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1 A BEAM ANALOGY PROCEDURE FOR STRENGTH OF INTERIOR SLAB-COLUMN CONNECTIONS OF UNBONDED POST-TENSIONED FLAT PLATES - PART B: COMPARISON WITH TEST RESULTS Shodolapo Olyemi Franklin 1,* & Solomon Olknle Ajayi 1 Department of Civil Engineering, University of Botswana, P. Bag 0061, Gaborone, Botswana Formerly, Department of Civil Engineering, University of Ibadan, Ibadan, Nigeria * franklinso@mopipi.b.bw ABSTRACT In an earlier stdy the athors proposed a beam analogy procedre to assess the nbalanced bending moment strength of nbonded post-tensioned interior slab-colmn connections withot shear reinforcement. In the present investigation the proposed approach is applied to a nmber of test reslts on post-tensioned slab connections reported in the literatre. It is fond that for connections sbjected to pre shear or gravity loading, good predictions of the pnching capacity are obtained; the mean vale of test to calclated pnching capacity is 0.98 with a standard deviation of For the case of the relatively few slab-colmn connections sbjected to shear and nbalanced moment loading, the predictions are qite conservative in most cases. The mean vale of test to calclated nbalanced moment strength is 1.09, however a higher standard deviation of 0.39 is obtained. It is conclded that more realistic tests on post-tensioned slab-colmn connections are needed in order to verify the general applicability of the beam analogy approach. In addition it is sggested that the method shold be refined to accont for the inflence of cracking on the torsion-shear interaction, in order to yield more consistent reslts. Keywords: Slab, colmn, beam, shear, bending, torsion, pnching, interaction 1. INTRODUCTION The adoption of nbonded post-tensioned flat slabs in bilding constrction sch as parking strctres, indstrial bildings and apartments offers several important economic advantages. For example sch slabs in the form of flat plates may be qite thin and flexible with span to depth ratios L/h of arond 45 to 50 being qite feasible. Frthermore sch slabs can cope with sitations of high live loads which together with long spans make the control of deflections particlarly significant. Here a proportion of the imposed vertical load can easily be conteracted or balanced by the prestressing. Unfortnately the slab-colmn jnction of nbonded post-tensioned flat plates has always been problematic for designers since sch strctres are qite ssceptible to a sdden pnching type of failre at the slab-colmn connection. This may on occasion initiate a progressive collapse throghot the strctre [1]. At an internal colmn of a flat slab strctre the application of pre gravity or shear loading may trigger pnching failre. However sch failres become a more critical design consideration when the slab-colmn connection is sbjected to transfer of shear and nbalanced bending moments. The shear stress distribtion in the slab arond the colmn becomes nonniform which effectively redces the shear strength at the jnction. Sch scenarios are qite common in bildings nder the inflence of horizontal wind loadings or earthqake effects []. One of the general methods that have been developed for the analysis and design of slab-colmn connections transferring shear and nbalanced moments is the beam analogy where the slab adjacent to the colmn is considered to act as beams rnning in two orthogonal directions framing into the colmn faces. The strips of slab that constitte the beams are sbjected to shear forces, bending moments and torsional moments and it is assmed that redistribtion of these actions can take place if necessary between the beams. The latter are assmed to be capable of developing their ltimate shear forces, bending moments and torsional moments. Frthermore interaction effects at the critical section close to the slab-colmn jnction are considered [3, 4]. Unfortnately the method jst referred to is difficlt to apply on accont of the large nmber of limiting combinations of bending, torsion and shear which have to be considered. In addition the method does not make any allowance for enhanced torsional shear strength de to two-way slab action. Park and Islam [5] proposed a relatively simpler beam analogy approach where the nbalanced bending moment strength is given by a single eqation. However they neglected any possible inflence of compressive membrane action on the flexral strength. Also the enhancing effects for shear and torsion were made by simply dobling the maximm vertical and torsional shear stresses on the critical faces near the slab-colmn jnction, in comparison to the limits normally sed for beams. Frthermore the beam analogy approach of Park and Islam was developed for reinforced concrete slabs. Conseqently Franklin and Ajayi [6] proceeding along the lines of Park and Islam 700

