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.
Trang 1OPTIMIZED USE OF THE OUTRIGGER SYSTEM TO STIFFEN THE COUPLED SHEAR WALLS IN TALL BUILDINGS
NAVAB ASSADI ZEIDABADI 1
, KAMAL MIRTALAE 1
* AND BARZIN MOBASHER 2
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 struc-ture 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.
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 out-rigger, 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 sig-nificantly 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)
Published online in Wiley Interscience (www.interscience.wiley.com) DOI:10.1002/tal.228
Accepted November 2002
* Correspondence to: Dr Kamal Mirtalae, Arizona Department of Transportation, Bridge Design Group, Mail Drop #631E, 205 South 17th Avenue, Phoenix, AZ 85007, USA
Trang 2showed 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 Fur-thermore, the best locations of the outrigger to minimize top drift or laminar shear and the effective parameters on the location are presented
Consider a coupled structural wall system in a fixed foundation stiffened by an outrigger and a beam
at level h sshown 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 stiff-ening 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:
(1)
(2)
l dy dx
hb
x
0
12
2
0
l dy dx
hb
s
0
12
2
0
Peripheral columns Coupled Shear Walls and Outrigger
Figure 1 Simplified plan of building
Trang 3where 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 h s , respectively and I b , 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
(3)
in which I = 2I1, where I1is second moment of area of each wall, and the axial forces in the walls in different sections are given respectively by
(4) (5)
where V srepresents 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
(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
dx
=
h
H
s x h
s
s
x
H
1=Ú 1
Ï Ì Ô Ó
Ô Ô
2 1
2 2
for for
hs
H
1
q
2
q
x
h
l
b
Figure 2 (a) Coupled shear walls stiffened by outrigger and internal beam (b) Substitute structure
Trang 4(8) where
(9)
(10)
To obtain the shear force in stiffening beam V s, consider the compatibility condition at its contra-flexure point:
(11)
in which E s I sis the flexural rigidity of the stiffening beam
Equating the corresponding terms of Equations (1) or (2) and Equation (11) at level h sgives the shear force of the stiffening beam thus; the shear forces will be
(12)
where q 1s and q 2s are the shear flows at level h s and S mis the relative flexural rigidity of the stiffening beam, defined as
(13)
In this investigation, the influence of the outrigger is considered as an unknown moment M hin the
location of the outrigger Moment M hcan also be represented by
(14)
The parameters F, c, l and are shown in Figure 2 By considering M h , the moment M ein Equations (7) and (8) can be given by
(15)
Therefore the complete solution of Equations (7) and (8) is
e
Ï Ì Ó
0
0
for for
s
M h=F c(2 +2 +l)
H
E I EI m
S S
b
=
V s=S Hq m 1s=S Hq m 2s
l dy dx
V b
S
s s
h s
2 0
13
2
0
g = 6I l3
hb I b
a2=gÊËl+ I ˆ¯
Al
d T
2 2 2 2 2
-a = -g
d T
2 1 2 2 1
-a = -g
Trang 5(16) (17)
The expression for laminar shear above and below the outrigger can be derived by using Equation (6):
(18) (19) For the external loads considered, the applied bending moment can be represented by
(20)
where P is considered the load at the top of the walls, and u and w are intensities of uniformly
dis-tributed and upper triangular disdis-tributed loads acting in the walls, respectively Thus, the complete solution of Equations (7) and (8) due to lateral loads is represented in following equations:
(21)
(22)
Consequently, the laminar shears are
(23)
(24)
The values of B1, B2, C1and C2can be determined by considering a set of boundary conditions
•At the top of the structure, x = H:
(25a)
•At the level of stiffening beam h s, boundary conditions are
(25b) (25c)
q h1( ) =s q h2( )s
T h1( ) +s V s+F i=T h2( )s
T H1( ) =0
dx
w H l
el
ÎÍ
˘
˚˙
dx
w H l
el
ÎÍ
˘
˚˙
H
2 = ¢2cosha + ¢2sinha + g2ÊË + 2 + 2 ˆ¯
H
1 = ¢1cosha + ¢1sinha + g2ÊË + 2+ 2 ˆ¯
q2M = -[B2asinhax+C2acoshax]
q1M = -[B1asinhax+C1acoshax]
T2M=B2coshax+C2sinhax+ g2M h
a
T1M=B1coshax+C1sinhax
Trang 6•At the base level, the laminar shear is given by
(25d) Solving Equations (25a)–(25d) gives the unknown integration constants:
(26)
(27) (28)
(29)
Likewise, the values of integration constants B¢1, B¢2, C¢1and C¢2can be determined The only alter-ation in boundary conditions which should be made is that Equalter-ation (25b) will be changed into the following equation:
(30)
Expressions for the constants B1¢, B2¢, C1¢ and C2¢ are given in Appendix 1
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
(31a) (31b) (31c) (31d) The lateral deflections due to the outrigger can be given as follows:
(32)
(33)
y EI
l
2
1 1
ÎÍ
˘
˚˙
g
y EI
B l
M
1
1 2
1
1
ÎÍ
˘
˚˙
a cosha a sinha d d
¢( ) = ¢ ( )
y h1 s y h2 s
y h1( ) =s y h2( )s
¢ ( ) =
y2( ) =0 0
T h1l( ) +s V s=T h2l( )s
M
h
s
s
1
2
1
=
-+ + +
Ê
Ê
l
l a
tanh cosh tanh
C2=0
1
tanha tanha
B1= -C1tanhaH
q2( ) =0 0
Trang 7The lateral deflections caused by external loads are
(34)
(35)
The values of K2, K3, d1, d2, d3, S(x), F(x), G(x) and Z(x) are given in Appendix 2.
