysis; (2) determination of the non-zero initial condi-tion and initial damage of the structural members caused by theblast loads, and (3) progressive collapse analysis with considera-tion of both the non-zero initial condition and damage in structuralmembers. A three-storey two-span RC frame is used as an exam-ple to demonstrate the efficiency and reliability of themethod. Thecommercial software LS-DYNA is used to perform the numericalcalculations. The commonly used alternative load pathmethod andthe direct simulation method are also used to analyse the progres-sive collapse of this example frame. Numerical results obtainedfrom the three approaches are compared. The reliability and ef-ficiency of the proposed method in the analysis of the structuralprogressive collapse to blast loadings are verified.

2. Critical blast scenarios for progressive collapse analysis of RCframes

As discussed above, besides completely destroying some keystructuralmembers, a blast load also causes damage to other struc-tural members and non-zero structural velocity and displacementwhen progressive collapse initiates. Therefore, a reliable structuralprogressive analysis should take into consideration the non-zeroinitiation conditions andpossible damage to other structuralmem-bers. Since the velocity, displacement, and the damage severityof the structure at the end of the blast-loading phase depend onthe blast scenarios, the critical blast scenarios that completely de-stroy some key structural members to cause progressive collapseneed be determined first. The corresponding structural displace-ment, velocity, and damage at the end of the blast-loading phasecan then be determined and used in the subsequent progressivecollapse analysis.The principles for selecting the critical blast scenarios (includ-

ing the location and charge weight) are as follows.a. The critical explosive location should be selected at places

where terrorist bombing or accidental explosion is possible.b. The critical charge weight is defined, in association with the

explosion center, as the minimum charge weight that will causecollapse to the columns that will be removed in the subsequentanalysis. This can be easily done by using many pressure–impulse(P–I) curves or design charts available in the literature for parti-cular structural columns.

Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703 1693

It should be mentioned that, for a given RC frame, there mightbe several critical blast scenarios, i.e., placing explosives at a fewpossible locations may all cause the collapse of the column underconsideration. In these cases, in order to get a full understandingof the progressive collapse resistant capacity of the RC frame, sev-eral progressive collapse analyses under different blast scenariosshould be carried out.

3. Derivation of the non-zero initial condition and initialdamage of adjacent structural members

3.1. Non-zero initial conditions

The initial condition is normally considered as the velocityand displacement of the structure or structure member at thebeginning of dynamic response analysis. Herein it is the velocityand displacement of the adjacentmembers at the time of completeloss of the key columns. It is also the beginning of the progressivecollapse analysis in the GSA and DoD guidelines.In order to derive the initial conditions of the adjacent mem-

bers, it is assumed that the progressive collapse begins at the endof the blast-loading phase. This is a reasonable assumption becausethe blast-loading duration is very short, usually of an order of mi-croseconds. To derive these initial conditions, the commonly usedequivalent single degree of freedom (SDOF) approach is used, asexplained in the following.

3.1.1. Damage modes and deflection shape functionBoth numerical and experimental studies indicate that the

damage modes of RC members under blast loading depend notonly on the blast load, but also on the properties of the structuralmember, such as the shear force and bending moment resistancecapacity of the member. In general, there are three failure modes,namely shear failuremode, flexural failuremode and the combinedshear and flexural failure mode. The exact damage mode andcorresponding deflection shape of a structural member underblast loading are normally predicted through detailed numericalanalysis or field tests. However, in practice, the structural memberdeflection shape is usually assumed when deriving the equivalentSDOF system [18], and this often leads to an acceptable predictionof the overall structural response.In this paper, to simplify the calculation, a plastic deflection

shape is assumed with the plastic mechanism for the structuralbeam or column. The deflection shape function is triangular, withtwo hinges at both ends plus a hinge at the mid-span, as shownin Fig. 1. It should be noted that in theory the use of an elasticdeflection shape is more appropriate because the consideredstructural members for calculations of initial conditions anddamage are assumed not to collapse by the blast loads. However,either elastic deflection or plastic deflection shape assumptionbrings in some error in deriving the equivalent SDOF system,especially in deriving the load–mass factor. In this study, the plasticdeflection shape is adopted only because of its simplicity andpopularity in practice. The assumption might lead to some error.If the displacement at the mid-span of the member is smax, the

deflection shape function of the member could be described by

s(y) = smax

(1−

yL/2

)(1)

where y is the distance measured from the mid-span of themember and L is the length of the member. Because of the veryshort duration of the blast load, the acceleration of themember canbe assumed as a constant during the loading phase. In this case, therelationship between the velocity and the displacement is linear.Therefore, the distribution of the initial velocity along the memberis

v(y) = vmax

(1−

yL/2

)(2)

Fig. 1. Assumed deflection shape of a uniformly loaded RC member.

where vmax is the velocity at the end of the blast-loading phase atthe mid-span of the structural element.

