A Simplified Approach for Seismic Calculation of a Tall Building Braced by Shear Walls and Thin-walled Open Section Structures

8/13/2019 A Simplified Approach for Seismic Calculation of a Tall Building Braced by Shear Walls and Thin-walled Open Section Structures

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Engineering Structures 29 (2007) 25762585

www.elsevier.com/locate/engstruct

A simplified approach for seismic calculation of a tall building braced byshear walls and thin-walled open section structures

Sid Ahmed Meftah, Abdelouahed Tounsi, Adda Bedia El Abbas

Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes, BP 89 Cite Ben Mhidi 22000 Sidi Bel Abbes, Algerie

Received 18 January 2006; received in revised form 2 November 2006; accepted 19 December 2006

Available online 22 February 2007

Abstract

In this paper an approximate hand-method for seismic analysis of an asymmetric building structure having constant properties along its height

is presented. The building is stiffened by a combination of shear walls and thin-walled open section structures. Based on the continuum technique

and dAlemberts principle, the governing equations of free vibration and the corresponding eigenvalue problem are derived. By applying the

Galerkin technique, a generalized method is proposed for the free vibration analysis of coupled vibration of a building braced by shear walls and

thin-walled open section structures. Simplified formulae are given to calculate the circular frequencies and internal forces of a building structure

subjected to earthquakes. The utility and accuracy of the method is demonstrated by an numerical example, in which the proposed method is

compared with finite element calculations.c2007 Elsevier Ltd. All rights reserved.

Keywords:Tall building; Eigenfrequency; Continuous approach; Thin-walled structures; Coupling vibration; Internal forces

1. Introduction

During an earthquake, damage to buildings is largely caused

by dynamic loads. Therefore, in order to design buildings

resistant to earthquakes, the dynamic characteristics of the

building must be known. The important characteristics, such

as circular frequencies and mode shapes, can be calculated

by numerical means such as the Finite Element Method

(FEM). While such methods are necessary for the final design,

approximate analyses are most helpful in preliminary designs.

A continuous approach to analysis for tall building structures

is available and has been used in the preliminary stages of the

design of high-rise structures subjected to lateral loading [14].

Over the years the method has been extended to an eigenvalue

problem, including free vibration and buckling analysis [59].

A generally asymmetric tall buildings may consist of any

combination of structural forms, such as frames, shear walls,

structural cores, coupled shear walls. These types of structure

have been widely analysed by many authors [511]. However,

Corresponding author.E-mail addresses:[email protected](S.A. Meftah),

[email protected](A. Tounsi), [email protected](A.B. El Abbas).

very few publications are available on the coupled vibrationcharacteristics of tall buildings braced by shear walls and

thin-walled open section structures [12]. Recently, Kuang and

Ng [13] presented an analytical method for the triply coupled

vibration of asymmetric shear wall structures. In their theory,

however, the flexural displacement and the offsets of the

flexural centre are determined with respect to the axes which

are perpendicular and parallel to the floor axis. The equations

are formulated in such a way that the principal axes of the walls

coincide in directions with those of the floor plan.

In this paper, a dynamic analysis of tall buildings braced by

shear walls and thin-walled open section structures as shown in

Fig. 1is presented. In such a structural configuration, the lateraldisplacements in two perpendicular (not necessary principal)

directions and the torsional rotation can no longer be treated

separately due to their coupling in the governing differential

equations of free vibration. Hence, if the flexural vibrations

in one direction are coupled with the torsional vibrations, the

resulting phenomenon is called double coupling; whereas if

the flexural vibrations in two mutually perpendicular directions

are all coupled with the torsional vibrations, it is referred to

as a triple coupling. Emphasis in the analysis is placed on

the lateraltorsional coupled vibration characteristics of the

0141-0296/$ - see front matter c2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2006.12.014
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S.A. Meftah et al. / Engineering Structures 29 (2007) 25762585 2577

Fig. 1. Floor plan of an asymmetric building braced by shear walls and thin-walled open cross section columns.

