The potential for progressive collapse of RC buildings can be estimated by column loss scenarios. The loss of either internal or external penultimate columns is among the most critical scenarios since the beam-slab substructures associated with the removed column becomes laterally unrestrained with two discontinuous edges. At large deformations, membrane behaviour of the associated slabs, consisting of a compressive ring of concrete around its perimeter and tensile membrane action in the central region, represents an important line of defence against progressive collapse.
Journal of Science and Technology in Civil Engineering NUCE 2018 12 (3): 10–22 ANALYTICAL MODEL FOR PREDICTING MEMBRANE ACTIONS IN RC BEAM-SLAB STRUCTURES SUBJECTED TO PENULTIMATE-INTERNAL COLUMN LOSS SCENARIOS Pham Xuan Data,∗, Trieska Yokhebed Wahyudib , Do Kim Anha a Faculty of Building and Industrial Construction, National University of Civil Engineering, 55 Giai Phong road, Hai Ba Trung district, Hanoi, Vietnam b Nanyang Technological University, Nanyang Avenue, 639798 Singapore Article history: Received 15 March 2018, Revised 28 March 2018, Accepted 27 April 2018 Abstract The potential for progressive collapse of RC buildings can be estimated by column loss scenarios The loss of either internal or external penultimate columns is among the most critical scenarios since the beam-slab substructures associated with the removed column becomes laterally unrestrained with two discontinuous edges At large deformations, membrane behaviour of the associated slabs, consisting of a compressive ring of concrete around its perimeter and tensile membrane action in the central region, represents an important line of defence against progressive collapse The reserve capacity can be used to sustain amplified gravity loads and to mitigate the progressive collapse of building structures In this paper, based on experimental observation of 1/4 scaled tests together with investigation of previous research works, an analytical model is proposed to predict the load-carrying capacity of beam-slab structures at large deformations Comparison with the test results shows that the analytical model gives a good estimation of the overall load-carrying capacity of the RC slabs by membrane actions Keywords: Membrane actions; compressive ring; penultimate columns; load-carrying capacity; laterally unrestrained slab c 2018 National University of Civil Engineering Introduction It has been experimentally observed that the ultimate load of laterally unrestrained two-way reinforced concrete slabs is significantly higher compared with the capacity calculated by yield-line analysis [1–8] The increase in the ultimate load is referred to as the contribution of membrane action which develops in slabs at large deformations Membrane action in a laterally-unrestrained slab can be explained in Fig After the formation of yield lines, the slab is divided into four independent parts which are connected together by the yield lines At large deformations the independent parts tend to move inwards under the action of increasing tensile forces at the centre of the slab, but are restrained from doing so by adjacent parts, creating a peripheral ring of compression supporting the ∗ Corresponding author E-mail address: phamxdatcdc@gmail.com (Dat, P X) 10 Dat, P X et al / Journal of Science and Technology in Civil Engineering central net of tensile forces The load-carrying capacity therefore comprises catenary action in the central region of the slabs and enhanced yield moment in the outer ring where in-plane compressive stresses occur The behaviour of laterally-unrestrained slabs Tension across Compression at large deformations has been extensively studied yield lines across yield lines by [3, 8–10] It has been shown that the overall load-carrying capacity of membrane actions was at least twice the yield-line capacity Recently, these mechanisms have been successfully applied to prevent the collapse of composite floors subjected to compartment fires in Europe through a Dat, P X./ Journal of Science and Technology in Civil Engineering Dat, P X./ Journal of Science and Technology in Civil Engineering simplified design method developed by [2] Nevertheless, most experimental and analytiNevertheless, most experimental and analytical works introduced so far Nevertheless, most experimental and analytical works introduced so far cal works introduced so far areare stillstill limited in aplimited in application, especially in terms of means to resist are still limited in application, especially in terms of means to resist progressive plication, especially in terms of means tocollapse resist The potential for progressive collapse of building progressive collapse The potential for progressive collapse of building structures can be estimated by column loss scenarios The loss of a progressive collapse The potential for progressive structures can be estimated by column loss scenarios The loss of a penultimate-internal is among the most critical scenarios since collapse of building structures can bepenultimate-internal estimated by column column is among the most critical scenarios since the beam-slab structures associated with the lost column become laterally beam-slab structures associated with the lost column become laterally column loss scenarios The lossunrestrained of a the penultimatewith two discontinuous slab edges As soon as the flexural unrestrained with two discontinuous slab edges As soon as the flexural internal column is among the most critical scenaraction in beams fails to carry Yield gravitylines loads which are Tension amplified zone by both action in beams fails to carry gravity loads which are amplified by both ios since the beam-slab structures associated with doubling-of-span and dynamic effects [11], the survival of the building doubling-of-span and dynamic effects [11],actions the survival of the building Figure1 Membrane in a laterally totally depends on the strength of membrane actions developed the lost column become laterallystructures unrestrained Figure Membrane actions in aactions laterally structureswith totally depends on the1.strength of membrane developed in the affected slabs as indicated in Fig 2(a) At floors above the first unrestrained two-way slabs [1] two discontinuous slab edges As soon flex- slabs asunrestrained slabs in as thethe affected indicated intwo Fig.way 2(a) At[1] floors above the first floor, gravity if the stiffness and strength of their compressive rings are ural action in beams fails to carry floor, ifloads the stiffness and strength of their compressive rings are insufficient to support catenary actionaction in thein deflected central area, area, insufficientand to dynamic support catenary the deflected central which are amplified by both doubling-of-span effects [11], the survival of the building tension forces from catenary action may pull in the perimeter ground tension of forces from catenary action may pull in affected the perimeter ground structures totally depends on the strength membrane actions developed in in the slabs as columns, leading to progressive collapse shown Fig 2(b) In this columns, leading to progressive collapse shown in Fig 2(b) In this indicated in Fig 2(a) At floors above the first floor, if the stiffness and strength of their comprespaper,paper, an experimental programme and anand analytical modelmodel to study the the an experimental programme an analytical to study behaviour of membrane actions in RC beam-slab systems will be be sive rings are insufficient to support catenary action in the deflected central area, tension forces from behaviour of membrane actions in RC beam-slab systems will discussed In the first part, the results