THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGS

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

Published online in Wiley Interscience (www.interscience.wiley.com). DOI:10.1002/tal.228

OPTIMIZED USE OF THE OUTRIGGER SYSTEM TO STIFFEN

THE COUPLED SHEAR WALLS IN TALL BUILDINGS

NAVAB ASSADI ZEIDABADI1, KAMAL MIRTALAE1* AND BARZIN MOBASHER2

1

Isfahan University of Technology, Isfahan, Iran; and Arizona Department of Transportation, Phoenix, Arizona, USA

2

Civil and Environmental Engineering Department, Arizona State University, Tempe, Arizona, USA

SUMMARY

Based on the conventional yet accurate continuum approach, a general analysis is presented for a pair of coupled

shear walls, stiffened by an outrigger and a heavy beam in an arbitrary position on the height. Subsequently, a

parametric study is presented to investigate the behavior of the structure. The optimum location of the outrigger

and the parameters affecting its position were also investigated. The results showed that the behavior of the structure can be significantly influenced by the location of the outrigger. It was also indicated that in most ordinary

cases the best location of the structure to minimize top drift is somewhere between 0·4 to 0·6 of the height of the

structure. Though this method is not a substitute for the finite element method, it gives an initial simple solution

to determine the size and position of outrigger, stiffening beam and coupled shear walls in the preliminary design

stages. Copyright © 2004 John Wiley & Sons, Ltd.

1.

INTRODUCTION

In modern residential tall buildings, lateral loads induced by wind or earthquake are often resisted by

a system of coupled shear walls. When a building increases in height, the stiffness of the structure

becomes more important. In addition, the depth of lintel beams connecting shear walls will usually be

confined by differences between floor-to-floor height and floor clear height, Hence, the coupling effect

of the connecting system may not be sufficient to provide the necessary lateral stiffness, and the tensile

bending stress and uplift forces may exceed the economical limits.

Different methods that can be used to overcome these problems may be the provision of an outrigger, addition of very stiff beams between walls or using both systems.

An outrigger is a stiff beam that connects the shear walls to exterior columns. When the structure

is subjected to lateral forces, the outrigger and the columns resist the rotation of the core and thus significantly reduce the lateral deflection and base moment, which would have arisen in a free core.

Several buildings with this type of bracing were built during the last three decades in North America,

Australia and Japan.

In some buildings with a pair of coupled shear walls to resist the lateral loads, floor slabs are

protruded from the shear walls to form balconies. At the outer edge of the balconies as shown in

Figure 1, the exterior columns are located to support the slabs. An outrigger can employ peripheral

columns to increase the overall stiffness of the structure and decrease the moments of the walls.

Numerous studies have been carried out on the analysis and behavior of outrigger structures (Coull

and Lao, 1988, 1989; Rutenburg and Eisenburg, 1990; Skraman and Goldaf, 1997). Moudarres (1984)

* Correspondence to: Dr. Kamal Mirtalae, Arizona Department of Transportation, Bridge Design Group, Mail Drop #631E, 205

South 17th Avenue, Phoenix, AZ 85007, USA

Copyright © 2004 John Wiley & Sons, Ltd.

Received December 2001

Accepted November 2002

10

N. A. ZEIDABADI ET AL.

Peripheral columns

Coupled Shear Walls and Outrigger

Figure 1. Simplified plan of building

showed that a top outrigger can reduce the lateral deflections in a pair of coupled shear walls. Using

the continuous medium method, Chan and Kuang (1989a, 1989b) conducted studies on the effect of

an intermediate stiffening beam at an arbitrary level along the height of the walls, and indicated that

the structural behavior of coupled shear walls could be significantly affected by particular positioning

of the stiffening beam. Afterwards, Coull and Bensmail (1991) as well as Choo and Li (1997) extended

Kuang and Chan’s method for two and multi-stiffening beams. Their studies also included both rigid

and flexible foundations for the structure.

In this paper, based on Chan and Kuang’s method, a continuum approach is designated to analyze

a pair of coupled shear walls, stiffened by an outrigger and an interior beam at an arbitrary location

on the height. A parametric study is used to investigate the influence of rigidities and locations of the

outrigger and interior beam on the lateral deflections and laminar shear forces in the structure. Furthermore, the best locations of the outrigger to minimize top drift or laminar shear and the effective

parameters on the location are presented.

2.

ANALYSIS

Consider a coupled structural wall system in a fixed foundation stiffened by an outrigger and a beam

at level hs shown in Figure 2. For analysis of the structure by continuum approach, the coupling beams

are replaced by continuous distribution of lamina with equivalent stiffness. It is also assumed that both

walls deflected equally throughout the height, so the points of contraflexure of the laminae and stiffening beam are at their mid-span points. If a hypothetical cut is made along the line of contraflexure,

the condition of vertical compatibility above and, below the outrigger leads to the following

equations:

l

dy1

hb 3

2

q1 dx 12 EIb

EA

l

Copyright © 2004 John Wiley & Sons, Ltd.

[Ú T dx + Ú

x

hs

1

hs

0

]

T2 dx = 0

dy2

hb 3

2 x

q2 T2 dx = 0

dx 12 EIb

EA Ú0

(1)

(2)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

11

OPTIMIZED USE OF OUTRIGGER SYSTEM

l

c

c

l

l

b/2

b

b/2

q1

Vs+F

h

F

F

H

x

q2

hs

(b)

(a)

Figure 2. (a) Coupled shear walls stiffened by outrigger and internal beam. (b) Substitute structure

where y1, q1, T1 and y2, q2, T2 are the lateral deflection, the laminar shear and the axial forces in the

walls in the section above and below level hs, respectively and Ib, E, A are second moment of area of

connecting beams, elastic modulus of walls and coupling beams and cross-section area of each wall.

The three successive terms represent the vertical deflection at the cut caused by slopes of the walls,

bending of laminae and axial deformation of the walls.

The general moment–curvature relationship of the walls is

d 2 y1

Ï

M

=

EI

+ T1l

e

ÔÔ

dx 2

Ì

2

Ô M = EI d y2 + T l

2

ÔÓ e

dx 2

for (hs £ x £ H )

(3)

for (0 £ x £ hs )

in which I = 2I1, where I1 is second moment of area of each wall, and the axial forces in the walls in

different sections are given respectively by

H

T1 = Ú q1dx

(4)

x

H

hs

hs

x

T2 = Ú q1dx + Vs + Ú q2 dx

(5)

where Vs represents the shear force in the stiffening beam.

By considering the equilibrium of a small vertical element of the continuous structure, it can be

shown that at any point along the height

q=

- dT

dx

(6)

By differentiating Equations (1) and (2) and combining with Equations (3) and (6), q and y can be

eliminated and then the governing equations for the axial forces in the walls can be given by

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

12

N. A. ZEIDABADI ET AL.

d 2 T1

- a 2 T1 = -g Me

dx 2

(7)

d 2 T2

- a 2 T2 = -g Me

dx 2

(8)

I

a 2 = g Êl + ˆ

Ë Al ¯

(9)

where

g =

6 Ib l

hb 3 I

(10)

To obtain the shear force in stiffening beam Vs, consider the compatibility condition at its contraflexure point:

l

dy2

VS b 3

2 hs

T2 dx = 0

dx 13Es Is EA Ú0

(11)

in which EsIs is the flexural rigidity of the stiffening beam.

Equating the corresponding terms of Equations (1) or (2) and Equation (11) at level hs gives the

shear force of the stiffening beam thus; the shear forces will be

Vs = Sm Hq1s = Sm Hq2 s

(12)

where q1s and q2s are the shear flows at level hs and Sm is the relative flexural rigidity of the stiffening

beam, defined as

Sm =

3.

h ES I S

H EIb

(13)

THE EFFECTS OF THE OUTRIGGER AND EXTERNAL LOADS

In this investigation, the influence of the outrigger is considered as an unknown moment Mh in the

location of the outrigger. Moment Mh can also be represented by

Mh = F(2c + 2l + l)

(14)

The parameters F, c, l and ഞ are shown in Figure 2. By considering Mh, the moment Me in Equations

(7) and (8) can be given by

Ï Me = 0

Ì

Ó Me = M h

for

for

hs £ x £ H

0 £ x £ hs

(15)

Therefore the complete solution of Equations (7) and (8) is

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

13

OPTIMIZED USE OF OUTRIGGER SYSTEM

T1M = B1 cosh a x + C1 sinh a x

T2 M = B2 cosh a x + C2 sinh a x +

(16)

g

Mh

a2

(17)

The expression for laminar shear above and below the outrigger can be derived by using Equation (6):

q1M = -[ B1a sinh a x + C1a cosh a x ]

(18)

q2 M = -[ B2a sinh a x + C2a cosh a x ]

(19)

For the external loads considered, the applied bending moment can be represented by

u

w

2

(2 H 3 - 3H 2 x + x 3 )

Me = Mel = P( H - x ) + ( H - x ) +

2

6H

(20)

where P is considered the load at the top of the walls, and u and w are intensities of uniformly distributed and upper triangular distributed loads acting in the walls, respectively. Thus, the complete

solution of Equations (7) and (8) due to lateral loads is represented in following equations:

T1l = B1¢ cosh a x + C1¢ sinh a h +

g Ê

u

wx

Mel + 2 + 2 ˆ

a2 Ë

a

a H¯

(21)

T2 l = B2¢ cosh a x + C2¢ sinh a h +

g Ê

u

wx

Mel + 2 + 2 ˆ

2 Ë

a

a

a H¯

(22)

Consequently, the laminar shears are

g dMel

w

q1l = - ÈÍ B1¢a sinh a + C1¢a cosh a x + 2 Ê

+ 2 ˆ ˘˙

Î

a Ë dx

a H¯˚

(23)

g dMel

w

q2 l = - ÈÍ B2¢a sinh a x + C2¢a cosh a x + 2 Ê

+ 2 ˆ ˘˙

Ë

dx

Î

a

a H¯˚

(24)

4.

