Irribaren 2011 Eng Struct (2)

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Engineering Structures 33 (2011) 28052820

Contents lists available at ScienceDirect

Engineering Structures

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

Investigation of the influence of design and material parameters in theprogressive collapse analysis of RC structuresB. Santaf Iribarren a,b, P. Berke b, Ph. Bouillard b, J. Vantomme a, T.J. Massart b,a Civil and Materials Engineering Department, Royal Military Academy (RMA), Av. Renaissance 30, 1000 Brussels, Belgiumb Building, Architecture and Town Planning Department (BATir) CP 194/2, Universit Libre de Bruxelles (ULB), Av. F.-D. Roosevelt 50, 1050 Brussels, Belgium

a r t i c l e i n f o

Article history:Received 26 February 2011Received in revised form20 May 2011Accepted 1 June 2011Available online 2 July 2011

Keywords:Progressive collapseReinforced concreteLayered beamStrain rate effectsFinite elements methodStructural dynamics

a b s t r a c t

This contribution deals with the modelling of reinforced concrete (RC) structures in the context ofprogressive collapse simulations. One-dimensional nonlinear constitutive laws are used to model thematerial response of concrete and steel. These constitutive equations are introduced in a layered beamapproach, in order to derive physically motivated relationships between generalised stresses and strainsat the sectional level. This formulation is used in dynamic progressive collapse simulations to study thestructural response of a multi-storey planar frame subjected to a sudden column loss (in the impulsiveloading range). Thanks to the versatility of the proposed methodology, various analyses are conductedfor varying structural design options and material parameters, as well as progressive collapse modellingoptions. In particular, the effect of the reinforcement ratio on the structural behaviour is investigated.Regarding the material modelling aspects, the influence of distinct behavioural parameters can beevaluated, such as the ultimate strain in steel and concrete or the potential material strain rate effectson the structural response. Finally, the influence of the column removal time in the sudden column lossapproach can also be assessed. Significant differences are observed in terms of progressive failure patternsfor the considered parametric variations.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

Progressive collapse is a situation in which a local failure ina structure leads to a load redistribution, resulting in an overalldamage to an extent disproportionate to the initial triggeringevent. Some examples of such a structural collapse occurred inthe last decades, such as the Ronan Point apartment building inLondon, which partially collapsed in 1968 due to a gas explosion,or the Murrah Federal Building in Oklahoma City, which wasdestroyed in 1995 following the explosion of a bomb truck. Owingto the catastrophic nature of its consequences, progressive collapsehas drawn an increasing interest in the civil engineering researchcommunity to derive new design rules.

Different simulation approaches dealing with the issue ofprogressive collapse can be found in the literature. This paper aimsat contributing to the one referred to as the alternate load pathapproach, which consists in considering stress redistributionsthroughout the structure following the loss of a vertical supportelement [13].

The recommendations for this approach presented by theUnited States Department of Defense (DoD) [2] and the General

Corresponding author.E-mail address: [email protected] (T.J. Massart).

Services Administration (GSA) [3] suggest the use of step-by-stepprocedures for linear static, nonlinear static and nonlinear dy-namic analyses. Other works propose static nonlinear calculationsaccounting for dynamic inertial effects via load amplificationfactors for steel structures [48]. The DoD and GSA guidelinesspecify a dynamic load amplification factor of 2 to account fordynamic effects in static computations for both reinforced con-crete and steel structures, which was considered to be highly con-servative by some authors working on steel structures [4,5,810]and on reinforced concrete frames [11]; while insufficient for oth-ers whose research is focused on steel frames [1214]. In order toobtain a systematic estimate of the dynamic load factors, equiv-alent static pushover procedures have been identified for steelstructures, based on energetic considerations [46,8]. An optimisa-tion approach based on nonlinear dynamic analyses was adoptedin [8] in order to determine the most appropriate values for thesefactors, by performing a parametric study on topological variablesfor regular steel frames.

Apart from these equivalent quasi-static approaches, whichconstitute a major part of the related literature, nonlinear dy-namic procedures were recently conducted for both reinforcedconcrete and steel structures, to a variable extent of complexity[912,1422]. While in most of the recent works the structuresare still modelled using 2D frames [9,10,12,14,19,20], full non-linear 3D dynamic computations with geometrically nonlinear

0141-0296/$ see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.06.005


2806 B. Santaf Iribarren et al. / Engineering Structures 33 (2011) 28052820

formulations are sometimes found in the literature related tosteel structures [18,22]. Nevertheless, such detailed approachesare scarce for reinforced concrete structures, partially due to thehigh complexity involved in the modelling of the sectional re-sponse of heterogeneous RC beams. Hence, most of the progres-sive collapse related literature is focused on steel structures [410,1214,18,2023]. Fewer contributions tackle the dynamic analysisof progressive collapse of reinforced concrete structures, amongwhich [11,1517,19,24]. Recent works as [1517] use an explicitfinite element code, which is adopted in the simulation of blast-inducedprogressive collapse analyses,where the blast can bemod-elled in a sophisticated way (as a pressure wave propagation in theair) or in a more simplified manner (as a pressuretime distribu-tion on the concerned elements). This computational technique ismainly suitable for modelling the dynamic response of concreteand steel during the loading stage, due to the small time steps re-quired by the explicit schemes. However, this event-modelling ap-proach may become prohibited for large and/or finely discretisedstructures, due to its computational cost. As a result, the mate-rial modelling is often based on simplified approaches in whichthe modelling of the reinforced concrete response needs a prioripostulated cross-sectional properties of beams, requiring an iden-tification of such generalised constitutive laws, as in [19]. In [24]the progressive collapse is simulated by using a plastic hinge for-mulation accounting for the steel bars yielding. A later work fromthe same author includes the steel bar fracture on a single beamtest [25].

The purpose here is to analyse the structural response of aRC frame subjected to a sudden column loss. Such an abruptmember removal results in an impulsive loading scenario. Thisevent independent approach is widely used in the context ofprogressive collapse simulation techniques [212,14,1822,26,27],with the aim of studying the structural resistance to the suddenloss of a primary load-bearing element, in contrast to the eventdependent approaches [1517]where the collapse triggering eventis modelled to analyse the structural response to a specific loadingscenario. According to [26], the sudden column loss approachconstitutes a useful design scenario for the assessment of structuralrobustness, since it offers an upper bound on the deformationsobtainedwith respect to an event dependent simulation approach.

The present contribution falls within the multilevel modellingof RC frames and uses a direct transition from the behaviour ofconcrete and steel at the constituent level to the response of themembers at the structural level. A multilayered beam formula-tion [2832] is adopted to this end, where the structural mem-bers sections are discretised through a finite number of layers forwhich one-dimensional constitutive relations are described. Thisapproach was used in the context of earthquake engineering forcyclic loading computations [29,30] or for the characterisation ofa beamcolumn connection macromodel for progressive collapsesimulations [19]. In [32] a layered model is developed to charac-terise the response of a simply supported RC slab to blast loadings.