of two ¼ scaled specimens which FigureFigure Membrane actions in a catenary action may pull in the perimeter ground columns, leading to progressive collapse shown discussed In the first part, the results of two ¼ scaled specimens in which Membrane actions in a were constructed and and tested under column loss to scenario are behaviour presented.In laterally unrestrained two-way Fig 2(b) In this paper, an experimental programme analytical model study the of were constructed andan tested under column loss scenario are presented.In laterally unrestrained two-way the second part, apart, simplified method to predict the overall load-carrying the second a simplified method to predict the overall load-carrying capacity of beam-slab systems is discussed capacity of beam-slab systems is discussed slabs [1] slabs [1] Figure Detail of a typical speci (a)a)Possible prevention byby mema) Possible prevention by membrane actions Possible prevention membrane actions (b) Possible modefailure b) Possible failure modemode b)failure Possible brane actions FigureFigure Collapse of building structures underunder a Penultimate-Internal column loss [7] Collapse of building structures a Penultimate-Internal column loss [7] Figure Collapse of building structures under a Penultimate-Internal column loss [7] Experimental programme Experimental programme 2.1 Design 2.1 Design 11 Two specimens have been built and to investigate the tensile membrane actionaction of RC Two specimens have designed, been designed, built tested and tested to investigate the tensile membrane of RC building structures under under a Penultimate-Internal (PI) column loss scenario The dimensions of theoftest are are building structures a Penultimate-Internal (PI) column loss scenario The dimensions thespecimens test specimens obtained by scaling down down to ¼ dimensions of a prototype building designed for gravity loading The design live load obtained by scaling to ¼ dimensions of a prototype building designed for gravity loading The design live load 2 2 detail of the test specimens can be summarized in Fig as is kN/m and the imposed dead load is kN/m The is kN/m and the imposed dead load is kN/m The detail of the test specimens can be summarized in Fig as Dat, P X et al / Journal of Science and Technology in Civil Engineering membrane actions in RC beam-slab systems will be discussed In the first part, the results of two 1/4 scaled specimens which were constructed and tested under column loss scenario are presented.In the second part, a simplified method to predict the overall load-carrying capacity of beam-slab systems is discussed Yield lines Tension zone Experimental programme Figure1 Membrane actions in a laterally 2.1 Design unrestrained two way slabs [1] Two specimens have been designed, built and tested to investigate the tensile membrane action of RC building structures under a Penultimate-Internal (PI) column loss scenario The dimensions of the test specimens are obtained by scaling down to 1/4 dimensions of a prototype building designed for gravity loading The design live load is kN/m2 and the imposed dead load is kN/m2 The detail of the test specimens can be summarized in Fig as well as Table Figure specimen [12] Figure3.3.Detail Detailof of a typical typical specimen [12] Table Summary on test specimens [12] Overall dimension (Aspect ratio) Top slab reinforcement Bottom slab reinforcement along X-direction Bottom slab reinforcement along Y-direction Notes PI-02 3000 × 4200 (a = 1.4) Φ3 at 50 (ρ = 0.44%) Φ3 at 100 (ρ = 0.22%) Φ3 at 100 (ρ = 0.22%) Isotropically reinforced PI-04 3000 × 3000 (a = 1.0) Φ3 at 100 (ρ = 0.22%) Φ3 at 100 (ρ = 0.22%) Φ3 at 50 (ρ = 0.44%) Orthotropically reinforced 2.2 Material properties Since the test specimens are scaled down by 1/4 from the prototype building, the diameter of reinforcing bars is also scaled down by a certain factor so that the reinforcement ratios in beams, 12 Dat, P X et al / Journal of Science and Technology in Civil Engineering slabs and columns of the specimens can be kept approximately the same as those of the prototype structure The plain round mild steel bar of mm in diameter, R3, is used for slab reinforcement The beams of the sub-assemblages are reinforced with R6, and the columns with 10 mm deformed bar (T10) In both beams and columns, R3 is used as transverse reinforcement The nominal yield strength of round bars and deformed bars is 320 N/mm2 and 460 N/mm2 , respectively The concrete used in the test specimen was a small-aggregate mix with a characteristic design strength of 30 MPa Due to the small thickness of slab (40 mm), chippings of mm are used instead of normal-size aggregate to prevent any congestion and honey combs due to inadequate compaction The concrete compressive test results are shown in Table Table Concrete compression test results [12] Sample No Weight (g) Max Load (kN) Cylinder Strength (MPa) Average 11393 11505 11407 11414 11820 11531 11512 499.9 615.9 478.8 436.2 621.4 482.5 528.9 28.3 34.9 27.1 24.7 35.2 27.3 29.6 2.3 Boundary condition Under penultimate column loss condition, the affected beam-slab substructures behave as laterally unrestrained due to two consecutive discontinuous edges Along the perimeter beams, however, the beam-column joints are rotationally restrained by the perimeter columns Therefore, a set of perimeter columns with one end pinned is designed to reasonably simulate the laterally yet rotationally restrained boundary condition As shown in Fig 4, the pin-ended columns allow the perimeter edges of specimens to move horizontally without any degree of restraint The lateral reaction at the pin connection may provide perimeter beam-column joints with sufficient rotational restraint 2.4 Loading method With a special emphasis on a uniformly distributed load applied onto the beam-slab substructures under column loss condition, a loading scheme is designed based on existing laboratory constraints to reasonably simulate the applied loads in a uniform manner A 200-ton actuator held by a reaction steel frame across the specimen is used to load the specimens to failure The load from the actuator is distributed equally to twelve point loads (Fig 5(a)) by means of loading trees (Fig 5(b)) Ball and socket joints between steel plates and steel rods are used to keep the loading system as vertical as possible when the test specimens deform excessively Finite element analysis (FEA) is employed to investigate the accuracy of the loading method The very small discrepancies of numerical predictions between the two cases indicate the reliability of the loading method Fig 6(a) shows the numerical models of square specimens with a plan dimension of m × m subjected to either uniformly distributed load of kN/m2 or 12 point loads of 0.75 kN The very small discrepancies of numerical results (bending moment diagram shown in Fig 6(b)) between 13 Dat, P X./ Journal of Science and Technology in Civil Engineering joints are rotationally restrained by the perimeter columns Therefore, a set of perimeter columns with one end pinned is designed to reasonably simulate the laterally yet rotationally restrained boundary condition.As shown in Fig.4, the pin-ended columns allow the perimeter edges of specimens to move horizontally without any degree of restraint The lateral reaction at etthe connection mayandprovide perimeter Dat, P X al pin / Journal of Science Technology in Civilbeam-column Engineering joints with sufficient rotational restraint free to move horizontally Figure Figure Supports and boundary conditioncondition [7] Supports and boundary [7] Loading 2.4.method Loading method With a special on a uniformly distributed load applied the onto beam-slab substructures under under With aemphasis