3.1.2. Maximum initial velocity and displacementIf the blast loads acting on the structural member are known,

the maximum initial velocity and displacement at the end of theblast-loading phase can be derived easily by analysing the equiv-alent SDOF system. According to Biggs’ approach [19], the equiva-lentmass, stiffness and load of the SDOF system can be determinedthrough the following formulae:

Me = KMMt (3)Mt = mL (4)

where Me is the equivalent mass and Mt is the total mass of thesystem, which is equal to the mass per unit lengthmmultiplied bythe total length of the member L. KM is the ratio of the equivalentmass to the total mass, which is related to the boundary conditionand deflection shape function of the member [19].

Fe(t) = KLFt(t) (5)Ft(t) = p(t)L (6)ke = KLk (7)

where Fe(t) is the equivalent load. Ft(t) is the total load of the sys-tem, which is equal to the load per unit length p(t) multiplied bythe total length of the member L. KL is the ratio of the equivalentload to the total load, which is also related to the boundary con-dition and the deflection shape function. Time t is the same in thetwo systems.Through Eqs. (3)–(7), all the parameters of the equivalent SDOF

system can be derived. Suppose that the equivalent blast load isFe(t); since the blast load is of very short duration, the equation ofmotion can be approximately written as

Fe(t) = Mea. (8)

Then

vtd =

∫ td0 Fe(t)dtMe

=IeMe

(9)

std =∫ td

0vdt =

∫ td0

∫ t0 Fe(t)dtdtMe

(10)

in which td is the loading duration. vtd and std are the velocity anddisplacement at time td, respectively. If the blast load is assumedto be triangular, the displacement can be finally derived as

std =23vtd td. (11)

Therefore, both the initial velocity and displacement can beobtained based on the above equations. It is worth noting that if

1694 Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703

Fig. 2. P–I curves for different damage degree D.

the blast load duration is so small that the member has no timeto deform during the blast-loading phase, the initial displacementwill be very small compared with the length of the member.In this case, for the purpose of simplification, this small initialdisplacement is ignored in the progressive collapse analysis of theRC frame.However, the velocity is not necessarily small, dependingon the blast-loading impulse, and it is always important to includeit in the analysis.

3.2. Initial damage

Initial damage is another very important parameter that shouldbe considered in the progressive collapse analysis of RC frames. Thedamage severity can be estimated by using the pressure–impulse(P–I) diagram for RC members [20–23]. The damage degree of themember is obtained by using the following proposed procedure.Suppose that the pressure–impulse diagram of a column devel-oped by the authors [20], as shown in Fig. 2, is available, the pro-cedure is as follows.a. Estimate the pressure and impulse acting on themember and

locate it in the pressure–impulse diagram in the P–I space.b. If the point is in the small damage range, for example, the

damage index D is smaller than 0.2, the initial damage of this RCmember is ignored.c. If the point is in the range of collapse (D > 0.8), it means that

the RC column is totally damaged and does not have or only hasminimum load-carrying capacity. In this case, the correspondingcolumn(s) are removed at the beginning of the progressive collapseanalysis.d. If the point is in other ranges, for example point A in Fig. 2,

the damage degree of this column is obtained by interpolation be-tween the two adjacent damage degrees DA and DB. It should benoted that the proper interpolation is done by deriving an inter-mediate P–I curve that has point A on it because of the nonlinear-ity of the P–I curve. In this study, a program is written in MATLABto derive this P–I curve. Fig. 2 shows the pressure–impulse curvecorresponding to the damage degree D.In order to model the initial damage of an RC member, one

should relate the above damage degree to the member materialstrength and stiffness degradation. In order to do this, a fewassumptions related to the member damage are made. First, it isassumed that damage only occurs in the concrete material. This isreasonable because, for an RCmember that still maintains a certainlevel of load-carrying capacity at the end of the blast-loadingphase, the steel bar is normally in the elastic range. This is becausethe blast-loading duration is very short, and damage to the RCmember in this loading phase is usually brittle failure. Therefore,the reinforcement is unlikely to enter the plastic deformationstage.Moreover, plastic deformation of steel bars is associatedwithlarge cracks in the concrete, which dramatically reduces the load-carrying capacity of the RC member. Second, it is assumed that the

damage of an RC member is limited to several damage zones. Thenumber and the location of the damage zones are dependent onthe damagemodes of the RCmember. If an RCmember is damagedprimarily by shear damage mode, two damage zones, each at oneend, are assumed. If the damage is primarily by flexural mode,one damage zone at the mid-span of the member is assumed. Thelength of the damage zone is assumed to be one fifth of thememberlength. In every damage zone, the damage degree is assumed tobe uniform. The occurrence of damage type, i.e., primarily shearor flexural damage, depends on the blast-loading duration andthe vibration period of the structural member. In this study, ifthe loading is quasi-static, the damage is assumed to be primarilyflexural failure; if the loading is of impulsive type, the damage isassumed to be primarily shear failure [20].The damaged concrete compressive strength and Young’s mod-

ulus for each damage zone are defined as

f ′c,dmg = KY f′

c (1− D) (12)

Edmg = KEE(1− D) (13)where f ′c and E are the yield compressive strength and the Young’smodulus of the undamaged concrete; f ′c,dmg and Edmg are the yieldcompressive strength and the Young’s modulus of the damagedconcrete, respectively. KY and KE are the modification factors usedto reduce the errors arising from the simplifications related to theassumptions of uniform loading and damage only to the concrete.In this study, however, both KE and KY are set to be 1, which, as willbe demonstrated, gives a very good representation of the effect ofstructural member damage on the progressive collapse. However,more examples with different blast-loading and structural damagescenarios need be analysed to derive more appropriate KY and KEvalues.

