Applied Mathematical Modelling 34 (2010) 2437–2451

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Applied Mathematical Modelling

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A simple mathematical model for approximate analysis of tall buildings

Reza Rahgozar a,*, Ali Reza Ahmadi b, Yasser Sharifi c,*

a Civil Engineering Department, University of Kerman, Kerman, P.O. Box 76169-133, Iranb Civil and Mechanical Engineering, International Center of High Technology, Mahan, P.O. Box 76315-117, Iranc Civil Engineering Department, University of Rafsanjan, Iran

a r t i c l e i n f o

Article history:Received 27 July 2008Received in revised form 2 November 2009Accepted 6 November 2009Available online 15 November 2009

Keywords:Tall buildingsFrame tubeShear coreBelt trussShear lag

0307-904X/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.apm.2009.11.009

* Corresponding authors. Tel./fax: +98 341 32200E-mail addresses: rahgozar@mail.uk.ac.ir (R. Rah

a b s t r a c t

The focus of this article is to present a new and simple mathematical model that may beused to determine the optimum location of a belt truss reinforcing system on tall buildingssuch that the displacements due to lateral loadings would generate the least amounts ofstress and strain in building’s structural members. The effect of belt truss and shear coreon framed tube is modeled as a concentrated moment applied at belt truss location, thismoment acts in a direction opposite to rotation created by lateral loads. The axial deforma-tion functions for flange and web of the frames are considered to be cubic and quadraticfunctions respectively; developing their stress relations and minimizing the total potentialenergy of the structure with respect to the lateral deflection, rotation of the plane section,and unknown coefficients of shear lag, the mathematical model is developed. The proposedmodel shows a good understanding of structural behavior; easy to use, yet reasonablyaccurate and suitable for quick evaluations during the preliminary design stage whichrequires less time. Numerical examples are given to demonstrate the ease of applicationand accuracy of the proposed modeled.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

In structures built at the beginning of 20th century, structural members were assumed to carry primarily the gravityloads. Today, due to advancements in structural designs/systems and high-strength materials, building weight is greatly re-duced and slenderness has increased, which has necessitated the consideration of lateral loads such as wind and earthquakein the design process. Lateral forces resulting from wind and seismic activities are now dominant in design considerations.Lateral displacement of such buildings must be strictly controlled, not only for occupants comfort, and safety, but also tocontrol secondary structural effects. Currently, there are many structural systems such as rigid frame, braced frame andshear-walled frame, frame-tube, braced-tube, bundled-tube and outrigger systems that can be used to enhance the lateralresistance in tall buildings [1].

In contrast to vertical loading, lateral load effects on a building increase exponentially with increase in its height. Duringthe last four decades engineers have developed several new framing schemes for tall buildings in order to minimize thematerial used. In general, frame tube structures are widely accepted as an economical system in high rise buildings overa wide range of building heights [1–19]. In its basic form, the system consists of closely spaced exterior columns alongthe periphery interconnected by deep spandrel beams at each floor. This produces a system of rigidly connected jointedorthogonal frame panels forming a rectangular tube which acts as a cantilevered hollow box according to the classical beamtheory (Fig. 1). The overturning moment of lateral loads is resisted by axial stresses in the columns of the four framed panels,whereas the shearing forces generated by the lateral loads are resisted by in-plane bending of beams and columns of two

. All rights reserved.

54.gozar), iran.mahan@gmail.com (A.R. Ahmadi), yasser_sharifi@yahoo.com (Y. Sharifi).

Fig. 1. Orthotropic membrane tube and distribution of axial stresses in framed tube.

2438 R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451

side frames. If the frame members are very rigid, then axial stresses in columns generated by the overturning moment maybe calculated using the common assumption as ‘‘plane sections remain plane”. Flexural and shear flexibilities of frame mem-bers, complicate the basic beam bending action of the framed tube cause of the shear lag phenomenon which has the effectof increasing axial stress in corner columns and decreasing the axial stress in the inner columns while reducing lateral stiff-ness of the structure.