special emphasis on a uniformly distributed load onto applied the beam-slab substructures olumn loss condition, a loading ascheme designed based on based existing constraints to reasonably simulatesimulate column loss condition, loadingisscheme is designed on laboratory existing laboratory constraints to reasonably e appliedthe loads in a loads uniform 200-tonAactuator by aheld reaction frame across specimen is used is used applied in amanner uniformAmanner 200-tonheld actuator by a steel reaction steel frametheacross the specimen load thetospecimens to failure.toThe load The fromload the from actuator distributed equally toequally twelvetopoint loads (Fig 5a)(Fig by 5a) by load the specimens failure the isactuator is distributed twelve point loads eans of loading (Fig.5b) and socket joints between plates andplates steel and rodssteel are used to keep loading means oftrees loading trees Ball (Fig.5b) Ball and socket joints steel between steel rods are usedthe to keep the loading stem as system verticalasasvertical possibleaswhen the when test specimens deform excessively possible the test specimens deform excessively Finite element is employed to investigate the accuracy of the loading The veryThe small Finiteanalysis element(FEA) analysis (FEA) is employed to investigate the accuracy of themethod loading method very small screpancies of numerical predictions between the two cases indicate reliability of the loading Fig 6(a)Fig 6(a) discrepancies of numerical predictions between the two cases the indicate the reliability of the method loading method ows theshows numerical models ofmodels square of specimens with a plan dimension of 3m x of 3m3m subjected to either to uniformly the numerical square specimens with a plan dimension x 3m subjected either uniformly 2 stributeddistributed load of kN/m or 12 point of 0.75 kN The very small discrepancies of numerical results (bending load of kN/m or loads 12 point loads of 0.75 kN The very small discrepancies of numerical results (bending Figure Supports and boundary condition [7] Figure Supports andcases boundary condition [7]3 and oment diagram in shown Fig 6(b)) between the two cases shown inshown Table and Figs indicated the reliability of momentshown diagram in Fig 6(b)) between the two in Table Figs indicated the reliability of 2.4 Loading method loading method Slightly better was accuracy was for obtained for the rectangular specimens e loadingthe method Slightly better accuracy obtained the rectangular specimens With a special emphasis on a uniformly distributed load applied onto the beam-slab substructures under column loss condition, a loading scheme is designed based on existing laboratory constraints to reasonably simulate the applied loads in a uniform manner A 200-ton actuator held by a reaction steel frame across the specimen is used to load the specimens to failure The load from the actuator is distributed equally to twelve point loads (Fig 5a) by means of loading trees (Fig.5b) Ball and socket joints between steel plates and steel rods are used to keep the loading system as vertical as possible when the test specimens deform excessively Finite element analysis (FEA) is employed to investigate the accuracy of the loading method The very small discrepancies of numerical predictions between the two cases indicate the reliability of the loading method Fig 6(a) shows the numerical models of square specimens with a plan dimension of 3m x 3m subjected to either uniformly distributed load of kN/m2 or 12 point loads of 0.75 kN The very small discrepancies of numerical results (bending moment diagram shown in Fig 6(b)) between the two cases shown in Table and Figs indicated the reliability of the loading method Slightly better accuracy was obtained for the rectangular specimens 12 loading loading positions Loading tree system (a)(a) Locations ofof12 positions (b) Loading tree system (a) Locations ofLocations 12 loading positions (b)(b)Loading tree system system Loading forseries PI series specimens Figure Figure Figure Loading forsystem PIfor series specimens [7] [7] [7] Loading system PI specimens the two cases shown in Table and Figs indicated the reliability of the loading method Slightly better accuracy was obtained for the rectangular specimens 2.5 Instrumentation The specimens are installed or mounted with measuring devices bothtree internally (a)test Locations of 12 loading positions (b) Loading system and externally (Fig 7) The concentrated loads by the actuator are measured by using an in-built load cell which Loading system for PI series specimens [7] is connected in series with Figure the actuator Vertical reaction forces and moments in eight supporting columns can be calculated through four strain gauges (SG-1,2,3,4) mounted on the opposing external surfaces of the columns as shown in Fig 7(a) At section where strain gauges are mounted, the axial 14 Dat, P X./ Journal of Science and Technology in Civil Engineering Dat, P Dat, X etP.al.X./ / Journal JournalofofScience Scienceand andTechnology TechnologyininCivil CivilEngineering Engineering (a) A uniform load(a) ofA1 uniform kN/m2 Numerical model of specimens load of kN/m (a) A uniform load of kN/m (b)(b) Equivalent setset of of 1212 point loads of of 0.75 kNkN Equivalent loads 0.75 (b) Equivalent set ofpoint 12 point loads of 0.75 kN Numerical model of specimens moment diagram in the two loading cases [7] NumericalFigure model6.ofBending specimens Figure Bending moment diagram in the two loading cases [7] Figure Figure Bending moment diagram the in the two loading cases [7] Table 3.Table Comparison of theofnumerical results between two loading Comparison the numerical results between the two loadingcases cases [7] [7] Table Comparison of the numerical results [7] between the two loading cases [7] Supports and boundary condition Uniform load load Uniform Figure Supports and boundary condition Uniform load[7] Central displacement Central displacement Axial force in edge columns Central displacement AxialAxial forceforce in corner in edge column columns Axial force in edge columns Axial force in corner column Axial force in corner column 2.5 Instrumentation 5.37 mm 5.37 mm 18.0 kN 5.37 mm 4.5 18.0kN kN 18.0 kN 4.5 kN 4.5 kN 12 loads 12point point loads 12 point loads 5.66 mm 18.2 kN 5.66 mm 4.2kN kN 18.2 5.66 mm 18.2 kN 4.2 kN 4.2 kN ErrorError Error 5.4 % 1.1% 5.4 % 6.7% 1.1% 5.4 % 6.7% 1.1% 6.7% The test specimens are installed or mounted with measuring devices both internally and externally (Fig 7) 2.5 The Instrumentation concentrated loads by the actuator are measured by using an in-built load cell which is connected in series with the actuator Vertical reaction forces and moments in eight supporting columns can be calculated through four strain The test specimens are installed or mounted with measuring devices both internally and externally (Fig 7) gauges (SG-1,2,3,4) mounted on the opposing external surfaces of the columns as shown in Fig 7(b) At section The where concentrated loadsare bymounted, the actuator are forces measured bymoments using anMin-built load cell which is connected in series with strain gauges the axial N1 and 1indicated in Fig 7(b) can be evaluated by steel the actuator Vertical reaction forces and moments in eight supporting columns can be calculated through four strain strains and cross-sectional properties as follows: gauges (SG-1,2,3,4) mounted on the opposing external surfaces of the columns as shown in Fig 7(b) At section N1 =NEsteel *As*(ε1+ ε2+ ε3+ ε4)/4 where strain gauges are mounted, the axial forces and moments M1indicated in Fig 7(b) can be evaluated by steel strains and cross-sectional properties as M1follows: = (Esteel*I* (ε3- εave.))