Proc Instn Ciu Engrs, Part 2,1982,73, Dec., 713-729 8563 A graphical method of comparing the swayresistance of tall building structures B STAFFORD SMITH, DSc, PhD, BSc, FICE* J C D HOENDERKAMP, BSc, MEng* M KUSTER MEngt A method is presented for the graphical determination and comparison of the total and maximum storey sway indices for tall building structures The method is based on the adoption of the equation for the deflexion of coupled shear walls as a generalized deflexion equation for most forms of tall building structural assemblies The total sway index and maximum storey sway index are expressed as equations,and graphically, as functions of two non-dimensional structural parameters The structure may consist of shear walls, coupled shear walls, rigid frames, braced frames or combinations of these, provided the structure does not twist The analyses are based on the assumption that structures are uniform with height; however, reasonable comparisons of sway resistance can also be made for nonuniform structures introduction In comparing the drift resistance of alternative structural schemes for a proposed high-rise building, there is a need for a rapid hand method of sway analysis that can be applied generallyto thewide variety of structural forms in current use This Paper presents such a method The method adopts a generalized approach by considering almost all high-rise bents, including rigid frames, braced frames,shear walls and coupled shear walls, as members of a structural family of cantilevers whose deflexion behaviour can be represented by the theory for coupled shear walls Two non-dimensional parameters are used to identify the deflexion behaviour of structures comprising a combination of these bents, as shown for example in Fig 1, providing the structural arrangement and loading are such that the structure does not twist The simple calculation of the two parameters for the structure allows reference to graphs todetermine the deflected form, the totalsway index and the storey sway index for three common types of horizontal loading: uniformly distributed, triangularly distributed and concentrated top loading The total sway index is defined as the lateral deflexion at the top of the structure divided by the totalheight The storey sway index isthe deflexion in a single storey divided by the storey height Written discussion closes 15 February 1983 for publication in Proceedings, Part * Department of Civil Engineering and Applied Mechanics, McGill University, Montreal t Lemieux, Royer and Partners, Consulting Engineers, Quebec 71 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved STAFFORD SMITH, HOENDERKAMP AND KUSTER Notation A , , A,, sectional area of column and bracing, respectively length of beam in coupled wall structure distance from centroid of column or wall to common centroid of bent UJh) elastic modulus c Cparameter (Id0 for shear rigidity of bent storey height total height of structure column number moment of inertia of wall and/or column etc moment of inertia of beam and column, respectively gross moment of inertia of structure structural parameter total sway factor maximum storey sway factor distance between centroids of walls and/or columns horizontal concentrated top load distance from column to brace connection with girder intensity of uniformly distributed lateral loading maximum intensity of triangularly distributed lateral loading distance measured from top of structure lateral deflexion structural parameters deflexion of storey-height segment due to racking shear rotation of column-girder joint The Authors have shown how coupled wall theory can be interpreted to represent the behaviour of rigid frames and braced frames as well as coupled walls,’ and to represent in addition the behaviour of structures combining these different types of bent.2 Because coupled wall theory is based on the assumption of uniformity of the structure with height, the method is more accuratefor structures which are close to thatcondition However, reasonably accurate results for deflexion and valid comparisons between the performances of alternative structural arrangements, may be obtained for non-uniform structures by this method General theory The analysis of deflexions and forces in coupled wall structures, Fig 2, has On the basis of the been available and used in practice for the past two