3.3. Validation

Numerical simulations are carried out to validate the aboveprocedures in deriving the initial condition and initial damage ofRC members. A typical RC column, which is extracted from the RCframe in Section 5, is analysed using LS-DYNA. This column is 3 mlong with a cross-section of 300mm×300mm. It has four verticalsteel bars, each having a diameter of 24 mm with the yield stress335 MPa. The stirrup is D10@200 with the yield stress 235 MPa.The finite elements and material model for the studied column

are exactly the same as those used in modelling the RC frame inSection 5, and will be described in detail there. In order to relatethe studied column to the actualmemberwithin the frame, columnconstraintswith higher fidelity are employed. As shown in Fig. 3(a),a footing and a head are included in the numericalmodel. The outervertical faces of the footing and the head were constrained againsthorizontal motions (i.e., in the x-direction and the y-direction)and the bottom face of the footing is constrained against verticalmotion (i.e., in the z-direction).The peak overpressure applied on the column is 2495 kPa,while

the reflected impulse is 3642 kPa ms. This is the same as theblast load acting on column C1 in the RC frame in Section 5. Sincethe critical standoff distance considered is 10 m, the blast loadis applied uniformly on one side of the RC column. Fig. 3(b) and(c) give the contours of the transverse velocity and the effectiveplastic strain of the column at the end of the positive phase ofthe blast load. As can be seen, the distribution of the transversevelocity along the column is approximately triangular, as assumedin Section 3.1. The damage zones are also at both ends. Since theblast load applied is in the impulsive range, this indicts that theassumption made in Section 3.2 is reasonable.In order to derive the initial velocity of the RC column using

the method discussed in Section 3.1, the RC column is simplifiedinto an SDOF system. The equivalent mass of the column is 369 kg;the equivalent stiffness is 1.13 × 108 N/m, the equivalent blastload is derived as 1437 kN, and the equivalent impulse is 2.10 kN s.

Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703 1695

Head

S

S

HRebar and ties

Concrete

Footing

a

b

c

Fig. 3. RC columnmodel and its transverse velocity and effective plastic strain contours at the end of the blast-loading phase. (a) boundary conditions, (b) transverse velocitycontours, (c) effective plastic strain contours.

1696 Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703

According to Eqs. (9)–(11), the initial velocity and displacement ofthe column are obtained as 5.52 m/s and 10.75 mm, respectively,as compared to 5.55m/s and 9.62mm derived from the numericalsimulation. These results indicate that using the equivalent SDOFsystem gives reliable displacement and velocity estimations of thecolumn at the end of the blast-loading phase.

4. Proposed method for progressive collapse analysis of RCframes under blast loading

The proposed method is based on the alternative load pathmethod in the GSA and DoD guidelines [6,7], but incorporates thenon-zero initial condition and initial damage of structural mem-bers in the analysis. The procedure of this method is as follows.a. Establish the finite element model of the RC frame.b. Select the critical blast scenarios from all possible cases

according to the above proposed principles; for each blast scenario,do Steps c to f.c. Prior to the removal of the key element, bring the model to

static equilibrium under the combination of dead loads and liveloads as defined in the GSA and DoD guidelines.d. From available P–I diagrams, assess the damage levels of all

members close to the explosion center, and calculate the initial ve-locity, displacement and the initial damage of the structural mem-bers that are not completely damaged by direct blast loading usingthe above proposedmethod; andmodify thematerial properties ofthese structural members according to the estimated damage de-grees from P–I diagrams.e. Remove those elements that are completely damaged by di-

rect blast loading instantaneously to perform progressive collapseanalysis with the non-zero initial velocity and displacement ap-plied to the structure.f. Continue the dynamic analysis until the structure reaches a

steady and stable condition or collapse.