/Rεave = (ε1+ ε2+ ε3+ ε4)/4 where Esteel, I, As, and R are elastic modulus N of1 = steel, area, Esteelmoment *As*(ε1of + εinertia, + ε3 + ε 4)/4 and radius of the hollow section, respectivelyε , ε , ε , ε , ε are the values recorded by SG-1,2,3,4 and average value of SGs-1,2,3,4.The ave (a) Arrangement of LVDTs Evaluation of reaction forces by strain gauges reactions and M1can = (E *I* (ε(b) εave ))/Rε ε3+ εby a)total Arrangement LVDTs b)steel Evaluation of reaction strain gauges 3-based avediagrams = (ε1+ εforces 2+ 4)/4in moment of beam-column joint be evaluated onof the illustrated 7(b) a) the Arrangement ofofLVDTs b) Evaluation reaction forces byFig.strain gauges where EsteelVertical , I, As, deformations and R are elastic modulus of steel, moment by of nine inertia, area, and radius of the hollow section, of Figure the test specimens were measured Variable Differential Transducers External instrumentations [7] Linear Figure(LVDT-1,2,3,4,5,6,7,8,9), External instrumentations [7] (LVDT) as shown in Fig 7(a) Readings from LVDT-5 was used to construct the loadrespectivelyε , ε , ε , ε , ε are the values recorded by SG-1,2,3,4 and average value of SGs-1,2,3,4.The reactions and ave Figure 7.1 External [7] displacement curve and theinstrumentations history of bending measured supporting columns Lateral the total moment of beam-column joint can be moments evaluated based onin the diagrams illustrated in deformations Fig 7(b) of the in x- and y- directions were measured by two other transducers installed on the top of columns: LVDTforcestest Nspecimen andIt moments M indicated Fig 7(b) can beaffect evaluated steel strains and cross-sectional 102 deformations of the testin specimens were byby nine Linear Variable Differential Transducers 01, Vertical was expected1 that these deformations may measured significantly the development of the peripheral properties as follows: compressive ring, and that of catenary action in the central region (LVDT) (LVDT-1,2,3,4,5,6,7,8,9), as shown in Fig 7(a) Readings from LVDT-5 was used to construct the loadN1 of = bending E steel ∗ A ε2 + ε3 +inεsupporting displacement curve and the history moments columns Lateral deformations of the s ∗ (ε1 +measured )/4 test specimen in x- and y- directions were measured by two other transducers installed on the top of columns: LVDTM1 = that (E steel ∗ I ∗deformations (ε3 − εave ))/Rε = (ε1 significantly + ε2 + ε3 + εthe ave affect )/4development of the peripheral 01, 02 It was expected these may compressive ring, of catenary action inof thesteel, central region.of inertia, area, and radius of the hollow where E , I, A ,and andthat R are elastic modulus moment steel s section, respectively ε1 , ε2 , ε3 , ε4 , εave are the values recorded by SG-1,2,3,4 and average value of SGs-1,2,3,4.The reactions and the total moment of beam-column joint can be evaluated based on the diagrams illustrated in Fig 7(b) Vertical deformations of the test specimens were measured by nine Linear Variable Differential 15 Dat, P X et al / Journal of Science and Technology in Civil Engineering Transducers (LVDT) (LVDT-1,2,3,4,5,6,7,8,9), as shown in Fig 7(a) Readings from LVDT-5 was used to construct the load-displacement curve and the history of bending moments measured in supporting columns Lateral deformations of the test specimen in x- and y- directions were measured by two other transducers installed on the top of columns: LVDT-01, 02 It was expected that these deformations maya)affect significantly ofb)Evaluation the peripheral compressive ring, and that of a) Arrangement of LVDTs of forcesbyby strain gauges Arrangement of LVDTsthe development b) Evaluation of reaction reaction forces strain gauges catenary action in the central region Figure External instrumentations Figure External instrumentations[7] [7] a) Before the test b) the test Before thetest test (b)After After the test a)(a)Before the b) After the test Figure Specimen PI-02 before and after the test [12] Figure Specimen PI-02 before and after the test [12] Figure Specimen PI-02to before andthe after the test [12] Two specimens and8.PI-04 are loaded failure displacement-controlled procedure with twotwo Two specimens PI-02 PI-02 and PI-04 are loaded to failure bybythe displacement-controlled procedure with Dat, P.mm X./ Journal ofvertical Science and Technology loading steps In the initial stage, the specimens are statically loaded with a loading step of After the loading steps In the initial stage, the specimens are statically loaded with a loading step of mm After the vertical central displacement reaches 50the mm,loading theare loading is mmtoward toward the the failure tensile membrane Two specimens PI-02 and PI-04 loaded toincreasedtoto2 2mm central displacement reaches 50 mm, stepstep is increased failure.Pure Pure tensile membrane action in the region central region which is defined by presence the presence of tensilestrain strainat at the the top surface ofofslab is observed in in action in the central which is defined by the of tensile top surface slab is observed Analytica failure by the displacement-controlled procedure the two at a central displacement of about 40 mm, depthofofRC RCslabs slabs As As the increases, the the the two tests at atests central displacement of about 40 mm, oneone depth thedisplacement displacement increases, central tension region expands significantly, resulting in huge in-plan bending moments throughout the specimens withtension two loading steps.significantly, In the initial stage,in the Compa central region expands resulting huge in-plan bending moments throughout the specimens slab, analysis specimens are statically a loading step Failure mode is loaded the most with important experimental observation as it is used to propose an analytical model for Failure mode is the most important experimental observation as it is used to propose an analytical model for three addition predicting the overall load-carrying capacity of the laterally-unrestrained beam-slab structure under a column loss of mm After the vertical central predicting the overall load-carrying capacitydisplacement of the laterally-unrestrained beam-slab structure under a column loss scenario With a relatively low slab reinforcement ratio of 0.2%, the failure of compressive ring due to concrete scenario With a relatively low slab reinforcement ratiotoof 0.2%, the failure of compressive ring due to concrete - Rotational reaches 50 mm, the loading step is increased crushing does not occur in the two specimens However, the failure mode appears at the final stage with two fullcrushing does not occur in the two specimens However, the failure mode appears at the final stage with two full- slab; depth cracksthe together with Pure bar fractures ofmembrane slabs and interior beams at the intersections of yield-lines This failure mm toward failure tensile depth cracks together with bar fractures of slabs and interior beams at the intersections of yield-lines This failure - Two interio mode can be observed very clearly in Specimen PI-02 demonstrated in Fig In combination with the horizontal action in observed the ofunrestrained central which is defined mode canmovement be veryregion clearly in Specimen PI-02 by demonstrated in Fig In combination with the horizontal edges, it is possible that the final failure is due to in-plane bending moment along the long movement ofunrestrained edges, it isat possible the final the presence of tensile strain the topthat surface of failure is due to in-plane bending moment along the long - Top reinfo span span centrelines of slab is observed in the two tests at a central disIt is predicte placement of about 40 mm, one depth of RC slabs capacity in t As the displacement increases, the central tension than that of