However, to the best knowledge of the authors, a fully mul-tilevel approach has not yet been applied for the detailed mod-elling of reinforced concrete structures subjected to progressivecollapse and hence constitutes a contribution to the study of thisphenomenon. A geometrically linear formulationwill be used here,which seems acceptable in the framework of RC structures con-sidering the rather small rotations involved due to the limitedductility of the sections. Nevertheless, the validity of such an as-sumption will be discussed in light of the obtained results. Thecomplexity and computational cost of the methodology adoptedhere should be balanced with the objectives of the study. Thepresent description enables a more realistic representation of thecross-sectional behaviour of reinforced concrete members where

axial load-bending moment interactions are considered in com-bination with material nonlinearities, potentially including mate-rial rate effects. It also avoids the need to postulate closed-formrelationships between generalised stresses and strains at the sec-tional level, preventing the related identification problem when-ever the design and/or material parameters need to bemodified orwhen a rate dependent approach is considered [33]. The complex-ity of quantifying the rotational capacity of RC members has beendemonstrated in the literature. The ductility as the ability of thestructure to redistribute moment and fail gradually is addressedin [34,35]. In particular, the concrete softening is considered as animportant component of the rotational capacity, which needs tobe taken into account in order to ensure the ductility of RC struc-tures. Themultilayered approach adopted here allows for a gradualstrength degradation as a consequence of the progressive failure ofthe constitutive layers, ensuring a proper account for the potentialductility of the RCmembers. Conversely, the advantages offered bythe present formulation are counterbalanced by the higher compu-tational cost required by a multiscale approach.

In this study, dynamic simulations are carried out on a two-dimensional representation of a reinforced concrete frame, mod-elled with multilayered beam elements and designed according toEurocode 2 requirements [36]. The response of the frame whensubjected to a sudden column loss, is then analysed. In order to as-sess the flexibility of this multiscale formulation, the influence ofparticular practical structural designs and material modelling op-tions in the structural robustness is analysed. A strain rate depen-dentmaterial approachwill be developed aswell to assess the levelof dependence of the structural response on material strain rateeffects.

The paper is organised as follows. In Section 2 the materialbehaviour of concrete and steel is described and the corresponding1D constitutive models are proposed. The multilayered beamdescriptionused for themodelling of the reinforcedbeamelementsis detailed, followed by the time integration scheme used forthe dynamic computations. The application of the proposedmethodology in the simulation of progressive collapse of RCstructures is presented in Section 3, where a planar framerepresenting a building facade is subjected to a sudden columnloss. A reference solution is obtained for a structure designedaccording to the Eurocodes. This section also presents the resultsfor varying material constitutive parameters of concrete and steel,namely the ultimate strains, which are varied in a realistic rangeof values obtained from the literature. Then, the influence ofthe reinforcement amount is analysed, by increasing the initialreinforcement ratios. As far as the technique of the sudden columnloss simulation is concerned, the effect of the column removal timein the structural failure pattern is also assessed. The influence of thelocation of the removed column is studied, aswell as the procedureadopted for the progressive collapse analysis (GSA vs. DoD). InSection 4 amaterial strain rate dependent formulation is describedand the related results are compared to those obtained via the rateindependent approach. Finally, a discussion is given in Section 5,along with some concluding remarks.

2. Rate independent modelling of RC structures

2.1. Material response of concrete and steel

The International Federation for Structural Concrete (fib) [37]describes the behaviour of concrete in compression as a stressstrain curve which depends on the concrete grade. The staticcompression curve is approximated by:

c,st

fc,st= k

2

1+ (k 2) for |c | < |c,lim| (1)

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B. Santaf Iribarren et al. / Engineering Structures 33 (2011) 28052820 2807

Fig. 1. Stressstrain curve of the proposed model for concrete in compression.

in which c,st is the static compressive stress and c thecompressive strain. A C30 concrete typewill be considered inwhatfollows with fc,st = 37.9 MPa the static compressive strength; k =1.882 the plasticity number and = c/c1, with c1 = 2.23hthe strain at maximum stress. The parameters k and c1 arecalculated as follows [37]:

k = Ecfc,st/ |c1| (2)

c1 = 1.60fc,st107

0.25 11000

(3)

with Youngs modulus Ec .In the present work, a bilinear stressstrain relationship is

adopted for the sake of simplicity, as suggested in Eurocode 2 [36].In Fig. 1 the simplified model used here is compared to theprescriptions of the fib, where the compressive stressstrain staticcurve is depicted. The assumed ultimate compressive strain isset to c,lim = 0.35% [36], after which the stress vanishes, inorder to represent the failure of concrete in compression. Thetensile contribution of concrete, being negligible with respect tothe compressive response,will be ignored for the sake of simplicity.

For the steel reinforcement, a 500 MPa yield strength steel isemployed. The simplified bilinear approach proposed by Eurocode2 [36] is used as well. The evolution of the yield stress is modelledadopting a linear hardening approximation. The ultimate strains,lim is here taken equal to 4%, which corresponds to a steelductility class between A and B [36]. In order to represent thefailure of steel, the stress is set to zero for strains exceeding thisvalue.

2.2. Constitutive equations: stress update algorithm

Based on the information reported in Section 2.1, a 1D elasto-plastic model is used for the behaviour of concrete and steel. Theplastic behaviour is defined by an evolution law, which dependson the plastic strain history parameter , defined in a one-dimensional approach as

=dt with = p (4)

with p the plastic strain. This parameter actually consists of twoterms, t and c controlling respectively the plastic flow in tensionand in compression. The plasticity criterion f is defined as

f ( , ) = t( t) for 0 c(c) for < 0 (5)

where is the stress and t and c the tensile and compressiveyield stresses, which depend on t and c respectively. If thisdependence is nonlinear, a local NewtonRaphson scheme is used

Table 1Material parameters of the simplified bilinear model: in accordance with [36,37].

Concrete Steelc (MPa) Ec (GPa) c,lim 0 (MPa) E (GPa) s,lim () (MPa)

37.9 17 0.35% 500 200 4% 0(1+1.5)

to determine the plastic state and the stress update. The stressupdate corresponding to an increment from state n to state n + 1can be written as

n+1 = n + E(n+1 pn+1) (6)with E the elastic modulus,n+1 the strain increment andpn+1the plastic strain increment.

