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Simplified torsion analysis for high-rise structures
Building andEnv#onment, Vol. 23, No. 2, pp. 153-158, 1988.
0360-1323/88 $3.00+0.00 © 1988 Pergamon Press plc.
Printed in Great Britain.
Simplified Torsion Analysis for High-rise Structures J. C. D. H O E N D E R K A M P * B. S T A F F O R D S M I T H : ' A simple approximate hand method of analysis is presented for determining the internal forces in multi-storey structures subject to torsional loading. The buildings may include plan-symmetric combinations of coupled walls, rigid frames, wall-frames, single shear walls, rigid frames with central shear walls and braced frames. The bending deformations in all individual structural members are taken into account as well as the axial shortening and lengthening of the columns. The method is based on the continuous medium analogy which enables the analysis to be reduced to simple closed formulae. It is restricted to structures with uniform geometry up the height and linear elastic behaviour of the structural members. It provides a simple and rapid means of estimating the internal forces in each individual structural element and it is appropriate to the preliminary stages of the design of proposed tall building structure.
(Tl)'o Torsion due to shear forces in continuous media of all bents Vbj Shear force in beam V~j Total shear force in column or wall Vi Total shear force in bent Vtj Complementary shear force in column w Intensity of uniformly distributed load w~ Maximum intensity of triangularly distributed horizontal load x Distance measured from top of structure y Horizontal deflection of structure z Location of bent Deflection parameter ~0 Rotation parameter Characteristic parameter 0 Rotation of structure in horizontal plane.
NOMENCLATURE Aj Cross-section area of column b Span of beam c Distance from neutral axis of column to common centroid of bent e Eccentricity of applied load E Modulus of elasticity EA~ Torsional parameter pertaining to axial stiffness of vertical members EAc 2 Flexural parameter pertaining to axial stiffness of vertical members E1 Flexural stiffness EI,o Warping stiffness Fig Gross flexural stiffness parameter Elo,g Gross warping stiffness parameter GA Racking shear rigidity GJ Torsional shear rigidity h Storey height H Total height of structure i Bent number j Column number k Deflection parameter ko Rotation parameter 1~,lr Distance between neutral axis of column and points of contraflexure in adjacent continuous medium Mbj Bending moment in beam Mcj Bending moment in column M~ Bending moment in bent P Horizontal concentrated load q General horizontal load Tj Axial force in column Tmj Shear force in bay (Th)" Incremental force in column (T1) Bending moment due to axial forces in vertical members of all bents (T l)' Storey shear force in continuous media of all bents (TI)~ Bending moment due to axial forces in vertical members of bent (TI)~ Storey shear force in continuous medium of bent "i" (Tl)o Bi-moment due to axial forces in vertical members of all bents
INTRODUCTION I N G E N E R A L high-rise structures comprise various types of lateral load-resisting structural elements with different load-deflection characteristics. If the building structure is subjected to lateral forces or torsion, a redistribution o f the horizontal load will occur between the various elements throughout the height of the building. The loads will flow between bents which deflect in a shear mode, e.g. rigid frames, a bending mode, e.g. shear walls, or elements which display a combination of both bending and shear modes of behaviour such as coupled walls. Consequently it is necessary to determine the complex distribution of load among the different structural assemblies in order to achieve a proper design. The method of torsion analysis presented in this paper adopts a generalized approach by considering many different types of high-rise bents as members of a single structural family of cantilevers whose behaviour can be represented by the continuous medium theory for coupled walls. The continuous medium method of analysis as applied to coupled walls has been available and used in practice for more than twenty years [1-3]. It has been shown how this coupled wall theory can be
* Ove Arup Partnership, London. t Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, Canada. 153
154
J. C. D. H o e n d e r k a m p and B. S t a f f o r d S m i t h
interpreted to represent the behaviour of other types of lateral load resisting elements such as rigid frames, braced frames and wall-frames [4, 5]. The Authors have also shown that the theory can be expanded to include the analysis of plan-symmetric multi-bent structures subject to symmetric lateral loads [6]. A torsion analysis for threedimensional buildings consisting of cantilevered cores and a set of identical coupled walls was presented earlier [7]. The continuous medium theory has been further extended to yield the rotations of symmetric multi-bent structures subject to torsion [8]. The method presented here allows for combinations of a wide variety of structural elements in current use. It will be shown in this paper that it can also be used to obtain the forces in the individual beams, walls, columns and bracing systems of each bent in the structure. The proposed method requires that for each bent three distinct structural parameters be evaluated which correspond to the major modes of behaviour. The parameters will in total account for the bending deformations in the vertical and horizontal members as well as the axial deformations in the walls, columns and bracing members. The rigidities of each bent are then transformed and summed to form rotation parameters for the total multi-bent structure. This is done by taking the relative location of the bents into account. In the continuous medium approach a representative differential equation written in terms of the characteristic parameters will yield closed form solutions for the loads resisted by each bent. These can then be further distributed to give the forces in the individual members for three distributions of torsional loading; uniformly distributed, triangularly distributed and a concentrated top loading. Because the coupled wall theory is based on the assumption of uniformity of the structure with height, the method is more accurate for structures which are close to that condition. However, valid comparisons between performances of alternative structural arrangements may be obtained for non-uniform structures by this method. The solutions are also of value in indicating important structural parameters in the design of three-dimensional structures.
Fig. 1. Plan--symmetric structure. lementary shear stiffness across the structure. The racking shear rigidity can be defined as the horizontal shear force to cause unit average slope in the storey height segment [111. (3) EAc 2. A flexural stiffness parameter expressing the contribution from the axial stiffness of the vertical members to the total moment resistance. In this calculation it is assumed that the transverse connecting member system is rigid, i.e. the vertical strain distribution across the bent is taken to be linear. The parameter expresses a summation of the second moment of area of the column sections about the common centroidal axis of the bent. The characteristic structural parameters take into account all bending deformations in the vertical and horizontal members of the bent as well as the axial deformations in the vertical members and any bracing systems. ROTATION OF SYMMETRIC STRUCTURES The deflection equation for planar coupled shear walls has been shown to be applicable to three-dimensional multi-bent structures subjected to torsion [8]. The three salient characteristic torsional stiffnesses representing the total structure are obtained by combining the lateral stiffnesses of each bent and their locations with respect to the axis of rotation, i.e. the x-axis as shown in Fig. 1. The torsional parameters are determined as follows : (1) Warping stiffness El, o = E E l , x z•.
