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The Eighth Asia-Pacific Conference on Wind Engineering,December 10–14, 2013, Chennai, India

Modal properties of twin buildings with structural coupling at various locations

K.T. Tse1, Jie Song 2

1 Assistant Professor of Department of Civil and Environment Engineering, HKUST, Hong Kong, P.R. China, [email protected]

2PhD candidate, Department of Civil and Environment Engineering, HKUST, Hong Kong, P.R. China, [email protected]

ABSTRACT There is an increasing trend that adjacent tall buildings are horizontally connected by links. In order to examine their dynamic properties, a 3-D analytical model of linked buildings for arbitrary location and number of link is developed in a matrix form. A parametric analysis is conducted using the model to investigate the effect of link height on modal properties. The results show that the link could modify structural modal properties significantly.

Keywords: Linked buildings, 3D analytical model, modal properties, structural coupling

1 Introduction With the development of aesthetic features, building materials, structural configuration, and construction technology, architectural design of tall buildings has becoming increasingly novel and spectacular in the recent decades, leading to the diversity of their exterior and dynamic behavior. Besides, more and more tall buildings are erected in relatively close proximity owing to the inadequate land availability in urban areas, especially in some metropolises. Consequently, there is a trend that tall buildings in close proximity are designed as linked buildings in many parts of the world, i.e. as a complex consisting of towers horizontally connected by links such as skybridges, skygarden, etc.

These linked buildings usually have considerable height to present grand appearance. As such, the wind-resistant design is one of the concerns for the realistic building design as wind loads will increase significantly with the rise of structural height. In order to predict wind-induced structural response, analytical model of the structure should be formed. There are several analytical models of linked buildings. For instance, two similar analytical models were adopted by Xu et al. (1999) and Richardson et al. (2011) to analyze seismic response of two adjacent buildings connected by damper device, respectively. In their model, the damper is modeled by a spring-damper system without mass. Obviously, the inertial force of link is not considered. Besides, the structural motion is constrained along the direction of damper, i.e. the model is a 1-D model. Recently, Lim’s group (Lim, 2009, Lim et al., 2011) developed a simplified six-degree-of-freedom (6-dof) model for two towers connected by a sky-bridge to examine the modal properties of the linked buildings. In order to deduce the model, the mode shapes of two towers are assumed to govern a power function. Although this assumption is widely acceptable for a single isolated building, the mode shapes of linked buildings could be affected by the link and hence they may not follow a power function, which will be illustrated in this paper.

As can be seen from the above discussion, even though dynamic properties of linked buildings have been studied by a few researchers through several proposed evaluation models,

Proc. of the 8th Asia-Pacific Conference on Wind Engineering – Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds)Copyright c© 2013 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: 978-981-07-8011-1doi:10.3850/978-981-07-8012-8 175 862

Proc. of the 8th Asia-Pacific Conference on Wind Engineering (APCWE-VIII)

they have not yet been investigated thoroughly. Moreover, there are a few limitations in the previous analytical models of linked buildings, e.g. neglecting the mass of the link in Xu’s model, addressing linked buildings with only one link and assuming the model shape of power function in Lim’s model. Therefore, this paper proposes a 3-D analytical model of linked buildings that could overcome the above limitations. In addition, the effect of the link location on structural modal properties is examined systematically by a parametric analysis. 2 Derivation of the analytical model A 3-D analytical matrix-form model of linked buildings that can take into account the effect of structural coupling of links is briefly outlined in this section. Fig. 1 is a sketch of two tall buildings (tower 1 and 2) with m floors connected by n links. Each tower floor of is modeled by a rigid diaphragm having three degrees-of-freedom, two translational motions along x and y and torsional motion about z. Each link is modeled by a beam with three degrees-of-freedom at each end, as shown in Fig. 2. In order to form the analytical evaluation model of the linked buildings, the structural-property matrices of each sub-part, i.e. the two towers and links, are formed at first. Later the total structural-property matrices are obtained by assembling them as a whole for the complex.