Eqs. (5) and (6) provide the set of constitutive expressions tobe linearised at each iteration in a return-mapping algorithm in ageneral fashion:

n+1 trialn+1 + Epn+1 = 0 (7)f (n+1, n+1) = 0 (8)with trialn+1 = n + En+1 the trial stress. By using (4), pcan be substituted in Eq. (7) and the constitutive problem canbe expressed in a residual form as a function of two variables( and ):

{R(n+1, n+1)} =n+1 trialn+1 + En+1f (n+1, n+1)

= 0. (9)

This system of equations is solved using a NewtonRaphsoniterative scheme. Based on the linearised form of Eq. (9), thecorrection at iteration (j+ 1) is computed asn+1n+1

j+1

= Jp1j Rj (10)where the Jacobian

Jpis defined as

Jp = {R}

{ , } (11)which can be used to evaluate the material tangent operator Hconsistent with the return-mapping algorithm:

H = . (12)

The constitutive parameters values used are shown in Table 1.

2.3. Layered beam formulation

A multilayered beam approach [2832] is used for modellingthe behaviour of the reinforced concrete members. Each cross-section where the beam response has to be computed i.e.the integration points of a finite element is discretised into afinite number of longitudinal layers where the 1D constitutiveequations for concrete and steel are applied. The cross-sectionalbehaviour of the element is thus directly derived by integration ofthe stressstrain response of the layers. According to Bernoullishypothesis the generalised stressesgen and strains Egen read:gen

= {N,M} (13)Egen

= {, } (14)with N the normal force, M the bending moment, the axialstrain and the curvature. Note that such averaged relationscan only be applied in a point-wise manner in classical structuralcomputations provided they do not exhibit overall softening, in



order to keep a well-posed description. If softening is obtained inthe beam response and unless a nonlocal or gradient type beamformulation is used, the corresponding dissipation at the structuralscale is computationally determinedby the element size [38,39]. Asa result, the structural discretisation will be chosen subsequentlyto provide localisation on a physically motivated beam length(size of the plastic hinges). This layered approach is summarised inFig. 2, showing that only the longitudinal reinforcements are takeninto account. The structural scale assumption (Bernoulli) remainsconsistent with this approach at the fine scale. As an additionalsimplifying assumption, a perfect bond is assumed between thelayers. The element-wise stresses {gen} are evaluated from thestrains {Egen} at each integration point of the beam element asfollows: first, the axial strains in each layer (i) are computed,

i = yi (15)where yi is the cross-sectional average vertical coordinate of layeri computed from the sectional centre of gravity. Then the layer-wise stresses (i.e. the axial stresses i) are obtained by applying the1D constitutive equations on each layer. For the layers containingthe steel reinforcements, the stresses in concrete and steel arecomputed separately and the layer average stress is obtaineddepending on the steel volume fraction of the considered layer. Thestress is computed at the mid-height of the layer, and assumed tobe constant over its thickness. Finally the cross-section generalisedstresses {gen} are evaluated by integrating the layer-wise one-dimensional stresses i through the cross-sectional area of thebeam:

N =

ii

M =

iyii(16)

with i the cross-sectional area of the layer. The related cross-sectional consistent tangent operator [Ht ] can be derived from thelayer-wise consistent tangent operatorsHi fromEq. (12) as follows:

[Ht ] = {gen}

{Egen} =

Hii

Hiyii

Hiyii

Hiy2ii

. (17)

Note that this approach allows for a gradual element-wise strengthdegradation as a consequence of the progressive failure of theconstitutive layers, for which the stress is set to zero whenthe failure criteria are fulfilled. It allows for a ductile sectionalresponse, which is proven to be an important factor for themoment redistribution in the structure [34,35]. This also meansthat rather complex nonlinear sectional responses exhibitingsoftening can be obtained even with simplified 1D constitutivelaws for the constituents (such as linear hardening) provided theyare followed by a sudden drop of the stress at the ultimate strain.The straightforwardness of the present formulation to describe theRC sectional behaviourwith respect to the closed-form approachesis pointed out.

2.4. Time integration scheme for the dynamic problem

An implicit Newmark scheme is adopted for the integration ofthe equations of motion in the simulations [40]. The discretisedequations describing the equilibrium in dynamics readf int ({q})+ [M] {q} = f ext (18)

withf intthe internal forces,

f extthe external forces and

[M] the mass matrix. In the present paper, [M] is consideredto be constant while the internal forces

f intare a nonlinear

function of the displacements {q}. To solve Eq. (18) within anincremental-iterative scheme, a linearisation of the equationsmust be performed. Writing equilibrium as a residual form, the

Fig. 2. Layered beam model: generalised stresses evaluation.

related structural Jacobianmatrix [J] is computed as the derivativeof the residual with respect to the displacements:

[J] = f int ({q})+ [M] {q} f ext

{q} . (19)

Using the notion of structural tangent operator [Kt ], whichdenotes the variation of the internal forces with respect to thedisplacements, the expression of the iteration matrix [J] is givenby:

[J] = [Kt ]+ [M] {q} {q} (20)

where [Kt ] is computed by applying the cross-sectional consistenttangent operator [Ht ] calculated in Eq. (17):

[Kt ] =e

Ve[B]t [Ht ] [B] dV (21)

and where [B] relates the generalised strains to the nodaldisplacements. Finally, the Newmark formulae allow computingthe remaining term {q}

{q} in Eq. (20), by defining relationshipsbetween the nodal displacements, velocities and accelerations.These formulae are expressed for a given time tn+1 as [40]:

{qn+1} = {qn} + (1 )t {qn} + t {qn+1}{qn+1} = {qn} +t {qn} +t2

12

{qn} +t2 {qn+1} (22)

with t the time step; and and the Newmark parametersassociated with the quadrature scheme, which determine thestability of the method [40]. Considering these equations, theiteration matrix results in

[J] = [Kt ]+ [M] 1t2

. (23)

A numerical damping of 5% is introduced in order to reducethe spurious high frequency vibrations in the structural response,which significantly increase the computational time in large-scalecalculations. The value of the Newmark algorithm parameters and are chosen such that this damping (affecting mainly thehigh frequency range) is provided in the simulation,while ensuringunconditional stability [40]:

= 12+ ; = 1

4

+ 1

2

2(24)

with > 0 the damping ratio, which is accordingly set to 0.05 inthis case.



Fig. 3. Five-storey frame under study.

Fig. 4. Description of the RC beam sections (values in mm).