BOUNDARY CONDITIONS

The values of B1, B2, C1 and C2 can be determined by considering a set of boundary conditions.

• At the top of the structure, x = H:

T1 ( H ) = 0

(25a)

• At the level of stiffening beam h , boundary conditions are

s

Copyright © 2004 John Wiley & Sons, Ltd.

T1 (hs ) + Vs + Fi = T2 (hs )

(25b)

q1 (hs ) = q2 (hs )

(25c)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

14

N. A. ZEIDABADI ET AL.

• At the base level, the laminar shear is given by

q2 (0) = 0

(25d)

Solving Equations (25a)–(25d) gives the unknown integration constants:

B1 = -C1 tanhaH

(26)

1

B2 = C1 Ê

- tanh ahˆ

Ë tanh ahs

¯

(27)

C2 = 0

(28)

-1

l ˆ

Ê

+

M

Ë 2l + 2c + l a 2 ¯ h

C1 =

Ê - K tanh aH + K - cosh ahs + cosh ah (tanh ah )ˆ

3

s

s

Ë 2

¯

tanh ahs

(29)

Likewise, the values of integration constants B¢1, B¢2, C¢1 and C¢2 can be determined. The only alteration in boundary conditions which should be made is that Equation (25b) will be changed into the

following equation:

T1l (hs ) + Vs = T2 l (hs )

(30)

Expressions for the constants B1¢, B2¢, C1¢ and C2¢ are given in Appendix 1.

5.

LATERAL DEFLECTION EQUATIONS

By integrating Equation (3) twice and using the compatibility condition represented in Equations

(31a)–(31d) the lateral deflection due to outrigger and external loads will be

y2 (0) = 0

(31a)

y2¢ (0) = 0

(31b)

y1 (hs ) = y2 (hs )

(31c)

y1¢(hs ) = y2¢ (hs )

(31d)

The lateral deflections due to the outrigger can be given as follows:

y1M = -

y2 M =

Copyright © 2004 John Wiley & Sons, Ltd.

1 È B1l

C1l

cosh a x + 2 sinh a x + xd 1 + d 2 ˘˙

EI ÍÎ a 2

˚

a

(32)

1 ÈÊ

gl

lB2

1 - 2 ˆ Mh x 2 - 2 cosh a x + d 3 ˘˙

Í

Ë

¯

EI Î

˚

a

a

(33)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

OPTIMIZED USE OF OUTRIGGER SYSTEM

15

The lateral deflections caused by external loads are

y1l =

gl ˆ

g l Ê ux 2

wx 3 ˆ ˘

1 ÈÊ

1

1

(

)

(

)

(

)

S

x

+

F

x

+

G

x

+

1

EI ÍÎË a 2 ¯

a

a2

a 2 Ë 2a 2 6a 2 H ¯ ˙˚

(34)

gl ˆ

g l Ê ux 2

wx 3 ˆ ˘

1 ÈÊ

1

(

)

(

)

S

x

+

Z

x

+

1

EI ÍÎË a 2 ¯

a2

a 2 Ë 2a 2 6a 2 H ¯ ˙˚

(35)

y2 l =

The values of K2, K3, d1, d2, d3, S(x), F(x), G(x) and Z(x) are given in Appendix 2.

6.

COMPATIBILITY EQUATION

In the aforementioned equations, all parameters related to external loads are determined. The parameters for the outrigger are also known, provided the moment due to outrigger, Mh is determined.

Moment Mh can be determined by a rotational computability equation. The pivot for this equation

is the intersection of the centeroidal axes of one wall with the outrigger. The rotational compatibility

equation can be given by

Ï Ê Mh ˆ h Ê M ˆ l3

¸

ÔË d ¯ s Ë h ¯

Ô 1

1 hs

(T2 l - T2 M )˝

y2¢ l (hs ) - y2¢ M (hs ) = Ì

+

Ú

3E0 I0

EA 0

Ô EAc

ÔC+l

Ó

˛

(36)

where E0, I0, C are elastic modulus of outrigger, second moment of area of outrigger between

centroidal axis and the edge of each wall, respectively. In the equation the terms on the left are rotations due to external loads and the outrigger moment respectively, and the successive terms, on the

right, are the axial deformation of the column, bending of the outrigger and axial deformation of the

wall.

By combining Equation (36) with Equations (17, 20, 22, 33, 35) moment Mh can be determined as

follows:

gl ˆ

l

g l Ê uhs

whs2 ˆ

ÈÊ

˘

(

)

(

)

1

S

¢

x

+

Z

¢

x

+

+ yTb ˙

ÍÎË a 2 ¯

a2

a 2 Ë a 2 2a 2 H ¯

˚

Mh =

Ï ÈÊ 1 - g l ˆ h - lBs¢¢ sinh ah ˘ + kh + wH + y Ê lB2¢¢ sinh ah + g l h ˆ ¸

Ì ÍË

s

s˙

s

s

s ˝

Ë a

a

˚

a2 ¯

a2 ¯˛

ÓÎ

(37)

in which

Tb =

B2¢l

C2¢l

(cosh ahs - 1)

sinh ahs +

a

a

3

g l Ï È -( H - hs )

H 3 hs ˘ w Ê H 3 hs H 2 hs2 hs4 ˆ

hs ¸

+

+

+ pÊ Hhs - ˆ ˝

+ 2 Ìu Í

+

+ 2 ˙+

Ë

Ë

¯

6

6 a ˚ H

2

4

2

2 ¯˛

a Ó Î

B2¢¢ =

Copyright © 2004 John Wiley & Sons, Ltd.

1

( B2 )

Mh

(38)

(39)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

16

N. A. ZEIDABADI ET AL.

EI

d (l + c) EAc

(40)

EI

l(l + c) EA

(41)

EI

l3

3E0 I0 a (l + c) H

(41)

k=

y=

w=

dS( x )

dZ ( x )

.

and Z ( x ) =

dx

dx

Having the outrigger moment Mh, the value of B1, B2 and C1 can be determined by using Equations

(26) through (29). The deflections and internal forces of the structure are given by

in which S ¢( x ) =

y( x ) = yl ( x ) - y M ( x )

(42)

q ( x ) = ql ( x ) - q M ( x )

(43)

T ( x ) = Tl ( x ) - TM ( x )

(44)

In tall building structures, one of the most important features that should be considered is the top drift,

therefore instead of y(x), YH is used in investigations. Consequently Equation (42) can be simplified

by

y H = ylH - y MH

(45)

in which ylH and yMH are top drifts due to external loads and the outrigger respectively.

7.

RELIABILITY OF THE METHOD

To ensure the reliability of the method, the deflection determined by this method was compared with

other methods such as the wide column method. The wide column method is one of the most reliable

methods for analyzing coupled shear walls (Stafford Smith and Coull, 1991; Tararath, 1988). The comparison is shown in Table 1. According to the table the results are very close.

8.

