Torsional Analysis

Copyright 2010 John Wiley & Sons, Ltd.

Torsional analysis of multi-storey building structures under horizontal load

K. A. Zalka*,

Department of Mechanics and Structures, Szent Istvn University, Budapest, Hungary

SUMMARY

New closed-form formulae are presented for the torsional analysis of asymmetrical multi-storey buildings braced by moment-resisting (and/or braced) frames, (coupled) shear walls and cores. The analysis is based on an analogy between the bending and torsion of structural systems. A closed-form solution is presented for the rotation of the building. The torsional behaviour is defi ned by three distinctive phenomena: warping torsion, Saint-Venant torsion and the interaction between the two basic modes. Accordingly, the formula for the maximum rotation of the building consists of three parts: the warping rotation is characterized by the warping stiffness of the bracing system, St Venant rotation is associated with the St Venant stiffness of the building and the third part is responsible for the interaction. It is demonstrated that the interaction between the warping and St Venant modes is always benefi cial, as it reduces the rotation of the structure. It is shown how the proposed formula for torsion can be used for the determination of the maximum defl ec-tion of multi-storey asymmetrical building structures. The results of a comprehensive accuracy analysis demonstrate the validity of the method. A worked example is given to show the ease of use of the proce-dure. Copyright 2010 John Wiley & Sons, Ltd.

1. INTRODUCTION

The torsional analysis of multi-storey building structures braced by frameworks, (coupled) shear walls and cores, subjected to lateral load represents a formidable task. Even the defl ection analysis of a single frameworka representative bracing elementleads to a complex problem. The main diffi -culty is caused by the fact that the framework develops both bending and shear deformations. This problem was recognized as early as in the 1940s, when L. Chitty (1947) published an excellent paper on parallel bars interconnected by crossbars. However, her solution was quite lengthy and fairly complicated for structural engineers to use for design purposes. Several attempts were subsequently made to produce simpler solutions (C sonka, 1950; B eck, 1956). In parallel with these developments, considerable attention was also paid to systems of frameworks and shear walls (R osman, 1960; D espeyroux, 1972). When several frameworks and (coupled) shear walls are put together to create a bracing system, the situation becomes even more diffi cult, even if the arrangement is symmetrical, mainly because of the interaction among the elements of the bracing system. Simplifi ed models were also introduced to enable easier treatment of the complex problem, which, occasionally produced methods of very limited range of application. K han and Sbarounis (1964), for example, showed that even for buildings whose primary lateral-resisting system consists of shear walls, the use of pure fl exural model was not appropriate. The combined fl exural and shear deformation in frame buildings was investigated by Blume (1968), who introduced a dimensionless parameter to monitor the sensi-tivity of the system to bending (or shear) deformation. Mir anda (199 9) developed an approximate method to estimate the maximum lateral deformation demands in multi-storey buildings responding primarily in the fundamental mode when subjected to earthquake ground motions. However, these (and a number of other) investigations only centred on the two-dimensional (lateral) problem.

THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGS

Published online 2 December 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/tal.665

* Correspondence to: Karoly A. Zalka, Department of Mechanics and Structures, Szent Istvn University, Budapest, Thkly t 74, Hungary, H-1146

E-mail: [email protected]

Struct. Design Tall Spec. Build. 22, 126143 (2013)


Because of the complexity of the torsional behaviour, not many authors deal with the problem. Considerable efforts have been made regarding the torsional behaviour of individual structural ele-ments (Council on Tall Buildings, 1978; Sea burg and Carter, 2003) but the global torsional behaviour of whole structural systems is a less cultivated area. There are some excellent publications that offer relatively simple solution for the global torsional problem (Council on Tall Buildings, 1978; I r w in, 1984; Co u ll and Wahab, 1993; H o e nderkamp, 1995; N a d jai and Johnson, 1998; H o w son and Rafezy, 2002; Schueller, 1990) but they are either still too complicated or of limited applicability, and neither of them is backed up with a comprehensive accuracy analysis.

To handle this three-dimensional problem in a simple way seems to be hopeless using conventional tools. However, by relying on an analogy between bending and torsion, a relatively simple solution can be produced. The aim of this paper is threefold: (a) to establish a new model for the analysis using this analogy; (b) to produce a simple closed-form solution for the rotation of a building that clearly shows the contribution of the different stiffness characteristics to the torsional resistance; and (c) to show how the proposed method can be used for the determination of the maximum defl ection of multi-storey asymmetrical building structures.

