Effective Beam Width for Flat Plate Systems with Edge Beams

Transcription

1 Zhang, J., Monteiro, P. J. M., and Morrison, H. F., Noninvasive surface measurement of corrosion impedance of reinforcing bar in concrete Part 1: Experimental result. ACI Materials Journal 98(2, pp Effective Beam Width for Flat Plate Systems with Edge Beams Young-Mi Park 1, a, Jae-Joo Lee 2,b and Sang Whan Han 3,c 1 Post doctoral Research Associate, Dept. of Architectural Engrg., Hanyang University, Seoul, Korea, Graduate Student, Dept. of Architectural Engrg., Hanyang University, Seoul, Korea, Professor, Dept. of Architectural Engrg., Hanyang University, Seoul, Korea, a cielmi@hanyang.ac.kr, b jjlee83@nate.com, c swhan82@hotmail.com ABSTRACT The purpose of this study is to propose a calibration factor for the effective beam width, which is able to incorporate the effect of lateral stiffness provided by the edge beams of perimeter moment-resisting frames. To verify the accuracy of the proposed calibration factor, this study compares lateral stiffness of the flat plate frames having edge beams estimated using the proposed calibration factor for the with that obtained from the experiment. This study shows that lateral stiffness of flat plate frames increases considerably due to the effect of edge beams, and the proposed calibration factor accurately reflects the effect of edge beams on lateral stiffness of flat plate frames. KEYWORDS: Effective beam width, Flat plate; Edge beam, Lateral load, Lateral stiffness 1. INTRODUCTION For tall buildings, flat plate frames have been often constructed with perimeter moment resisting frames and shear walls. For predicting lateral drifts and slab design moments in flat plate frames under lateral loads, effective beam width method ( has been widely used. In the, slabs in the flat plate frame are represented by an effective beam having the same depth as the slab and an effective beam width (=effective width coefficient slab width. The don t, however, represent the edge beams of perimeter moment-resisting frames. In this study, a factor calibrating the effective beam width is proposed for incorporating the effect of edge beams in the. For verifying the accuracy of the proposed calibration factor, this study compares lateral stiffness estimated using the proposed calibration factor for the with lateral stiffness obtained from the experiment. 2. EFFECTIVE BEAM WIDTH COEFFICIENT 2.1 Effective of edge beams on flat plate frames Under lateral loads, the lateral stiffness and torsional stiffness of the flat plate frames increase due to the effect of edge beams. The existence of edge beam enables the economical design on lateral load 713

2 analysis. Thus, the effect of edge beams has to be incorporated in approximate method for flat plate frames. For example, in the, effective beam width [Eq. (1] has to be calibrated to incorporate the effect of edge beams. Eq. (1, for interior frame is proposed by Hwang and Moehle ( For interior frames: b ( c l ν = 2 + /3 /(1 (1 1 1 where c1 and l 1 = column and slab dimension in the loading direction, and ν = Poissons ratio. l 1=6m H=3.5m Exterior Frame l 2=6m Interior Frame t Exterior Frame l 2 /2=3m Slab thickness : 0.2m Column : 0.5m 0.5m c 2 c 1 Figure 1. Model frame of flat plate system 2.2 Effective beam width In this study, a calibration factor, γ is proposed for incorporating the effect of edge beams in the effective beam width, b [Eq. (1]. Edge beams in the flat plate frames behave as a torsional element. Thus, cross-sectional torsional stiffness constant ( C and cross-sectional area ( A of the torsional element in the ACI 318 (2008 are used. In this study, lateral stiffness of flat plate frames with or without edge beams is estimated and compared. For this purpose, a model frame as shown in Fig. 1 is considered. The ranges of beam width and depth are considered from 0.3m to 0.5m and from 0.5m to 0.8m, respectively. Commercial software Midas-GENw (2008 is used. Solid element is used for columns, beams and slabs. Fig. 2 shows the lateral stiffness ratio ( K / K with respect to the ratio of the Cross-sectional torsional stiffness constant ( C/ C, and the ratio of the sectional areas ( A/ A, respectively. It is noted that prime( denotes the properties for flat plate frames without edge beams. Fig. 2 shows K / K is affected by the properties of edge beams. As shown in Fig. 2, K / K is more dispersed with respect to C/ C than to A/ A. Thus, this study proposes an equation for calculating the calibration factor (γ with respect to A/ A. Calibration factor is estimated using the following procedure. For a given A/ A, lateral stiffness ( K, K of the flat plate frame having edge beams is estimated using FE analysis as well as with effective beam width in Eq. (1. Since conventional does not account for the effect of edge beams, K deviates from K. For obtaining accurate lateral stiffness using, the effective beam width [Eq. (1] is calibrated, and 714