structures. Based on the continuum technique and dAlemberts

principle, the governing equations of free vibration are derived,

which consist of a set of differential equations of two lateral

flexure vibrations coupled by Vlasov torsion vibration [14]

(i.e. St Venant + warping torsion vibrations). By employing

the Galerkin approach, a generalized method for solution of the

eigenvalue equations of the problem is proposed for analysingthe lateraltorsional coupled vibration of buildings braced

by shear walls and thin-walled open section structures. The

analysis is extended to calculate the base internal forces arising

from earthquakes. The proposed method is simple and accurate

enough to be used at the concept design stage in particular. It

can be useful to verify the results of the FEM where the time-

consuming produce of handling all the data can always be a

source of error.

2. Statement of the problem

We consider a building structure which contains a

combination of lateral load-resisting systems formed by shear

walls and thin-walled open section structures as shown in Fig. 1.

The arrangement of the stiffening system is either symmetrical

or arbitrary.

The arrangement of the lateral load-resisting systems is

identical at each floor. The stiffnesses at every storey are also

identical, the masses of the individual floors and their horizontal

distribution are the same.The walls and thin-walled open section structures will

deform predominantly in flexural mode.

The P -Delta effect induced by the second-order overturning

moment is neglected in this study. However, this effect must

be taken into account in the final analysis stage of tall building

structures.

The proposed method is applicable only to buildings whose

vertical bracing elements develop no or only negligible axial

deformation.

Our aim is to develop a simple approximate expression for

the calculation of the eigenfrequencies and seismic loads of the

considered building structures.


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2578 S.A. Meftah et al. / Engineering Structures 29 (2007) 25762585

3. Method of analysis

3.1. Basic assumptions, approach

We assume that the material behaves in a linearly elastic

manner.

The floor slabs of the building are very rigid in-plane but

very flexible out-of plane, such that the whole structure is

assumed to deflect as a rigid section displacement.

ConsiderFig. 1which shows typical asymmetric shear walls

and thin-walled open section structures of total height H. In the

analysis, the structure is considered as an equivalent flexural

cantilever, which is located at the centre of flexural rigidity

O. Under the action of lateral loading, the flexural cantilever

beam may undergo deformations of both lateral flexure and

torsion. The vertical z -axis is chosen over the structural height

and through the centre of flexural rigidity, O. The co-ordinate

(xC,yC)represents the position of the geometric centre of the

floor plan,C, in thex O yco-ordinate system.

It is also assumed that the structure has a uniformlydistributed mass m, flexural stiffnesses E Ix and E Iy in the x

andy directions, respectively, coupling stiffnessE Ix y , warping

torsion stiffness E I and St Venant torsion constant G J along

the structural height. Stiffness E Ix y is the flexural coupling

stiffness between the flexural motions in the x andy directions.

3.2. Governing differential equations

Based on dAlemberts principle, the governing equations of

free vibration of the structure can be derived by substituting

inertial forces into the equations of static equilibrium given by

E Ix 4

u(z, t)z4

+E Ix y 4

v(z, t)z4

+ m2

t2[u(z, t) yC(z, t)]= 0 (1)

E Iy4v(z, t)

z4 +E Ix y

4u(z, t)

z4

+ m2

t2[v(z, t)+xC(z, t)]= 0 (2)

and

E I 4(z, t)

z4 G J

2(z, t)

z2

m2

t2

yCu(z, t)xcv(x , t) R

2(z, t)

=0 (3)

where u and v are deflections of the centre O in x and y

directions, respectively, and is angle of rotation of the floor

plan about the point O at the height z (0 z H). R is the

inertial radius of gyration given in theAppendix.Details of the

calculation of flexural and torsional stiffnesses and geometric

properties are also given in theAppendix.

Eqs.(1) and (2)describe that the sum of all forces applied

on the floor plan in x and y directions should be equal to zero;

Eq.(3) describes that the sum of all moments about the centre

O applied on the floor plan should be equal to zero.