Although large frameworks and even whole buildings are now routinely analysed using computer packages, the proposed method may be useful from several aspects. It helps the structural engineer to understand the complex three-dimensional behaviour, and thus enables the manipulation of the stiffnesses and the location of the bracing units in such a way that optimum structural arrangement is achieved. The proposed method may also prove to be useful at the preliminary design stage when quick checks are needed with different structural arrangements. Its usefulness cannot be over-emphasized for checking the results of a fi nite element (computer-based) analysis when the input procedure involves tens of thousands of data and mishandling one datum may have catastrophic consequences.

The continuum method will be used for the analysis, and it will be assumed that

(1) The structures are at least four storeys high and the storey heights are identical.(2) The fl oor slabs of the building have great in-plane and small out-of-plane stiffness.(3) The structures are subjected to uniformly distributed lateral load.(4) The structures develop small deformations and their material is linearly elastic.

2. TORSIONAL BEHAVIOUR AND BASIC CHARACTERISTICS

As with thin-walled bars, multi-storey building structures react to torsion by utilizing their torsional resistance. As with thin-walled bars, the torsional resistance of multi-storey buildings originates from two sources. The warping stiffness is associated with the in-plane bending stiffness of the individual bracing units, which is activated by their moment arm (perpendicular distance) measured from the shear centre of the bracing system. This phenomenon is best demonstrated by the torsional behaviour of a single I-column on a fi xed base and with a free upper end, whose warping stiffness EI is cal-culated by multiplying the (in-plane) bending stiffness of its fl anges and the square of the perpendicu-lar distance of the fl anges from the shear centre of the column (Figure 1(a)):

EI EI h E tb h E tb hflange = = =2 2 12 4 2 242 3 2 3 2

(1)

where E is the modulus of elasticity of the material of the column. Point O marks the shear centre of the column and z-axis passing through the shear centre is the axis of rotation.

The St Venant torsional stiffness of the bracing system is associated with the in-plane shear stiff-ness of the bracing units, which is activated by its moment arm (perpendicular distance) measured from the shear centre of the bracing system. For its demonstration and using the same I-column as above, assume that the fl anges are pierced with big openings of rectangular shape (they are in fact frames). The St Venant torsional stiffness K is calculated by multiplying the shear stiffness of the

Struct. Design Tall Spec. Build. 22, 126143 (2013)DOI: 10.1002/tal

TORSIONAL ANALYSIS OF BUILDINGS 127


fl anges (i.e. the frames) and the square of the perpendicular distance of the fl anges from the shear centre of the column (Figure 1(b)):

K K h Kh = =2 2 22 2

(2)

where K is the shear stiffness of the fl anges. It is easy to see that in building structures, the fl oor slabs of the building (with their great in-plane stiffness) play the role of the web of the I-column in making the bracing elements (the fl anges) work together.

It is also clear that, apart from the distance of the bracing units from the shear centre of the build-ing, the bending and shear stiffnesses of the individual bracing units play an important role, so these characteristics are given fi rst. The continuum model is applied where the stiffness characteristics are considered continuously distributed over the height of the structure.

In the case of a framework (the most characteristic bracing unit) two types of bending stiffness are considered. The local bending stiffness of the framework is the sum of the bending stiffnesses of the columns:

EI E Ic c ii

n

=

=

,1

(3)

where Ic,i is the second moment of area of the ith column, n is the number of columns and E is the modulus of elasticity of the framework.

The global bending stiffness of a framework is defi ned by

EI E A tg c i ii

n

=

=

, 21

(4)

where Ac,i is cross-sectional area of the ith column and ti is the distance of the ith column from the centroid of the cross-sections.

M

z

b

h

t

M

z

b

h

)b)a

O O

Figure 1. Rotation of an I-column on a fi xed base. (a) with solid fl anges; and (b) with fl anges with openings.

K. A. ZALKA


128


The shear stiffness of the framework (distributed over the height) is given as

K K r K KK Kb b

c

c b= =

+ (5)

The above shear stiffness basically depends on two terms. Term Kb is the stiffness of the beams (distributed over the height):

K EIl hb

b i

ii

n

=

=

2 61

1,

(6)

where Ib,i is the second moment of area of the ith beam, h is the storey height and li is the bay between the ith and (i + 1)th columns. Term Kc represents the stiffness of the columns:

K EIhc

c=

122

(7)

where Ic is the second moment of area of the columns as in Equation (3). Finally,

rK

K Kc

c b=

+ (8)

in Equation (5) is a reduction factor.To avoid some overrepresentation of the columns of the framework (they appear in the formulae

of both the local bending stiffness and the shear stiffness), the local bending stiffness should be reduced by factor r (Hegedus and Kollr, 1999). Accordingly, instead of Equation (3), the following expression will be used as the local bending stiffness of the framework:

EI rEI rE Ic c ii

n

= =

=

,1

(9)

Shear walls are frequently used for bracing purposes. In terms of the above three stiffness charac-teristics, only their EI value is of fi nite magnitude and it corresponds to the local bending stiffness of a framework. A shear wall can be modelled for the analysis of a system consisting of shear walls and frameworks as a special framework whose resistance against shear defl ection and global bending defl ection is infi nitely great. In this way, the analysis can be carried out in a relatively simple way as all the bracing units are frameworks.