In the aforementioned equations, all parameters related to external loads are determined The
para-meters for the outrigger are also known, provided the moment due to outrigger, M his determined
Moment M hcan 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
(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 rota-tions 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 M hcan be determined as follows:
(37)
in which
(38)
(39)
¢¢ = ( )
B
1
l
H
s s
+ È-( - ) + +
Î
˚
˙ + ÊË + + ˆ¯+ ÊË - ˆ¯
Ï Ì Ó
¸
˝
˛
2
2
1
g
M
l
l
h
b
s s
=
-Ê
Ë ˆ¯ ¢( ) + ¢( ) - ÊË +
ˆ
¯+
È ÎÍ
˘
˚˙
-Ê
Ë ˆ¯
-¢¢
È ÎÍ
˘
˚˙+ + +
¢¢
+
Ê
1
2 1
2 2
2
2
2
g
g
g
g a
Ï Ó
¸
˛
¢ ( ) - ¢ ( ) =
Ê
Ë ˆ¯
+
Ê
Ë ˆ¯
Ï Ì Ô Ó Ô
¸
˝ Ô
˛
Ô +
Ú
M
EA
M h
h s
c
h s
3
0 0
0
3
l
l
y EI
l
H l
2 2 3 2
1
= ÈÊË - ˆ¯ ( ) + ( ) - ÊË + ˆ¯ ÎÍ
˘
˚˙
g
g
y EI
l
H l
2 2 3 2
1
= ÈÊË - ˆ¯ ( ) + ( ) + ( ) - ÊË + ˆ¯ ÎÍ
˘
˚˙
g
g
Trang 8(41)
(41)
Having the outrigger moment M h , the value of B1, B2and C1 can be determined by using Equations (26) through (29) The deflections and internal forces of the structure are given by
(42) (43) (44)
In tall building structures, one of the most important features that should be considered is the top drift,
therefore instead of y(x), Y His used in investigations Consequently Equation (42) can be simplified by
(45)
in which y lH and y MHare top drifts due to external loads and the outrigger respectively
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 com-parison is shown in Table 1 According to the table the results are very close
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:
(46)
(47)
T0=uH g2T0
a2 *
y H=y lH-y MH
T x( ) = ( ) -T x l T M( )x
q x( ) = ( ) -q x l q M( )x
y x( ) = ( ) -y x l y M( )x
dx
( ) = ( )
¢( ) = ( )
dx
a l
=
+
EI
3 0 0
3
y l
= +
EI
k l
= +
EI
Trang 9where q*, T0* and y H* 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
EI y
4
*
Table 1 Contrasting the solutions gained by the presented method with those determined by the wide column
method (equal frame)
Wide column Presented
(m) * 10 -2
q *
0.0 0.2 0.4 0.6 0.8 1.0
Sm= 0
Sm= 1 2 5
1 0
a H = 3
Figure 3 Variation of laminar shear with height in a structure with internal beam but without outrigger
Trang 10values of S m 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 S mand k* enhances the stiffness of the structure, as
it is obvious from the figures that by increasing S mand k* the curves become nearer Thus, it is
sug-q*
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.0 0.2 0.4 0.6 0.8 1.0
Sm=10, 5, 2, 1, 0
k * = y * = 20
a H = 3
hs/H =.5
w = 05
q*max 0.10 0.15 0.20 0.25 0.30 0.35 0.40
hs
0.0 0.2 0.4 0.6 0.8 1.0
Sm= 10, 5, 2, 1, 0
a H = 3
w = 05
Figure 4 Variation of laminar shear with height in a structure with outrigger
Figure 5 Effect of outrigger location on maximum laminar shear
Trang 11gested that S mand 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 illus-trated 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
100 Y*
hs
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Sm= 10, 5, 2, 1, 0
k* = y* = 20
a H = 3
Sd= 9
w = 05
100Y*
hs
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.0
a H = 3
w = 05
k* = 100, 50, 20, 10, 5, 1 Figure 6 Effect of outrigger location on top drift for different relative flexural rigidities of the internal beam
Figure 7 Effect of relative axial rigidity of the columns on top drift for different locations of the outrigger