advances in structural engineering -...

Conceptual Design of Tall and UnconventionallyShaped Structures: A Handy Analytical Method

by

Alberto Carpinteri , Giuseppe Lacidogna and Sandro Cammarano

Reprinted from

Advances in Structural EngineeringVolume 17 No. 5 2014

MULTI-SCIENCE PUBLISHING CO. LTD.5 Wates Way, Brentwood, Essex CM15 9TB, United Kingdom

1. INTRODUCTIONTall buildings have always been the symbol ofsupremacy of the nations engaged in their construction.Since its first appearance, this structural typology hasmet approval in the public eye. Nowadays, they havebecome a worldwide architectural phenomenon, even forthat countries regarded as less advanced, which howeverare demonstrating a fast industrial growth. As a matter offact, many tall buildings are built in Asia countries, suchas China, Korea, Japan and Malaysia since the economiccapability and technological progress underlie thedevelopment of such innovative architectural works.

Historically, the early reason of growing in heightwas commercial, having to compensate for the lack ofspace and natural light in a urban densely populatedland (Taranath 1988). However the higher the building,the more sensitive it became to lateral actions coming

Advances in Structural Engineering Vol. 17 No. 5 2014 767

Conceptual Design of Tall and Unconventionally

Shaped Structures: A Handy Analytical Method

Alberto Carpinteri , Giuseppe Lacidogna* and Sandro CammaranoDepartment of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Torino, Italy

(Received: 28 April 2013; Received revised form: 23 February 2014; Accepted: 28 February 2014)

Abstract: Nowadays high-rise buildings are a worldwide architectural phenomenon.In the last decades, next to economics, municipal regulations and politics, aestheticshas got a leading role in planning and design of these structures, giving rise to bizarreshapes, from diagrid systems to twisted, tapered and tilted ones. In their structuraldesign, the choice of an appropriate model able to thoroughly identify the keyparameters governing the response of the structure as well as the force flow actingwithin the stiffening members is all along a crucial factor. In this paper a three-dimensional formulation is proposed to evaluate the lateral load distribution ofexternal actions in tall buildings, in which the geometry of the stiffeners can varyalong the height. This method takes into account any combination of bracings,including elements with open thin-walled cross-section, which are analysed in theframework of Timoshenko-Vlasov’s theory of sectorial areas. In order to evaluate theeffectiveness and the suppleness of the formulation, comparisons with otherapproaches derived from the literature and numerical examples regarding newarchitectural trends are carried out.

Key words: structural behaviour, modelling methods, tall buildings, lateral load distribution, thin-walled cross-section, unconventionally shaped building, twisted structure, tapered structure.

from wind and earthquakes. Without lateral stiffeners,the dimensions of the structural elements increased sothat they couldn’t be longer a satisfactory solution froman architectural point of view. In addition, it constituteda limit on the evolution in height of these revolutionaryconstructions. The 17-storey (64 m) MonadnockBuilding in Chicago, being an impressive structure inwhich the resistant mechanism relies on heavy masonrywalls, is the symbol of this issue (Taranath 2005).

Subsequently, the conventional load-bearing systemswere substituted by new technologies which reduced thedepth and width of the structural members at buildingperimeter. The first result was a steel frame structure.This typology was followed by several systemsdesigned to absorb and distribute the horizontal loadaccording to their own stiffness. At this stage frames,braced frames, shear walls and interactive frame – shear

*Corresponding author. Email address: [email protected]; Fax: +390110904999; Tel: +390110904871.

(306 m) London Bridge Tower, also known as Shard ofGlass, which is a pyramidal shaped building, now thetallest structure in Europe.

Even if the increase of the complexity of the forms isbalanced by powerful computers and several multi-function Finite Element (FE) software, the choice of anappropriate model able to thoroughly identify the keyparameters governing the response of the structure aswell as the force flow acting within the stiffeningmembers remains crucial.

On the one hand, FE programs can evaluate theconstruction in its entirety, reaching an high degree ofaccuracy. Indeed they could model any detail, givingthe idea that nothing gets lost. Nevertheless this skillcan hide some drawbacks (Howson 2006;Steenbergen and Blaauwendraad 2007). During thedesign stage, it’s very difficult to assess the resistantcontributions coming from different stiffeners as wellas handle an enormous amount of data. In effect,especially during the phase of evolution of theconcept, the former could cause time-consumingmisunderstandings; the latter could be a source forerrors. Moreover, the great number of input andoutput data does not support a clear explanation of thestructural mechanism and does not allow the designersto identify the distribution of the external forcesamong the stiffening members.

On the other hand, based on some carefully chosenhypotheses, simplified procedures could represent a validalternative in the early stage of conceptual design, beingcharacterised by some advantages, such as a faster datapreparation and a more transparent method of analysis,which make the process less liable to unexpected errors.In addition, unlike FE simulations, the limited degree ofaccuracy is balanced by the capability to provide acomprehensive picture of the structural behaviour and togain knowledge of the key parameters governing theresponse of the building. In any case, being reciprocallycomplementary instruments, both approaches can lendsupport to the engineer’s judgment: in the early stages,approximate methods evaluate the basic characteristics ofthe project; in the final ones, FE models can conduct amore thorough computation.

In this paper numerical procedures for the definitionof the stiffness matrix of vertical bracings used in tallbuildings, whose geometry can vary along the height,are proposed. In this way, the work by Carpinteri (1985)can be easily extended to encompass unusually shapedstructures, such as tapered or twisted buildings. In orderto evaluate the effectiveness and the suppleness of themethod, comparisons with other approaches derivedfrom the literature and numerical examples regardingnew architectural trends are carried out.

768 Advances in Structural Engineering Vol. 17 No. 5 2014

Conceptual Design of Tall and Unconventionally Shaped Structures: A Handy Analytical Method

wall combinations appeared (Coull 1972; Heidebrecht1973).

Later, designers concluded that the building could betreated in a holistic manner, giving rise to various othermodels which increased its lateral resistance without anexcessive use of structural materials. For this reason, thetraditional resistant schemes were gradually replaced bya global approach. The structure was considered as avertical cantilever or a system of cantilevers on theground, having all the required lateral stiffness allocatedto the perimeter of the building. This shrewdness aimsto increase the structural depth of lateral load-resistingsystem components and, thereby, their resistantcontribution. In this direction, systems organised toreduce the overturning moment of the cantilever schemeand transfer the reduced moment to outer membersthrough extremely rigid elements were devised. At thesame time, closely spaced columns and deep spandrelbeams rigidly connected together and smeared on theperimeter of the structure gave rise to a single three-dimensional element similar to a large tube (Coull 1971,1977; Khan 1974). According to these outlooks anddepending on the height of the construction, severalsolutions, such as outrigger, framed-tube, bundled-tubeand tube-in-tube systems were realised.