Coupled shear walls can also be considered special frameworks (with some difference), and the characteristics given above for frameworks above can also be used (with some modifi cation). The determination of the local and global bending stiffnesses is identical to those of frameworks. Because of their slightly different behaviour, instead of Equations (5) and (6), the following equation should be used for the determination of the shear stiffness:

K K r K K

K Kb b

c

c b

* *= =

+ (10)

In Equation (10) Kb* is the modifi ed beam stiffness as

K EI l s l s

l h EIl GA

bb i i i i

ib i

i b i

=

+ + +

+

+6

1 12

12

12

32

,

,

,

(( ) ( ) )

1

1n

(11)

where G is the modulus of elasticity of shear of the beams, Ab,i is the cross-sectional area of the ith beams, si, si+1 are the width of the ith and (i + 1)th wall section, li* is the distance between the ith and




(i + 1)th wall sections, is a constant depending on the shape of the cross section of the beams ( = 1.2 for rectangular cross sections).

In-fi lled frameworks and frameworks with cross-bracing can also be part of a bracing system. When their bending stiffness is calculated, their local bending stiffness is calculated directly as the sum of the bending stiffnesses of the columns, with r = 1, i.e. EI = EIc. The calculation of the global bending stiffness EIg is identical to that of the rigid frames, i.e. according to Equation (4). Their shear stiffness K should be determined according to the different types of bracing. Ready-to-use formulae are available in structural engineering monographs, e.g. in Stafford Smith and Coull (1991) and Zalka (2 0 00).

In addition to the stiffnesses of the bracing units, their distance from the shear centre is also needed.The location of the shear centre is defi ned as the centre of stiffnesses of the bracing units.

The stiffness of each bracing unit is defi ned as the reciprocal of the top (in-plane) defl ection of the unit:

Sy Hj j

= ( )1

(12)

With the stiffnesses of the units, the calculation of the location of the shear centre is carried out in the co-ordinate system x y, whose origin lies in the upper left corner of the plan of the building and whose axes are aligned with the sides of the building (Figure 2):

x

S x

Sy

S y

S

j jm

jm

j jm

jmo o

= =

1

1

1

1

, (13)

where xj and yj are the perpendicular distance of the jth bracing unit from x and y and m is the number of bracing units. For the calculation of the location of the shear centre, only the in-plane stiffness of the frameworks is taken into account. Once the location of the shear centre is determined, coordinate system x y has fulfi lled its role and the new coordinate system x y is established with its origin in the shear centre (Figure 2).

Detailed explanation of the phenomena discussed and the terms used in this section are to be found in Zalka (2009).

xc

O

x

y y

x

yo

xo = t1 = t tj

t2

EI1 = EIEIg,1 = EIgK1 = K EIj

EIg,j Kj

tm

EIm, EIg,m, Km

EI2, EIg,2, K2

C

w

xj

1

2

m

j

Figure 2. Plan arrangement of the bracing system.

K. A. ZALKA


130


3. TORSIONAL ANALYSIS

Knowing the stiffness characteristics of the individual bracing units as well as their perpendicular distance from the shear centre, it is now possible to carry out the torsional analysis of the bracing system of the building. The torsional analysis is based on an analogy well known in the stress analysis of thin-walled structures in bending and torsion (Vlasov, 1961; Kollbr u n n er and Basler, 1969). According to the analogy, translations, bending moments and shear forces correspond to rotation, warping moments and torsional moments, respectively. It follows from the analogy that the results of the defl ection analysis of a system of frameworks, (coupled) shear walls and cores can be used for the torsional analysis if the characteristic stiffnesses of the defl ection analysis are matched with stiffnesses that characterize the torsional problem. The governing differential equation of the defl ec-tion of the bracing system (Zalka, 2009 ) assumes the form

=

y y

w

EIaz

22

21 (14)

The solution to this differential equation is

y zw

E I IH z z wz

KswEIK s

H z Hg

( ) =+( )

+

( ) +3 4 22 2 36 24 2

cosh s iinhcosh

z

H

1 (15)

where

w wq= (16)

with w being the intensity of the horizontal load, q being an apportioner determining how much the base unit takes of the total horizontal load, and

