Contents lists available at ScienceDirect

Engineering Structures

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

New analytical calculation models for compressive arch action in reinforcedconcrete structures

Xinzheng Lua,⁎, Kaiqi Linb, Chenfeng Lic, Yi Lid

a Key Laboratory of Civil Engineering Safety and Durability of Ministry of Education, Tsinghua University, Chinab Beijing Engineering Research Center of Steel and Concrete Composite Structures, Tsinghua University, Chinac College of Engineering, Swansea University, Swansea SA1 8EN, UKd Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, China

A R T I C L E I N F O

Keywords:Reinforced concrete frameProgressive collapseCompressive arch actionCalculation modelSlab effect

A B S T R A C T

Research challenges associated with progressive collapse of reinforced concrete (RC) structures have attractedgrowing attention from researchers and industries worldwide, since the 1995 explosion at the Murrah FederalBuilding in Oklahoma City. The compressive arch action (CAA), as a favorable mechanism to provide thestructural resistance to progressive collapse under a column removal scenario, has been extensively studied usingboth experimental and theoretical approaches. However, the existing prediction models for the CAA resistanceare either too complicated or in need of additional information like the peak deformation of the specimen.Another major weakness in the previous CAA calculation models is the negligence of the slab effect, which cancontribute significantly to the structural resistance. In this study, based on the finite element analysis of 50progressive collapse tests reported in the literature and 217 newly designed beam-slab substructures, explicit andeasy-to-use CAA calculation models are developed for RC frame beams with and without slabs. The proposedmodels are validated against both experimental and numerical results with a mean absolute error being less than10%. The findings from this study can serve to provide a quantitative reference for practical design of RC framestructures against progressive collapse.

1. Introduction

Progressive collapse study has become an important research fron-tier since the collapse of Alfred P. Murrah Federal Building in the 1995Oklahoma City bombing attack [1]. Current research outcomes indicatethat a structure can resist the progressive collapse through two majormechanisms, i.e., the beam mechanism at small deformations and thecatenary mechanism at large deformations [2–10]. As such, fully uti-lizing these two resistance mechanisms has become the primary ob-jective of the current progressive collapse design codes [11,12]. Ac-cording to the existing experimental observations, compressive archaction (CAA) is commonly found in reinforced concrete (RC) beams atsmall deformations [5–10,13–19]. Quantitatively, CAA has been foundto improve the structural resistance by 30–150% as indicated in theexisting studies [10,14]. Therefore, an accurate and easy-to-use calcu-lation model for CAA is of significant value for achieving a rationalprogressive collapse design.

An illustration of CAA is shown in Fig. 1, where the middle columnfails and loses its load carrying capacity. The unbalanced gravity loadfrom the structure above, which is originally carried by the failed

column, is subsequently transformed into a concentrated load P on thebeam-column joint. At the early stage of loading, the structural re-sistance to progressive collapse is provided by the flexural capacities(i.e., M1 & M2) of the frame beams (Fig. 1a). As the displacement in-creases, cracking of the concrete will cause a migration of the neutralaxis, accompanied by an in-plane expansion of the specimen. When thisexpansion is restrained by the boundaries at the beam ends, CAA will beformed in the beams which will in turn enhance the strength of thespecimens as shown in Fig. 1b [20].

Many theoretical investigations have been performed to evaluatethe CAA resistance, among which Park & Gamble proposed one of themost widely accepted CAA calculation model [21]. The Park & Gam-ble’s model was validated by Su et al. [14] and Qian et al. [10] withtheir experimental tests. Afterwards, Yu & Tan [22] and Kang & Tan[23] further updated the Park & Gamble’s model, by calculating itera-tively the force–displacement relationship of RC beams at the CAAstage. The existing literature shows that the computational models ofPark & Gamble [21], Yu & Tan [22] and Kang & Tan [23] are suffi-ciently accurate when calculating the CAA resistance of RC framebeams. However, further studies are still required due to the following

https://doi.org/10.1016/j.engstruct.2018.04.097Received 24 August 2017; Received in revised form 26 April 2018; Accepted 30 April 2018

⁎ Corresponding author.E-mail address: luxz@tsinghua.edu.cn (X. Lu).

Engineering Structures 168 (2018) 721–735

0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

T

reasons:

(1) The abovementioned models are not suitable for practical use inprogressive collapse design: (a) the Park & Gamble’s model [21]requires the peak displacement of the CAA, which cannot be easilyobtained for real structures; (b) the models proposed by Yu & Tan[22] and Kang & Tan [23] require iterative calculations of the re-lative depth of the equivalent compression zone, making the asso-ciated computation infeasible without dedicated computer pro-gramming.

(2) In real RC frame structures, the frame beams and slabs are alwayscast together to carry the dead and live loads. According to Renet al. [7] and Lu et al. [8], the presence of slab can significantlyimprove the progressive collapse resistance at the CAA stage.However, existing models can only calculate the CAA of framebeams. In the presence of slabs, reinforcing steels on beam sections1–4 (Fig. 1b) do not yield simultaneously, which violates the fun-damental assumptions of the existing models. Hence, when thesemodels are used for evaluating the CAA resistance of the framebeams with the slab effect, significant errors would be expected.

In this study, we simplified the boundary conditions as fixed ends inorder to provide an explicit solution for calculating the CAA resistanceof RC beams under interior column removal scenarios. Such simplifi-cation also helped to simplify the prediction equations, making theproposed model more practical and feasible for practical engineeringapplications. Note that in the alternate path (AP) design method asspecified in most of the existing progressive collapse design codes[11–12], a column removal means removing the clear height of theselected column between the lateral restraints at both column ends. Inreal situations, the interior beams or beam-slab substructures are al-ways surrounded by the peripheral structural components (i.e., beams,columns and slabs of the adjacent bays). These components will providesufficient restraints to the deformation of the boundaries. It is thereforerational to assume such fixed boundaries when considering the interiorcolumn removal case in an RC structure.

To overcome the limitations of the existing calculation models forCAA, we established a series of finite element (FE) models fully vali-dated against a large database of experimental outcomes. For both

beam and beam-slab specimens, their sectional stress-strain distribu-tions, key design parameters and the corresponding sensitivities wereanalyzed using the validated FE models. Following the experimentaland numerical analysis, we proposed explicit and easy-to-use CAAcalculation models for RC beams with and without slabs. Comparisonwith the published experimental results of other researchers confirmsthat the newly developed models can accurately produce the pro-gressive collapse resistance of RC beams (with and without slabs) at theCAA stage. The computational procedure of the new models is simpleand easy to implement. The findings from this study can serve to pro-vide a quantitative reference for practical design of RC frame structuresagainst progressive collapse.

