lateral torsion.pdf

Engineering Structures 31 (2009) 1903–1915

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Engineering Structures

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Coupled lateral–torsional frequencies of asymmetric, three-dimensionalstructures comprising shear-wall and core assemblies with stepwisevariable cross-sectionB. Rafezy a, W.P. Howson b,∗a Sahand University of Technology, PO. Box 51335/1996, Tabriz, Iranb Cardiff School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA, UK

a r t i c l e i n f o

Article history:Received 14 April 2008Received in revised form5 September 2008Accepted 16 January 2009Available online 11 April 2009

Keywords:Thin-walled structuresAsymmetric structuresTorsional vibrationCouplingWall-core structuresContinuummechanicsThree-dimensional models

a b s t r a c t

A simple and accurate model for asymmetric, three-dimensional wall-core structures is developed thatenables any desired natural frequency to be determined by a method which guarantees that no naturalfrequencies can be missed. The model assumes that the primary walls and cores run in two orthogonaldirections and that their propertiesmay vary in a stepwise fashion at one ormore storey levels. A vectorialapproach is used to generate the governing differential equations for coupled flexural-torsional motionthat are finally incorporated into an exact dynamic stiffness matrix (exact finite element) that can modelany uniform segment of the structure. A model of the original structure can then be assembled in theusualway. Since themass of each segment is assumed to be uniformly distributed, it is necessary to solve atranscendental eigenvalue problem,which is accomplished using theWittrick–Williams algorithm.Whenthe structure can be represented realistically by a uniform cantilever, solutions can be found easily, byhand. A parametric study comprising five, three-dimensional, asymmetric wall-core structures is givento compare the accuracy of the current approach with that of a full finite element analysis.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The type of mathematical model developed herein can beclassified as a simplified globalmodel. This implies that the originalstructure is treated holistically, but simplified prior to analysis sothat only its dominant characteristics are retained. The resultingmodel is solved exactly so that no additional accuracy is lostin solution. The use of such models can be quite compelling inappropriate circumstances [1], such as preliminary design, whenthe concept may be evolving rapidly, or when it is necessaryto check solutions developed elsewhere. Moreover, the followingreview of related work serves to highlight the growing popularityof such techniques.Approximate methods have recently been developed that

can deal with the vibration of asymmetric three-dimensionalstructures, in which the translational and torsional modes ofvibration are coupled. Kuang and Ng [2,3] considered the problemof doubly asymmetric, proportional structures inwhich themotionis dominated by shear walls. For the analysis, the structure is

∗ Corresponding author. Tel.: +44 0 2920 874263; fax: +44 0 2920 874597.E-mail addresses: [email protected], [email protected] (B. Rafezy),

[email protected] (W.P. Howson).

0141-0296/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.01.024

replaced by an equivalent uniform cantilever whose deformationis coupled in flexure and warping torsion. The same authorsextended this concept to the case of wall-frame structures byallowing for bending and shear. In this case however, the walland frame systems are independently proportional, but result in anon-proportional structural form [4]. In a recent publication, theyhave extended their work to tall building structures comprisingframes, walls, structural cores and coupled walls [5]. As in theirprevious work, they have replaced the structure with a uniformcantilever, derived the governing differential equation for freevibration and then solved the corresponding eigenvalue problemusing a generalisedmethod based on the Galerkin technique.Wall-frame structures have also been addressed by Wang et al. [6],who used an equivalent eccentricity technique that is appropriatefor non-proportional structures. However, the analysis is limitedto finding the first two coupled natural frequencies of uniformstructures with singly asymmetric plan form.Hand methods have also received considerable attention and

are particularly suitable for check calculations. In recent papersby Zalka [7,8], such a method is presented which can deal withthe three-dimensional frequency analysis of buildings braced byframeworks, coupled shear-walls and cores.In a relatively recent publication, Potzta and Kollar [9] replaced

the original structure by an equivalent sandwich beam that can

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1904 B. Rafezy, W.P. Howson / Engineering Structures 31 (2009) 1903–1915

model both slender and wide structures consisting of frames,trusses and coupled shear walls. In a subsequent paper, analternative approach is adopted in which the natural frequenciesof the replacement beam are solved approximately. This, togetherwith other simplifying assumptions, leads to simple formulae fordetermining the required natural frequencies [10]. Reference [10]also includes a useful tabulated summary of related work by thefollowing authors [7,11–18].The most recent contribution has been made by Rafezy

et al. [19] who presented a simple, accurate model for the cal-culation of natural frequencies of asymmetric, three dimensionalframe structures whose properties may vary through the height ofthe structure in a stepwise fashion at one or more storey levels.Their stiffness formulation enables the structure to be modelled asa stepped shear-torsion cantilever which yields the lower naturalfrequencies for medium to tall structures surprisingly accurately.The methods developed in the references above offer solutions

of varying accuracy, depending on the assumptions employed.Surprisingly, apart from the latter paper, none of them allowsfor step changes of properties along the height of the structure,despite the fact that this is almost inevitably the case in practicalbuilding structures of reasonable height. This study therefore seeksto extend the concept of the paper by Rafezy et al. [19] to wall-corestructures.

2. Problem statement

The class of building structure considered herein comprises twosets of orthogonal shear walls that are additionally stiffened bycores whose principal axes run parallel to the same orthogonaldirections. Since walls and cores deflect predominantly in aflexural configuration, it is assumed that they obey Bernoulli–Eulerbending theory that allows for bending deformation but notshearing deformation. In the case of an asymmetric arrangementof walls and cores, torsional effects are produced and may becomesignificant or even critical in tall buildings. In addition, wallsand cores in buildings do not warp freely as they are restrainedagainst warping at foundation level, thus the effect of warpingshould be taken into account in addition to St. Venant torsion. Thewarping-restrained torsion is often referred to as Vlasov’s torsionand can lead to longitudinal stresses in the walls and cores thatare sometimes greater than longitudinal stresses due to overallbending of the structure. In this study both the St. Venant andwarping rigidity of the cores are taken into account, but the St.Venant torsional component of the walls, which is dependent oncircular shear flows within individual wall elements, is small incomparison and can therefore be ignored.The underlying approach adopted with the model is to dissect

the original building structure into segments, by cutting throughthe structure horizontally at those storey levels corresponding tochanges in storey properties. Thus the storeys contained withina segment between any two adjacent cut planes are identical. Atypical segment is then considered in isolation. Initially, one of thetwo orthogonal plan directions is selected. If there is a primarywall running in this direction it is replaced by an equivalentflexural beam. If there is a core that has a component of flexurein this direction, it is replaced by an equivalent flexure–torsionbeam located on the core’s original shear centre. In each case, thesubstitutemembers have uniformly distributedmass and stiffness.The flexural beam only allows for bending deformation, while theflexure–torsion beam allows for bending deformation, St. Venantand warping torsion. In turn, each additional wall or core thatcontributes to the structural action in the current direction isreplaced by its own substitute beam and the effect of all thesebeams is summed tomodel the effect of the original structure. Thisleads directly to the differential equation governing the motion

Fig. 1. Typical floor plan of an asymmetric three-dimensional wall-core structure.O and C denote the locations of the rigidity andmass centres, respectively. The floorsystem EFGH is considered to be rigid in its plane.

of the segment in the chosen direction. The same procedure isthen adopted for all the cores and those walls that contributeto the structural action in the orthogonal direction. Once bothequations are available it requires little effort to write down thesubstitute expressions for the coupled torsional motion. The threeequations thus formed are subsequently solved exactly and posedin dynamic stiffness form. The resulting coupled flexure–torsionbeam element can then be used to reconstitute the originalstructure by assembling the dynamic stiffness matrices for theindividual segments in the usual manner.It is clear from the element formulation that the final model

has a transcendental dependence upon the frequency parameter.The required natural frequencies are therefore determined bysolving the model using an exact technique, based on theWittrick–Williams algorithm, that can be arrested after achievingany desired accuracy and which also ensures that no naturalfrequencies can be missed.