2 1= + = +a b s ab

, (17)

aK

EI

a

c

a

b

b KEI

fa

b

aII

b

g

j

jj

f

j

jj

fj

jj

f

jj

jj

=

+

+

=

+

=

=

= =

+

1

1 1

2

2 2

1

,

, == =+ +KK

cII

j

jj

g j

g j

1 1, ,,

(18)

Maximum defl ection develops at z = H:

y y H wHE I I

wHKs

wEIK s

H HHg

maxsinh

cosh= ( ) =

+( ) + +

4 2

2 2 38 21 1

(19)

In the above equations, stiffnesses EI, EIg and K are those of the base unit, and f is the number of frameworks/coupled shear walls. The base unit is the bracing unit whose K/EI ratio is the greatest. The bracing units are numbered in such a way that the base unit is always given number 1 and then, for the sake of simplicity, the subscript 1 is dropped, hence the stiffness characteristics of the base unit are EI = EI1, EIg = EIg,1 and K = K1 (Figure 2). Details of the derivation and full explanation of the terms in the above equations are given in Zalka (2009).

All w e have to do now is to identify the torsional characteristics that are analogous to the bending (EI and EIg) and shear (K) stiffnesses with the defl ection analysis.

Stiffness EI is the local bending stiffness with the defl ection analysis. The corresponding stiffness with the torsional analysis is the local warping stiffness:




EI EIt = 2 (20)where t is the perpendicular distance of the base unit from the shear centre (Figure 2).

Stiffness EIg is the global bending stiffness with the defl ection analysis. The corresponding stiffness with the torsional analysis is the global warping stiffness:

EI EI tg g = 2 (21)

Stiffness K is the shear stiffness with the defl ection analysis. The corresponding stiffness with the torsional analysis is the warping shear stiffness:

K Kt = 2 (22)With the above analogous characteristics, the governing differential equation of torsion assumes

the form

=

22

21m

EIaz

(23)

The solution is given by

zm

E I IH z z mz

K smEIK s

H zg

( ) =+( )

+

(3 4 22 2 36 24 2

cosh )) +

H z

Hsinh

cosh1

(24)

Maximum rotation develops at z = H:

max

sinhcosh

= ( ) =+( ) +

+H mHE I I

mHK s

mEIK s

H HHg

4 2

2 2 38 21

1 (25)Instead of the lateral load on the base unit (w) in Equations (14) and (19), Equations (23)(25)

contain the torsional moment m that the base unit takes of the total torsional moment. Its value is determined as follows.

Each of the bracing units takes torsional moment according to their torsional stiffness. The torsional stiffness of the jth unit is defi ned as

S S t ty Hj j j

j

j , = = ( )

22

(26)

where tj is the perpendicular distance of the jth bracing unit from the shear centre and yj(H) is the (in-plane) top defl ection of the jth unit. Thus, the torsional apportioner related to the base unit assumes the form

q S

S jm

=

,1

(27)

where S is the torsional stiffness of the base unit

S ty H

t S = ( ) =2

2 (28)

and m is the number of bracing units. The torsional moment the base unit takes is, therefore,

m m q wx qt c= = (29)

where mt = wxc is the total torsional moment on the bracing system.

K. A. ZALKA


132


Equivalents of coeffi cients and s in Equations (14), (15) and (19) also have to be established for use in Equations (23)(25). Careful investigation of Equations (17) and (18) shows that if the torsional equivalentsstiffness (moment arm)2are substituted for the stiffnesses in Equations (15) and (19), the moment-arms drop out of the formulae. It follows that the coeffi cients defi ned by Equations (17) and (18) remain unchanged and should be used for the torsional analysis as well.

The evaluation of Equations (24) and (25) using the rotational data of 126 bracing systems ranging in height from 4 to 80 storeys (cf Validation of the continuum model later on) leads to the following observations:

(1) The torsional behaviour of the building can be separated into three distinctive parts. The bending stiffness of the individual bracing units (activated through rotation around the shear centre) is associated with warping torsionfi rst term in Equation (25). The shear stiffness of the bracing units (activated through rotation around the shear centre) results in pure, St Venant-type torsionsecond term in Equation (25). Because of the different (bending-type and shear-type) rotation shapes (Figure 1), there is an interaction between the two modes, defi ned by the third term in Equation (25). Figure 3 shows the characteristic types of rotation of a 40-storey building braced by frameworks.

(2) The effect of interaction between the warping and St Venant modes is always benefi cial as it reduces the rotation of the structure.