(1)
(2) Torsional stiffness STRUCTURAL
PARAMETERS BENTS
OF PLANAR
It has been shown that the continuous medium theory for coupled shear walls is appropriate for the deflection analysis of single bent structures such as rigid frames, braced frames, wall-frames, rigid frames with centrally located walls [9] and braced frames with multi-storey bracing systems [10]. In the analysis of these types of structures the bents are represented by three distinct structural parameters which are defined as follows : (1) El. This term expresses the flexural rigidity of all vertical members in the bent which are continuous up the total height of the structure. It is obtained by assuming that all horizontal members and bracing systems are cut. (2) GA. This represents the racking shear rigidity of the bent. It is a measure of the vertical shear stiffness between the vertical components, as induced by the transverse connecting member system, and of the comp-
GJ ~- Z G A , × V .
(2)
(3) Rotational stiffness pertaining to the axial rigidity of the vertical members EA~ = Y, EAc~ x z~
(3)
in which Eli is the flexural rigidity of bent "i" and z is the distance measured from the origin. Similarly for equations (2) and (3). The rotations up the height of the structure are given by well a 6 ~ H ) + 24 i H ) 1
+~
[ 1 -- (x/H) 2
(~k~
cosh [(kaH)o (1 - x/H)]-- 1
-~
(k~H)~cosh (k~H)o
(k~H)o [sinh (kaH)o - sinh (kax)o]~ ]
(k~H)~
;j (4)
where w x e represents the torsion per unit height which
Simplified Torsion Analysis f o r High-rise Structures consists of a uniformly distributed lateral load w, acting at a distance e from the origin along the z-axis, H is the total height of the structure and x is the distance measured from the top. A gross warping parameter is defined as
155
M , is unknown and thus (TI), must be calculated separately. (TI), is a bending moment due to axial forces in the vertical members. The generic fourth order differential equation for a single bent subjected to a general horizontal load of intensity q [12], is given by
EI~,g = ~,EIg, x z~,
(5)
1 y~V _ (ka)~y" = ~ {q,(x)- M,(x)fl,},
EIg, = EI~+ EAc~.
(6)
where y, indicates the deflections in bent "i" and can be expressed as
(13)
where
Two characteristic rotation parameters are expressed as
Yi = z, x 0.
ct2 = aJ/EIo.
(7)
This allows the equations for the bending moment and load intensity for the bent to be rewritten as
k~ = EI~,a/EA,o.
(8)
and
The rotation equation is written in terms of two characteristic non-dimensional parameters, (aH)o and ko, which completely govern the twisted shape of the structure. FORCES IN THE MULTI-BENT STRUCTURE
The equations for the bi-moments and torque in a multi-bent structure subject to a torsional load w ' e , are given by wex 2 2 = EI~O"+ (TOo,
(9)
wex = EI~O" + (TO'o,
(10)
and
where EIoO" and EI~O" are forces in the continuous medium model which can be obtained from the second and third derivatives of the rotation equation, equation (4). The bi-moments due to the axial forces in the vertical members of the structure, (TOo, are obtained by substituting the second derivative of equation (4) into equation (9).
(14)
M, = Eli x yT +(TI),,
(15)
M7 = ql,
(16)
q, = EI, x yl v + (Tl):'.
(17)
and since
An additional characteristic parameter is defined as t? = GA,/EAc?.
(18)
Substituting equations (15) and (17) into equation (13) leads to a second order differential equation for (Tl),. Further substituting equation (14) and considering the boundary conditions of zero axial force in the columns at the top and zero shear force in the continuous medium at the base of the structure will yield an expression for the bending moments in bent "i" caused by the axial forces in the vertical members.
2
[
well (Tl), = EIo~-~x (EAc~ x z3 (C, - C2) sinh (fix),
-- (C3 + C4) cosh (fix), + ~
+ C3
fcosh [(kan)o(1 - x/ H)] + (k~n)o sinh (kax)o ] ]
we [sinh ( k a H ) o - (kaH)o (Tl)o = k~02 [_ ~ x sinh (kax)o (k~tx)2 . ] -cosh(k~tx)o+~+ 1 .
(19) in which (ll)
The torsion (Tl)'o in the structure corresponding to horizontal racking forces, which are complementary to the shear forces in the continuous media of all bents, is given by the first derivative of equation (11). The four continuous functions, EI~O", El, off", (TOo and (T/)~ represent the forces in the total multi-bent structure that must be distributed to the individual bents.
(19a)
1 + C~(k~n)~ C2 = (fill), cosh (fill),' 1
(19b)
1
C3 = (flH)~
(k~ - 1) (kctH) 2'
(fill), C, = ( k g - 1) (kaH)2o{(flH)? - (kaH)g}"
(19c) (19d)
There exists a special case for which fl, = fl, i.e. all bents have identical values for fl, then
FORCES IN A SINGLE BENT
The bending moment equation for a single bent in a structure subject to torsion can be expressed as Mt = E l i x z , × O " +(Tl)i
Ct = (C3 + C,) tanh (fill),,
(12)
in which Eli x zl x 0" represents the sum of the bending moments in the vertical members of bent "i" and can easily be obtained from the second derivatives of equation (4). The total bending moment carried by the bent,
Ci = C2 = 0,
(20a) 1
C3 = - Ca = (kaH)02 ,
(20b)
and equation (19) can be simplified to (T0'=
EAci2 x zi---o,Tl) ( EA~,
(21)
J. C. D. Hoenderkamp and B. Stafford Smith
156
_
_
I-
COMMON CENTROIDAL AXIS
bers replaced by continuous media. This method can be applied to wall-frames, rigid frames, coupled walls as well as rigid frames with central walls. The assumed linear strain distribution across the bent yields an equation for the axial force in column '~/" at a distance c~ from the common centroidal axis.