3. Numerical simulations for rate independent behaviour

3.1. Description of the RC planar frame subjected to column loss

Fig. 3 shows the structure under study, consisting of a two-dimensional representation of a five-storey eight-bay RC frame.The choice of this simplifying assumption (2D analysis) is justifiedby the conservative nature of the results obtainedwith such an ap-proachwith respect to full 3D computations, inwhich the transver-sal resistance would be considered. The floor span and height are6 m and 3 m respectively, except for the ground floor which is 6 mhigh. The span in the perpendicular direction is 6 m. The columnsand beams dimensions are 450450mm2 and 600450mm2 re-spectively. The envelope (vertical distance between the steel barsand the nearest section edge) is set to 5 cm for all the consideredreinforcement schemes. The slab is 20 cm thick and an additionalconcrete layer of 10 cm is assumed. The floor does not provide anyresistance. The loads applied to the floor are those correspondingto the slab and beamweights as well as the live loads. The columnsare subjected to their gravitational load. The minimal amount ofreinforcement required for the beam sections has been obtainedby way of the structural design analysis software Diamonds 2010by Buildsoft [41], according to Eurocode 2 [36]. Three different sec-tions have been obtained accordingly: one for the columns and twofor the beams. Fully continuous bottom reinforcement is providedin the beams and 2/3 of the top reinforcement is made contin-uous as well, in order to ensure beam-to-beam continuity acrossthe columns, which according to the GSA is essential for resistingthe load reversals that follow the loss of a primary support [3]. Al-though transversal reinforcement is not explicitly represented, thestructures under study will be considered to be properly designedagainst shear failure, as prescribed by the GSA [3], so that flexuraleffects are predominant in the beam response. The sectional di-mensions and the reinforcing details are indicated in Fig. 4.

The purpose is to analyse the structural response of the framewhen subjected to the sudden loss of its rightmost bottom column,

Table 2Loads applied to the beams (excluding self-weight).

Loads (kN/m) Dead Live Total GSA Total DoD

Floors 43.2 18 47.7 52.2Roof 43.2 6 44.7 46.2

which would correspond to a blast or impact loading scenarioresulting in a full column removal. The sudden column losstechnique is considered to offer a reasonable balance betweenthe usefulness of the results and the computational effort to paywith respect to event dependent approaches where the triggeringevent is explicitly modelled [26], including for instance horizontalloading due to the triggering event. This approximation is usedhere to keep the computational time acceptable, and to allowcomparisonwith other approaches using this technique.Moreover,in [26] it was observed that the sudden column loss offered anupper bound on the structural deformation with respect to amore detailed approach where the blast loads were explicitlymodelled. In practical terms, the sudden column loss techniqueemployed here consists in erasing the column from the initialstructural topology and replacing it by its resultant forces, appliedwith the regular loads in a first static loading step. The columnresultant forces are subsequently removed (in a given removaltime determined by the nature of the initial event: impact, blast...)while the regular loads are kept constant. This procedure preservesthe real loading sequence. The GSA guidelines [3] specify a loadingcombination of

Total Loads = Dead Loads+ 0.25 Live Loadsto be applied for the dynamic simulation of progressive collapse.Therefore, 25% of the live loads are here applied. The resulting loadsare indicated in Table 2.

The loading history used in the simulations is shown in Fig. 3:all the loads are applied to the structure in a sufficiently large timet0 = 60 s for the loading to be considered static, since it preventsinertial effects from developing, as well as rate effects if they wereconsidered. Then the member forces (R) are decreased to zero in agiven removal time tr and the response of the structure is observedover a time span of 2 s. Unless specified otherwise, the time stepemployed ist = 5 ms and the removal time tr is also set to 5 msto simulate the instantaneous column loss representing the effectof a blast [42].

Two-noded Bernoulli beam elements with three integrationpoints are used. A linear interpolation is used for the axialdisplacement,while aHermite polynomial interpolation is used forthe deflection. The cross-sectional discretisation i.e. the thicknessof the layers in the multilayered approach is taken equal to1 cm. All computations are performed using a geometricallylinear description. Since the energy dissipation depends onthe discretisation adopted due to the softening nature of theproblem [38,39], here the beam elements have been given a lengthequal to 77% of the sectional height, so that localisation occurs

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Table 3Description of the tests.

Case Reinforcement c,lim (%) s,lim (%) tr (ms) Removed column GSA/DoD

A Reference 0.35 4 5 Lateral GSAB Reference 0.5 4 5 Lateral GSAC Reference 0.35 10 5 Lateral GSAD Reference 0.5 10 5 Lateral GSAE 1.5 reference 0.35 4 5 Lateral GSAF Reference 0.35 4 500 Lateral GSAG Reference 0.35 4 5 Interior GSAH Reference 0.35 4 5 Lateral DoDI Reference 0.35 4 5 Interior DoD

(a) After 200 ms. (b) After 300 ms. (c) After 420 ms.

Fig. 5. Case A: response of the reference structure to a sudden column removal.

on a region comparable to the corresponding characteristic plastichinge lengthwheremost of the permanent beam rotation is knownto be concentrated [34,35].

To show the flexibility of the multilayered beam formulationto account for variations of the simulation parameters, as opposedto the approaches based on a priori postulated macroscopic cross-sectional response of the RC beams, the influence of particulardesign options and of material parameters in the rate independentstructural response is analysed. In the first set of computations, thematerial parameters affecting the ultimate behaviour of concreteand steel are varied over the commonly used range found inthe literature, to see their influence at the structural level. Notethat these material variations, which may have a direct effecton the cross-sectional response, are naturally incorporated at thestructural level thanks to the multilayered approach. This is incontrast with other simplified approaches in which the derivationof closed-form relationships between generalised stresses andstrains requires an offline identification process whenever thematerial parameters are changed. The second set focuses on thevariation of the design parameters. First, the effect of the amountof steel reinforcement is investigated. Then, the influence of theloading options is assessed: the value of the column removal timetr is varied, so that the inertial effects in the overall response of thestructure can be analysed. The influence of the topological locationof the failing column is also investigated. Finally, the influence ofthe procedure chosen for progressive collapse verification (GSA vs.DoD) is studied. Table 3 summarises the values of these parametersfor the different computations that constitute the present study.

3.2. Reference solution (case A)

As previouslymentioned, the structure under consideration hasbeen designed to meet the requirements of the Eurocodes 2 interms of minimal reinforcement amounts. Only 32 mm diameter

Table 4Interpretation of the structural failure symbols.

Symbol Interpretation

Yielding of the steel rebars Crushing of concrete in less than 1/3 of the section Crushing of concrete in more than 1/3 of the section Failure of the steel rebars

bars have been used for the reference design (see details in Fig. 4),which results in a total reinforcement ratio of 1.6% for the columns,0.6%0.9% for the beams in tension and 0.6% for the beams incompression. Fig. 5 depicts the results of the reference test atdifferent times, counting from the onset of the column removal. Aclose-up view of the rightmost side of the structure is provided forthe sake of clarity. The true-scale deformed shape of the structureis shown. The interpretation of the symbols employed to describethe failure pattern of the structure is summarised in Table 4:black dots () represent the plastification of the steel bars atthe considered element; triangles () indicate that crushing ofconcrete in compression has occurred in less than one third of theelement section (less than 1/3 of the concrete layers has reachedthe strain limit); squares () stand for concrete crushing in morethan one third of the section; and crosses () represent the failureof the steel rebars and thus the final failure of the section. Notethat the failure of a layer (reaching the ultimate strain in concreteor steel) does not stop curvature increment, but it eliminates thelayer contribution to the cross-sectional strength. This allows for agradual strength degradation through the section. This progressivedegradation is also depicted for various sections of the structure:both for steel and for concrete, white indicates the areas thatfall within the elastic range; red represents the zones havingreached the plastic domain; and black denotes material failure(the corresponding layer-wise stress has dropped to zero after theultimate strain is reached). Tensile concrete is not depicted. It must



Fig. 6. Case B: response for c,lim = 0.5%.

be noted that the fully damaged elements are not removed fromthe topology. This means that the load transfer from potential fullyfailed elements to the neighbouring elements is not represented,since here the prime purpose is to study the ability of the structureto withstand the sudden column loss. As a result, the simulationis lead until a failure situation is detected, and the resultingmechanism is not modelled.