associated theory the characteristicdeflexion equation hasbeen shown to be6 in which E l a’ i = GA/x k2 = E I d x EA,C? (2) (3) where I i is the second moment of area of wall i about its own centroidal axis, Ai and ci are the section area of wall i and the distance from its centroid to the 71 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved SWAY RESISTANCE O F TALL BUILDING STRUCTURES Fig Multi-bent plan symmetric structure +- axis l Fig Coupled shear wall structure 71 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved STAFFORD SMITH, HOENDERKAMP AND KUSTER common centroidal axis of the coupled wall areas, respectively, and I , is the moment of inertia of the coupled walls behaving fully compositely about their common centroidalaxis, that is I, = Ii+ c A,cZ (4) G A symbolizes the racking shear rigidity of the structure which, for coupled walls, is givenby (5) G A = 12~1,12/hb3 in which I , , b and h are the moment of inertia, length and vertical spacing, respectively, of the connecting beams, and l is the distance between the centroidal axes of the walls Taking boundary conditions of fixity at the base and zero external moment and shear at the top, the solutionof equation (1) can be shown to be + cosh(kaH)(l - 31 x / H ) - - (kaH){sinh(kaH) - sinh(kax)} ( ~ c z Hcosh(kati) )~ This is the characteristic deflexion equation for coupled shear walls It has been shown'.' that equation (6) represents also the deflexion of rigid frames, braced frames and symmetrically loaded symmetrical combinations of these Equation (6) is written in terms of the two characteristic non-dimensional parameters EH and k that completely govern the deflected shape of the structure The shape consists partly of a bending mode, i.e with concavity downwind, in the lower region, and a shear mode, i.e with concavity upwind, in the upper region The relative magnitudes of the two modes and hence, the overall configuration of the structure, depend on the values of aH and k In extreme cases the shape willbe entirely flexural or entirely shear The total sway index, ytodH, may be found from equation (6) by setting I 0-1 1.0 J I 2.0 4.0 6.0 10 aH 20 40 60 100 Fig Total sway factorKt-uniforrnlydistributed load 71 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved SWAY RESISTANCE O F TALL BUILDING STRUCTURES X = and dividing through by H cosh(kaH) - - (kaH)sinh(kaH) ( k ~ t Hcosh(kaH) )~ The magnitude of the storey sway is given by the derivative of equation (6) H ) + (kaH)cosh(kax) + -sinh(kaH)(l( k ~-z Hx /cosh(kaH) )~ The maximum value of the storey sway occurs at the level for which the second derivative of equation (6)is zero [ (Gy d2Y o = - -dx2 - + (k2 - 1) (ki& { - x/H) + (kaH)sinh(kax) + cosh(kaH)(l (kaH)' cosh(kaH) The solution of x/H from equation (9) for particular values of a H and k determines the level at which the maximum storey sway occurs for a structure The value of x/H is then substituted into equation (8) to determine the magnitude of the maximum storey sway Equations (7) and (8) can be rewritten as wH3 = EI, 0.1 0.11 1.0 2.0 6.0 4.0 20 , I I I 10 40 60 I 100 aH Fig Maximum storey sway factor K,-uniformlydistributed load 71 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved STAFFORD SMITH, HOENDERKAMP AND KUSTER 0.8 1.0 1.o I 2.0 I I I I I I 4.0 6.0 10 aH 20 40 60 I 100 Fig Location of maximum storey sway-uniformly distributed load Both equations (10) and (11) contain a distribution term, K , preceded by a term that governs the magnitude of the distribution for a particular structure The distribution terms K , and K , are functions of parameters aH and k Values of K , and K , have been solved for practical ranges of values of aH and k, and plotted in Figs and Therefore, the total sway and maximum storey sway of a structure can be obtained by calculating the appropriate values of aH and k, then using these and Figs and to obtain K , and K , which are then substituted into equations (10) and (11) Fig 5, gives the values of ( x / H )for the maximum storey sway in a structure Equations (7H11) relate to structures subjected to uniformly distributed loading Equations and graphs for triangularly distributedloadingand concentrated top loading are given in Appendix The threeloading cases and combinations of them can be used to approximate the distributions of wind loading and static equivalent earthquake