PARAMETRIC STUDY

It is useful to express the equations representing the internal forces and deflections of the structure in

non-dimensional form to enable a parametric study. The value of top drift in addition to laminar shear

and axial forces of the walls can be given in dimensionless form. These values under uniform load

are given in the following equations:

Copyright © 2004 John Wiley & Sons, Ltd.

q = uH

g

q*

a2

(46)

T0 = uH

g

T0 *

a2

(47)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

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OPTIMIZED USE OF OUTRIGGER SYSTEM

Table 1. Contrasting the solutions gained by the presented method with those determined by the wide column

method (equal frame)

aH

Sm

3·5

3·5

3·5

3·5

3·93

2·94

2·94

2·94

2·94

2·94

2·94

2·94

2·91

2·91

2·91

2·91

2·31

0

0

0

0·707

0

0·707

0·707

hs (m)

w

H (m)

33·6

33·6

33·6

67·2

67·2

99

51

51

51

51

51

51

-5

67·2

67·2

67·2

67·2

67·2

99

99

99

99

99

99

99

5·2 * 10

0·135

0·135

5·78 * 10-2

5·78 * 10-2

8 * 10-5

8 * 10-5

6·66 * 10-6

3·45 * 10-5

3·45 * 10-5

1·59 * 10-2

3·78 * 10-2

k

y

Wide column

method

(m) * 10-2

0·164

0·113

0·13

0·164

0·164

0·11

0·11

0·247

0·158

0·158

0·158

0·158

0·133

0·106

0·106

0·133

0·133

9·69 * 10-2

9·69 * 10-2

0·161

0·121

0·121

0·121

0·121

1·079

1·548

1·36

1·75

1·635

8·30

5·40

7·175

5·638

6·129

5·916

6·24

Presented

method

(m) * 10-2

1·0839

1·557

1·361

1·749

1·634

8·34

5·43

7·22

5·654

6·167

5·93

6·26

1.0

0.8

Sm=0

5 2 S m=1

10

x/H

0.6

0.4

aH = 3

0.2

0.0

0 .0 0

h s /H = .5

0 .0 5

0.10

0.15

0 .2 0

0 .2 5

0 .3 0

0 .3 5

q*

Figure 3. Variation of laminar shear with height in a structure with internal beam but without outrigger

yH =

uH 4

yH *

EI

(48)

where q*, T0* and yH* are the value of dimensionless laminar shear, axial force of the walls and top

drift, respectively, as shown in Appendix 3.

In Figures 3 and 4 the variation of laminar shear on the height of the structure with an internal beam

without outrigger and an outrigger along with an internal beam are shown, respectively. According to

the figures, the effect of the outrigger on laminar shear is substantial, provided an internal beam is not

used in the structure, and this effect is nominal when an internal beam is used, especially for large

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

18

N. A. ZEIDABADI ET AL.

1.0

k* = y* = 20

aH=3

w = .05

hs/H =.5

x/H

0.8

Sm=10, 5, 2, 1, 0

0.6

0.4

0.2

0.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

q*

Figure 4. Variation of laminar shear with height in a structure with outrigger

1.0

k* = y * = 20

aH=3

w = .05

hs/H =.5

0.8

hs/H

0.6

Sm= 10, 5, 2, 1, 0

0.4

0.2

0.0

0.10

0.15

0.20

0.25

0.30

0.35

0.40

q*max

Figure 5. Effect of outrigger location on maximum laminar shear

values of Sm. Figure 5 shows the influence of the outrigger on maximum laminar shear. It can be seen

that the best location to minimize the laminar shear is 0.4 of the height from the bottom.

The effect of the outrigger and the internal beam position on top drift for different relative flexural

rigidity of the internal beam and relative axial rigidity of the columns is shown in Figures 6 and 7

respectively. The figures indicate that increasing Sm and k* enhances the stiffness of the structure, as

it is obvious from the figures that by increasing Sm and k* the curves become nearer. Thus, it is sug-

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

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OPTIMIZED USE OF OUTRIGGER SYSTEM

1.0

k* = y* = 20

aH=3

w = .05

Sd= .9

0.9

0.8

0.7

hs/H

0.6

Sm= 10, 5, 2, 1, 0

0.5

0.4

0.3

0.2

0.1

0.0

1.5

2.0

2.5

3.0

3.5

4.0

100 Y*

Figure 6. Effect of outrigger location on top drift for different relative flexural rigidities of the internal beam

1.0

*

y = 20

aH=3

w = .05

0.9

0.8

Sd= .9

0.7

hs/H

0.6

*

k = 100, 50, 20, 10, 5, 1

0.5

0.4

0.3

0.2

0.1

0.0

1.5

2.0

2.5

3.0

100Y

3.5

4.0

*

Figure 7. Effect of relative axial rigidity of the columns on top drift for different locations of the outrigger

gested that Sm and k* not exceed their economical limits. In other words only stiffening the internal

beam or just fortifying the columns is not always an economical way to control the top drift of the

structure.

In Figure 8 the effect of outrigger relative flexural rigidity and the location of the outrigger are illustrated. It can be seen that by stiffening the outrigger top drift decreases. Figure 9 shows the influence

of outrigger location on top drift for different parameters of coupled shear walls, aH. The figures

indicate that the influence of the outrigger is decreased when aH augments. Figures 10 and 11 show

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

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N. A. ZEIDABADI ET AL.

hs / H

1.0

0.9

k * = 20

0.8

y = 20

aH=3

0.7

S d= .9

*

0.6

w = 0, .05, .1, .2, .5, 1

0.5

0.4

0.3

0.2

0.1

0.0

1.0

1.5

2.0

2.5

100 Y

3.0

3.5

4.0

*

Figure 8. Effect of relative flexural rigidity of the outrigger on top drift for different locations of the outrigger

1.0

k* = y * = 20

w = .05

Sd= .9

0.9

0.8

hs / H

0.7

0.6

aH = 8, 7, 6, 5, 4, 3, 2

0.5

0.4

0.3

0.2

0.1

0.0

0

1

2

3

100 Y

4

5

6

*

Figure 9. Effect of parameter of coupled shear walls aH on top drift for different locations of outrigger

the effect of outrigger location on resistant moment due to outrigger for different parameters of coupled

shear walls aH and relative flexural rigidity of the outrigger w. These figures indicate that the

maximum resistant moment occurs when the location of the outrigger is from 0.2 to 0.4 the height of

the structure. These figures also show that when aH increases the amount of resistant moment

decreases and when relative flexural regidity of the outrigger decreases, resistant moment Mh also

increases. Figures 12 and 13 show the effect of outrigger location on axial forces of the wall for

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

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OPTIMIZED USE OF OUTRIGGER SYSTEM

0.58

aH = 2

0.56

k = .05 S d = .9

0.54

hs / H

y = .05 S m = 2

0.52

aH = 3

0.50

aH

0.48

0.46

0.0

0.2

0.4

=4

0.6

0.8

1.0

w

Figure 10. Optimum outrigger location for different relative flexural rigidities of the outrigger

0.56

0.54

aH

hs / H

aH

aH

=6

7

0.50

=

0.52

=5

0.48

0.46

0.44

0.42

0.40

0.0

0.2

0.4

0.6

0.8

1.0

w

Figure 11. Optimum outrigger location for different relative flexural rigidities of the outrigger

different relative flexural rigidity of the stiffening beam and outrigger respectively. These figures

show that when Sm increases the amount of axial force of the wall T0 also increases, and when w

increases T0 decreases. From Figures 3 to 13 it can be seen that the slopes of the curves are not very

large. As a result, small movements in the location of the outrigger do not affect the behavior of the

structure significantly.

In Figures 10 and 11 the influence of outrigger relative flexural rigidity on the best location of the

outrigger to minimize top drift is shown. It can be seen that the optimum locations of outrigger in

coupled shear walls are different from those in ordinary cores. Besides, the state of curves is also different especially for a large aH. The best location to minimize top drift in ordinary shear walls can

be found in Coull and Lao (1988, 1989) and Rutenburg and Eisenburg (1990).

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

22

N. A. ZEIDABADI ET AL.

0.56

aH = 2

hs / H

0.54

0.52

0.50

aH = 3

0.48

aH

=4

0.46

0.44

0.0

0.2

0.4

0.6

0.8

1.0

k

Figure 12. Optimum outrigger location for different relative rigidities of the columns

w = .05 S d = .9

0.56

y = .05 Sm = 2

hs / H

0.52

0.48

0.44

0.40

0.36

0.0

aH = 5

aH = 6

aH = 7

0.2

0.4

0.6

0.8

1.0

k

Figure 13. Optimum outrigger location for different relative rigidities of the columns

The optimum location of the outrigger with respect to relative axial rigidity of the columns to minimize top drift is shown in Figures 12 and 13. These figures indicate that when k increases the best

location of the outrigger goes downwards. Figure 14 shows the influence of Sm on the best location of

the structure. The figure illustrates that Sm has a nominal effect on the best location of the structure,

or the best location of the outrigger is not virtually affected by the rigidity of the beam. It can be

concluded from Figures 10–14 that the best location of the outrigger is often between 0·4 to 0·6 of

the height from the bottom.

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

23

OPTIMIZED USE OF OUTRIGGER SYSTEM

0.60

0.58

aH = 2

hs / H

0.56

aH = 3

aH = 4

0.54

aH = 5

aH = 6

0.52

aH = 7

aH = 8

0.50

0.48

0

2

4

6

8

10

Sm

Figure 14. Optimum outrigger location for different relative flexural rigidities of the walls

9.

CONCLUSION

On the basis of continuum approach, a method is derived for analyzing the structural behavior of

coupled shear walls stiffened by an internal beam and an outrigger in a haphazard location along the

height of the structure. The beneficial effect of the outrigger on the structural behavior and lateral

deflections of coupled shear walls is investigated.

