Based on the conventional yet accurate continuum approach, a general analysis is presented for a pair of coupled shear walls, stiffened by an outrigger and a heavy beam in an arbitrary position on the height. Subsequently, a parametric study is presented to investigate the behavior of the structure. The optimum location of the outrigger and the parameters affecting its position were also investigated. The results showed that the behavior of the structure can be significantly influenced by the location of the outrigger. It was also indicated that in most ordinary cases the best location of the structure to minimize top drift is somewhere between 0·4 to 0·6 of the height of the structure. Though this method is not a substitute for the finite element method, it gives an initial simple solution to determine the size and position of outrigger, stiffening beam and coupled shear walls in the preliminary design stages. Copyright © 2004 John Wiley Sons, Ltd.
THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGS Struct Design Tall Spec Build 13, 9–27 (2004) Published online in Wiley Interscience (www.interscience.wiley.com) DOI:10.1002/tal.228 OPTIMIZED USE OF THE OUTRIGGER SYSTEM TO STIFFEN THE COUPLED SHEAR WALLS IN TALL BUILDINGS NAVAB ASSADI ZEIDABADI1, KAMAL MIRTALAE1* AND BARZIN MOBASHER2 Isfahan University of Technology, Isfahan, Iran; and Arizona Department of Transportation, Phoenix, Arizona, USA Civil and Environmental Engineering Department, Arizona State University, Tempe, Arizona, USA SUMMARY Based on the conventional yet accurate continuum approach, a general analysis is presented for a pair of coupled shear walls, stiffened by an outrigger and a heavy beam in an arbitrary position on the height Subsequently, a parametric study is presented to investigate the behavior of the structure The optimum location of the outrigger and the parameters affecting its position were also investigated The results showed that the behavior of the structure can be significantly influenced by the location of the outrigger It was also indicated that in most ordinary cases the best location of the structure to minimize top drift is somewhere between 0·4 to 0·6 of the height of the structure Though this method is not a substitute for the finite element method, it gives an initial simple solution to determine the size and position of outrigger, stiffening beam and coupled shear walls in the preliminary design stages Copyright © 2004 John Wiley & Sons, Ltd INTRODUCTION In modern residential tall buildings, lateral loads induced by wind or earthquake are often resisted by a system of coupled shear walls When a building increases in height, the stiffness of the structure becomes more important In addition, the depth of lintel beams connecting shear walls will usually be confined by differences between floor-to-floor height and floor clear height, Hence, the coupling effect of the connecting system may not be sufficient to provide the necessary lateral stiffness, and the tensile bending stress and uplift forces may exceed the economical limits Different methods that can be used to overcome these problems may be the provision of an outrigger, addition of very stiff beams between walls or using both systems An outrigger is a stiff beam that connects the shear walls to exterior columns When the structure is subjected to lateral forces, the outrigger and the columns resist the rotation of the core and thus significantly reduce the lateral deflection and base moment, which would have arisen in a free core Several buildings with this type of bracing were built during the last three decades in North America, Australia and Japan In some buildings with a pair of coupled shear walls to resist the lateral loads, floor slabs are protruded from the shear walls to form balconies At the outer edge of the balconies as shown in Figure 1, the exterior columns are located to support the slabs An outrigger can employ peripheral columns to increase the overall stiffness of the structure and decrease the moments of the walls Numerous studies have been carried out on the analysis and behavior of outrigger structures (Coull and Lao, 1988, 1989; Rutenburg and Eisenburg, 1990; Skraman and Goldaf, 