(3) The effect of interaction signifi cantly becomes smaller as the height of the structures increases.(4) The effect of interaction is roughly constant over the height of the structure (Figure 3(c)).To conclude the investigation of the torsional behaviour, some special cases will now be considered

as their analysis leads to extremely simple solutions in many practical cases.Case A: The horizontal elements of the bracing system (including connecting beams in the frame-

works and the fl oor slabs) have negligibly small bending stiffness.This case is characterized by Kb 0 (for the frameworks). Consequently, the shear stiffness of the

system tends to zero (K 0), which leads to a 0 and b 0 and 0. The governing differential Equation (23) simplifi es to

=

m

EIt

(30)

and the solutions for the rotation and the top rotation assume the form

z/H

b) c)a)

d)

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

1z/H

z/H

z/H

-

Figure 3. Typical rotation shapes. (a) warping; (b) St Venant; (c) interaction; and (d) combined.




=

m

EIH z zt 3 4

6 24 (31)

and

max = ( ) =H m HEIt

4

8 (32)

where EI is the local warping stiffnesses. This case is identifi ed in Figure 3(a). The use of Equation (32) should be considered when the shear stiffness of the bracing units are very small and/or when the bracing system consists of shear walls only. It should be noted that in this case, mt is the total torsional moment and I represents the sum of the warping stiffnesses of the shear walls.

Case B: Bracing systems comprising multi-bay, low-rise frameworks tend to develop predomi-nantly St Venant-type rotation and the effect of the warping stiffness becomes insignifi cant.

This case is characterised by a 0 and b and governing differential Equation (23) cannot be used directly. However, after some rearrangement, the original derivation leads to

=

m

K (33)

where K = Kbt2. This differential equation, together with the boundary conditions (0) = 0 and (0) = 0, lead to the rotation and the top rotation as

=

mz

K

2

2 (34)

and

max = ( ) =H mHK2

2 (35)

The characteristic rotation shape is shown in (Figure 3(b)). It is certainly worth considering the use of Equation (35) when the building is relatively low and the bracing system only consists of (mainly multi-bay) frameworks.

Case C: The structure is relatively very slender (with great height/width ratio). The structure de-velops predominantly (global) warping rotation. The second and third terms in Equations (24) and (25) tend to be by orders of magnitude smaller than the fi rst term and the solutions for the rotation and the top rotation effectively become

=

+( )

m

E I IH z z

g

3 4

6 24 (36)

and

max = ( ) =+( )H

mHE I Ig

4

8 (37)

This case is illustrated in Figure 3(a). It is interesting to note that both Case (A) and Case (C) are characterized by warping-type rotation.

K. A. ZALKA


134


4. STRUCTURES WITH VARYING STIFFNESS OVER THE HEIGHT

In many practical cases, the stiffness of the bracing system varies over the height (Figure 4(a)). The structural designer likes to have a quick assessment of such situations as to the signifi cance of the effect of the variation of stiffness. A lower bound is readily obtained if the calculation is carried out using the smallest value of stiffness (I1 at the top of the structure; Figure 4(a)). In a similar way, an upper bound is easily available using the biggest value of stiffness (Io at the bottom of the structure; Figure 4(a)). The difference between the two results shows if a more accurate calculation is warranted.

For a more accurate calculation, a decision has to be made about the distribution of the stiffness over the height (shown by dashed line in Figure 4(b)). The distribution can be given as

I I zfzm

=

1 (38)

where m can assume 1, 2, 3 and 4.Using the above law for the distribution of the stiffness, a reduction factor can be derived examin-

ing the behaviour of the column under a compressive force (Dinnik, 1929 and 1932; Timoshenko, 196 8 ). The results o f this derivation (values of reduction factor rI) are given in Figure 5 and in Table 1 as a function of I1/Io.

The question remains: which distribution should be used? It depends on the actual situation, but according to the graphs in Figure 5, as a rule, the distribution of the stiffness has no great effect on the situation. As the subject of the investigation in this paper is torsion, m = 2 is recommended as the behaviour of the built-up column (Figure 4(c)) in bending is the closest to that of a bracing system in torsion.

b)a)

z

x

H

f

Io

Iz

I1

Io

I1

c)

x

y

x

z

Figure 4. Building with stiffness varying over the height.




Once the reduction factor is obtained, all the stiffnesses should be modifi ed according to

I r II= 0 (39)

The stiffnesses modifi ed in this manner should be used in Equations (20)(22) to obtain the torsional characteristics that now take into account the variation of stiffness.