(EAc) i . . . . T~ - ( ~ c 2 ~ I (11)i ,
--_T,_ !
F
where (Tl)i is obtained from equation (19). The shear force (T/)~ is distributed between the individual continua of the bays according to a procedure similar to distributing the shear stress across the depth of a beam. These individual shear forces are uniform across the width of the bays.
~(EAc)~
r;,j- ( ~ . ~1
(24)
C
(Tl);,
(25)
,..1
Fig. 2. Multi-bay "continuous" bent. where (Tl)o is given by equation (11). It is shown by equation (21) that the "axial" moment for the multi-bent structure, (T1)o, is distributed to the individual bents in proportion to their axial stiffness parameters, EAc 2, and their distances from the centre of rotation, the x-axis. The result of this is that the floor rotations in the vertical planes of the bents, due to axial deformations in the columns, are proportional to the distances from the axis of symmetry of the structure. This assumption was made earlier for the deflection analysis of multi-bent structures [6, 8, 9, 11]. The shear force equation for a single bent is given by the first derivative of equation (15), i.e.
V, = Elgy;" + (Tl);,
V,j = T~I~+ T~lr,
(26)
in which l~ and lr are the distances between the neutral axis of the column and the points of contraflexure in the adjacent continuous media. T't and T~ are obtained from equation (25). The total shear force in a column or wall is the sum of the racking shear force from equation (26) and a part of the single curvature shear force Eliyi", thus
(22)
where Eliyi" represents the sum of the shear forces due to single curvature flexure in the vertical members of bent " i " and can easily be obtained from the third derivative of equations (4) and (14). The shear force (TI); corresponding to racking of the bent is given by the first derivative of equation (19). In the special case where all bents have identical values o f / L the equation for this shear force simplifies to
EAc~ x zi ( TI) ; ( Tl)'o, EA,~
where (Tl)~ can be obtained from equation (19) and Y~(EAc)ex is a summation of the products of the excluded column areas and their distances to the common centroidal axis of the bent. The shear forces in the continuous media must be transferred to the vertical members. It can easily be shown [11] that the complementary shear force in the column can be taken as
V,.j = Ely;"+ V~j,
(27)
in which EIj represents the flexural rigidity of a single column or wall. The total bending moment in a floor to floor column is a combination of two components. At the top of the column /''~h
M,.j ( t o p ) = EIjy:'-~)V,~,.
(28)
At the bottom of the column (23)
in which (T1)'o is the torsion due to racking shear forces in the bents and is given by the first derivative of equation (11). The two components for bending, EIiy;' and (Tl)~, together with the components for shear, ELy," and (Tl)~, represent the four continuous functions of the forces in bent "i". These must be distributed further to the individual horizontal and vertical members as well as the bracing systems in that bent.
M,.j (bottom) = E l y ; ' -
~ V,:j,
(29)
where h indicates the storey height. It is assumed that the forces in the storey height columns and walls are best represented by the mid-storey values from the continuous functions.
Beams The shear force in a beam at level x can be expressed as
FORCES IN S T R U C T U R A L E L E M E N T S
Columns The procedure for the distribution of the continuous forces in a bent to individual members will be demonstrated for the structure shown in Fig, 2. It represents the left half of a multi-bay bent with the horizontal mem-
Vhj =
~xx+h/2T~,i dx
(30)
~-h/2
where T ' j is the shear force in the continuous medium in the bay in question and is obtained from equation (25). Alternatively, this force may be estimated as follows
Simplified Torsion Analysis for Hiyh-rise Structures
(T2)'
(T2)' Fig. 3. Single storey continuum.
Vb: = T~: x h.
(31)
For the calculation of the beam shear force this way, it is suggested that the value of T~,j be obtained at the floor level. The maximum bending moment in the girder may be approximated as b Mbj = Vbj x ~,
(32)
where b represents the length of the beam. In this equation it is assumed that the point of contraflexure occurs at midspan. This is correct only if the beam-column joint rotations are equal across a floor level. For other cases the bending moments in the beams may be estimated by distributing the difference between the column moments above and below the floor level.
Braced frames The bending moment equation for a single braced bent in a structure is given by equation (12). If the beamcolumn joints consist of pinned connections, the flexural stiffness of the columns is negligible and the bending moments, Eliy~', as well as the shear forces, EI~y~", may be taken to be zero. The axial forces in the columns can be obtained from equation (19). The complementary shear force (T/)' in a braced frame is given by the first derivative of equation (19) and is carded by the continuous medium as the horizontal component of axial forces in the bracing system. It represents the storey shear and is to be evaluated at mid-storey level. The shear force (Th)' is carried by the continuous medium as the vertical component of the axial forces in the braces. It represents the incremental axial forces in the columns and is obtained as follows h
(Th)' = (Tl)' x 7
(33)
where l represents the width of the braced bay. Figure 3 shows the shear forces in a single storey segment in the continuum. The forces in the x- and k-bracings can be approximated by treating the braced bay as a truss.
ACCURACY An approximation exists in the stage of combining a set of bents into a single structure represented by only three rotational properties ; EI~, GJand EAo. A rigorous torsion analysis of multi-bent structures consisting of different types of bents is extremely complex and unsuit-
157
able for a rapid hand method of analysis. In determining the characteristic rotation parameters, ~02 and kg, for the total structure, it is assumed that the rotations of the bents in their vertical planes are constrained at each floor level, i.e. the slope of the floor, as a result of axial deformations in the columns, at each bent is proportional to its distance from the x-axis. It should be noted that this is not identical to the shape of the deflection profile, which includes the slope caused by shear. A study has been made of various structures of different heights with a wide range of values for the characteristic non-dimensional parameters. The tests have shown that when a number of identical bents are combined, the maximum induced error will be less than 5%. The t-values for all bents are identical in this case. The inclusion of single shear walls will not alter the accuracy of the analysis since the t-value for a shear wall does not exist. Additional tests have shown that when combining bents of similar type, the maximum induced error will be 10% in the worst cases and substantially less in the majority of structures. When bents of different types are combined in a single structure the following can be used as a guide to the accuracy of the continuous medium analysis of tall building structures : if the ratio fl(max)/fl(min) < 3.0 the error in the proposed method of analysis will be less than 10%. fl(max) and fl(min) are the two bents in the structure with the largest and smallest t-values, respectively. For ratios larger than 3.0, it is mathematically possible for the error to exceed 10% but, for all the practical sized structures devised by the Authors, the errors were well within 10% [4,6, 11].
