Stiffness Reduction Factor for Flat Slab Structures under Lateral Loads

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1 TECHNICAL NOTES Stiffness Reduction Fctor for Flt Slb Structures under Lterl Lods Sng-Whn Hn, Ph.D., P.E. 1 ; Young-Mi Prk 2 ; nd Seong-Hoon Kee 3 Abstrct: Effective bem width model EBWM hs been widely used for predicting lterl drifts nd slb moments in flt slb structures under lterl lods. As the slb moment due to lterl lods increses, slb stiffness decreses due to crck formtion. The ccurcy of the EBWM strongly depends on how the reduced slb stiffness is estimted. For this purpose, this study developed equtions for clculting slb stiffness reduction fctor by conducting nonliner regression nlysis using stiffness reduction fctors estimted from collected test results. The slb stiffness reduction fctor is defined s rtio of reduced slb stiffness due to crck formtion to the stiffness of the uncrcked slb section. For verifying the proposed eqution, the lterl stiffness of two slb column connection specimens tested by the writers ws compred with the lterl stiffness clculted using the EBWM with the proposed stiffness reduction fctor. Further, two flt plte specimens hving two continuous spns were lso considered for verifying the proposed equtions for. DOI: / ASCE ST X CE Dtbse subject hedings: Slbs; Bems; Stiffness; Experimenttion; Drift; Lterl lods. Introduction Flt slb structures hve often been nlyzed by two-dimensionl frme models, such s the equivlent frme model EFM nd the effective bem width model EBWM. Under lterl lods, flt slbs cn be modeled conveniently by the EBWM. In the EBWM, the flt slb is modeled s n effective bem hving the sme depth s the slb nd n effective bem width =effective width fctor slb width. As the level of pplied slb moments due to lterl lods increse, more crcks propgte in the slbs. Thus, the reduction in slb stiffness due to the effect of crcks should be reflected in the nlysis of flt slb structures. Vnderbilt nd Corley 1983 recommended stiffness reduction fctor of 1/3 for the equivlent frme model. As effective slb stiffness cn be underestimted by using constnt =1/3, Hwng nd Moehle 1993 proposed n eqution for clculting stiffness reduction fctor for squre slbs with squre columns. Grossmn 1997 proposed fctor K d for estimting effective bem width degrdtion ccording to the level of lterl drifts. It is noted, however, tht K d represents lterl stiffness degrdtion, which is not the sme s the effective bem width degrdtion of slbs. Luo nd Durrni 1995,b used 1 Professor, Dept. of Architecturl Engineering, Hnyng Univ., Seoul , Kore. E-mil: swhn82@hotmil.com 2 Grdute Student, Dept. of Architecturl Engineering, Hnyng Univ., Seoul, Kore. 3 Grdute Student, Dept. of Civil nd Environmentl Engineering, Univ. of Texs, Austin, TX Note. This mnuscript ws submitted on Jnury 15, 2008; pproved on Jnury 20, 2009; published online on My 15, Discussion period open until November 1, 2009; seprte discussions must be submitted for individul ppers. This technicl note is prt of the Journl of Structurl Engineering, Vol. 135, No. 6, June 1, ASCE, ISSN /2009/ /$ the effective moment of inerti I e of bems specified in ACI to clculte reduced slb stiffness. For clculting I e, the mount of slb reinforcement hs to be given. However, t the beginning stge of design, the mount of slb reinforcement is unknown. Tble 1 summrizes the existing stiffness reduction fctors. The objective of this study is to propose equtions for clculting the slb stiffness reduction fctor in the EBWM with respect to pplied moment M normlized by crcking moment M cr. For this purpose, this study collected test results of 20 interior nd 10 exterior slb-column connection specimens. Stiffness Reduction Fctor for Flt Slbs Min Vribles for Stiffness Reduction Fctor As the moment