3. Theory

Consider the hypothetical layout of a typical floor plan ofthe asymmetric, three-dimensional wall-core structure shown inFig. 1. The walls run in two orthogonal directions and the cores’principal axes are parallel to these directions. It is assumed thatthe rigidity centre of the structure, O, at each floor level lies on avertical line that runs through the height of the structure.It is now assumed that the origin of the co-ordinate system

is located at the rigidity centre, O, with the x and y co-ordinatesrunning parallel to the walls. The z-axis then runs vertically fromthe base of the building and coincides with the rigidity axis. PointC(xc, yc) denotes the centre of mass at a typical floor level. It isassumed that the floor system is rigid in its plane and that thecentre of mass at each level lies on a vertical line, the mass axis,that runs through the height of the structure.When the rigidity andmass axes of a structure do not coincide, the lateral and torsionalmotion of the building will always be coupled in one or moreplanes.During vibration, the displacement of the mass centre at any

time t in the x–y plane can be determined as the result of a puretranslation followed by a pure rotation about the rigidity centre,see Fig. 2. During the translation phase the rigidity centre movesto O′ and the mass centre C moves to C ′, displacements in eachcase of u(z, t) and v(z, t) in the x and y directions, respectively.During rotation, the mass centre moves additionally from C ′ to C ′′,an angular rotation of ϕ(z, t) about O′. The resulting translations,(uc, vc) of the mass centre in the x and y directions, respectively,areuc(z, t) = u(z, t)− ycϕ(z, t) and (1a)vc(z, t) = v(z, t)+ xcϕ(z, t). (1b)

B. Rafezy, W.P. Howson / Engineering Structures 31 (2009) 1903–1915 1905

Fig. 2. Coupled translational-torsional vibration of the structure. O and C move toO′ and C ′ , respectively, during translation and C ′ moves additionally to C ′′ duringrotation about O′ .

Fig. 3. Typical segment formed by cutting the structure through planes EkFkGkHkand Ek+1Fk+1Gk+1Hk+1 that correspond to the kth and k + 1th changes in storeyproperties. (Some lines have been omitted/included for clarity.).

More generally, it is clear that the displacements of a typical point(xi, yi) are given by Eqs. (1a) and (1b) when c = i.The structure is now divided into segments along the z axis by

notionally cutting the structure along horizontal planes at thosestorey levels corresponding to changes in storey properties. Fig. 3shows a typical segment formed by cutting the structure throughplanes EkFkGkHk and Ek+1Fk+1Gk+1Hk+1 that correspond to the kthand k+ 1th changes in storey properties. The number of storeys inany one segment can vary from one to the total number of storeysin the structure if it is uniform throughout its height. However, inany one segment each storey must have the same properties.We now consider a typical segment in isolation and seek to

replace each primary wall by a substitute flexural beam thatreplicates its in-plane motion. We also substitute each core witha flexure–torsion beam which can undergo flexural deflection aswell as St. Venant and warping torsion. We start by considering atypical wall, wall i, that runs parallel to the y–z plane, see Fig. 1.This wall is replaced by a single substitute beam, beam i, shownin Fig. 4. This beam is a two-dimensional flexural beam of length Land has uniformly distributedmass and flexural stiffness. Themassand elastic axes therefore coincide with the local z-axis and theelastic axis is only permitted bending deformation vi(z, t) in they direction, where z and t denote distance from the local origin

Fig. 4. Co-ordinate system and positive sign convention for the substitutetwo-dimensional flexural beam in the local y–z plane. (a) Member convention.(b) Element convention.

and time, respectively. Likewise, we consider a typical core, core i,and replace it by a single flexure–torsion substitute beam, beami, shown in Fig. 5, which is then located at the shear centre ofthe former core. This beam is a three-dimensional flexure–torsionbeam of length L and has uniformly distributed mass, flexural andtorsional stiffness. The elastic axis therefore coincides with thelocal z-axis and is permitted bending deformation vi(z, t) in the ydirection and torsional rotation ϕ(z, t). Themass axis runs parallelto the elastic axis and is not generally coincident with the elasticaxis. The distances of the local mass axis of core i from its ownelastic axis are denoted by exi and eyi in the x and y directions,respectively.The equations of motion for the substitute flexural beam can be

developed from Fig. 4 by considering a typical elemental length ofthe beam, δz, and employing D’Alembert’s principle. The dynamicequilibrium equations for the motion of the element in the y–zplane may then be written as

∂Qwyi(z, t)∂z

δz = mwyi∂2vi(z, t)∂t2

δz (2a)

or

∂

∂z

(−EIwyi

∂3vi(z, t)∂z3

)= mwyi

∂2vi(z, t)∂t2

(2b)

where E is Young’smodulus,Qwyi(z, t) is the shear force,mwyi is theuniformly distributed mass per unit length and EIwyi is the flexuralrigidity of the element for motion in the y direction.With a similar argument, the motion for the substitute

flexure–torsion beam can be studied by considering a typicalelemental length of the beam, δz, from Fig. 5. The equations ofmotion governing the element when translating in the y–z planemay be written as follows using Eq. (1b) and noting that exi is theeccentricity between the elastic andmass axes of the original core i.

∂Qcyi(z, t)∂z

δz = mci∂2

∂t2(vi(z, t)+ exiϕ(z, t))δz (3a)

or

∂

∂z

(−EIcyi

∂3vi(z, t)∂z3

)= mci

∂2

∂t2(vi(z, t)+ exiϕ(z, t)) (3b)

where Qcyi(z, t) is the shear force, mci is the uniformly distributedmass per unit length and EIcyi is the flexural rigidity of the elementfor motion in the y direction.If the equivalent procedure is carried out for all of the i walls

that run parallel to the y–z plane and all of the cores, the dynamic


Fig. 5. Co-ordinate system and positive sign convention for the substitute three-dimensional flexural-torsion beam (a) Member and element convention for the x–zplane. (b) Member and element convention for the y–z plane.

equilibrium equation for motion in the y–z plane may be writtenas

∂

∂z

nwy∑i=1

−EIwyi∂3vi(z, t)∂z3

+∂

∂z

nc∑i=1

−EIcyi∂3vi(z, t)∂z

=

nwy∑i=1

mwyi∂2vi(z, t)∂t2

+

nc∑i=1

mci∂2

∂t2(vi(z, t)+ exiϕ(z, t)) (4)

where nwy and nc are the number of walls and cores, respectively.Noting that EIwyi and EIcyi are constant over the length of the

member and substituting for vi(z, t) and vi(z, t) from Eq. (1b) withc replaced by i givesnwy∑i=1

−EIwyi∂4(v(z, t)+ xiϕ(z, t))

∂z4

+

nc∑i=1

−EIcyi∂2(v(z, t)+ xiϕ(z, t))

∂z2

−

nwy∑i=1

mwyi∂2(v(z, t)+ xiϕ(z, t))

∂t2

−

nc∑i=1

mci∂2(v(z, t)+ (xi + exi)ϕ(z, t))