CONCLUSIONS A generalized hand method for torsion analysis is presented for high-rise structures consisting of combinations of rigid frames, braced frames, coupled walls, wall-frames and single shear walls. The method allows a rapid assessment of internal forces in the beams, walls, columns and bracing members of each bent in the structure. It requires the calculation of two non-dimensional characteristic parameters (~H)0 and k~ which characterize the performance of the total structure subject to torsional loading. The method of analysis is restricted to structures in which the plan arrangement is symmetric. The theory is based on the assumption of, and therefore is most accurate for, structures that are uniform throughout their height. It may be used also, however, for practical structures whose properties vary with height, but with less accurate results. The information obtained from this method should give the design engineer an easy means of comparing the suitability of alternative structural proposals, in addition to providing initial structural data for a more accurate computer analysis, or allowing a check on the reasonableness of the final output of a computer analysis. Acknowledgement--The Authors wish to acknowledge the support given to this research by the Natural Sciences and Engineering Research Council of Canada.
J. C. D. Hoenderkamp and B. Stafford Smith
158
REFERENCES l. 2. 3. 4. 5. 6. 7. 8. 9. I0. 11. 12.
APPENDIX:
H. Beck, Contribution to the analysis of coupled shear walls. Proc. Am. Conc. Inst. 59, 1055-1070 (1962). R. Rosman, Approximate analysis of shear walls subject to lateral loads. Proc. Am. Conc. Inst. 61, 717-734 (1964). A. Coull and J. R. Choudhury, Stresses and deflections in coupled shear walls. Proc. Am. Const. Inst. 64, 65 72 (1967). B. Stafford Smith, M. Kuster and J. C. D. Hoenderkamp, A generalized approach to the deflection analysis of braced frame, rigid frame and coupled wall structures. Can. J. Cir. Engng. 8, 230-240 (1981). B. Stafford Smith and J. C. D. Hoenderkamp, Simple deflection analysis of planar wall-frames. Proc. Asian Regional Conf. on Tall Buildings and Urban Habitat, 33-41. Kuala Lumpur (1982). B. Stafford Smith, M. Kuster and J. C. D. Hoenderkamp, Generalized method for estimating the drift in high-rise structures. Jnl. of the Structural Division, ASCE. 110, 1549-1562 (1984). A. Coull, Interactions between coupled shear walls and cantilever cores in three-dimensional regular symmetrical cross-wall structures. Proc. Instn. Cir. Engrs. Lond. 55, 827-840 (1973). J . C . D . Hoenderkamp and B. Stafford Smith, Approximate rotation analysis for plan-symmetric high-rise structures. Proe. Inst. Cir. Engrs. Lond. 83, 755-767 (1987). B. Stafford Smith, J. C. D. Hoenderkamp and M. Kuster, A graphical method of comparing the sway resistance of tall building structures. Proc. Inst. Cir. Engrs. Lond. 73, 713-729 (1982). J . C . D . Hoenderkamp, A generalized hand method of analysis for tall building structures subject to lateral loads. Ph.D. Thesis, McGill University, Montreal, Canada (1983). J.C.D. Hoenderkamp and B. Stafford Smith, Simplified analysis of symmetric tall building structures subject to lateral loads. Proc. Third International Conf. on Tall Buildings, 28-36. Hong Kong (1984). D.P. Abergel, Deflection solutions of special coupled wall structures by differential equations. M.Eng. Thesis, McGill University, Montreal, Canada (1981).
Two other loading cases which may be required in practice are : (1) A torsional load, P x e, resulting from a concentrated horizontal load P at the top of structure located a distance e from the origin along the z-axis. The rotations up the height of the structure are
PeH3y1
o(x):
1
ALTERNATIVE LOADING CASES
b-
l f x~
l f x~ 3
1
[1
l[x~2
l(x~3_
1-x/n h
(k:
cosh [(kaH)o(1 - x/ H)] - I (kotH)g cosh (k~H)o
+
{ 1/(k~H)o - (kaH)o/2} [sinh (k~H)o - sinh (kax)o!~] (k~H)~ cosh kaH)o JJ
(36)
The bending moments in bent " i " due to axial forces in the columns are given by
1
+ tmJ + k02-1
f l -xlH sinh (kctx)o - s i n h (kotH)o~] ×l ~ + (kctH)~ cosh (k~tH)o J [
f
( ~'" l t )i = ~w~eHZ x (EAc~ x zi) I ( C ] + C5) sinh (flx)i (34)
The bending moments in bent " i " due to axial forces in the columns are given by (TI), = ~
x (EAc~ x zf) - C 2 sinh (fix), + ~
f (kaH)o sinh (kax)o ~] -cosh(k~H)o - •j j' (35)
+ U 4 ~t
in which C2 and C4 are given by equations (19b) and (19d) respectively. If the/1-values for each bent are equal, then C2 and C4 are given by equations (203) and (20b). (2) A torsion load, wi x e, resulting from a triangularly distributed load with intensity w] at the top down to zero at the base of the structure and located a distance e from the origin along the z-axis. The rotations up the height of the structure are
""
w,eH4 [ ~ 0 l llx ~ I ( x ' ] ' _ 1 (xX] ' tx)=-~-g -8tH)+~\H] 120\HJ
_ [cosh [(k~H)o(1 -x/H)] +U4~ ( cosh (kc~H)o
{ I/(kaH)o -- (kotH)o/2} sinh (kax)0~] cosh (kotH)o JJ'
(37)
in which C , C3 and C4 are given by equations (19a), (19c) and (19d) respectively, C5 -
C 3- 1/2 + C4(kotH)o { 1/(kaH)o - (kocH)o/2}
(/~H), cosh (/~H),
(37a)
If the fl-values for each bent are equal, then C] = C5 = 0 and C3 and C4 are given by equation (20b).