loading as specified in many building codes For top sway and storey sway, the results for combinations of loading may be superposed However, for maximum storey sway, superposition gives an inaccurate result because the maximum values occur at different levels t ‘l ‘2 Fig Rigid framestorey-heightsegment 71 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved SWAY RESISTANCE O F TALL BUILDING STRUCTURES Determination of parameters aH and k The accuracy of predicting the behaviour of a wide range of structures by the single generalized equation depends on the proper determination of aH and k2 for each type of structure Generally, the parameter calculations are based on equations (2) and (3) Rigid frames 10 The racking shear rigidity of a rigid frame bent is evaluated using the assumption that all column-to-girder joints at a floor level rotate equally (Fig 6) Then GA = 12E h[l/C + 1/G] In which C = (I&), where I , is the moment of inertia of each column in the bent, passing through a particular floor level, h is the storey height, and G = (I& where I , is the moment of inertia of each girder in the bent at that floor level and l is its span 11 EI for the frame is given by c (EI,) and the gross moment of inertia I , for the rigid frame bent is I , 1I , + c A , c El = = (13) (14) where I , , A, and c are respectively the moment of inertia, sectional area and distance from the common centroid of the column sectional areas of each column Fig Types of bracing: (a) single X-braced; (b) double X-braced; (c) K-braced; (d) full storey-height knee bracing 71 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved STAFFORDSMITH,HOENDERKAMPANDKUSTER Braced frames 12 Three of the most common types of braced frames are considered: Xbraced, K-braced and full storey height knee-braced bents For the calculation of the shearrigidity, hinges are assumed at the nodalpoints of the truss (a) Single X-braced (Fig 7(a)) where h is the storey height; the bay width; A , , the cross-sectional area of the column, and A d , the cross-sectional area of the brace ( b ) Double X-braced (Fig 7(b)) 2hl’E GA = + (d3/Ad) (h3/Ac) (c) K-braced (Fig 7(c)) hlZE GA = 2[(h3/Ac) + (d3/Ad)1 (6) Full storey-height knee bracing (Fig 7(d)) GA = 2hE (hz1/6r,) + (h3/uzAc)+ (d3/u2Ad) where I , is the moment of inertia of the girder and u is the distance from the column to the brace connection with the girder 13 Inthe case of a single-bay braced frame with non-moment resistant column splices the moment of inertia of a column I , is effectively zero The reand k Z ,infinity and unity respectively, are unsuitable for use in sulting values of CL’ equations (6)-(8) It is recommended therefore that, in these cases only, I , is assigned a fictitious small value equal to 0X)01A,12/2 This gives a value for kZ of 1.001 and a large but consistent value for a’, which allows equations ( H ) to operate c / ,’, / ‘ / / b , I b / I -I //// l Fig Interacting wall and column 720 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved SWAY RESISTANCE O F T A L L B U I L D I N G S T R U C T U R E S Fig Plan of singly-symmetric multi-bent structure while causing a negligible additional errorin the result 14 If the column splices and the column-to-girder connections in X- and K-braced frames are moment resistant, the characteristic parameters a’ and kZ should be evaluated by the procedure outlined for rigid frames The shear rigidity of such a braced frame is the sum of the shear rigidities of the rigid frame and braced frame modes of behaviour (equations (12) and (13, (16)or (17)) Coupled walls 15 The shear stiffness for a coupled shear wall, Fig 2, is given by equation (5) while the gross moment of inertia is given byequation (4),in which Wall and column 16 The racking shear rigidity of a column rigidly connected to a shear wall as shown in Fig 6EIb GA = {(l lh + rX1 + 2r + S)} (21) where r = b/l, S = (B - 3r - l)/@+ 2), in which B = (6E1JEIb)/(1/h).If a column is connected to each side of the wall, C A is the sum of G A values evaluated for the two sides The structural parameters E l and E l , are calculated as for a rigid frame The assumed linear distribution of axial strain in the vertical members across the bent will yield’better results for structures with walls located near the common centroidal axis than for walls whichare off-centre Multi-bent structures 17 The