∂t2= 0 (5)

where xi and xi are the distances of wall i and core i from therigidity centre, O, respectively. Since O is the centre of rigidity,∑nwyi=1 EIwyixi +

∑nci=1 EIcyixi = 0 and Eq. (5) can be simplified to

EIy∂4v(z, t)∂z4

+my∂2v(z, t)∂t2

+ xcmy∂2ϕ(z, t)∂t2

= 0 (6)

in which

EIy =nwy∑i=1

EIwyi +nc∑i=1

EIcyi (7a)

and

myxc =nwy∑i=1

mwyixi +nc∑i=1

mci(xi + exi)

wheremy =nwy∑i=1

mwyi +nc∑i=1

mci. (7b)

Since the total mass of the segment contributes to its vibration,including the mass of the walls running in the x direction andthe rigid diaphragms, my should be replaced by m, where m isthe equivalent distributed mass over the height of the segment.Therefore

EIy∂4v(z, t)∂z4

+m∂2v(z, t)∂t2

+mxc∂2ϕ(z, t)∂t2

= 0. (8)

In an identical fashion, the dynamic equilibrium relationship formotion in the x–z plane can be written as

EIx∂4u(z, t)∂z4

+m∂2u(z, t)∂t2

−myc∂2ϕ(z, t)∂t2

= 0. (9)

Finally, it should be noted that the cores and walls runningparallel to the x–z and y–z planes also provide the torsionalstiffness of the building. Thus the required equation for torsioncan be developed from a consideration of the torsional equilibriumabout O, which yieldsnwy∑i=1

−EIwyixi∂4(v(z, t)+ xiϕ(z, t))

∂z4

+

nc∑i=1

−EIcyixi∂4(v(z, t)+ xiϕ(z, t))

∂z4

−

nwy∑i=1

mwyixi∂2(v(z, t)+ xiϕ(z, t))

∂t2

−

nc∑i=1

mci(xi + exi)∂2(v(z, t)+ (xi + exi)ϕ(z, t))

∂t2

−

[nwx∑j=1

−EIwxjyj∂4(u(z, t)− yjϕ(z, t))

∂z4

+

nc∑j=1

−EIcxjyj∂4(u(z, t)− yjϕ(z, t))

∂z4

−

nwx∑j=1

mwxjyj∂2(u(z, t)− yjϕ(z, t))

∂t2


−

nc∑j=1

mcj(yj + eyj)∂2(u(z, t)− (yj + eyj)ϕ(z, t))

∂t2

]

+

nc∑i=1

−EIwci∂4ϕ(z, t)∂z4

+

nc∑i=1

GJci∂2ϕ(z, t)∂z2

= 0 (10)

where EIwci and GJci are the warping and St. Venant torsionalrigidity of core i about its own shear centre, respectively. Eq. (10)can be simplified to

EIw∂4ϕ(z, t)∂z4

− GJ∂2ϕ(z, t)∂z2

+myxc∂2v(z, t)∂t2

−mxyc∂2u(z, t)∂t2

+ Ig∂2ϕ(z, t)∂t2

= 0 (11)

where

EIw =nwy∑i=1

EIwyix2i +nwx∑j=1

EIwxjy2j +nc∑i=1

EIcyix2i

+

nc∑j=1

EIcxjy2j +nc∑i=1

EIwci (12a)

GJ =nc∑i=1

GJci (12b)

Ig =nwy∑i=1

mwyix2i +nwx∑j=1

mwxjy2j

+

nc∑i=1

mci(xi + exi)2 +nc∑j=1

mcj(yj + eyj)2. (12c)

EIw is thewarping torsional rigidity of thewalls and cores about theflexural rigidity centre O; GJ is the sum of the St. Venant torsionalrigidity of the cores about their own shear centres and Ig is the polarsecond moment of mass about the rigidity centre O.As before, the total mass of the cores and walls running in the x

and y directions, as well as that of the rigid diaphragms, shouldbe taken into account. Thus Eqs. (8), (9) and (11) can finally berearranged and written in the following form

EIx∂4u(z, t)∂z4

+m∂2u(z, t)∂t2

−myc∂2ϕ(z, t)∂t2

= 0 (13a)

EIy∂4v(z, t)∂z4

+m∂2v(z, t)∂t2

+mxc∂2ϕ(z, t)∂t2

= 0 (13b)

EIw∂4ϕ(z, t)∂z4

− GJ∂2ϕ(z, t)∂z2

−myc∂2u(z, t)∂t2

+mxc∂2v(z, t)∂t2

+mr2m∂2ϕ(z, t)∂t2

= 0 (13c)

where rm is the polarmass radius of gyration of the structure aboutthe rigidity centre O. Eqs. (13a)–(13c) are the required differentialequations of motion.The minimum requirements for Eqs. (13) to be valid are that

the structural system should contain either a single core or a singlewall running in each of the x and y directions.

4. Eigenvalue problem

Eqs. (13) are now solved and posed in dynamic stiffnessform. Although each equation was developed individually froma consideration of the planar flexural and flexure–torsion beamsof Figs. 4 and 5, they now describe the motion of a three-dimensional, flexure–torsion coupled beam whose co-ordinatesystem and sign convention are shown in Fig. 6. This beam (exactfinite element) will replace a typical segment of the original,

unde

flec

ted

defl

ecte

d

unde

flec

ted

defl

ecte

d

a

b

Fig. 6. Co-ordinate system and positive sign convention for forces anddisplacements of the three-dimensional flexure–torsion coupled beam. (a)Memberand element convention for the x–z plane. (b) Member and element convention forthe y–z plane.

asymmetric, three-dimensional wall-core structure. The whole ofthe original structure can then be reconstituted by assembling theexact finite elements corresponding to each segment in the usualway.Eqs. (13) are solved on the assumption of harmonic motion, so

that the instantaneous displacements can be written as

u(z, t) = U(z) sinωt (14a)v(z, t) = V (z) sinωt (14b)ϕ(z, t) = Φ(z) sinωt (14c)

where U(z), V (z) and Φ(z) are the amplitudes of the sinusoidallyvarying displacements and ω is the circular frequency.Substituting Eqs. (14) into Eqs. (13) and re-writing in non-

dimensional form gives

U ′′′′(ξ)− β2xω2U(ξ)+ ycω2β2xΦ(ξ) = 0 (15a)

V ′′′′(ξ)− β2yω2V (ξ)− xcω2β2yΦ(ξ) = 0 (15b)


Φ ′′′′(ξ)− α2ϕΦ′′(ξ)− ω2β2ϕΦ(ξ)

+ ycω2β2x

γ 2xU(ξ)− xcω2

β2y

γ 2yV (ξ) = 0 (15c)

where

α2ϕ =GJEIwL2, (16a)

β2x =mL4

EIx, (16b)

β2y =mL4

EIy, (16c)

β2ϕ = r2mmL4

EIw(16d)

γ 2x =EIwEIx, (16e)

γ 2y =EIwEIy

and (16f)

ξ =zL. (16g)

Eqs. (15) can be re-written in the following matrix formD4 − ω2β2x 0 ycω2β2x

0 D4 − ω2β2y −xcω2β2y

ycω2β2x

γ 2x−xcω2

β2y

γ 2yD4 − α2ϕD

2− ω2β2ϕ

[U(ξ)V (ξ)Φ(ξ)