factors As m region expands significantly, resulting in huge inthe plan bending moments throughout the specimens Figure Failure mode of Specimen PI-02 [12] Figure Failure mode of Specimen PI-02 [12] Failure mode is the most important experidevelopment of membrane action is more significant and the load-ca mental observation as it is used to propose an analaterally-unrestrained slab at large deflection forms a self-equilibrating forces at of thethe outer ring and tensile membrane forces in the central re lytical model for predicting the overall load-carrying capacity laterally-unrestrained beam-slab plastic behaviour and simplifying the stress distribution into rectangul structure under a column loss scenario With a relatively low slab reinforcement of 0.2%, the into rectangul plastic behaviour and simplifying ratio the stress distribution stresses theinyield be simplified into in-plane stress distr failure of compressive ring due to concrete crushing does notalong occur thelines twocan specimens However, of Element cracks results together in Eq (1), with bar fractures of the failure mode appears at the final stage with two full-depth slabs and interior beams at the intersections of yield-lines This failure mode can be observed very clearly in Specimen PI-02 demonstrated in Figs and In combination with the horizontal movement of unrestrained edges, it is possible that the final failure is due to in-plane bending moment along the long span 16 Dat, P X et al / Journal of Science and Technology in Civil Engineering Analytical model Compared to the analysis of a simply supported slab, analysis of a beam-and-slab substructure requires three additional factors to be considered as follows: Rotational restraint along the perimeter edges of the slab; Two interior beams in the centre line; and Top reinforcement along the interior beams at the centrelines of the slab It is predicted that the enhancement of load-carrying capacity in the beam-and-slab substructure is greater than that of the simply supported slab due to these factors As more reinforcement is provided in the slab, the development of membrane action is more significant and the load-carrying capacity of RC slab is greater A laterally-unrestrained slab at large deflection forms a self-equilibrating mechanism with compressive membrane forces at the outer ring and tensile membrane forces in the central region indicated in Fig 10 Assuming rigid-plastic behaviour and simplifying the stress distribution into rectangular stress block, in Fig 10 Assuming rigid-plastic behaviour and simplifying the stress distribution into rectangular stress block,the variation of membrane stresses along the yield lines can be simplified into in-plane stress distribution in Fig 10 Considering equilibrium of Element results in Eq (1), L nL kbKT C C E A D kbKT C Element S S f D F C B T (k/(1+k))v ((nL)²+l²/4) nL (1/(1+k))v ((nL)²+l²/4) T bKT T +T bKT C T bKT Compression Reinforcement in long span=T Moment=M 0 nL Element Element Element Reinforcement in short span=T Moment=µM Tension Figure 10 AssumedFigure in-plane [1, membrane 2] 10.membrane Assumed forces in-plane forces [1, 2] T1 + T3 sin φ = (C − T ) T = bK(T 0,top + T 0,bot )(L − 2nL) nL L C E T2 = f 1.1K(Ttop+Tbot)l/2 (L/2-nL)/cosf cosf L/2 sin f F 1.1T f L S T T bKT 0,bot 1+k (nL)2 + l2 (1) (2) (3) /2 Failure Mode T = T b,top cosf L/2-(L/2-nL)/cosf 17 Figure 12 Analysis of membrane action in failure mode for RC beam-slab structure [1] (4) Dat, P al / Journal of Science Technology in Civil Engineering Dat, P X X./etJournal of Science and and Technology in Civil Engineering Dat, P X./ Journal of Science and Technology in Civil Engineering kbKT l2 0,bot Technology in Civil Dat, P X./ Journal T1 + T3 C =of Science (nL)2are+Engineering (5) (1) and From Fig 9, there equations (2)-(7), where j = (C - T2 ) T21 + Tsin + Fig k 9, there are equations (1) From (2)-(7), where sin j = (C - T2 ) +2T(3T0,top + T0,bot )(L - 2nL) T1 T =1 bK (2) From L: largest span of slab where (1) Fig 9, there are2rectangular equations (2)-(7), sin j = (C - T2 ) bKT 0,bot (2) T1 = L: largest span bK (T0,top + T0,bot )(L - 2nL) ofl rectangular slab (nL)of + (k − 1) (6) S = (2)(3) L: l:largest of rectangular bKT T = bKo(,Tbot0,topỉ + T0,botư )(L - 2nL spanspan rectangular slab slab )l tan ϕ l:shortest shortest span of4 rectangular slab T2 =1 bKT ỗ ổ ÷ ö( nL ) +2 l (3) o ,bot ố ổ1ỗ+1k ứử ữ ( nL ) +l42 T2 = bKT b: parameter magnitude of membrane force l: shortest span rectangular slab (3) nL ofdefining 2o ,bot ỗố + k÷ ø ( nL )2 + b: parameter defining magnitude of membrane force T2 = sin φ = (7) è1+ k ø parameter defining magnitude of membrane b: k: parameter defining magnitude of membrane force force l2 (4) k: parameter magnitude of membrane force T3 = Tb ,top (nL)2 +defining magnitude of membrane force (4) k: n: T3 = Tb ,top parameter defining parameter defining yield line (4) T = T n: parameter defining yield line b , top (5) where kbKTo,Fig there l (2)–(7), ö From are Eqs L: largest span of rectangular slab; l: shortest span bot æ 10, parameter definingyield-line yield line (5) n: φ: C = kbKTo,bot angle defining pattern ỉ 1÷ ư(nL) + 2l ç C = ( nL ) + (5) kbKT φ: angle defining yield-line + k l ofCrectangular force; k: parameter defining èỉ slab; øư b: parameter defining magnitude of membrane pattern = 2o,bot ỗ ỗố + k÷ ÷ø (nL)2 + φ: KT angle defining yield-line pattern inline; top steel per unit width inyield-line the shorter pattern; span 0,top: force of èmembrane 1+ k ø n: parameter defining magnitude force; yield ϕ: angle defining KT , : force in top steel per unit width in the shorter span top KT , : force in top steel per unit width in the shorter span top bKT l (6) in the 2 per unit o in , bot top steel width span; KT 0,bot : force inwidth bottom steel per unit KTshorter , : force in bottom steel per unit in the shorter bKT 0,top :1 force (6) SKT == , bot (nL ) 2+ l ( k - 1) KT0 0bot ,bot: force in bottom steel per unit width in the shorter bKTo,obot l (k - 1(6) 2) + ( S nL ) KT , : force in bottom steel per unit width in the shorter span bot tan j width in the shorter span (nL ) T+b,top4(k: force S = tan - 1) in top interior span beam steel span tan jj 22 Substituting into Eq (1) gives Eq (8), TTb,top : force in beam steel intop topinterior interior top: force Tb,topb:,force in top interior beambeam steel steel (7) nLnL (7)(7) SinSin j= Substituting into Eq (1) gives Eq nL Substituting into Eqgives (1) gives Eq (8), (8), 4na (1into − 2n) Sin jj == Substituting Eq (1) Eq (8), l k = + (8) 2l 2 2+ l 2 2 (1 - 2n ) + 4n a na 24 na (1 n ) (nL) ++444 na (1 n ) (8) (8) (8) 1+ + 2 k = 1kk+== (nL )) (nL n a +a44 na + 41 n2+ 2 (a) Failure Mode [2] Concrete com- (b) Failure Mode [1] Fracture of represion failure in the corner of the inforcement across the centre of slab slab Figure 10 Three possible failure modes (c) Failure Mode [1] Fracture of reinforcement across the intersection of yield lines Figure10 10 Three Threepossible possiblefailure failuremodes modes Figure Figure 11 Three possible failure modes The magnitude of parameter k can be obtained through Eq (8) The value of parameter b can be obtained by The magnitude ofparameter parameter canbebeobtained obtained through Eq.the (8).critical The value value of parameter can be by The magnitude can Eq (8) The can are be obtained obtained by considering the failure of modes of slab.k kDepending on howthrough and where section is formed, bthere three Thefailure magnitude ofmodes parameter kDepending canstage be shown obtained through Eq The value parameter canthree be considering failure slab.TMA onhow how and where the(8) critical section is there considering thethefailure modes ofof Depending on and critical section