sway analysis is applicable to structures consisting of combinations of the described types of bent, provided the structure does not twist under horizontal loading Such structures include plan symmetrical arrangements of parallel bents subjected to symmetrical loading, as shown for example in Fig In these cases, the in-plane rigidity of the slabs constrains the translations of the bents to be identical at each floor level 18 The evaluation of overall parameters c? and kZ for the total structure is made in these cases by calculating first the stiffness parameters G A , x E l , , E l , and EA, c’ for each individual bent The overall parameters for a structure consisting of n bents is then given by 721 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved STAFFORD SMITH, HOENDERKAMP W = 30.0 kN/m A N D KUSTER height Fig 10 Plan of multi-bent example structure a’ = k’ = f: GAY^ C W,) (22) f: ( E I g ) i1(EA, c’) (23) If any of the bents is a core or a shear wall of inertia, say, I , , it may be notionally taken as a pair of non-coupled walls each of inertia I,/2 so that its contributionsto the numerator and denominator of equation (22) are, respectively, zero and E l , , and to thenumerator and denominator of equation (23) are, respectively, E I , and zero The term EI, in equations ( H ) refers to the whole structure and is ob- ” tained, therefore, from 1(H,)for the set of bents Examples 19 The procedure for the deflexion analysis for tall buildings is given in Appendix Three examples demonstrate how the method may be used to compare the effectiveness of alternative structural solutions for a proposed high-rise building The floor plan of the 24-storey, 84.0 m structure in Fig 10 shows a symmetrical arrangement consisting of two types of bent: A and B A uniformly distributed Table Structuralparameters of individual bents (equation number given in brackets) Type of bent I GA, N Coupled walls Shear wall Rigid frame Braced frame X log 2.535 ( ) 0.354 (12) 0,441 (17) Ic E I , Nm’ X 1OIZ 11 EAc’, Nm2 X 10lZ 1.783 (20) 1.496 0.384 0.332 (19) 0.267 0.004 (13) I EI,, Nm2 X 10l2 2.115(4) 0.267 1.500 (14) 0.384 (14) Table Sway results (equation number given in brackets; fig number givenin square brackets) ~ Example 5.465 1.336 0.152 0.186 111758 4.78 111443 0.343 37.443 1.0021 0.284 0.349 11747 11.24 11608 3.036 1.181 0.269 0.328 11740 11.36 11607 0.477 0.807 722 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved SWAY RESISTANCE OF TALL BUILDING STRUCTURES ' 9.25 m ' Fig 11 Coupled wall bent Fig 12 Rigid and braeed frame bents 723 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved STAFFORD SMITH, HOENDERKAMP AND KUSTER lateral load of 3.0 X 104 N/m and elastic moduli E of 2.5 X 10” N/m’ for concrete and 2.0 X 10” N/m2 for steel are assumed The storey height is 3.5 m and the concrete walls have a thickness of 0.25 m Table shows the four salient structural parameters for the four different types of assembly used in the examples followed by the equation number in brackets The non-dimensional parameters aH and k for each example are found in Table together with the final results for the sway (a) Example The interior assemblies, A, in the first structure consist of coupled walls as shown in Fig 11 The moment of inertia of the connecting beams is 5.4 X 10-3 m4 The exterior single shear walls, B, have a thickness of 0.25 m (b) Example A steel solution comprises a rigid frame and a braced frame (Fig 12) In the rigid frame, A: for the exterior columns I , = 4.0 X 10-3 m4 and A , = 5.0 X 10-’ m’; for the interior columns, I , = 6.0 X lO-’ m4, and A , = 6.0 X 10-’ m4; for the beams I , = 1.0 X 10-3 m4 In the braced frame, B: for the columns, A , = 6.0 X 10-’ m’; for the bracings A , = 0.3 X 10-’ m2 (c) Example In this mixed solution the rigid frame used in example is combined with the concrete shear wall of example 20 The examples illustrate the ease with which the sway resistance of different proposed structures can be compared All these cases are below the usual limiting sway index of approximately 1/450 Although the method is strictly accurate only for structures with uniform properties upthe height-an unlikely case in practice-a reasonable estimate of the total sway index for a structurethat reduces up the height can be made by analysing a uniform structure with properties of about 85% of those at the base of