]= 0

(17)

in which D = d/dξ .Eq. (17) can be combined into one equation by eliminating

either U, V orΦ to give the twelfth-order differential equation∣∣∣∣∣∣∣∣D4 − ω2β2x 0 ycω2β2x

0 D4 − ω2β2y −xcω2β2y

ycω2β2x

γ 2x−xcω2

β2y

γ 2yD4 − α2ϕD

2− ω2β2ϕ

∣∣∣∣∣∣∣∣W (ξ) = 0 (18)

whereW = U , V orΦ .The solution of Eq. (18) is found by substituting the trial solution

W (ξ) = eaξ to yield the characteristic equation∣∣∣∣∣∣∣∣τ 2 − ω2β2x 0 ycω2β2x

0 τ 2 − ω2β2y −xcω2β2y

ycω2β2x

γ 2x−xcω2

β2y

γ 2yτ 2 − α2ϕτ − ω

2β2ϕ

∣∣∣∣∣∣∣∣W (ξ) = 0 (19)

where τ = a2.Eq. (19) is a sixth order equation in τ and it can be proven

(Appendix) that it always has three positive and three negative realroots. Let these six roots be τ1, τ2, τ3, −τ4, −τ5 and −τ6, whereτj (j = 1, 6) are all real and positive. Therefore the twelve roots ofEq. (19) can be obtained as

α,−α β,−β γ ,−γ iδ,−iδiη,−iη iµ,−iµ (20a)

where

α =√τ1, β =

√τ2, γ =

√τ3, δ =

√τ4,

η =√τ5 and µ =

√τ6 (20b)

and i =√−1.

It follows that the solution of Eq. (18) is of the form

W (ξ) = C1 coshαξ + C2 sinhαξ + C3 coshβξ + C4 sinhβξ+ C5 cosh γ ξ + C6 sinh γ ξ + C7 cos δξ + C8 sin δξ+ C9 cos ηξ + C10 sin ηξ + C11 cosµξ + C12 sinµξ. (21)

Eq. (21) represents the solution forU(ξ), V (ξ) andΦ(ξ), since theyare related via Eq. (18). Hence they can be written individually as

U(ξ) = Cu1 coshαξ + Cu2 sinhαξ + C

u3 coshβξ + C

u4 sinhβξ

+ Cu5 cosh γ ξ + Cu6 sinh γ ξ

+ Cu7 cos δξ + Cu8 sin δξ + C

u9 cos ηξ + C

u10 sin ηξ

+ Cu11 cosµξ + Cu12 sinµξ (22a)

V (ξ) = Cv1 coshαξ + Cv2 sinhαξ + C

v3 coshβξ + C

v4 sinhβξ

+ Cv5 cosh γ ξ + Cv6 sinh γ ξ

+ Cv7 cos δξ + Cv8 sin δξ + C

v9 cos ηξ + C

v10 sin ηξ

+ Cv11 cosµξ + Cv12 sinµξ (22b)

Φ(ξ) = C1 coshαξ + C2 sinhαξ + C3 coshβξ + C4 sinhβξ+ C5 cosh γ ξ + C6 sinh γ ξ+ C7 cos δξ + C8 sin δξ + C9 cos ηξ + C10 sin ηξ+ C11 cosµξ + C12 sinµξ. (22c)

The relationship between the constants Cuj , Cvj and Cj (j = 1, 12)

also follows from Eq. (18) as

Cu2j−1 = tuj C2j−1 and (23a)

Cu2j = tuj C2j (j = 1, 6) (23b)

Cv2j−1 = tvj C2j−1 and (23c)

Cv2j = tvj C2j (j = 1, 6) (23d)

where

tuj =−ycω2β2xτ 2j − ω

2β2x, (24a)

tvj =xcω2β2yτ 2j − ω

2β2y(j = 1, 2, 3, 4, 5, 6). (24b)

Following the sign convention of Fig. 6(a) and (b), expressions forthe bending rotations θx(ξ), θy(ξ) and the gradient of the twistΦ ′(ξ) are easily established as

θx(ξ) =1LdU(ξ)dξ

, (25a)

θy(ξ) =1LdV (ξ)dξ

and (25b)

Φ ′(ξ) =1LdΦ(ξ)dξ

. (25c)

The corresponding bending moments Mx(ξ), My(ξ), the bi-moment B(ξ), lateral shear forces Qx(ξ), Qy(ξ) and torsionalmoment T (ξ) are likewise easily determined from the appropriatestress/strain relationships as

Mx(ξ) =−EIxL2d2U(ξ)dξ 2

, (26a)

My(ξ) =−EIyL2d2V (ξ)dξ 2

and (26b)

B(ξ) =EIwL2d2Φ(ξ)dξ 2

(26c)


Qx(ξ) =−EIxL3d3U(ξ)dξ 3

, (26d)

Qy(ξ) =−EIyL3d3V (ξ)dξ 3

(26e)

T (ξ) =−EIwL3d3Φ(ξ)dξ 3

+GJLdΦ(ξ)dξ

. (26f)

The nodal displacements and forces can now be defined in themember co-ordinate system of Fig. 6(a) and (b), as followsAt ξ = 0 U = U1, θx = θ1x, V = V1,

θy = θ1y, Φ = Φ1, Φ ′ = Φ ′1 (27a)At ξ = 1 U = U2, θx = θ2x, V = V2,

θy = θ2y, Φ = Φ2, Φ ′ = Φ ′2 (27b)At ξ = 0 Qx = −Q1x, Mx = M1x, Qy = −Q1y,

My = M1y, T = −T1, B = −B1 (27c)At ξ = 1 Qx = Q2x, Mx = −M2x, Qy = Q2y,

My = −M2y, T = T2, B = B2. (27d)Then the nodal displacements can be determined from Eqs. (22)

and (25) as[d1d2

]=

E1 E2 0 00 0 E1R1 E2R2E1Ch E2C E1Sh E2SE1R1Sh −E2R2S E1R1Ch E2R2C

[CoCe]

(28)

where

d1 =

U1V1Φ1θ1xθ1yΦ ′1

, d2 =

U2V2Φ2θ2xθ2yΦ ′2

,

Co =

C1C3C5C7C9C11

, Ce =

C2C4C6C8C10C12

,

E1 =

[tu1 tu2 tu3tv1 tv2 tv31 1 1

], E2 =

[tu4 tu5 tu6tv4 tv5 tv61 1 1

]

R1 =1L

[α 0 00 β 00 0 γ

]R2 =

1L

[δ 0 00 η 00 0 µ

]

Ch =

[coshα 0 00 coshβ 00 0 cosh γ

],

Sh =

[sinhα 0 00 sinhβ 00 0 sinh γ

],

C =

[cos δ 0 00 cos η 00 0 cosµ

], S =

[sin δ 0 00 sin η 00 0 sinµ

].