isofformed, formed, therebare are three possible modes of the slab atslab the in Fig 10where [1-2] the The typical failure modes are indicated by possible failure modes theslab slab atthe theTMA TMA stage shown in Fig.10 10 [1-2] The typical failure modes are indicated by obtained considering the modes slab Depending how where the critical section formation ofbylarge cracks across the shorter span of of the slab in resulting in[1-2] the on fracture ofand thefailure reinforcement as in Fig possible failure modes ofof the atfailure stage shown Fig The typical modes are indicated by formation of large cracks across theshorter shorter span the[2] slab resulting in theTMA fracture of the the reinforcement as in 10(b) andof10(c) Nevertheless, recent test by Bailey etthe al showed that compression failure due to large in-plane formation large cracks across the span ofof slab resulting the fracture of reinforcement as[1, in Fig Fig is formed, there are three possible failure modes of the slab atin the stage shown in Fig 11 2] 10(b) and 10(c) Nevertheless, recenttest testby byBailey Bailey [2]showed showed that compression failure indicated due in-plane compressive force at the slab perimeter edge can also beetet counted as anotherthat possible mode of failure in Fig 10(b) and 10(c) Nevertheless, recent alal[2] compression failure due to to large large The typical failure modes are indicated formation cracks across shorter span in-plane of Fig the 10(a) compressive force slabperimeter perimeter edgecan canby also becounted countedof aslarge another possible modethe of compressive force at at thethe slab edge also be as another possible mode of failure failure indicated indicatedin in Fig slab 10(a).resulting in the fracture of the reinforcement as in Fig 11(b) and 11(c) Nevertheless, recent test 10(a) Failure Mode by Bailey et al [2] showed that compression failure due to large in-plane compressive force at the slab Failure Failure Mode 1in-plane compressive forces at the slab perimeter edge govern the slab failure, the magnitude of IfMode large perimeter edge can also be counted as another possible mode of failure indicated in Fig 11(a) membrane forces in-plane which are reflected byforces parameter can be determined equilibrium slab edge section large compressive thebslab slab perimeter edgefrom govern the slab slaboffailure, the magnitude If If large in-plane compressive forces atatthe perimeter edge govern the the magnitude of of Assuming the maximum of the compressive block is limited to from 0.45 of averagefailure, effective depth, membranethat forces which aredepth reflected by parameterstress b can be determined equilibrium of slab edgethe section membrane forces which are reflected by parameter b can be determined from equilibrium of slab edge section Failure Mode following equation can be obtained Eq (9), Assuming that the maximum depth of the compressive stress block is limited to 0.45 of average effective depth, the Assuming that the maximum depth of the compressive stress block is limited to 0.45 of average effective depth, the following equation obtained Eq (9), ỉ bebe +d cancan ỉ d(9), ỉ K + 1ưư If large compressive edge govern the slab failure, following obtained Eq (9) the magỗ 0.67 bequation = in-plane f 0.45 T ỗslab perimeter ỗ 1forces2 ữat-the ữữ cu dứ2 ố byổ2K ỗ ổ forces +ứ ữứ1 ử b can be determined from equilibrium nitude of membrane which parameter kKT 21++dreflected ốổ ổỗdd1are (9) b = o ổỗố çç 0.67 f cu 0.45 ư÷ - T0ỉçK + ư÷ư÷ ÷÷÷the of slab section Assuming maximum compressive stress blockspan; is limited to bd1 =edge 0ố.67 T0 ỗdố2depth ỗ è the ÷ø-span; where is effective depth offreinforcement in shorter is effective depth of reinforcement in longer KT(9) cu 0.45that kKT ữof ỗ ứ ứ o kKTper span; incan is force inaverage steel unit width in theèthe shorter T0equation isèforce steel per unit width in(9), the longer span; fcu is ø ø è ø o effective 0.45 of depth, following be obtained Eq where d1 is effective depth of reinforcement in shorter span; d2 is effective depth of reinforcement in longer span; KT0 compressive cube strength where d1 is in effective depth reinforcement in shorter is effective of reinforcement longer span;fcuKT is force steel per unitofwidth in the shorter span;span; T0 isd2force in steeldepth per unit width in the in longer span; is0 + width in the longer span; fcu is is compressive force in steelcube perstrength unit width in the 1shorter span; T0 is dforce + din steel perK unit b= 0.67 fcu 0.45 − T0 (9) compressive cube strength kKT 2 o 18 Dat, P X et al / Journal of Science and Technology in Civil Engineering where d1 : effective depth of reinforcement in shorter span; d2 : effective depth of reinforcement in longer span; KT: force in steel per unit width in the shorter span; T : force in steel per unit width in the longer span; fcu : compressive cube strength To predict the magnitudes of membrane forces in failure mode 2, a free body diagram as shown in Fig 12 is analyzed It is assumed that all reinforcement along the critical section (line EF) is at ultimate stress, which is approximately 10 percent greater than the yield stress According to Hayes [3], this is a reasonable assumption since the mode of failure is by fracture of reinforcement Hence, taking moment about E gives b= [1.1l2 K(T 0,top + T 0,bot )/8 + 1.1T l/2]/K (10) AT 0,bot + BT 0,bot + CT 0,bot − D(T 0,top + T 0,bot ) The derivation for parameter b in failure mode is also introduced by analyzing the free body diagram of the critical section in the slab Since the critical section is assumed to be at the intersection of yield lines, the free body diagram will be as shown in Fig 13 b= (1 + k)(3.3T 0,bot + 13.2T /l) KT [3k2 + 4n2 a2 (2k2 + k − 1)] Failure Mode nL.sinf (11) Failure Mode nL.sinf nL nL C C (l/2).cosf (l/2).cosf E E S 1.1KT0bot.l/2 S (nL).sinf /(3(1+k)) 1.1KT0bot.l/2 f (nL).sinf /(3(1+k)) f T2 T2 (nL).sinf /(3(1+k)) (nL).sinf /(3(1+k)) Figure 12.13 Analysis of membrane action in failure Figure 13.13 Analysis of membrane action in failure Figure Analysis of membrane action Figure in failure Analysis of membrane action in failu mode for RC beam-slab structure [1] mode for RC beam-slab structure [1, 2] mode for RC beam-slab structure [1, 2]mode for RC beam-slab structure [1, 2] where A, B, C, and D are defined as follows The detailed derivation of Eqs (1), (9), (10), (11) can be found in reference [1, 2] After the parameter b for all possible failure modes has been obtained, the correct failure mode can be determined Since this is an upper bound or an unsafe approach, the failure mode that gives the smallest b is deemed to be the correct failure mode Table shows the comparison between parameter b obtained from the three possible failure modes It can be concluded that failure mode is the correct failure mechanism as it gives the smallest parameter, b, for both specimens This is in line with the test results of Specimen PI-02, as shown in Fig 19 Dat, P X et al / Journal of Science and Technology in Civil Engineering 1 1+k k2 B= 1+k A= l2 (L/2 − nL l2 l2 − (nL)2 + − (nL)2 + , 8n nL 1+k nL2 k l2 L L nL − (nL)2 + , D= − nL − 3(1 + k) 4 C= l2 (k − 1) 16n Table Comparison of parameter b Specimen PI-02 PI-04 Failure Mode 9.53 11.30 5.32 5.20 2.07 4.70 Once the membrane forces are defined, the corresponding load-carrying capacity of slabs by means of enhancement factor can be calculated The contribution of membrane forces and the increase in bending resistance in enhancing the load-carrying capacity of the slab are calculated using Eq (12) e = (e1m + e1b ) − (e1m + e1b ) − (e2m + e2b ) + 2µa2 (12) The subscript “m” and “b” in above Eq (12) denote the enhancement in load-carrying capacity due to membrane forces and