From the structural results in Fig. 5 it can be observed thatfailure progresses as follows: first the yielding of the tensile steelrebars occurs, followed by the crushing of concrete. Finally, thetensile rebars reach the ultimate strain causing the failure of thebeam at the corresponding section. The progressive failure of theframe is restrained to the external bay initially supported by theremoved column. Although failure does not spread beyond thisregion, themaximum allowable extent of collapse according to theGSA is exceeded since it should be confined to the structural baysdirectly associatedwith the instantaneously removed verticalmemberin the floor level directly above the instantaneously removed verticalmember [3]. In the present case, all the floors above the failingcolumn are included in the collapse region. As shown in Fig. 5, thecomplete failure of the exterior bay occurs 420 ms after the onsetof the column removal. The amount of continuous reinforcementhere provided seems insufficient to prevent progressive collapsefor instantaneous removal of an exterior column.

3.3. Influence of the material parameters

3.3.1. Ultimate strain of concrete and steel (cases B, C, D)For a given, fixed design, it is important to assess the influence

of material parameters for which varying values are found in theliterature. The choice of suchmaterial parametersmay indeed playan important role in the resultant structural failure pattern. Hence,the sudden column loss is performed for different values of theultimate strains of steel and concrete respectively. The referenceultimate strain of concrete in compression previously used isc,lim = 0.35%, which corresponds to the most widely adoptedvalue for unconfined concrete [36,37]. However, the confiningeffect of the steel stirrups, which are not explicitly modelled in thepresent study, would induce an increase in the strain limit whichdepends on the lateral compressive stress and is often calculatedas a function of the ratio of shear reinforcement [43]. Hence, theuse of a higher value for the concrete strain limit is motivated inthe present parametric study since it may result in an enhancedstructural resistance to progressive collapse. On the other hand, astrain limit of s,lim = 4% is adopted for steel as a reference value,which would correspond to a ductility between classes A and Bdescribed in Eurocode 2 [36]. Higher values for this parametermaybe used, matching a higher ductility class. The set of computationscarried out is the following (Table 3): in the first one (case B) theultimate strain of concrete is increased to c,lim = 0.5% whilethe ultimate strain of steel is kept unvaried; in the second one(case C) the steel ultimate strain adopted is s,lim = 10% (resultingin a ductility class C according to Eurocode 2) and the concretestrain limit is kept equal to c,lim = 0.35%; finally, the thirdcomputation (case D) is conducted for increased values of bothlimit strains: c,lim = 0.5% in concrete and s,lim = 10% in steel.The results of the three computations are shown in Figs. 68.

The change of the ultimate strain of concrete by itself doesnot influence the global response, as can be seen in Fig. 6. It hasa slightly delaying effect in the concrete crushing initiation. Thecrushed concrete area in the shown sections is slightly smaller thanin case A. The complete failure of the rightmost bay takes placebarely 10 ms later than in the reference test.

On the other hand, the increase of the strain limit of steel(Fig. 7) causes a significant delay on the appearance of failedsections. For instance, the final collapsed pattern is reached 885msafter the onset of the column removal, which doubles the timerequired with respect to the reference case A. Note that the failingsections havemore than one third of their concrete surface crushedbefore the failure of the tensile steel rebars takes place, which


Fig. 7. Case C: response for s,lim = 10%.

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shows an increase in the efficiency of the RC sections since abetter exploitation of the concrete in compression is achieved. Thisresult indicates the important role that this parameter plays in thestructural response.

If both parameters are increased simultaneously (Fig. 8), thestructural collapse is prevented from occurring. The increasedductility of both materials leads to a structural strength gain,favouring the load redistributions. Fig. 11(a) shows the verticaldisplacement history of the node located at the connection withthe removed column, as a function of the parameter combinations.The cross indicates the moment of collapse of the rightmost bay,when it occurs. The increase of the ultimate strain for steel leadsto a remarkably higher ductility at the structural scale, since thedisplacement at the moment of structural failure doubles the oneobtained for the reference case A. For this particular case, resultsshow that a computation including large displacements would beuseful, contrarily to the previous tests where the displacementsobtained can be considered small enough to neglect catenaryeffects.

3.4. Influence of the design parameters

3.4.1. Reinforcement ratio (case E)The loss of the lateral ground column induces a sudden reversal

of the bending moment in the central region of the supportedbeams, as well as a high increase of the bending moment atthe connections with the adjacent internal columns. If thesevalues of the bending moment were to be used for an elasticdesign of the collapse resistant structure, the originally calculatedreinforcement ratios would have to be multiplied by a factor 3 or4 in certain sections of the beams. A lower increase is howevertested in this study, in order to remain in a reasonable rangeof reinforcement amount, which additionally allows taking intoaccount the load redistributions linked to the plastification ofconcrete and/or steel. Thus, the alternative design now consideredis obtained by replacing the 32 mm diameter bars by 40 mmdiameter ones, resulting in a reinforcing area of approximately 1.5times the original one. The value of the resulting reinforcementratios is now between 1% and 1.4% for the beams and 2.5%for the columns, falling within a range of values found in theliterature [15,17]. The response of the structure to the suddencolumn loss is shown in Fig. 9. As opposed to the reference results,this new design corresponding to an increased reinforcement areaprevents collapse from happening. Plastification of the steel rebarsis however observed in a few sections. The extra reinforcementapplied, although much inferior to the one theoretically requiredfor ensuring a collapse resistant (elastic) design, provides thenecessary strength enhancement in the beam connections forthe load redistributions to develop. Fig. 11(b) shows the verticaldisplacement of the node located just above the removed columnconnexion for the reference test and the test with increasedreinforcement.

3.4.2. Influence of the column removal time (case F)In this section, the interest is shifted towards the inertial effects

in the progressive failure scheme of the structure. A larger columnremoval time is applied in order to represent a longer durationof the initial triggering event: tr = 500 ms is now adopted, theorder of magnitude of the time in which a column would beremoved in a low velocity impact event. This result is reportedin Fig. 10. It shows that inertial effects play a major role insuch a progressive collapse analysis: higher removal times, andsubsequently lower accelerations, can lead to no collapse after thecolumn loss. For the studied case, only plastification of the steelbars occurs at certain points. The vertical displacement historyof the top node of the removed column is depicted in Fig. 11(c),

Fig. 8. Case D: response for c,lim = 0.5% and s,lim = 10%.