(29)Hence the vector of constants [CoCe]T can be determined fromEq. (28) as

[CoCe

]=


−1 [d1d2

]. (30)

In similar fashion the vector of nodal forces can be determinedfrom Eqs. (26) and (27) as

[p1p2

]=

0 0 Q1 Q2M1 M2 0 0−Q1Sh Q2S −Q1Ch −Q2C−M1Ch −M2C −M1Sh −M2S

[CoCe]

(31)

where

p1 =

Q1xQ1yT1M1xM1yB1

, (32a)

p2 =

Q2xQ2yT2M2xM2yB2

, (32b)

Q1 =

tu1α3Bx tu2β

3Bx tu3γ3Bx

tv1α3By tv2β

3By tv3γ3By

α3Eo − αF β3Eo − βF γ 3Eo − γ F

(32c)

Q2 =

−tu4δ3Bx −tu5η3Bx −tu6µ

3Bx−tv4 δ

3By −tv5η3By −tv6µ

3By−δ3Eo − δF −η3Eo − ηF −µ3Eo − µF

(32d)

M1 =

−tu1 (α2Ax) −tu2 (β2Ax) −tu3 (γ 2Ax)−tv1 (α2Ay) −tv2 (β

2Ay) −tv3 (γ2Ay)

−α2Do −β2Do −γ 2Do

, (32e)

M2 =

tu4 (δ2Ax) tu5 (η2Ax) tu6 (µ

2Ax)tv4 (δ

2Ay) tv5 (η2Ay) tv6 (µ

2Ay)δ2Do η2Do µ2Do

(32f)

where

Ax =EIxL2, (33a)

Ay =EIyL2, (33b)

Bx =EIxL3, (33c)

By =EIyL3

(33d)

Do =EIwL2, (33e)

Eo =EIwL3, (33f)

F =GJL. (33g)

Thus the required stiffnessmatrix can be developed by substitutingEqs. (30) into Eqs. (31) to give[p1p2

]=

0 0 Q1 Q2M1 M2 0 0−Q1Sh Q2S −Q1Ch −Q2C−M1Ch −M2C −M1Sh −M2S

×


−1 [d1d2

](34)


or

p = kd. (35)

The stiffness relationship of Eq. (35) is general and can beused in the normal way to assemble more complex forms.The required natural frequencies of the resulting structure aredetermined by evaluating its overall dynamic stiffness matrix ata trial frequency ω∗ and using the Wittrick–Williams algorithm toestablish howmanynatural frequencies have been exceededbyω∗.This clearly provides the basis for a convergence procedure thatcan yield the required natural frequencies to any desired accuracy.The corresponding mode shapes can then be recovered by anyappropriate method [20].

5. Wittrick–Williams algorithm

The Wittrick–Williams algorithm [21,22] has been availablefor over thirty years and has received considerable attention. Thealgorithm states that

J = J0 + s{K} (36)

where J is the number of natural frequencies of the structureexceeded by some trial frequency, ω∗, Jo is the number of naturalfrequencies that would still be exceeded if all members wereclamped at their ends so as to make D = 0 and s{K} is the signcount of the dynamic structure stiffness matrix K. s{K} is definedin reference [22] and is equal to the number of negative elementson the leading diagonal of the upper triangular matrix obtainedfrom K, when ω = ω∗, by the standard form of Gauss eliminationwithout row interchanges.From the definition of Jo, it can be seen that

Jo =∑Jm (37)

where Jm is the number of natural frequencies of a member,with its ends clamped, which have been exceeded by ω∗, andthe summation extends over all members of the structure. Insome cases it is possible to determine the value of Jm for anindividual member symbolically using a direct approach [20] thatgives an analytical expression for Jm. However this is impracticalin the present case due to the algebraic complexity. Instead, Jmis evaluated using an argument based on Eq. (37) that appliesthe Wittrick–Williams algorithm [22] in reverse. The procedurecorresponds to the one originally proposed by Howson andWilliams [23] and is described as follows.Consider an element which has been isolated from its

neighbours by clamping its ends. Treating this member as acomplete structure, it is evident that the required value of Jm couldbe evaluated if its natural frequencies were known. Unfortunatelythis simple structure can rarely be solved easily. We thereforeseek to establish a different set of boundary conditions (other thanclamped–clamped) which admit a simple solution from which thesolution for the clamped–clamped case can be deduced. This ismost easily achieved in the present case by imposing what will bedefined as simply supported boundary conditions, i.e. at

ξ = 0 and ξ = 1, U = V = Φ = 0 andMx = My = B = 0.

(38)

The stiffness relationship for a single member subject to theseboundary conditions can then be obtained by deleting appropriaterows and columns from Eq. (35) to leave

M1xM1yB1M2xM2yB2

=

K4,4 K4,5 K4,6 K4,10 K4,11 K4,12K5,4 K5,5 K5,6 K5,10 K5,11 K5,12K6,4 K6,5 K6,6 K6,10 K6,11 K6,12K10,4 K10,5 K10,6 K10,10 K10,11 K10,12K11,4 K11,5 K11,6 K11,10 K11,11 K11,12K12,4 K12,5 K12,6 K12,10 K12,11 K12,12

θ1x

θ1y

Φ ′1θ2x

θ2y

Φ ′2

(39)

or

pss = kssdss (40)

where the Ki,j are the remaining elements of k with their originalrow i and column j subscripts and kss is the required 6× 6 matrixfor this simple one member structure.Application of the Wittrick–Williams algorithm [22] to this

simple structure gives

Jss = Jm + s{kss} or (41)Jm = Jss − s{kss} (42)

where Jss is the number of natural frequencies of the simplysupported member that lie below the trial frequency ω∗, Jm is therequired number of clamped–clamped natural frequencies of themember lying below ω∗, s{kss} is the number of negative elementson the leading diagonal of k∆ss , where k

∆ss is the upper triangular

matrix obtained by applying the usual form of Gauss eliminationto kss.The evaluation of s{kss} is clearly straightforward and the

problem thus lies in determining Jss.Based on Eqs. (22) and (26a)–(26c), the boundary conditions of

Eq. (38) are satisfied by assuming solutions for the displacementsU(ξ), V (ξ) andΦ(ξ) of the form

U(ξ) = Ck sin(kπξ), (43a)V (ξ) = Dk sin(kπξ) and (43b)Φ(ξ) = Ek sin(kπξ) (k = 1, 2, 3, . . .) (43c)

where Ck, Dk and Ek are constants.Substituting Eqs. (43) into Eq. (17) gives(kπ)4 − ω2β2x 0 ycω2β2x

0 (kπ)4 − ω2β2y −xcω2β2y

ycω2β2x

γ 2x−xcω2

β2y

γ 2y(kπ)4 + α2ϕ(kπ)

2− ω2β2ϕ

×

[CkDkEk

]= 0 (44)

in which ω represents the coupled natural frequencies of themember with simply supported ends. The non-trivial solution ofEq. (44) is obtained when∣∣∣∣∣∣∣∣∣(kπ)4 − ω2β2x 0 ycω2β2x

0 (kπ)4 − ω2β2y −xcω2β2y

ycω2β2x

γ 2x−xcω2

β2y

γ 2y(kπ)4 + α2ϕ(kπ)

2− ω2β2ϕ

∣∣∣∣∣∣∣∣∣ = 0. (45)

Eq. (45) is a cubic equation in ω2 and yields three positivevalues of ω for each value of k. It is then possible to calculate Jssby substituting progressively larger values of k until all of thosenatural frequencies lying below ω∗ have been accounted for. OnceJss is known, Jm can be calculated from Eq. (42).