3.3. Eigenvalue equation

Since the motion of the structure in free vibration at

any point of the structural height z is harmonic and the

corresponding deflection shape is independent of time t, the

displacement and the torsional rotation may be expressed in the

form

u(z, t)

v(z, t)

(z, t)

=

u( )

v()

( )

sin t (4)

in which = z/H, is the circular frequency, and the mode

shape vector is

D( )=

u( )

v()

( )

. (5)

By substituting Eq. (4) into Eqs. (1)(3) and carrying out

the necessary differentiation, the eigenvalue equations of the

problem can be obtained:

E Ix

m H4

4u( )

4 +

E Ix y

m H4

4v()

4 2 [u( ) yC( )]= 0 (6)

E Iy

m H4

4v()

4 +

E Ix y

m H4

4u( )

4 2 [v()+xC( )]= 0 (7)

E I

H4

4( )

z4

G J

H2

2( )

2

+ m2yCu( )xcv() R

2( )

=0. (8)

The boundary conditions of the above equations are:

D( )= D( )= 0 at =0 and

D

( )= 0 and

u( )= v ( )= E I( )G J( )= 0 at =1.

(9)

3.4. The Galerkin method

Based on the Galerkin technique [15] a method of solution

is proposed for solving the eigenvalue problem given in

Eqs.(6)(8). According to the principle of the Galerkin method,

the solution of the eigenvalue equations for a continuous

structural system can be expressed in a form of a linear

combination of arbitrarily selected shape functions u ( ), v()and ( ), i.e.,

u( )

v()

( )

=

i

ai f( )

bi g( )

ci k( )

(10)

in whichai , bi andci are the constants corresponding to thei th

vibration mode.

When using the Galerkin method with the chosen shape

functions there is no need to satisfy differential equation which

define the eigenvalue problem. But it is necessary and sufficient

to satisfy only the geometric boundary condition. Such shape

functions are classified as admissible functions.


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3.5. Uncoupled vibration

The uncoupled natural frequencies of lateral flexural

vibrations in the two directions (not necessarily the principal

directions) and St Venant warping torsion about the vertical

axis of the structures, represented by x , y and , can be

determined from the solutions of the frequency equations:

1+cosh jcos j =0 (j = x ,y) (11)

1+

1+

4

2()2

cosh 1cos 2

+2

2()sinh 1sin 2 = 0 (12)

where

1 =

()2 +

4

4 +

2

2 and

2 =

()

2 +4

4

2

2 .

(13)

The solutions of Eq. (11) are given by Timoshenko [16] as

follows:

(1)x =(1)y =1.875

(2)x =(2)y =4.694

(3)x =(3)y =7.855

(4)x =(4)y =10.996

(i)x =(i)y =

i

1

2

(i =5, 6, 7, . . . n. . .).

(14)

The uncoupled natural frequencies can then be obtained using

x =2x

x, y =

2y

y(15)

in which x and y are the characteristic parameters, defined

by

2x =m H4

E Ix, 2y =

m H4

E Iy. (16)

The solution of the frequency equation (12)give the values of

(i)

(i = 1, 2 . . . n. . .), the characteristic parameter and

given as:

2

=m H4

E IwR2, 2 =

G J H2

E I. (17)

The associated mode shapes, u( ), v() and ( ) for each

vibration mode are determined by

u(i )( )

v(i )( )

(i)( )

=

a(i )

b(i)

c(i )

N(i)x ( ) N(i )y ( ) N(i ) ( ) (18)

where a, b and c are indeterminate constants and N(i )( ) are

the non-normalized shape functions(N = f, g, k),

N(i)j = cosh

(i )j cos

(i )j

cosh (i)j +cos

(i)j

sinh (i)j +sin

(i )j

(sinh

(i )

j sin

(i)

j ) j = x y (19)

N(i)

= cosh

(i )

1cos

(i)

2

(i)

1

2cosh

(i)

1+(i)

2

2cos

(i )

2

(i)

1

2sinh

(i)

