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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
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
(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).
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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).
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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.
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Fig. 4. Validation of the FE models.
Fig. 5. Parametric study of the peak CAA displacement.
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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.
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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.
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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.
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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.
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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]).
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