Only later, next to economics, municipal regulationsand politics, aesthetics got a leading role in planning anddesign of these structures. In this way, changes in theexterior form of the constructions were supported by theemerging architectural trends and by the developmentsin structural analysis techniques, made possible by theadvent of high-speed digital computers. Diagrid systemsconfirm the breakthrough in the idea of tall building,since almost all vertical elements are eliminated infavour of diagonal members able to carry, at the sametime, gravity loads and lateral actions. Theirtriangulated configuration uniformly arranged on theentire façade enables to model groundbreaking shapes.Furthermore, this scheme assures an unexpectedbending and shear rigidity, since the diagonal elementswork only axially, minimising in this way the sheardeformations (Ali 2007). The 30 St. Mary Axe inLondon, a diagrid structure also known as the Swiss ReBuilding, demonstrates that the current architecture hasforsaken prismatic forms, to embrace curved ones.

Anyway, developments regarding the design of high-rise and irregular buildings are described by acontinuously evolving process. As a matter of fact, allover the world, further bizarre shapes, such as twisted,tapered or tilted, have been commissioned and, in somecases, already built. Glaring examples are the HSBTurning Torso, a twisted skyscraper of 54 storeys(190 m) in Malmo (Sweden), and the 66-storey

2. STIFFNESS MATRIX FOR BRACINGSWITH VARIABLE CROSS-SECTION

The computation of the stiffness matrix for prismaticmembers is well-known. The corresponding analyticalmethod can be easily implemented in a computerprogram to evaluate the contribution of the mainresistant schemes to the horizontal strengthening ofhigh-rise buildings.

According to the new architectural trends, in the caseof bracings with variable cross-section, the computationof the stiffness matrix is more complex than other cases.For this purpose, in this section, appropriate methodsable to analyse stiffeners whose geometry varies alongthe height are proposed. In particular, analyticalformulations for tapered and twisted bracings, havingthin-walled closed or open sections, are derived.

2.1. Tapered and Twisted Bracings with ClosedSection (Warping Negligible)

Because of the nature of the problem and the type ofstructure involved which can be easily assimilated to aplanar shear wall for each principal direction of inertia,it’s advantageous to consider its plane behaviour andcompute the floor displacements starting from theapplied loads.

It’s well-known that a unitary force applied to the i-th level gives rise to displacements of all the levels:these values constitute the i-th column of thecompliance matrix of the stiffener.

In the case of 2-storey shear wall, the coefficients ofthe compliance matrix D are:

D D D11

13

121 11

12

2

13 2

= = +h

EJ

h h

EJ, ,

(1)

being hi and Ji the storey height and the second ordermoment of inertia of the i-th level (from bottom to top)respectively (Figure 1).

Since some geometrical characteristics related to thetwo levels of the shear wall have been considereddifferent each other, for the definition of the term D22

it’s more convenient to take into consideration twostructural schemes: scheme 1 shows, for the specificload condition, the first floor constrained; scheme 2, freefrom additional constraints, shows at the first level theresultant system of forces due to the initial loading case.The first four components of the term D22 concern thescheme 2: the first two describe the displacement of thefirst level, while the third and fourth are the consequentrigid displacements of the second level. The last term ofD22 is related to the scheme 1 and represents thedisplacement of the second level being the first oneconstrained.

The procedure can be easily extended to consider Nfloors, each having its own storey height hi and its ownmoment of inertia Ji (Figure 2).

The generalised term Dij (with j ≤ i), representing thedisplacement of the i-th level due to the application of aunitary force to the j-th level, can be evaluated througha recursive process. For a given load condition appliedto the j-th level, the resultant system of forces at the firstlevel is estimated. Then the displacement of the i-thlevel for this load case is deduced:

D D Dh

EJ

h h

EJ

h h

EJ12 21 2213

1

2 12

1

2 12

13 2 2

= = + + +,

,+ +h h

EJ

h

EJ1

22

1

23

23



1.h2

1

12nd floor

1st floor

h2

D21

D12

D221

D11

h1

1

Scheme 1 Scheme 2

(a) (b)

Figure 1. (a) Evaluation of the terms belonging to the compliance matrix of the shear wall; (b) Schemes for the computation of the

displacement D22

(2)

The same computation is repeated considering theresultant system of forces at the k-th level (k = 2, 3, .., j).A complete expression of Dij is given by the sum of allcontributions.

By means of the Eqn 3 the computation of the lowertriangular part of the N × N matrix D is executed.Exploiting its symmetry, as proved by Betti’s theorem,the upper triangular part is completed.

The same method can be extended to assess itstorsional behaviour. In this case, neglecting the warpingof the section, the generic term of the N × N torsionalcompliance matrix Dϑ is expressed by means of thetorsional moment of inertia Jt and the shear modulus G.

(4)Dij j

ikj k

kϑ ,

t,

.== =∑ ∑1 1

h

GJ

(3)

Dh

EJ

h

EJij ji

kj k

k

j k k

k

= +−( )

+= =∑ ∑1 1

3 2

3

1 1

2

( )

.+ +−

−( )

h

EJ

h

EJk

k

j k k

ki k

2

2

1 11 1

h h

h h

13

1

1 12

1

12

1

1 1

3

1 1

2

2

1 1

EJ EJ

EJ EJ1

+−( )

+

+ +−

j

j( )

−( )1 1

1i.

The last step is the evaluation of the 3N × 3N stiffnessmatrix K* of the generic bracing, in its own coordinatesystem. Its structure is block diagonal, constituted by the2N × 2N stiffness matrix K*

d related to the localdisplacements u and v and the N × N stiffness matrix K*

ϑ

related to the rotation. Each of them is obtained byinverting the corresponding compliance matrix.

(5)

(6)

In the case of twisted bracings, the structure of thestiffness matrix is no longer block diagonal, since thesub-matrix K*

d becomes full. The computation of itscomponents follows the same approach used for taperedstructures, taking into account the increasing rotation ofthe sections from the ground to the top (details of theprocedure are in Appendix).