2. Validation of the Park & Gamble’s model

2.1. The Park & Gamble’s model

The Park & Gamble’s model [21] is based on the deformationcompatibility and force equilibrium of RC beams under a concentratedload (Fig. 2a). Considering the isolated beam model shown in Fig. 2b,the progressive collapse resistance (P) at CAA stage can be expressed as:

= + −P M M Nδβl

2( )1 2

(1)

where δ is the peak displacement corresponding to the peak load; M1

and M2 are the moments at the beam ends; N is the axial force inducedby CAA; l is the total length of the two-span beam; β is the ratio betweenthe net span and the total span l. Note that M1, M2 and N can be derivedby calculating the resultant forces at the corresponding cross sections.

In the Park & Gamble’s model, the relative depths of the compres-sion zones at Sections 1 and 2 are obtained by solving the equations ofdeformation compatibility and force equilibrium:

′ = − − ⎛⎝

+ ⎞⎠

+′− − ′ +

′c h δ βl

δε t

lT T C C

f β b2 4 42

1.7

2s s

c 1 (2)

= − − ⎛⎝

+ ⎞⎠

−′− − ′ +

′c h δ βl

δε t

lT T C C

f β b2 4 42

1.7

2s s

c 1 (3)

where ′c and c are the relative depth of the compression zone atSections 1 and 2, respectively; h and b are the height and width of the

Fixed boundary Fixed boundaryFailed column

PM1

M2

M1

41 2 3

(a) Resistance mechanism of RC beams at initial loading stage

Fixed boundary Fixed boundaryFailed column

PM1

M2

M1

N N

Compressive arching zone Tensile cracks

1

2 3

4

(b) Resistance mechanism of RC beams at CAA stage Fig. 1. CAA in RC beams at small deformation stage under concentrated load (Note: 1, 2, 3, 4 denote various beam sections).

X. Lu et al. Engineering Structures 168 (2018) 721–735

722

beam, respectively; ε is the axial compressive strain of the beam; t is thelongitudinal movement of the boundary; ′Cc and Cc are the resultantforces in concrete at Sections 1 and 2; ′T and T are the tensile forces inreinforcement at Sections 1 and 2, respectively; ′Cs and Cs are thecompressive forces in reinforcement at Sections 1 and 2, respectively; fc'is the concrete cylinder strength; β1 is the ratio of the depth of theequivalent rectangular stress block to the depth of neutral-axis, as de-fined in ACI 318–14 [24].

In order to calculate the force contributions from concrete and re-inforcement, it was assumed by Park & Gamble that the critical sectionshave reached their strengths when a full plasticity is developed at thecritical sections of the beam (i.e., the peak point at the CAA stage) [21].Note that under certain circumstances, the above assumption may notbe very valid as the compressive reinforcement at Section 2 may notyield at the peak point of CAA. However, since the Park & Gamble’smodel has been proven to be able to provide accurate predictions of theload carrying capacities of RC beams under CAA [10,14,21], this as-sumption is nonetheless adopted in the calculation procedure of the RCbeam specimens for its simplicity. Hence, the moments M1, M2 and theaxial force N can be calculated by combining Eqs. (2) and (3) with thestress-strain relationships of the reinforcement and concrete at the ul-timate state, after which the CAA resistance P of the RC beams can thenbe readily calculated following Eq. (1). The expressions ofM1,M2 and Nat the beam ends are obtained as:

= + − = ′ + −N C C T f β cb C T0.85c s c 1 s (4)

= ′ ′ − ′ + ′ − ′ + ′ − ′M f β c b h β C h d T h d0.85 (0.5 0.5 c ) (0.5 ) (0.5 )1 c 1 1 s (5)

= ′ − + − ′ + − ′M f β cb h β c C h d T h d0.85 (0.5 0.5 ) (0.5 ) (0.5 )s2 c 1 1 (6)

where d' is the thickness of the concrete cover.

2.2. The database of experimental tests

Many experiments have been reported to study progressive collapseof RC frame beams under different column removal scenarios. In orderto validate the calculation models for CAA, a total of 50 RC progressivecollapse specimens, tested by other researchers under a mid-columnremoval scenario, were collected from the literature. Details of these RCspecimens are listed in Appendix A [5–7,10,13–17,25–26], in which 45are beam specimens and 5 are beam-slab specimens. In addition to thetest specimens included in Appendix A, we also reviewed a number ofother RC progressive collapse tests found in the literature (e.g., Lu et al.[8], Qian et al. [10], Choi & Kim [18], Sadek et al. [19], Qian & Li [27],Prasad & Hutchinson [28], and Dat & Tan [29]). However, comparingto the illustrations shown in Figs. 1 and 2, these additional tests typi-cally have different boundary conditions or deformation modes, andhence they are not considered in this work.

2.3. Validation of the Park & Gamble’s model

Following the Park & Gamble’s model [21], the CAA resistance P iscomputed for all 45 beam specimens given in Appendix A, and the

Fig. 2. The Park & Gamble’s model (Note: 1, 2, 3, 4 denote various beam sections).

X. Lu et al. Engineering Structures 168 (2018) 721–735

723

calculated results are compared with the corresponding experimentalmeasurements. Note that the Park & Gamble’s model requires the dis-placement δ at the peak resistance of CAA, for which the experimentallymeasured values are adopted in the calculation. The comparisons areshown in Fig. 3a, where the mean absolute error (MAE) is 10.97% witha standard deviation of 0.128. This confirms that given the peak dis-placement δ corresponding to the peak resistance of CAA, the Park &Gamble’s model is sufficiently accurate in calculating the CAA re-sistance P.

However, for real-world structures, it is often hard, if not im-possible, to obtain the δ value through experiments, making the Park &Gamble’s model impractical in engineering practice. To resolve thissituation, Park & Gamble [21] recommended =δ h0.5 for use in theircalculation procedure. The CAA resistances calculated using the as-sumption =δ h0.5 are shown in Fig. 3b, where the MAE is 20.47% witha standard deviation of 0.158. This suggests that the prediction ob-tained with =δ h0.5 significantly underestimates the CAA resistance,and the reduced prediction accuracy poses a major constraint to thepractical use of the Park & Gamble’s model.

3. The FE model

Finite element analyses of the 50 RC beam and beam-slab testspecimens were conducted to predict their CAA resistances under themid-column removal scenario based on OpenSees [30]. The fiber-beamelement was adopted in these FE analyses, since its feasibility and ac-curacy in simulating progressive collapse of RC beams have been con-firmed in many previous research works [2–4,31–35]. The FE model ofevery test specimen shared the similar modeling strategies in OpenSees.Specifically, displacement-based beam-column elements were used tomodel the RC beams. Five Gauss integration points were consideredalong each element. The beam section was defined using the fibersection in OpenSees. Reinforcement fibers were arranged at the exactpositions as those in the test specimens according to the drawingsprovided in the literature. Concrete01 Material and Steel02 Materialwith 2% strain hardening were used to model the concrete and re-inforcing steel, respectively, in OpenSees [33]. Among the existingexperimental studies, very limited data could be found to quantify thebar-slip behaviors within the beams. As such, the bar-slip behaviorbetween the reinforcement and the concrete were simulated by redu-cing the elastic stiffness of the reinforcement [36]. Moreover, in most ofthe relevant experimental tests, column stubs are always considered toremain at the column removal location of the specimen as specified inthe AP method in DoD 2016 [12]. Such column stubs, containing muchmore reinforcement than the beams, often deform elastically during thetests. Hence, the mid beam-column joints in the FE models weremodeled using elastic beam-column elements in OpenSees.