A number of simplified analysis methods have been developed. Khan and Amin [2] suggested that for very preliminarydesign purposes, the shear lag effects may be approximated by treating the framed tube structure as a pair of equivalentchannels each with an effective flange width of not more than half the width of the web panel or more than 10% of the build-ing height. Chan et al. [3] proposed to evaluate shear lag effects in cantilevered structures with solid shear walls as web pan-els and rigidly jointed beam-column frames as flange panels by assuming the distribution of axial displacements across thewidth of flange panels to be of either parabolic or hyperbolic cosine shape. Coull and Bose [4] and Coull and Ahmed [5] devel-oped an orthotropic membrane analogy of transforming the framework panels into equivalent orthotropic membranes eachwith elastic properties so chosen to represent the axial and shear behavior of the actual framework. Khan and Stafford Smith[6] have also developed an orthotropic membrane analogy for simplified analysis of framework panels by using finite ele-ment analysis to determine the equivalent elastic properties of the membranes. Subsequently, Ha et al. (1987) further devel-oped the orthotropic membrane analogy to include shear deformation of frame members and deformation of beam-columnjoints in the derivation of equivalent elastic properties. Their membrane analogy is more refined than others and hence it ismore accurate.

As an innovative and efficient structural system, the outrigger system comprises of a central core, including either bracedframes or shear walls, with horizontal ‘‘outrigger” trusses or girders connecting the core to external columns as shown inFigs. 2 and 3. Furthermore, in most cases, external columns are interconnected by an exterior belt girder. The outriggersand belt girder should be at least one and often two stories deep to realize adequate stiffness. Thus, they are generally posi-tioned at plant levels to reduce the obstruction they create. Thus the 42-storey-high, First Wisconsin Center, with its steelstructure (Fig. 4) in Milwaukee, the 88 storey-high Jin Mao Building with its composite structure in Shanghai, and the tallestbuilding in the world, the 101-storey-high Taipei building with its composite structure are excellent examples of this system[8].

When a tall building is subjected to lateral forces, tie-down action of the belt truss restrains bending of the shear core byintroducing a point of inflection in its deflection curve. This reversal in curvature reduces the lateral movement at the top.The belt trusses function as horizontal fascia stiffeners and engage the exterior columns, which are not directly connected tothe outrigger belt truss. If a building is to have one or more floors devoted to mechanical equipment, rather than lease space,large belt or outrigger trusses can be placed in the perimeter, one storey in height [1,9].

In this paper a simple mathematical model for calculation of stresses in columns of combined framed tube, shear core,and belt truss system is presented. The belt truss and shear core are considered as a bending spring with constant rotationalstiffness which acts as concentrated moment at its positioned level (Fig. 5). Stress distributions in flange-frame and

Fig. 2. Schematic plan of combined system.

Fig. 3. Outriggers system.

R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451 2439

web-frame systems are assumed to be cubic and quadratic functions respectively as shown in Fig. 1. Here total potential en-ergy minimized to derive equations that predict structural deformation. It can be noted that for a given system the magni-tude of beneficial effect of tying down the exterior columns to the core is a function of two distinct characteristics, thestiffness of the equivalent spring and the magnitude of the rotation of cantilever at the spring location due to external loads.The stiffness of the equivalent spring, for example, is at a minimum when located at the top and a maximum when at thebottom. The stain energy that can be stored in the spring is a function of stiffness and the rotation of the cantilever at itslocation. The rotation of the free cantilever for the loading varies from a maximum value at the top to zero at the base. There-fore, from the point of view of spring stiffness alone it is desirable to locate the belt truss at the base, whereas from a con-sideration of rotation, the converse is true. It is obvious that the optimum location is somewhere between. For the sake ofsimplicity, a closed form solution to obtain the optimum location for belt truss can be derived using principles of calculus;compatibility equation for rotation at the belt truss is formed, then the optimum location of belt truss can be obtained by

Fig. 4. First Wisconsin Center Milwaukee, USA

Fig. 5. Behavior of shear core and belt truss system.