1997) Moudarres (1984) * Correspondence to: Dr Kamal Mirtalae, Arizona Department of Transportation, Bridge Design Group, Mail Drop #631E, 205 South 17th Avenue, Phoenix, AZ 85007, USA Copyright © 2004 John Wiley & Sons, Ltd Received December 2001 Accepted November 2002 10 N A ZEIDABADI ET AL Peripheral columns Coupled Shear Walls and Outrigger Figure Simplified plan of building showed that a top outrigger can reduce the lateral deflections in a pair of coupled shear walls Using the continuous medium method, Chan and Kuang (1989a, 1989b) conducted studies on the effect of an intermediate stiffening beam at an arbitrary level along the height of the walls, and indicated that the structural behavior of coupled shear walls could be significantly affected by particular positioning of the stiffening beam Afterwards, Coull and Bensmail (1991) as well as Choo and Li (1997) extended Kuang and Chan’s method for two and multi-stiffening beams Their studies also included both rigid and flexible foundations for the structure In this paper, based on Chan and Kuang’s method, a continuum approach is designated to analyze a pair of coupled shear walls, stiffened by an outrigger and an interior beam at an arbitrary location on the height A parametric study is used to investigate the influence of rigidities and locations of the outrigger and interior beam on the lateral deflections and laminar shear forces in the structure Furthermore, the best locations of the outrigger to minimize top drift or laminar shear and the effective parameters on the location are presented ANALYSIS Consider a coupled structural wall system in a fixed foundation stiffened by an outrigger and a beam at level hs shown in Figure For analysis of the structure by continuum approach, the coupling beams are replaced by continuous distribution of lamina with equivalent stiffness It is also assumed that both walls deflected equally throughout the height, so the points of contraflexure of the laminae and stiffening beam are at their mid-span points If a hypothetical cut is made along the line of contraflexure, the condition of vertical compatibility above and, below the outrigger leads to the following equations: l dy1 hb q1 dx 12 EIb EA l Copyright © 2004 John Wiley & Sons, Ltd [Ú T dx + Ú x hs hs ] T2 dx = dy2 hb x q2 T2 dx = dx 12 EIb EA Ú0 (1) (2) Struct Design Tall Spec Build 13, 9–27 (2004) 11 OPTIMIZED USE OF OUTRIGGER SYSTEM l c c l l b/2 b b/2 q1 Vs+F h F F H x q2 hs (b) (a) Figure (a) Coupled shear walls stiffened by outrigger and internal beam (b) Substitute structure where y1, q1, T1 and y2, q2, T2 are the lateral deflection, the laminar shear and the axial forces in the walls in the section above and below level hs, respectively and Ib, E, A are second moment of area of connecting beams, elastic modulus of walls and coupling beams and cross-section area of each wall The three successive terms represent the vertical deflection at the cut caused by slopes of the walls, bending of laminae and axial deformation of the walls The general moment–curvature relationship of the walls is d y1 Ï M = EI + T1l e ÔÔ dx Ì Ô M = EI d y2 + T l ÔÓ e dx for (hs £ x £ H ) (3) for (0 £ x £ hs ) in which I = 2I1, where I1 is second moment of area of each wall, and the axial forces in the walls in different sections are given respectively by H T1 = Ú q1dx (4) x H hs hs x T2 = Ú q1dx + Vs + Ú q2 dx (5) where Vs represents the shear force in the stiffening beam By considering the equilibrium of a small vertical element of the continuous structure, it can be shown that at any point along the height q= - dT dx (6) By differentiating Equations (1) and (2) and combining with Equations (3) and (6), q and y can be eliminated and then the governing equations for the axial forces in the walls can be given by Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 12 N A ZEIDABADI ET AL d T1 - a T1 = -g Me dx (7) d T2 - a T2 = -g Me dx (8) I a = g Êl + ˆ Ë Al ¯ (9) where g = Ib l hb I (10) To obtain the shear force in stiffening beam Vs, consider the compatibility condition at its contraflexure point: l dy2 VS b hs T2 dx = dx 13Es Is EA Ú0 (11) in which EsIs is the flexural rigidity of the stiffening beam Equating the corresponding terms of Equations (1) or (2) and Equation (11) at level hs gives the shear force of the stiffening beam thus; the shear forces will be Vs = Sm Hq1s = Sm Hq2 s (12) where q1s and q2s are the shear flows at level hs and Sm is the relative flexural rigidity of the stiffening beam, defined as Sm = h ES I S H EIb (13) THE EFFECTS OF THE OUTRIGGER AND EXTERNAL LOADS In this investigation, the influence of the outrigger is considered as an unknown moment Mh in the location of the outrigger Moment Mh can also be represented by Mh = F(2c + 2l + l) (14) The parameters F, c, l and ഞ are shown in Figure By considering Mh, the moment Me in Equations (7) and (8) can be given by Ï Me = Ì Ó Me = M h for for hs £ x £ H £ x £ hs (15) Therefore the complete solution of Equations (7) and (8) is Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 13 OPTIMIZED USE OF OUTRIGGER SYSTEM T1M = B1 cosh a x + C1 sinh a x T2 M = B2 cosh a x + C2 sinh a x + (16) g Mh a2 (17) The expression for laminar shear above and below the outrigger can be derived by using Equation (6): q1M = -[ B1a sinh a x + C1a cosh a x ] (18) q2 M = -[ B2a sinh a x + C2a cosh a x ] (19) For the external loads considered, the applied bending moment can be represented by u w (2 H - 3H x + x ) Me = Mel = P( H - x ) + ( H - x ) + 6H (20) where P is considered the load at the top of the walls, and u and w are intensities of uniformly distributed and upper triangular distributed loads acting in the walls, respectively Thus, the complete solution of Equations (7) and (8) due to lateral loads is represented in following equations: T1l = B1¢ cosh a x + C1¢ sinh a h + g Ê u wx Mel + + ˆ a2 Ë a a H¯ (21) T2 l = B2¢ cosh a x + C2¢ sinh a h + g Ê u wx Mel + + ˆ Ë a a a H¯ (22) Consequently, the laminar shears are g dMel w q1l = - ÈÍ B1¢a sinh a + C1¢a cosh a x + Ê + ˆ ˘˙ Î a Ë dx a H¯˚ (23) g dMel w q2 l = - ÈÍ B2¢a sinh a x + C2¢a cosh a x + Ê + ˆ ˘˙ Ë dx Î a a H¯˚ (24) BOUNDARY CONDITIONS The values of B1, B2, C1 and C2 can be determined by considering a set of boundary conditions • At the top of the structure, x = H: T1 ( H ) = (25a) • At the level of stiffening beam h , boundary conditions are s Copyright © 2004 John Wiley & Sons, Ltd T1 (hs ) + Vs + Fi = T2 (hs ) (25b) q1 (hs ) = q2 (hs ) (25c) Struct Design Tall Spec Build 13, 9–27 (2004) 14 N A ZEIDABADI ET AL • At the base level, the laminar shear is given by q2 (0) = (25d) Solving Equations (25a)–(25d) gives the unknown integration constants: B1 = -C1 tanhaH (26) B2 = C1 Ê - ahˆ Ë ahs ¯ (27) C2 = (28) -1 l ˆ Ê + M Ë 2l + 2c + l a ¯ h C1 = Ê - K aH + K - cosh ahs + cosh ah (tanh ah )ˆ s s Ë ¯ ahs (29) Likewise, the values of integration constants B¢1, B¢2, C¢1 and C¢2 can be determined The only alteration in boundary conditions which should be made is that Equation (25b) will be changed into the following equation: T1l (hs ) + Vs = T2 l (hs ) (30) Expressions for the constants B1¢, B2¢, C1¢ and C2¢ are given in Appendix LATERAL DEFLECTION EQUATIONS By integrating Equation (3) twice and using the compatibility condition represented in Equations (31a)–(31d) the lateral deflection due to outrigger and external loads will be y2 (0) = (31a) y2¢ (0) = (31b) y1 (hs ) = y2 (hs ) (31c) y1¢(hs ) = y2¢ (hs ) (31d) The lateral deflections due to the outrigger can be given as follows: y1M = - y2 M = Copyright © 2004 John Wiley & Sons, Ltd È B1l C1l cosh a x + sinh a x + xd + d ˘˙ EI ÍÎ a ˚ a (32) ÈÊ gl lB2 - ˆ Mh x - cosh a x + d ˘˙ Í Ë ¯ EI Î ˚ a a (33) Struct Design Tall Spec Build 13, 9–27 (2004) OPTIMIZED USE OF OUTRIGGER SYSTEM 15 The lateral deflections caused by external loads are y1l = gl ˆ g l Ê ux wx ˆ ˘ ÈÊ 1 ( ) ( ) ( ) S x + F x + G x + EI ÍÎË a ¯ a a2 a Ë 2a 6a H ¯ ˙˚ (34) gl ˆ g l Ê ux wx ˆ ˘ ÈÊ ( ) ( ) S x + Z x + EI ÍÎË a ¯ a2 a Ë 2a 6a H ¯ ˙˚ (35) y2 l = The values of K2, K3, d1, d2, d3, S(x), F(x), G(x) and Z(x) are given in Appendix COMPATIBILITY EQUATION In the aforementioned equations, all parameters related to external loads are determined The parameters for the outrigger are also known, provided the moment due to outrigger, Mh is determined Moment Mh can be determined by a rotational computability equation The pivot for this equation is the intersection of the centeroidal axes of one wall with the outrigger The rotational compatibility equation can be given by Ï Ê Mh ˆ h Ê M ˆ l3 ¸ ÔË d ¯ s Ë h ¯ Ô 1 hs (T2 l - T2 M )˝ y2¢ l (hs ) - y2¢ M (hs ) = Ì + Ú 3E0 I0 EA Ô EAc ÔC+l Ó ˛ (36) where E0, I0, C are elastic modulus of outrigger, second moment of area of outrigger between centroidal axis and the edge of each wall, respectively In the equation the terms on the left are rotations due to external loads and the outrigger moment respectively, and the successive terms, on the right, are the axial deformation of the column, bending of the outrigger and axial deformation of the wall By combining Equation (36) with Equations (17, 20, 22, 33, 35) moment Mh can be determined as follows: gl ˆ l g l Ê uhs whs2 ˆ ÈÊ ˘ ( ) ( ) S ¢ x + Z ¢ x + + yTb ˙ ÍÎË a ¯ a2 a Ë a 2a H ¯ ˚ Mh = Ï ÈÊ - g l ˆ h - lBs¢¢ sinh ah ˘ + kh + wH + y Ê lB2¢¢ sinh ah + g l h ˆ ¸ Ì ÍË s s˙ s s s ˝ Ë a a ˚ a2 ¯ a2 ¯˛ ÓÎ (37) in which Tb = B2¢l C2¢l (cosh ahs - 1) sinh ahs + a a g l Ï È -( H - hs ) H hs ˘ w Ê H hs H hs2 hs4 ˆ hs ¸ + + + pÊ Hhs - ˆ ˝ + Ìu Í + + ˙+ Ë Ë ¯ 6 a ˚ H 2 ¯˛ a Ó Î B2¢¢ = Copyright © 2004 John Wiley & Sons, Ltd ( B2 ) Mh (38) (39) Struct Design Tall Spec Build 13, 9–27 (2004) 16 N A ZEIDABADI ET AL EI d (l + c) EAc (40) EI l(l + c) EA (41) EI l3 3E0 I0 a (l + c) H (41) k= y= w= dS( x ) dZ ( x ) and Z ( x ) = dx dx Having the outrigger moment Mh, the value of B1, B2 and C1 can be determined by using Equations (26) through (29) The deflections and internal forces of the structure are given by in which S ¢( x ) = y( x ) = yl ( x ) - y M ( x ) (42) q ( x ) = ql ( x ) - q M ( x ) (43) T ( x ) = Tl ( x ) - TM ( x ) (44) In tall building structures, one of the most important features that should be considered is the top drift, therefore instead of y(x), YH is used in investigations Consequently Equation (42) can be simplified by y H = ylH - y MH (45) in which ylH and yMH are top drifts due to external loads and the outrigger respectively RELIABILITY OF THE METHOD To ensure the reliability of the method, the deflection determined by this method was compared with other methods such as the wide column method The wide column method is one of the most reliable methods for analyzing coupled shear walls (Stafford Smith and Coull, 1991; Tararath, 1988) The comparison is shown in Table According to the table the results are very close PARAMETRIC STUDY It is useful to express the equations representing the internal forces and deflections of the structure in non-dimensional form to enable a parametric study The value of top drift in addition to laminar shear and axial forces of the walls can be given in dimensionless form These values under uniform load are given in the following equations: Copyright © 2004 John Wiley & Sons, Ltd q = uH g q* a2 (46) T0 = uH g T0 * a2 (47) Struct Design Tall Spec Build 13, 9–27 (2004) 17 OPTIMIZED USE OF OUTRIGGER SYSTEM Table Contrasting the solutions gained by the presented method with those determined by the wide column method (equal frame) aH Sm 3·5 3·5 3·5 3·5 3·93 2·94 2·94 2·94 2·94 2·94 2·94 2·94 2·91 2·91 2·91 2·91 2·31 0 0·707 0·707 0·707 hs (m) w H (m) 33·6 33·6 33·6 67·2 67·2 99 51 51 51 51 51 51 -5 67·2 67·2 67·2 67·2 67·2 99 99 99 99 99 99 99 5·2 * 10 0·135 0·135 5·78 * 10-2 5·78 * 10-2 * 10-5 * 10-5 6·66 * 10-6 3·45 * 10-5 3·45 * 10-5 1·59 * 10-2 3·78 * 10-2 k y Wide column method (m) * 10-2 0·164 0·113 0·13 0·164 0·164 0·11 0·11 0·247 0·158 0·158 0·158 0·158 0·133 0·106 0·106 0·133 0·133 9·69 * 10-2 9·69 * 10-2 0·161 0·121 0·121 0·121 0·121 1·079 1·548 1·36 1·75 1·635 8·30 5·40 7·175 5·638 6·129 5·916 6·24 Presented method (m) * 10-2 1·0839 1·557 1·361 1·749 1·634 8·34 5·43 7·22 5·654 6·167 5·93 6·26 1.0 0.8 Sm=0 S m=1 10 x/H 0.6 0.4 aH = 0.2 0.0 0 h s /H = 0 0.10 0.15 0 0 q* Figure Variation of laminar shear with height in a structure with internal beam but without outrigger yH = uH yH * EI (48) where q*, T0* and yH* are the value of dimensionless laminar shear, axial force of the walls and top drift, respectively, as shown in Appendix In Figures and the variation of laminar shear on the height of the structure with an internal beam without outrigger and an outrigger along with an internal beam are shown, respectively According to the figures, the effect of the outrigger on laminar shear is substantial, provided an internal beam is not used in the structure, and this effect is nominal when an internal beam is used, especially for large Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 18 N A ZEIDABADI ET AL 1.0 k* = y* = 20 aH=3 w = 05 hs/H =.5 x/H 0.8 Sm=10, 5, 2, 1, 0.6 0.4 0.2 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 q* Figure Variation of laminar shear with height in a structure with outrigger 1.0 k* = y * = 20 aH=3 w = 05 hs/H =.5 0.8 hs/H 0.6 Sm= 10, 5, 2, 1, 0.4 0.2 0.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 q*max Figure Effect of outrigger location on maximum laminar shear values of Sm Figure shows the influence of the outrigger on maximum laminar shear It can be seen that the best location to minimize the laminar shear is 0.4 of the height from the bottom The effect of the outrigger and the internal beam position on top drift for different relative flexural rigidity of the internal beam and relative axial rigidity of the columns is shown in Figures and respectively The figures indicate that increasing Sm and k* enhances the stiffness of the structure, as it is obvious from the figures that by increasing Sm and k* the curves become nearer Thus, it is sug- Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 19 OPTIMIZED USE OF OUTRIGGER SYSTEM 1.0 k* = y* = 20 aH=3 w = 05 Sd= 0.9 0.8 0.7 hs/H 0.6 Sm= 10, 5, 2, 1, 0.5 0.4 0.3 0.2 0.1 0.0 1.5 2.0 2.5 3.0 3.5 4.0 100 Y* Figure Effect of outrigger location on top drift for different relative flexural rigidities of the internal beam 1.0 * y = 20 aH=3 w = 05 0.9 0.8 Sd= 0.7 hs/H 0.6 * k = 100, 50, 20, 10, 5, 0.5 0.4 0.3 0.2 0.1 0.0 1.5 2.0 2.5 3.0 100Y 3.5 4.0 * Figure Effect of relative axial rigidity of the columns on top drift for different locations of the outrigger gested that Sm and k* not exceed their economical limits In other words only stiffening the internal beam or just fortifying the columns is not always an economical way to control the top drift of the structure In Figure the effect of outrigger relative flexural rigidity and the location of the outrigger are illustrated It can be seen that by stiffening the outrigger top drift decreases Figure shows the influence of outrigger location on top drift for different parameters of coupled shear walls, aH The figures indicate that the influence of the outrigger is decreased when aH augments Figures 10 and 11 show Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 20 N A ZEIDABADI ET AL hs / H 1.0 0.9 k * = 20 0.8 y = 20 aH=3 0.7 S d= * 0.6 w = 0, 05, 1, 2, 5, 0.5 0.4 0.3 0.2 0.1 0.0 1.0 1.5 2.0 2.5 100 Y 3.0 3.5 4.0 * Figure Effect of relative flexural rigidity of the outrigger on top drift for different locations of the outrigger 1.0 k* = y * = 20 w = 05 Sd= 0.9 0.8 hs / H 0.7 0.6 aH = 8, 7, 6, 5, 4, 3, 0.5 0.4 0.3 0.2 0.1 0.0 100 Y * Figure Effect of parameter of coupled shear walls aH on top drift for different locations of outrigger the effect of outrigger location on resistant moment due to outrigger for different parameters of coupled shear walls aH and relative flexural rigidity of the outrigger w These figures indicate that the maximum resistant moment occurs when the location of the outrigger is from 0.2 to 0.4 the height of the structure These figures also show that when aH increases the amount of resistant moment decreases and when relative flexural regidity of the outrigger decreases, resistant moment Mh also increases Figures 12 and 13 show the effect of outrigger location on axial forces of the wall for Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 21 OPTIMIZED USE OF OUTRIGGER SYSTEM 0.58 aH = 0.56 k = 05 S d = 0.54 hs / H y = 05 S m = 0.52 aH = 0.50 aH 0.48 0.46 0.0 0.2 0.4 =4 0.6 0.8 1.0 w Figure 10 Optimum outrigger location for different relative flexural rigidities of the outrigger 0.56 0.54 aH hs / H aH aH =6 0.50 = 0.52 =5 0.48 0.46 0.44 0.42 0.40 0.0 0.2 0.4 0.6 0.8 1.0 w Figure 11 Optimum outrigger location for different relative flexural rigidities of the outrigger different relative flexural rigidity of the stiffening beam and outrigger respectively These figures show that when Sm increases the amount of axial force of the wall T0 also increases, and when w increases T0 decreases From Figures to 13 it can be seen that the slopes of the curves are not very large As a result, small movements in the location of the outrigger not affect the behavior of the structure significantly In Figures 10 and 11 the influence of outrigger relative flexural rigidity on the best location of the outrigger to minimize top drift is shown It can be seen that the optimum locations of outrigger in coupled shear walls are different from those in ordinary cores Besides, the state of curves is also different especially for a large aH The best location to minimize top drift in ordinary shear walls can be found in Coull and Lao (1988, 1989) and Rutenburg and Eisenburg (1990) Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 22 N A ZEIDABADI ET AL 0.56 aH = hs / H 0.54 0.52 0.50 aH = 0.48 aH =4 0.46 0.44 0.0 0.2 0.4 0.6 0.8 1.0 k Figure 12 Optimum outrigger location for different relative rigidities of the columns w = 05 S d = 0.56 y = 05 Sm = hs / H 0.52 0.48 0.44 0.40 0.36 0.0 aH = aH = aH = 0.2 0.4 0.6 0.8 1.0 k Figure 13 Optimum outrigger location for different relative rigidities of the columns The optimum location of the outrigger with respect to relative axial rigidity of the columns to minimize top drift is shown in Figures 12 and 13 These figures indicate that when k increases the best location of the outrigger goes downwards Figure 14 shows the influence of Sm on the best location of the structure The figure illustrates that Sm has a nominal effect on the best location of the structure, or the best location of the outrigger is not virtually affected by the rigidity of the beam It can be concluded from Figures 10–14 that the best location of the outrigger is often between 0·4 to 0·6 of the height from the bottom Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 23 OPTIMIZED USE OF OUTRIGGER SYSTEM 0.60 0.58 aH = hs / H 0.56 aH = aH = 0.54 aH = aH = 0.52 aH = aH = 0.50 0.48 10 Sm Figure 14 Optimum outrigger location for different relative flexural rigidities of the walls CONCLUSION On the basis of continuum approach, a method is derived for analyzing the structural behavior of coupled shear walls stiffened by an internal beam and an outrigger in a haphazard location along the height of the structure The beneficial effect of the outrigger on the structural behavior and lateral deflections of coupled shear walls is investigated Parametric study shows that an outrigger can significantly reduce the lateral deflection of the structure The study also shows that the position of the outrigger can substantially affect the behavior and lateral deflection of the structure Even though an outrigger can be very effective on lateral deflection, the effect of the outrigger on laminar shear is nominal, if an internal beam is used in the structure Furthermore, the investigation indicates that the axial stiffness of columns as well as flexural stiffness of the outrigger has a significant effect on the outrigger location, but the influence of flexural rigidity of the internal beam is nominal Finally, the study shows that the best location of the outrigger is usually somewhere between 0.4 to 0.6 of the height of the structure from the bottom REFERENCES Chan HC, Kuang JS 1989a Stiffened coupled shear walls Journal of Engineering Mechanics, ASCE 115(4): 689–703 Chan HC, Kuang JS 1989b Elastic design charts for stiffened coupled shear walls Journal of Structural Engineering, ASCE 115(2): 247–267 Choo BS, Li GQ 1997 Structural analysis of multi-stiffened coupled shear walls on flexible foundations Computers and Structures 64(1–4): 837–848 Coull A, Bensmail L 1991 Stiffened coupled shear walls Journal of Structural Engineering, ASCE 117(8): 2205–2223 Coull A, Lao WHO 1988 Outrigger braced structures subjected to equivalent static seismic loading In Proceedings of 4th International Conference on Tall Buildings, Hong Kong, 1988; 395–401 Coull A, Lao WHO 1989 Analysis of multi-outrigger-braced tall building structures Journal of Structural Engineering, ASCE 115(7): 1811–1816 Moudares FR 1984 Outrigger-braced coupled shear walls Journal of Structural Engineering, ASCE 110(12): 2871–2890 Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 24 N A ZEIDABADI ET AL Rutenburg A, Eisenburg M 1990 Stability of outrigger-braced tall building structures In Proceedings of 5th International Conference on Tall Buildings, Hong Kong, 1990; 881–892 Straman J, Goldaf E 1997 Outrigger braced structures in concerete In Seventh International Conference on Computing in Civil Engineering, Seoul, Korea, 1997; 933–938 Stafford Smith B, Coull A 1991 Tall Building Structures, Analysis and Design Wiley: Chichester Taranath BS 1988 Structural Analysis of Tall Buildings McGraw Hill: London APPENDIX K1 = -g (u + w) a cosh aH K2 = cosh ahs - Sm sinh ahs K3 = sinh ahs - SmaH cosh ahs K = - Sm g a2 C2¢ = ÈuH ( H h ) + w Ê H - h - ˆ + PH ˘ - s s ÍÎ ˙˚ 2Ë a2 ¯ g a2 ÈuH + Ê wH ˆ + P - Ê w ˆ ˘ ÍÎ Ë ¯ Ë a H ¯ ˙˚ C1¢ = B2¢ tanhahs - z z1 B1¢ = K1 - C1¢ tanhaH B2¢ = z (K1 K2 - K - C2¢ sinh ahs ) + z 1z z cosh ahs + z ahs z = - ahs aH z = K2 aH - K3 z = K1 ahs - C2 APPENDIX gl l C1l d = - ÈÍÊ - ˆ Mh hs + ( B1 - B2 ) sinh ahs + cosh ahs ˘˙ Ë ¯ a a Î ˚ a gl l lB2 C1l d = - ÈÍ Ê - ˆ Mh hs2 + ( B1 - B2 ) cosh ahs + + sinh ahs + d 1hs ˘˙ Ë ¯ Î ˚ a a a a d3 = Copyright © 2004 John Wiley & Sons, Ltd lB2 a2 Struct Design Tall Spec Build 13, 9–27 (2004) OPTIMIZED USE OF OUTRIGGER SYSTEM S( x ) = 25 u( x - Hx + H x ) w( x - 10 H x + 20 H x ) p( x - Hx ) + 24 120 H F( x ) = [( B1¢ - B2¢ ) sinh ahs + (C1¢ - C2¢ ) cosh ahs ]( x - hs ) G(x ) = B1¢(cosh ahs - cosh ax ) + C1¢(sinh ahs - sinh ax ) + B2¢ (1 - cosh ahs ) + C2¢ (ax - sinh ahs ) Z ( x ) = B2¢ (1 - cosh ax ) + C2¢ (ax - sinh ax ) APPENDIX * - y*MH y*H = ylH y*MH = ya Mh* q1* = q1*l - q1*M q2* = q2*l - q2*M q *M1 = q1a Mh* q *M = q2 a Mh* where q1a = - K (b1¢ sinh Kx + c1¢ cosh Kx ) Sd q2 a = - K (b2¢ sinh Kx ) Sd b1 c1 ya = - ÈÍ cosh K + sinh K + N1 + N2 ˘˙ ÎK ˚ K sinh Kxs c1 N1 = - ÈÍ(1 - Sd )xs + (b1 - b2 ) + cosh Kxs ˘˙ K K Î ˚ N2 = (1 - Sd ) xs2 (b1 - b2 ) c1 + sinh Kxs + cosh Kxs K K2 Ê- + S ˆ d Ë d ¯ c1 = cosh Kxs - K2 K + K3 + cosh Kxs + cosh Kxs Kxs Kxs b1 = c1 K Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) 26 N A ZEIDABADI ET AL ˆ b2 = c1 Ê Ë Kxs - K ¯ Mh* = h1 h2 È (1 - xs ) xs ˘ xs h1 = b2¢ sinh Kxs + c2¢ (cosh Kxs - 1) + Sd Í + ˙ + (1 - Sd ) L p + d ¢ - Sd Ê ˆ Ë ¯ 6 K K Î ˚ h2 = (1 - Sd )xs - b2 b2 sinh Kxs + Kxs + y Ê sinh Kxs + Sd xs ˆ + w Ë ¯ K K Lp = (xs - 3xs2 + 3xs ) d ¢ = - b2¢ sinh Kxs + c2¢ (1 - cosh Kxs ) b2¢ = z D1 + z D2 z cosh Kxs + z Kxs c2¢ = D1 = Sd K2 (- Sm Sd )(1 - xs ) (- Sd ) - Sdz + sinh Kxs K K cosh K K2 D2 = - Sd Kxs Sd - K cosh K K K = aH x= x H xs = hs H The values of y*1H, q1* and T1* are presented in Moudares (1984) NOTATION A B1, B2, B1¢, B2¢ b c C1, C2, C1¢, C2¢ Cross-section area of each wall Integration constants Clear span length of coupling beam Distance between centroidal axis and edge of each wall Integration constants Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) OPTIMIZED USE OF OUTRIGGER SYSTEM d E E0, Es F H h hs I1 I Ib, Is, I0 l Me Mel Mh p q qil, qiM Sm T Til, TiM u Vs w x y yH a, g l w, y 27 Distance between the columns Elastic modulus of walls and coupling beams Elastic moduli of outrigger and stiffening beam, respectively Axial resistant force of columns Total height of structure Height of story Location of outrigger and stiffening beam from bottom Second moment of area of each wall Total second moment of area of walls equal to 2I1 Second moment of area of connecting beams, stiffening beam and outrigger Distance between centroidal axes of walls Applied moment Applied moment due to external load Resistant moment caused by outrigger Concentrated load at top of structure Laminar shear in equivalent medium Laminar shear in section i due to external loads and outrigger, respectively Relative flexural rigidity of stiffening beam Axial force of each wall Axial force in each wall caused by external loads and outrigger, respctively Intensity of uniformly distributed load Shear force of stiffening beam Maximum intensity of triangular distributed load Height coordinate Lateral deflection of walls Lateral deflection of walls at top level Structural parameter Clear length of outrigger Dimensionless parameters of structure Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9–27 (2004) [...]... an internal beam is used in the structure Furthermore, the investigation indicates that the axial stiffness of columns as well as flexural stiffness of the outrigger has a significant effect on the outrigger location, but the influence of flexural rigidity of the internal beam is nominal Finally, the study shows that the best location of the outrigger is usually somewhere between 0.4 to 0.6 of the. .. Bensmail L 199 1 Stiffened coupled shear walls Journal of Structural Engineering, ASCE 117(8): 2205–2223 Coull A, Lao WHO 198 8 Outrigger braced structures subjected to equivalent static seismic loading In Proceedings of 4th International Conference on Tall Buildings, Hong Kong, 198 8; 395 –401 Coull A, Lao WHO 198 9 Analysis of multi -outrigger- braced tall building structures Journal of Structural Engineering,... to 0.6 of the height of the structure from the bottom REFERENCES Chan HC, Kuang JS 198 9a Stiffened coupled shear walls Journal of Engineering Mechanics, ASCE 115(4): 6 89 703 Chan HC, Kuang JS 198 9b Elastic design charts for stiffened coupled shear walls Journal of Structural Engineering, ASCE 115(2): 247–267 Choo BS, Li GQ 199 7 Structural analysis of multi-stiffened coupled shear walls on flexible foundations... Effect of relative axial rigidity of the columns on top drift for different locations of the outrigger gested that Sm and k* not exceed their economical limits In other words only stiffening the internal beam or just fortifying the columns is not always an economical way to control the top drift of the structure In Figure 8 the effect of outrigger relative flexural rigidity and the location of the outrigger. .. locations of outrigger the effect of outrigger location on resistant moment due to outrigger for different parameters of coupled shear walls aH and relative flexural rigidity of the outrigger w These figures indicate that the maximum resistant moment occurs when the location of the outrigger is from 0.2 to 0.4 the height of the structure These figures also show that when aH increases the amount of resistant... rigidities of the columns The optimum location of the outrigger with respect to relative axial rigidity of the columns to minimize top drift is shown in Figures 12 and 13 These figures indicate that when k increases the best location of the outrigger goes downwards Figure 14 shows the influence of Sm on the best location of the structure The figure illustrates that Sm has a nominal effect on the best... location of the outrigger to minimize top drift is shown It can be seen that the optimum locations of outrigger in coupled shear walls are different from those in ordinary cores Besides, the state of curves is also different especially for a large aH The best location to minimize top drift in ordinary shear walls can be found in Coull and Lao ( 198 8, 198 9) and Rutenburg and Eisenburg ( 199 0) Copyright © 2004... and stiffening beam from bottom Second moment of area of each wall Total second moment of area of walls equal to 2I1 Second moment of area of connecting beams, stiffening beam and outrigger Distance between centroidal axes of walls Applied moment Applied moment due to external load Resistant moment caused by outrigger Concentrated load at top of structure Laminar shear in equivalent medium Laminar shear. .. increases the amount of axial force of the wall T0 also increases, and when w increases T0 decreases From Figures 3 to 13 it can be seen that the slopes of the curves are not very large As a result, small movements in the location of the outrigger do not affect the behavior of the structure significantly In Figures 10 and 11 the influence of outrigger relative flexural rigidity on the best location of the outrigger. .. Moudares FR 198 4 Outrigger- braced coupled shear walls Journal of Structural Engineering, ASCE 110(12): 2871–2 890 Copyright © 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 9 27 (2004) 24 N A ZEIDABADI ET AL Rutenburg A, Eisenburg M 199 0 Stability of outrigger- braced tall building structures In Proceedings of 5th International Conference on Tall Buildings, Hong Kong, 199 0; 881– 892 Straman