Typically, the load-displacement relationships obtained through theFE analyses of Specimens B1 [6] and S6 [7] are plotted in Fig. 4a and b,respectively, where the blue lines indicate the FE results and the1 redlines denote the experimental measurements. For all 45 beam speci-mens and 5 beam-slab specimens listed in Appendix A, the numericallypredicted CAA resistances are compared with the experimental mea-surements in Fig. 4c, where the MAE is found to be 8.34% with astandard deviation of 0.097. It is evident that the accuracy of the pre-sent FE model is better than that of the Park & Gamble’s model, whichjustifies the use of this FE model to carry out comprehensive parametricstudies presented for both beam and beam-slab specimens in the sub-sequent sections.

Moreover, based on the above FE models, a parametric study isconducted to evaluate the influence of the boundary stiffness on thepeak resistance of an RC beam under CAA. Two indexes (i.e., Ra and Rr)proposed by Yu & Tan [37] are adopted, of which, Ra is defined as theratio between the axial stiffness of the boundary and the EA/l of the RCbeam, and Rr is the ratio between the rotational stiffness of theboundary and the EI/l of the RC beam. The results indicate that theboundary of an RC beam can be assumed as fixed ends when Ra isgreater than 2 and Rr is greater than 10.

4. The new CAA calculation model for beam specimens

As discussed in Sections 2, the practical application of the Park &Gamble’s model relies on an erroneous and oversimplified assumptionof the peak displacement =δ h0.5 , which introduces significant error tothe CAA resistance prediction and limits the usefulness of the model.Through the FE parametric study, we aim to find a better way toevaluate the δ value, and significantly improve the prediction accuracyof the Park & Gamble’s model.

According to the FE results of the 45 beam specimens, three factorsare found to affect the value of δ: (1) beam length l (2) beam height hand (3) concrete cylinder strength fc'. Specimen B1 (Fig. 4a) tested byChu et al. [6] was taken as a prototype to study these key influencingfactors. The values of l, h and fc' are varied one at a time in the FE modelof Specimen B1 within the range of 1m≤ l≤ 8m, 0.1 m≤ h≤ 0.5 mand 10MPa≤ fc'≤ 50MPa, respectively. The results of the parametricstudy are presented in Fig. 5.

It is evident from the parametric study that δ is approximatelyproportional to l2, inversely proportional to h, and linearly reducedwith the increase of fc'. Note that the influence of fc' can be neglectedcompared to that of l2 and h. Therefore, l2/h is adopted in this study topredict δ. Note that with respect to the units, the term l2/h also results

Fig. 3. Calculated CAA resistance from the Park & Gamble’s model versus experimental results.

1 For interpretation of color in Fig. 4, the reader is referred to the web version of thisarticle.

X. Lu et al. Engineering Structures 168 (2018) 721–735

724

Fig. 4. Validation of the FE models.

Fig. 5. Parametric study of the peak CAA displacement.

X. Lu et al. Engineering Structures 168 (2018) 721–735

725

in a rational unit for predicting the peak displacement. For the afore-mentioned 45 beam specimens, the values of δ and l2/h due to thevalidated FE model are collated and regressed in Fig. 6a. The regressionresult indicates that δ can be estimated by l2/h with a satisfactorycorrelation coefficient (i.e., R2= 0.9459). The equation for δ from theregression is thus:

=δ l h0.00050 /2 (7)

In the proposed equation (Eq. (7)) estimating δ, the influence of thebeam stiffness is considered automatically. For a unit width of the re-inforced concrete beam, both the flexural stiffness and axial stiffness aredetermined by three parameters: (1) the elastic modulus, E; (2) thelength of the beam specimen, l; (3) the height of the beam cross section,h. Note that for most of the RC beams in engineering practice, theelastic modulus is mainly determined by the Young’s modulus of theconcrete (Ec), which is related to the compressive strength of concrete.Fig. 5c shows that the compressive strength of concrete has limitedinfluence on δ. Hence, only the length (l) and height (h) of the beam areused to express the peak displacement (δ) from the regression analysisand the stiffness of the beam is considered implicitly in Eq. (7).

Substituting Eq. (7) into Eqs. (1)–(6), the CAA resistance of RCbeams can be predicted. The calculation results are compared with theexperimental ones in Fig. 6b, where the MAE is 8.85% with a standarddeviation of 0.109. More detailed results, including the calculated peakresistances and corresponding boundary axial forces, can also be foundin Table A.1 in Appendix A. For the CAA, quantification of the peakaxial force could help gain a better understanding of the collapse me-chanisms of RC structures. It should be noted, however, that theboundary axial force of an RC substructure under CAA is very sensitiveto the actual boundary stiffness. Moreover, as indicated by many ex-isting experimental and analytical results [5,10,14,17,22], theboundary axial force may not reach the peak value when the RC beamreaches the peak resistance under CAA. Notwithstanding, further stu-dies are required to evaluate the axial force in an RC beam under CAA.

The results indicate that the proposed model provides a much moreaccurate prediction of the CAA resistance of an RC beam compared tothe prediction using the oversimplified assumption =δ h0.5 . Comparedto the iterative models proposed by Yu & Tan [22] and Kang & Tan[23], the proposed new model is explicit and completely avoids thetime-consuming iterations, making it more practical and feasible forengineering applications. In addition, as δ in Eq. (7) is estimated basedon the FE parametric study, the uncertainties associated with the ex-perimental measurement of δ can be eliminated, leading to even smaller

errors in predictions from the proposed model than the original Park &Gamble’s model (Fig. 3a).

5. The new CAA calculation model for RC beam-slab specimens

As reviewed earlier in Section 1, a major weakness of the Park &Gamble’s model is the negligence of the slab effects. In real structures,the slab is normally cast together with the underneath beams, and it canprovide additional resistance to progressive collapse of the structure.The interaction between the beam and the slab and its effect on the CAAresistance can be examined through experimental tests and numericalanalyses of the beam-slab specimens. For a beam-slab specimen, thesectional stress-strain distributions of both reinforcement and concreteare more complicated than those of a beam specimen. Fig. 7a shows thefree-body diagram of the symmetrical half of the beam-slab sub-structure, while the internal forces at Sections 1 and 2 are illustrated inFig. 7b. The notations in Fig. 7 are explained in Table 1. Compared withthe beam specimens, the major differences in the beam-slab specimensare:

(1) The beam-slab specimen has a T-shaped cross section instead of arectangular one.