the actual structure Avalid comparison of the sway resistance can then be made from the corresponding maximum sway indices of the equivalent uniform structures 21 It is significant that the maximum storey sway index in a tall building is at the level where d2y/dx2 is equal to zero, i.e at the point of contraflexure in the deflexion curve Below that level the structure has a flexural shape with concavity downwind and, above that level, it has ashear mode shapewith concavity upwind Thus, the structureof example with a point of contraflexure at aboutone-fifth of the height from the base has a predominantly shear-mode configuration, while examples and 3, each with a point of contraflexure above mid-height have a stronger flexural configuration; these reflect the high flexural stiffnesses of the shear walls Comments on combining bents 22 An approximation exists in the procedure for lumping a set of different bents as defined by equations (22) and (23) The accuracy of the lumping depends on the similarity of the free deflexioncharacteristics of each bent A study hasbeen made of various structures of different heights with a wide range of values of the parameters aH and k2 In nearly all cases the approximate results for deflexion compared closely with stiffness matrix computer analyses The tests have shown that when combining bents of a similar type the maximum induced error will be about 10% in the worst cases and substantially less in the majority of structures When bents of different types are combined in a single structure it issuggested that 724 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved SWAY RESISTANCE O F TALL BUILDING STRUCTURES an additional parameter 1is calculated for each individual bent where This parameter is not necessary for single shear walls as they will not induce additional errors 23 If the ratio I, ,JAB < 10 the error of the proposed method will be less than 10%.For a I-ratio larger than 10, it is mathematically possible for the error to exceed 10% but, for all the practical sized structures devised by the Authors in this range, the errors were well within 10% In example AA = 2.36 X 10-4 and 1, = 1.15 X 10-3 giving a ratio of4.86 Conclusions 24 A generalized hand method forsway analysis is presented for high-rise structures comprising rigid frames, braced frames, coupled walls and shear walls, or combinations of these The method allows a rapid assessment of a structure's adequacy in sway resistance, as well as an easy comparison between the suitability in sway resistance of different structural proposals for a high-rise building It is necessary to calculate two non-dimensional parameters, aH and k, for the structure, which characterize its sway performance The parameters may be substituted in formulae, or used to refer to graphs, to obtain solutions for the deflected shape, the total sway index and the maximum storey sway index 25 Complex high-rise structures including flexural bents (shear walls, cores, braced frames) and shear bents (rigid frames) as well as flexural-shear bents (coupled walls) may be analysed by the method Complex structures with a mixture of bent types deflectin a flexural mode in the lower part (concavity downwind), and in a shear mode in the upper part (concavity upwind) with a point of contraflexure at the junction The more flexurally dominant the structure the more extensive the flexural region and thehigher the point of contraflexure 26 The method is restricted to structures in which the plan arrangement and horizontalloading are symmetrical so that thestructure does not twist The method is based on the assumption of, and therefore is accurate only for, structures that are uniform throughout their height It may be used however, to obtain the approximate sway in and comparisons of sway between, practical structures whose properties vary withheight Acknowledgement 27 The Authors wish to record their appreciation of the Natural Sciences and Engineering Research Council of Canada for their financial support for this research project Appendix : Procedure for the deflexion analysisof tall buildings Structural parameters 28 For each individual bent calculate to 21 E l , , GA, EAc' and EI, using equations 12 Non-dimensional parameters aH and kZ 29 For a single bent structure use equations (2) and(3) For a symmetric combination of bents refer to equations (22) and (23).Also find EI,for the total structure(= EIJ 725 c" Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved STAFFORD SMITH, HOENDERKAMP AND KUSTER Sway results for single bent structuresand multi-bent structures ( a ) Using diagrams 30 For a uniformly distributed loud, W , enter the graphsin Figs and with aH and kZ to obtain K,and K, which are then substituted into equations (10) and (ll), respectively Fig yields the locationof the maximum storey sway 31 For a concentrated top loud, P , use Figs 13 and 14 to obtain K, and K, which are substituted into equations (25) and (27), respectively For this type of loading the maximum storey sway is invariably at the top 32 For a triangularly distributed loud, w l , Figs 15 and 16 yield K,and K , which are tobe substituted into equations (29) and (31), respectively The location of the maximum storey sway is obtained from Fig 17 4.0 60 10 aH 20 30 60 100 60 100 Fig 13 Total sway factor &-concentrated topload I 1.0 2.0 4.0 I I I 1- 6.0 10 20 40 lYH Fig 14 Maximum storey sway factor K,-concentratedtop load 726 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved SWAY RESISTANCE O F TALL B U I L D I N G STRUCTURES ( b ) Using dejlexionformulae 33 For a uniformly distributed load, W , refer to equations (7H9) Equation (9) for the location of the maximum storey sway has to be solved by trial and error This can be avoided by using the diagramin Fig instead 34 For a concentrated top load, P , use equations (26) and (28) to obtain K,and K,which are then to be substituted into equations (25) and (27), respectively For a concentrated top load the maximum storey sway isinvariably at the top of the structure 35 For a triangularly distributed load, W,,equations (30) and (32) yield K, and K, which are tobe substituted into equations (29) and (31), respectively The location of the maximum storey sway is obtained by solving equation (33) by trial and error This canbe avoided if the diagram in Fig 17 is used instead 20 4.0 6.0 10 20 60 40 100 aH Fig 15 Total sway factor K,triangularly 0.11 1.0 distributed load I I I I I I I 2.0 4.0 6.0 10 20 40 60 I 100 aH Fig 16 Maximum storey sway factorK,-triangularlydistributed load 727 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved STAFFORD SMITH, HOENDERKAMP AND 1.01 1.0 2.0 6.0 4.0 KUSTER 10 I I 20 40 60 I 100 OH Fig 17 Location of maximum storey sway-triangularly distributed load Appendix 2: Alternative loading cases Concentrated load P at the top 36 The total sway index is where sinh(kaH) l The distribution factor K , can be rapidly evaluated by using the graphin Fig 13 37 The maximum storey sway for this type of loading occurs at the top of the structure regardless of the structuralparameters (g) m.r PH2 =-xK, EL in which Fig 14 can be used to find K , Triangularly distributed load,w l ( l - x/H), where wl indicates the maximum intensity at the top of the structure 38 The total sway index is where the distributionfactor K 11 1 +r k - { m ‘-120 (kaH)/Z}sinh(kaH) - + cosh(kaH) + { l/(kaH) ( k ~ r Hcosh(kaH) )~ - 728 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved SWAY RESISTANCE O F T A L L B U I L D I N G S T R U C T U R E S K , may be evaluated using Fig 15 39 The maximum storey sway W, = H3 EI, X K, in whichthe distribution factor K,mustbe evaluated at the point of inflection +m [(kaH)2 { -E+T('")'+)' K ' =-!+!(.?)3-.L(.?)4 H 24 H (kaH)* - sinh(kaH)(l - x/H) + { I/(kaH) - (kaH)/2}cosh(kax) (kaH)3 cosh(kaH) Values of K,can quickly be found in Fig 16 40 The location of the maximum storey sway isdefined by the following condition d2Y -dx2 =o=- (x/H - 1) (iy -1 X +={m - (H)I - {l/(kaH) - (kaH)/2}sinh(kax) + cosh(kaH)(I - x/H) (kaH)' cosh(kaH) Fig 17 may be used to find values for x/H References STAFFORDSMITHB et al A generalized approach to the deflection analysis of braced frame, rigid frame and coupled wall structures Can J Ciu Engng 1981.8, June, NO.2, 23&240 STAFFORDSMITHB et al Generalized method for estimating the drift in high-rise structures J Struct Diu Am Soc Civ Engrs Submitted for publication BECKH Contribution to the analysis of coupled shear walls Proc Am Concr Inst.,1962, 59, Aug., NO.8,1055-1070 ROSMAN R.Approximate analysis of shear walls subject to lateral loads Proc Am Concr Inst., 1964,61, June, No 6,717-734 COULL A and CHONDHURY J R Stresses and deflections in coupled shear walls Proc Am Concr Inst.,1967,64, Feb., No 2,65-72 KUSTERM A parameter study of tall building structures McGill University, Montreal, MEng thesis, 1978 729 Downloaded by [ UNIVERSITY OF CAMBRIDGE] on [29/07/16] Copyright © ICE Publishing, all rights reserved [...]