5. PRACTICAL APPLICATION

Multi-storey buildings under horizontal load never develop torsion only. When the bracing system of the building is doubly symmetrical, the shear centre of the bracing system (O) and the centre of the plan of the building (C) coincide (Figure 6(a)). Under horizontal loadrepresented by its resultant F

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

rI

I1 Io

m=1

m=2

m=3

m=4

Figure 5. Values for reduction factor rI.

Table 1. Reduction factor rI for the variation of stiffness.

I1/Io 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

rI m = 1 0.657 0.710 0.756 0.797 0.835 0.871 0.905 0.939 0.970 1.0m = 2 0.547 0.645 0.715 0.771 0.820 0.862 0.901 0.936 0.969 1.0m = 3 0.508 0.622 0.694 0.762 0.813 0.861 0.899 0.935 0.969 1.0m = 4 0.487 0.610 0.684 0.758 0.809 0.858 0.898 0.935 0.969 1.0

a) Symmetrical arrangement b) Asymmetrical arrangement

O O C

C

FF

M xc

Figure 6. (a) Symmetrical; and (b) asymmetrical arrangement.

K. A. ZALKA


136


in Figure 6the building develops lateral displacement and no rotation occurs. This case is fully investigated in Zalka (2009).

When the building is asymmetrical, the shear centre of the bracing system and the centroid of the plan of the building do not coincide (Figure 6(b)). The external load passing through the centroid causes two things: lateral displacement in the plane of the load and rotation around the shear centre (Figure 7(a)). The behaviour of the building is best analysed by transferring the load to the shear centre. This procedure results in a horizontal load passing through the shear centre and a torsional moment M = Fxc, where xc is the distance between the shear centre and the centroid (Figure 7). Force F develops lateral displacements only (vo in Figure 7(b)) and torsional moment M develops rotation () around the shear centre (Figure 7(c)), which causes additional displacement (v). At any given location, the total displacement is obtained by adding up the two components:

v v v= +o (40)

The maximum displacement of the building develops at the top at a corner of the plan of the build-ing (point A in Figure 7) and, making use of the angle of rotation, is obtained from

v v H v xmax max= ( ) = +o (41)where xmax is the distance of the corner point (where maximum defl ection occurs) from the shear centre. The fi rst term (vo) on the right-hand side of Equation (40) can be obtained using Equation (19) and the angle of rotation is determined by Equation (25).

6. VALIDATION OF THE CONTINUUM MODELACCURACY ANALYSIS OF 126 SYSTEMS

It is essential for any respectable approximate method to examine its range of validity and accuracy. The continuum model for the defl ection analysis has already been thoroughly investigated and proved to be a reliable one (Zalka, 2009): a comprehensive accura cy analysis involving 270 test structures measured the accuracy of Equation (19), the very formulae on which the present proposed method is based.

As for the proposed method itself, the results of another accuracy analysis is given in this section. The aim of the accuracy analysis was to validate Equation (25) derived for the maximum rotation. The results obtained using the approximate formula were compared to the results of the Finite Element solution. The AXIS VM fi nite element package (Axis, 2003) was used for the comparison, whose results were considered exact.

The top rotation of fourteen bracing systems (Figure 8) under uniformly distributed torsional load, using 11 individual bracing units was calculated. The height of the structures varied between 4 and 80 storeys in eight steps (4, 10, 16, 22, 28, 34, 40, 60 and 80 storeys), creating 126 test cases. The

O C

F

M=Fxc

v

= +O

F

O

vo vAA A

xc

b) c)a)

xmax

xmax

Figure 7. Displacements. (a) v: maximum displacement; (b) vo: displacement due to force F; (c) v: displacement due to torsional moment M.




24 m

a) F5+F5

O F5 F5 12 OF5 F5F5

24 m

b) F5+F5+F5

24 m

c) F5+F5+W3+W3

O F5 F5

W3

W3

24 m

d) F5+F5+F5+W2+W2

F5 F5

W2

W2

O F5

W2

W2

12 O

F5 F7

e) F5+F7

O W1 W1

f) F3+F3+F3+F3+W1+W1

F3 F3

F3

O

W1

F7

g) F6+F7+W1+W4

F6

W4

24 m

6

6 O

W1

h) F6+W1+W4+W4

F6

W4

12 m 6 6

W4

O

W1

i) W1+W1+W4+W4

W1

W4

12 m 6 6

W4

F7F7 F7 F7

O

F7 F7F7F7 O F7

312 = 36 m

j) F7+F7+F7+F7 k) F7+F7+F7+F7+F7

412 = 48 m

F6

l) F6+F10

24 m 56 = 30 m 24 m

6

6

6

6

18 F10

F5 F7 F7 F7 F7 F5

O

O

m) F5+F7+F7+F7+F7+F5 n) F5+W4+W4+F1+F1+U

F5

W4

F1

F1

U O

W4

F3

Figure 8. Bracing systems for the accuracy analysis.