(37b)
0360-1323/88 $3.00+0.00 © 1988 Pergamon Press plc.
Printed in Great Britain.
Simplified Torsion Analysis for High-rise Structures J. C. D. H O E N D E R K A M P * B. S T A F F O R D S M I T H : ' A simple approximate hand method of analysis is presented for determining the internal forces in multi-storey structures subject to torsional loading. The buildings may include plan-symmetric combinations of coupled walls, rigid frames, wall-frames, single shear walls, rigid frames with central shear walls and braced frames. The bending deformations in all individual structural members are taken into account as well as the axial shortening and lengthening of the columns. The method is based on the continuous medium analogy which enables the analysis to be reduced to simple closed formulae. It is restricted to structures with uniform geometry up the height and linear elastic behaviour of the structural members. It provides a simple and rapid means of estimating the internal forces in each individual structural element and it is appropriate to the preliminary stages of the design of proposed tall building structure.
(Tl)'o Torsion due to shear forces in continuous media of all bents Vbj Shear force in beam V~j Total shear force in column or wall Vi Total shear force in bent Vtj Complementary shear force in column w Intensity of uniformly distributed load w~ Maximum intensity of triangularly distributed horizontal load x Distance measured from top of structure y Horizontal deflection of structure z Location of bent Deflection parameter ~0 Rotation parameter Characteristic parameter 0 Rotation of structure in horizontal plane.
NOMENCLATURE Aj Cross-section area of column b Span of beam c Distance from neutral axis of column to common centroid of bent e Eccentricity of applied load E Modulus of elasticity EA~ Torsional parameter pertaining to axial stiffness of vertical members EAc 2 Flexural parameter pertaining to axial stiffness of vertical members E1 Flexural stiffness EI,o Warping stiffness Fig Gross flexural stiffness parameter Elo,g Gross warping stiffness parameter GA Racking shear rigidity GJ Torsional shear rigidity h Storey height H Total height of structure i Bent number j Column number k Deflection parameter ko Rotation parameter 1~,lr Distance between neutral axis of column and points of contraflexure in adjacent continuous medium Mbj Bending moment in beam Mcj Bending moment in column M~ Bending moment in bent P Horizontal concentrated load q General horizontal load Tj Axial force in column Tmj Shear force in bay (Th)" Incremental force in column (T1) Bending moment due to axial forces in vertical members of all bents (T l)' Storey shear force in continuous media of all bents (TI)~ Bending moment due to axial forces in vertical members of bent (TI)~ Storey shear force in continuous medium of bent "i" (Tl)o Bi-moment due to axial forces in vertical members of all bents
INTRODUCTION I N G E N E R A L high-rise structures comprise various types of lateral load-resisting structural elements with different load-deflection characteristics. If the building structure is subjected to lateral forces or torsion, a redistribution o f the horizontal load will occur between the various elements throughout the height of the building. The loads will flow between bents which deflect in a shear mode, e.g. rigid frames, a bending mode, e.g. shear walls, or elements which display a combination of both bending and shear modes of behaviour such as coupled walls. Consequently it is necessary to determine the complex distribution of load among the different structural assemblies in order to achieve a proper design. The method of torsion analysis presented in this paper adopts a generalized approach by considering many different types of high-rise bents as members of a single structural family of cantilevers whose behaviour can be represented by the continuous medium theory for coupled walls. The continuous medium method of analysis as applied to coupled walls has been available and used in practice for more than twenty years [1-3]. It has been shown how this coupled wall theory can be
* Ove Arup Partnership, London. t Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, Canada. 153
154
J. C. D. H o e n d e r k a m p and B. S t a f f o r d S m i t h
interpreted to represent the behaviour of other types of lateral load resisting elements such as rigid frames, braced frames and wall-frames [4, 5]. The Authors have also shown that the theory can be expanded to include the analysis of plan-symmetric multi-bent structures subject to symmetric lateral loads [6]. A torsion analysis for threedimensional buildings consisting of cantilevered cores and a set of identical coupled walls was presented earlier [7]. The continuous medium theory has been further extended to yield the rotations of symmetric multi-bent structures subject to torsion [8]. The method presented here allows for combinations of a wide variety of structural elements in current use. It will be shown in this paper that it can also be used to obtain the forces in the individual beams, walls, columns and bracing systems of each bent in the structure. The proposed method requires that for each bent three distinct structural parameters be evaluated which correspond to the major modes of behaviour. The parameters will in total account for the bending deformations in the vertical and horizontal members as well as the axial deformations in the walls, columns and bracing members. The rigidities of each bent are then transformed and summed to form rotation parameters for the total multi-bent structure. This is done by taking the relative location of the bents into account. In the continuous medium approach a representative differential equation written in terms of the characteristic parameters will yield closed form solutions for the loads resisted by each bent. These can then be further distributed to give the forces in the individual members for three distributions of torsional loading; uniformly distributed, triangularly distributed and a concentrated top loading. Because the coupled wall theory is based on the assumption of uniformity of the structure with height, the method is more accurate for structures which are close to that condition. However, valid comparisons between performances of alternative structural arrangements may be obtained for non-uniform structures by this method. The solutions are also of value in indicating important structural parameters in the design of three-dimensional structures.
Fig. 1. Plan--symmetric structure. lementary shear stiffness across the structure. The racking shear rigidity can be defined as the horizontal shear force to cause unit average slope in the storey height segment [111. (3) EAc 2. A flexural stiffness parameter expressing the contribution from the axial stiffness of the vertical members to the total moment resistance. In this calculation it is assumed that the transverse connecting member system is rigid, i.e. the vertical strain distribution across the bent is taken to be linear. The parameter expresses a summation of the second moment of area of the column sections about the common centroidal axis of the bent. The characteristic structural parameters take into account all bending deformations in the vertical and horizontal members of the bent as well as the axial deformations in the vertical members and any bracing systems. ROTATION OF SYMMETRIC STRUCTURES The deflection equation for planar coupled shear walls has been shown to be applicable to three-dimensional multi-bent structures subjected to torsion [8]. The three salient characteristic torsional stiffnesses representing the total structure are obtained by combining the lateral stiffnesses of each bent and their locations with respect to the axis of rotation, i.e. the x-axis as shown in Fig. 1. The torsional parameters are determined as follows : (1) Warping stiffness El, o = E E l , x z•.