1+() sin

(i)

2

sinh

(i )

1

(i)

1

(i)

2

sin (i)

2

. (20)

3.6. Internal forces about the floor axis

In the response modal analysis of buildings subjected to

earthquakes an equivalent load is determined in each mode of

vibration. For spatial vibration[17]

f(i )xf(i )y

f(i )

=

10 {D( )}

T [M] d{l}10 {D( )}

T [M] {D( )} dm[M] {D( )}SAi (21)

where f(i)x and f

(i )y are the horizontal distributed forces in

the x and y directions, respectively, f(i) is the resultant

moment about thez-axis,SAi is the spectral acceleration (which

depends on the period of vibration, damping and ground peak

acceleration), and {l} is the influence vector, which representsthe direction of excitation. Matrix[M]is given by

[M] =

1 0 yc0 1 xc

yc xc R2 x 2c y

2c

. (22)

The total horizontal load, which is identical to the base shear

forces, is obtained by integrating Eq.(26)over the height of the

building as

V(i)xV(i)y

V(i)

= H

1

0

f(i)xf(i)y

f(i)

d . (23)

The base overturning moment can be obtained with

M(i )yM(i )xB(i )

= H2

10

f(i)xf(i)y

f(i)

d. (24)

3.7. Internal forces according to the principal direction axis

The design value of the shear forces and the base overturning

moment can be calculated according to the principal axis


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xOy (Fig. 1) defined by the angle from the x0y axis

given by

=Arctg

2IxyIx Iy

2

. (25)

The shapes modes for free vibration of the building structure

according to the principal axis are given by:

D( )

=

u(i)p ( )

v(i)p ( )

(i )p ( )

=

cos sin 0sin cos 0

0 0 1

u(i )p ( )

v(i )p ( )

(i)p ( )

(26)

and the location of the geometric centre Cin the x Oy co-

ordinate system is:

x C =x Ccos + yCsin (27a)

yC = yCcos xCsin . (27b)

The design shear base loads are determined when the

earthquake ground motion is in the principal axis direction. For

each mode this forces are given by

V(i )x

V(i )y

V(i )

= H

10

10

D( )

T[M] d{l}1

0 {D( )}T [M] {D( )} d

m[M]

D

( )

SAi d (28)and the base overturning moment

M(i )y

M(i )x

B(i )

= H2

10

10

D( )

T[M] d{l}1

0 {D( )}T [M] {D( )} d

m[M]D( )

SAi d (29)

where M is the mass matrix about the principal direction axis

written as

[M]

= 1 0 y

c

0 1 xc

yc xc R

2 x 2c y2c

. (30)

4. Computation procedure

4.1. Natural frequencies and associated mode shapes

The procedure of computation for determining natural

frequencies and associated mode shapes of tall buildings braced

by shear walls and thin-walled open section structures in

coupled vibration is presented as follows.

Step 1: Calculate lateral and torsional stiffnesses E Ix , E Iy ,

E Ix y ,E Iand G Jusing Eqs.(A.2)and(A.3)in theAppendix,

and geometric properties xc,yc and R using Eqs.(A.1),(A.2)

and(A.4).

Step 2: Calculate the uncoupled frequencies (i)x ,

(i)y and

(i)

using Eqs.(15)and(12).

Step 3: Determine the coupled frequencies using

2

x

2

yc

2

2Y 2 xc

2

yc2 xc

2 R2(2

2)

(i)

=0 (31)

where

=4

x;y

and =

m H4

E Ix y(32)

and the associate mode shapes using

u(i)p ( )

v(i)p ( )

(i )

p ( )

=

a(i )p

b(i )p

c

(i )

p

N(i)x ( ) N(i)

y ( ) N(i)

( )

(33)

where the vibration mode number i = 1, 2, 3, . . . n. . .and the

shape number p = 1, 2, 3; the constants a(i )p , b

(i)p , c

(i )p satisfy

the following relations:

a(i )p

c(i )p

=

(i )2

p ((i)2

y yc + (i )xc

(i )2

p yc)

(i)2

(i )2

y (i )2

x +(i)2

y (i)2

p +(i )2

x (i )2

p (i)4

p

(34)

and

b(i)p

c(i )

p

=

(i )2

p ((i)2

x xc+ (i )yc

(i )2

p xc)

(i)2

(i )2

y (i )2

x +(i)2

y (i)2

p +(i )2

x (i )2

p (i)4

p

.