Fig. 9. Case E: response for 50% increased reinforcement.

Fig. 10. Case F: response for a column removal time tr = 500 ms.

which has also been computed for two other removal times tr= 250 ms and tr = 350 ms. This illustrates to which extent theinertial effects affect the structural response and is consistent withthe statement from [26], claiming that the sudden column loss

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a b

c

Fig. 11. Vertical displacement of the node on top of the removed column as a function of (a) the material parameters; (b) the reinforcement amount; (c) the removal time.

idealisation (considering an instantaneous column removal) offersan upper bound on the deformations obtained through an eventdependent approach.

3.4.3. Influence of the removed column location (case G)Finally, the effect of the topological location of the failing

member is assessed by testing the sudden loss of an interior groundcolumn. The material and modelling parameters are the referenceones. Fig. 12 summarises the response of the frame to the loss ofthe third ground column from the right.

Contrarily to the reference caseA (Fig. 5), the frame resists to thesudden column loss. Although yielding of the steel rebars or evenconcrete crushing take place in some sections, complete failure isnot observed in any of the beams. The damaged area in this caseis also restrained to both bays directly associated with the failingcolumn. The fact that an interior support is now removed resultsin an increased ability of the frame to redistribute loads amongthe remaining elements at both sides. This is made possible by thecontinuous bottom reinforcement provided in the beams, which isable to accommodate the bending moment reversal caused by thecolumn removal.

3.4.4. Influence of the progressive collapse procedure (GSA vs. DoD)(cases H, I)

The DoD guidelines [2] propose a loading combination thatincludes 50% of the live loads instead of 25% as suggested by theGSA [3]. Thus the total downward loads applied are

Total Loads = Dead Loads+ 0.5 Live Loadsand are indicated in Table 2. This analysis procedure is nowadopted to simulate the sudden column loss. The results for anexterior column removal (case H) are depicted in Fig. 13. The totalfailure of the rightmost bay takes place about 100 ms earlier thanin the reference case (Fig. 5). Apart from this fact, no significantdifference is found between both results.

On the contrary, if an interior column removal is considered(case I), as shown in Fig. 14, the results differ substantially

from the ones previously obtained via the GSA specifications(Fig. 12). The collapse of the two bays directly associated withthe removed column occurs now that 50% of the live loads areapplied. Fig. 15 depicts the displacement history for cases H and I.This indicates that the results obtained in a sudden loss analysismay differ depending on the procedure standards considered,even for the most widely used methods, which shows theimportance of the procedure used for the simulation of progressivecollapse.

4. Investigation of the strain rate effects

4.1. Strain rate effects in RC

For a wide range of triggering events (impact, blast...),progressive collapse is a dynamic process involving rather highstrain rates. As a result, the global response of a part or theentire structure is likely to entail dynamic effects, as well asnonlinear effects in the material response. Reinforced concretestructures subjected to high loading rates are expected to havea different response than when loaded statically. For instance,concrete shows a strongly rate dependent behaviour, with bothcompressive and tensile strengths increasing significantlywith thestrain rate [37,44,45], as confirmed by experimental tests carriedout bymeans of the Split Hopkinson Bar [46,47]. Steel also presentsa strain rate sensitive behaviour, although to a lower extent thanconcrete [48]. Since the progressive collapse related structuralbehaviour considerably depends on stress redistributions, thismaterial strain rate dependence, leading to a enhancement in thelocal strength with increasing strain rates, might also induce adifferent overall response of the whole structure. Themultilayeredbeam formulation can help in assessing the need for such amodelling tool. It indeed allows for the incorporation of therate effects at the sectional level linking the generalised stresses(N and M) to the generalised strains ( and ) and their rates( and ) implicitly; thereby avoiding a complex identification ofclosed-form rate dependent sectional evolution laws.

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(a) After 300 ms. (b) After 2000 ms.

Fig. 12. Case G: response to the sudden loss of an interior column.


Fig. 13. Case H: response to the DoD approach (50% of the live loads).

(a) After 300 ms. (b) After 450 ms.

Fig. 14. Case I: response to the DoD approach for an interior column removal.

Numerous references on the rate dependent behaviour ofconcrete can be found in the literature [4951], where the strengthenhancement at the material level is studied and modelled inthe context of small-scale structural computations. However, theinfluence of the viscous effects at the global scale has not yetbeen investigated in contributions related to structural progressivecollapse. Here, the rate dependent effects are incorporated from

the rate dependence at the material level using the layered beamapproach.

4.2. Rate dependent response of concrete and steel

The International Federation for Structural Concrete (fib) [37]describes the rate dependent behaviour of concrete in compression

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(a) Exterior column loss. (b) Interior column loss.

Fig. 15. Vertical displacement of the node on top of the removed column as a function of the analysis procedure (GSA vs. DoD).

by means of a rate dependent modulus of elasticity and the so-called dynamic increase factors (DIF) on strength, representingthe ratio of dynamic to static strength, as determined by Malvarand Crawford [45]. The effect of strain rate on Youngs modulusand the dynamic increase factors are estimated by Malvar andCrawford [45]:

EcEc,st

=

st

0.026(25)

DIF = fcfc,st

=

st

1.026sfor 30 s1

s

st

1/3for > 30 s1

(26)

where Ec is the compressive Youngs modulus at strain rate ;Ec,st the compressive static Youngs modulus; fc the compressivestrength at strain rate ; fc,st the static compressive strength(37.9 MPa for a C30 concrete); a strain rate between 3 105 s1and 300 s1; st the compressive static strain rate of 3 105 s1;log(s) = 6.156s 2; s = 1/(5+ 9fc,st/107).

Based on these dynamic increase factors, a strain rate depen-dence is introduced both in the elastic and in the plastic domain,to match the fib description in terms of peak stress, strain at peakstress and ultimate strain. This allows preserving the crushing en-ergy (area under the compressive stressstrain curve) as shown inFig. 16(a), where the simplifiedmodel used here is compared to theprescriptions by the fib. The compressive stressstrain curves for aC30 concrete type are depicted for various strain rates. The rate de-pendent curves suggested by the fib have been constructed follow-ing the static stressstrain expression based on relations (1)(3),and the DIFs from Eq. (26). A rate dependent Youngs modulus ishence adopted, an important feature due to its influence in the loadredistributions involved in progressive collapse analyses. The ulti-mate compressive strain (0.35%) corresponding to the crushingstrain of concretewill be considered constant independently of thestrain rate, as no evolution of this parameterwas evidenced in [52].The dynamic increase factors for a C30 concrete type obtainedwiththe proposed model are in good agreement with the prescriptionsof the fib [37], as observed in Fig. 17(a).