2 proposed a modified beam analogy procedre applicable to nbonded post-tensioned flat slab -colmn connections withot shear reinforcement. However, in contrast to the method of Park and Islam, their beam analogy takes into accont the enhancing effects of compressive membrane action in relation to torsion, shear and flexre inclsive in a more comprehensive and logical fashion. In the present stdy the athors approach is compared with available test reslts on post-tensioned flat slabs in order to ascertain its validity and seflness.. METHODOLOGY.1 Smmary of the Proposed Approach The strength of a slab-colmn connection sbjected to pre vertical or shear loading is given by the expression 0.3f pc Vo.68 d 1 f c c d c d (1) The modified beam analogy for the nbalanced bending moment strength of the slab-colmn connection may be smmed p by the eqation M m 1 m H T o pc c d 1.68 c dd 0.5V K K c d H T x y0.5 f c o 10f 1 f c pc 0.3f f c V 1 V o AB CD 1 () where the bending moment contribtion is listed first, followed by the vertical shear contribtion in sqare brackets and lastly the torsional moment terms in that order. The modification factor for the compressive membrane enhancement, (1+ H/T o ) has been applied to the vertical shear and torsional moment contribtions as shown in Eqation (). It is also applied in the expressions for the flexral moments m and m, however this is not immediately explicit. A complete description of all the terms and symbols in Eqations (1) and () is given in [6] and will not be repeated here.. Post-tensioned slabs and models tilized for present stdy Over the past 5 6 decades nmeros tests have been condcted to simlate the pnching phenomenon in flat slabcolmn jnctions. However tests on nbonded post-tensioned flat slabs or plates are still relatively few in contrast to their reinforced concrete conterparts. Mention mst be made of the pioneering tests of Scordelis et al [7] and other investigators [8 10] which largely inflenced American and British thinking on the pnching problem. More recently several other tests of varying significance tilizing post-tensioned flat slab models have been condcted [11 16]. Unfortnately several of these models did not adeqately simlate the bondary conditions existing in prototype strctres and additionally had span/depth ratios which were nrepresentative of those normally occrring in practice. Frthermore a large nmber of the tests condcted were intended to stdy the post-pnching behavior of prestressed flat slabs [17]. There is a scarcity of realistic tests reported in the literatre on post-tensioned slab-colmn models sbject to moment transfer loading. A notable exception in this regard is the tests of Franklin and Long [18] who employed models extending to mid-span which as a conseqence cold develop compressive membrane in- plane forces that normally arise in continos strctres. Also the increase in tendon forces at failre recorded for these models are approximately of the same magnitde as observed in mlti-panel strctres. More significantly, since a nmber of the models were statically indeterminate, meaningfl load-deflection relationships were obtained and redistribtion of moments cold take place as in mlti-panel strctres. Other advantages that arise from the se of sch models have been noted by Franklin [17]. In the present investigation the models of Scordelis et al [7], Grow and Vanderbilt [8], Smith and Brns [9] and Regan [10] have been selected. These models have been categorized as being sbject to pre gravity or shear loading. For combined shear and transfer of moment loading, the tests of Franklin and Long [18] in addition to the tests reported by Hawkins [19] have been tilized. The latter models of Hawkins have been inclded to spplement the relatively few tests involving moment transfer loading. 701

3 3. RESULTS AND DISCUSSION Table 1 shows a comparison of the predicted pnching capacities sing the proposed modified beam analogy formla as given by Eqation (1) with the experimental loads for slabs sbjected to pre shear or vertical loading. The agreement between calclated and experimental pnching capacities can be seen to be generally good despite the wide range of variables fond in the tests. The mean ration of test to predicted pnching capacity, V T /V o is 0.98 with a standard deviation of Investigator Specimen Concrete strength, f c (N/mm ) Scordelis et al [7] Grow and Vanderbilt [8] Smith and Brns [9] Table 1. Comparison of predicted pnching capacities with test reslts Effective Average Pnching depth, d concrete capacity in (mm) prestress, test, V T Predicted pnching capacity, V o (KN) V T /V o f pc (N/mm ) (KN) S S S S S S S S G G G G G G G G S S S Regan [10] DT DT DT DT DT DT DT DT EL EL EL EL EL Predicted pnching capacities for several of the models of Scordelis et al [7] are nsafe. For these lift slabs the majority of the specimens developed the fll moment capacities at the perimeter of the slab collars. Also the experimental load-deflection plots sggest a prononced inflence of flexral effects on the pnching failre. Hence the possibility of flexral failre by the formation of yield lines in the slab shold be considered. Model S5 of this test series had a very low effective depth of the prestressing tendons ( 0.5 x slab thickness) which is certainly ntypical of most lift slab strctres. Flexral effects also appear to predominate in the tests of Grow and Vanderbilt [8]. In fact tendon stress increases of the order of 00 % were observed for tendons passing throgh the colmn in some cases. For the tests of Smith and Brns [9], model S1 had no bonded reinforcement over the colmn and jst prior to failre the initial crack widened considerably in lie of new ones forming. In the other two specimens of this series all the bars passing throgh the 70