The matrix K*d, whose coefficients are referred to the

coordinate system of the ground level, is hencecomposed by four N × N sub-matrices:

(7)

A general procedure for a N-storey structureassimilated to a three-dimensional shear wall that tapersand twists at the same time along the height may bedeveloped. Therefore, the displacement of the i-th level,in X direction, due to the loading vector Fj = {Fx,j Fy,j}applied to the j-th level, is defined by two contributionsDx,ij and Dxy,ij:

(8a)

Dh

EJ

h

EJ, ,

x,

v v

ij j

i

k

j k

k

j k k

k

= +−

= =∑ ∑( )

1 1

3 2

3

1 1

2

+

( )

+ +

−

v v

h

EJ

h

EJ, ,

k

k

j k k

k

2

2

1 111 1

3

1

2

3

i k

k

k

j

k− +

+ +

( )

( )cos

u

α

h

EJ,

−−+

+ +−

( )

( )

1

2

1 1

2

2

k k

k

k

k

j k

h

2EJ

h

EJ

,

,

u

u

hh

EJ,

k

k

i k ku

sin ,

( )

( )

−1 1

2α

Kd* x

*xy*

xy*

y*

=

k k

k k.

KK O

O K* d

*

*=

ϑ

.

KO

Od* u

*

v*

=

k

k,



Figure 2. Multi-storey shear wall having different geometrical

characteristics for each floor

N-1

N

i

j

2

1

J2 h2

hNJN

h1 l1

l2

lj

li

lN

J1

In the same way, the generic term Dy,ij is given by thefollowing expression:

By means of the Eqns 8, the computation of the lowertriangular part of the 2N × 2N matrix D of thedisplacements is executed. Due to the symmetry, the uppertriangular part is completed. Once defined, by inversion,

Dh

EJ

h

EJy,

v vij j

i

k

j k j k k=

= =∑ ∑ +

−( )

1 1

3 2

3

1 1

2,k ,k

+

+−( )

+

v v

h

EJ

h

EJk j k k2

2

1 1

,k ,k

−( )

( )

++

1 1

3

2

i k ksin

u

α

h

EJ

3k

,k

11 1

2

1 1

2

2

j k k

k j

−( )

+

+−

+

h

2EJ

h

EJ

u

u

,k

,k

kk k

ki k

( )

( )

( )

−

h

EJu

cos .

,k

1 12

α

Dh

EJ

h

EJxy,

v vij j

i

k

j k j k k= +

−= =∑ ∑

( )

1 1

3 2

3

1 1

2,k ,k

+−( )

+

+ v v

h

EJ

h

EJk j k k2

2

1 1

,k ,k

( )

+−

− +

−

1 1

3

3

1 1

i k

k j k

u

h

EJ,k

(( )+

+−( )

+

h

2EJ

h

EJ

h

EJ

k

k j k k

2

2

2

1 1

u

u u

,k

,k ,k

( )

−1 1

i k k kcos sin .α α

the 2N × 2N sub-matrix K*d, according to the Eqn 6, the

complete 3N × 3N stiffness matrix K*is obtained.In order to evaluate the effectiveness of the Eqns 3

and 8, two comparisons regarding tapered and twistedbeams are performed. In the first case, a thin-walledhollow section cantilever is analysed through a FiniteElement program, in which the structure is modelled bymeans of thin shell elements. Information concerningthe geometrical dimensions and the mechanicalproperties are shown in Figure 3(a), whereas the resultsin terms of transversal displacements are highlighted inTable 1. In the second case, the transversaldisplacements of a twisted beam are acquired from thepaper by Zupan and Saje (2006). The scheme of themodel as well as the geometrical and mechanicalproperties are indicated in Figure 3(b); Table 2 reportsthe comparison of the acquired results. In both cases,subdividing the beams in 40-50 sub-elements, Eqns 3and 8 lead to solutions with an adequate degree ofaccuracy. Besides, such segmentation proves to beplausible in high-rise buildings, being the number offloors equal or, at most, greater.

2.2. Tapered Bracings with Thin-Walled OpenSection (Warping Prevalent)

A numerical procedure for the definition of the stiffnessmatrix of tapered thin-walled open section bracings intheir local coordinate system is now derived. For thesestructures, the process of tapering along the height refersto a vertical axis passing through the barycentre of thesection. Since the centroid and the shear centre do notcoincide, the location of the latter varies section bysection (Figure 4).

As in the previous cases, the expression of the stiffnessmatrix K* is obtained by the inversion of the 3N × 3Ncompliance matrix D. For this purpose, the computationof the coefficients of the above matrix is executed by



fx

fy

1.1

0.32

12

E = 29 × 106 kN/m2

v = 0.22fy

fx

152

0.5

0.5

1

0.1

0.1

E = 30 × 106 kN/m2

v = 0.18

(a) (b)

Figure 3. (a) Tapered hollow; (b) Twisted doubly symmetrical section cantilever

(8b)

(8c)

means of the Principle of Virtual Work, in which thecontribution of the Bimoment action is considered:

(9)

where the apex f stands for the fictional system ofinternal forces and r for the real system ofdisplacements.

1( ) ( ) ,f r

z⋅ = +

∫η M

M

EIB

B

EIdz( )

( )( )

( )f

rf

r

ω

The proposed method is based on the assumptionsthat the shear effects on very slender structures arenegligible and the Bimoment action is evaluatedsupposing the torsional rigidity GJt equal to zero. Bymeans of the Eqn 9, further coefficients arise, so that Dbecomes a full matrix. This means that the torsionalbehaviour is connected to the flexural one, as well as theforces acting along a principal direction give rise also todisplacements in the other direction. This behaviour isdue to the variation of location of the shear centre alongthe longitudinal axis, which consequently affects thedefinition of the resultant actions on the generic level.

For the analysis, we suppose to apply the localcoordinate system to the shear centre of the ground level.The actions, applied to the generic floor according to thiscoordinate system, show an eccentricity compared to theshear centre of the same level. This scheme involvesfurther torsional actions on the generic section, whichhave to be taken into account in the study (Figure 5).

In this way, each local force, at the same time, causesdisplacements in its principal direction, according to theflexural behaviour, rotations, due to the additionaltorsional component, and displacements in the otherprincipal direction, derived from the contribution of theBimoment action.