For steel, Youngs modulus is assumed to be rate independent,as observed by Malvar and Crawford [48]. The DIFs correspondingto the yield stress for steel obey the following expression accordingto [48]:

DIFsteel = fyfy,st =

st

(27)

with fy the yield stress at strain rate ; fy,st the static yield stress(here 500 MPa); a strain rate between 104 and 225 s1; st

the static strain rate of 104 s1; = 0.074 0.040fy/414. Asfor concrete, the model is adjusted to match the results in [48] interms of DIFs. The ultimate strain, which is not observed to varyas a function of the strain rate [48], is here taken equal to 4%. Inorder to represent the failure of steel, the stress is set to zero forstrains exceeding this value, independently of the value of strainrate. Fig. 16(b) shows the rate dependent stressstrain curves. TheDIFs are depicted in Fig. 17(b).

4.3. Rate dependent constitutive model

A natural way to include the strain rate effects in a materialmodel is the introduction of viscous terms in the constitutivelaw. For instance, Pedersen et al. [49] developed a rate dependentmacroscopic material model based on the microscopic andmesoscopic behaviour of concrete. In the present contribution,strain rate dependent one-dimensional constitutive laws forconcrete and steel are used in the layers of the layered beammodel.Perzynas viscoplastic model [53,54] is adopted for the behaviourof both constituents. The viscoplastic strain rate vp is expressed asa function of the yield function f defined in Eq. (5):

vp = 1

f0

N dfd

(28)

where 0 is the initial yield stress; f is the yield function whichaccounts for the overstress ( ), since here the applied stress canexceed the yield stress contrarily to the rate independent plasticcase; and N are viscosity parameters, with N 1 an integernumber; and denotes the Macaulay brackets. Note that theparameter depends on the strain rate () to match the materialresponse reported in the literature in terms of DIFs [37,45,48].Rewriting Eq. (28) in an incremental fashion, the residual in Eq. (9)now reads:

{R(n+1, n+1)}

=n+1 trialn+1 + E(n+1)n+1(n+1)n+1 tn+1

f (n+1, n+1)

0

N dfd

n+1

(29)where Eq. (4) and the approximation = /t have been used,with t the time step. Note that now the rate dependence alsoneeds to be introduced in the Youngsmodulus E() for concrete, toobtain a strain at peak stress that matches the previsions reportedin [37], as shown in Fig. 16(a). The values of the viscoplastic modelparameters are given in Table 5. The rate dependent parametersEc(), c() and s() are depicted in Fig. 18. A perturbationtechnique is used to evaluate the material tangent operator Hconsistent with the return-mapping algorithm. This avoids the

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(a) Concrete in compression. (b) Steel.

Fig. 16. Rate dependent response of the proposed model.

(a) Concrete in compression [45]. (b) Steel [48].

Fig. 17. Dynamic increase factors.

(a) Youngs modulus Ec() in concrete. (b) Viscoplastic parameter c in concrete.

(c) Viscoplastic parameter s in steel.

Fig. 18. Rate dependent constitutive parameters.




Fig. 19. Rate dependent response for a column removal time of tr = 5 ms.

Table 5Viscoplastic model parameters.

Concrete SteelEc (GPa) Nc c (s1) Ns s (s1)

Ec() (Fig. 18(a)) 1 c() (Fig. 18(b)) 1 s() (Fig. 18(c))

analytical derivation of the constitutive equations with respect tothe strain , a nontrivial task due to the strain rate dependence ofthe parameters Ec(), c() and s().

The introduction of the previous rate dependent constitutiveequations (, ) in the layered beam formulation provides ratedependent relations between the beam generalised stresses andstrains. To obtain such relationships, the layer-wise strain rates arecomputed from the generalised strain rates {Egen} = , asi = yi . (30)

An analytical version of such a rate dependent multilayeredapproach was used in [32], where the strain rate dependence isintroduced in a simply supported RC slab under blast loading bymultiplying the layer-wise stresses i(i) by their correspondingdynamic increase factors DIF(i). Contrarily to the present work,the rate dependence of concrete in the elastic region was howevernot taken into account.

At the structural scale, the equilibrium equations now read:f int ({q}, {q})+ [M] {q} = f ext (31)

wheref intis also a function of the velocities {q}:

f int ({q}, {q}) =e

Ve[B]t

gen

Egen

, {Egen} dV . (32)

Note that the structural tangent matrix [Kt ] from Eq. (21) nowcontains inherent rate dependent damping terms stemming fromthe rate dependence of the internal forces. This means that thestrain rate dependent nature of thematerial constitutive equationsprovides a certain amount of damping at the structural level, asopposed to rate independent approaches.

4.4. Rate dependent simulation of progressive collapse

The sudden column loss simulation is now performed using thepreviously described strain rate dependent approach. This is donefor two different removal times tr = 5 ms and tr = 500 ms,which determine the extent of the strain rate effects in the struc-tural response. Results are shown in Figs. 19 and 20 respectively.

For instance, if a removal time tr = 5 ms is applied, the materialstrength-enhancing strain rate effects prevent collapse from hap-pening, as opposed to the rate independent case (case A) wherethe entire rightmost bay collapses. It must be noted that, for thestudied reinforcement arrangement, this structural enhancementis mostly provided by the rate dependence of the steel bars, sincethe amount of reinforcement is similar at both sides (upper andlower) for the sections considered here. The contribution of theconcrete rate dependence to this improved structural response isthus lower. For a removal time of tr = 500 ms, the structure doesnot collapse either, as it was also the case in the rate inde-pendent approach (case F). The comparison of both results withrespect to the rate independent case in terms of vertical displace-ments is shown in Fig. 21. It can be observed that taking the strainrate effects in consideration results in a structural integrity im-provement. If the column removal is performed in a larger times-pan tr = 500 ms, the strain rate effects become less significant.In this case, the difference between the rate dependent and the rateindependent approaches is lower: the final response patterns aresimilar. The classically adopted rate independent approach thusleads to a more important damage evolution as the removal timedecreases, due to the increase of the inertial effects: while theinertial effects favour the propagation of the failure in the struc-ture, the strain rate effects limit the degree of failure. Never-theless, the results obtained confirm that the use of the suddencolumn loss idealisation (where an instantaneous removal of thecolumn is assumed) to obtain an upper bound on the deformationdemands [26] is also applicable to the rate dependent case.