2440 R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451

differentiating deflection equation and set to zero. Details of the minimization process for obtaining rotations and lateral dis-placements are explained in Sections 2 and 3. Here the building is modeled as a cantilever beam with a spring at a distancewhere belt truss is located. All results obtained are compared with those obtained using SAP 2000 software so to investigateefficiency and accuracy of the proposed analytical procedure.

R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451 2441

2. Simulating for outrigger system (framed tube, shear core and belt truss)

In this section a simple mathematical model which can predict stress distribution and displacement profile for a com-bined system of framed tube, shear core, and belt truss is presented. Kwan [14] has proposed a model for analysis of framedtube structures. In his model a number of assumptions are made in describing the frame tube system using equivalent ortho-tropic plates. He developed relationships for stress analysis and calculation of the system’s stiffness. Work of Kwan [14] wasa useful introduction to research on the combined of frame tube, shear core, and belt truss system. In this research, a newmodel for analysis of the outrigger system is presented based on the followings assumptions:

(a) Floor slabs are considered to be rigid diaphragms.(b) Distribution of axial deformation in flange and web are according to be cubic and quadratic functions respectively.(c) The effect of belt truss and shear core on framed tube structure is modeled as a concentrated moment applied at belt

truss location, this moment acts in a direction opposite to rotation created by lateral loads.(d) Stress concentrations occurring at the location of belt truss are neglected.(e) Spacing of beams and columns are uniform throughout the building height.(f) All beams and columns are uniform along the building height.(g) Core is fully fixed at the base.

Using above assumptions, frame tube is modeled similar to a beam with a box cross section. Let the axial displacements inthe flange and web panels, denoted respectively by Wf and Ww, be approximated by the following equations, [14]

Wf ¼ /a ð1� bÞ þ byb

� �2� �

; ð1Þ

Ww ¼ /a ð1� aÞ xaþ a

xa

� �3� �

; ð2Þ

where / is the rotation of the plane section joining the four corners of the tubular structure which initially lie on the samehorizontal plane; a and b are dimensionless shear lag coefficients representing the degrees of shear lag in the web and flangepanels respectively, 2b and 2a are the width of flange and web panel dimensions and x and y are coordinates of any pointalong the height of structure (Fig. 1).

The axial strains in the flange, ef and web, ew of the panels are given by Kwan [14] can be expressed by the followingequations:

ef ¼@Wf

@z; ew ¼

@Ww

@z: ð3Þ

Similarly the shear strains in the flange, cyz and web, cxz panels are given, respectively by Kwan [14]

cyz ¼@Wf

@y; cxz ¼

@u@zþ @Ww

@x: ð4Þ

Utilizing these equations, the strain energy can be calculated by the following relation proposed by Kwan [14]:

Pe ¼Z H

0

Z b

�btf ðEf e2

f þ Gf c2yzÞdydzþ

Z H

0

Z a

�atwðEwe2

w þ Gwc2xzÞdxdzþ

Z H

02EmAke2

k dz: ð5Þ

In Eq. (5), Ef , Gf , Ew and Gw are modulus of elasticity and equivalent shear modulus of flange and web frames respectively, Em

is modulus f elasticity of four corner columns, Ak is the cross sectional area of each corner columns, ek is the axial strain ofeach corner columns, tf and tw are equivalent thickness for flange and web frames and H is height of the frame. The potentialenergy due to concentrated, homogenous and triangle distributed loads is calculated by using the following equationsrespectively [14]:

Pp ¼ �PuðHÞ; ð6Þ

Pp ¼ �Z H

0UuðzÞ; ð7Þ

Pp ¼ �Z H

0T

zH

uðzÞ: ð8Þ

In Eqs. (6)–(8), uðzÞ is the lateral displacement of structure at height z from the base of building, and P, U and T, are concen-trated load, intensity of homogenous distributed load, intensity of triangle load, respectively. Also the potential energy dueto the concentrated moment, Khc created by the action of belt truss and shear core on the frame tube can be expressed as:

Pp ¼ Kh2c ; ð9Þ

where K is the rotational stiffness of belt truss and hc is the rotation of frame at level of belt truss respectively.