Parametric study shows that an outrigger can significantly reduce the lateral deflection of the structure. The study also shows that the position of the outrigger can substantially affect the behavior and

lateral deflection of the structure. Even though an outrigger can be very effective on lateral deflection,

the effect of the outrigger on laminar shear is nominal, if an internal beam is used in the structure.

Furthermore, the investigation indicates that the axial stiffness of columns as well as flexural stiffness

of the outrigger has a significant effect on the outrigger location, but the influence of flexural rigidity

of the internal beam is nominal. Finally, the study shows that the best location of the outrigger is

usually somewhere between 0.4 to 0.6 of the height of the structure from the bottom.

REFERENCES

Chan HC, Kuang JS. 1989a. Stiffened coupled shear walls. Journal of Engineering Mechanics, ASCE 115(4):

689–703.

Chan HC, Kuang JS. 1989b. Elastic design charts for stiffened coupled shear walls. Journal of Structural

Engineering, ASCE 115(2): 247–267.

Choo BS, Li GQ. 1997. Structural analysis of multi-stiffened coupled shear walls on flexible foundations.

Computers and Structures 64(1–4): 837–848.

Coull A, Bensmail L. 1991. Stiffened coupled shear walls. Journal of Structural Engineering, ASCE 117(8):

2205–2223.

Coull A, Lao WHO. 1988. Outrigger braced structures subjected to equivalent static seismic loading. In

Proceedings of 4th International Conference on Tall Buildings, Hong Kong, 1988; 395–401.

Coull A, Lao WHO. 1989. Analysis of multi-outrigger-braced tall building structures. Journal of Structural

Engineering, ASCE 115(7): 1811–1816.

Moudares FR. 1984. Outrigger-braced coupled shear walls. Journal of Structural Engineering, ASCE 110(12):

2871–2890.

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

24

N. A. ZEIDABADI ET AL.

Rutenburg A, Eisenburg M. 1990. Stability of outrigger-braced tall building structures. In Proceedings of 5th

International Conference on Tall Buildings, Hong Kong, 1990; 881–892.

Straman J, Goldaf E. 1997. Outrigger braced structures in concerete. In Seventh International Conference on

Computing in Civil Engineering, Seoul, Korea, 1997; 933–938.

Stafford Smith B, Coull A. 1991. Tall Building Structures, Analysis and Design. Wiley: Chichester.

Taranath BS. 1988. Structural Analysis of Tall Buildings. McGraw Hill: London.

APPENDIX 1

K1 =

-g (u + w)

a 4 cosh aH

K2 = cosh ahs - Sm sinh ahs

K3 = sinh ahs - SmaH cosh ahs

K 4 = - Sm

g

a2

C2¢ =

ÈuH ( H h ) + w Ê H 2 - h 2 - 2 ˆ + PH ˘

- s

s

ÍÎ

˙˚

2Ë

a2 ¯

g

a2

ÈuH + Ê wH ˆ + P - Ê w ˆ ˘

ÍÎ

Ë 2 ¯

Ë a 2 H ¯ ˙˚

C1¢ =

B2¢ tanhahs - z 3

z1

B1¢ = K1 - C1¢ tanhaH

B2¢ =

z 1 (K1 K2 - K 4 - C2¢ sinh ahs ) + z 1z 3

z 1 cosh ahs + z 2 tanh ahs

z 1 = 1 - tanh ahs tanh aH

z 2 = K2 tanh aH - K3

z 3 = K1 tanh ahs - C2

APPENDIX 2

gl

l

C1l

d 1 = - ÈÍÊ 1 - 2 ˆ Mh hs + ( B1 - B2 ) sinh ahs +

cosh ahs ˘˙

Ë

¯

a

a

Î

˚

a

1

gl

l

lB2 C1l

d 2 = - ÈÍ Ê 1 - 2 ˆ Mh hs2 + ( B1 - B2 ) 2 cosh ahs + 2 + 2 sinh ahs + d 1hs ˘˙

Ë

¯

2

Î

˚

a

a

a

a

d3 =

Copyright © 2004 John Wiley & Sons, Ltd.

lB2

a2

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

OPTIMIZED USE OF OUTRIGGER SYSTEM

S( x ) =

25

u( x 4 - 4 Hx 3 + 6 H 2 x 2 ) w( x 5 - 10 H 2 x 3 + 20 H 3 x 2 ) p( x 3 - 3 Hx 2 )

+

24

120 H

6

F( x ) = [( B1¢ - B2¢ ) sinh ahs + (C1¢ - C2¢ ) cosh ahs ]( x - hs )

G(x ) = B1¢(cosh ahs - cosh ax ) + C1¢(sinh ahs - sinh ax ) + B2¢ (1 - cosh ahs ) + C2¢ (ax - sinh ahs )

Z ( x ) = B2¢ (1 - cosh ax ) + C2¢ (ax - sinh ax )

APPENDIX 3

* - y*MH

y*H = ylH

y*MH = ya Mh*

q1* = q1*l - q1*M

q2* = q2*l - q2*M

q *M1 = q1a Mh*

q *M 2 = q2 a Mh*

where

q1a = -

K

(b1¢ sinh Kx + c1¢ cosh Kx )

Sd

q2 a = -

K

(b2¢ sinh Kx )

Sd

b1

c1

ya = - ÈÍ 2 cosh K + 2 sinh K + N1 + N2 ˘˙

ÎK

˚

K

sinh Kxs c1

N1 = - ÈÍ(1 - Sd )xs + (b1 - b2 )

+ cosh Kxs ˘˙

K

K

Î

˚

N2 = (1 - Sd )

xs2 (b1 - b2 )

c1

+

sinh Kxs + cosh Kxs

2

K

K2

Ê- 1 + S ˆ

d

Ë d

¯

c1 =

cosh Kxs

- K2 tanh K + K3 + cosh Kxs + cosh Kxs tanh Kxs

tanh Kxs

b1 = c1 tanh K

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

26

N. A. ZEIDABADI ET AL.

1

ˆ

b2 = c1 Ê

Ë tanh Kxs - tanh K ¯

Mh* =

h1

h2

3

È 1 (1 - xs )

xs ˘

xs

h1 = b2¢ sinh Kxs + c2¢ (cosh Kxs - 1) + Sd Í + 2 ˙ + (1 - Sd ) L p + d ¢ - Sd Ê 2 ˆ

Ë

¯

6

6

K

K

Î

˚

h2 = (1 - Sd )xs -

b2

b2

sinh Kxs + Kxs + y Ê sinh Kxs + Sd xs ˆ + w

Ë

¯

K

K

Lp =

1 3

(xs - 3xs2 + 3xs )

6

d ¢ = - b2¢ sinh Kxs + c2¢ (1 - cosh Kxs )

b2¢ =

z 1 D1 + z 2 D2

z 1 cosh Kxs + z 2 tanh Kxs

c2¢ =

D1 =

Sd

K2

(- Sm Sd )(1 - xs ) (- Sd )

- Sdz 2

+

sinh Kxs

K

K cosh K

K2

3

D2 =

- Sd tanh Kxs Sd

- 2

K 3 cosh K

K

K = aH

x=

x

H

xs =

hs

H

The values of y*1H, q1* and T1* are presented in Moudares (1984).

NOTATION

A

B1, B2, B1¢, B2¢

b

c

C1, C2, C1¢, C2¢

Cross-section area of each wall

Integration constants

Clear span length of coupling beam

Distance between centroidal axis and edge of each wall

Integration constants

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

OPTIMIZED USE OF OUTRIGGER SYSTEM

d

E

E0, Es

F

H

h

hs

I1

I

Ib, Is, I0

l

Me

Mel

Mh

p

q

qil, qiM

Sm

T

Til, TiM

u

Vs

w

x

y

yH

a, g

l

w, y

27

Distance between the columns

Elastic modulus of walls and coupling beams

Elastic moduli of outrigger and stiffening beam, respectively

Axial resistant force of columns

Total height of structure

Height of story

Location of outrigger and stiffening beam from bottom

Second moment of area of each wall

Total second moment of area of walls equal to 2I1

Second moment of area of connecting beams, stiffening beam and outrigger

Distance between centroidal axes of walls

Applied moment

Applied moment due to external load

Resistant moment caused by outrigger

Concentrated load at top of structure

Laminar shear in equivalent medium

Laminar shear in section i due to external loads and outrigger, respectively

Relative flexural rigidity of stiffening beam

Axial force of each wall

Axial force in each wall caused by external loads and outrigger, respctively

Intensity of uniformly distributed load

Shear force of stiffening beam

Maximum intensity of triangular distributed load

Height coordinate

Lateral deflection of walls

Lateral deflection of walls at top level

Structural parameter

Clear length of outrigger

Dimensionless parameters of structure

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

Published online in Wiley Interscience (www.interscience.wiley.com). DOI:10.1002/tal.228

OPTIMIZED USE OF THE OUTRIGGER SYSTEM TO STIFFEN

THE COUPLED SHEAR WALLS IN TALL BUILDINGS

NAVAB ASSADI ZEIDABADI1, KAMAL MIRTALAE1* AND BARZIN MOBASHER2

1

Isfahan University of Technology, Isfahan, Iran; and Arizona Department of Transportation, Phoenix, Arizona, USA

2

Civil and Environmental Engineering Department, Arizona State University, Tempe, Arizona, USA

SUMMARY

Based on the conventional yet accurate continuum approach, a general analysis is presented for a pair of coupled

shear walls, stiffened by an outrigger and a heavy beam in an arbitrary position on the height. Subsequently, a

parametric study is presented to investigate the behavior of the structure. The optimum location of the outrigger

and the parameters affecting its position were also investigated. The results showed that the behavior of the structure can be significantly influenced by the location of the outrigger. It was also indicated that in most ordinary

cases the best location of the structure to minimize top drift is somewhere between 0·4 to 0·6 of the height of the

structure. Though this method is not a substitute for the finite element method, it gives an initial simple solution

to determine the size and position of outrigger, stiffening beam and coupled shear walls in the preliminary design

stages. Copyright © 2004 John Wiley & Sons, Ltd.