5. Progressive collapse analysis of an RC frame: Comparisonand verification

5.1. Numerical model

The software LS-DYNA is utilized to carry out the progressivecollapse analysis of the example frame structure, as shown in Fig. 4.The frame has two bays with a span of 6 m each in the x-direction,and 3 m in the y-direction. The storey height is 3 m for all levels.The dimensions of all the columns are 300 mm× 300mm, and thebeams are 200 mm × 300 mm. All the columns and beams have2% longitudinal reinforcement with the yield stress 335 MPa andø10@200 mm hoop reinforcement with the yield stress 235 MPa.The slab is 150 mm thick with a dimension of 6 m × 3 m. Thelongitudinal reinforcement is also 2% and the yield stress of thesteel bar is 335 MPa.Solid elements (50 mm cubes) with a single integration point

are used to model the column, beam and slab. The shell elementis utilized to model the rigid ground. The numerical convergencestudy shows that further decrease of the mesh size only has a littleeffect on the numerical results but leads to a much longer calcula-tion time. Therefore, a mesh size of 50 mm is used in the study.ThematerialmodelMAT_CONCRETE_DAMAGE (MAT_72) avail-

able in LS-DYNA is used in the present study to model the con-crete [24]. This model has been used to analyse concrete subjectedto impulsive loading successfully [20]. Different strain rate effectscan be implemented for tension and compression to simulate thedesired rate effects. The simulated crack patterns using this con-crete damage model also agree well with the experimental obser-vations [24,25].The material model MAT_PLASTIC_KINEMATIC (MAT_003) is

used to model the steel. It is an elastic–plastic material model

Fig. 4. Sketch of the RC frame.

Table 1Material properties of concrete.

Compressive strength Young’s modulus Poisson’s ratio Density

24 MPa 23000 MPa 0.2 2500kg/m3

Table 2Material properties of steel.

Strength Young’s modulus Poisson’s ratio Steel ratio

335 MPa 200000 MPa 0.3 2%

with strain rate effect. Thematerial modelMAT_RIGID (MAT_20) isused to model the rigid ground. The contact between the structuremembers and rigid ground is also considered using the *CONTACTmodel available in the software [24]. Thematerial properties of theconcrete and steel used in the model are given in Tables 1 and 2.Many empirical relations are available in the literature to con-

sider the strain rate effect on concrete material properties. In thispaper, the K&C model, which is an improvement of the CEB modelbased on test results, is adopted [26,27]. The effect of strain rate onthe concrete and steel strength is typically represented by a param-eter, namely the dynamic increase factor (DIF ). It is the ratio of thedynamic-to-static strength versus strain rate. In the K&C model,the DIF values for compressive and tensile strengths are definedseparately.The DIF of the tensile strength is given by the following

equations:

TDIF =ftdfts=

(ε̇d

ε̇ts

)δfor ε̇d ≤ 1 s−1 (14)

TDIF =ftdfts= β

(ε̇d

ε̇ts

)1/3for ε̇d > 1 s−1 (15)

where ftd is the dynamic tensile strength at the strain rate ε̇d, fts isthe static tensile strength at the strain rate ε̇ts (ε̇ts = 10−6 s−1), andlogβ = 6δ − 2, in which δ = 1/(1+ 8f ′c /f

′co), f

′co = 10 MPa, and f

′c

is the static uniaxial compressive strength in MPa. In compression,the empirical formulae are given as

CDIF =fcdfcs=

(ε̇d

ε̇cs

)1.026αfor ε̇d ≤ 30 s−1 (16)

CDIF =fcdfcs= γ (ε̇d)

1/3 for ε̇d > 30 s−1 (17)

where fcd is the dynamic compressive strength at the strain rate ε̇d,ε̇cs = 30× 10−6 s−1, log γ = 6.156α − 0.49, α = (5+ 3fcu/4)−1,

Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703 1697

fcs is the static compressive strength, and fcu is the static cubecompressive strength in MPa.For steel, the dynamic increase factor (DIF ) is given as [28]

DIF =(

ε̇

10−4

)α(18)

α = 0.074− 0.040fy414

(19)

where ε̇ is the strain rate of the steel bar in s−1 and fy is the steel baryield strength in MPa. This formulation is valid for steel bars withyield stress between 290 and 710 MPa and for strain rate between10−4 s−1 and 225 s−1.In order to simulate the progressive collapse process of the RC

frame, the so-called erosion algorithm is used. This algorithm isemployed to capture the physical fracture process of the mate-rial if no significant reverse loading occurs to the fractured el-ements [29]. There may be a variety of criteria governing theerosion of the material, such as principal stress, principal strain,shear strain, pressure, and so on. Xu and Lu [29] used the princi-pal tensile strain as the erosion criterion for reinforced concrete.Themaximum principal tensile strain at failure is assumed as 0.01.Their simulation results for concrete spallation show a consistentcomparison with the relevant experimental observations. Unos-son [30] adopted numerical erosion based on a shear strain cri-terion to simulate the penetration and perforation of three typesof high-performance concrete (HPC) targets. The maximum shearstrain at failure they used is 0.8 to 0.9.It must be emphasized here that the erosion technique is in-

troduced to overcome the large distortion problem in numericalsimulations. It has no solid physical background. The erosion cri-teria must be used with caution, as early and premature erosion ofmaterial can lead to incorrect model predictions, and significantlyincrease the mesh-size dependency of the calculation [31,32].Therefore, the limiting value for the erosion criteria, i.e., the maxi-mumvalue of each damage criterion at failure, cannot be too small.Otherwise, incorrect model predictions might occur.In this study, two erosion criteria, i.e., principal strain and