... limiting sway index of approximately 1/450 Although the method is strictly accurate only for structures with uniform properties upthe height-an unlikely case in practice -a reasonable estimate of the total sway index for a structurethat reduces up the height can be made by analysing a uniform structure with properties of about 85% of those at the base of the actual structure Avalid comparison of the sway resistance... lumping a set of different bents as defined by equations (22) and (23) The accuracy of the lumping depends on the similarity of the free deflexioncharacteristics of each bent A study hasbeen made of various structures of different heights with a wide range of values of the parameters aH and k2 In nearly all cases the approximate results for deflexion compared closely with stiffness matrix computer analyses... this range, the errors were well within 10% In example 2 AA = 2.36 X 10-4 and 1, = 1.15 X 10-3 giving a ratio of4 .86 Conclusions 24 A generalized hand method forsway analysis is presented for high-rise structures comprising rigid frames, braced frames, coupled walls and shear walls, or combinations of these The method allows a rapid assessment of a structure's adequacy in sway resistance, as well as an... shapewith concavity upwind Thus, the structureof example 2 with a point of contraflexure at aboutone-fifth of the height from the base has a predominantly shear-mode configuration, while examples 1 and 3, each with a point of contraflexure above mid-height have a stronger flexural configuration; these reflect the high flexural stiffnesses of the shear walls Comments on combining bents 22 An approximation... location of the maximum storey sway has to be solved by trial and error This can be avoided by using the diagramin Fig 5 instead 34 For a concentrated top load, P , use equations (26) and (28) to obtain K,and K,which are then to be substituted into equations (25) and (27), respectively For a concentrated top load the maximum storey sway isinvariably at the top of the structure 35 For a triangularly... height It may be used however, to obtain the approximate sway in and comparisons of sway between, practical structures whose properties vary withheight Acknowledgement 27 The Authors wish to record their appreciation of the Natural Sciences and Engineering Research Council of Canada for their financial support for this research project Appendix 1 : Procedure for the deflexion analysisof tall buildings... STRUCTURES an additional parameter 1is calculated for each individual bent where This parameter is not necessary for single shear walls as they will not induce additional errors 23 If the ratio I, ,JAB < 10 the error of the proposed method will be less than 10%.For a I-ratio larger than 10, it is mathematically possible for the error to exceed 10% but, for all the practical sized structures devised by the Authors... an easy comparison between the suitability in sway resistance of different structural proposals for a high-rise building It is necessary to calculate two non-dimensional parameters, aH and k, for the structure, which characterize its sway performance The parameters may be substituted in formulae, or used to refer to graphs, to obtain solutions for the deflected shape, the total sway index and the maximum... storey sway index 25 Complex high-rise structures including flexural bents (shear walls, cores, braced frames) and shear bents (rigid frames) as well as flexural-shear bents (coupled walls) may be analysed by the method Complex structures with a mixture of bent types deflectin a flexural mode in the lower part (concavity downwind), and in a shear mode in the upper part (concavity upwind) with a point of. .. publication 3 BECKH Contribution to the analysis of coupled shear walls Proc Am Concr Inst.,1962, 59, Aug., NO.8,1055-1070 4 ROSMAN R.Approximate analysis of shear walls subject to lateral loads Proc Am Concr Inst., 1964,61, June, No 6,717-734 5 COULL A and CHONDHURY J R Stresses and deflections in coupled shear walls Proc Am Concr Inst.,1967,64, Feb., No 2,65-72 6 KUSTERM A parameter study of tall building