increase in bending resistance, respectively On the other hand, the subscript “1” and “2” in the Eq (12) indicate the enhancement from element (the trapezoid section) and element (the triangular section) of the slabs in Fig 10, respectively The detailed derivation of Eq (12) can be found in references [1, 2] Since the enhancement factor obtained by using Eq (12) is originally derived for the case of a simply supported slab, the factor cannot be readily used to calculate the increase of load-carrying capacity of RC beam-slab structure Instead, the enhancement factor can only be applied to the loadcarrying capacity of positive yield line of RC beam-slab structure The load-carrying contribution from the negative yield line is initially assumed to remain constant after yielding Nevertheless, as observed during the test, concrete crushing at the bottom face of interior beam-column joints (Fig 14) cause the bottom face of concrete to flake off and reduce the effective depth (d) and the negative beam and slab moment capacity of the section (Mbeam,neg & m sneg ) Hence, as deflection increases, the loadcarrying contribution from the negative yield line is not constant, but decreasing This phenomenon is also reflected in the variation of column bending moments during the test as shown in Fig 15 Hence, it can be assumed that the decreasing slope for negative yield line capacity at large deflection is the same as the decreasing slope of column moments (r) obtained during the tests Assuming that the total load-carrying capacity (Ptot ) of the slab is equal to the linear summation of the positive yield line load (P pos ) and the negative yield line load (Pneg ), the total load-carrying capacity of the RC beam-slab structure can be expressed mathematically as Eq (13) While the enhancement factor (e) can be obtained analytically, the reduction factor (r) is obtained empirically Hence, the simple method to predict the load-deflection relationship at TMA stage becomes semiempirical Ptot = e × P pos + r × Pneg (13) 20 RC beam-slab structure The load-carrying contribution from the negative yield line is initially assumed to remain beam-column joints (Fig 12) cause the bottom face of concrete to flake off and reduce the effective depth (d) and the constant after yielding Nevertheless, as observed during the test, concrete crushing at the bottom face of interior negative beam and slab moment capacity of the section (Mbeam, neg & m , off).and Hence, as deflection increases, the loadbeam-column joints (Fig 12) cause the bottom face of concrete to flakes neg reduce the effective depth (d) and the carrying contribution from the negative yield line is not constant, but decreasing This is also reflected negative beam and slab moment capacity of the section (Mbeam, neg& ms,neg) Hence, as phenomenon deflection increases, the loadin the variation of column bending moments during the test as shown in Fig 13 Hence, it can be assumed the carrying contribution from the negative yield line is not constant, but decreasing This phenomenon is also that reflected decreasing slope for negative yield line capacity at large deflection is the same as the decreasing slope of column in the variation of column bending moments during the test as shown in Fig 13 Hence, it can be assumed that the Dat, P X ettests al.line / Journal of Science anddeflection Technology Engineering moments (r) obtained during the decreasing slope for negative yield capacity at large is in theCivil same as the decreasing slope of column moments (r) obtained during the tests load capacity (kN/m 2) Figure 12 Concrete crushing atatthe bottom Figure 13.Declining 13.Decliningtrend trendof of bendinginincolumns columns[12] [12] Figure Concretecrushing crushingat thebottom bottom face face of Figure 14.12 Concrete the face of Figure 15 Declining trend bending of bending in columns beam-column joint [12] beam-columnjoint joint[12] [12] beam-column [12] Dat, P X./ Journal of Science and Technology in Civil Engineering Assuming that the total load-carrying of the the slab slab isis equal equal toto the the linear linearsummation summationofofthe the tot)) of Assuming that the total load-carrying capacity capacity (P (Ptot positive yield line load (Ppos) and the negative yield line load (Pneg), Load-Displacement the total load-carrying capacity of the RC beamRelationship of Specimen PI-02 positive yield line load (Ppos) and the negative yield line load (Pneg), the total load-carrying capacity of the RC beamslabValidation structure can be expressed mathematically as Eq (13) While the enhancement factor (e) can be obtained of the analytical model with the experimental results 20 slab structure can be expressed mathematically as Eq (13) While the enhancement factor (e) can be obtained analytically, the reduction factor (r) is obtained empirically Hence, the simple method to predict the load-deflection 15 Hence, the simple method to predict the load-deflection analytically, the reduction factor (r) is obtained empirically relationship TMA stage semi-empirical Fig 16 atand Fig 17 becomes show the load-deflection curves obtained from both the test results and the relationship at TMA stage becomes semi-empirical 10 tes t res ul ts a na l ytica l model analytical for Specimen PI-02 (13) and PI-04, respectively Since the semi-analytical model is Ptotmodel = e´P pos + r ´ Pneg = e ´ P + r ´ P (13) based onPrigid, perfectly plastic behaviour, it is not intended to predict the elastic and elastic-plastic tot pos neg Validation Analytical Results 50 100 figures 150 200 at large 250 300 350 behaviour at of thethe initial stage Model of the with tests.theItExperimental can be seen from both that deformation, central displacement (mm) Validation of the Analytical Model with the Experimental Results Dat, P X./ Journalmodel of Science andproduces Technology in Civil Engineering the semi-analytical good agreement with the tests results Fig 14 and Fig 15 show the load-deflection curves obtained from both the test results and the analytical load capacity (kN/m 2) load capacity (kN/m 2) Figure 14 Comparison between test results analytical modelrigid, for Specimen PI-02analytical [12] plastic model forFig Specimen andshow PI-04, Since the semi-analytical model isandbased on 14 andPI-02 Fig 15 therespectively load-deflection curves obtained from both the test results andperfectly the Load-Displacement Relationship of Specimen PI-02 Load-displacement Relationship of Specimen PI-04the behaviour, itSpecimen is not intended to predict the elastic and elastic-plastic behaviour at the initial stage of tests Itplastic can be model for PI-02 and PI-04, respectively Since the semi-analytical model is based on rigid, perfectly 20 seen from both that attolarge deformation, the elastic-plastic semi-analytical model produces good agreement withIt the t res ul ts behaviour, it isfigures not intended predict theteselastic and behaviour at the initial stage of the tests cantests be 20 15 results a na l ytica l model seen from both figures that at large deformation, the semi-analytical model produces good agreement with the tests 15 10 results 0 50 100 150 200 250 300 10 a na l yti ca l model experi mental tes t 0 350 20 40 60 80 100 120 140 central displacement (mm) central displacement (mm) load capacity (kN/m 2) Figure 14 Comparison between test results and analytical model for Specimen PI-02 [12] Figure 16 Comparison between test results and Figure 17 Comparison between test results Figure 15 Comparison between test results and analytical model for Specimen PI-04 and [12] of Specimen PI-04 Conclusion andanalytical Future Works analyticalLoad-displacement model for Relationship Specimen PI-02 [12] model for Specimen PI-04 [12] A new