shear strain, are adopted. The element will be deleted if eitherone of the two erosion criteria is met. Limiting values for boththe principal strain and shear strain erosion criterion are carefullyselected. First, the range of the limiting value is decided accordingto the available references in the literature. For the principal straincriterion, the limiting value is set to be 0.10 initially, which isten times the limiting value for principal tensile strain criterionin [29]. The limiting value for the shear strain criterion is setto be 0.8 according to [30]. In order to get reasonable values ofthese two erosion criteria, several calculations are carried out; ongradually increasing the limiting values of these two criteria fromthe numerical results, the limiting value of the principal straincriterion for erosion is decided to be 0.15 and the shear straincriterion to be 0.9. Further increasing these valueswill lead to largedistortion of the numerical elements, while decreasing the valuesmay result in premature eroding of the materials in the structuralmodel.

5.2. Benchmark analysis

In order to verify the proposed method of progressive collapseanalysis of RC frames, a benchmark progressive analysis of theexample RC frame is carried out by using the direct simulationmethod. In the analysis, blast loads acting on the front face of theRC frame are directly applied to the structure.The blast scenario considered is a detonation on the ground sur-

face at a distance of 10m from the centre column in front of the RCframe. The blast load estimation formulae and pressure–impulse

Fig. 5. Pressure–impulse diagram of a column in the RC frame.

diagram of the RC column developed by the authors [20,33] areused to determine the critical TNT charge weight that only knocksoff the centre column (key column).For a standalone rectangular column, the reflected pressure and

impulse at the base are estimated by [33]

PrF (0) ={1.936+ 0.402 ln (b)+ [4.833

+ 1.980 ln (b)]e−0.65Z}Ps0F (0.5 ≤ Z ≤ 10) (20)

IrF (0) ={2.154+ 0.291 ln (b)+ [136.554

+ 65.001 ln (b)]e−6Z}IsF (0.5 ≤ Z < 1) (21)

IrF (0) ={1.452+ 0.287 ln (b)+ [3.221

+ 1.577 ln (b)]e−0.65Z}IsF (1 ≤ Z ≤ 10) (22)

where PrF (0) and IrF (0) are the reflected pressure and impulserespectively at the base of the column, b is thewidth of the column,Z is the scaled distance defined by the charge weight and standoffdistance in m/kg1/3. Ps0F and IsF are the incident pressure andimpulse at the same point as PrF (0), respectively. They could beeasily obtained from the design charts in TM5-1300 [18].It should bementionedhere that Eqs. (20)–(22) can only be used

to predict the reflected pressure and impulse at the base of the RCcolumn that is exactly in front of the explosive charge center witha zero degree incident angle. For other columns, the equivalentstandoff distance is used to take into account the incident angleeffect [18].In this study, the blast load is assumed to be uniform on each

column and all equal to the blast load at the base of the respectivecolumn. However, as the RC frame is of three storeys, the top floorcolumn is up to 6–9mabove the ground. Therefore, for the columnsin a different storey, the height effect is considered. The formulaeto estimate the reflected pressure and impulse at height hp are [33]

PrF (hp) = PrF (0)− 31.53Z−2.64h2pPrF (hp) ≥ 0 (kPa) (0.5 ≤ Z ≤ 10)

(23)

IrF (hp) = IrF (0)− (49.86Z−2.82)h2pIrF (hp) ≥ 0 (kPa) (0.5 ≤ Z ≤ 10)

(24)

where hp is the height at the base of each column measured fromthe ground surface. hp is zero for the first-storey columns andequal to the storey height for the second-storey and third-storeycolumns.The pressure–impulse diagram of the RC column can be

obtained using the formulae in [20]. The P–I diagrams are repre-sented by Eq. (25), in which P0 and I0 are the pressure and im-pulsive asymptotes corresponding to each critical damage degree

1698 Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703

Table 3Pressure and impulsive asymptotes for P–I curves obtained from the formulae in [20].

D = 0.2 D = 0.5 D = 0.8P0 (kPa) I0 (kPa ms) P0 (kPa) I0 (kPa ms) P0 (kPa) I0 (kPa ms)

Numerical results 750 1690 1000 2190 1300 3450

Table 4Calculated blast loads acting on the columns.

Column C1 & C3 C2 C4 & C6 C5 C7 & C9 C8

Peak pressure (kPa) 2495 3639 2117 3072 1362 1936Impulse (kPa ms) 3642 4062 3062 3164 1900 1369

of the column. The pressure and impulsive asymptotes dependon the column dimensions, longitudinal and hoop reinforcementratio, and concrete and reinforcement strength. Table 3 lists theestimated pressure and impulse asymptotes for the column un-der consideration. The corresponding P–I curves are shown inFig. 5. More detailed information regarding the P–I diagrams canbe found in [20].

(P − P0)(I − I0) = 12(P0/2+ I0/2)1.5. (25)With the P–I diagrams and the standoff distance, using Eqs.