(1)
(2) Torsional stiffness STRUCTURAL
PARAMETERS BENTS
OF PLANAR
It has been shown that the continuous medium theory for coupled shear walls is appropriate for the deflection analysis of single bent structures such as rigid frames, braced frames, wall-frames, rigid frames with centrally located walls [9] and braced frames with multi-storey bracing systems [10]. In the analysis of these types of structures the bents are represented by three distinct structural parameters which are defined as follows : (1) El. This term expresses the flexural rigidity of all vertical members in the bent which are continuous up the total height of the structure. It is obtained by assuming that all horizontal members and bracing systems are cut. (2) GA. This represents the racking shear rigidity of the bent. It is a measure of the vertical shear stiffness between the vertical components, as induced by the transverse connecting member system, and of the comp-
GJ ~- Z G A , × V .
(2)
(3) Rotational stiffness pertaining to the axial rigidity of the vertical members EA~ = Y, EAc~ x z~
(3)
in which Eli is the flexural rigidity of bent "i" and z is the distance measured from the origin. Similarly for equations (2) and (3). The rotations up the height of the structure are given by well a 6 ~ H ) + 24 i H ) 1
+~
[ 1 -- (x/H) 2
(~k~
cosh [(kaH)o (1 - x/H)]-- 1
-~
(k~H)~cosh (k~H)o
(k~H)o [sinh (kaH)o - sinh (kax)o]~ ]
(k~H)~
;j (4)
where w x e represents the torsion per unit height which
Simplified Torsion Analysis f o r High-rise Structures consists of a uniformly distributed lateral load w, acting at a distance e from the origin along the z-axis, H is the total height of the structure and x is the distance measured from the top. A gross warping parameter is defined as
155
M , is unknown and thus (TI), must be calculated separately. (TI), is a bending moment due to axial forces in the vertical members. The generic fourth order differential equation for a single bent subjected to a general horizontal load of intensity q [12], is given by
EI~,g = ~,EIg, x z~,
(5)
1 y~V _ (ka)~y" = ~ {q,(x)- M,(x)fl,},
EIg, = EI~+ EAc~.
(6)
where y, indicates the deflections in bent "i" and can be expressed as
(13)
where
Two characteristic rotation parameters are expressed as
Yi = z, x 0.
ct2 = aJ/EIo.
(7)
This allows the equations for the bending moment and load intensity for the bent to be rewritten as
k~ = EI~,a/EA,o.
(8)
and
The rotation equation is written in terms of two characteristic non-dimensional parameters, (aH)o and ko, which completely govern the twisted shape of the structure. FORCES IN THE MULTI-BENT STRUCTURE
The equations for the bi-moments and torque in a multi-bent structure subject to a torsional load w ' e , are given by wex 2 2 = EI~O"+ (TOo,
(9)
wex = EI~O" + (TO'o,
(10)
and
where EIoO" and EI~O" are forces in the continuous medium model which can be obtained from the second and third derivatives of the rotation equation, equation (4). The bi-moments due to the axial forces in the vertical members of the structure, (TOo, are obtained by substituting the second derivative of equation (4) into equation (9).
(14)
M, = Eli x yT +(TI),,
(15)
M7 = ql,
(16)
q, = EI, x yl v + (Tl):'.
(17)
and since
An additional characteristic parameter is defined as t? = GA,/EAc?.
(18)
Substituting equations (15) and (17) into equation (13) leads to a second order differential equation for (Tl),. Further substituting equation (14) and considering the boundary conditions of zero axial force in the columns at the top and zero shear force in the continuous medium at the base of the structure will yield an expression for the bending moments in bent "i" caused by the axial forces in the vertical members.
2
[
well (Tl), = EIo~-~x (EAc~ x z3 (C, - C2) sinh (fix),
-- (C3 + C4) cosh (fix), + ~
+ C3
fcosh [(kan)o(1 - x/ H)] + (k~n)o sinh (kax)o ] ]
we [sinh ( k a H ) o - (kaH)o (Tl)o = k~02 [_ ~ x sinh (kax)o (k~tx)2 . ] -cosh(k~tx)o+~+ 1 .
(19) in which (ll)
The torsion (Tl)'o in the structure corresponding to horizontal racking forces, which are complementary to the shear forces in the continuous media of all bents, is given by the first derivative of equation (11). The four continuous functions, EI~O", El, off", (TOo and (T/)~ represent the forces in the total multi-bent structure that must be distributed to the individual bents.
(19a)
1 + C~(k~n)~ C2 = (fill), cosh (fill),' 1
(19b)
1
C3 = (flH)~
(k~ - 1) (kctH) 2'
(fill), C, = ( k g - 1) (kaH)2o{(flH)? - (kaH)g}"
(19c) (19d)
There exists a special case for which fl, = fl, i.e. all bents have identical values for fl, then
FORCES IN A SINGLE BENT
The bending moment equation for a single bent in a structure subject to torsion can be expressed as Mt = E l i x z , × O " +(Tl)i
Ct = (C3 + C,) tanh (fill),,
(12)
in which Eli x zl x 0" represents the sum of the bending moments in the vertical members of bent "i" and can easily be obtained from the second derivatives of equation (4). The total bending moment carried by the bent,
Ci = C2 = 0,
(20a) 1
C3 = - Ca = (kaH)02 ,
(20b)
and equation (19) can be simplified to (T0'=
EAci2 x zi---o,Tl) ( EA~,
(21)
J. C. D. Hoenderkamp and B. Stafford Smith
156
_
_
I-
COMMON CENTROIDAL AXIS
bers replaced by continuous media. This method can be applied to wall-frames, rigid frames, coupled walls as well as rigid frames with central walls. The assumed linear strain distribution across the bent yields an equation for the axial force in column '~/" at a distance c~ from the common centroidal axis.
(EAc) i . . . . T~ - ( ~ c 2 ~ I (11)i ,
--_T,_ !