(35)

It is seen from Eq.(21)that this equation is a cubic equation for

coupled frequency 2. In the i -th vibration mode, the solution

of Eq.(21)will give three values of coupled frequency: (i )1

(i )2

(i)3 . Each of the three frequencies for a given mode

(i)p

corresponds to a particular pattern of vibration.

Step 4: Determine the base shear forces and overturning

moments.

The base shear forces in the y and x directions are calculated

from Eqs.(23)and(28)for the principal axis directions. In the

same way, the base overturning moments are calculated fromEqs. (24) and (29). Finally, the design value of the internal

forces can be calculated by combining the modal response,

using the method of Square Root of Squares (SRSS).

5. Numerical example

In order to verify the accuracy of the mechanical concept of

the proposed method, a building braced by either shear wall and

angle type thin-walled open section structures is analysed. The

floor plan given inFig. 2 is that of a 25 storey building. The

modulus of elasticity is E = 25,000 MN/m2, shear modulus

G = 10,420 MN/m2, the storey height is h = 3 m, the total

height of the building is H = 75 m and the thickness of the


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Fig. 2. Floor plan of example structure.

Fig. 3. Modelling procedure using the FEM (SAP 2000[18]).

Fig. 4a. Vibration frequencies and associated mode shape for the first mode.


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Fig. 4b. Vibration frequencies and associated mode shape for the second mode.

shear wall and the thin-walled angle cross sections are equal to

0.3 m. The thickness of the floor slab is 0.15 m. Assuming that

the weight per unit volume of the building is 25 kN/m2, the

mass density per unit length is m = 114,365 t/m.

It is required to determine the natural frequencies of the

first three modes for coupled lateral flexural-warping torsion

vibration and the associated mode shapes.

Step 1: The geometrical properties of the structure are given in

Table A.1. Ix ,Iy ,Ix y,I and Jare determined using Eqs.(A.2)

and(A.3)in theAppendix.

Ix =65.30 m4, Iy =70.69 m

4, Ix y = 32.55 m4,

J =0.508 m4 and I = 7588.09 m6.

The location of the geometric centre C of the floor is

determined from formulae(A.2)and(A.4)

xc = 6.222 m and, yc = 7.577 m.

Step 2: The characteristic structural parameters are calculated

by employing Eqs.(16),(17)and(22):

= 4.446 s2, x =1.448 s1, y =1.431 s

1,

= 0.396 s1

and =1.914 s1

.

By employing Eq. (15) and solving Eq. (12) the uncoupled

natural frequencies can then be determined.

Step 3: The coupled natural frequencies can be determined by

solving the cubic equation(21)for each vibration mode. The

results are shown in Table A.2,these are later compared with

those obtained by employing a FEM analysis package SAP

2000 [18], using a fine mesh model, subdivided into a large

number of shell type finite elements as shown inFig. 3.Using

Eqs. (23)(25),the three first natural mode shapes of coupled

vibration for the example structure are determined and plotted

inFig. 4.

Step 4: The seismic forces are determined using the response

Modal Analysis. The calculation was carried according to

spectral acceleration shown inFig. 5.The base shear forces and

overturning moments of the three firsts modes in the case of

different earthquakes excitations are given in Tables A.3and

A.4.

6. Conclusions

A generalized hand method for seismic analysis of

asymmetric structures braced by shear walls and thin-walled

open section columns is presented. Based on the continuum


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Fig. 4c. Vibration frequencies and associated mode shape for the third mode.