(20)–(22), the critical charge weight can be determined such thatthe generated blast pressure and impulse will just cause the keycolumn to collapse. This standoff distance and the correspondingexplosion weight are then considered as the critical explosionscenario, which represents theminimum explosion threat to causepossible progressive collapse of the structure. In this study, asshown in Fig. 5, the critical charge weight is determined to be1000 kg with the standoff distance 10 m from the centre columnin front of the frame. The corresponding blast loads acting on theRC columns calculated from Eqs. (20)–(24) are given in Table 4.It should be noted that the uplifting loads on the beams and

slabs are neglected because the explosive is located 10 m awayfrom the structure. In this case, the blast load acting on the top andbottomsurface of the beam is almost the same; the overall upliftingload is therefore very small. However, for the case when theexplosive is near the frame, the uplifting loads on the beams andslabs must be considered since they generate an uplifting initialvelocity and displacement, which is probably another importantfactor that needs be addressed in progressive collapse analysis.In the benchmark analysis, from 0 ms to 100 ms, the combi-

nation of dead loads and live loads as defined in GSA guidelinesis gradually applied to the frame. According to the GSA guide-lines, when carrying out the nonlinear dynamic progressive col-lapse analysis of buildings, the static load as defined below shouldbe applied on the structure first.Load = DL+ 0.25LL (26)in which DL is the self-weight and LL is the live load of the struc-ture. In this paper, the live load considered is 4 kN/m2. The weightof the infill walls, which are not modelled, is also applied on thebeams. The value is 80 kN/m2.At t = 100 ms, after applying the static load to the structure,

all the blast loads acting on the column of the RC frame are appliedto the structure. Fig. 6 shows the collapse process of the RC framesimulated in the benchmark analysis through the direct simulationmethod. From the figure one can see that, at t = 150 ms, thekey column collapses. At this moment, it is obvious to see that theadjacent columns also suffer a certain level of damage. At aboutt = 400 ms, the combination of the vertical load and transverseload damages the other two first-floor columns heavily and thetwo columns begin to collapse. At t = 500 ms, the second-floorcolumns begin to fail due to the pulling force of the connectedbeams. The whole frame goes down rapidly and collapses to theground at about t = 800 ms.

5.3. Alternative load path method

The alternative load path method included in the GSA and DoDguidelines allows for four levels of analysis, namely linear elasticstatic analysis, nonlinear static analysis, linear elastic dynamicanalysis and nonlinear dynamic analysis. In this section, the GSAnonlinear dynamic analysis, which is known as the most accurate,is carried out to obtain the progressive collapse process of theestablished RC frame. The procedure can be briefly summarized inthe following [6].a. Establish the finite element model of the RC frame.b. Prior to the removal of the key element, bring the model to

static equilibriumunder the combination of the dead loads and liveloads as defined in Eq. (26).c. With the model stabilized, remove the appropriate key

element instantaneously.d. Continue the dynamic analysis until the structure reaches a

steady and stable condition or collapse.Fig. 7 gives the numerical results of the GSA nonlinear

dynamic analysis. Since the first two steps of applying thecombined dead load and live load are the same as the abovebenchmark analysis, the pictures from 0 ms to 100 ms arenot presented herein. The figure shows that, after the removalof the key column, the slabs and columns above the removedkey column go down rapidly until t = 350 ms because ofthe dynamic effects and the redistribution of the load path.Afterwards, the displacement response becomes slower and thewhole structure stabilizes without total collapse, indicating thatthe GSA nonlinear dynamic analysis procedure overestimates thestructural capacity against progressive collapse because it neglectsthe non-zero initial condition and damage in the structural mem-bers in the progressive collapse analysis.

5.4. The proposed method

Using the proposed method, the initial velocity and initialdamage of the other structural members are derived from themethod proposed in Section 3. The maximum initial velocities anddisplacements of the adjacent RC columns are given in Table 5. Ascan be seen, the maximum initial displacement of all the columnsis only 8.06 mm, which is only 0.3% of the column height. This isbecause of the very short duration of the blast load. As modellingof these initial displacements is very time consuming and theyare also rather small, the initial displacements of the columnsare neglected in the analysis. Moreover, because the upliftingforces acting on the beams and slabs are neglected, only the blastload in the transverse direction acting on the beam is considered.However, because the RC floor with high in-plane stiffness andlarge mass constrains the possible response of the beam in thetransverse direction, both the initial velocity and displacement ofthe beam are very small at the end of the blast-loading phase, so

Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703 1699

Fig. 6. Collapse process of the RC frame from the benchmark analysis (direct simulation).

Fig. 7. Response process of the RC frame from the GSA nonlinear dynamic analysis.

they are also neglected in the analysis. Therefore, only damage andthe initial velocity of columns are considered.The blast loads acting on the columns listed in Table 4

are plotted in the pressure–impulse diagram of the RC columngenerated in Section 5.2, as shown in Fig. 8. The damage degrees of

the columns are obtained from the method proposed in Section 3.They are also given in Table 5. The damage level of each column isused to modify the column properties accordingly.With the modified column properties, the progressive collapse

analysis is carried out by removal of the key column for dynamic

1700 Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703

Table 5Initial damage degree and maximum initial velocity and displacement of RC columns.