in the concrete member increses, its flexurl stiffness decreses due to crcks. The following eqution for effective inerti moment I e is specified in the ACI section ACI 2005, Eq. 9-8 : I e = M 3 cr M I g + 1 M cr M 3 I cr 1 where I g =moment of inerti of the gross concrete section nd I cr =moment of inerti of the crcked section including reinforcement. As shown in Eq. 1, I e becomes smller with incresing pplied moment M. It is worth noting tht for clculting I cr in Eq. 1, the sectionl properties of the crcked concrete section nd the mount of reinforcement re necessry. The mount of reinforcement, however, is not determined t the beginning stge of design. Fig. 1 shows the reduction in stiffness K exp /K FEM with respect to M /M cr for three flt plte slb column connection specimens S1, S2, nd S3 tested by Morrison et l. 1983, where JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2009 / 743

2 Tble 1. Summries of Previously Proposed Stiffness Reduction Fctors Source Stiffness reduction fctor Vnderbilt nd Corley 1983 =1/3 for equivlent frme model lterl drift of 0.2% Moehle nd Diebold /10 scle two-story frme with flt pltes ws tested for evluting stiffness reduction fctor of 1/3 for equivlent frme model t lterl drift of 0.2% Pn nd Moehle 1988 four slb-column connections were tested =1/3 for equivlent frme model nd effective bem width model t lterl drift of 0.2% Hwng nd Moehle /10 scle flt-plte frme with nine pnels were tested: Grossmn 1997 Luo nd Durrni 1995, b =5 c l 0.1 L where c=dimension of squre column dimension; l=length of squre slb; nd L=service live lod lb/ft 2 effective bem width l 2 considering slb stiffness reduction bsed on the test results of Hwng nd Moehle 1993 : l 2 =K d 0.3l 1+c 1 l 2 l 1 + c 2 c 1 d 2 0.9h K FP where c 1 nd c 2 =column dimension; l 1 nd l 2 =slb length; d=effective depth of slb; h=slb thickness; k d =stiffness reduction fctor t the trgeted drift index: fctors 1.1, 1.0, 0.8, nd 0.5 for the drifts 1/ 800, 1/400, 1/200, nd 1/100, respectively; nd K FP =effect fctor ccording to connection: fctors 1.0, 0.8, nd 0.6 for interior, exterior, nd corner supports, respectively stiffness reduction fctor reflecting grvity lods reduced slb stiffness is clculted using the effective inerti moment I e : I e = M cr M 3 I g + 1 M cr M 3 I cr, I g = l 2h 3 12 V g, = A c fc where V g /4A c fc =grvity-sher rtio; I g =gross moment of inerti of the effective width slb; I cr =moment of inerti of crcked effective width slb; nd M =pplied moment Fig. 1. Effect of reinforcement yielding on stiffness reduction 744 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2009

3 K exp =ctul lterl stiffness estimted from the test nd K FEM = lterl stiffness from the elstic finite element nlysis without considering crcks. Specimen S3 hd the most mount of flexurl reinforcement on the slb mong the specimens, wheres Specimen S1 hd the lest reinforcement. Stiffness reduction is significntly ffected by the level of pplied moments, wheres the mount of reinforcement does not ffect the stiffness reduction fctor, K exp /K FEM until slb reinforcement yields. Thus, this study ttempted to propose equtions for the slb stiffness reduction fctor with respect to M /M cr. Slb Stiffness Reduction Fctor In the EBWM, slb stiffness is reduced by the stiffness reduction fctor, which ccounts for the effect of crcks. Slb stiffness in the EBWM is clculted using effective bem width =slb width l 2 effective bem width fctor nd slb depth. The following equtions re effective bem width fctors proposed by Bnchick 1987, which re used in this study for clculting the slb stiffness before crcks occur: i = 5 c l 1 1 l 2 4 l for interior frmes 2 e = 3 c l 1 1 l 2 8 l for exterior frmes 3 where c 1 nd l 1 =column dimension nd the slb spn length in the loding direction, respectively; l 