6. Special case: Uniform structures

When all the storeys of a wall-core building can be consideredto be identical, the whole building may be modelled as a singleflexure–torsion beam that is clamped at one end and free at theother. The end conditions for such a beam are

d1 = 0; (46a)p2 = 0. (46b)

The natural frequencies of such a beam can be found easily byhand if the St. Venant torsional rigidity of the cores is ignored.Such a course of action is considered in the Numerical ResultsSection, where its effect has been assessed through a parametric


study. The study shows that the St. Venant torsional rigidity makeslittle contribution towards the overall behaviour of structurescomprising open cores andmay safely be ignored inmost practicalcases. When this is the case, Eq. (46b) can be written in thefollowing form using Eq. (26)U ′′(ξ = 1)V ′′(ξ = 1)Φ ′′(ξ = 1)U ′′′(ξ = 1)V ′′′(ξ = 1)Φ ′′′(ξ = 1)

= 0. (47)

If Eqs. (47) and (46a) are substituted into Eqs. (28), suitablydifferentiated, it is clear that the condition for non-trivial solutionsis∣∣∣∣∣∣∣E1 E2 0 00 0 E1R1 E2R2

E1R21Ch −E2R21C E1R22Sh −E2R

22S

E1R31Sh E2R31S E1R32Ch −E2R32C

∣∣∣∣∣∣∣ = 0. (48)

Noting that the St. Venant torsional rigidity of the cores has beenignored, it is easy to show that the three positive and three negativeroots of Eq. (19) are symmetrical and therefore

R1 = R2 and E1 = E2. (49)

Hence Eq. (48) can be written in the following form∣∣∣∣∣∣∣E1 0 0 00 E1 0 00 0 E1 00 0 0 E1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣I 0 0 00 R1 0 00 0 R21 00 0 0 R31

∣∣∣∣∣∣∣∣∣∣∣∣∣∣I I 0 00 0 I ICh C Sh SSh −S Ch C

∣∣∣∣∣∣∣=0. (50)It is clear in this form that only the right-handdeterminant canpassthrough zero for non-trivial solutions. Noting that the elementsof the right hand determinant all are diagonal matrices, it can besimplified to∣∣∣∣∣∣∣I 0 I 00 I 0 I0 0 −C− Ch −S− Sh0 0 S− Sh −C− Ch

∣∣∣∣∣∣∣ = 0 (51)

or

|I+ CCh| = 0. (52)

The solution of Eq. (52) can be calculated to any desiredaccuracy, although for illustrative purposes the solutions are givenapproximately as

α or β or γ = 1.875, 4.694, 7.855, 10.996 and(k−

12

)π for k = 5, 6, 7, . . . .

(53)

Substituting Eq. (53) into Eq. (19) and solving for ω yields therequired natural frequencies. In addition, it is interesting to notethat by combiningω and τ into a single parameter, an even simplersolution procedure can be obtained as follows. Let

b = τ/ω (54)

then Eq. (19) can be written in the following form∣∣∣∣∣∣b2 − β2x 0 ycβ2x0 b2 − β2y −xcβ

2y

ycβ2ϕ −xcβ2ϕ r2m(b2− β2ϕ)

∣∣∣∣∣∣W (ξ) = 0. (55)

Eq. (55) is a cubic equation in the frequency parameter b2 and itcan be proven [24] that it always has three positive roots. Let theseroots be b21, b

22 and b

23. The natural frequency of the building can

Fig. 7. Floor plan of all structures considered in Section 8.

then be calculated from

ωj =τ

bj(j = 1, 2, 3). (56)

Substituting Eq. (53) into Eq. (56) gives

ω(1)j =

3.5156bj

, (57a)

ω(2)j =

22.0336bj

, (57b)

ω(3)j =

61.7010bj

, (57c)

ω(4)i =

120.9120bj

(j = 1, 2, 3) (57d)

and(ω(k)j =

(k− 0.5)2π2

bj. . . k = 5, 6, 7, . . .

)(j = 1, 2, 3). (57e)

7. Examples

The work of this section consolidates the foregoing theory byperforming a parametric study on five wall-core structures ofvarying height and comparing the lower natural frequencies withthose obtained from a full finite element analysis. The structures,which have 5, 10, 20, 40 and 60 storeys, respectively, all have thesamedoubly asymmetric floor plan and equal storey heights of 3m.Each structure comprises three walls in the y direction (W1-W3),threewalls in the x direction (W4-W6) and two cores (C1-C2), eachofwhich are connected to each other by typical rigid diaphragms ateach floor level with the arrangement shown in Fig. 7. In the 5, 10and 20 storey buildings, the properties of the structural elementsdo not change along the height of the structure, so each structurecan be modelled using a single substitute beam element and thenatural frequencies canbedetermined from the theory of Section6.In the 40 and 60 storey buildings, the properties of the structuralelements change in a step-wise fashion every 20 storeys. Table 1shows the the thickness of the walls and cores of all the buildingson different floor levels.For simplicity in determining member masses, half the mass

of the walls and cores framing into and emanating from a floordiaphragm, together with the mass of the diaphragm is stated asan equivalent uniformly distributed floor mass at that storey level.Thus the centre of mass is at the geometric centre of the floor plan.


Table 1Thickness of the walls and cores for each building.

Building height (storeys) Floors Thickness of the walls and cores

5 1st to 5th 0.20 m10 1st to 10th 0.20 m20 1st to 20th 0.20 m40 1st to 20th 0.25 m

21st to 40th 0.20 m60 1st to 20th 0.30 m

21st to 40th 0.25 m41st to 60th 0.20 m

Table 2Properties of Core C1 of Fig. 7.

Thickness of the core Icx1 (m4) Icy1 (m4) Iwc1 (m6) Jc1 (m4)

0.20 m 2.250 14.416 14.175 0.0320.25 m 2.820 18.030 17.719 0.0630.30 m 3.390 21.654 21.263 0.108

Table 3Properties of the substitute flexure–torsion beam calculated from Eq. (7), theequivalent equations for motion in the x–z plane and Eq. (12).

Thickness of the walls and cores Ix (m4) Iy (m4) Iw (m6) J (m4)

0.20 m 36.013 27.466 4263.60 0.0640.25 m 45.040 34.350 5331.01 0.1250.30 m 54.074 41.244 6397.21 0.216

This corresponds precisely to the automatic idealisation process inETABS [25] and additionally only requires the totalmass of the floorto be converted into the equivalent uniformly distributed mass ofthe member in the substitute beam approach. Arbitrarily the massis assumed to have a constant value of 360 kg/m2 at each floorlevel, even where the stiffness properties of the member change.

The shearmodulus and Young’smodulus for allmembers are takento be G = 9× 109 N/m2 and E = 2× 1010 N/m2, respectively.The properties of Core C1 in Fig. 7, with regard to its own shear

centre, are given in Table 2. The properties of Core C2 can thereforebe obtained from the same table by exchanging the values of Ixand Iy.The location of the centre of rigidity, O, should satisfy the

following equations as discussed in Section 3.

3∑i=1

EIwyixi +2∑i=1

EIcyixi = 0 (58a)

3∑j=1

EIwxjyj +2∑j=1

EIcxjyj = 0. (58b)

Since all the structures have the same doubly asymmetric floorplan and the thickness of the walls and cores in the 40 and 60storey buildings change in the same ratio, it is clear that the rigiditycentre at each floor level lies in a vertical line through the building.Utilising Eqs. (58a) and (58b), the eccentricities in the x and ydirections can be calculated as

xc = 6.79 m, yc = 2.89 m.

The properties of the substitute beam for different thicknessesof the walls and cores have been obtained using Eq. (7), theequivalent equations for motion in the x–z plane and Eq. (12) andare given in Table 3.The distributed mass of the substitute beam (smeared from

the diaphragm) and the polar mass radius of gyration of thediaphragms about the shear centre can be calculated as follows

m = 18× 30× 360/3 = 64, 800 kg/m

r2m =182 + 302

12+ 6.792 + 2.892 = 156.46 m2.