Column C1 & C3 C4 & C6 C5 C7 & C9 C8

Initial damage degree D 0.65 0.53 0.61 0.00 0.00Maximum initial velocity (m/s) 5.85 4.91 5.07 3.05 2.20Maximum initial displacement (mm) 8.06 6.70 4.93 4.03 1.47

Fig. 8. Pressure–impulse diagram of the column and different blast loads oncolumns.

analysis with nonzero initial velocities applied to respectivecolumns. This is done as a full restart analysis in LS-DYNA. It shouldbe noted that,when applying the initial velocity on the RC columns,it is very time consuming to apply the initial velocity to everynode according to the deflection shape function. For simplificationpurposes, the column is divided into five segments, and eachsegment is assumed to have the same initial velocity, equal to thelargest velocity in this segment.The results of this analysis are shown in Fig. 9. In the figure, it

is clear that, at about t = 400 ms, the damage in the first-floorcolumns is severe, and these columns start to collapse. At t =500 ms, the second-floor columns begin to fail due to the pulling

force of the connected beams, and thewhole frame collapses to theground at about t = 800 ms.

5.5. Comparison and discussion

The numerical results of the GSA nonlinear dynamic analysisand the proposed method are compared with those of the bench-mark analysis to verify the accuracy and reliability of the proposedmethod.One can clearly see from Figs. 6, 7 and 9 that the frame does not

collapse in the GSA nonlinear dynamic analysis, while it collapsesto the ground almost at the same time in the benchmark analysisand the proposed method. This is because, in the GSA nonlineardynamic analysis, the catenary effect of the beams will produce aforce to balance the gravity load after removing the centre columnand therefore resist the collapse of the frame. However, in theproposed method, by considering the initial damage and the non-zero initial condition of the structure, the other first-floor columnsalso fail at about t = 400 ms, which is almost the same as that inthe benchmark analysis. The damage and collapse of the adjacentcolumns accelerate the collapse of the structural members, leadingto the total collapse of the frame. These observations clearlyindicate the overestimation of the GSA nonlinear dynamic analysisprocedure on the capacity of the frame in resisting progressivecollapse because it neglects the initial damage and non-zeroinitiation conditions generated by the blast load.Fourmain response quantities at the key nodes and elements of

the structural model are also extracted from the numerical resultsand compared with each other. They are the following.

Fig. 9. Collapse process of the RC frame from the proposed method.

Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703 1701

a. Vertical and transverse displacement at node N1.b. Vertical accelerations at node N1.c. Vertical velocity at node N1.d. Stress in element E1.

N1 is the node at the middle of the beam–column joint abovethe first-storey front center column; E1 is the beam element thatmodels the horizontal steel bar on the bottom side of the jointabove the center column. Their exact locations are indicated inFig. 4.Fig. 10 shows the comparison of the vertical displacement at

node N1 during the collapse process of the frame from differentapproaches. As can be seen, the result from the proposed methodis very close to that of the benchmark analysis. That from the GSAnonlinear dynamic analysis is also similar initially, but it becomesa constant after t = 400 ms, indicating that the structure does notcollapse. As also shown, when the vertical displacement reachesabout negative 3 m, the node touches the ground and some slightrebounds may occur.Fig. 11 shows the comparison of the transverse displacement

at node N1 during the collapse process of the frame from thethree analyses. As shown, the transverse displacement from theGSA nonlinear dynamic analysis is smaller than those from thebenchmark analysis and the proposed method. This is becausethe GSA nonlinear dynamic analysis neglects the initial non-zeroconditions in the structure. The dynamic effect of the suddenremoval of the center column enables the frame to vibrate anddeform mainly in the vertical direction, but the collapse of thefront side of the frame causes a slight rotation of the whole frame,leading to a small displacement in the positive y-direction at nodeN1. The result from the proposed method agrees reasonably wellwith that from benchmark analysis. The slight overestimation ofthe lateral displacement by the proposed method might be causedby the simplified way that the initial velocity is applied on thestructure. As described above, when applying the initial velocity ona column, the column is divided into five segments, and a constantinitial velocity equal to the largest velocity in this segment is used.This results in a slight overestimation of the initial velocity ofeach column, and hence in the lateral displacement of the frame.Nonetheless, as shown in the figure, the difference is relativelysmall.The comparison of the vertical velocities at node N1 from dif-

ferent analyses is shown in Fig. 12. As shown, the vertical velocityincreases quickly with the removal of the center column becauseof the dynamic effect of the redistribution of the load. The veloc-ity increment rate slows down, indicating a smaller acceleration.This is because, after the damage or removal of the center column,the two beams on both sides of the collapsed column act as a ‘‘longbeam’’. The catenary effect of the ‘‘long beam’’ generates a force tobalance the gravity load and therefore reduces the collapse acceler-ation of the frame. The velocity estimated from the GSA nonlineardynamic analysis oscillates around zero after t = 400 ms. Thosefrom the benchmark analysis and the proposedmethod rapidly in-crease again at about t = 300 ms until the structural member im-pacts the ground. This increase in velocity is caused by the damageof the ‘‘long beam’’ or the adjacent columns. The impact betweenthe structural member and the ground immediately changes thedirection of the vertical velocity at node N1, indicating that somerebounding occurs. The result from the proposed method againagrees well with that from the benchmark analysis.Fig. 13 compares the vertical acceleration at node N1. It shows