K. A. ZALKA


138


bays of the one-, two- and three-bay reinforced concrete rigid frames were 6 m and the storey height was 3 m (F1, F3, F5, F6, F7 and F10 in Figure 8). The rectangular cross sections of the columns (in metres) were 0.4 0.4 for F1, F3, F6 and F7; 0.4 0.7 for F5 and 0.4 2.0 for F10. The rectangular cross sections of the beams (in metres) were 0.4 0.4 for F1, F5, F7 and F10; 0.4 1.0 for F3 and 0.4 0.7 for F6. The cross sections of the shear walls were 0.3 12.0 for W1; 0.2 2.0 for W2; 0.2 6.71 for W3 and 0.3 6.0 for W4. The wall sections for the U-core were h = 4.0 and b = 4.0 with a wall thickness of t = 0.3. The modulus of elasticity for the structures was E = 25 kN/mm2.

The cross sections of the beams, columns and shear wall were chosen in such a way that the struc-tures covered a wide range of stiffnesses. The torsional shapes represented predominant warping-type, mixed warping-type and St Venant-type, and predominant St Venant-type deformation. The summary of the results (range of error, average absolute error and maximum error) is given Table 2.

The results summarized in Table 2 demonstrate the performance of the proposed method. It should be emphasized that the proposed method produced conservative estimates in every test case. The error range of the method was between 0% and 25%. In the 126 cases, the average difference between the results of the proposed analytical method and the fi nite element solution was around 9%.

7. WORKED EXAMPLE

Calculate the maximum defl ection of the 28-storey building shown in Figure 9, subjected to a uni-formly distributed lateral load of intensity w = 1.0 kN/m2 in direction y. The building is braced by two one-bay frameworks (F1), one two-bay framework (F5), two shear walls (W4) and a U-core (U). The FE computer analysis resulted in ymax = 404 mm, and this result is to be checked. The bracing

Table 2. Accuracy of the proposed method related to the maximum rotations of the 126 bracing systems of frameworks and shear walls (Figure 8).

MethodRange of error

(%)Average absolute

error (%)Maximum error

(%)Proposed method (Equation (25)) 0 to 25 9 25

6 m

6 m

6 m

1 : F1

O

yo = t1 = t3 = 6 m

6 : F1

2 : F5

3 : W4

5 : W4

4 : U

x

x

t5 = t6 = 6 m

6 m 6 m 6 m

yo y

xo = t2 = 13.4 t4 = 4.886

C

xc = 1.4 e = 1.714 h = 4.0

w

O4

Figure 9. Layout and geometrical characteristics for the worked example.




units are numbered as shown in Figure 9. General data for the bracing units are given in the previous section and the stiffness and other characteristics are summarized in Table 3.

The maximum defl ection of the building consists of two parts. The defl ection of the shear centre is calculated fi rst, then the additional defl ection due to the rotation around the shear centre is added (cf. Figure 7).

The defl ection of the shear centre comes from a lateral defl ection analysis when the horizontal load is acting through the shear centre and causes defl ection only. The horizontal load is resisted by frame-work F5 and core U. The two perpendicular frameworks (F1) and two shear walls (W1) have insig-nifi cant resistance in the direction of the external load and therefore they can safely be neglected. Framework F5 is chosen as the base unit (as no other framework is available). In using the stiffnesses of framework F5 and core U (second and last rows in Table 3), the load share and the load on frame-work F5 are

q SS S

w qw=+

=

+= = = =

2

2 4

0 6860 686 1 882

0 2671 0 2671 24 6 41.. .

. . . kN m

Coeffi cients a, b , and s are calculated using Equations (17) and (18), bearing in mind that the number of frameworks is now one:

aK

EIb K

EIf

g= = = = = =

66 947504000

0 000133 66 947807 285

1 0 08293. . ..

.

= + = = + = + =a b s ab

0 2882 1 1 0 0001330 08293

1 0016. ..

.

With the above coeffi cients, the top defl ection of the shear centre is calculated from Equation (19):

vo =

+

6 41 848 5 04807 10

6 41 842 66947 1 0016

6 41 8072

4

6

2

2..

..

. 99066947 1 0016

1 0 2882 84 24 224 2

1 02 3

+ ( )( )

=.

. sinh .cosh .