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The total potential energy is just the sum of the potential energy of the applied force and the strain energy of the struc-ture. Having obtained the expression for the total potential energy, the governing differential equation can be the be derivedby minimizing the total potential energy with respect to /, u and shear lag coefficients yields four differential equations withpartial derivatives of first and second order. Minimization of the total potential energy with respect to / yields the governingequation for /, which may be interpreted as the moment equilibrium equation and expressed in the following form:

M ¼ EI@/@Z

; ð10Þ

where ðEIÞ effective bending stiffness of the tubular structure and M is the overturning moment of the lateral load at positionz. This equation is not easy to solve because ðEIÞ varies with the height and is dependent on other unknowns. Nevertheless, ifthe effect of the variation on ðEIÞ with height on the bending rotation / is assumed negligible, then / can be evaluated bydirect integration as follows:

/ ¼ 1EI

Z z

0M dz ð11Þ

Likewise, minimization of the total potential energy with respect to u yields the following governing equation for u:

S ¼ 4Gwtwað@u=@zþ /Þ; ð12Þ

where S is the shear at a section with position z. This is actually the horizontal shear equilibrium equation. From this equa-tion, u can be determined by direct integration as follows:

u ¼Z z

0

S4Gwtwa

� /

� �dz: ð13Þ

Replacing parameters / and u in the potential energy relation and using the relations proposed by Kwan [14] for a and b asmulti terms functions with unknown coefficients of a1, a2, b1 and b2

a ¼ a1 1� zH

� �2þ a2 2

zH

� �� z

H

� �2� �

; ð14Þ

b ¼ b1 1� zH

� �2þ b2 2

zH

� �� z

H

� �2� �

: ð15Þ

Minimizing the total potential energy and applying boundary conditions that at the top of the structure, the axial stressesare equal to zero, which leads to da=dz ¼ 0 and db=dz ¼ 0 at z = H, it can be shown that the polynomial functions for a and bcan each be expressed in terms of only unknown coefficients. The parameters a1, a2, b1 and b2 in Eqs. (14) and (15) for con-centrated, homogenous and triangle distributed have been given by Kwan [14]. In this paper the same notations are used forconcentrated moments due to the belt truss, these parameters are calculated as follows:

a1 ¼35½8B1 � 3nwB2�

21n2wB5 þ 40nwB4 þ 160B1

; ð16Þ

a2 ¼35½8B1 þ 3nwB3�

21n2wB5 þ 40nwB4 þ 160B1

; ð17Þ

b1 ¼15½56B1 � 5nf B2�

5n2f B5 þ 40nf B4 þ 672B1

; ð18Þ

b2 ¼15½56B1 þ 5nf B3�

5n2f B5 þ 40nf B4 þ 672B1

: ð19Þ

In Eqs. (16)–(19), the coefficients nw, nf , B1, B2, B3, B4 and B5 are as follows:

nw ¼GwC2

Ewa2 ; nf ¼Gf C

2

Ef b2 ; ð20Þ

B1 ¼ 6C2 � 15CH þ 10H2

B2 ¼ 12C2 � 49CH þ 42H2

B3 ¼ �12C2 þ 49CH � 70H2 þ 35H3=C

B4 ¼ 108C2 � 273CH þ 182H2

B5 ¼ 48C2 � 140CH þ 105H2

0BBBBBB@

1CCCCCCA; ð21Þ

where C is the distance of belt truss from the base of structure.Hence, the displacement of flange and web frames can be calculated from Eqs. (1), (2), (14), and (15), differentiating these

equations and multiplying the results by equivalent modulus of elasticity for each panel, the related equations of axial stressdistribution are derived as follows:

R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451 2443

ðrzÞf ¼ Efd/dz

a ð1� bÞ þ byb

� �2� �

; ð22Þ

ðrzÞw ¼ Ewd/dz

a ð1� aÞ xaþ a

xa

� �3� �

: ð23Þ

In Eqs. (22) and (23), ðrzÞf and ðrzÞw are axial stresses in flange and web frames without the effect of belt truss respectively