that the proposed method gives consistent prediction of accel-eration responses as the benchmark analysis. The GSA approachunderestimates the acceleration responses. Fig. 14 compares thestress time histories in element E1 from different analyses. Asshown, initially, the stress in the element E1 is negative; the steelbar is in compression. This is because, after application of the dead

Fig. 10. Comparison of the vertical displacements at node N1 from differentanalyses.

Fig. 11. Comparison of the transverse displacements at node N1 from differentanalyses.

Fig. 12. Comparison of the vertical velocities at node N1 from different analyses.

load and live load, negativemoment occurs at the joint section, andthe steel reinforcement that is at the bottom side of the beam willbe in compression. After the collapse of the center column, the twobeams above the center column will work as a ‘‘longer one’’. Thesection of the joint, which is at the middle of the ‘‘long beam’’, willexperience a positive moment. Thus, the stress in element E1 is intension. In the benchmark analysis and the proposed method, thetension stress in the element reduces suddenlywhen the structuralmember reaches the ground, whereas that from the GSA analysisremains almost a constant after t = 400 ms because the structurestabilizes. The figure also shows that the proposedmethod and thebenchmark analysis give similar predictions of stress in elementE1.

1702 Y. Shi et al. / Engineering Structures 32 (2010) 1691–1703

Fig. 13. Comparison of the vertical accelerations at node N1 from differentanalyses.

Fig. 14. Comparison of the stress in element E1 (steel reinforcement) fromdifferentanalyses.

The above observations indicate that the proposed methodgives accurate predictions of progressive collapse of frame struc-tures. Compared to the GSA nonlinear dynamic analysis, the addi-tional calculation effort of the proposedmethod in determining thenon-zero velocity and damage of each structural member is mini-mum, as the initial damage can be determined from available P–Idiagrams and the initial velocity from an SDOF analysis, but theresults are more accurate. On the other hand, compared to the di-rect simulations, the proposed method gives comparable results,but the computational time and computer memory requirementare substantially less since the progressive analysis is a free vibra-tion analysis, and therefore the mesh size used can be larger thanthat in direct simulations.It should be also noted that other numerical simulations with

different frame examples have also been carried out to further ver-ify the reliability of the proposedmethod. One example consideredis a simple RC frame, which is similar to the one presented pre-viously in the paper. The difference is the column dimension andthe concrete compressive strength. Herein the column dimensionconsidered is 250× 250 mm and the concrete strength is 20 MPa.In this example, when progressive collapse analyses of the frameare carried out using the proposedmethod, GSAnonlinear dynamicanalysis and direct simulationmethod, the numerical results all in-dicate that the simple frame finally collapses under the same blastscenario. However, it is also very obvious that both the collapseprocess and the typical dynamic responses of the simple RC framederived from the proposed method analysis and the direct simu-lation method analysis are almost the same, while they are verydifferent from the one obtained from the GSA nonlinear dynamic

analysis [34]. All these examples demonstrate the accuracy of us-ing the proposedmethod in the prediction of structural progressivecollapse induced by explosive loadings.

6. Conclusion

In this paper, a new procedure for progressive collapse analysisof RC frames is proposed. It is based on the alternative load pathmethod in the GSA and DoD guidelines, but with modifications byincluding the inevitable non-zero initial conditions and damage inthe structuralmembers caused by the direct blast load. Themethoduses P–I diagrams to estimate the damage to structural membersby the direct blast load, and the equivalent SDOF approach toestimate the velocity and displacement of structure members atthe end of the blast-loading phase. A three-storey two-bay RCframe is analysed to demonstrate the efficiency and reliability ofthe proposedmethod. It is found that the proposedmethod gives asimilar prediction of the frame collapse process to that of the directsimulation of the structure response to blast load. Because theproposed method does not require a comprehensive modelling ofthe structure, it therefore substantially reduces the computationaltime and computer memory requirements. Compared to the GSAnonlinear dynamic analysis method, the proposed method givesbetter predictions of the structural progressive collapse withminimum additional effort in determining the non-zero initialconditions and damage in structural members at the end of blastloading phase when progressive collapse starts.

Acknowledgements

The authors would like to acknowledge the financial supportfrom theNational Natural Science Foundation of China under Grantnumber 50638030 and 50528808, the National Key TechnologiesR&D Program of China under Grant number 2006BAJ13B02,the Key Project of Tianjin Application Basis and ForefrontTechnology Research Program of China under grant number08JCZDJC19500, and the Australian Research Council under Grantnumber DP0774061 for carrying out this research.

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