Fig.1 An example of linked tall buildings Fig. 2 Model of link

The structural-property matrices of each tower depend on the specific structural configuration, which can be obtained by FEM model readily. Accordingly, the mass matrix MT and stiffness matrix KT for two towers without link can be packaged as

mm

Tower

JM

MJ

M

M

662

2

2

1

1

1

0

0

×

=M

mmT

yT

x

yyy

xxx

zzTy

Tx

yyy

xxx

Tower

KKK

KK

KK

KKK

KK

KK

66222

22

22

111

11

11

000

000

×

=

θθθθ

θ

θ

θθ

θ

θ

K

(1)

where M1, M2, J1, and J2 are mass matrix and moment of inertia of tower 1 and 2, respectively; Ktrs (t=1 or 2, s, r=x, y or ) is the stiffness matrix of tower t.

Assuming that each link is a fixed-fixed beam without out-plane stiffness as shown in Fig. 2, its stiffness matrix Kl with respect to the degrees-of-freedom of the link {xL1, yL1, L1, xL2, yL2, L2}T can be expressed as (Przemieniecki, 1985)

FLx2 MLy1

MLy2

x

y

xL2

yL2

L2

FLy1 FLy2

Link

xL1

yL1

L1

FLx1

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( ) ( )

( ) ( )

2 2

2 2

0 0 0 00 12 6 0 12 60 6 4 0 6 2

0 0 0 00 12 6 0 12 60 6 2 0 6 4

a a

b b b b

b b b bl

a a

b b b b

b b b b

k kk l k l k l k lk l k k l k

Kk k

k l k l k l k lk l k k l k

β β

β β

−−

+ − −=

−− − −

− − +

(2)

where l is the length of link; axial stiffness ka=EA/l; bending stiffness kb=EI'/l, in which I' =I/(1+ ), =12 EI/(GAl2); is a correction factor, for rectangular beam, =1.2.

The above stiffness matrix Kl is formed for each link. In order to assemble it to the stiffness matrix of towers, which is formed at the mass center of each floor, it should be transferred to the mass center of the related floor. According to the relationship shown in Fig. 3, the stiffness matrix of the link with respect to the degrees-of-freedom at the related mass center of the related floor can be derived as (Huang, 2001)

Tl lK H K H′ = (3)

Fig. 3 Relationship between the link and tower 1

where H is the transformation matrix, 1 1

1 1

1

2 2 2

2 2

6 6

1 sin1 cos 0

0 10 1 sin

0 1 cos1

rr

HH

H rr

αα

αα

×

−

= =

−

(4)

here H1 and H2 are transfer matrices for tower 1 and 2, respectively; rt is the distance between the center of the floor of tower t and link end; and is the inclined angle, as shown in Fig. 3.

Assuming that all n links are identical, the total stiffness of all links KL can be simply packaged as

[ ] nnnlL UK 66 ×⊗′=K (5)

where Un is a n×n unit matrix; ⊗ is Kronecker tensor product. Now the stiffness matrices of all parts have been obtained, then the total stiffness

matrix of the complex Ktotal can be assembled by

E ETtotal Tower L= +K K (6)

where E is a total location indicator matrix to indicate the relationship of the location between links and towers, which is defined by

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1

1

1

2

2

2 6 6

0 00

0 0E

0 00

0 0m n

E

E

E

EE

E×

= (7)

Here, Et (t=1 or 2) is a location indicator matrix to show the relative location relationship between links and each tower. Taking tower 1 for example, E1=[e1, e2,…, ep,…, en]m×n, and each column of E1 is a vector as

{ }0, 0, 1 ,0, 0 Tpe = (8)

Since each link is only connected to a particular story of tower 1, the elements of ep are zero except for the ith element equal to one, which means that the ith floor of tower 1 is connected with the pth link. For example, if there is a three-story link connected with tower 1 at the top three slabs, E1 can be formed as

1

3

0 0 0

0 0 01 0 00 1 00 0 1

m

E

×

= (9)

In this way, the matrix E1 and E2 can be formed. If two towers are twin, we have E2=E1, then the whole location matrix E can be simplified as

6 1E U E= ⊗ (10)

where U6 is a 6×6 unit matrix. Similarly, lumping the mass of the link to the connected floors of the two towers

equally, the total mass matrix of the complex Mtotal can be assembled as

( ) ( )2 21 2 ,1 ,

1 = E 1,1, ,1,1, ,..., E2

Ttotal T L T L L ndiag r r diag m m= + + × × ⊗ ×M M M M (11)

where mL,p (p=1,2,…,n) is the mass of the pth link. Then the equation of the undamped free vibration for the linked buildings can be

expressed readily as

D D 0total total+ =M K (12)

where D is the displacement response vector, D={D1x, D1y, D1 , D2x, D2y, D2 }T, and Dts={dts1, dts2, …, dtsm}T (t=1 or 2, s, r=x, y or ). Then, the modal properties of the linked buildings can be readily obtained by solving the characteristic equation as follows

( )2 0total totalω− Φ =K M (13)

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where is structural frequency and is the related mode shape. In this way, the total structural-property matrices for the complex can be formed

systematically, irrespective of the link location and irrespective of the number of links. Subsequently, modal properties of the complex can be calculated by solving the corresponding characteristics equation. 3 Validation of the analytical model To evaluate the accuracy of the above analytical matrix-form model of linked tall buildings, it is profitable to simulate the same complex with a detailed FEM to capture comprehensive structural properties on the linked buildings and then calculate the “exact” structural modal properties. In this section, therefore, an example of linked tall buildings with two identical towers is simulated by FEM. In this example, each tower is a 40-storey, 160 m tall reinforced concrete framed-tube structure with a uniform square floor plan of 30 m by 30 m throughout its height, as shown in Fig. 4. At the top two floors, there is a two-story link with span of 15m and width of 18m. Due to the symmetrical configuration of each tower, there is no static and dynamic coupling within each building since the centers of mass and stiffness of each floor are coincident with the geometric center of each floor. Making use of the mass and stiffness matrix of each tower provided by the FEM model, the total structural properties of linked buildings can be assembled according to the approach in section 2. Subsequently, the modal properties of linked buildings can be computed by solving the related characteristic equation. Meanwhile, the FEM also can output structural modal properties directly. Therefore, the accuracy of the analytical 3-D matrix-form model can be estimated by comparing modal properties calculated by these two approaches.

Fig. 4 FEM model of the linked buildings

Table 1 shows the comparison of the natural periods of the two approaches. Clearly, The natural periods calculated by these two do not show any noticeable differences, which gives compelling evidence that the matrix-form model of linked buildings can provide equal accuracy as FEM. As is known, one of drawbacks of the FEM for dynamic analysis is its time-consuming modeling and calculation. For the subsequent modal and dynamic analysis,

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therefore, we can replace the time-consuming FEM model with the analytical model, which can save a lot of time and make the issue more manageable.

Table 1 Natural periods calculated by the matrix-form model and FEM model

Analysis method Mode

1 2 3 4 5 6 7 8 9 10

Analytical model (s) 4.182 4.182 3.368 1.779 1.656 1.553 1.553 1.417 1.234 0.743

FEM model (s) 4.184 4.184 3.334 1.779 1.675 1.553 1.553 1.417 1.229 0.747

Difference (%) -0.03 -0.03 1.02 -0.02 -1.15 -0.03 -0.03 0.01 0.42 -0.48

4 Effects of link location on modal properties A link could be considered at any height for realistic linked buildings. In order to examine the effect of link location on the structural modal properties, a sensitivity analysis of linked buildings with respect to the link height is conducted here. As the modal properties in the x direction are decoupled with those in the y and directions, they can be examined independently. In addition, the effect of the link location on the modal properties in the x direction is similar to that on that in the y and directions, so here we just present the effect of the link on the modal properties in the x direction.