For the case of a 2-storey bracing, the diagrams offlexural moment and bimoment, which are taken intoaccount as the contribution of the real system ofdisplacements in Eqn 9, are reported in Figure 6. The



StopCtop

Cground

Sground

Z

Figure 4. Thin-walled open section bracing, which tapers with

respect to the centroidal axis

Table 1. Free end displacement of a tapered cantilever

N. of levels Centroidal force fx = 10 kN Centroidal force fy = 10 kN

x disp. [m] Err. [%] y disp. [m] Err. [%]5 2.840E – 03 1.626E + 01 1.662E – 02 5.090E + 0010 2.572E – 03 5.302E + 00 1.614E – 02 2.097E + 0020 2.482E – 03 1.611E + 00 1.597E – 02 1.015E + 0030 2.457E – 03 5.628E – 01 1.592E – 02 7.009E – 0140 2.444E – 03 6.836E – 02 1.590E – 02 5.520E – 01FEM 2.443E – 03 1.581E – 02

Error [%] = (Present model – FEM)/FEM × 100

Table 2. Free end displacement of a twisted cantilever

N. of unitary force fx unitary force fy

levels

x disp. [m] Err. [%] y disp. [m] Err. [%] x disp. [m] Err. [%] y disp. [m] Err. [%]10 5.5001E – 03 1.43E + 00 1.4805E – 03 –1.39E + 01 1.4805E – 03 –1.39E + 01 1.6720E – 03 –4.06E + 0050 5.4426E – 03 3.72E – 01 1.6730E – 03 –2.66E + 00 1.6730E – 03 –2.66E + 00 1.7295E – 03 –7.62E – 01100 5.4344E – 03 2.20E – 01 1.6960E – 03 –1.32E + 00 1.6960E – 03 –1.32E + 00 1.7377E – 03 –2.89E – 01150 5.4316E – 03 1.68E – 01 1.7036E – 03 –8.82E – 01 1.7036E – 03 –8.82E – 01 1.7405E – 03 –1.28E – 01200 5.4302E – 03 1.42E – 01 1.7074E – 03 –6.61E – 01 1.7074E – 03 –6.60E – 01 1.7419E – 03 –4.75E – 02

Zupan et al. 5.4224E - 03 1.7187E - 03 1.7187E - 03 1.7427E - 03

Error [%] = (Present model - Zupan et al.)/(Zupan et al.) × 100

same diagrams, in which the generic action issubstituted by a unitary load, allow to identify thecontribution of the fictional system of forces.

Thus, after performing the calculations, the genericexpression for the compliance matrix is

(10)

in which only Dx, Dy and Dϑ are symmetrical sub-matrices. In addition, it should be noted that the sub-matrices belonging to the lower triangular part of D arerelated to those of the upper part by means of thetranspose operation.

Once obtained the stiffness matrix by the inversion ofEqn 10, the last step focuses on the addition of the

D

D D D

D D D

D D D

=

x x

xyT

y y

xT

yT

xy ϑ

ϑ

ϑ ϑ ϑ

component related to the torsional rigidity GJt,previously neglected. It may be easily computedthrough the Eqn 4, which defines the correspondingcompliance matrix. Then it is inverted and added to theN × N sub-matrix related to the rotations. Hence theexpression of the matrix K* for tapered thin-walled opensection bracings is completed.

Exploiting the numerical results obtained byEisenberger (1995) on the structural behaviour of atapered thin-walled open section beam subjected to aconcentrated torsional action, the capabilities of thepresent method can be verified. The scheme proposedfor the comparison is a cantilever beam of 40 cm,whose geometrical characteristics are shown in Figure7. The mechanical properties are the Young’s modulusE = 21 × 105 kg/cm2 and the shear elastic modulusG = 8.05 × 105 kg/cm2. The external load is defined bya concentrated torque Mz = 300 kg cm, applied to thefree end of the structure. The comparison of the results



Figure 5. Local coordinate system for a tapered thin-walled open section bracing

Sground Sground

°CStop

°Cey

y* y*x*x*

ex

Fy

Fx

Mz

Stop

Figure 6. Main diagrams for the computation of the compliance matrix of a 2-storey bracing

Staticscheme

Flexuralmoment

Flexuralmoment

Bi-moment

Bi-moment

Static scheme

F2

M2

M2hM1

M22h

F1F2h

F1h

M1h

F22h

F2e2

F1e1F2e2 h

F1e1 hF2h (e2 + e1)

h

h

in terms of rotation of the free end of the cantilever arehighlighted in Table 3. In particular, in addition to thesolutions acquired through the present method, the table includes the results obtained subdividing thebeam in sub-elements of equal length, each havingconstant geometrical properties. In this case, for eachelement, the equation of torsional equilibrium related tothin-walled open section beams (Vlasov 1961) is solvedanalytically, employing the following boundaryconditions: rotation and its derivative equal to zero atthe clamped end; Bimoment action equal to zero at thefree end; continuity conditions for the rotation, itsderivative and the Bimoment action at the intermediatesections. As it can be seen observing the table ofresults, the degree of accuracy of the proposedapproach is good, if the procedure is applied to the caseof high-rise buildings, being the per cent error lowerthan 7 %.

3. NUMERICAL EXAMPLESThe developed numerical procedures may be easilyadapted to the analytical method proposed by Carpinteriet al. (2010, 2012), which allows to analyse the lateralload distribution of external actions in tall buildings,

stiffened by different types of vertical bracings.In order to highlight the adaptability of the Eqns 8

and 9, two numerical examples which take into accounthigh-rise buildings stiffened by twisted or taperedbracings are performed. Both of them are theoreticalsince any structural details have not been provided bythe project managers.

The first model concerns the 54-strorey HSB TurningTorso, design by Calatrava in Malmo (Sweden). It is atwisted skyscraper reaching 190 m of height with arotation from the base to the top of 90 degrees (Figure 8).It is assumed that the lateral stiffening relies on twoconcentric bracings: the innermost element exhibits acircular hollow section which tapers upwards byreducing its thickness from 2.5 to 0.4 m; the outermosthas a mono symmetrical section which twistsanticlockwise around its shear centre. Since the latterdoes not coincide with the barycentre, further torsionalactions are expected in the computation.

It is assumed that both the cantilevers are made ofconcrete, whose Young’s modulus is 4.5 × 104 and 2.5 × 104 MPa for the circular and mono symmetricalsection respectively, whereas the Poisson ratio is 0.18for both. The influence of creep and shrinkage is nottaken into account in the analysis. The member cross-section properties are given in Figure 9 and Table 4.