2 =slb spn length perpendiculr to the loding direction; nd =Poisson s rtio. Before proposing the slb stiffness reduction fctor, this study investigted the reltionship between the slb stiffness reduction fctor nd the lterl stiffness reduction fctor. The lterl stiffness reduction fctor is the rtio between the lterl stiffness K l, ks of the connection clculted using reduced slb stiffness to the lterl stiffness clculted without considering slb stiffness reduction K l,ks. For this purpose, the slb column connections shown in Fig. 2 were considered, which hd severl different combintions of slb spect rtio l 2 /l 1, column spect rtio c 2 /c 1, nd column to spn rtio c 1 /l 1, where c 2 =column dimension perpendiculr to c 1. The slb stiffness in the connections ws reduced by between 0 nd 100%, nd then the corresponding lterl stiffness of the connection ws estimted using the EBWM. As shown in Fig. 2, slb stiffness reduction does not lwys equl the lterl stiffness reduction K l, ks /K l,ks. In prticulr, with incresing c 2 /c 1 nd decresing c 1 /l 1, the lterl stiffness reduction fctor devites more significntly from the slb stiffness reduction fctor. It is seen tht lterl stiffness reduction is s lrge s two times the slb stiffness reduction when c 2 /c 1 =2.0, c 1 /l 1 =0.05, nd the slb stiffness reduction=0.2. Thus, this study determined the slb stiffness reduction fctor by using the following procedure. First, this study collected test results of the slb column connection nd flt plte frme specimens obtined under qusi-sttic cyclic lterl lods, from which the lterl stiffness of ech specimen ws estimted. Then the lterl stiffness of ech specimen ws clculted using the EBWM with clibrted slb stiffness. Clibrtion of slb stiffness ws repeted until the difference between the lterl stiffness clculted using EBWM with the clibrted slb stiffness nd the ctul lterl stiffness of the specimen is within 2%. The clibrtion fctor ws then recorded in the dtbse, which becomes the slb stiffness reduction fctor. Nonliner regression nlysis ws conducted using stiffness reduction fctors stored in the dtbse to mke equtions for with respect to M /M cr. Slb Stiffness Reduction Fctor for Interior nd Exterior Slb Column Connections To propose n eqution for clculting the stiffness reduction fctor for interior flt slbs, this study collected tests results from 20 interior slb column connections under qusi-sttic cyclic lods. The informtion on these 20 specimens is listed in Tble 2. It is noted tht ll specimens listed in Tble 2 re isolted interior slb column connections, except for the specimens tested by Robertson nd Durrni 1990, which hd two bys. As mentioned erlier, the stiffness reduction fctor vries significntly ccording to the chnge in M /M cr, so tht this study proposed n eqution with respect to M /M cr. The lterl stiffness of ech test specimen is estimted by connecting pek-to-pek displcement of the lst cycle t ech loding step Hwng nd Moehle 1993; Robertson nd Durrni 1990, which is depicted in Fig. 3. Fig. 4 shows the slb stiffness reduction fctor obtined using test results. Nonliner regression nlysis ws conducted to propose the following eqution: = M 0.5 M cr M 0.5 M cr t interior slb column connections The slb stiffness reduction fctor clculted using the proposed eqution is lso plotted in Fig. 4. The coefficient of vrition for nd the correltion fctor re 0.33 nd 0.82, respectively. For the stiffness reduction fctor for exterior flt slbs, test results of 10 exterior slb column connections under cyclic lterl lods were collected, nd re summrized in Tble 3. Fig. 5 shows the ctul slb stiffness reduction fctor using the test results nd the following eqution obtined from nonliner regression nlysis: = M 0.5 M cr M 0.5 M cr t exterior slb column connections The coefficient of vrition