In the sensitivity analysis, the structural configuration of the two towers and link is the same as that of the case in the validation section except for the link elevation. By changing the link from the bottom to the top, the modal properties of linked building with link at any location can be obtained.

Fig. 5 Variation of structural period with link height

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Fig. 6 Mode shape of the symmetric mode 1, x1-sym for several cases with different height

of link

Fig. 7 Mode shape of the antisymmetric mode 1, x1-asym, for several cases with different

height of link

The variation of structural periods of the first four modes in the x direction with the link location is shown in Fig. 5, in which the vertical axial is the elevation of the link top. The corresponding structural mode shape can be sorted as two categories: symmetric modes, e.g. x1-sym and x2-sym, indicating that the mode shapes of the two towers are identical, and antisymmetric modes, e.g. x1-asym and x2-asym, denoting that the mode shapes of the two towers are opposing. The mode shapes of the first symmetric mode x1-sym and the first antisymmetric mode x1-asym for several representative link locations, such as 160m, 140m, 120m, 80m, 40m, and 20m are presented in Fig.6 and 7, respectively, where the value in the legend is the elevation of each link top.

As can be seen from Fig. 3, the period of the symmetric modes, e.g. modes x1-sym and x2-sym, increased with the link elevation, while that of antisymmetric mode, such as modes x1-asym and x2-asym, shows a complicate variation with the height. For the symmetric mode, the link has no effect on the mode shape, as can be seen from Fig. 5, since the mode shapes of two towers are identical so that the link at any height just sways simultaneously with the two towers and the stiffness of the link has no effect on the mode shape. However, the increase of the link mass will make the structure flexible and hence the related period of the symmetric mode increases with the rise of the link. For antisymmetric modes, however, the stiffness of the link can synchronize or constrain the motion of the two towers, as can be seen in Fig. 7, leading to that the period is smaller than that of the related symmetric mode. In addition, at lower elevations, the period decreases with the link location, while at higher elevation, the period increases with the link location. Therefore, there is a link elevation, e.g. about 120m for mode x 1-asym, that can provide the minimum period, i.e. the maximum frequency. It should be noted from Fig. 7 that the mode shape of the antisymmetric mode varied noticeably with the increase of the link height, changing from a linear form to a hyperbolic one due to the constraining effect of link. It is not appropriate, obviously, to assume that the mode shape can be expressed by a power function in this case.

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5 Conclusions A 3-D matrix-form analytical model of two towers connected with links was developed systematically, which addresses the modal properties of linked buildings with multiple links at any arbitrary location easily. Making use of the model, the effect of link location on the modal properties was examined. The results show that the link can increase periods of symmetric mode due to the additional mass of link, whereas those of antisymmetric mode could be decreased due to the constraining effect of link, which can provide a location that has the minimum period for the antisymmetric mode. Besides, the mode shape of antisymmetric mode changed from a linear form to a hyperbolic form with the increase of the link height, which should be given attention. References Huang, K. (2001), Static, seismic and wind-resistant analysis of linked two towers, Zhejiang University,

Hangzhou.

Lim, J. (2009), Structural coupling and wind-induced response of twin tall buildings with a skybridge, Colorado State University.

Lim, J. and B. Bienkiewicz, et al. (2011), “Modeling of structural coupling for assessment of modal properties of twin tall buildings with a skybridge,” Journal of Wind Engineering and Industrial Aerodynamics 99 (5): 615-623.

Przemieniecki, J. S. (1985). Theory of matrix structural analysis, Dover Publications.

Richardson, A. and K. K. Walsh, et al. (2013), “Closed-form equations for coupling linear structures using stiffness and damping elements,” Structural Control and Health Monitoring 20 (3): 259-281.

Xu, Y. L. and Q. He, et al. (1999), "Dynamic response of damper-connected adjacent buildings under earthquake excitation," Engineering Structures 21 (2): 135-148.

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