F
where (Tl)i is obtained from equation (19). The shear force (T/)~ is distributed between the individual continua of the bays according to a procedure similar to distributing the shear stress across the depth of a beam. These individual shear forces are uniform across the width of the bays.
~(EAc)~
r;,j- ( ~ . ~1
(24)
C
(Tl);,
(25)
,..1
Fig. 2. Multi-bay "continuous" bent. where (Tl)o is given by equation (11). It is shown by equation (21) that the "axial" moment for the multi-bent structure, (T1)o, is distributed to the individual bents in proportion to their axial stiffness parameters, EAc 2, and their distances from the centre of rotation, the x-axis. The result of this is that the floor rotations in the vertical planes of the bents, due to axial deformations in the columns, are proportional to the distances from the axis of symmetry of the structure. This assumption was made earlier for the deflection analysis of multi-bent structures [6, 8, 9, 11]. The shear force equation for a single bent is given by the first derivative of equation (15), i.e.
V, = Elgy;" + (Tl);,
V,j = T~I~+ T~lr,
(26)
in which l~ and lr are the distances between the neutral axis of the column and the points of contraflexure in the adjacent continuous media. T't and T~ are obtained from equation (25). The total shear force in a column or wall is the sum of the racking shear force from equation (26) and a part of the single curvature shear force Eliyi", thus
(22)
where Eliyi" represents the sum of the shear forces due to single curvature flexure in the vertical members of bent " i " and can easily be obtained from the third derivative of equations (4) and (14). The shear force (TI); corresponding to racking of the bent is given by the first derivative of equation (19). In the special case where all bents have identical values o f / L the equation for this shear force simplifies to
EAc~ x zi ( TI) ; ( Tl)'o, EA,~
where (Tl)~ can be obtained from equation (19) and Y~(EAc)ex is a summation of the products of the excluded column areas and their distances to the common centroidal axis of the bent. The shear forces in the continuous media must be transferred to the vertical members. It can easily be shown [11] that the complementary shear force in the column can be taken as
V,.j = Ely;"+ V~j,
(27)
in which EIj represents the flexural rigidity of a single column or wall. The total bending moment in a floor to floor column is a combination of two components. At the top of the column /''~h
M,.j ( t o p ) = EIjy:'-~)V,~,.
(28)
At the bottom of the column (23)
in which (T1)'o is the torsion due to racking shear forces in the bents and is given by the first derivative of equation (11). The two components for bending, EIiy;' and (Tl)~, together with the components for shear, ELy," and (Tl)~, represent the four continuous functions of the forces in bent "i". These must be distributed further to the individual horizontal and vertical members as well as the bracing systems in that bent.
M,.j (bottom) = E l y ; ' -
~ V,:j,
(29)
where h indicates the storey height. It is assumed that the forces in the storey height columns and walls are best represented by the mid-storey values from the continuous functions.
Beams The shear force in a beam at level x can be expressed as
FORCES IN S T R U C T U R A L E L E M E N T S
Columns The procedure for the distribution of the continuous forces in a bent to individual members will be demonstrated for the structure shown in Fig, 2. It represents the left half of a multi-bay bent with the horizontal mem-
Vhj =
~xx+h/2T~,i dx
(30)
~-h/2
where T ' j is the shear force in the continuous medium in the bay in question and is obtained from equation (25). Alternatively, this force may be estimated as follows
Simplified Torsion Analysis for Hiyh-rise Structures
(T2)'
(T2)' Fig. 3. Single storey continuum.
Vb: = T~: x h.
(31)
For the calculation of the beam shear force this way, it is suggested that the value of T~,j be obtained at the floor level. The maximum bending moment in the girder may be approximated as b Mbj = Vbj x ~,
(32)
where b represents the length of the beam. In this equation it is assumed that the point of contraflexure occurs at midspan. This is correct only if the beam-column joint rotations are equal across a floor level. For other cases the bending moments in the beams may be estimated by distributing the difference between the column moments above and below the floor level.
Braced frames The bending moment equation for a single braced bent in a structure is given by equation (12). If the beamcolumn joints consist of pinned connections, the flexural stiffness of the columns is negligible and the bending moments, Eliy~', as well as the shear forces, EI~y~", may be taken to be zero. The axial forces in the columns can be obtained from equation (19). The complementary shear force (T/)' in a braced frame is given by the first derivative of equation (19) and is carded by the continuous medium as the horizontal component of axial forces in the bracing system. It represents the storey shear and is to be evaluated at mid-storey level. The shear force (Th)' is carried by the continuous medium as the vertical component of the axial forces in the braces. It represents the incremental axial forces in the columns and is obtained as follows h
(Th)' = (Tl)' x 7
(33)
where l represents the width of the braced bay. Figure 3 shows the shear forces in a single storey segment in the continuum. The forces in the x- and k-bracings can be approximated by treating the braced bay as a truss.
ACCURACY An approximation exists in the stage of combining a set of bents into a single structure represented by only three rotational properties ; EI~, GJand EAo. A rigorous torsion analysis of multi-bent structures consisting of different types of bents is extremely complex and unsuit-
157
able for a rapid hand method of analysis. In determining the characteristic rotation parameters, ~02 and kg, for the total structure, it is assumed that the rotations of the bents in their vertical planes are constrained at each floor level, i.e. the slope of the floor, as a result of axial deformations in the columns, at each bent is proportional to its distance from the x-axis. It should be noted that this is not identical to the shape of the deflection profile, which includes the slope caused by shear. A study has been made of various structures of different heights with a wide range of values for the characteristic non-dimensional parameters. The tests have shown that when a number of identical bents are combined, the maximum induced error will be less than 5%. The t-values for all bents are identical in this case. The inclusion of single shear walls will not alter the accuracy of the analysis since the t-value for a shear wall does not exist. Additional tests have shown that when combining bents of similar type, the maximum induced error will be 10% in the worst cases and substantially less in the majority of structures. When bents of different types are combined in a single structure the following can be used as a guide to the accuracy of the continuous medium analysis of tall building structures : if the ratio fl(max)/fl(min) < 3.0 the error in the proposed method of analysis will be less than 10%. fl(max) and fl(min) are the two bents in the structure with the largest and smallest t-values, respectively. For ratios larger than 3.0, it is mathematically possible for the error to exceed 10% but, for all the practical sized structures devised by the Authors, the errors were well within 10% [4,6, 11].