ðrBz Þf ¼ Efd/dz

a ð1� bÞ þ byb

� �2� �

� EfKhc

ðEIÞ0a ð1� bMÞ þ bM

yb

� �2� �

; ð24Þ

ðrBz Þw ¼ Ewd/dz

a ð1� aÞ xaþ a

xa

� �3� �

� EwKhc

ðEIÞ0a ð1� aMÞ

xaþ aM

xa

� �3� �

: ð25Þ

The axial stresses in flange, ðrBz Þf and web, ðrBZ Þw frames with the effect of belt truss are given by Eqs. (24) and (25).Where aM and bM are shear lag coefficients of web and flange panels and ðEIÞ0 is the equivalent bending stiffness of concen-trated moment created by belt truss and shear core. At upper level of the structure, / is not zero and effect of this parametermust be considered for axial stress distribution. However, the terms related to / in comparison to the terms forming Eqs.(22)–(25) are negligible in a primary analysis [14].

In the combined system axial stresses which are expressed by Eqs. (24) and (25) are dependent on the unknown param-eter d/=dz. Thus, by replacing the axial stresses of frame tube, before superimposing the Eqs. (22) and (23), in moment equi-librium of the structure equation (26), the amount of d/=dz ¼ M=EI is determined

M ¼ EId/dz¼Z a

�a2twðrzÞwxdxþ

Z b

�b2tf ðrzÞf adyþ 4Akrka: ð26Þ

In Eq. (26) rk is the axial stress of each corner columns in the frame tube system. By equating EIðd/=dzÞ with externalmoments created by the three types of loading: considered here, the equivalent bending stiffness of ðEIÞ can be calculatedusing Eq. (27) and ðEIÞ0 for concentrated moment using Eq. (28), i.e.

EI ¼ 43

Ewtwa3 1� 25a

� �þ 4Ef tf a2b 1� 2

3b

� �þ 4EmAka2; ð27Þ

ðEIÞ0 ¼ 43

Ewtwa3 1� 25aM

� �þ 4Ef tf a2b 1� 2

3bM

� �þ 4EmAka2: ð28Þ

The equivalent stiffness of the twisting spring which includes the effect of belt truss on frame tube is calculated by mul-tiplying the columns cross sectional area of flange frame by distance between the two frames and subtracting ðEIÞ0=C asfollows:

K ¼ ð2aÞ2Ef

2C1þ 2b

s

� �Ac �

ðEIÞ0

C; ð29Þ

where 2a is distance between flange frames, s is column spacing and Ac is the sum of columns cross sectional areas of flangeframe. Thus, the rotation of combined system (h), at belt truss level for concentrated (P), homogenous (U) and triangle loaddistribution (T), can be calculated from the following equations:

hCp ¼ðEIÞ0

ðEIÞ0 þ KCPEI

HC � C2

2

!þ P

4Gwtwa

" #; ð30Þ

hCd¼ ðEIÞ0

ðEIÞ0 þ KCUEI

12

H2C � 12

HC2 þ 16

C3� �

þ U4Gwtwa

ðH � CÞ� �

; ð31Þ

hCt ¼ðEIÞ0

ðEIÞ0 þ KCTEI

13

H2C � 14

HC2 þ 124

C4

H

!þ T

4GwtwaC2� C2

2H

!" #: ð32Þ

3. Lateral displacement of combined system

The lateral displacement, u of the system can be evaluated by first, substituting the of values ðEIÞ and ðEIÞ0 from Eqs. (27)and (28) into Eq. (11), then the result substituted into Eq. (13), lateral displacement of the structure subjected to concen-trated, homogenous and triangle distributed loading and concentrated moment are determined. For calculating the lateraldisplacement of combined system (u), it has been assumed that ðEIÞ and ðEIÞ0 are constant along the height and are equalin measure to the base dimension of the structure. Displacement of combined system can be estimated at a distance of z fromthe base of the structure subjected to the following loads:

2444 R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451

Concentrated loading:

u ¼ PEI

12

Hz2 � 16

z3� �

þ P4Gwtwa

z�KhCp z2

2ðEIÞ0for z < C; ð33Þ

u ¼ PEI

12

Hz2 � 16

z3� �

þ P4Gwtwa

z�KhCp z2

2ðEIÞ0�

KhCp C2ðEIÞ0

ðz� CÞ for z P C: ð34Þ

Homogenous distributed loading:

u ¼ UEI

14

H2z2 � 16

Hz3 þ 124

z4� �

þ U4Gwtwa

Hz� 12

z2� �

� KhCdz2

2ðEIÞ0for z < C; ð35Þ

u ¼ UEI

H2z2

4� Hz3

6þ z4

24

!þ U

4GwtwaHz� z2

2

� �� KhCd

2ðEIÞ0ðz2 � Cðz� CÞÞ for z P C: ð36Þ

Triangle distributed loading:

u ¼ TEI

16

H2z2 � 112

Hz3 þ 1120

z5

H

� �þ T

4GwtwaHz2� 1

6z3

H

� �� KhCt z

2

2ðEIÞ0for z < C; ð37Þ

u ¼ TEI

H2z2

6� Hz3

12þ z5

120H

!þ T

4GwtwaHz2� z3

6H

� �� KhCt

2ðEIÞ0ðz2 � Cðz� CÞÞ for z P C: ð38Þ

4. Examples and comparisons with computer analysis

Numerical examples are given to demonstrate the ease of application and accuracy of the proposed approximate method.Two types of high-rise buildings such as 45 and 55 storey buildings subjected to homogenous distributed load are investi-

Fig. 6. Model of numerical example.

Fig. 7. Displacement of 45 stories combine system subjected to homogeneous distributed loading (B.L. = H/6).

Fig. 8. Displacement of 45 stories combine system subjected to homogeneous distributed loading (B.L. = H/2).

Fig. 9. Displacement of 45 stories combine system subjected to homogeneous distributed loading (B.L. = 3H/4).

R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451 2445

gated as shown in Fig. 6. The following specifications are used in numerical examples for each of the buildings: plan dimen-sion of building = 30 � 35 m (length of web panel of frame tube = 30 m; length of flange panel of frame tube = 35 m); storeyheight = 3 m; column spacing = 2.5 m; thickness of shear core panels = 250 mm; cross-sectional area of beams and col-umns = 57,500 mm4; width of beams and columns = 600 mm; beams and columns moments of inertia = 3,174,000,000 mm4;modulus of elasticity = 204,000 MPa; shear modulus of elasticity = 78,420 MPa and Poisson ratio = 0.25.

Fig. 10. Stress distribution in web of 45 stories combine system subjected to homogeneous distributed loading (B.L. = H/6).

Fig. 11. Stress distribution in web of 45 stories combine system subjected to homogeneous distributed loading (B.L. = H/2).

Fig. 12. Stress distribution in web of 45 stories combine system subjected to homogeneous distributed loading (B.L. = 3H/4).

2446 R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451

All structures have been analyzed and the results of displacement and axial stresses in flange and web of each buildingsubjected to homogenous distributed loading from proposed approximate method are compared with the results of SAP2000 software. After verifying the results, optimum location for the belt truss is obtained by satisfying the objectives ofdecreasing displacement and smoothing axial stress in frame tube.

Fig. 13. Stress distribution in flange of 45 stories combine system subjected to homogeneous distributed loading (B.L. = H/6).

Fig. 14. Stress distribution in flange of 45 stories combine system subjected to homogeneous distributed loading (B.L. = H/2).

Fig. 15. Stress distribution in flange of 45 stories combine system subjected to homogeneous distributed loading (B.L. = 3H/4).

R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451 2447

Variation in belt position for a 45 and 55 storey buildings, stress distribution and the maximum displacements are studiedhere. Displacement, stress distribution in flange and web frames for a combined system of 45 stories are shown in Figs. 7–15and 55 stories in Figs. 16–24. As shown in Figs. 7–9, 16–18, when the belt truss reaches 1/6 of frames height, the displace-ments at highest level of building reach a minimum amount. Comparison of the results between SAP 2000 [20] and the

Fig. 16. Displacement of 55 stories combine system subjected to homogeneous distributed loading (B.L. = H/6).