1.

INTRODUCTION

In modern residential tall buildings, lateral loads induced by wind or earthquake are often resisted by

a system of coupled shear walls. When a building increases in height, the stiffness of the structure

becomes more important. In addition, the depth of lintel beams connecting shear walls will usually be

confined by differences between floor-to-floor height and floor clear height, Hence, the coupling effect

of the connecting system may not be sufficient to provide the necessary lateral stiffness, and the tensile

bending stress and uplift forces may exceed the economical limits.

Different methods that can be used to overcome these problems may be the provision of an outrigger, addition of very stiff beams between walls or using both systems.

An outrigger is a stiff beam that connects the shear walls to exterior columns. When the structure

is subjected to lateral forces, the outrigger and the columns resist the rotation of the core and thus significantly reduce the lateral deflection and base moment, which would have arisen in a free core.

Several buildings with this type of bracing were built during the last three decades in North America,

Australia and Japan.

In some buildings with a pair of coupled shear walls to resist the lateral loads, floor slabs are

protruded from the shear walls to form balconies. At the outer edge of the balconies as shown in

Figure 1, the exterior columns are located to support the slabs. An outrigger can employ peripheral

columns to increase the overall stiffness of the structure and decrease the moments of the walls.

Numerous studies have been carried out on the analysis and behavior of outrigger structures (Coull

and Lao, 1988, 1989; Rutenburg and Eisenburg, 1990; Skraman and Goldaf, 1997). Moudarres (1984)

* Correspondence to: Dr. Kamal Mirtalae, Arizona Department of Transportation, Bridge Design Group, Mail Drop #631E, 205

South 17th Avenue, Phoenix, AZ 85007, USA

Copyright © 2004 John Wiley & Sons, Ltd.

Received December 2001

Accepted November 2002

10

N. A. ZEIDABADI ET AL.

Peripheral columns

Coupled Shear Walls and Outrigger

Figure 1. Simplified plan of building

showed that a top outrigger can reduce the lateral deflections in a pair of coupled shear walls. Using

the continuous medium method, Chan and Kuang (1989a, 1989b) conducted studies on the effect of

an intermediate stiffening beam at an arbitrary level along the height of the walls, and indicated that

the structural behavior of coupled shear walls could be significantly affected by particular positioning

of the stiffening beam. Afterwards, Coull and Bensmail (1991) as well as Choo and Li (1997) extended

Kuang and Chan’s method for two and multi-stiffening beams. Their studies also included both rigid

and flexible foundations for the structure.

In this paper, based on Chan and Kuang’s method, a continuum approach is designated to analyze

a pair of coupled shear walls, stiffened by an outrigger and an interior beam at an arbitrary location

on the height. A parametric study is used to investigate the influence of rigidities and locations of the

outrigger and interior beam on the lateral deflections and laminar shear forces in the structure. Furthermore, the best locations of the outrigger to minimize top drift or laminar shear and the effective

parameters on the location are presented.

2.

ANALYSIS

Consider a coupled structural wall system in a fixed foundation stiffened by an outrigger and a beam

at level hs shown in Figure 2. For analysis of the structure by continuum approach, the coupling beams

are replaced by continuous distribution of lamina with equivalent stiffness. It is also assumed that both

walls deflected equally throughout the height, so the points of contraflexure of the laminae and stiffening beam are at their mid-span points. If a hypothetical cut is made along the line of contraflexure,

the condition of vertical compatibility above and, below the outrigger leads to the following

equations:

l

dy1

hb 3

2

q1 dx 12 EIb

EA

l

Copyright © 2004 John Wiley & Sons, Ltd.

[Ú T dx + Ú

x

hs

1

hs

0

]

T2 dx = 0

dy2

hb 3

2 x

q2 T2 dx = 0

dx 12 EIb

EA Ú0

(1)

(2)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

11

OPTIMIZED USE OF OUTRIGGER SYSTEM

l

c

c

l

l

b/2

b

b/2

q1

Vs+F

h

F

F

H

x

q2

hs

(b)

(a)

Figure 2. (a) Coupled shear walls stiffened by outrigger and internal beam. (b) Substitute structure

where y1, q1, T1 and y2, q2, T2 are the lateral deflection, the laminar shear and the axial forces in the

walls in the section above and below level hs, respectively and Ib, E, A are second moment of area of

connecting beams, elastic modulus of walls and coupling beams and cross-section area of each wall.

The three successive terms represent the vertical deflection at the cut caused by slopes of the walls,

bending of laminae and axial deformation of the walls.

The general moment–curvature relationship of the walls is

d 2 y1

Ï

M

=

EI

+ T1l

e

ÔÔ

dx 2

Ì

2

Ô M = EI d y2 + T l

2

ÔÓ e

dx 2

for (hs £ x £ H )

(3)

for (0 £ x £ hs )

in which I = 2I1, where I1 is second moment of area of each wall, and the axial forces in the walls in

different sections are given respectively by

H

T1 = Ú q1dx

(4)

x

H

hs

hs

x

T2 = Ú q1dx + Vs + Ú q2 dx

(5)

where Vs represents the shear force in the stiffening beam.

By considering the equilibrium of a small vertical element of the continuous structure, it can be

shown that at any point along the height

q=

- dT

dx

(6)

By differentiating Equations (1) and (2) and combining with Equations (3) and (6), q and y can be

eliminated and then the governing equations for the axial forces in the walls can be given by

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

12

N. A. ZEIDABADI ET AL.

d 2 T1

- a 2 T1 = -g Me

dx 2

(7)

d 2 T2

- a 2 T2 = -g Me

dx 2

(8)

I

a 2 = g Êl + ˆ

Ë Al ¯

(9)

where

g =

6 Ib l

hb 3 I

(10)

To obtain the shear force in stiffening beam Vs, consider the compatibility condition at its contraflexure point:

l

dy2

VS b 3

2 hs

T2 dx = 0

dx 13Es Is EA Ú0

(11)

in which EsIs is the flexural rigidity of the stiffening beam.

Equating the corresponding terms of Equations (1) or (2) and Equation (11) at level hs gives the

shear force of the stiffening beam thus; the shear forces will be

Vs = Sm Hq1s = Sm Hq2 s

(12)

where q1s and q2s are the shear flows at level hs and Sm is the relative flexural rigidity of the stiffening

beam, defined as

Sm =

3.

h ES I S

H EIb

(13)

THE EFFECTS OF THE OUTRIGGER AND EXTERNAL LOADS

In this investigation, the influence of the outrigger is considered as an unknown moment Mh in the

location of the outrigger. Moment Mh can also be represented by

Mh = F(2c + 2l + l)

(14)

The parameters F, c, l and ഞ are shown in Figure 2. By considering Mh, the moment Me in Equations

(7) and (8) can be given by

Ï Me = 0

Ì

Ó Me = M h

for

for

hs £ x £ H

0 £ x £ hs

(15)

Therefore the complete solution of Equations (7) and (8) is

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

13

OPTIMIZED USE OF OUTRIGGER SYSTEM

T1M = B1 cosh a x + C1 sinh a x

T2 M = B2 cosh a x + C2 sinh a x +

(16)

g

Mh

a2

(17)

The expression for laminar shear above and below the outrigger can be derived by using Equation (6):

q1M = -[ B1a sinh a x + C1a cosh a x ]

(18)

q2 M = -[ B2a sinh a x + C2a cosh a x ]

(19)

For the external loads considered, the applied bending moment can be represented by

u

w

2

(2 H 3 - 3H 2 x + x 3 )

Me = Mel = P( H - x ) + ( H - x ) +

2

6H

(20)

where P is considered the load at the top of the walls, and u and w are intensities of uniformly distributed and upper triangular distributed loads acting in the walls, respectively. Thus, the complete

solution of Equations (7) and (8) due to lateral loads is represented in following equations:

T1l = B1¢ cosh a x + C1¢ sinh a h +

g Ê

u

wx

Mel + 2 + 2 ˆ

a2 Ë

a

a H¯

(21)

T2 l = B2¢ cosh a x + C2¢ sinh a h +

g Ê

u

wx

Mel + 2 + 2 ˆ

2 Ë

a

a

a H¯

(22)

Consequently, the laminar shears are

g dMel

w

q1l = - ÈÍ B1¢a sinh a + C1¢a cosh a x + 2 Ê

+ 2 ˆ ˘˙

Î

a Ë dx

a H¯˚

(23)

g dMel

w

q2 l = - ÈÍ B2¢a sinh a x + C2¢a cosh a x + 2 Ê

+ 2 ˆ ˘˙

Ë

dx

Î

a

a H¯˚

(24)

4.