Table 4aCoupled natural frequencies (Hz) of the 5-storey building from the continuum and FEMmodels. The frequency reduction factor of Eq. (59), rf = 0.8416.

Freq. no. 3D flexure–torsion beam(Eq. (35))

3D flexure–torsion beam(Eq. (57a)) (GJ = 0)

Modified 3D flexure–torsion beam(Eq. (59))

ETABS (FEM) Difference %

(1) (2) (3) = (1)× rf (4) (1)−(4)(4)

(2)−(4)(4)

(3)−(4)(4)

1 5.7715 5.7694 4.8573 4.5111 27.94 27.89 7.672 8.1179 8.1170 6.8320 6.0192 34.87 34.85 13.503 11.4532 11.4499 9.6390 8.4687 35.24 35.20 13.82

Av. 32.68 32.65 11.67

Table 4bCoupled natural frequencies (Hz) of the 10-storey building from the continuum and FEMmodels. The frequency reduction factor of Eq. (59), rf = 0.9106.





(1) (2) (3) = (1)× rf (4) (1)−(4)(4)

(2)−(4)(4)

(3)−(4)(4)

1 1.4434 1.4423 1.3144 1.2883 12.04 11.95 2.022 2.0295 2.0293 1.8481 1.7813 13.93 13.92 3.753 2.8645 2.8625 2.6084 2.5100 14.12 14.04 3.92

Av. 13.37 13.31 3.23

Table 4cCoupled natural frequencies (Hz) of the 20-storey building from the continuum and FEMmodels. The frequency reduction factor of Eq. (59), rf = 0.9522.





(1) (2) (3) = (1)× rf (4) (1)−(4)(4)

(2)−(4)(4)

(3)−(4)(4)

1 0.3615 0.3606 0.3442 0.3421 5.67 5.41 0.622 0.5074 0.5073 0.4831 0.4792 5.88 5.86 0.823 0.7173 0.7156 0.6830 0.6756 6.17 5.92 1.10

Av. 5.91 5.73 0.85


Table 4dCoupled natural frequencies (Hz) of the 40-storey building from the continuum and FEMmodels. The frequency reduction factor of Eq. (59), rf = 0.9752.


3D flexure–torsion beam(Eq. (35)) (GJ = 0)



(1) (2) (3) = (1)× rf (4) (1)−(4)(4)

(2)−(4)(4)

(3)−(4)(4)

1 0.1011 0.1002 0.0986 0.0977 3.48 2.56 0.912 0.1410 0.1410 0.1375 0.1373 2.69 2.69 0.153 0.2005 0.1988 0.1955 0.1937 3.51 2.63 0.94

Av. 3.23 2.63 0.67

Table 4eCoupled natural frequencies (Hz) of the 60-storey building from the continuum and FEMmodels. The frequency reduction factor of Eq. (59), rf = 0.9833.


3D flexure–torsion beam(Eq. (35)) (GJ = 0)

Modified 3D flexure–torsion beam ETABS (FEM) Difference %

(1) (2) (3) = (1)× rf (4) (1)−(4)(4)

(2)−(4)(4)

(3)−(4)(4)

1 0.0492 0.0481 0.0483 0.0474 3.80 1.48 1.962 0.0678 0.0677 0.0666 0.0666 1.80 1.65 0.003 0.0975 0.0956 0.0958 0.0940 3.72 1.70 1.89

Av. 3.11 1.61 1.28

8. Numerical results

Column 2 of Tables 4a–4e shows the first three coupled naturalfrequencies (Hz) of the 5,10, 20, 40 and 60 storey buildings,respectively, obtained from the proposed three-dimensionalflexure–torsion beam theory. The third column in each table showsthe natural frequencies when the St. Venant torsional rigidity ofthe cores is ignored. (These figures can easily be obtained usingthe theory of Section 6 and Eq. (57a) in the case of the 5, 10 and20 storey buildings since they have uniform properties throughouttheir height.) In the fourth column, the results of Column 2have been modified by multiplying the natural frequencies by areduction factor rf , which allows for the fact that a proportionof the mass of the building is concentrated at floor levels andis therefore not uniformly distributed over the height of thebuilding, as assumed in the derivation of the proposed model. Itis clear that this assumption has considerable influence on shortbuildings, although it can normally be ignored when dealing withtall buildings. Zalka [8] suggests that the reduction factor rf can becalculated as

rf =√

nn+ 2.06

(59)

where n is the number of storeys in the building.Finally, the fifth column in each table shows the results of a full

finite element analysis of the original structures obtained using thevibration programme ETABS, in which the automatic idealisationprocess was utilised that assumes uniformly distributed mass onrigid floor diaphragms. Relevant comparisons aremade in columnssix to eight.

9. Discussion

The results in Table 4a for the five storey building are the leastaccurate of those presented due to the considerable differencebetween the concentrated and distributed mass of the floors overthe height of the structure. However, the difference betweenthe model results and those of the finite element analysis stilllie below 12% when the reduction factor of Eq. (59) is applied.As the number of storeys increases, the difference between theresults becomes significantly less, as shown in Tables 4b–4e. Theresults for the 20, 40 and 60 storey buildings would appear to beperfectly satisfactory, even without applying the reduction factor.Comparison of the results in columns 2 and3of each table indicates

Fig. 8. Graphs of the difference between the averaged results from the proposedmodel for the three cases indicated in Tables 4a–4e and those from the full finiteelement analysis of the original structures.

that the St. Venant rigidity of open cores can safely be ignored. Theomission of this effect might have greater influence in structurescomprising closed cores and this requires further investigation.Finally, Fig. 8 shows the difference between the results from thefull finite element analysis of the original structures and those fromthe three forms of the proposed model i.e. with GJ, without GJand modified. This suggests that the proposed model is likely tobe satisfactory for buildings with 10 to 60 storeys, depending onthe accuracy required, although more investigation is required forshorter and taller structures.

10. Conclusions

A simple and accuratemodel has been developed for calculatingthe lower natural frequencies corresponding to overall modesof vibration of medium and tall building structures. Within thisscope it can encompass many geometric configurations rangingfrom uniform structures with doubly symmetric floor plans todoubly asymmetric ones with step changes of member propertiesat any number of storey levels. The model has been developedon the assumption of uniformly distributed mass and stiffnessand thus necessitates the solution of a transcendental eigenvalueproblem. This can be solved to any desired accuracy by use ofthe Wittrick–Williams algorithm, which also guaranties that nonatural frequencies can be missed. When all the storeys within abuilding can be considered to be identical, the required solutions


can be found easily by hand. Results of a parametric study showthat the model is likely to yield results of sufficient accuracy forengineering calculations when the number of storeys is greaterthan about ten and less than about sixty. As is inevitably the casewhenusing simplifiedmodels, their accuracy should be thoroughlychecked prior to use against datum results for the class of structurebeing considered.