(2) Due to the slab reinforcement, over-reinforced failure occurs atSections 1 and 4 (i.e., the concrete reaches the crush strain beforethe tensile reinforcement yields) [7]. On the contrary, at Sections 2and 3, the tensile reinforcement yields at the early stage of theloading procedure but the compressive reinforcement might notyield at the peak resistance of CAA. The assumption in the Park &Gamble’s model that both concrete and reinforcement reach theirultimate state is no longer applicable for a beam-slab specimen. As aresult, the forces M1, M2 and N at the beam ends in Eq. (1) cannotbe directly calculated.

(3) For the reasons given in (1) and (2) above, the proposed model forpredicting the peak displacement δ of CAA, i.e. Eq. (7), althoughbeing very accurate for the regular beam specimen, is not suitablefor the T-shaped beam-slab specimen.

In order to provide a preliminary view on the limitation of the Park& Gamble’s model (which assumes yielding of all reinforcements), wedirectly applied the standard calculation procedure of their model tothe beam-slab specimens (S2 to S6) tested by Ren et al. [7] and com-pared the peak resistance predictions with the experimental measure-ments. As shown in Fig. 8, the MAE is 20.40%, and for four fifths of the

Fig. 6. Validation of the proposed model for CAA calculation of RC beam specimens.

X. Lu et al. Engineering Structures 168 (2018) 721–735

726

specimens the errors exceed 20%, which makes the Park & Gamble’smodel unfeasible for the beam-slab specimens.

5.1. Parametric study of beam-slab specimens using validated FE model

The validation described in Section 3 has proved that the proposedFE model is capable of simulating the behavior of beam-slab specimensat the CAA stage with a good accuracy. Due to the limited number of

the beam-slab experiments in the literature, we performed a compre-hensive parametric study of the beam-slab specimens using the vali-dated FE model, where a series of RC beam-slab substructures withcommonly used dimensions were designed according to the ChineseCode for the Design of Concrete Structures [38]. The design parameters

Fig. 7. Diagram of the beam-slab specimen at CAA stage.

Table 1Description of notations in Fig. 7.

Notation Description

T'SST Tensile force in top slab reinforcement at Section 1T'SBT Tensile force in top beam reinforcement at Section 1T'SSB Tensile force in bottom slab reinforcement at Section 1C'SBB Compressive force in bottom beam reinforcement at Section 1C'C Compressive force in concrete at Section 1c' Height of concrete compressive zone at Section 1s Height of concrete compressive zone in slab at Section 1TSSB Tensile force in bottom slab reinforcement at Section 2TSBB Tensile force in bottom beam reinforcement at Section 2CSST Compressive force in top slab reinforcement at Section 2CSBT Compressive force in top beam reinforcement at Section 2CC Compressive force in concrete at Section 2c Height of concrete compressive zone at Section 2d' Thickness of concrete coverhf Slab thicknesshb Beam height minus slab thicknessbf Slab widthbb Beam widthln Net span

Fig. 8. Comparison between the experimental results and the Park & Gamble’smodel for beam-slab specimens.

Table 2Design parameters of different beam-slab substructures.

Design parameter Value range

Span length/m 4 6 8Beam height/m 0.4, 0.5 0.5, 0.6, 0.7 0.7, 0.8, 0.9, 1.0Beam width/m 0.2, 0.25 0.2, 0.25, 0.3, 0.35 0.2, 0.3, 0.4, 0.5Slab thickness/m 0.08, 0.12, 0.16 0.08, 0.12, 0.16 0.12, 0.16Slab width/m 1.33, 2.67, 4.0 2.0, 4.0, 6.0 2.67, 5.34, 8.0

X. Lu et al. Engineering Structures 168 (2018) 721–735

727

of these beam-slab substructures are listed in Table 2. The 6-story and4-bay RC frame used by Ren et al. [7], Lu et al. [8] and Lin et al. [32] istaken as the prototype building. The dead load and live load on thefloor are 5.0 kN/m2 and 2.0 kN/m2, respectively. For each individualdesign, only one design parameter is varied according to the valuesgiven in Table 2. In total, 217 models (designated as model number1–217) with different span lengths, beam heights, beam widths, slabthicknesses and slab widths are derived based on the MOHURD 2010code [38] and analyzed using the validated FE model.

5.2. Analysis of typical cross sections

For all 217 beam-slab substructures, Sections 1 and 2 are chosen asthe typical cross sections as shown in Fig. 7b. According to the previousanalysis, at the point of peak CAA resistance: (1) the bottom beam re-inforcement at Section 1 yields under compression, while the depth ofthe compressive zone of concrete, the resultant forces of the slab re-inforcement and the top beam reinforcement are unknown; (2) thebottom beam reinforcement at Section 2 yields under tension, while the

depth of the concrete compressive zone, the resultant forces of the slabreinforcement and the top beam reinforcement are also unknown.Hence, to calculate the CAA resistance of the beam-slab substructures,the following parameters need to be specified:

(1) Parameters for Section 1:(a) The depth of the concrete compressive zone in the slab (i.e., s).

Note that s < 0 indicates that the concrete compressive zonelocates only at the beam web section.

(b) The tensile force of the bottom slab reinforcement (i.e., T'SSB).(c) The tensile forces of the top slab and the top beam reinforce-

ments (i.e., T'SST and T'SBT). For simplicity, it is assumed that thetop slab and the top beam reinforcements are at the sameheight. Assuming that the plane sections remain plane, thestrains of the top slab and the top beam reinforcement alsoremain the same.

(2) Parameters for Section 2:(a) The depth of the concrete compressive zone in the slab (i.e., c).

Note that c > hf indicates that part of the concrete compressive

Fig. 9. Stress levels of reinforcement at Sections 1 and 2 of the beam-slab FE models.

X. Lu et al. Engineering Structures 168 (2018) 721–735

728

zone locates at the beam web section.(b) The tensile force of the bottom slab reinforcement (i.e., TSSB).(c) The compressive forces of the top slab and the top beam re-

inforcements (i.e., CSST and CSBT). Assuming that the planesections remain plane, the strains of the top slab and the topbeam reinforcements also remain the same.

Let σs = κify denote the stress levels of the reinforcement at differentsections, where κi is the reinforcement stress coefficient at Section i andfy is the yield stress of the reinforcement. The stresses of the re-inforcement at Sections 1 and 2 of the 217 beam-slab FE models at thepeak resistance of CAA are shown in Fig. 9.