3 With edge beams Without edge beams C/C A/A Figure 2. Lateral stiffness ratios with respect to ratio of cross-sectional torsional stiffness constant ( C/ C and cross-sectional areas ( A/ A Lateral stiffness ratio (K/K, % 160 Standard deviation:2.25 Coefficient of correlation:0.94 Standard deviation:1.13 Coefficient of correlation:0.98 Calibration factor (γ γ = 0.5 A A Calibration factor (γ γ = (0.5 A A + 0.2( l 2 l 1 l 1 /l 2 =1.0 (with beam Area ratio(a/a Slab aspect ratio(l 1 /l 2 (a without slab aspect ratio( l 1 (b with slab aspect ratio( l 1 Figure 3. Calibration factor the lateral stiffness of the frame is re-estimated. This procedure is repeated until the difference between K and K is within a tolerance. Fig. 3(a shows calibration factor γ with respect to A/ A. The following equation for γ is obtained from the regression analysis. γ = 0.5 A/ A (2 The effective beam width may be also affected by the aspect ratios of slabs and columns ( l 1 and c1/ c 2. The stiffness ratio, K / K is inversely proportion to l 1. Fig. 3(b shows γ with respect to l 1, marked by a solid circle, from which a modified equation for γ is obtained; γ = (0.5 A/ A + 0.2( l / l (3 2 1 Unlike the case of slab aspect ratios, no clear relationship between column aspect ratios ( c 1 / c 2 and K / K can be found. The difference between K and K is less than 5%, thus this study neglects the effect of column aspect ratios on calibration factor (γ. In summary, effective beam width with the proposed γ is expressed as follows; at exterior connections for interior frame γ b= (0.5 A/ A + 0.2( l / l (2 c + l /3/(1 ν (

4 3. VERIFICATION OF PROPOSED EFFECTIVE BEAM WIDTH To verify the proposed effective beam width ( b in Eq. (4, this study compares lateral stiffness obtained from the with that obtained from the experiment (Moehle and Diebold The effective beam widths for the interior and exterior connections are determined by Eq.(1 and (4, respectively. This study considers actual lateral stiffness at a drift ratio of 1/800 (=0.125% and 1/200 (=0.5%. At a drift ratio of 1/800, lateral stiffness K is estimated without considering the effect of cracks whereas, at a drift ratio of 1/200, the effect of cracks is incorporated in the by using stiffness reduction factor, β [Eq. (5], which was proposed by Hwang and Moehle (2000. ( L β = 5 c/ l 0.1 / /3 (5 where c = dimension of a square column dimension, l = length of a square slab, and L = live load. 18 Experimental results Lateral stiffness (kn/mm with Eq.(1 and (4 (a 1/800 with Eq.(1 Test specimens with Eq.(1 and (4 (b1/200 Figure 4. Lateral stiffness at a drift ratio of 1/800 and 1/200 At a drift ratio of 1/800 in shown Fig. 4, K obtained using the with Eq. (1 and (4 is as close as 97% of the actual lateral stiffness whereas K obtained using the only with Eq.(1 underestimates the actual lateral stiffness approximately by 25%. At a drift ratio of 1/200, K estimated from the with Eq. (1 and (4 is 9% less than the actual lateral stiffness obtained from the experiment, whereas the with Eq. (1 underestimates the actual lateral stiffness by 28%. with Eq.( CONCLUSIONS (1 Lateral stiffness and torsional stiffness of flat plate frames increases due to the effect of edge beams. It is observed that the without considering the effect of edge beams significantly underestimates actual lateral stiffness of the flat plate frames having edge beams. (2 This study proposes an equation for calculating the calibration factor (γ for the effective beam width with respect to A/ A. (3 The proposed calibration factor is verified by comparing lateral stiffness estimated from the proposed effective beam width and from experimental test of 2-story flat plate frame with edge beams. 716

5 ACKNOWLEDGEMENTS This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (KRF D01140 and SRC/ERC (R REFERENCES ACI committee 318, Building code requirements for reinforced concrete (ACI Farmington Hills (MI: American Concrete Institute. Hwang, S.J. and Moehle, J.P., Models for laterally loaded slab-column frames, ACI Structural Journal, 97(2: MIDAS IT, Midas GENw users manual, Korea. Moehle, J.P. and Diebold, J.W., Lateral load response of flat plate frame, Journal of Structural Engineering, ASCE, 111(10: