A graphical method of analysis is presented for preliminary design of outrigger trussbraced highrise shear wall structures with nonfixed foundation conditions subject to horizontal loading. The method requires the calculation of six structural parameters: bending stiffness for the shear wall, bending and racking shear stiffnesses for the outrigger, an overall bending stiffness contribution from the exterior columns, and rotational stiffnesses for the shear wall and column foundations. The method of analysis employs a simple procedure for obtaining the optimum location of the outrigger up the height of the structure and a rapid assessment of the influence of the individual structural elements on the lateral deflections and bending moments of the highrise structure. It is concluded that all six stiffnesses should be included in the preliminary analysis of a proposed tall building structure as the optimum location of the outrigger as well as the reductions in horizontal deformations and internal forces in the structure can be significantly influenced by all the structural components. Copyright © 2004 John Wiley Sons, Ltd.
THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGS Struct Design Tall Spec Build 13, 7387 (2004) Published online in Wiley Interscience (www.interscience.wiley.com) DOI:10.1002/tal.235 SHEAR WALL WITH OUTRIGGER TRUSSES ON WALL AND COLUMN FOUNDATIONS J C D HOENDERKAMP Department of Architecture and Building, Eindhoven University of Technology, Eindhoven, The Netherlands SUMMARY A graphical method of analysis is presented for preliminary design of outrigger truss-braced high-rise shear wall structures with non-fixed foundation conditions subject to horizontal loading The method requires the calculation of six structural parameters: bending stiffness for the shear wall, bending and racking shear stiffnesses for the outrigger, an overall bending stiffness contribution from the exterior columns, and rotational stiffnesses for the shear wall and column foundations The method of analysis employs a simple procedure for obtaining the optimum location of the outrigger up the height of the structure and a rapid assessment of the influence of the individual structural elements on the lateral deflections and bending moments of the high-rise structure It is concluded that all six stiffnesses should be included in the preliminary analysis of a proposed tall building structure as the optimum location of the outrigger as well as the reductions in horizontal deformations and internal forces in the structure can be significantly influenced by all the structural components Copyright â 2004 John Wiley & Sons, Ltd INTRODUCTION The outrigger-braced high-rise shear wall shown in Figure consists of a centrally located wall with two equal-length trusses positioned at a distance x from the top of the structure The outriggers connect the shear wall to columns in the faỗade of the structure This way the horizontally applied load will force the wall to behave compositely with the exterior structure by introducing axial forces in the columns These forces form a restraining moment which is in the oposite direction to the bending moment from the horizontal load This effect will decrease the bending moments in the wall from outrigger level down to the base and it will reduce the horizontal deflections of the structure The bending moments in the shear wall and axial forces in the columns are resisted by foundation structures on piles which are indicated by translational springs Combined with the laterally applied loading the restraining moment will force a triple curvature in the wall up the height and double curvature in the trusses as shown in Figure In the horizontal deflection analysis of outrigger-braced shear walls on fixed foundations it has been shown (Stafford Smith and Coull, 1991; Stafford Smith and Salim, 1981) that the wall can be represented by a single flexural stiffness parameter The outriggers were assumed to be prismatic members rigidly connected to the wall and hinge connected to the exterior columns The resulting single curvature behaviour of these flexural members was represented by a single bending stiffness parameter * Correspondence to: J C D Hoenderkamp, Department of Architecture and Building, Structural Design Group, Eindhoven University of Technology, P.O Box 513, Route 5600 MB Eindhoven, The Netherlands E-mail: j.c.d.hoenderkamp@bwk.tue.nl Copyright â 2004 John Wiley & Sons, Ltd Received January 2003 Accepted February 2003 74 J C D HOENDERKAMP outrigger x outrigger h H facade column shear wall facade column piles b c c b ground level deflected shape Figure Shear wall with outrigger trusses It was further taken that the columns are pin connected to a fixed foundation and could thus be represented by a parameter which represents the axial stiffnesses of the columns only With three stiffness parameters representing the wall, outriggers and exterior columns it was possible to combine them in a single dimensionless parameter which allowed a graphical procedure to obtain the optimum location of the outriggers such that they would cause the largest reduction in horizontal deflection at the top of the structure Figure also shows simplified models for the foundations of the central shear wall and exterior columns In the structural analysis the individual elements are only subjected to forces resulting from horizontal loading on the structure The foundations under the pin-connected faỗade columns are taken to be piled foundations which act in a vertical direction only They are modelled as linear springs The shear wall foundation is only subjected to a bending moment The net axial load on this foundation as a result of lateral loading is zero as it is positioned in the centre of a symmetric structure The foundation of the wall can thus be modelled by a rotational spring with a rotational spring constant The outriggers consist of trusses of which the deformations due to shear cannot be ignored The shear stiffness of the trusses in addition to their flexural stiffness must be included in the analysis In order to allow the truss to be subjected to flexural deformations it will be necessary to assume the structural floors of the building to be connected to the trusses at the exterior columns and shear wall locations only; i.e the floor structure is not part of the outrigger structure but it will cause the shear wall and exterior columns to have identical rotations at all floor levels Further simplifying assumptions are: the structure behaves linear elastically; the sectional properties of the shear wall, exterior columns and outriggers are uniform througout their height or length; and the distribution of the lateral loading is uniformly distributed along the height of the structure A compatibility equation is to be developed for the rotations in the wall and outrigger at the intersection of the neutral line of the outriggers and the face of the shear wall This leads to an expression Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 75 SHEAR WALL WITH OUTRIGGER TRUSSES for the restraining moment allowing the reduction in horizontal deflection at the top to be determined Maximizing this reduction will yield the optimum location of the outrigger SHEAR WALL The rotations in the shear wall are the result of a uniformly distributed horizontal load w, and a restraining moment M caused by a reverse action of the outriggers They will cause rotations at outrigger level due to bending in the wall and rotation of the wall foundation as presented in Figure For the shear walls it will be assumed that plane sections remain plane in bending so that the rotations at the shear wall centre line are identical to those at the face of the wall 2.1 Rotations due to lateral loading w The rotation in the reinforced concrete shear wall at level x can quite simply be expressed as follows: q s ; b ;w = w( H - x ) EIs (1) where H is the total height of the structure, x is the distance measured from the top of the stucture and EIs is the flexural stiffness of the shear wall The rotation of the wall due to the rotation of its foundation is wH 2Cs q s ;Cs ;w = (2) in which Cs is the rotational stiffness of the shear wall foundation 2.2 Rotations due to restraining moment M The rotation in the shear wall at level x as a result of bending in the wall is given by q s ;b ;M = - qs;b;w qs;b;M M ( H - x) EIs (3) qs;Cs;w qs;Cs;M x H EIs EIs Cs Cs Figure Rotations in shear wall Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 76 J C D HOENDERKAMP The rotation of the shear wall and its foundation are M Cs q s ;Cs ;M = - (4) It is noted here that the rotations caused by restraining moment M are in the opposite direction to those by horizontal load w OUTRIGGER The rotations in the outrigger due to the restraining moment are obtained by splitting up this action as shown in Figure 3, where the outriggers have been separated from the braced frame for clarity The restraining moment in the structure due to outrigger action is the product of the axial force in the exterior columns and the distance between them: M = Fa (2 b + 2c) = Fa (2l) (5) in which Fa is the restraining force in the exterior columns, l is the distance from the exterior column to the centre line of the shear wall and b is the length of the flexible outrigger measured from the faỗade column to the outrigger/shear wall interface Considering the free body diagram of a single outrigger, then Fa Ơ b = Fr Ơ h (6) where Fr represents the complementary shear forces in the outriggers and h is the vertical dimension of the outrigger The free body diagram in the centre of Figure allows an expression for the restraining moment on the shear wall to be written as M = Fa c + Fr h Fr Fa Fr Fr outrigger Fa Fa Fr x Fr Fa Fa outrigger (7) Fr Fr h Fa H Fr shear wall b c c b Figure Free body diagram restraining forces Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 77 SHEAR WALL WITH OUTRIGGER TRUSSES qr;b;Fr Fr/2 Fr/2 Fr/2 Fr/2 (a) qr;s;Fr Fr h Fr b a (b) Figure Bending and shear deformations in truss Substituting Equations (6) and (7) into Equation (5) will lead to an expression for the effective restraining moment causing bending and shear deformations in the outriggers as shown in Figure 2Fr h = M a (8) where a dimensionless parameter a= 3.1 l b (9) Rotations due to restraining force Fr The double curvature bending in the outrigger as shown in Figure 4(a) is caused by axial strain in the top and bottom cords The rotations in the truss at both ends are identical because the structural floors are assumed to be connected to the outrigger at the columns and shear wall only The rotation due to bending at the outrigger/wall interface is given by q r ;b ;Fr = Fr h Mb = (12 EIr b) 24aEIr (10) where EIr is the bending stiffness of the outrigger, which can be obtained as follows: EIr = EAr h 2 (11) where Ar is the cross-sectional area of the top and bottom chords of the rigger The rotation due to racking shear results from strain in the diagonals as is indicated in Figure 4(b) and can be expressed as Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 78 J C D HOENDERKAMP q r ;s ;Fr = Fr h M = h GAi haGAr (12) It has been shown earlier (Hoenderkamp and Snijder, 2000) that the racking shear stiffness of a rigger, GAr, is the sum of the individual racking shear stiffnesses of all the bracing segments in that rigger So for the outrigger structure n GAr = GAi (13) i =1 where n represents the total number of segments in the two outriggers and GAi is the racking shear stiffness of a single segment of width a For several types of bracing systems the racking shear stiffness is given Appendix A 3.2 Rigid body rotations of outrigger The wide-column behaviour of the shear wall due to bending caused by horizontal load w will result in rigid body rotations of the outriggers as shown in Figure This reverse rotation is given by d s ;b ;w c q r ;b ;w = - ấ = - (q r ;b ;w ) ậ b b (14) Substituting Equation (1) into Equation (14) yields 3 ẽ w( H - x ) c q r ;b ;w = - è ể EIs b (15) An additional reverse rotation in the trusses occurs due to the rotation of the wall foundation when subjected to horizontal loading and is shown in Figure This rigid body rotation for the outrigger can be expressed as follows: q r ;Cs ;w = - ấ ậ d s ;Cs ;w c = - (q s ;Cs ;w ) b b (16) qs;b;w qr;b;w qr;b;w d s;b;w b d s;b;w c c b Figure Rigid body rotation of outriggers due to bending in shear wall Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 79 SHEAR WALL WITH OUTRIGGER TRUSSES qs;Cs;w qr;Cs;w qr;Cs;w d s;Cs;w d s;Cs;w c b c b Figure Rigid body rotation of outriggers due to rotation of wall foundation substituting Equation (2) into Equation (16) yields ẽ wH c q r ;Cs ;w = - è ể 2Cs b (17) Restraining moment M will also cause rigid body rotations of the outriggers The shear wall will deflect in the opposite direction as shown in Figures and The rigid body rotation of the outriggers will now be in the clockwise direction For bending in the wall: q r ;b ;M = - ấ ậ c d s ;b ;M = - (q s ; b ;M ) b b (18) Substituting Equation (3) into Equation (18) yields M ( H - x) c q r ;b ;M = ẽè ể EIs b (19) For rotation of the shear wall foundation: q r ;Cs ;M = - ấ ậ d s ;Cs ;M c = - (q r ;Cs ;M ) b b (20) Substituting Equation (4) into Equation (20) yields M c q r ;Cs ;M = ẽè ể Cs b 3.3 (21) Rotations due to restraining force Fa The restraining forces in the exterior columns will cause two more rigid body rotations of the outriggers: one resulting from shortening and lengthening of the exterior columns and another due to vertical displacements in the column foundations Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 80 J C D HOENDERKAMP The first outrigger rotation can be defined as the column change in length divided by the length of the outrigger This leads to q r ;a ;Fa = Fa ( H - x ) bEAc (22) where Ac is the cross-sectional area of the exterior column Defining an overall stiffness parameter for the exterior column structure to be EIc = EAc l (23) and substituting Equations (5) and (23) into Equation (22) will yield the following expression: q r ;a ;Fa = M ( H - x )a EIc (24) The second outrigger rotation can be defined as the foundation displacement divided by the length of the outrigger, which yields q r ;Cc ;Fa = Fa k aM = b 2l k (25) where k represents the translational stiffness of the column foundation Defining an overall rotational stiffness parameter for the combined column foundations as Cc = l k (26) and substituting Equation (26) into Equation (25) leads to the following simplified expression: q r ;Cc ;Fa = aM Cc (27) HORIZONTAL DEFLECTION Compatibility in rotation at level x requires that the rotations in the shear wall, qs, and outrigger, qr, at their interface are identical Substituting for the individual rotations yields the following expression: 3 w( H - x ) wH M ( H - x ) M ẽ w( H - x ) c ẽ wH c + = -è -è EIs 2Cs EIs Cs ể EIs b ể 2Cs b Mb M M ( H - x ) c ẽ M c aM ( H - x ) a M + + + ẽè + +è + 24aEIr haGAr ể EIs b ể Cs b EIc Cc (28) Simplifying leads to w( H - x ) wH H-x H-x b 1 + = M ẽè + + + + + 2 EIs 2Cs EIc ể EIs 24a EIr GAr Cs Cc Copyright â 2004 John Wiley & Sons, Ltd (29) Struct Design Tall Spec Build 13, 7387 (2004) SHEAR WALL WITH OUTRIGGER TRUSSES 81 Setting two characteristic flexibility parameters as H H + EIs EIc (30) b 1 + + + 24a EIr GAr Cs Cc (31) S1 = and S2 = leads to the following expression for the restraining moment: 3 H ẽ w( H - x ) wH áẽ M=è + è ( ) 2Cs ể H - x S1 + HS2 ể EIs (32) The horizontal deflection at the top of the building can be now obtained as follows: ytop = wH wH M ( H - x ) MH + EIs 2Cs EIs Cs (33) where the first two terms on the right-hand side represent the free horizontal deflections of the shear wall at the top due to bending of the wall and rotation of its foundation as a direct result of lateral loading The third term is due to the restraining moment M which causes a reverse deflection and rotation at outrigger level, resulting in a deflection reduction at the top The last term represents a horizontal deflection at the top of the wall due to reverse foundation rotation OPTIMUM LOCATION OF OUTRIGGER The reduction in horizontal deflection at the top of the structure due to restraining moment M is represented by the last two terms on the right-hand side of Equation (33) and can be expressed as 2 Há ẽH - x yred = M è + Cs ể EIs (34) This reduction is maximized by differentiating it w.r.t x, setting it equal to zero and solving for x After simplifying this leads to d ẩẽ - 3x - x áẽ = + è1 - x - x + x + è (g H ) dx ẻể (g H ) ể1 - x + w (35) in which two characteristic non-dimensional parameters for the outrigger-braced shear wall structure have been set as follows: gH = Copyright â 2004 John Wiley & Sons, Ltd HCs EIs (36) Struct Design Tall Spec Build 13, 7387 (2004) 82 J C D HOENDERKAMP 0.2 Rigger Location, x/H 0.3 0.4 Values of g H > 100 0.5 10 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Values of w Figure Optimum location of outrigger and w= S2 S1 (37) A dimensionless location parameter for the optimum position of the outrigger has been set as x= x H (38) Figure shows a graphical presentation of optimum outrigger locations up the height of a shear wall braced structure on wall and column foundations as a function of two non-dimensional characteristic parameters, gH and w EXAMPLE The structural floor plan of a 29-storey, 87 m high building in Figure shows the arrangement of four identical shear walls with horizontal steel trusses on both sides Each truss is m long, has a single storey height of m and comprises five X-braced segments The building is subjected to a uniformly distributed lateral load of 1ã6 kN/m2 The flexural stiffness of the reinforced concrete shear wall EIs = 1ã5 Ơ 109 kNm2 For the horizontal trusses: top and bottom chords Ar = 1ã78 Ơ 10-2 m2, diagonals Ad = 9ã726 Ơ 10-3 m2 and for the exterior columns Ac = 3ã12 Ơ 10-2 m2 The elastic modulus of steel E = 2ã1 Ơ 108 kN/m2 The rotational stiffness of the shear wall foundation Cs = 2ã0 Ơ 108 kNm and the translational stiffness of the column foundations k = 4ã0 Ơ 105 kN/m The detailed calculations are shown here for a single shear wall with one outrigger on both sides The flexural stiffness of the outrigger structure is given by Equation (11): Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 83 SHEAR WALL WITH OUTRIGGER TRUSSES Shear walls with outriggers 9m 9m 9m 9m 9m 9m 9m 9m w Figure Structural floor plan EIr = EAr h 2 = 21 Ơ 10 Ơ 178 Ơ 10 -2 Ơ 32 = 1682 Ơ 10 kNm The racking shear stiffness is the sum for 10 segments Equation (A.1) yields GAr = 10 2a h Ơ 18 2Ơ3 EA = Ơ 21 Ơ 10 Ơ 9726 Ơ 10 -3 = 9272 Ơ 10 kN d d3 (18 + 32 ) The global second moment of area of the exterior columns can be obtained from Equation (23): EIc = EAc l = Ơ 21 Ơ 10 Ơ 312 Ơ 10 -2 Ơ ấ + = 2388 Ơ 10 kNm ậ An overall rotational stiffness parameter for the foundations under the exterior columns is given by Equation (26): Cc = l k = Ơ 1352 Ơ Ơ 10 = 1458 Ơ 10 kNm The dimensionless parameter a in Equation (9) yields a= l 135 = = 15 b The characteristic parameters S1 and S2 are given by Equations (30) and (31): S1 = H H 87 87 -1 + = + = 9443 Ơ 10 -8 (kNm) EIs EIc 15 Ơ 10 2388 Ơ 10 Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 84 J C D HOENDERKAMP b 1 + + + 2 C C 24a EIr GAr s c 1 -1 = + + + = 3775 Ơ 10 -8 (kNm) 8 24 Ơ 15 Ơ 1682 Ơ 10 Ơ 15 Ơ 9272 Ơ 10 20 Ơ 10 1458 Ơ 10 S2 = The characteristic non-dimensional parameters for the structure can now be obtained from Equations (36) and (37): gH = HCs 87 Ơ 20 Ơ 10 = = 1160 EIs 15 Ơ 10 w= S2 3775 Ơ 10 -8 = = 0400 S1 9443 Ơ 10 -8 From the diagram in Figure it can now quite easily be determined that the optimum location of the outrigger will be at x/H = 0ã34 Locating the outrigger at a mid-storey level nearest to the theoretical optimum location, i.e 28ã5 m from the top, and using Equation (32) yields the restraining moment: 3 H ẽ w(H - x ) wH ẽ M=è + è 2Cs ể (H - x )S1 + HS2 ể EIs 3 18 Ơ 87 ẽ 87 ẽ18(87 - 285 ) =è + è ( ) Ơ 20 Ơ 10 ể 87 - 285 Ơ 9443 Ơ 10 -8 + 87 Ơ 3775 Ơ 10 -8 ể Ơ 15 Ơ 10 = 1592 Ơ 10 kNm This is a 23ã4% reduction in the bending moment at the base of the shear wall It is not the maximum possible moment reduction for the frame; this will occur with the outrigger at a lower position The reduction occurs between outrigger level and the base of the structure The horizontal deflections and reductions at the top of the structure are given by Equation (33): wH wH M (H - x ) MH + EIs 2Cs EIs Cs 18 Ơ 87 18 Ơ 87 1592 Ơ 10 (87 - 2852 ) 1592 Ơ 10 Ơ 87 = + 8 Ơ 15 Ơ 10 Ơ 20 Ơ 10 Ơ 15 Ơ 10 20 Ơ 10 m = 00859 + 00296 - 00358 - 00069 = 00728m Ytop = The total reduction in horizontal deflection is 37ã0% The values in Table give percentage reductions of lateral deflections at the top of the structure and of bending moments in the shear wall for the optimum outrigger location This location, xopt, is defined as the position for which the deflection reduction has a maximum value The bottom row gives the horizontal deflections at the top of the structure The percentages given in parentheses indicate values obtained from computer stiffness matrix analyses In the column marked Flexible all stiffness parameters have been given finite values In the third column the rotational stiffness of the shear wall foundation has been given an infinite value, the other stiffnesses remaining unchanged Similar reasoning has been applied to columns 4, and Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 85 SHEAR WALL WITH OUTRIGGER TRUSSES Table Structural performance w gH xopt (m) % Mred % yred ytop (mm) Flexible Cs = Cc = GAr = Cs = Cc = GAr = 0ã400 11ã6 28ã5 23ã4 (23ã5) 37ã0 (37ã1) 72ã8 0ã347 25ã5 18ã9 (19ã0) 34ã6 (34ã7) 56ã2 0ã327 11ã6 31ã5 25ã7 (25ã8) 39ã8 (39ã9) 69ã6 0ã231 11ã6 34ã5 29ã4 (29ã5) 44ã3 (44ã5) 64ã3 0ã105 34ã5 27ã1 (27ã2) 45ã7 (45ã9) 46ã7 The results show that increasing the stiffness of the exterior column foundations and the racking shear of the outrigger yields larger reductions for the horizontal deflection and bending moments in the shear wall Increasing the stiffness of the shear wall foundation, however, will cause the outrigger to be less effective The last column assumes infinite stiffnesses for the foundations and the racking shear of the outrigger This can lead to seriously underestimating the deflections CONCLUSIONS The analysis of shear wall structures with outriggers on non-fixed foundations subjected to lateral loading is similar to that of outrigger structures on fixed foundations The method requires six structural parameters: bending stiffness for the wall (EIs), bending and racking shear stiffnesses for the outrigger (EIr and GAr), an overall bending stiffness contribution from the exterior columns (EIc) and rotational stiffnesses from the shear wall and column foundations (Cs and Cc) In the analysis of a shear wall with outrigger trusses on non-fixed foundations, the structural properties can be combined into two characteristic flexibility parameters: S1 for the vertical structural elements, and S2 representing the outrigger and foundation structures The simplified model for analysis yields two characteristic non-dimensional parameters: w and gH, which can be used in a diagram to determine the optimum level of the outrigger structure Increasing the rotational stiffness of the shear wall foundation will move the optimum location of the outrigger higher up the structure but will not improve its efficiency Increasing the stiffness of the foundations of the exterior columns will lower the optimum location of the outrigger structure and improve its performance w.r.t horizontal deflection and moment reduction Increasing the racking shear stiffness of the outriggers will also lower the optimum location of the outriggers and improve its performance The reductions in horizontal deflections and bending moments of the shear wall are influenced by all structural parameters It is therefore suggested that all stiffnesses be included in the preliminary design of a proposed tall building structure APPENDIX A: RACKING SHEAR STIFFNESS OF BRACING SYSTEMS Figure shows several types of bracing that can be used in the vertical and horizontal trusses of the structure The racking shear stiffnesses of these bracing systems are given for a standard segment with a length a and a height h All connections are taken to be pinned with exceptions for K- and kneebracings where the braces are pin connected at the top to continuous beams of length a It should be noted that the vertical members not have any influence on the racking shear stiffness of the segment Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) 86 J C D HOENDERKAMP Ar Ad d Ad d h Ad Ad Ar (b) (a) a a m e m Ar, Ir Ar d h Ad Ad Ar, Ir Ar a Ad d (c) (d) a Figure Typical truss bracings The racking shear stiffness for an X-braced segment as shown in Figure 9(a) is given by GAi ;X = 2a h EAd d3 (A.1) where Ad is the cross-sectional area of a diagonal and d represents its length The racking shear stiffness of the K-braced segment in Figure 9(b) is GAi ;K = a hE 2d a3 + Ar Ad (A.2) in which Ar is the cross-sectional area of the horizontal member Bracing systems with a single diagonal, N-bracing, as shown in Figure 9(c) have the following racking shear stiffness: GAi ;N = a hE d a3 + Ad Ar (A.3) The racking shear stiffness of a full-height knee-braced segment as indicated in Figure 9(d) can be expressed as GAi ;KB = Copyright â 2004 John Wiley & Sons, Ltd m hE d m3 m2e2h2 + + Ad Ar aIr (A.4) Struct Design Tall Spec Build 13, 7387 (2004) SHEAR WALL WITH OUTRIGGER TRUSSES 87 in which m is the horizontal distance between column and cord connections to bracing, e is the horizontal distance between tops of knee-braces and Ir indicates the second moment of area for the horizontal member ACKNOWLEDGEMENTS The author wishes to record his appreciation of H H Snijder, M C M Bakker and M R Trouw for their contributions to this project REFERENCES Hoenderkamp JCD, Snijder HH 2000 Simplified analysis of faỗade rigger braced high-rise structures Structural Design of Tall Buildings 9: 309319 Stafford Smith B, Coull A 1991 Tall Building Structures Wiley: New York Stafford Smith B, Salim I 1981 Parameter study of outrigger-braced tall building structures Journal of the Structural Division, ASCE 107(ST10): 20012013 Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 7387 (2004) [...]... contribution from the exterior columns (EIc) and rotational stiffnesses from the shear wall and column foundations (Cs and Cc) In the analysis of a shear wall with outrigger trusses on non-fixed foundations, the structural properties can be combined into two characteristic flexibility parameters: S1 for the vertical structural elements, and S2 representing the outrigger and foundation structures The simplified... deflections 7 CONCLUSIONS The analysis of shear wall structures with outriggers on non-fixed foundations subjected to lateral loading is similar to that of outrigger structures on fixed foundations The method requires six structural parameters: bending stiffness for the wall (EIs), bending and racking shear stiffnesses for the outrigger (EIr and GAr), an overall bending stiffness contribution from... stiffness of the exterior column foundations and the racking shear of the outrigger yields larger reductions for the horizontal deflection and bending moments in the shear wall Increasing the stiffness of the shear wall foundation, however, will cause the outrigger to be less effective The last column assumes infinite stiffnesses for the foundations and the racking shear of the outrigger This can lead... columns will lower the optimum location of the outrigger structure and improve its performance w.r.t horizontal deflection and moment reduction Increasing the racking shear stiffness of the outriggers will also lower the optimum location of the outriggers and improve its performance The reductions in horizontal deflections and bending moments of the shear wall are influenced by all structural parameters... Ytop = The total reduction in horizontal deflection is 37ã0% The values in Table 1 give percentage reductions of lateral deflections at the top of the structure and of bending moments in the shear wall for the optimum outrigger location This location, xopt, is defined as the position for which the deflection reduction has a maximum value The bottom row gives the horizontal deflections at the top of the... column marked Flexible all stiffness parameters have been given finite values In the third column the rotational stiffness of the shear wall foundation has been given an infinite value, the other stiffnesses remaining unchanged Similar reasoning has been applied to columns 4, 5 and 6 Copyright â 2004 John Wiley & Sons, Ltd Struct Design Tall Spec Build 13, 738 7 (2004) 85 SHEAR WALL WITH OUTRIGGER TRUSSES. .. = Copyright â 2004 John Wiley & Sons, Ltd 2 m 2 hE d 3 m3 m2e2h2 + + Ad Ar 6 aIr (A.4) Struct Design Tall Spec Build 13, 738 7 (2004) SHEAR WALL WITH OUTRIGGER TRUSSES 87 in which m is the horizontal distance between column and cord connections to bracing, e is the horizontal distance between tops of knee-braces and Ir indicates the second moment of area for the horizontal member ACKNOWLEDGEMENTS The... RACKING SHEAR STIFFNESS OF BRACING SYSTEMS Figure 9 shows several types of bracing that can be used in the vertical and horizontal trusses of the structure The racking shear stiffnesses of these bracing systems are given for a standard segment with a length a and a height h All connections are taken to be pinned with exceptions for K- and kneebracings where the braces are pin connected at the top to continuous... characteristic non-dimensional parameters: w and gH, which can be used in a diagram to determine the optimum level of the outrigger structure Increasing the rotational stiffness of the shear wall foundation will move the optimum location of the outrigger higher up the structure but will not improve its efficiency Increasing the stiffness of the foundations of the exterior columns will lower the optimum location... moment at the base of the shear wall It is not the maximum possible moment reduction for the frame; this will occur with the outrigger at a lower position The reduction occurs between outrigger level and the base of the structure The horizontal deflections and reductions at the top of the structure are given by Equation (33): wH 4 wH 3 M (H 2 - x 2 ) MH + 8 EIs 2Cs 2 EIs Cs 4 3 18 Ơ 87 18 Ơ 87 1592