BOUNDARY CONDITIONS

The values of B1, B2, C1 and C2 can be determined by considering a set of boundary conditions.

• At the top of the structure, x = H:

T1 ( H ) = 0

(25a)

• At the level of stiffening beam h , boundary conditions are

s

Copyright © 2004 John Wiley & Sons, Ltd.

T1 (hs ) + Vs + Fi = T2 (hs )

(25b)

q1 (hs ) = q2 (hs )

(25c)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

14

N. A. ZEIDABADI ET AL.

• At the base level, the laminar shear is given by

q2 (0) = 0

(25d)

Solving Equations (25a)–(25d) gives the unknown integration constants:

B1 = -C1 tanhaH

(26)

1

B2 = C1 Ê

- tanh ahˆ

Ë tanh ahs

¯

(27)

C2 = 0

(28)

-1

l ˆ

Ê

+

M

Ë 2l + 2c + l a 2 ¯ h

C1 =

Ê - K tanh aH + K - cosh ahs + cosh ah (tanh ah )ˆ

3

s

s

Ë 2

¯

tanh ahs

(29)

Likewise, the values of integration constants B¢1, B¢2, C¢1 and C¢2 can be determined. The only alteration in boundary conditions which should be made is that Equation (25b) will be changed into the

following equation:

T1l (hs ) + Vs = T2 l (hs )

(30)

Expressions for the constants B1¢, B2¢, C1¢ and C2¢ are given in Appendix 1.

5.

LATERAL DEFLECTION EQUATIONS

By integrating Equation (3) twice and using the compatibility condition represented in Equations

(31a)–(31d) the lateral deflection due to outrigger and external loads will be

y2 (0) = 0

(31a)

y2¢ (0) = 0

(31b)

y1 (hs ) = y2 (hs )

(31c)

y1¢(hs ) = y2¢ (hs )

(31d)

The lateral deflections due to the outrigger can be given as follows:

y1M = -

y2 M =

Copyright © 2004 John Wiley & Sons, Ltd.

1 È B1l

C1l

cosh a x + 2 sinh a x + xd 1 + d 2 ˘˙

EI ÍÎ a 2

˚

a

(32)

1 ÈÊ

gl

lB2

1 - 2 ˆ Mh x 2 - 2 cosh a x + d 3 ˘˙

Í

Ë

¯

EI Î

˚

a

a

(33)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

OPTIMIZED USE OF OUTRIGGER SYSTEM

15

The lateral deflections caused by external loads are

y1l =

gl ˆ

g l Ê ux 2

wx 3 ˆ ˘

1 ÈÊ

1

1

(

)

(

)

(

)

S

x

+

F

x

+

G

x

+

1

EI ÍÎË a 2 ¯

a

a2

a 2 Ë 2a 2 6a 2 H ¯ ˙˚

(34)

gl ˆ

g l Ê ux 2

wx 3 ˆ ˘

1 ÈÊ

1

(

)

(

)

S

x

+

Z

x

+

1

EI ÍÎË a 2 ¯

a2

a 2 Ë 2a 2 6a 2 H ¯ ˙˚

(35)

y2 l =

The values of K2, K3, d1, d2, d3, S(x), F(x), G(x) and Z(x) are given in Appendix 2.

6.

COMPATIBILITY EQUATION

In the aforementioned equations, all parameters related to external loads are determined. The parameters for the outrigger are also known, provided the moment due to outrigger, Mh is determined.

Moment Mh can be determined by a rotational computability equation. The pivot for this equation

is the intersection of the centeroidal axes of one wall with the outrigger. The rotational compatibility

equation can be given by

Ï Ê Mh ˆ h Ê M ˆ l3

¸

ÔË d ¯ s Ë h ¯

Ô 1

1 hs

(T2 l - T2 M )˝

y2¢ l (hs ) - y2¢ M (hs ) = Ì

+

Ú

3E0 I0

EA 0

Ô EAc

ÔC+l

Ó

˛

(36)

where E0, I0, C are elastic modulus of outrigger, second moment of area of outrigger between

centroidal axis and the edge of each wall, respectively. In the equation the terms on the left are rotations due to external loads and the outrigger moment respectively, and the successive terms, on the

right, are the axial deformation of the column, bending of the outrigger and axial deformation of the

wall.

By combining Equation (36) with Equations (17, 20, 22, 33, 35) moment Mh can be determined as

follows:

gl ˆ

l

g l Ê uhs

whs2 ˆ

ÈÊ

˘

(

)

(

)

1

S

¢

x

+

Z

¢

x

+

+ yTb ˙

ÍÎË a 2 ¯

a2

a 2 Ë a 2 2a 2 H ¯

˚

Mh =

Ï ÈÊ 1 - g l ˆ h - lBs¢¢ sinh ah ˘ + kh + wH + y Ê lB2¢¢ sinh ah + g l h ˆ ¸

Ì ÍË

s

s˙

s

s

s ˝

Ë a

a

˚

a2 ¯

a2 ¯˛

ÓÎ

(37)

in which

Tb =

B2¢l

C2¢l

(cosh ahs - 1)

sinh ahs +

a

a

3

g l Ï È -( H - hs )

H 3 hs ˘ w Ê H 3 hs H 2 hs2 hs4 ˆ

hs ¸

+

+

+ pÊ Hhs - ˆ ˝

+ 2 Ìu Í

+

+ 2 ˙+

Ë

Ë

¯

6

6 a ˚ H

2

4

2

2 ¯˛

a Ó Î

B2¢¢ =

Copyright © 2004 John Wiley & Sons, Ltd.

1

( B2 )

Mh

(38)

(39)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

16

N. A. ZEIDABADI ET AL.

EI

d (l + c) EAc

(40)

EI

l(l + c) EA

(41)

EI

l3

3E0 I0 a (l + c) H

(41)

k=

y=

w=

dS( x )

dZ ( x )

.

and Z ( x ) =

dx

dx

Having the outrigger moment Mh, the value of B1, B2 and C1 can be determined by using Equations

(26) through (29). The deflections and internal forces of the structure are given by

in which S ¢( x ) =

y( x ) = yl ( x ) - y M ( x )

(42)

q ( x ) = ql ( x ) - q M ( x )

(43)

T ( x ) = Tl ( x ) - TM ( x )

(44)

In tall building structures, one of the most important features that should be considered is the top drift,

therefore instead of y(x), YH is used in investigations. Consequently Equation (42) can be simplified

by

y H = ylH - y MH

(45)

in which ylH and yMH are top drifts due to external loads and the outrigger respectively.

7.

RELIABILITY OF THE METHOD

To ensure the reliability of the method, the deflection determined by this method was compared with

other methods such as the wide column method. The wide column method is one of the most reliable

methods for analyzing coupled shear walls (Stafford Smith and Coull, 1991; Tararath, 1988). The comparison is shown in Table 1. According to the table the results are very close.

8.

