Shear wall with outrigger trusses on wall and column foundations (p 73 87)

A graphical method of analysis is presented for preliminary design of outrigger trussbraced highrise shear wall structures with nonfixed foundation conditions subject to horizontal loading. The method requires the calculation of six structural parameters: bending stiffness for the shear wall, bending and racking shear stiffnesses for the outrigger, an overall bending stiffness contribution from the exterior columns, and rotational stiffnesses for the shear wall and column foundations. The method of analysis employs a simple procedure for obtaining the optimum location of the outrigger up the height of the structure and a rapid assessment of the influence of the individual structural elements on the lateral deflections and bending moments of the highrise structure. It is concluded that all six stiffnesses should be included in the preliminary analysis of a proposed tall building structure as the optimum location of the outrigger as well as the reductions in horizontal deformations and internal forces in the structure can be significantly influenced by all the structural components. Copyright © 2004 John Wiley Sons, Ltd.

SHEAR WALL WITH OUTRIGGER TRUSSES ON WALL AND

COLUMN FOUNDATIONS

J C D HOENDERKAMP

Department of Architecture and Building, Eindhoven University of Technology, Eindhoven, The Netherlands

SUMMARY

A graphical method of analysis is presented for preliminary design of outrigger truss-braced high-rise shear wall structures with non-fixed foundation conditions subject to horizontal loading The method requires the calcula-tion of six structural parameters: bending stiffness for the shear wall, bending and racking shear stiffnesses for the outrigger, an overall bending stiffness contribution from the exterior columns, and rotational stiffnesses for the shear wall and column foundations.

The method of analysis employs a simple procedure for obtaining the optimum location of the outrigger up the height of the structure and a rapid assessment of the influence of the individual structural elements on the lateral deflections and bending moments of the high-rise structure It is concluded that all six stiffnesses should

be included in the preliminary analysis of a proposed tall building structure as the optimum location of the out-rigger as well as the reductions in horizontal deformations and internal forces in the structure can be significantly influenced by all the structural components Copyright © 2004 John Wiley & Sons, Ltd.

1 INTRODUCTION The outrigger-braced high-rise shear wall shown in Figure 1 consists of a centrally located wall with

two equal-length trusses positioned at a distance x from the top of the structure The outriggers connect

the shear wall to columns in the façade of the structure This way the horizontally applied load will force the wall to behave compositely with the exterior structure by introducing axial forces in the columns These forces form a restraining moment which is in the oposite direction to the bending moment from the horizontal load This effect will decrease the bending moments in the wall from out-rigger level down to the base and it will reduce the horizontal deflections of the structure The bending moments in the shear wall and axial forces in the columns are resisted by foundation structures on piles which are indicated by translational springs Combined with the laterally applied loading the restraining moment will force a triple curvature in the wall up the height and double curvature in the trusses as shown in Figure 1

In the horizontal deflection analysis of outrigger-braced shear walls on fixed foundations it has been shown (Stafford Smith and Coull, 1991; Stafford Smith and Salim, 1981) that the wall can be repre-sented by a single flexural stiffness parameter The outriggers were assumed to be prismatic members rigidly connected to the wall and hinge connected to the exterior columns The resulting single cur-vature behaviour of these flexural members was represented by a single bending stiffness parameter Published online in Wiley Interscience (www.interscience.wiley.com) DOI:10.1002/tal.235

Copyright © 2004 John Wiley & Sons, Ltd Received January 2003

* Correspondence to: J C D Hoenderkamp, Department of Architecture and Building, Structural Design Group, Eindhoven University of Technology, P.O Box 513, Route 7 5600 MB Eindhoven, The Netherlands.