CONCLUSIONS A generalized hand method for torsion analysis is presented for high-rise structures consisting of combinations of rigid frames, braced frames, coupled walls, wall-frames and single shear walls. The method allows a rapid assessment of internal forces in the beams, walls, columns and bracing members of each bent in the structure. It requires the calculation of two non-dimensional characteristic parameters (~H)0 and k~ which characterize the performance of the total structure subject to torsional loading. The method of analysis is restricted to structures in which the plan arrangement is symmetric. The theory is based on the assumption of, and therefore is most accurate for, structures that are uniform throughout their height. It may be used also, however, for practical structures whose properties vary with height, but with less accurate results. The information obtained from this method should give the design engineer an easy means of comparing the suitability of alternative structural proposals, in addition to providing initial structural data for a more accurate computer analysis, or allowing a check on the reasonableness of the final output of a computer analysis. Acknowledgement--The Authors wish to acknowledge the support given to this research by the Natural Sciences and Engineering Research Council of Canada.
J. C. D. Hoenderkamp and B. Stafford Smith
158
REFERENCES l. 2. 3. 4. 5. 6. 7. 8. 9. I0. 11. 12.
APPENDIX:
H. Beck, Contribution to the analysis of coupled shear walls. Proc. Am. Conc. Inst. 59, 1055-1070 (1962). R. Rosman, Approximate analysis of shear walls subject to lateral loads. Proc. Am. Conc. Inst. 61, 717-734 (1964). A. Coull and J. R. Choudhury, Stresses and deflections in coupled shear walls. Proc. Am. Const. Inst. 64, 65 72 (1967). B. Stafford Smith, M. Kuster and J. C. D. Hoenderkamp, A generalized approach to the deflection analysis of braced frame, rigid frame and coupled wall structures. Can. J. Cir. Engng. 8, 230-240 (1981). B. Stafford Smith and J. C. D. Hoenderkamp, Simple deflection analysis of planar wall-frames. Proc. Asian Regional Conf. on Tall Buildings and Urban Habitat, 33-41. Kuala Lumpur (1982). B. Stafford Smith, M. Kuster and J. C. D. Hoenderkamp, Generalized method for estimating the drift in high-rise structures. Jnl. of the Structural Division, ASCE. 110, 1549-1562 (1984). A. Coull, Interactions between coupled shear walls and cantilever cores in three-dimensional regular symmetrical cross-wall structures. Proc. Instn. Cir. Engrs. Lond. 55, 827-840 (1973). J . C . D . Hoenderkamp and B. Stafford Smith, Approximate rotation analysis for plan-symmetric high-rise structures. Proe. Inst. Cir. Engrs. Lond. 83, 755-767 (1987). B. Stafford Smith, J. C. D. Hoenderkamp and M. Kuster, A graphical method of comparing the sway resistance of tall building structures. Proc. Inst. Cir. Engrs. Lond. 73, 713-729 (1982). J . C . D . Hoenderkamp, A generalized hand method of analysis for tall building structures subject to lateral loads. Ph.D. Thesis, McGill University, Montreal, Canada (1983). J.C.D. Hoenderkamp and B. Stafford Smith, Simplified analysis of symmetric tall building structures subject to lateral loads. Proc. Third International Conf. on Tall Buildings, 28-36. Hong Kong (1984). D.P. Abergel, Deflection solutions of special coupled wall structures by differential equations. M.Eng. Thesis, McGill University, Montreal, Canada (1981).
Two other loading cases which may be required in practice are : (1) A torsional load, P x e, resulting from a concentrated horizontal load P at the top of structure located a distance e from the origin along the z-axis. The rotations up the height of the structure are
PeH3y1
o(x):
1
ALTERNATIVE LOADING CASES
b-
l f x~
l f x~ 3
1
[1
l[x~2
l(x~3_
1-x/n h
(k:
cosh [(kaH)o(1 - x/ H)] - I (kotH)g cosh (k~H)o
+
{ 1/(k~H)o - (kaH)o/2} [sinh (k~H)o - sinh (kax)o!~] (k~H)~ cosh kaH)o JJ
(36)
The bending moments in bent " i " due to axial forces in the columns are given by
1
+ tmJ + k02-1
f l -xlH sinh (kctx)o - s i n h (kotH)o~] ×l ~ + (kctH)~ cosh (k~tH)o J [
f
( ~'" l t )i = ~w~eHZ x (EAc~ x zi) I ( C ] + C5) sinh (flx)i (34)
The bending moments in bent " i " due to axial forces in the columns are given by (TI), = ~
x (EAc~ x zf) - C 2 sinh (fix), + ~
f (kaH)o sinh (kax)o ~] -cosh(k~H)o - •j j' (35)
+ U 4 ~t
in which C2 and C4 are given by equations (19b) and (19d) respectively. If the/1-values for each bent are equal, then C2 and C4 are given by equations (203) and (20b). (2) A torsion load, wi x e, resulting from a triangularly distributed load with intensity w] at the top down to zero at the base of the structure and located a distance e from the origin along the z-axis. The rotations up the height of the structure are
""
w,eH4 [ ~ 0 l llx ~ I ( x ' ] ' _ 1 (xX] ' tx)=-~-g -8tH)+~\H] 120\HJ
_ [cosh [(k~H)o(1 -x/H)] +U4~ ( cosh (kc~H)o
{ I/(kaH)o -- (kotH)o/2} sinh (kax)0~] cosh (kotH)o JJ'
(37)
in which C , C3 and C4 are given by equations (19a), (19c) and (19d) respectively, C5 -
C 3- 1/2 + C4(kotH)o { 1/(kaH)o - (kocH)o/2}
(/~H), cosh (/~H),
(37a)
If the fl-values for each bent are equal, then C] = C5 = 0 and C3 and C4 are given by equation (20b).
(37b)