model to estimate the load-deflection relationship of laterally-unrestrained RC beam-slab structure at TMA stage has been proposed The model predicts that the slabs will fail due to fracture of reinforcement along the intersection of yield lines, which is similar to the failure mechanism observed in the tests Comparison with the test results also shows that the semi-analytical model gives a very good estimation of the overall load-carrying capacity of RC slabs at large deflections However, the number of experiments was slightly less (only two samples in this article) 20 15 10 Conclusion and future works a na l yti ca l model experi mental tes t Nevertheless, the simple model to predict the behaviour of RC beam-slab structure at TMA stage presented in this paper has not yet been extended to incorporate the decreasing negative yield line capacity analytically In addition, a safe maximum value for the central displacement can be further determined Finally, the experimental tests together with the simple analysis presented in this paper are designed to address the internal penultimate column 140 loss scenario The effectiveness of TMA under external penultimate column loss scenario should be further investigated.Also study more about numerical simulation and research by other authors to verify the analytical model 20 40 estimate 60 80 load-deflection 100 120 A new model to the relationship of laterally-unrestrained RC beam-slab central displacement (mm) structure at TMA stage has been proposed The model predicts that the slabs will fail due to fracture of Acknowledgements The study is presented in this paper was financially supported by National Foundation For Science and reinforcement along the intersection of yield lines, which similar to the failure mechanism observed Figure 15 Comparison between test results and analytical model for Specimen PI-04 [12]Technology Development (NAFOSTED), Vietnam through Grant #107.01-2016.07 The financial support is greatly appreciated The authors would like tosemi-analytical thank Professor Tan Kang Hai (NTU, Singapore) for his helpful comments on the and tests Comparison with the test results also shows that the model gives a very in Conclusion Future Works the first draft of this paper A new model to estimate the load-deflection relationship of laterally-unrestrained RC beam-slab structure at References good of the load-carrying capacity TMA stage hasestimation been proposed The model predictsoverall that the slabs will fail due to fracture of reinforcement along theof RC slabs at large deflections However, the intersection of yield lines, which is similar to the failure mechanism observed in the tests Comparison with the test number oftheexperiments was less (only twocapacity samples in this article) results also shows that semi-analytical model gives a very slightly good estimation of the overall load-carrying of RC slabs at large deflections However, the number of experiments was slightly less (only two samples in this article) Nevertheless, the simple model to predict the behaviour of RC beam-slab structure at TMA stage Nevertheless, the simple model to predict the behaviour of RC beam-slab structure at TMA stage presented in presented has not yet been extended to Inincorporate the decreasing negative yield line this paper has not yet in been this extended paper to incorporate the decreasing negative yield line capacity analytically addition, a safe maximum value for the central displacement can be further determined Finally, the experimental tests together with theanalytically simple analysis presented in thisaddition, paper are designed a to address themaximum internal penultimate column capacity In safe value for the central displacement can be further loss scenario The effectiveness of TMA under external penultimate column loss scenario should be further investigated.Also study more about numerical simulation and research by other authors to verify the analytical model determined Finally, the experimental tests together with the simple analysis presented in this paper Acknowledgements The study presented in this paper was financially supported by National Foundation For Science and Technology Development (NAFOSTED), Vietnam through Grant #107.01-2016.07 The financial support is greatly appreciated The authors would like to thank Professor Tan Kang Hai (NTU, Singapore) for his helpful comments on the first draft of this paper 21 References Dat, P X et al / Journal of Science and Technology in Civil Engineering are designed to address the internal penultimate column loss scenario The effectiveness of TMA under external penultimate column loss scenario should be further investigated Also study more about numerical simulation and research by other authors to verify the analytical model Acknowledgements The study presented in this paper was financially supported by National Foundation For Science and Technology Development (NAFOSTED), Vietnam through Grant #107.01-2016.07 The financial support is greatly appreciated The authors would like to thank Professor Tan Kang Hai (NTU, Singapore) for his helpful comments on this paper References [1] Bailey, C G (2001) Membrane action of unrestrained lightly reinforced concrete slabs at large displacements Engineering Structures, 23(5):470–483 [2] Bailey, C G., Toh, W S., and Chan, B M (2008) Simplified and advanced analysis of membrane action of concrete slabs ACI Structural Journal, 105(1):30–40 [3] Hayes, B (1968) Allowing for membrane action in the plastic analysis of rectangular reinforced concrete slabs Magazine of Concrete Research, 20(65):205–212 [4] Kemp, K O (1967) Yield of a square reinforced concrete slab on simple supports, allowing for membrane forces The Structural Engineer, 45(7):235–240 [5] Park, R and Gamble, W L (1998) Reinforced concrete slabs John Wiley & Sons [6] Park, R (1964) Tensile membrane behaviour of uniformly loaded rectangular reinforced concrete slabs with fully restrained edges Magazine of Concrete Research, 16(46):39–44 [7] Pham, X D (2009) Tensile membrane action in preventing progressive collapse of RC building structure subjected to a column removal PhD First Year Report, Nanyang Technological University, Singapore [8] Sawczuk, A and Winnicki, L (1965) Plastic behavior of simply supported reinforced concrete plates at moderately large deflections International Journal of Solids and Structures, 1(1):97–111 [9] Brotchie, J F and Holley, M J (1971) Membrane action in slabs International Symposium on the Cracking, Deflection, and Ultimate Load of Concrete Slab Systems - American Concrete Institute, 30: 345–377 [10] Mitchell, D and Cook, W D (1984) Preventing progressive collapse of slab structures Journal of Structural Engineering, 110(7):1513–1532 [11] Sasani, M., Bazan, M., and Sagiroglu, S (2007) Experimental and analytical progressive collapse evaluation of actual reinforced concrete structure ACI Structural Journal, 104(6):731–739 [12] Wahyudi, T Y (2010) Tensile membrane action at reinforced concrete slabs Final Year Report, School of Civil and Environmental Engineering, Nanyang Technological University 22 ... T+b,top4(k: force S = tan - 1) in top interior span beam steel span tan jj 22 Substituting into Eq (1) gives Eq (8), TTb,top : force in beam steel intop topinterior interior top: force Tb,topb:,force... Central displacement AxialAxial forceforce in corner in edge column columns Axial force in edge columns Axial force in corner column Axial force in corner column 2.5 Instrumentation 5.37 mm 5.37... magnitude of membrane force force l2 (4) k: parameter magnitude of membrane force T3 = Tb ,top (nL)2 +defining magnitude of membrane force (4) k: n: T3 = Tb ,top parameter defining parameter defining