Appendix. The roots of Eq. (19)

The nature of the roots of the characteristic Eq. (19) isinvestigated in this appendix. For this purpose, Eq. (19) is re-written again for convenience.∣∣∣∣∣∣∣∣τ 2 − ω2β2x 0 ycω2β2x

0 τ 2 − ω2β2y −xcω2β2y

ycω2β2x

γ 2x−xcω2

β2y

γ 2yτ 2 − α2ϕτ − ω

2β2ϕ

∣∣∣∣∣∣∣∣W (ξ) = 0. (A.1)

Since α2ϕ , β2x , β

2y , β

2ϕ , γ

2x , γ

2y , xc and yc are all real constants, the

coefficients in Eq. (A.1) are all real. It will be convenient to notethat the determinantal part of Eq. (A.1) is a 6th order polynomialfunction f (τ ) so that

f (τ ) =

∣∣∣∣∣∣∣A(τ ) 0 D(τ )0 B(τ ) E(τ )

1γ 2xD(τ )

1γ 2yE(τ ) C(τ )

∣∣∣∣∣∣∣ (A.2)

in which

A(τ ) = τ 2 − ω2β2x , (A.3)

B(τ ) = τ 2 − ω2β2y , (A.4)

C(τ ) = τ 2 − α2ϕτ − ω2β2ϕ (A.5)

D(τ ) = ycω2β2x , (A.6)

E(τ ) = −xcω2β2y . (A.7)

The quantity f (τ ) is a smooth, continuous function and it is easyto show that f (τ )→∞ as τ →±∞. Additionally f (τ ) is negativewhen τ = 0, as shown below.Substituting τ = 0 in Eq. (A.2) gives

f (0) =

∣∣∣∣∣∣∣∣−ω2β2x 0 ycω2β2x0 −ω2β2y −xcω2β2y

ycω2β2x

γ 2x−xcω2

β2y

γ 2y−ω2β2ϕ

∣∣∣∣∣∣∣∣ (A.8)

or

f (0) = −ω2β2x

(ω4β2yβ

2ϕ −

1γ 2yx2cω

4β2y

)

+ ycω2β2x

(1γ 2xycω2β2x β

2y

). (A.9)

Eq. (A.9) can be simplified to

f (0) = −(1/r2m)(r2m − x

2c − y

2c )β

2x β2yβ2ϕω6 (A.10)

in which rm is the polar mass radius of gyration about the flexuralrigidity centre O and can be related to the polar mass radiusof gyration about the centre of mass, rmc , through the followingequation

r2m = r2mc + x

2c + y

2c . (A.11)

Fig. A.1. (a) Graph of A(τ ) versus τ ; (b) Graph of B(τ ) versus τ .

Therefore

(r2m − x2c − y

2c ) > 0. (A.12)

The right hand side of Eq. (A.10) is the product of six positiveparameters multiplied by a negative sign and therefore f (0) isalways negative.Consider now themore general form of f (τ ). From Eq. (A.3) it is

clear that A(τ ) is a second order equation in terms of τ and alwayshas one positive (x1) and one negative (x2) root, as follows

x1,2 = ±ωβx. (A.13)

In similar fashion, B(τ ) has one positive (y1) and one negative (y2)root given by

y1,2 = ±ωβy. (A.14)

Before discussing the roots of f (τ ) = 0, it is useful to calculatethe quantity f (τ )when τ = x1. Then

f (x1) = −1γ 2xD2(x1)B(x1). (A.15)

Since γ 2x and D2(x1) always have positive values, the sign of

f (x1) only depends on the sign of B(x1). Similarly

f (y1) = −1γ 2yE2(y1)A(y1) (A.16)

and the sign of f (y1) only depends on the sign of A(y1).Now Eq. (A.3) and Fig. A.1(a) show that

A(τ ) < 0 when x2 < τ < x1 and (A.17a)A(τ ) > 0 when τ > x1 or τ < x2. (A.17b)

Similarly, Eq. (A.4) and Fig. A.1(b) show that

B(τ ) < 0 when y2 < τ < y1 and (A.18a)B(τ ) > 0 when τ > y1 or τ < y2. (A.18b)

We now wish to consider the two cases in which x1 < y1 andx1 > y1.Fig. A.1(a) and A.1(b) show that when x1 < y1, B(x1) < 0 and

A(y1) > 0 so that f (x1) > 0 and f (y1) < 0.Similarly, when x1 > y1, B(x1) > 0 and A(y1) < 0 so that

f (x1) < 0 and f (y1) > 0.If λ1 represents the minimum value of x1 and y1, i.e. λ1 =

Min[x1, y1], and λ2 represents the maximum value of x1 and y1,i.e. λ2 = Max[x1, y1], then

f (0) < 0, (A.19a)f (λ1) > 0, (A.19b)f (λ2) < 0 and (A.19c)f (∞) > 0. (A.19d)

This implies that there are at least three positive real roots ofthe function f (τ ) in the intervals (0, λ1), (λ1, λ2) and (λ2,∞). SeeFig. A.2.


Fig. A.2. Graph of f (τ ) versus τ .

Using an identical argument, the roots of f (τ ) for negativevalues of τ can be defined as

f (0) < 0, (A.20a)f (λ3) > 0, (A.20b)f (λ4) < 0 and (A.20c)f (−∞) > 0 (A.20d)

where λ3 represents the maximum value of x2 and y2, i.e. λ3 =Max[x2, y2] and λ4 represents the minimum value of x2 and y2,i.e. λ4 = Min[x2, y2]. This implies that there are at least threenegative real roots of the function f (τ ) in the intervals (−∞, λ4),(λ4, λ3) and (λ3, 0).Since Eq. (A.1) is a sixth order equation in terms of τ , it has been

proven that it will always have three negative and three positivereal roots.

References

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[7] Zalka KA. A simplified method for calculation of the natural frequencies ofwall-frame buildings. Eng Struct 2001;23(12):1544–55.

[8] Zalka KA. Global structural analysis of buildings. London: E&FN Spon; 2001.[9] Potzta G, Kollar LP. Analysis of building structures by replacement sandwichbeams. Internat J Solids Struct 2003;40:535–53.

[10] Tarjan G, Kollar LP. Approximate analysis of building structures with identicalstories subjected to earthquakes. Internat J Solids Struct 2004;41:1411–33.

[11] Skattum SK. Dynamic analysis of coupled shear walls and sandwich beams.California Institute of Technology; 1971.

[12] Basu AK. Seismic design charts for coupled shearwalls. J Struct Eng ASCE 1983;109(2):335–52.

[13] Rosman R. Stability and dynamics of shear wall frame structures. Build Sci1974;9:55–63.

[14] Rutenberg A. Approximate natural frequencies for coupled shear walls. EarthqEng Struct Dyn 1975;4:95–100.

[15] Smith BS, Crowe E. Estimating periods of vibration of tall buildings. J Struct EngASCE 1986;112(5):1005–19.

[16] Smith BS, Yoon YS. Estimating seismic base shears of tall wall-frame buildings.J Struct Eng ASCE 1991;117(10):3026–41.

[17] Kopecsiri A, Kollar LP. Approximate seismic analysis of building structures bythe continuummethod. Acta Tech Acad Sci Hung 1999;108(3–4):417–46.

[18] Kopecsiri A, Kollar LP. Simple formulas for the analysis of symmetric (plane)bracing structures subjected to earthquakes. Acta Tech Acad Sci Hung 1999;108(3–4):447–73.

[19] Rafezy B, Zare A, Howson WP. Coupled lateral–torsional frequencies ofasymmetric, three dimensional frame structures. Internat J Solids Struct 2007;44(1):128–44.

[20] Howson WP. A compact method for computing the eigenvalues andeigenvectors of plane frames. Adv Eng SoftwWorkstations 1979;1(4):181–90.

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[22] Wittrick WH, Williams FW. A general algorithm for computing naturalfrequencies of elastic structures. Quart J Mech Appl Math 1971;24(3):263–84.

[23] Howson WP, Williams FW. Natural frequencies of frames with axially loadedTimoshenko members. J Sound Vib 1973;26(5):503–15.

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