The FE results indicate that the bottom beam reinforcement yieldsunder compression at Section 1 when the beam-slab substructures reachtheir peak CAA resistance. The top slab reinforcement yields undertension, while the yielding status of the bottom slab reinforcement

Fig. 10. Approximation for stress level of the reinforcement.

Fig. 11. Comparison of the predicted δ by Eq. (26) and the FE results.

X. Lu et al. Engineering Structures 168 (2018) 721–735

729

depends on the ratio between the area of the upper slab and the un-derneath beam (i.e., bbhb/bfhf). The relation between σs/fy and bbhb/bfhfis shown in Fig. 10a (σs is the stress of the reinforcement), from whichthe following equations can be derived:

⎧⎨⎩

= ⩾= − <

κ b h b hκ b h b h b h b h

1.0, when / 0.453.13 / 0.41, when / 0.45

1SSB b b f f

1SSB b b f f b b f f (8)

Similarly, for Section 2, the following equations can be derived fromFig. 10b:

⎧⎨⎩

= − + ⩾= <

κ b h b h b h b hκ b h b h

1.25 / 1.75, when / 0.61.0, when / 0.6

2SSB b b f f b b f f

2SSB b b f f (9)

= =κ κ 0.62SST 2SBT (10)

In order to calculate the resultant moments and axial forces at thebeam ends, the depths of the concrete compressive zone (i.e., c and s)need to be specified. For the axially restrained beam-slab substructureswith fixed boundaries as shown in Fig. 7a, the following relation can bederived through deformation compatibility of the beams:

′ + = + + = −c c h s c h δ2b (11)

Eq. (11) can also be re-written as:

+ = − − = −s c h h δ h δ2 2b f (12)

According to the cross sectional equilibrium illustrated in Fig. 7b:

′ + ′ − ′ + ′ + ′ = + + − +C C T T T C C C T T( ) ( )C SBB SST SBT SSB C SST SBT SSB SBB

(13)

The reinforcement stress analyses (Fig. 9a) indicates that in mostcases the bottom slab reinforcement at Section 1 is under compression,which implies that the concrete at the beam web is also under com-pression. Therefore, according to Fig. 7 and ACI 318-14 [24], it can bederived that:

′− = ′ + ′ − ′C C f h b f β b s f β b c0.85 0.85C C c b b c 1 f c 1 f (14)

Substituting Eq. (14) into Eq. (12) yields:

′ + ′ − ′ =′ + ′ + ′ − ′ + + − −

f h b f β b s f β b cT T T C C C T T

0.85 0.85c b b c 1 f c 1 f

SST SBT SSB SBB SST SBT SSB SBB (15)

Based on the above, the resultant force of the reinforcement can bewritten as:For Section 1:

′ = ′ = ′ =T f A T f A C f A, ,SST yst st SBT ybt bt SBB ybb bb

′ = ⎧⎨⎩

⩾

− <T

f A b h b h

f A b h b h b h b h

when / 0.45

(3.13 / 0.41) when / 0.45SSBysb sb b b f f

ysb sb b b f f b b f f (16)

For Section 2:

= = =T f A C f A C f A, 0. 6 , 0. 6SBB ybb bb SST yst st SBT ybt bt

= ⎧⎨⎩

⩽

− + >T

f A b h b h

f A b h b h b h b h

when / 0.6

( 1.25 / 1.75) when / 0.6SSBysb sb b b f f

ysb sb b b f f b b f f (17)

where fy and A are the yield stress and the reinforcement area, re-spectively.Eq. (15) can also be re-written as:

− =′ + ′ + ′ − ′ + + − − − ′

′s c

T T T C C C T T f h bf β b0.85

SST SBT SSB SBB SST SBT SSB SBB c b b

c 1 f (18)

The heights of the concrete compression zone at different sections canthen be obtained by combining Eqs. (12) and (18):

= − +′ + ′ + ′ − ′ + + − − − ′

′s h δ T T T C C C T T f h b

f β b2 4 1.7f SST SBT SSB SBB SST SBT SSB SBB c b b

c 1 f

(19)

= − −′ + ′ + ′ − ′ + + − − − ′

′c h δ T T T C C C T T f h b

f β b2 4 1.7f SST SBT SSB SBB SST SBT SSB SBB c b b

c 1 f

(20)

Substituting s and c into the following equations gives the concreteresultant compressive forces at Sections 1 and 2:

′ = ′ + ′C f h b f β b s0.85C c b b c 1 f (21)

= ′C f β b c0.85C c 1 f (22)

As both concrete and reinforcement forces are specified, the resultantforces at Sections 1 and 2 (N, M1 and M2) can be calculated as:

= ′ + ′ − ′ + ′ + ′N C C T T T( )C SBB SST SBT SSB (23)

= − ′ − + + ′ +′ + ′ + ′ − ′ + ′ + ′−

M f β b s h h β s f h b hT T C h d T h d h

0.85 ( 0.5 0.5 ) 0.5( )(0.5 ) (0.5 )

1 c 1 f b 1 c b b f

SST SBT SBB SSB f (24)

= ′ − ++ + − ′ − + ′−

M f β b c h β cC C T h d T h d h0.85 (0.5 0.5 )

( )(0.5 ) (0.5 )2 c 1 f 1

SST SBT SBB SSB f (25)

Finally, the progressive collapse resistance of the beam-slab

Fig. 12. Validation of the proposed model beam-slab specimens.

X. Lu et al. Engineering Structures 168 (2018) 721–735

730

substructures at the CAA stage can be determined by substituting N, M1

and M2 into Eq. (1).

5.3. Predicting the peak displacement of CAA for beam-slab substructures

Similar to the calculation procedure for RC beams, the calculation ofbeam-slab substructures also requires the peak displacement (i.e., δ) ofCAA in Eq. (1) to be determined. Following a similar regression analysispresented in Figs. 5 and 6, the geometries of all 217 beam-slab modelsare used to regress δ and the following equation is obtained:

⎜ ⎟⎜ ⎟ ⎜ ⎟= × ⎛⎝

× ⎛⎝

⎞⎠

+ ⎞⎠

× ⎛⎝ +

⎞⎠

δ bb

lh h

0.000276 0.0023 0.98750.5

f

b

2

f b (26)

where hb, bb, bf and hf are defined as in Table 1, and l is the total span ofthe two-span beam. As shown in Fig. 11, a good agreement is observedbetween the predicted δ due to Eq. (26) and the FE results.