4 colmn yielded long before failre. Franklin et al [0] and Franklin [17] have sggested that all the three models of the series failed by the formation of a yield line mechanism. The predicted pnching capacities for the tests of Regan [10] are conservative and generally satisfactory with the exception of models DT3 and EL1 which failed in bending with yielding of the longitdinal reinforcement and crshing of the concrete across the fll width at mid-span. It shold be noted here however that all the models of Regan were only prestressed longitdinally and had ordinary reinforcement in the transverse direction. The models were intended to represent the regions arond intermediate colmn spports of prestressed slab bridges. Since the vast majority of the tests ended in pnching failres and all the test specimens contained non-prestressed bonded reinforcement, it can be inferred that the latter helped to control cracking and allowed redistribtion to occr, with the reslt that the fll ltimate capacity was attained at the critical sections. In Figre 1 vales of test pnching capacities V T are plotted against the predicted vales V o. It is obvios that the linear relationship shown (represented by the eqation V T = V ) is a reasonable approximation to the reslts. Figre 1. Plot of test verss predicted pnching capacities for models nder pre shear loading Table shows the comparison of the predicted nbalanced bending moment strength M with the measred reslt M T for the models of Franklin and Long [18] and Hawkins [19]. The proportions transferred by flexre, shear and torsion are also indicated. Also shown in Table is a comparison of the predicted ltimate shear force V based on the proposed beam analogy procedre with the measred vales V T. The overall mean ratio V T /V is 0.89 with a standard deviation of 0.18 while if the reslt of specimen 7B is ignored, the mean vale of the ratio M T /M is 1.09 with a standard deviation of The nsafe predictions of ltimate shear force for models 1, 3 and 4 of Hawkins [19] are likely de to the fact that the amont of bonded reinforcement over the colmn region was insfficient to provide the redistribtion of actions necessary for the slabs to attain their maximm shear capacity. However this factor shold also be copled with the nsal lateral loading techniqe adopted, the nrealistic span to depth ratios of the specimens and the interaction of the several variables investigated in the tests [18]. The nsafe predictions of the pnching shear force for models 5B and 3M of Franklin and Long are probably de to the fact that model 5B failed on accont of the formation of an overall flexral mechanism and that there was an absence of bonded reinforcement along the east-west colmn centreline for model 3M. The nsafe prediction of nbalanced bending moment strength for model 3M of Franklin and Long (i.e. M T /M < 1) is probably cased by the absence of bonded reinforcement as noted previosly. In fact Franklin and Long conclded that extra bonded reinforcement shold be provided at all critical locations to ensre that a strctre has sfficient strength and dctility. However this omission shold also prodce a similar otcome in the reslt of 703

5 model 5B; the reason for M T /M of this model being as high as 1.31 is ncertain at this stage. Models 1, 3 and 4 of Hawkins developed large deflections prior to failre by pnching, bt apparently moments did not increase all the way to failre for all three specimens ths explaining the nsafe predictions for nbalanced bending moment strengths. In Figre for the combined loading cases examined, vales of the ratio of measred to predicted nbalanced bending moment strength M T /M are plotted against vales of the ratio of measred to predicted ltimate shear capacity V T /V. It can be seen from Figre that the predictions of nbalanced bending moment strength are conservative for several of the models shown. In fact the linear relationship shown (represented by the eqation M T /M = V T /V ) may not be satisfactory de to the factors discssed earlier. Additional realistic tests on posttensioned slab-colmn connections may be needed to spplement the relatively few test reslts existing in the literatre in order to assist in the verification of the applicability of the crrent beam analogy procedre. A frther consideration which has not been discssed ths far bt is worth mentioning at this stage is the inflence of cracking on the torsion-shear interaction. The proposed beam analogy as exemplified by Eqation () for the nbalanced bending moment strength M does not take this into accont. This aspect if considered wold almost certainly modify the torsional strength contribtion in Eqation (1) and conseqently affect the M T /M ratios for all tests incorporating combined loading. Table. Comparison of predicted shear strength and nbalanced moment strength with test reslts Average Pnching V o (1+H/T o ) Predicted V T /V concrete shear force (KN) ltimate prestress, in test, V T shear force f pc (N/mm ) (KN) V (KN) Investigator Specimen Concrete strength, f c (N/mm ) Franklin and Long [18] Hawkins [19] Table (contined) Specimen Eccentricity e/l 1B B B B M M M Measred nbalanced moment, M T (KNm) Portion of M transferred by varios actions listed below Flexre Shear Torsion Predicted nbalanced strength, M (KNm) M T /M 1B B B B M M M