PARAMETRIC STUDY

It is useful to express the equations representing the internal forces and deflections of the structure in

non-dimensional form to enable a parametric study. The value of top drift in addition to laminar shear

and axial forces of the walls can be given in dimensionless form. These values under uniform load

are given in the following equations:

Copyright © 2004 John Wiley & Sons, Ltd.

q = uH

g

q*

a2

(46)

T0 = uH

g

T0 *

a2

(47)

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

17

OPTIMIZED USE OF OUTRIGGER SYSTEM

Table 1. Contrasting the solutions gained by the presented method with those determined by the wide column

method (equal frame)

aH

Sm

3·5

3·5

3·5

3·5

3·93

2·94

2·94

2·94

2·94

2·94

2·94

2·94

2·91

2·91

2·91

2·91

2·31

0

0

0

0·707

0

0·707

0·707

hs (m)

w

H (m)

33·6

33·6

33·6

67·2

67·2

99

51

51

51

51

51

51

-5

67·2

67·2

67·2

67·2

67·2

99

99

99

99

99

99

99

5·2 * 10

0·135

0·135

5·78 * 10-2

5·78 * 10-2

8 * 10-5

8 * 10-5

6·66 * 10-6

3·45 * 10-5

3·45 * 10-5

1·59 * 10-2

3·78 * 10-2

k

y

Wide column

method

(m) * 10-2

0·164

0·113

0·13

0·164

0·164

0·11

0·11

0·247

0·158

0·158

0·158

0·158

0·133

0·106

0·106

0·133

0·133

9·69 * 10-2

9·69 * 10-2

0·161

0·121

0·121

0·121

0·121

1·079

1·548

1·36

1·75

1·635

8·30

5·40

7·175

5·638

6·129

5·916

6·24

Presented

method

(m) * 10-2

1·0839

1·557

1·361

1·749

1·634

8·34

5·43

7·22

5·654

6·167

5·93

6·26

1.0

0.8

Sm=0

5 2 S m=1

10

x/H

0.6

0.4

aH = 3

0.2

0.0

0 .0 0

h s /H = .5

0 .0 5

0.10

0.15

0 .2 0

0 .2 5

0 .3 0

0 .3 5

q*

Figure 3. Variation of laminar shear with height in a structure with internal beam but without outrigger

yH =

uH 4

yH *

EI

(48)

where q*, T0* and yH* are the value of dimensionless laminar shear, axial force of the walls and top

drift, respectively, as shown in Appendix 3.

In Figures 3 and 4 the variation of laminar shear on the height of the structure with an internal beam

without outrigger and an outrigger along with an internal beam are shown, respectively. According to

the figures, the effect of the outrigger on laminar shear is substantial, provided an internal beam is not

used in the structure, and this effect is nominal when an internal beam is used, especially for large

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

18

N. A. ZEIDABADI ET AL.

1.0

k* = y* = 20

aH=3

w = .05

hs/H =.5

x/H

0.8

Sm=10, 5, 2, 1, 0

0.6

0.4

0.2

0.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

q*

Figure 4. Variation of laminar shear with height in a structure with outrigger

1.0

k* = y * = 20

aH=3

w = .05

hs/H =.5

0.8

hs/H

0.6

Sm= 10, 5, 2, 1, 0

0.4

0.2

0.0

0.10

0.15

0.20

0.25

0.30

0.35

0.40

q*max

Figure 5. Effect of outrigger location on maximum laminar shear

values of Sm. Figure 5 shows the influence of the outrigger on maximum laminar shear. It can be seen

that the best location to minimize the laminar shear is 0.4 of the height from the bottom.

The effect of the outrigger and the internal beam position on top drift for different relative flexural

rigidity of the internal beam and relative axial rigidity of the columns is shown in Figures 6 and 7

respectively. The figures indicate that increasing Sm and k* enhances the stiffness of the structure, as

it is obvious from the figures that by increasing Sm and k* the curves become nearer. Thus, it is sug-

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

19

OPTIMIZED USE OF OUTRIGGER SYSTEM

1.0

k* = y* = 20

aH=3

w = .05

Sd= .9

0.9

0.8

0.7

hs/H

0.6

Sm= 10, 5, 2, 1, 0

0.5

0.4

0.3

0.2

0.1

0.0

1.5

2.0

2.5

3.0

3.5

4.0

100 Y*

Figure 6. Effect of outrigger location on top drift for different relative flexural rigidities of the internal beam

1.0

*

y = 20

aH=3

w = .05

0.9

0.8

Sd= .9

0.7

hs/H

0.6

*

k = 100, 50, 20, 10, 5, 1

0.5

0.4

0.3

0.2

0.1

0.0

1.5

2.0

2.5

3.0

100Y

3.5

4.0

*

Figure 7. Effect of relative axial rigidity of the columns on top drift for different locations of the outrigger

gested that Sm and k* not exceed their economical limits. In other words only stiffening the internal

beam or just fortifying the columns is not always an economical way to control the top drift of the

structure.

In Figure 8 the effect of outrigger relative flexural rigidity and the location of the outrigger are illustrated. It can be seen that by stiffening the outrigger top drift decreases. Figure 9 shows the influence

of outrigger location on top drift for different parameters of coupled shear walls, aH. The figures

indicate that the influence of the outrigger is decreased when aH augments. Figures 10 and 11 show

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

20

N. A. ZEIDABADI ET AL.

hs / H

1.0

0.9

k * = 20

0.8

y = 20

aH=3

0.7

S d= .9

*

0.6

w = 0, .05, .1, .2, .5, 1

0.5

0.4

0.3

0.2

0.1

0.0

1.0

1.5

2.0

2.5

100 Y

3.0

3.5

4.0

*

Figure 8. Effect of relative flexural rigidity of the outrigger on top drift for different locations of the outrigger

1.0

k* = y * = 20

w = .05

Sd= .9

0.9

0.8

hs / H

0.7

0.6

aH = 8, 7, 6, 5, 4, 3, 2

0.5

0.4

0.3

0.2

0.1

0.0

0

1

2

3

100 Y

4

5

6

*

Figure 9. Effect of parameter of coupled shear walls aH on top drift for different locations of outrigger

the effect of outrigger location on resistant moment due to outrigger for different parameters of coupled

shear walls aH and relative flexural rigidity of the outrigger w. These figures indicate that the

maximum resistant moment occurs when the location of the outrigger is from 0.2 to 0.4 the height of

the structure. These figures also show that when aH increases the amount of resistant moment

decreases and when relative flexural regidity of the outrigger decreases, resistant moment Mh also

increases. Figures 12 and 13 show the effect of outrigger location on axial forces of the wall for

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

21

OPTIMIZED USE OF OUTRIGGER SYSTEM

0.58

aH = 2

0.56

k = .05 S d = .9

0.54

hs / H

y = .05 S m = 2

0.52

aH = 3

0.50

aH

0.48

0.46

0.0

0.2

0.4

=4

0.6

0.8

1.0

w

Figure 10. Optimum outrigger location for different relative flexural rigidities of the outrigger

0.56

0.54

aH

hs / H

aH

aH

=6

7

0.50

=

0.52

=5

0.48

0.46

0.44

0.42

0.40

0.0

0.2

0.4

0.6

0.8

1.0

w

Figure 11. Optimum outrigger location for different relative flexural rigidities of the outrigger

different relative flexural rigidity of the stiffening beam and outrigger respectively. These figures

show that when Sm increases the amount of axial force of the wall T0 also increases, and when w

increases T0 decreases. From Figures 3 to 13 it can be seen that the slopes of the curves are not very

large. As a result, small movements in the location of the outrigger do not affect the behavior of the

structure significantly.

In Figures 10 and 11 the influence of outrigger relative flexural rigidity on the best location of the

outrigger to minimize top drift is shown. It can be seen that the optimum locations of outrigger in

coupled shear walls are different from those in ordinary cores. Besides, the state of curves is also different especially for a large aH. The best location to minimize top drift in ordinary shear walls can

be found in Coull and Lao (1988, 1989) and Rutenburg and Eisenburg (1990).

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

22

N. A. ZEIDABADI ET AL.

0.56

aH = 2

hs / H

0.54

0.52

0.50

aH = 3

0.48

aH

=4

0.46

0.44

0.0

0.2

0.4

0.6

0.8

1.0

k

Figure 12. Optimum outrigger location for different relative rigidities of the columns

w = .05 S d = .9

0.56

y = .05 Sm = 2

hs / H

0.52

0.48

0.44

0.40

0.36

0.0

aH = 5

aH = 6

aH = 7

0.2

0.4

0.6

0.8

1.0

k

Figure 13. Optimum outrigger location for different relative rigidities of the columns

The optimum location of the outrigger with respect to relative axial rigidity of the columns to minimize top drift is shown in Figures 12 and 13. These figures indicate that when k increases the best

location of the outrigger goes downwards. Figure 14 shows the influence of Sm on the best location of

the structure. The figure illustrates that Sm has a nominal effect on the best location of the structure,

or the best location of the outrigger is not virtually affected by the rigidity of the beam. It can be

concluded from Figures 10–14 that the best location of the outrigger is often between 0·4 to 0·6 of

the height from the bottom.

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

23

OPTIMIZED USE OF OUTRIGGER SYSTEM

0.60

0.58

aH = 2

hs / H

0.56

aH = 3

aH = 4

0.54

aH = 5

aH = 6

0.52

aH = 7

aH = 8

0.50

0.48

0

2

4

6

8

10

Sm

Figure 14. Optimum outrigger location for different relative flexural rigidities of the walls

9.

CONCLUSION

On the basis of continuum approach, a method is derived for analyzing the structural behavior of

coupled shear walls stiffened by an internal beam and an outrigger in a haphazard location along the

height of the structure. The beneficial effect of the outrigger on the structural behavior and lateral

deflections of coupled shear walls is investigated.