5.4. Validation of the accuracy of the proposed model

Based on the δ values predicted by Eq. (26), the calculation proce-dure presented in Section 5.2 (i.e., Eqs. (19)–(25)) can be followed tocalculate the CAA resistance of the beam-slab specimens tested by Renet al. [7]. Specimens S6 is taken as an example to illustrate the calcu-lation procedure of the proposed model, and the details of which areprovided in Appendix B. For all 217 beam-slab substructures designedin this study, the comparison between the proposed model and the FEresults is provided in Fig. 12a, where the MAE of prediction is 6.25%with a standard deviation of 0.064. For the five one-way beam-slabspecimens (S2 to S6) tested by Ren et al. [7], the calculation results ofthe proposed model are also compared with the FE and experimentalresults in Fig. 12b, where the MAE of prediction error is 9.46%. Moredetailed results, including the calculated peak resistances and the cor-responding boundary axial forces of the five beam-slab specimens, canalso be found in Table A.2 in Appendix A. These comparisons confirmthat the proposed explicit model is able to obtain accurately the CAAresistance of the beam-slab specimens.

6. Conclusion

Compressive arch action serves as a major mechanism to resistprogressive collapse in RC frames at the small deformation stage. Anaccurate and efficient model to calculate the CAA resistance has sig-nificant value to the progressive collapse design of RC structures.Existing prediction models can neither meet the engineering practicedemands, nor calculate the CAA resistance of the beam-slab specimens.

Based on an experimental database of 45 beam specimens and 5 beam-slab specimens, and the FE analyses of 45 beam specimens and 217beam-slab substructures, explicit and practical calculation models forCAA in RC beams with and without slabs are proposed in this study. Themain conclusions are:

(1) This study proposes a simple and explicit model capable of calcu-lating the CAA resistance of RC beams based solely on the geo-metric and reinforcement information. The equations to predict thepeak displacement of CAA is proposed based on the regression ofthe FE results of 45 specimens tested by other researchers. Moreimportantly, the proposed model can be easily mastered by prac-ticing engineers in designing RC structures against progressivecollapse.

(2) A set of 217 beam-slab substructures are designed within the mostpractically used ranges of span length, beam height, beam width,slab thickness and slab width. The sectional stresses of concrete andreinforcement of the beam-slab substructures under CAA are pro-posed based on the FE results of these beam-slab substructures. Theoutcome of this study can help researchers gain a better under-standing of the CAA in RC structures when considering the slabeffects. Moreover, these results provide a foundation for developingthe calculation model for CAA for the beam-slab substructures.

(3) The explicit CAA resistance calculation model, as well as the cor-responding peak displacement prediction method, for beam-slabsubstructures is proposed. Comparisons with both the FE and ex-perimental results confirm that the proposed explicit model is ableto accurately calculate the CAA resistance of RC beam-slab sub-structures.

It should be noted that the proposed model is developed based onthe assumptions of fixed boundary conditions at both beam ends. Itshould also be recognized that a flexible boundary can notably influ-ence the progressive collapse resistance of RC structures. Future studiesare therefore needed to investigate such boundary effects, based on theoutcomes of the present study. Correspondingly, the proposed δ pre-diction method and resistance calculation model should also be updatedin order to take into account different boundary conditions.

Acknowledgement

The authors are grateful for the financial support received from theNational Natural Science Foundation of China (No. 51778341), and theEuropean Community's Seventh Framework Programme (Marie CurieInternational Research Staff Exchange Scheme, Grant No. 612607).

Appendix A

A set of 50 tests considering a mid-column removal scenario are collected from published literature, of which 45 are beam specimens[5–7,10,13–17,25–26] and 5 are beam-slab specimens [7].

X. Lu et al. Engineering Structures 168 (2018) 721–735

731

TableA.1

Summaryof

beam

specim

ens[5–7

,10,13

–17,25

–26].

Referen

ceNo.

Specim

enBo

unda

rystiffne

ss/

kN/m

aBe

amsection

Totalleng

thl/

mm

Long

itud

inal

reinforcem

ent

Materialprop

erties

Cap

acityof

CAA

Ana

lyticalresult

Heigh

th/

mm

Width

b/mm

Top

Mid

Bottom

f c'(cylin

der)

/MPa

f y/M

PaDisp.

/mm

Force/kN

Axial

force/

kNPe

akresistan

ce/

kN

Chu

etal.[

6]1

B1Not

give

n17

085

4000

2Φ8+

1Φ6

/2Φ

835

.20

370(Φ

8),

460(Φ

6)42

.65

26.93

154.36

23.29

Ren

etal.[

7]2

B3Not

give

n20

085

4000

2Φ8+

1Φ6

/2Φ

835

.20

450(Φ

8),

430(Φ

6)33

.33

41.00

209.88

38.31

Suet

al.[

14]

3A1

1×

106

300

150

2700

2Φ12

/2Φ

1225

.84

350

48.00

168.00

353.45

171.41

4A2

1×

106

300

150

2700

3Φ12

/3Φ

1228

.24

350

56.40

221.00

380.39

213.97

5A3

1×

106

300

150

2700

3Φ14

/3Φ

1431

.20

340

76.40

246.00

408.91

256.85

6A4

1×

106

300

150

2700

2Φ12

/1Φ

1423

.04

350(Φ

12),

340(Φ

14)

65.00

147.00

320.77

148.99

7A5

1×

106

300

150

2700

3Φ12

/2Φ

1226

.48

350

70.70

198.00

360.73

189.34

8A6

1×

106

300

150

2700

3Φ14

/2Φ

1428

.64

340

69.20

226.00

384.78

224.50

9B1

1×

106

300

150

4200

3Φ14

/3Φ

1418

.56

340

100.00

125.00

267.55

125.36

10B2

1×

106

300

150

5700

3Φ14

/3Φ

1419

.28

340

102.00

82.90

270.14

86.88

11B3

1×

106

300

150

5700

3Φ14

/2Φ

1421

.12

340

85.50

74.70

294.36

78.69

12C1

1×

106

200

100

2700

2Φ12

/2Φ

1215

.92

350

33.70

60.90

105.54

59.24

13C2

1×

106

200

100

2700

2Φ12

/2Φ

1216

.80

350

33.50

64.90

111.12

60.22

14C3

1×

106

200

100

2700

2Φ12

/2Φ

1216

.32

350

28.70

68.60

108.08

59.69

Che

net

al.[

26]

15KLJ

-31.37

5×

106

290

145

2700

2Φ12

/1Φ

1425

.20

354(Φ

12),

341(Φ

14)

55.10

189.00

339.47

164.16

16KLJ

-61.37

5×

106

305

160

2700

3Φ12

/3Φ

1424

.88

354(Φ

12),

341(Φ

14)

67.10

235.00

383.92

225.65

17KLJ

-91.37

5×

106

294

144

2700

3Φ14

/2Φ

2028

.80

341(Φ

14),

340(Φ

20)

51.74

259.00

384.76

248.65

Niu

etal.[

25]