Concerning the load, only wind actions are consideredaccording to the formulas indicated by the ItalianTechnical Regulations (Ministero delle Infrastrutture2008), which follow the same method contained inEurocode 1 (European Committee for Standardization2002). Therefore, the wind action may be reduced to asystem of concentrated static loads, applied to thebarycentre of the pressure distribution. The size, shapeand dynamic properties of the building as well as theregion and the altitude of the location affect thecomputation of the intensity of the action. For the sake ofsimplicity, none of the mentioned properties has been



Mz

9

1

1

4

8

18

Figure 7. Tapered thin-walled open section cantilever

(Eisenberger 1995)

Table 3. Free end rotation [rad] of a tapered thin-walled open section cantilever

Present N. of AnalyticalN. of levels model Err. [%] equations solution Err. [%]

10 5.081E – 04 1.83E + 01 10 5.073E – 04 1.82E + 01

30 4.671E – 04 8.79E + 00 30 4.670E – 04 8.78E + 00

50 4.604E – 04 7.24E + 00 50 4.604E – 04 7.24E + 00

100 4.557E – 04 6.14E + 00 100 4.557E – 04 6.14E + 00

150 4.542E – 04 5.79E + 00 150 4.542E – 04 5.79E + 00

200 4.534E – 04 5.62E + 00 200 4.534E – 04 5.61 E + 00Eisenberger 4.293E-04 FEM 4.296E-04

Error [%] = (Computed – Eisenberger)/Eisenberger × 100

considered. Therefore, a wind pressure equal to 390.62N/m2 acts along the principal directions X and Y.

The results of the analysis are presented in Figures 10and 11. As regards Figures 10(a) and 10(b), thedisplacements along the X and Y direction are reported,whereas Figure 10(c) shows the rotations at the floorlevels. Similarly, in Figure 11 the lateral loaddistribution of the external actions between thestiffeners highlights the resistant contribution of thetwisted element compared to the tapered one. Inparticular, the former plays a predominant role in the toppart of the building, whereas, in the bottom part, thelatter constitutes the main horizontal stiffening.

It should be noted a discontinuity next to theconstraint due to the different law of variation whichcharacterises the bracings. Such difference leads to anexchange of high interaction forces in the bottom part ofthe building, which modifies the trend of the shear.

The second numerical example is focused on theanalysis of a conical structure conceived by NormanFoster in 1989 for the city of Tokyo (Japan). TheMillennium Tower is a high-rise building composed by170 storeys, which correspond to a total height of 840metres (Figure 12).

The present model is imaginary, because only apreliminary design has been performed until now.Therefore no details on the floor layout or on thehorizontal stiffening has been found. Consequently thefollowing structural choices as well as geometricalproperties may represent a valid proposal for itspractical construction.

The proposed horizontal stiffening is composed byseven thin-walled open section shear walls which taperupwards, until the 170th floor (Figure 13). In particular,the inner section reaches the top level with adimensional reduction of 80 per cent, whereas theothers, defined by different heights corresponding tothe 50th, 60th, 70th, 80th, 90th and 100th floor, show areduction of 40 per cent. Nevertheless, in all cases the



Figure 8. HSB Turning Torso by Calatrava, Malmo (from

www.flickr.comphotosmiltoncorrea)

Y

15.6

25

10.6

25 15.6

10 X14.2 10.6

12.5

Top floorGround floor

11.4

Y

X16.4

0.2

Figure 9. Presumed floor plan of the HSB Turning Torso

(Dimensions in metres)

Table 4. Cross-section properties of the bracings

Circular hollow section Mono symmetrical section

Second moment Jxx [m4] 2229.89 (B) 209.35 (T) 2070.00 (B)Second moment Jyy |m4] 2229.89 (B) 209.35 (T) 4007.09 (B)Torsional rigidity Jt [m4 ] 4459.79 (B) 418.70 (T) 3940.03Global coordinate xs of the shear 0.00 0.00centre [m]Global coordinate ys of the shear 0.00 0.00centre [m]Angle ϕ [°] * 0.00 0.00

(B) At the base of the building. (T) At the top of the building. * Rotation of the local coordinate system with respect to the global coordinate one.



55

50

45

40

35

30

25

20

15

5

00 1000 2000 3000

TX (kN)

10

Total Tapered Twisted

55

50

45

40

35

30

25

20

15

5

00–2000 600040002000

MZ (kNm)

10

Flo

orF

loor

(a)

(c)

55

50

45

40

35

30

25

20

15

5

00 1000 2000 3000

Ty (kN)

10



Flo

or

(b)

54

45

36

27

18

0E + 00 4E − 03 8E − 03

ux (m)

1.2E − 02 1.6E − 02

0E + 00 4E − 03 8E − 03

uy (m)

1.2E − 02 1.6E − 02

0E + 00 2E − 06 4E − 06

ϑ (rad)

6E − 06 8E − 06

Flo

or

9

0

54

45

36

27

18

Flo

orF

loor

9

0

54

45

36

27

18

9

0

(a)

(b)

(c)

Figure 10. Displacements of the floors in the global coordinate

system. Translation in: (a) X direction; (b) Y direction; and (c)

rotation

Figure 11. Loading distribution between the tapered and twisted

element: shear in: (a) X direction; (b) Y direction; and (c) torsional

moment

thickness of the walls remains constant.It should be noted that the tapering law of the

elements, whose reference axis is the barycentre ofeach section, is not the same. Further detailscharacterise the model: in the case of ‘C’-shapedbracings, the same top section has been considered,

whereas, for the inner bracing, between the 130th and170th floor, the initial section has been reduced to across-shaped one. The levels, which correspond to astructural discontinuity, are shown in Figure 13; thegeometrical dimensions of the cross-sections arereported in Table 5 and Figure 14.

The material properties are described by a Young’smodulus equal to 5.0 × 104 MPa for the ‘C’-shapedbracings and 7.0 × 104 MPa for the inner element; thePoisson ratio for the entire structural skeleton is 0.18.The creep and shrinkage effects are excluded from theanalysis. The same load condition is taken into accountfor this numerical example, with a wind pressure of390.62 N/m2 applied to the lateral surface of thebuilding.