Parametric study shows that an outrigger can significantly reduce the lateral deflection of the structure. The study also shows that the position of the outrigger can substantially affect the behavior and

lateral deflection of the structure. Even though an outrigger can be very effective on lateral deflection,

the effect of the outrigger on laminar shear is nominal, if an internal beam is used in the structure.

Furthermore, the investigation indicates that the axial stiffness of columns as well as flexural stiffness

of the outrigger has a significant effect on the outrigger location, but the influence of flexural rigidity

of the internal beam is nominal. Finally, the study shows that the best location of the outrigger is

usually somewhere between 0.4 to 0.6 of the height of the structure from the bottom.

REFERENCES

Chan HC, Kuang JS. 1989a. Stiffened coupled shear walls. Journal of Engineering Mechanics, ASCE 115(4):

689–703.

Chan HC, Kuang JS. 1989b. Elastic design charts for stiffened coupled shear walls. Journal of Structural

Engineering, ASCE 115(2): 247–267.

Choo BS, Li GQ. 1997. Structural analysis of multi-stiffened coupled shear walls on flexible foundations.

Computers and Structures 64(1–4): 837–848.

Coull A, Bensmail L. 1991. Stiffened coupled shear walls. Journal of Structural Engineering, ASCE 117(8):

2205–2223.

Coull A, Lao WHO. 1988. Outrigger braced structures subjected to equivalent static seismic loading. In

Proceedings of 4th International Conference on Tall Buildings, Hong Kong, 1988; 395–401.

Coull A, Lao WHO. 1989. Analysis of multi-outrigger-braced tall building structures. Journal of Structural

Engineering, ASCE 115(7): 1811–1816.

Moudares FR. 1984. Outrigger-braced coupled shear walls. Journal of Structural Engineering, ASCE 110(12):

2871–2890.

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

24

N. A. ZEIDABADI ET AL.

Rutenburg A, Eisenburg M. 1990. Stability of outrigger-braced tall building structures. In Proceedings of 5th

International Conference on Tall Buildings, Hong Kong, 1990; 881–892.

Straman J, Goldaf E. 1997. Outrigger braced structures in concerete. In Seventh International Conference on

Computing in Civil Engineering, Seoul, Korea, 1997; 933–938.

Stafford Smith B, Coull A. 1991. Tall Building Structures, Analysis and Design. Wiley: Chichester.

Taranath BS. 1988. Structural Analysis of Tall Buildings. McGraw Hill: London.

APPENDIX 1

K1 =

-g (u + w)

a 4 cosh aH

K2 = cosh ahs - Sm sinh ahs

K3 = sinh ahs - SmaH cosh ahs

K 4 = - Sm

g

a2

C2¢ =

ÈuH ( H h ) + w Ê H 2 - h 2 - 2 ˆ + PH ˘

- s

s

ÍÎ

˙˚

2Ë

a2 ¯

g

a2

ÈuH + Ê wH ˆ + P - Ê w ˆ ˘

ÍÎ

Ë 2 ¯

Ë a 2 H ¯ ˙˚

C1¢ =

B2¢ tanhahs - z 3

z1

B1¢ = K1 - C1¢ tanhaH

B2¢ =

z 1 (K1 K2 - K 4 - C2¢ sinh ahs ) + z 1z 3

z 1 cosh ahs + z 2 tanh ahs

z 1 = 1 - tanh ahs tanh aH

z 2 = K2 tanh aH - K3

z 3 = K1 tanh ahs - C2

APPENDIX 2

gl

l

C1l

d 1 = - ÈÍÊ 1 - 2 ˆ Mh hs + ( B1 - B2 ) sinh ahs +

cosh ahs ˘˙

Ë

¯

a

a

Î

˚

a

1

gl

l

lB2 C1l

d 2 = - ÈÍ Ê 1 - 2 ˆ Mh hs2 + ( B1 - B2 ) 2 cosh ahs + 2 + 2 sinh ahs + d 1hs ˘˙

Ë

¯

2

Î

˚

a

a

a

a

d3 =

Copyright © 2004 John Wiley & Sons, Ltd.

lB2

a2

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

OPTIMIZED USE OF OUTRIGGER SYSTEM

S( x ) =

25

u( x 4 - 4 Hx 3 + 6 H 2 x 2 ) w( x 5 - 10 H 2 x 3 + 20 H 3 x 2 ) p( x 3 - 3 Hx 2 )

+

24

120 H

6

F( x ) = [( B1¢ - B2¢ ) sinh ahs + (C1¢ - C2¢ ) cosh ahs ]( x - hs )

G(x ) = B1¢(cosh ahs - cosh ax ) + C1¢(sinh ahs - sinh ax ) + B2¢ (1 - cosh ahs ) + C2¢ (ax - sinh ahs )

Z ( x ) = B2¢ (1 - cosh ax ) + C2¢ (ax - sinh ax )

APPENDIX 3

* - y*MH

y*H = ylH

y*MH = ya Mh*

q1* = q1*l - q1*M

q2* = q2*l - q2*M

q *M1 = q1a Mh*

q *M 2 = q2 a Mh*

where

q1a = -

K

(b1¢ sinh Kx + c1¢ cosh Kx )

Sd

q2 a = -

K

(b2¢ sinh Kx )

Sd

b1

c1

ya = - ÈÍ 2 cosh K + 2 sinh K + N1 + N2 ˘˙

ÎK

˚

K

sinh Kxs c1

N1 = - ÈÍ(1 - Sd )xs + (b1 - b2 )

+ cosh Kxs ˘˙

K

K

Î

˚

N2 = (1 - Sd )

xs2 (b1 - b2 )

c1

+

sinh Kxs + cosh Kxs

2

K

K2

Ê- 1 + S ˆ

d

Ë d

¯

c1 =

cosh Kxs

- K2 tanh K + K3 + cosh Kxs + cosh Kxs tanh Kxs

tanh Kxs

b1 = c1 tanh K

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

26

N. A. ZEIDABADI ET AL.

1

ˆ

b2 = c1 Ê

Ë tanh Kxs - tanh K ¯

Mh* =

h1

h2

3

È 1 (1 - xs )

xs ˘

xs

h1 = b2¢ sinh Kxs + c2¢ (cosh Kxs - 1) + Sd Í + 2 ˙ + (1 - Sd ) L p + d ¢ - Sd Ê 2 ˆ

Ë

¯

6

6

K

K

Î

˚

h2 = (1 - Sd )xs -

b2

b2

sinh Kxs + Kxs + y Ê sinh Kxs + Sd xs ˆ + w

Ë

¯

K

K

Lp =

1 3

(xs - 3xs2 + 3xs )

6

d ¢ = - b2¢ sinh Kxs + c2¢ (1 - cosh Kxs )

b2¢ =

z 1 D1 + z 2 D2

z 1 cosh Kxs + z 2 tanh Kxs

c2¢ =

D1 =

Sd

K2

(- Sm Sd )(1 - xs ) (- Sd )

- Sdz 2

+

sinh Kxs

K

K cosh K

K2

3

D2 =

- Sd tanh Kxs Sd

- 2

K 3 cosh K

K

K = aH

x=

x

H

xs =

hs

H

The values of y*1H, q1* and T1* are presented in Moudares (1984).

NOTATION

A

B1, B2, B1¢, B2¢

b

c

C1, C2, C1¢, C2¢

Cross-section area of each wall

Integration constants

Clear span length of coupling beam

Distance between centroidal axis and edge of each wall

Integration constants

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

OPTIMIZED USE OF OUTRIGGER SYSTEM

d

E

E0, Es

F

H

h

hs

I1

I

Ib, Is, I0

l

Me

Mel

Mh

p

q

qil, qiM

Sm

T

Til, TiM

u

Vs

w

x

y

yH

a, g

l

w, y

27

Distance between the columns

Elastic modulus of walls and coupling beams

Elastic moduli of outrigger and stiffening beam, respectively

Axial resistant force of columns

Total height of structure

Height of story

Location of outrigger and stiffening beam from bottom

Second moment of area of each wall

Total second moment of area of walls equal to 2I1

Second moment of area of connecting beams, stiffening beam and outrigger

Distance between centroidal axes of walls

Applied moment

Applied moment due to external load

Resistant moment caused by outrigger

Concentrated load at top of structure

Laminar shear in equivalent medium

Laminar shear in section i due to external loads and outrigger, respectively

Relative flexural rigidity of stiffening beam

Axial force of each wall

Axial force in each wall caused by external loads and outrigger, respctively

Intensity of uniformly distributed load

Shear force of stiffening beam

Maximum intensity of triangular distributed load

Height coordinate

Lateral deflection of walls

Lateral deflection of walls at top level

Structural parameter

Clear length of outrigger

Dimensionless parameters of structure

Copyright © 2004 John Wiley & Sons, Ltd.

Struct. Design Tall Spec. Build. 13, 9–27 (2004)

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