Fig. 5. Spectral accelerations.

approach and dAlemberts principle, the governing differential

equation on free vibration and the corresponding eigenvalue

equation coupled lateral flexuralSt Venant and warping torsion

vibration have been derived. The Galerkin method is applied for

solving the eigenvalue equation. A computational procedure is

presented to determine the natural frequencies and associated

mode shapes in coupled vibration. This study was next

extend to determine the design internal forces of asymmetric

structures braced by shear walls and thin-walled open section

columns subjected to earthquake excitation. The results from

the proposed method are in good agreement with those from

a comprehensive package program for analysis of the building

structure stage and for final analysis.

Appendix

For buildings of rectangular plan shape and subjected to a

uniformly distributed mass at floor level, the radius of gyration

is obtained from

R =

L2 + B 2

12 +x 2C+y

2C

. (A.1)

The location of the centre of flexural rigidity for a general

asymmetric shear wall/open thin-walled cross section structure

as shown inFig. 1is given by:

xO =

q

xqE Iy,q

q

E Iy,q, yO =

q

yqE Ix ,q

q

E Ix ,q(A.2)


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Table A.1

Flexural and torsional moment of inertia of shear wall and thin-walled angle cross section structures

Bent x y Ix (m4) Iy (m

4) Ixy (m4) J(m4) I (m

6)

1 23 1 4.0157 4.0157 2.3933 0.072 1304.13

2 24 12 0.0135 5.4 0 0.054 109.906

3 23 23 4.0157 4.0157 2.3933 0.072 2085.63

4 1 23 4.0157 4.0157 2.3933 0.072 1064.715 2.25 2.25 45.598 45.598 27.320 0.162 2013.32

6 15 15 7.645 7.645 7.626 0.0763 1010.38 65.304 70.690 32.553 0.5083 7588.094

Table A.2

Comparison of circular frequencies (rad/s) of the proposed method and FEM analysis

Frequencies 1(1) 2

(1) 3(1) 1

(2) 2(2) 2

(3) 1(3) 2

(3) 3(3)

Proposed method 1.648 1.787 4.620 10.197 11.147 28.806 28.490 31.193 80.600

FEM analysis 1.628 1.976 5.091 9.912 11.919 28.128 26.563 31.851 75.973

Table A.3

Comparaison of the internal forces under the x O yaxis

Mode Vx (kN) Vy (kN) My (kN m) Mx (kN m)

Present FEM Present FEM Present FEM Present FEM

1 951.73 1090.09 777.68 745.14 51,863.16 60,431.83 42,378.38 41,338.41

2 1611.70 1829.57 1234.85 1221.19 25,268.72 28,498.63 19,411.17 19,158.35

3 887.75 1083 654.44 661.28 8,314.62 10,064.02 6,497.30 5,977.06

SRSS 2071.58 2389.25 1599.35 1576.02 58,287.46 67,568.20 47,063.10 45,952.49

Table A.4

Comparison of the internal forces under the principal direction axis

Mode Vx (kN) My (kN m)Present FEM Present FEM

1 1581.75 1566.39 86,183.97 86,887.16

2 2536.96 2581.42 39,812.39 40,319.50

3 1359.50 1430.89 13,010.68 13,159.43

SRSS 3284.25 3341.37 95,822.65 96,686.15

and the flexural and warping properties of the equivalent

cantilever are

Ix =q

Ix,q , Iy =q

Iy,q , Ix y =q

Ix y,q ,

I =

q

[( xq xO )2Iy,q + ( yq yO )

2Ix,q ]

J =

q

Jq (A.3)

where Ix ;qIy,qIx y,q , I,q and Jq are the flexural and warping

properties of theq-th shear wall/thin-walled open cross section.

As shown inFig. 1the location of the geometric centreCof

the uniform floor slabs in the co-ordinate system x O y is given

by

xC = xC xO , yC = yC yO . (A.4)

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Documents

A Simplified Approach for Seismic Calculation of a Tall Building Braced by Shear Walls and Thin-walled Open Section Structures