Consequently, results concerning the displacementsalong the principal directions of the global coordinatesystem are highlighted in Figure 15, whereas Figure 16reports the load distribution between the main verticalmembers. A clear difference between the two principaldirections is observed with respect to the shear



Figure 12. Millennium Tower by Sir Norman Foster (from

www.fosterandpartners.com)

80th floor80th floor70th floor60th floor50th floor

765 m

840 m

585 m

450 m

4

2

6

5

3

7

1

405 m360 m315 m270 m225 m

90th floor90th floor

100th floor

100th floor

130th floor

130th floor

170th floor

Top floor

Ground floor 50th floor 60th floor 70th floor

Figure 13. Hypothetical scheme of the horizontal stiffening for the Millennium Tower: the structural members taper upwards, each having

its own tapering law, and reach different heights

Table 5. Cross-section properties of the thin-walled open section bracings

Open section shear wall N. 1 2, .., 5 6, 7

Ground floor Top floor Ground floor Top level Ground floor Top levelSecond moment Jxx [m4] 55605.64 66.29 82.30 18.06 82.30 18.06Second moment Jyy [m4] 23830.99 66.29 1484.39 325.80 1484.39 325.80Warping constant Jω [m6] 6325059.37 – 5906.30 471.68 5906.30 471.68Torsional rigidity Jt [m4] 179.58 18.21 5.45 3.29 5.45 3.29Global coordinate xc of the 0.00 0.00 –20.15 (2) –20.15(2) 0.00 0.00barycentre [m]Global coordinate yc of the 0.00 0.00 10.88 (2) 10.88 (2) 30.38 (6) 30.38 (6)barycentre [m]Angle ϕ [°] * 0.00 0.00 0.00 0.00 0.00 0.00

(2) Open section shearwall N.2; (6) Open section shear wall N.6. * Rotation of the local coordinate system with respect to the global coordinate one.



C6

C5

40

40

1.5

C3C4

C1

C7

X

Y

C2 20

6 0.8

(a)

(b)

Figure 14. Hypothetical scheme of the horizontal stiffening for the Millennium Tower: (a) global coordinate system XY; (b) geometrical

dimensions of the cross-sections at the ground floor (in metres)

170

153

136

119

102

85

68

51

34

17

0−10000 −5000 0 5000 10000 15000

Tx (kN)

Flo

or

TotalSw N. 1Sw N. 2-7

170

153

136

119

102

85

68

51

34

17

0−5000 0 5000 10000 15000

Ty (kN)

Flo

or

TotalSw N. 1Sw N. 2-7

(a) (b)

Figure 16. Loading distribution between the inner (Sw N.1) and the ‘C’-shaped bracings (Sw N.2-7), Shear trend in: (a) X direction;

and (b) Y direction

Figure 15. Displacements of the Millennium Tower in: (a) X direction; and (b) Y direction

(a) (b)

00

34

68

102

136

170

0.2 0.4 0.6 0.8

ux (m)

Flo

or

00

34

68

102

136

170

0.2 0.4 0.6 0.8uy (m)

Flo

or

distribution: on the one hand, as reported in Figure 16(b),the flexural stiffness of the inner section along the Yaxis is so large that the contribution of the ‘C’-shapedbracings is almost negligible; on the other hand, withregard to the X axis, remarkable discontinuities areevident due to the different heights of the ‘C’-shapedelements. Indeed, along this direction, since the latterexhibit a flexural stiffness quite comparable to that ofthe inner member, high interaction forces ariseallowing the ‘C’-shaped sections to absorb, in thebottom part of the building, about 25 per cent of totalshear.

These findings may suggest to the designer thepossibility of considering further structural arrangementor different solutions, such as outrigger systems ortubular elements, in order to avoid to concentrate themost of the load on a single huge bracing.

In summary, the previous figures demonstrate thecapabilities of the analytical method in the evaluation ofthe gross displacements as well as in the detection of thedistribution of the external forces between the mainvertical bracings which concur to stiffen high-risebuildings. The method may be used to find out the optimalconfiguration of the structural members, which allows toachieve the best performance in presence of static windloads. As a matter of fact, the choice of different heightswith respect to the ‘C’-shaped elements has been drivenby the need of reducing the displacements, withoutcompromising the living space of the floors.

Thus, analytical methods prove to be adequate in theearly stage of the conceptual design of so complexconstructions. With the qualities of a quick datapreparation and a more transparent evaluation of theresults, they may play a decisive role in support of thedesigner’s judgment.

4. CONCLUSIONIn the design of the last high-rise buildings, innovativeand bizarre shapes have come to the fore due to theemerging aesthetical factor which has assumed aleading role in this field. Tapered, twisted, tiltedstructures as well as profiles defined by a doublecurvature have been realised hither and thither in theworld, being the Shard Glass, the 30 St. Mary Axe(London - England) and the HSB Turning Torso(Malmo - Sweden) the more representative of all.

The evolution of the shape has required, in parallelwith the well-known Finite Element Method, to findnew analytical techniques able to thoroughly identifythe key parameters which govern the structuralbehaviour of more and more complex buildings. Forthis purpose, in the present paper, a three-dimensionalformulation, which evaluates the structural behaviourof high-rise buildings stiffened by unconventionallyshaped bracings, is proposed. In particular, numerical

techniques focused on the computation of the stiffnessmatrices of tapered and twisted bracings arehighlighted. In order to evaluate the effectiveness andthe suppleness of the model, comparisons with otherapproaches derived from the literature and twonumerical examples regarding new architectural trendsare carried out. The obtained results confirm thesignificant contribution of the proposed method to thedesigner’s judgment in the early stages of the project ofvery complex buildings.

ACKNOWLEDGEMENTSThe financial support provided by the Ministry ofUniversity and Scientific Research (MIUR) for thePhD scholarship “Tall buildings constructed withadvanced materials: a global approach for the analysisunder static and dynamic loads”, is gratefullyacknowledged.

REFERENCESAli, M.M. and Moon, K.S. (2007). “Structural developments in tall

buildings: current trends and future prospects”, Architectural

Science Review, Vol. 50, No. 3, pp. 205–223.

Capurso, M. (1981). “Sul calcolo dei sistemi spaziali di

controventamento, parte 1”, Giornale del genio Civile, Vol. 1-2-

3, pp. 27–42. (in Italian)

Carpinteri, Al. and Carpinteri, An. (1985). “Lateral load distribution

between the elements of a three-dimensional civil structure”,

Computers and Structures, Vol. 21, No. 3. pp. 563–580.

Carpinteri, A., Lacidogna, G. and Puzzi, S. (2010). “A global

approach for three-dimensional analysis of tall buildings”, The

Structural Design of Tall and Special Buildings, Vol. 19, No. 5,

pp. 518–536.

Carpinteri, A., Corrado, M., Lacidogna, G. and Cammarano, S.

(2012). “Lateral load effects on tall shear wall structures of

different height”, Structural Engineering and Mechanics, Vol.

41, No. 3, pp. 313–337.

Coull, A. and Subedi, N.K. (1971). “Framed-tube structures for

high-rise buildings”, Journal of the Structural Division, ASCE,

Vol. 91, No. 8, pp. 2097–2105.

Coull, A. and Irwin, A.W. (1972). “Model investigation of shear wall

structures”, Journal of the Structural Division, ASCE, Vol. 98,

No. 6, pp. 233–1237.