for nd the correltion fctor re 0.26 nd 0.80, respectively. It is noted tht the pplied moment M cn be clculted by considering grvity nd lterl forces. Verifiction of the Proposed Equtions For verifying the proposed stiffness reduction fctor, this study considered one interior nd one exterior slb column connection specimens RI-50 nd RE-50 tested by the writers. The test ws conducted under qusi-sttic cyclic lods Hn et l. 2006, Prk et l Fig. 6 shows the hysteretic curves of specimens RI-50 nd RE-50. Specimens RI-50 nd RE-50 hd similr level of the grvity lod expressed by grvity sher rtio V g / V c, which is 0.50, where V g =grvity lod pplied t the slb column connection nd V c =design sher strength of specimens where the strength reduction fctor is JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2009 /

4 Fig. 2. Reltionship between the slb stiffness reduction fctor nd the lterl stiffness reduction fctor 746 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2009

5 Tble 2. Dimensions, Mteril Properties, Reinforcement Rtio, nd Grvity Lod of Interior Connection Specimens Dimensions cm Reinforcement rtio % Slb Column Aspect rtio Slb top Slb bottom Grvity lod Source Morrison et l Robertson nd Durrni 1990 Pn nd Moehle 1988 Zee nd Moehle 1984 Frhey et l Lbel f c f y l 1 l 2 h d ve c 1 c 2 H l 2 /l 1 c 2 /c 1 c 1 /l 1 MP MP within outside within outside S S S S S I C SE S SO LL L INT V g kn V g /V c Fig. 3. Lterl stiffness using pek-to-pek points JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2009 / 747

6 Figs. 7 nd b show the envelope curves of interior nd exterior slb column connection specimens RI-50 nd RE-50, respectively, which re extrcted from the hysteretic curves. Fig. 7 lso shows the curves constructed using EBWM nd EFM with existing slb stiffness reduction fctors. As shown in Figs. 7 nd b, the lterl stiffness estimted using the EBWM with the proposed slb stiffness reduction fctor is the closest to the ctul lterl stiffness of the specimens mong those obtined using EBWM nd EFM with existing slb stiffness reduction fctors. This study lso compred the lterl stiffness of Specimens 2C nd 6LL hving two continuous spns Robertson nd Durrni 1990 with tht obtined using the EBWM with the proposed stiffness reduction fctors. The only difference between Specimens 2C nd 6LL is the level of grvity lods. The grvity sher rtios V g / V c, =0.75 of the interior nd exterior connections of Specimen 6LL were 0.72 nd 0.59, respectively, wheres the grvity sher rtio V g / V c of the interior connection nd exterior connections of Specimen 2C were 0.29 nd 0.24, respectively. Fig. 8 shows tht the envelope curves obtined using the EBWM with the proposed stiffness reduction fctor ccurtely mtches the ctul envelope curve extrcted from the hysteretic curves obtined from the test in the wide rnge of drift rtios. Conclusions This study proposed stiffness reduction fctor bsed on the test results of flt plte slb column connections nd flt plte frmes. The following conclusions re obtined from this study. 1. The stiffness reduction fctor is significntly ffected by the level of pplied moments, wheres the stiffness reduction Tble 3. Dimensions, Mteril Properties, Reinforcement Rtio, nd Grvity Lod of Exterior Connection Specimens Source Robertson nd Durrni 1990 Luo et l Hwkins et l Lbel Dimensions cm Fig. 4. Stiffness reduction fctor for interior connections Slb Column Aspect rtio Slb top Slb bottom f c f y l 1 /2 l 2 h d ve c 1 c 2 H l 2 /l 1 c 2 /c 1 c 1 /l 1 MP MP within Reinforcement rtio % outside within outside Grvity lod V g kn V g /V c 9E C S LL L IE E E E / JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2009