E-mail: j.c.d.hoenderkamp@bwk.tue.nl

It was further taken that the columns are pin connected to a fixed foundation and could thus be rep-resented by a parameter which represents the axial stiffnesses of the columns only With three stiff-ness parameters representing the wall, outriggers and exterior columns it was possible to combine them in a single dimensionless parameter which allowed a graphical procedure to obtain the optimum location of the outriggers such that they would cause the largest reduction in horizontal deflection at the top of the structure

Figure 1 also shows simplified models for the foundations of the central shear wall and exterior columns In the structural analysis the individual elements are only subjected to forces resulting from horizontal loading on the structure The foundations under the pin-connected façade columns are taken

to be piled foundations which act in a vertical direction only They are modelled as linear springs The shear wall foundation is only subjected to a bending moment The net axial load on this foundation

as a result of lateral loading is zero as it is positioned in the centre of a symmetric structure The foun-dation of the wall can thus be modelled by a rotational spring with a rotational spring constant The outriggers consist of trusses of which the deformations due to shear cannot be ignored The shear stiff-ness of the trusses in addition to their flexural stiffstiff-ness must be included in the analysis In order to allow the truss to be subjected to flexural deformations it will be necessary to assume the structural floors of the building to be connected to the trusses at the exterior columns and shear wall locations only; i.e the floor structure is not part of the outrigger structure but it will cause the shear wall and exterior columns to have identical rotations at all floor levels Further simplifying assumptions are: the structure behaves linear elastically; the sectional properties of the shear wall, exterior columns and outriggers are uniform througout their height or length; and the distribution of the lateral loading is uniformly distributed along the height of the structure

A compatibility equation is to be developed for the rotations in the wall and outrigger at the inter-section of the neutral line of the outriggers and the face of the shear wall This leads to an expression

x

H

h

column wall column

outrigger outrigger

deflected shape

Figure 1 Shear wall with outrigger trusses

for the restraining moment allowing the reduction in horizontal deflection at the top to be determined Maximizing this reduction will yield the optimum location of the outrigger

2 SHEAR WALL

The rotations in the shear wall are the result of a uniformly distributed horizontal load w, and a restrain-ing moment M caused by a reverse action of the outriggers They will cause rotations at outrigger

level due to bending in the wall and rotation of the wall foundation as presented in Figure 2 For the shear walls it will be assumed that plane sections remain plane in bending so that the rotations at the shear wall centre line are identical to those at the face of the wall

2.1 Rotations due to lateral loading w

The rotation in the reinforced concrete shear wall at level x can quite simply be expressed as follows:

(1)

where H is the total height of the structure, x is the distance measured from the top of the stucture and

EIsis the flexural stiffness of the shear wall The rotation of the wall due to the rotation of its foun-dation is

(2)

in which Csis the rotational stiffness of the shear wall foundation

2.2 Rotations due to restraining moment M

The rotation in the shear wall at level x as a result of bending in the wall is given by

(3)

qs b M

s

M H x EI

qs C w

s

s

wH C

2

2

qs b w

s

w H x EI

6

qs;b;w qs;b;M qs;Cs;w qs;Cs;M

EIs EIs

x

H

Figure 2 Rotations in shear wall

The rotation of the shear wall and its foundation are

(4)

It is noted here that the rotations caused by restraining moment M are in the opposite direction to those

by horizontal load w.

3 OUTRIGGER The rotations in the outrigger due to the restraining moment are obtained by splitting up this action

as shown in Figure 3, where the outriggers have been separated from the braced frame for clarity The restraining moment in the structure due to outrigger action is the product of the axial force in the exte-rior columns and the distance between them:

(5)

in which F a is the restraining force in the exterior columns, l is the distance from the exterior column

to the centre line of the shear wall and b is the length of the flexible outrigger measured from the

façade column to the outrigger/shear wall interface Considering the free body diagram of a single outrigger, then

(6)

where F r represents the complementary shear forces in the outriggers and h is the vertical dimension

of the outrigger The free body diagram in the centre of Figure 3 allows an expression for the restrain-ing moment on the shear wall to be written as

(7)

M=2F c a +2F h r

F a¥b=F r¥h

M=F a(2b+2c) =F a( )2l

qs C M

s

s

M C

Fa

b c c b

h

outrigger outrigger

shear wall

H

x

Figure 3 Free body diagram restraining forces

Substituting Equations (6) and (7) into Equation (5) will lead to an expression for the effective restrain-ing moment causrestrain-ing bendrestrain-ing and shear deformations in the outriggers as shown in Figure 4

(8) where a dimensionless parameter

(9)