Fig. 17. Displacement of 55 stories combine system subjected to homogeneous distributed loading (B.L. = H/2).

Fig. 18. Displacement of 55 stories combine system subjected to homogeneous distributed loading (B.L. = 3H/4).

2448 R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451

proposed approximate method when belt truss is located at various locations of building height are listed in Tables 1 and 2. Itcan be seen that for 45 and 55 storey building when the belt truss is located at H/6 height from the base, the proposedapproximate method overestimates the maximum axial stress by 7.5% and 7% and underestimates the maximum lateral dis-placement by 1.2% and 1.5% respectively. The results show that the percentage of error is low and acceptable. The mainsources of errors between the proposed approximate method and SAP 2000 are as follows:

Fig. 19. Stress distribution in web of 55 stories combine system subjected to homogeneous distributed loading (B.L. = H/6).

Fig. 21. Stress distribution in web of 55 stories combine system subjected to homogeneous distributed loading (B.L. = 3H/4).

Fig. 20. Stress distribution in web of 55 stories combine system subjected to homogeneous distributed loading (B.L. = H/2).

R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451 2449

(1) All closely spaced perimeter columns tied at each floor level by deep spandrel beams considered to form a tubularstructure, this approximation is modified to behaves as an orthotropic membranes so that the difference in responsealong the width and height are taken into account.

Fig. 22. Stress distribution in flange of 55 stories combine system subjected to homogeneous distributed loading (B.L. = H/6).

Fig. 23. Stress distribution in flange of 55 stories combine system subjected to homogeneous distributed loading (B.L. = H/2).

Fig. 24. Stress distribution in flange of 55 stories combine system subjected to homogeneous distributed loading (B.L. = 3H/4).

Table 1Comparison of the results between sap and the proposed approximate method for a 45 storey building subjected to homogeneous distributed loading.

Position of belt truss fromthe base of building

Percentage of error in displacementat the top of building

Percentage of error in cornercolumns of flange panel

H6

1.2 7.5H2

2 8.63H4

3 10

2450 R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451

Table 2Comparison of the results between sap and the proposed approximate method for a 55 storey building subjected to homogeneous distributed loading.

Position of belt trussfrom the base of building

Percentage of error in displacementat the top of building

Percentage of error in cornercolumns of flange panel

H6

1.5 7H2

2.4 8.53H4

3 9.5

R. Rahgozar et al. / Applied Mathematical Modelling 34 (2010) 2437–2451 2451

(2) The order of interpolant selected in approximating the axial displacement in flange and web panels of the frame tube.(3) Equivalencing the elastic properties of the frame tube, modeled as plate and shear core, and belt truss with a simple

rotational spring.

5. Conclusions

A simple hand calculations method for approximate analysis of framed tube, shear core and belt truss systems in high risebuildings subjected to lateral loads such as wind and earthquake was proposed. In this paper, a primary non-continuousstructure with a set of simple assumptions is modeled as a continuous structure with orthotropic plates. In the proposedmethod, the distribution of axial deflections in each panels of flange or web are considered independently. Closed form solu-tions are obtained, from which the effects of various parameters on the overall structural behavior can be readily evaluated.So, the shear lag in each panel of flange and web of the frame are calculated separately. It is revealed that for 45 and 55 storeybuilding when the belt truss is located at H/6 height from the base, the proposed approximate method overestimates themaximum axial stress by 7.5% and 7% and underestimates the maximum lateral displacement by 1.2% and 1.5% respectively.The main sources of errors are explained in Section 4. The proposed model shows a good understanding of structural behav-ior; easy to use, yet reasonably accurate and suitable for quick evaluations during the preliminary design stage which re-quires less time. Numerical examples demonstrate the ease of application and accuracy of the proposed modeled whichis applicable to a combined system of framed tube, shear core and belt truss over a wide range of building heights.

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