7 fctor does not vry with respect to the mount of reinforcement until the slb reinforcement yields. Thus, this study proposed equtions for clculting stiffness reduction fctor with respect to the level of pplied moment, M, normlized by crcking moment M cr. 2. The lterl stiffness reduction fctor is not lwys the sme s the slb stiffness reduction fctor. Thus, the slb stiffness reduction fctor cnnot be developed directly from the lterl stiffness reduction fctor. 3. This study conducted nonliner regression nlysis to propose equtions for clculting the slb stiffness reduction fctor, which re shown s follows: Fig. 5. Stiffness reduction fctor for exterior connections Fig. 6. Hysteretic curves = M 0.5 M cr M 0.5 M cr 0.5 M 0.5 t interior slb column M M cr M t exterior slb column connections cr 4. It is shown tht the EBWM with the proposed stiffness reduction fctor ccurtely predicts the lterl stiffness of two JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2009 / 749

8 The views expressed re those of the writers, nd do not necessrily represent those of the sponsors. Fig. 7. Envelope curves for RI-50 nd RE-50 Fig. 8. Envelope curves for flt pltes with two continuous spns slb column connection specimens nd two flt plte specimens hving two continuous spns. Acknowledgments The writers cknowledge the finncil support provided by the Kore Reserch Foundtion D01140 nd SRC/ERC R References Americn Concrete Institute ACI Building code requirements for structurl concrete nd commentry, ACI Committee 318, Detroit. Bnchik, C. A Effective bem width coefficients for equivlent frme nlysis of flt-plte structures. ME thesis, Univ. of Cliforni, Berkeley, Clif. Frhey, D. N., Adin, M. A., nd Ynkelevsky, D. Z RC flt slb-column subssemblges under lterl loding. J. Struct. Eng., 119 6, Grossmn, J. S Verifiction of proposed design methodologies for effective width of slbs in slb-column frmes. ACI Struct. J., 94 2, Hn, S. W., Kee, S.-H., Prk, Y.-M., Lee, L.-H., nd Kng, T. H.-K Hysteretic behvior of exterior post-tensioned flt plte connections. Eng. Struct., 28 14, Hwkins, N. M., Wong, C. F., nd Yng, C. H Slb-edge column connections trnsferring high intensity reversing moment norml to the edge of the slb. Progress Rep. on NSF Project No. ENV , SM 78-1, Dept. of Civil Engineering, Univ. of Wshington, Settle. Hwng, S. J., nd Moehle, J. P An experimentl study of fltplte structures under verticl nd lterl lods. Rep. No. UCB/ EERC-93-03, Univ. of Cliforni, Berkeley, Clif. Luo, Y. H., nd Durrni, A. J Equivlent bem model for flt-slb buildings. Prt I: Interior connection. ACI Struct. J., 92 1, Luo, Y. H., nd Durrni, A. J. 1995b. Equivlent bem model for flt-slb buildings. Prt II: Exterior connection. ACI Struct. J., 92 2, Luo, Y. H., Durrni, A. J., nd Conte, J. P Equivlent frme nlysis of flt plte buildings for seismic loding. J. Struct. Eng., 120 7, Moehle, J. P., nd Diebold, J. W Lterl lod response of flt plte frme. J. Struct. Eng., , Morrison, D. G., Hirsw, I., nd Sozen, M. A Lterl-lod test of RC slb column connection. J. Struct. Eng., , Pn, A. P., nd Moehle, J. P Reinforced concrete flt pltes under lterl loding: An experimentl study including bixil effects. Rep. No. UCB/EERC88/16, College of Engineering, Univ. of Cliforni, Berkeley, Clif. Prk, Y.-M., Hn, S. W., nd Ryu, J.-H Comprison of seismic behviors of interior joints in PT nd RC flt plte systems. Key Eng. Mter., , Robertson, I. N., nd Durrni, A. J Seismic response of connections in indeterminte flt-slb subssemblies. Structurl Reserch t Rice, Rep. No. 41, Dept. of Civil Engineering, Rice Univ., Houston. Vnderbilt, M. D., nd Corley, W. G Frme nlysis for concrete buildings. Concr. Int.: Des. Constr., 5 12, Zee, H. L., nd Moehle, J. P Behvior of interior nd exterior flt plte connections subjected to inelstic lod reversls. Rep. No. UCB/EERC-84/07, College of Engineering, Univ. of Cliforni, Berkeley, Clif. 750 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / JUNE 2009