3.1 Rotations due to restraining force F r

The double curvature bending in the outrigger as shown in Figure 4(a) is caused by axial strain in the top and bottom cords The rotations in the truss at both ends are identical because the structural floors are assumed to be connected to the outrigger at the columns and shear wall only The rotation due to bending at the outrigger/wall interface is given by

(10)

where EI ris the bending stiffness of the outrigger, which can be obtained as follows:

(11)

where A ris the cross-sectional area of the top and bottom chords of the rigger

The rotation due to racking shear results from strain in the diagonals as is indicated in Figure 4(b) and can be expressed as

EI r=EA h r 2

2

q

a

r b F

r

r

F h

EI b

Mb EI

; ; =

Â

2

a =l

b

2F h r = M a

b

a

h

Fr/2

Fr/2

Fr/2

Fr/2

Fr

Fr

(a)

(b)

qr;b;Fr

qr;s;Fr

Figure 4 Bending and shear deformations in truss

It has been shown earlier (Hoenderkamp and Snijder, 2000) that the racking shear stiffness of a rigger,

GA r, is the sum of the individual racking shear stiffnesses of all the bracing segments in that rigger

So for the outrigger structure

(13)

where n represents the total number of segments in the two outriggers and GA iis the racking shear

stiffness of a single segment of width a For several types of bracing systems the racking shear

stiff-ness is given Appendix A

3.2 Rigid body rotations of outrigger

The wide-column behaviour of the shear wall due to bending caused by horizontal load w will result

in rigid body rotations of the outriggers as shown in Figure 5 This reverse rotation is given by

(14)

Substituting Equation (1) into Equation (14) yields

(15)

An additional reverse rotation in the trusses occurs due to the rotation of the wall foundation when subjected to horizontal loading and is shown in Figure 6 This rigid body rotation for the outrigger can be expressed as follows:

(16)

s C w

s C w

s

s

s

b

c b

; ;

; ;

; ;

= -ÊË ˆ¯= - ( )

qr b w

s

w H x EI

c b

Ì Ó

¸

˝

˛

6

s b w

r b w

b

c b

; ;

; ;

; ;

= -ÊË ˆ¯= - ( )

GA r GA i i

n

=

=

 1

q

a

r s F

r

r

F h

h GA

M

h GA

Â

2

d s;b;w

d s;b;w

qs;b;w

c

c

b

b

Figure 5 Rigid body rotation of outriggers due to bending in shear wall

substituting Equation (2) into Equation (16) yields

(17)

Restraining moment M will also cause rigid body rotations of the outriggers The shear wall will deflect

in the opposite direction as shown in Figures 5 and 6 The rigid body rotation of the outriggers will now be in the clockwise direction For bending in the wall:

(18) Substituting Equation (3) into Equation (18) yields

(19) For rotation of the shear wall foundation:

(20) Substituting Equation (4) into Equation (20) yields

(21)

3.3 Rotations due to restraining force F a

The restraining forces in the exterior columns will cause two more rigid body rotations of the outrig-gers: one resulting from shortening and lengthening of the exterior columns and another due to ver-tical displacements in the column foundations

qr C M

s

s

M C

c b

; ; =ÏÌ ¸˝

s C M

r C M

s

s

s

b

c b

; ;

; ;

; ;

= -ÊË ˆ¯= - ( )

qr b M

s

M H x EI

c b

; ; =ÏÌ ( - )¸˝

s b M

s b M

b

c b

; ;

; ;

; ;

= -ÊË ˆ¯= - ( )

qr C w

s

s

wH C

c b

; ; = -ÏÌ Ó

¸

˝

˛

2

2

ds;Cs;w

ds;Cs;w

qs;Cs;w

c

c

b

b

Figure 6 Rigid body rotation of outriggers due to rotation of wall foundation

The first outrigger rotation can be defined as the column change in length divided by the length of the outrigger This leads to

(22)

where A cis the cross-sectional area of the exterior column

Defining an overall stiffness parameter for the exterior column structure to be

(23) and substituting Equations (5) and (23) into Equation (22) will yield the following expression:

(24)

The second outrigger rotation can be defined as the foundation displacement divided by the length of the outrigger, which yields