5. Discussion and concluding remarks

A multilayered beam element was developed to simulate thedynamic response of RC structures subjected to a sudden columnloss. Due to the quasi-instantaneous character of the columnremoval (which takes place in a single time step, i.e. 5ms), a loadingscenario in the impulsive range is considered here. A nonlinearmodel was adopted for the material behaviour of concrete andsteel. This multilevel approach allows for a direct transitionbetween the material constitutive equations and the response atthe structural level, providing physically motivated relationshipsfrom the generalised stresses to the generalised strains. Moreimportantly, a strain rate dependent material formulation hasalso been developed. The rate dependence is introduced in boththe elastic and the plastic behaviours. The sudden column lossmethodology has been applied for a structure designed accordingto the Eurocodes prescriptions. A study was conducted by varying

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Fig. 20. Rate dependent response for a removal time tr = 500 ms.

Fig. 21. Vertical displacement of the node on top of the removed column: ratedependent vs. rate independent approach.

some design and material parameters in a commonly used rangeto analyse their effect on the obtained collapse patterns. Thefollowing conclusions may be drawn:

1. As expected, the design options have an important effect onthe progressive collapse response of a structure: an increaseof 50% on the initial reinforcement amount shows a significantincreased resistance to the sudden column loss in the studiedstructure.

2. While the ultimate strain of concrete does not appear to bedecisive in the progressive collapse mechanism, the ultimatestrain of steel might be determinant to the point that collapsecan be avoided by increasing the value of this parameter in arealistic range. Moreover, higher values of the ultimate strainof steel lead to a much more ductile structural response.

3. The column removal time is also determinant in the structuralresponse, since the inertial effects are directly related to thisvariable: for higher removal times the accelerations involved inthe process are smaller and thus the extent of structural damageis lower. This confirms that the sudden column loss idealisationrepresents a useful design scenario for the assessment ofstructural robustness [26].

4. The location of the removed element also has an effect onthe propagation of collapse: the removal of an interior columnleads to lower degrees of structural damage, due to the supportprovided by the neighbouring column at both sides.

5. The procedure used for the simulation of the sudden columnloss (GSA or DoD) has a significant effect on the results: theDoD procedure, which considers 50% of the live loads, mightlead to conservative designs with respect to the GSA guidelines,

which suggests including only 25% of the live loads in thecomputations.

6. The strain rate effects in the sudden column loss simulationhave been proven to be potentially significant at the structurallevel. The strength enhancementwhich stems from themateriallevel results in a structural integrity improvement for thestudied structure. This structural enhancement is mostlyprovided by the rate dependence of the steel bars due to theparticular reinforcement arrangement adopted here.

Various simplifications have been adopted for the presentstudy, for which a discussion is presented next. First, a planar rep-resentation of the structure has been employed as an initial ap-proach to the problem. Note that the single column removal ina two-dimensional representation would correspond to remov-ing a whole row of columns in 3D since the transversal resistanceis not taken into account. This simplified approach might thuslead to conservative results. The extension to a three-dimensionalapproach seems to be the logical next step in order to obtain amore realistic structural response. However, this extension is non-trivial both from the computational cost viewpoint, and from theperspective of structural complexities (representation of torsionaleffects, stiffening effects of the floors...). Secondly, the geometri-cally linear formulation adopted here is partially justified by thesmall-range displacements obtained in most of the computations.Nevertheless, a large displacement analysis is foreseen for forth-coming works even though this issue is out of scope of the presentpaper. Third, the shear reinforcements have not been explicitlymodelled. An implicit, phenomenological way of taking them intoaccount has been considered by increasing the concrete ultimatestrain. The results however showed no significant differences inthe structural response. Moreover, the Bernoulli beam approxima-tion adopted in the present formulation is unable to account for theshear deformation, since the cross-sections remain plane and per-pendicular to the longitudinal axis of the beam. This choice, whichimplies that shear failure is not considered, was justified by thefact that the structures under study are considered to be properlydesigned against shear failure (as prescribed by the GSA [3]) sothat flexural effects are predominant in the beam response. In or-der to obtain a more complete approach, in which different failuremodes could be incorporated, the upgrade to a Timoshenko beamelement formulation would be recommended as in [30] where theformulation of a Timoshenko multifiber beam element was elabo-rated and applied for a small-scale analysis. The proper modellingof shear failure would however require in addition a shear failurecriterion based on the layered beam approach results, a nontrivialextension. Finally, the failed elements are not removed from thetopology and thus remain attached to the structure after reach-ing complete failure, keeping transferring stresses to the intactelements in the vicinity. As a consequence, the impact of failingmembers on adjacent elements is not included, ignoring the cor-responding dynamic load transfer. The sudden column loss ap-proximation was used in the present study, with the purpose ofanalysing the progressive spread of damage in the structure af-ter localised failure has occurred, since all the potential sources ofcollapse initiation cannot be investigated. A detailed characterisa-tion of the triggering event would offer the most realistic solution,in which the full dynamic character of the triggering event wouldbe included, incorporating potential horizontal components of thistriggering event. However, the computational time requiredwouldbe prohibitive, due to the small step size needed for a thoroughsimulation of a short-time duration event. The sudden column losstechnique is considered to offer a reasonable balance between theusefulness of the results and the computational effort.

In spite of the aforementioned simplifying assumptions, the ef-fects observed with the present approach are supposed to remain

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general and are expected to be reproduced in a more completeapproach. Moreover, the versatility of the present multilevelmethodology must be underlined: it provides a more realistic rep-resentation of the cross-sectional response of reinforced concretememberswhere axial load-bendingmoment interactions are natu-rally included, avoiding the need to postulate approximate closed-form relationships between generalised stresses ans strains. Itoffers an increased flexibility with respect to other approaches,for which any material and/or design parametric variation wouldrequire an offline identification process to derive analytical ex-pressions. In the present formulation, these variations are natu-rally introduced in the structural response. Furthermore, it can beextended to other fine-scale constitutive laws, including a differ-entmaterial response and/or additionalmodelling features such asdamage, a particularlymeaningful feature of thematerial responseunder cyclic loading. Additionally, the presentmultilevel approachcould be considered as a complement to other simplified method-ologies such as the one proposed in [5,6], in order to overcomethe high computational effort involved in the proposed detailedapproach. In these contributions, the authors present a multilevelframework for progressive collapse assessment of structures sub-ject to sudden column loss which allows including various levelsof structural modelling sophistication: both detailed and simpli-fied approaches can be combined according to the level of struc-tural idealisation considered. Instead of using a nonlinear dynamicfinite element analysis, the nonlinear static response is only anal-ysed and the dynamic effects are evaluated in a simplifiedmanner.Such a strategy which allows a significant reduction of the com-putational time could be combined with the material and/or de-sign sensitivity issues studied in the present analysis (potentiallyincluding the material rate dependence) to yield additional practi-cal rules for design.

Acknowledgements

The research presented in this paper is supported by theBelgian Ministry of Defence (project DY-03) for the first author.Financial support from the F.R.S.-FNRS under grants 1.5.032.09.F and 2.4.513.09. F (FRFC) for the last author and under grant1.2.093.10. F for the second author is gratefully acknowledged.

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