.. . . .079 0 337 0 027 0 389+ = m

The location of the shear centre is needed for the rotation analysis. As the bracing system has an axis of symmetry (x), one coordinate of the shear centre is readily available. The other coordinate is obtained as the centroid of the stiffnesses, using Equation (13), so the shear centre coordinates are as follows:

x

S x

S

S xS S

yi i

m

i

mom= =

+=

+=

1

1

4 4

4 2

1 882 18 2861 882 0 686

13 40. .. .

. , oo m= 6 0.

Knowing the location of the shear centre, the (perpendicular) distances of the bracing units from the shear centre (tj) and the arm of the load (xc) can be established as given in Figure 9.

Table 3. Stiffness and other characteristics of the bracing units for the worked example.

Bracing unit K (MN) r () Ic (m4)

EI (MNm2)

EIg (MNm2)

b (1/m2)

y (m) (Equation (19))

S (1/m) (Equation (12))

F1 28.444 0.8 0.004267 85.333 72 000 0.333 4.92 0.203F5 66.947 0.941 0.0343 807.29 504 000 0.008 1.457 0.686W4 1 5.4 135 000 1.106 0.904U 1 11.245 281 125 0.531 1.882

K. A. ZALKA


140


All the six bracing units participate in resisting torsion. Core U is the only three-dimensional item, but because it lies on the symmetry axis, it can only utilize its second moment of area with respect to axis x. Framework F1 is the base unit (as it has the greatest b = K/EI ratio). Its stiffness character-istics are calculated using Equations (20)(22):

EI EIt = = = 2 2 6 485333 6 3 072 10. kNm

EI EI tg g = = = 2 6 2 6 472 10 6 2592 10 kNm

K Kt = = = 2 2 6 228444 6 1 024 10. kNm

The torsional moment share on the base unitEquation (27)is calculated using the torsional stiffnesses of the bracing units

S S t S S S , , ,. . , . . .1 2 2 6 2 26 0 203 7 308 13 4 0 686 123 2= = = = = = =

S S S , , ,. . , . . .3 5 2 4 26 0 904 32 544 4 886 1 882 44 929= = = = =

as

q S

S im

= =

+ + + + +=

,.

. . . . . .

1

7 3087 308 123 2 32 544 44 929 32 544 9 308

0..0295

With apportioner q and the distance between the centroid and the shear centre xc, the torsional moment share on the base unit is given by Equation (29):

m wx qc= = = 24 1 4 0 0295 0 9912. . . kNm m

Coeffi cients a, b , and s for the calculation of the rotation are obtained using Equations (17) and (18), bearing in mind that the number of frameworks is now three (F1, F5 and F1):

aK

EI

II

II

II

II

II

KK

II

KK

g

g

g

g

g=

+ +

+ +=

+1

1

28 44472000

1 0 0322

3

3

2

2

3

3

.. 2229

0 0034132 8820 16

1

1 0 032290 003413

2844466947

10 000.

..

..

.+

+ += 222

b KEI

fII

KK

II

KK

=

+ +=

+1

2844485333

3

1 0 032290 003413

28444669

2

2

3

3

.. 447

10 16613

+= .

= + = = = + = + =a b s ab

0 16635 0 408 1 1 0 000220 16613

1 0013. . ..

.

With the above coeffi cients, the top rotation around the shear centre is calculated from Equation (25):

max =

+

0 991 848 25 95 10

0 991 842 1 024 10 1 0013

0 9

4

6

2

6 2.

..

. .

. 991 3 072 1066947 1 0013

1 0 408 84 34 2734 27

6

2 2

+ ( )(

..

. sinh .cosh . ))

= + =1 0 00238 0 00340 0 0001 0 00568. . . .




The maximum defl ection of the building can now be obtained using Equation (41):

v v xmax max . . .= + = + =o m 0 389 0 00568 13400 0 454

The exact solution using the FE package is vmax = 0.404 m.

8. CONCLUSIONS

An analogy between the bending and torsion of bars makes it possible to carry out a simple torsional analysis of multi-storey building structures braced by frameworks, (coupled) shear walls and cores. Application of the analogy leads to a closed form formula for the rotation of the building.

The torsional behaviour is defi ned by three distinctive phenomenon: warping torsion, St Venant torsion and the interaction between the two modes. The separation of the basic (warping and St Venant) modes enables the structural designer to identify the key contributors to the torsional resist-ance of the building and to achieve optimum structural performance. The interaction between the warping and St Venant modes is always benefi cial as it reduces the rotation of the building. Its effect may be signifi cant for low-rise structures but becomes rapidly negligible as the height of the structure increases.

Using the formula for the rotation of the building, the maximum defl ection of any asymmetric multi-storey buildings can be readily determined.

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