Coull, A. and Bose, B. (1977). “Simplified analysis of framed-tube

structures”, Journal of the Structural Division, ASCE, Vol. 101,

No. 11, pp. 2223–2240.

Eisenberger, M. (1995). “Nonuniform torsional analysis of variable

and open cross-section bars”, Thin-Walled Structures, Vol. 21,

No. 2, pp. 93–105.

Eurocode 1 (2002). Actions on Structures, General Actions,

Densities, Self-Weight, Imposed Loads for Buildings, European

Committee for Standardization, Brussels, Belgium.

Heidebrecht, A.C. and Stafford Smith, B. (1973). “Approximate

analysis of tall wall-frame structures”, Journal of the Structural

Division, ASCE, Vol. 99, No. 2, pp. 199–221.



Howson, W.P. (2006). “Global analysis: back to the future”, The

Structural Engineer, Vol. 84, No. 3, pp. 18–21.

Khan, F.R. (1974). Tubular Structures for Tall Buildings, Handbook

of Concrete Engineering, Van Nostrand Reinhold Company, New

York, USA.

Lee, J., Bang, M. and Kim, J.Y. (2008). “An analytical model for

high-rise wall-frame structures with outriggers”, The Structural

Design of Tall and Special Buildings, Vol. 17, No. 4,

pp. 839–851.

Ministero delle Infrastrutture (2008). “DM 14/01/2008: Nuove

norme tecniche per le costruzioni”, Gazzetta Ufficiale,

04.02.2008, No.29. (in Italian)

Steenbergen, R.D.J.M. and Blaauwendraad, J. (2007). “Closed-form

super element method for tall buildings of irregular geometry”,

International Journal of Solids and Structures, Vol. 44, No. 17,

pp. 5576–5597.

Taranath, S.B. (1988). Structural Analysis and Design of Tall

Buildings, McGraw-Hill, New York, USA.

Taranath, S.B. (2005). Wind and Earthquake Resistant Buildings,

Marcel Dekker, New York, USA.

Timoshenko, S. (1936). Theory of Elastic Stability (1st Ed.),

McGraw-Hill, New York, USA.

Vlasov, V. (1961). Thin-Walled Elastic Beams (2nd Ed.),

(Jerusalem: Israeli Program for scientific translation), US Science

Foundation, Washington, USA.

Zupan, D. and Saje, M. (2006). “The linearized three-dimensional

beam theory of naturally curved and twisted beams: The strain

vectors formulation”, Computer Methods in Applied Mechanics

and Engineering, Vol. 195, No. 33–36, pp. 4557–4578.

APPENDIX

COMPUTATION OF THE STIFFNESSMATRIX FOR TWISTED CLOSED SECTION

BRACINGSHere the procedure for the computation of the stiffnessmatrix of twisted elements is described in detail. Theobtained expressions can be easily extended to considervertical bracings defined by N consecutive levels.

Referring to the case of a 2-storey shear wall, aprincipal coordinate system for each floor is defined, sothat the system XY is related to the ground level whilethe system (X*Y*)i is related to the i-th level. Accordingto this arrangement, all the coordinate systems show thesame origin (Figure A.1). With regards to the first level,let F1

* be the 2-vector representing the shear-loadingalong the principal axes of the local coordinate system(X*Y*)1 and F1 the 2-vector representing the shear-loading along the axes of the coordinate system XY, sothat

(A.1)

where N1 represents the orthogonal matrix oftransformation from the system XY to the local system(X*Y*)1 and α1 is the rotation angle between the Y axisof the ground level and the Y*

1 axis of the first level.

FF

F1* u,1

v,1

=

=

−

cos

sin

sin

cos

α

α

α

α1

1

1

1

=

F

FF

x,1

y,1

1 1N ,



C-C

C C

BB

A A

B-B

A-A

Y*2

Y*1 X*1

XO

O

O

Y, Y*0

X, X*0

Y, X*2

X

α2

α1

Y*i

Y*i

X*i

X*i

O

O

Y

Y

Fy

My

Y

X

XFx

Mx

αi

αi

(a) (b)

Figure A.1. Model of a twisted bracing: (a) each level showing its own local coordinate system (X*Y*)i; (b) scheme for the definition of the

local components of the external forces and flexural moments

Likewise, the displacement vector δ*1 related to the local

coordinate system is connected with the displacementvector δ1 related to the coordinate system XY throughthe same matrix N1. The local displacements of the firstfloor due to forces placed at the same level along thelocal axes are expressed by means of the principalmoments of inertia, as follows:

(A.2a, b)

Taking into account the Eqn A.1 for the actions andthat corresponding to the displacements, the Eqns A.2,referred to the coordinate system XY, are given by

(A.3b)

which, in a concise form, become

(A.4a)

(A.4b)

In the same manner, the rigid displacements of thesecond floor due to the same load condition, in thecoordinate system (X*Y*)1, are defined as

(A.5b)δv,21 v,11

3

uv,1

3

u

3

u

h

EJF

h

EJ

h

EJF= + = +

δ

2 3 2 vv,1,

(A.5a)δ δu,21 u,11

3

vu,1

v vu,1

h

EJF

h

3EJ

h

2EJF= + = +

2

3 3

,

δy,11 xy,11 x,1 y,11 y,1

D F D F= + .

δx,11 x,11 x,1 xy,11 y,1

D F D F= + ,

δy,11

v ux,1

h

3EJ

h

EJF= −

3 3

1 13cos sinα α ++

+ ( ) + ( )

sin cos

h

3EJ

h

EJv

3

u

3

1

2

1

2

3α α F

y,1,

(A.3a)

δx,11

v ux,1

h

3EJ EJF= ( ) + ( )

+

3

1

2 3

1

2

3cos sin

α αh

cos sin+ −

h

3EJ

h

EJv

3

u

3

1 13α α FF

y,1,

δ δu

vu v

uv

h

EJF

h

EJF

, , , ,, .

11

3

1 11

3

13 3= =

and, with regards to the coordinate system of the groundlevel, as

Their synthetic form is expressed as follows

(A.7a)

(A.7b)

As regards the displacements δ2 due to the loading F2,it’s convenient to consider three different contributions.The first two, related to the coordinate system (X*Y*)1,depend on the resultant system of forces at the first level:a shear-loading F2

(1) equal in modulus to F2 and a 2-vectorM(1), representing the flexural moments, equal to F2h. Thelast contribution refers to the coordinate system (X*Y*)2

and describes the displacement of the second floor due toF2 having considered the first level constrained. It shouldbe noted that the aforementioned actions are referred to thecoordinate system of the ground level.