18FA

1Not

give

n30

015

057

002Φ

14/

2Φ14

15.85

395

75.54

55.73

234.38

66.87

19FB

1Not

give

n30

015

057

003Φ

14/

2Φ14

12.57

395

79.08

64.04

185.87

71.57

20FB

2Not

give

n30

015

057

003Φ

14/

3Φ14

15.95

395

110.38

75.60

235.92

89.17

21FC

1Not

give

n30

015

057

003Φ

16/

2Φ16

13.74

333

59.51

84.93

203.14

78.15

22FC

2Not

give

n30

015

057

003Φ

16/

3Φ16

13.73

333

104.02

104.45

203.03

92.14

Yu&

Tan[13]

23S1

1.06

×10

525

015

057

502Φ

10+

1Φ13

/2Φ

1031

.20

511

78.00

41.64

89.87

42.67

24S2

1.06

×10

525

015

057

503Φ

10/

2Φ10

31.20

511

73.00

38.38

41.15

33.65

25S3

4.29

×10

525

015

057

503Φ

13/

2Φ10

38.20

511(Φ

10),

494(Φ

13)

74.40

54.47

259.84

62.32

26S4

4.29

×10

525

015

057

503Φ

13/

2Φ13

38.20

494

81.00

63.22

275.87

71.86

27S5

4.29

×10

525

015

057

503Φ

13/

3Φ13

38.20

494

74.50

70.33

275.87

81.89

28S6

4.29

×10

525

015

057

503Φ

16/

2Φ13

38.20

494(Φ

13),

513(Φ

16)

114.40

70.33

275.87

83.90

29S7

4.29

×10

525

015

045

503Φ

13/

2Φ13

38.20

494

74.40

82.82

218.74

92.57

30S8

4.29

×10

525

015

033

503Φ

13/

2Φ13

38.20

494

45.90

121.34

210.95

132.35

Yu&

Tan[37]

31F1

-CD

4.29

×10

525

015

057

503Φ

13/

2Φ13

27.54

488

87.00

51.10

276.91

66.68

32F2

-MR

4.29

×10

525

015

057

503Φ

13/

2Φ13

27.54

488

51.00

62.84

276.91

66.68

Sasani

etal.[

15]

33P1

Not

give

n19

019

041

705Φ

9.5

/2Φ

9.5

41.00

516

92.43

71.79

426.83

68.69

Qianet

al.[

10]

34P1

Not

give

n18

010

040

002Φ

10/

2Φ10

19.90

437

35.75

31.60

113.42

30.72

35P2

Not

give

n14

080

2800

2Φ10

/2Φ

1020

.80

437

32.85

35.50

75.74

31.31

(con

tinuedon

next

page)

X. Lu et al. Engineering Structures 168 (2018) 721–735

732

TableA.1

(con

tinued)

Referen

ceNo.

Specim

enBo

unda

rystiffne

ss/

kN/m

aBe

amsection

Totalleng

thl/

mm

Long

itud

inal

reinforcem

ent

Materialprop

erties

Cap

acityof

CAA

Ana

lyticalresult

Heigh

th/

mm

Width

b/mm

Top

Mid

Bottom

f c'(cylin

der)

/MPa

f y/M

PaDisp.

/mm

Force/kN

Axial

force/

kNPe

akresistan

ce/

kN

Farhan

gVesaliet

al.

[16]

361

Not

give

n18

018

044

002Φ

10/

2Φ10

30.50

620

49.00

40.50

302.38

39.39

372

Not

give

n18

018

044

002Φ

10/

2Φ10

27.00

620

44.00

35.70

268.81

35.89

383

Not

give

n18

018

044

002Φ

10/

2Φ10

30.00

620

50.00

41.40

298.68

39.10

394

Not

give

n18

018

044

003Φ

10/

3Φ10

26.00

620

54.00

40.10

258.86

41.67

405

Not

give

n18

018

044

003Φ

10/

3Φ10

29.50

620

54.00

41.60

293.70

41.91

416

Not

give

n18

018

044

003Φ

10/

3Φ10

30.00

620

52.00

39.40

298.68

42.21

Aloglaet

al.[

17]

42SS

11×

105

250

150

5750

3Φ10

/2Φ

1019

.35

510

101.00

34.00

128.28

39.50

43SS

21×

105

250

150

5750

3Φ10

2Φ10

2Φ10

19.35

510

96.80

37.90

128.28

39.50

44SS

31×

105

250

150

5750

3Φ10

2Φ10

2Φ10

19.86

510

86.80

37.20

128.97

45.68

45SS

41×

105

250

150

5750

3Φ10

2Φ10

2Φ10

19.86

510

91.40

36.70

128.97

43.83

aTh

ebo

unda

rystiffne

ssis

used

tocalculatethebo

unda

rymov

emen

tt,which

isrequ

ired

inEq

s.(2)an

d(3)in

thePa

rk&

Gam

ble’smod

el.

TableA.2

Summaryof

beam

-slabspecim

ens[7].

Referen

ceSp

ecim

enBe

amsection

Beam

reinforcem

ent

Slab

section

Slab

reinforcem

ent

Totalleng

thl/mm

Materialprop

erties

Cap

acityof

CAA

Ana

lyticalresult

h/mm

ab b/m

mTo

pBo

ttom

h f/m

mb f/m

mLo

ngitud

inal

andtran

sverse

f c'(cylin

der)/M

Paf y/M

PaDisp./m

mFo

rce/kN

Axial

force/kN

Peak

resistan

ce/k

N

Top

Bottom

Ren

etal.[

7]S6

170

852Φ

8+

1Φ6

2Φ8

5020

00Φ6@

190

Φ6@

190

4000

35.20

390(Φ8),

387(Φ6),

370(Φ

10)

42.00

47.33

741.70

44.56

S217

085

2Φ8+

1Φ6

2Φ8

5068

5Φ6@

190

Φ6@

190

4000

35.20

40.00

47.29

326.84

40.17

S320

085

2Φ8+

1Φ6

2Φ8

5020

00Φ6@

190

Φ6@

190

4000

35.20

38.00

75.49

784.63

64.57

S417

085

3Φ10

2Φ10

5020

00Φ6@

190

Φ6@

190

4000

35.20

37.00

47.17

728.01

47.09

S517

085

2Φ8+

1Φ6

2Φ8

7520

00Φ6@

160

Φ6@

160

4000

35.20

32.00

56.72

1297

.90

63.39

aDefi

nition

sof

h,b b,h

fan

db f

aregive

nin

Fig.

7an

dTa

ble1.

X. Lu et al. Engineering Structures 168 (2018) 721–735

733

Appendix B

Detailed CAA resistance calculation procedure for the beam-slab specimen (Specimen S6 of Ren et al. [7]).

References

[1] Sozen MA, Thornton CH, Corley WG, Mlakar PF. The Oklahoma City bombing:structure and mechanisms of the Murrah building. J Perform Constr Facil1998;12(3):120–36.

[2] Li Y, Lu XZ, Guan H, Ye LP. An improved tie force method for progressive collapseresistance design of reinforced concrete frame structures. Eng Struct2011;33(10):2931–42.