6 Figre. Vales of test to predicted nbalanced moment strength plotted against vales of test to predicted ltimate shear force 4. CONCLUSIONS The reslts of the present investigation involving a comparison of the beam analogy approach previosly developed with test data reveal the following: a) Predicted vales of pnching capacity for pre shear or gravity loading sitations are in very good agreement with reslts of model tests on nbonded post-tensioned slab-colmn specimens incorporating a wide range of variables. b) For cases of combined shear and transfer of moment loadings, predicted vales of nbalanced bending moment strengths are generally conservative. In the relatively few cases where the predictions are nsafe, this can be attribted to the provision of insfficient amont of ordinary bonded reinforcement at all critical locations. c) The athors approach for the calclation of the nbalanced bending moment strength takes no accont of the inflence of cracking on the torsion-shear interaction. Conseqently the proposed eqation for nbalanced bending moment strength may reqire some modification to inclde this effect. 5. REFERENCES [1]. Hawkins, N.M. and Mitchell, D., 1979, Progressive Collapse of Flat Plate Strctres, Jornal American Concrete Institte, Vol. 77, No. 1, pp []. Park, R. and Gamble, W.L., 000, Reinforced Concrete Slabs, nd Edition, John Wiley and Sons, New York, pp [3]. Hawkins, N.M., 1971, Shear and Moment Transfer between Concrete Flat Plates and Colmns, Progress Report on National Science fondation Grant No. GK 16375, Dept. of Civ. Engrg., University of Washington, Seattle, U.S.A. [4]. Hawkins, N.M., 1974, Shear Strength of Slabs with Moments Transferred to Colmns, Shear in Reinforced Concrete, Vol., Pblication SP-4, American Concrete Institte, Detroit, Mich., pp [5]. Park, R. and Islam, S., 1976, Strength of Slab-Colmn Connections with Shear and Unbalanced Flexre, Jornal Strct. Division, ASCE, Vol. 10, No.ST9, pp [6]. Franklin, S.O. and Ajayi, S.O., 01, A Beam Analogy Procedre for Strength of Interior Slab-Colmn Connections of Unbonded Post-tensioned Flat Plates Part A: Development of the Method, International Jornal of Research and Reviews in Applied Sciences, ARPA Press, Vol. 13, Isse 1, pp [7]. Scordelis, A.C., Lin, T.Y. and May, H.R., 1958, Shearing Strength of Prestressed Lift Slabs, Jornal American Concrete Institte, Vol. 55, No. 4, pp [8]. Grow, J.B. and Vanderbilt, M.D., 1967, Shear Strength of Prestressed Lightweight Aggregate Concrete Flat Plates, Jornal Prestressed Concrete Institte, Vol. 1, No. 4, pp

7 [9]. Smith, S.W. and Brns, N.H., 1974, Post-tensioned Flat Plate to Colmn Connection Behavior, Jornal Prestressed Concrete Institte, Vol. 19, No. 4, pp [10]. Regan, P.E., 1985, The Pnching Resistance of Prestresed Concrete Slabs, Proc. Instn. Civ. Engs., London, Part, Vol. 79, pp [11]. Roschke, P.N. and Inoe, M., 1991, Effects of Banded Post-tensioning in Prestressed Concrete Flat Slabs, Jornal of Strct. Engr., ASCE, Vol. 117, No., pp [1]. Gardener, N.J. and Kallage, M.R., 1998, Pnching Shear Strength of Continos Post-tensioned Concrete Flat Plates, ACI Materials Jornal, Vol. 95, No. 3, pp [13]. Melo, G.S.S.A. and Regan, P.E., 1998, Post-pnching Resistance of Connections between Flat Slabs and Interior Colmns, Magazine of Conc. Res., Vol. 50, No. 4, pp [14]. Ramos, A.M.P. and Lcio, V.J.G., 000, Pnching of Prestressed Flat Slabs Experimental Analysis, Proceedings of the International Workshop on Pnching Shear Capacity of Reinforced Concrete Slabs, Stockholm, Sweden, pp [15]. Silva, R.J.C., Regan, P.E. and Melo, G.S.S.A., 007, Pnching of Post-tensioned Slabs Tests and Codes, ACI Strctral Jornal, Vol. 104, No., pp [16]. Ramos, A.P. and Lcio, V.J.G., 008, Post-pnching Behavior of Prestressed Concrete Flat Slabs, Magazine of Conc. Res., Vol. 60, No. 4, pp [17]. Franklin, S.O., 010, On Yield Line Estimates of the Pnching Strength of Fll Panel Unbonded Posttensioned Flat Slabs at Internal Colmns, ARPN Jornal of Engineering and Applied Sciences, Vol. 5, No. 1, pp [18]. Franklin, S.O. and Long, A.E., 198, The Pnching Behavior of Unbonded Post-tensioned Flat Plates, Proc. Instn. Civ. Engs., London, Part, Vol. 73, pp [19]. Hawkins, N.M., 1981, Lateral Load Resistance of Unbonded Post-tensioned Flat Plate Constrction, Jornal Prestressed Concrete Institte, Vol. 6, No. 1, pp [0]. Franklin, S.O., Cleland, D.J. and Long, A.E., 198, A Flexral Method for the Prediction of the Pnching Capacity of Unbonded Post-tensioned Flat Slabs at Internal Colmns, Proc. Instn. Civ. Engs., London, Part, Vol. 73, pp