The terms related to the shear-loading F2(1) are defined

by the Eqns A.5, by means of the components Fx,2 andFy,2 applied to the first floor.

δy,21 xy,21 x,1 y,21 y,1

D F D F= + ,

δx,21 x,21 x,1 xy,21 y,1

D F D F= + .

(A.6b)

δy,21

3

v

3

v1 1

3

u

3

h

EJ

h

EJcos sin

h

EJ

h=

+

−

+ +

3 2

3

α α

22EJcos sin

F

u1 1

x,1

+

α α

+

+

( ) +

+ +

h

EJ

h

EJsin

h

EJ

h

EJ

3

v

3

v1

3

u

3

3 2

3 2

2α

uu

y,1F

( )

cos

.

α1

2

(A.6a)

δx,21

3

v

3

v1

3

u

3

h

EJ

h

EJcos

h

EJ

h

EJ

=

+

( ) +

+ +

3 2

3 2

2α

uu

x,1F

h

( )

+

+

sin

α1

2

33

v

3

v1 1

3

u

3

u

EJ

h

EJcos sin

h

EJ

h

EJ

3 2

3 2

+

+

− +

α α

cos sin

,

α α1 1

Fy,1



(A.8a)

(A.8b)

The bending contribution of the force F2 at the firstlevel is expressed by the components My and Mx, fromwhich the local flexural moments Mv and Mu may bedefined.

(A.9a)

(A.9b)

Therefore, with respect to the system (X*Y*)1, thelocal displacements of the second level due to theapplied moments Mv and Mu at the first level are easilycomputed:

(A.10a)

(A.10b)

The last contribution, which considers the first floorconstrained, is given by the Eqns A.2 through the

δv,21

(1)2

u

2

uu

3

u

3

u

M =h

EJ

h

EJM

h

EJ

h

EJ

( ) +

= +

2

2

+( )−F F

x,2 y,2sin cos .α α

1 1

δu,21

(1)2

v

2

vv

3

v

3

v

M =h

EJ

h

EJM

h

EJ

h

EJ

( ) +

= +

2

2

+( )F F

y,2x,cos sin ,

2 1 1α α

M M M

h F F

u y x

x,2 y,2

= − +

= − +( )sin cos

sin cos .

α α

α α

1 1

1 1

M M M

h F F

v y x

x,2 y,2

= +

= +( )cos sin

cos sin ,

α α

α α

1 1

1 1

δ

α

v,21 2

3

u

3

u

x,2

Fh

EJ

h

EJ

F

( )

sin

1

3 2( ) = +

−11 1+( )F

y,2cos .α

δ

α

u,21 2

3

v

3

v

x,2

Fh

EJ

h

EJ

F F

( )

cos

1

1

3 2( ) = +

+yy,2

sin ,α1( )

rotation angle α2 between the Y axis of the ground leveland the Y2

∗ axis of the second level.

(A.11a)

(A.11b)

Thus, the components of the displacement vector δ2

in the coordinate system XY are defined by means of thecoefficients Dx,22, Dy,22 and Dxy,22:

(A.12a)

(A.12b)

where

Dh

3EJ

h

EJ

h

3EJ

y,22v v

v

= +

( ) +

+

3 3

1

2

3

2 sinα

ssin

cos

α

α

2

2 3 3

1

2 3

2( ) + +

( ) +

h

3EJ

h

EJ

h

3EJ

u u

u

( )cos ,α

2

2

Dh

3EJ

h

EJ

h

3EJ

x,22v v

v

= +

( ) +

+

3 3

1

2

3

2 cosα

ccos

sin

α

α

2

2 3 3

1

2 3

2( ) + +

( ) +

h

3EJ

h

EJ

h

3EJ

u u

u

( )sin ,α

2

2

δy,22 xy,22 x,2 y,22 y,2

D F D F= + ,

δx,22 x,22 x,2 xy,22 y,2

D F D F= + ,

δ

α α

v,22 2

3

uv,2

3

ux,2 y,2

F =h

EJF

h

3EJF F

( )

= − +

3

2 2sin cos(( ).

δ

α α

u,22 2

3

vu,2

3

vx,2 y,2

F =h

EJF

h

3EJF F

( )

= +(3

2 2cos sin )),



As a result, the compliance matrix related to thedisplacements may be assembled and, by inversion, givesrise to the stiffness matrix, whose coefficients are referred tothe coordinate system of the ground level. In this case, thefull stiffness matrix is composed by four 2 × 2 sub-matrices.

(A.13)

NOTATIONB(r) real bimoment actionB(f) fictional bimoment actionD compliance matrix of the shear wallDij displacement of the i-th floor as a consequence

of a load applied to the j-th floorDϑ compliance matrix related to the rotationsDx,ij displacement of the i-th floor in the X direction

as a consequence of a load, acting along the Xdirection, applied to the j-th floor

Dy,ij displacement of the i-th floor in the Y directionas a consequence of a load, acting along the Ydirection, applied to the j-th floor

Dxy,ij displacement of the i-th floor in the X (Y)direction as a consequence of a load, acting along

Kk k

k kd* x

*xy*

xy*

y*

=

Dh

3EJ

h

EJ

h

3EJ

h

EJ

xy,22v v

u u

= +

+

− +

3 3

3 3

2

2

+

+

−

cos sinα α1 1

3 3h

3EJ

h

3EJv u

cos sin .α α

2 2

the Y (X) direction, applied to the j-th floorDyϑ,ij displacement of the i-th floor in the Y direction as

a consequence of a torque applied to the j-th floorE Young’s modulusFj loading vector whose components are Fx and Fy,

acting along the directions of the coordinatesystem XY of the ground level, applied to the j-th level

G shear modulushi storey heightI second moment of inertiaIω warping constantJi second moment of inertia of the shear wallJt torsional rigidity à la de Saint VenantK* stiffness matrix of a vertical bracingku

* stiffness matrix related to the displacements inthe local direction u

kv* stiffness matrix related to the displacements in

the local direction vKd

* stiffness matrix related to the displacements inthe local coordinate system

kx* stiffness matrix related to the displacements in

the local direction X of the coordinate system ofthe ground level

ky* stiffness matrix related to the displacements in

the local direction Y of the coordinate system ofthe ground level

kxy* stiffness matrix related to the displacements in

the local direction Y (X) of the coordinatesystem of the ground level, as a consequence ofa load vector acting along the X (Y) direction

Kϑ* stiffness matrix related to the rotations

N number of floorsM(f) fictional flexural momentM(r) real flexural momentη(r) real displacement



advances in structural engineering -...

Documents