[3] Li Y, Lu XZ, Guan H, Ye LP. An energy-based assessment on dynamic amplificationfactor for linear static analysis in progressive collapse design of ductile RC framestructures. Adv Struct Eng 2014;17(8):1217–25.

[4] Li Y, Lu XZ, Guan H, Ye LP. Progressive collapse resistance demand of RC framesunder catenary mechanism. ACI Struct J 2014;111(5):1225–34.

[5] Yu J, Tan KH. Structural behavior of RC beam-column sub-assemblages under amiddle column removal scenario. J Struct Eng 2012;139(2):233–50.

[6] Chu MJ, Zhou YL, Lu XZ, Li Y. An experimental study on one-way reinforced con-crete beam-slab substructures for resisting progressive collapse. China Civil Eng J2016;49(2):31–40. in Chinese.

[7] Ren PQ, Li Y, Lu XZ, Guan H, Zhou YL. Experimental investigation of progressive

collapse resistance of one-way reinforced concrete beam-slab substructures under amiddle-column-removal scenario. Eng Struct 2016;118:28–40.

[8] Lu XZ, Lin KQ, Li Y, Guan H, Ren PQ, Zhou YL. Experimental investigation of RCbeam-slab substructures against progressive collapse subject to an edge-column-removal scenario. Eng Struct 2017;149:91–103.

[9] Yi WJ, He QF, Xiao Y, Kunnath SK. Experimental study on progressive collapse-resistant behavior of reinforced concrete frame structures. ACI Struct J2008;105(4):433–9.

[10] Qian K, Li B, Ma JX. Load-carrying mechanism to resist progressive collapse of RCbuildings. J Struct Eng 2015;141(2):04014107.

[11] General Services Administration (GSA). Alternate path analysis and design guide-lines for progressive collapse resistance. Washington, DC; 2016.

[12] Department of Defense (DoD). Design of structures to resist progressive collapse.Unified facility criteria, UFC 4-023-03, Washington, DC; 2016.

[13] Yu J, Tan KH. Special detailing techniques to improve structural resistance againstprogressive collapse. J Struct Eng 2013;140(3):04013077.

[14] Su YP, Tian Y, Song X. Progressive collapse resistance of axially-restrained framebeams. ACI Struct J 2009;106(5):600–7.

[15] Sasani M, Werner A, Kazemi A. Bar fracture modeling in progressive collapseanalysis of reinforced concrete structures. Eng Struct 2011;33(2):401–9.

X. Lu et al. Engineering Structures 168 (2018) 721–735

734

[16] FarhangVesali N, Valipour H, Samali B, Foster S. Development of arching action inlongitudinally-restrained reinforced concrete beams. Constr Build Mater2013;47:7–19.

[17] Alogla K, Weekes L, Augusthus-Nelson L. A new mitigation scheme to resist pro-gressive collapse of RC structures. Constr Build Mater 2016;125:533–45.

[18] Choi H, Kim J. Progressive collapse-resisting capacity of RC beam-column sub-as-semblage. Mag Concr Res 2011;63(4):297–310.

[19] Sadek F, Main JA, Lew HS, Bao Y. Testing and analysis of steel and concrete beam-column assemblies under a column removal scenario. J Struct Eng2011;137(9):881–92.

[20] Rankin GIB, Long AE. Arching action strength enhancement in laterally-restrainedslab strips. Proc Inst Civil Eng-Struct Build 1997;122(4):461–7.

[21] Park R, Gamble WL. Reinforced concrete slabs. John Wiley & Sons; 2000.[22] Yu J, Tan KH. Analytical model for the capacity of compressive arch action of re-

inforced concrete sub-assemblages. Mag Concr Res 2014;66(3):109–26.[23] Kang SB, Tan KH. Analytical study on reinforced concrete frames subject to com-

pressive arch action. Eng Struct 2017;141:373–85.[24] American Concrete Institute (ACI). Building code requirements for structural con-

crete (ACI 318-14) and commentary (318R-14). Detroit, MI; 2014.[25] Niu JX, Su YP, Wang HX. Tentative studies on collapse resistance capacity of re-

inforced concrete frame beam. In: Proceeding of the 20th national conference onstructural engineering. Beijing: Engineering Mechanics Press; 2011. p. 377–82 [inChinese].

[26] Chen MH, Song XS, Su YP. Influence of reinforcement ratio on ultimate bearingcapacity of rein-forced concrete frame beams: test study on arch action. J NatDisasters 2010;1:44–8. in Chinese.

[27] Qian K, Li B. Slab effects on response of reinforced concrete substructures after loss

of corner column. ACI Struct J 2012;109(6):845–55.[28] Prasad S, Hutchinson TC. Evaluation of older reinforced concrete floor slabs under

corner support failure. ACI Struct J 2014;111(4):839–49.[29] Dat PX, Tan KH. Experimental study of beam–slab substructures subjected to a

penultimate-internal column loss. Eng Struct 2013;55:2–15.[30] Mazzoni S, McKenna F, Scott MH, Fenves GL. The open system for earthquake

engineering simulation (OpenSees). User command-language manual. Berkeley:Pacific Earthquake Engineering Research Center. University of California; 2006.

[31] Ren PQ, Li Y, Guan H, Lu XZ. Progressive collapse resistance of two typical high-riseRC frame shear wall structures. J Perform Constr Facil 2015;29(3):04014087.

[32] Lin KQ, Li Y, Lu XZ, Guan H. Effects of seismic and progressive collapse designs onthe vulnerability of RC frame structures. J Perform Constr Facil2017;31(1):04016079.

[33] Bao Y, Kunnath SK, El-Tawil S, Lew HS. Macromodel-based simulation of pro-gressive collapse: RC frame structures. J Struct Eng 2008;134(7):1079–91.

[34] Talaat M, Mosalam KM. Modeling progressive collapse in reinforced concretebuildings using direct element removal. Earthq Eng Struct Dyn 2009;38(5):609–34.

[35] Kazemi-Moghaddam A, Sasani M. Progressive collapse evaluation of MurrahFederal Building following sudden loss of column G20. Eng Struct 2015;89:162–71.

[36] Pan WH, Tao MX, Nie JG. Fiber beam-column element model considering re-inforcement anchorage slip in the footing. B Earthq Eng 2017;15(3):991–1018.

[37] Yu J, Tan KH. Experimental and numerical investigation on progressive collapseresistance of reinforced concrete beam column sub-assemblages. Eng Struct2013;55:90–106.

[38] Ministry of Housing and Urban-Rural Development of the People’s Republic ofChina (MOHURD). Code for design of concrete structures. GB50010-2010, Beijing,China; 2010 [in Chinese].

X. Lu et al. Engineering Structures 168 (2018) 721–735

735