(25)

where k represents the translational stiffness of the column foundation Defining an overall rotational

stiffness parameter for the combined column foundations as

(26) and substituting Equation (26) into Equation (25) leads to the following simplified expression:

(27)

4 HORIZONTAL DEFLECTION

Compatibility in rotation at level x requires that the rotations in the shear wall, q s, and outrigger,

qr, at their interface are identical Substituting for the individual rotations yields the following expression:

(28)

Simplifying leads to

(29)

w H x

EI

wH

C M

H x EI

H x EI

b

EI h GA C C

w H x

EI

wH C

M H x EI

M C

w H x EI

c b

wH C

c b Mb

EI

M

h GA

M H x EI

c b

M C

c b

M H x EI

24

Ì Ó

¸

˝

-Ï Ì Ó

¸

˝

˛

C c

qr C F a

c

c a

M C

C c= 2l2k

a

c a

F k b

M k

2l2

c

a

M H x EI

EI c= 2EA cl2

qr a F

a

c

a

F H x bEA

Setting two characteristic flexibility parameters as

(30)

and

(31)

leads to the following expression for the restraining moment:

(32)

The horizontal deflection at the top of the building can be now obtained as follows:

(33)

where the first two terms on the right-hand side represent the ‘free’ horizontal deflections of the shear wall at the top due to bending of the wall and rotation of its foundation as a direct result of lateral

loading The third term is due to the restraining moment M which causes a reverse deflection and

rota-tion at outrigger level, resulting in a deflecrota-tion reducrota-tion at the top The last term represents a hori-zontal deflection at the top of the wall due to reverse foundation rotation

5 OPTIMUM LOCATION OF OUTRIGGER

The reduction in horizontal deflection at the top of the structure due to restraining moment M is

represented by the last two terms on the right-hand side of Equation (33) and can be expressed as

(34)

This reduction is maximized by differentiating it w.r.t x, setting it equal to zero and solving for x.

After simplifying this leads to

(35)

in which two characteristic non-dimensional parameters for the outrigger-braced shear wall structure have been set as follows:

(36)

g H HC

EI

s

s

=

d

dx x x x

x x

2

Ï Ì Ó

¸

˝

Ï

È Î

˚

˙ =

y M H x

EI

H C

red

= ÏÌ - + ¸˝

2

y wH EI

wH C

M H x EI

MH C

top

M w H x

EI

wH C

H

H x S HS

+ Ï

Ì Ó

¸

˝

˛ ( - ) +

Ï Ó

¸

˛

S b

EI r h GA r C s C c

24

S H EI

H EI

(37)

A dimensionless location parameter for the optimum position of the outrigger has been set as

(38)

Figure 7 shows a graphical presentation of optimum outrigger locations up the height of a shear wall braced structure on wall and column foundations as a function of two non-dimensional characteristic parameters, gH and w

6 EXAMPLE The structural floor plan of a 29-storey, 87 m high building in Figure 8 shows the arrangement of four identical shear walls with horizontal steel trusses on both sides Each truss is 9 m long, has a single storey height of 3 m and comprises five X-braced segments The building is subjected to a uniformly distributed lateral load of 1·6 kN/m2

The flexural stiffness of the reinforced concrete shear wall EI s= 1·5 ¥ 109

kNm2

For the horizontal trusses: top and bottom chords A r= 1·78 ¥ 10-2m2

, diagonals A d= 9·726 ¥ 10-3m2

and for the exterior columns A c= 3·12 ¥ 10-2m2

The elastic modulus of steel E= 2·1

¥ 108

kN/m2

The rotational stiffness of the shear wall foundation C s= 2·0 ¥ 108

kNm and the

trans-lational stiffness of the column foundations k= 4·0 ¥ 105

kN/m

The detailed calculations are shown here for a single shear wall with one outrigger on both sides The flexural stiffness of the outrigger structure is given by Equation (11):

x x H

=

w =S

S

2 1

0.2

0.3

0.4

0.5

0.6

0.7

Values of w

> 100 10 6 4 3 2 Values of g H

Figure 7 Optimum location of outrigger