Tall Building Structures

TALL BUILDING STRUCTURES

Course content: 1.

Introduction

2.5 hrs.

2.

Coupled planar walls

3.5 hrs.

3.

Coupled nonplanar walls

3.5 hrs.

4.

Coupling effects of beams and slabs

3.5 hrs.

5.

Finite element/strip analysis

3.5 hrs.

6.

Wall-frame interaction

3.5 hrs.

7.

Framed tube structures

3.5 hrs.

8.

Outrigger braced structures

3.5 hrs.

9.

Shear lag effects in buildings

3.5 MS.

10.

Other topics/revision (tahe announced)

2.5 MS. _... ---.... --33 MS.

1

RECO~NDEDTEXTBOOKS

1.

F. K Kong et al. (Edited), Handbook of Structural Concrete, Pitman Books Ltd., 1983, Chapters 37 and 38.

2.

I.Struct.E./I.C.E., Manual for the Design of Reinforced Concrete Building Structures, The Institution of StructuralEngineers, 1985.

3.

B. S. Taranath, Structural Analysis and Design of Tall Buildings, McGraw. Hill Book Co., 1988.

4.

I.A MacLeod, Analytical Modelling of Structural Systems, Ellis Horwood, 1990.

5.

B. Stafford Smith and A. Coull, Tall Building Structures: Analysis and Design, John Wiley & Sons Inc., 1991.

6.

T.Balendra, Vibration of Buildings to Wind and Earthquake Loads, Springer-Verlag,. 1993.

2

1. Introduction

1".1 Dermition From the structural engineer's point ofview, a tall building may be defined as one in which lateral forces due to wind or earthquake play an important or dominant role in the structural design. Low-rise buildings are generally designed to resist gravitational .loads, and the influence of lateral forces only checked subsequently, since most building codes allow some overstress due to the transient nature ofthe lateral forces. However, the structure of a high-rise building must be designed to resist both horizontal as well as vertical forces, and an optimum system sought to minimize the influence ofthe former. It is difficult to define a high-rise building in terms of height or number of storeys. The importance of the horizontal loads relative to the vertical loads is mainly a· factor of the slenderness ratio, i e. height to width ratio, ofthe building and is thus dependent also on the lateral dimensions of the building. Nevertheless, very roughly speaking, one may say that a low-rise building ranges from 1 to I (j storeys. A medium-rise building probably ranges· from 10 to 20 storeys, and· a high-rise building is one that has more than 20 storeys. As buildings get to more than 50 storeys high, the lateral stiffuess (the lateral stiffuess requirement is often given in terms of a maximum allowable lateral deflection of 1/500 ··of height) would very likely become more critical than the lateral strength; such b:uildings

may be regarded as super-high-rise buildings.

1.2 Loadings Both vertical and horizontal loads need to be considered. Vertical loads comprise of dead load and live load. Dead load is determined from the designed member sizes and the material densities. Member sizes depend on the span lengths of the structural components; basically, longer span structures are more bulky and therefore heavier. Live load (also referred to

3

as imposed load) is .dependent on the intended function of the building. For small loaded areas, the effects of a concentrated live load need to be considered. For multi-span structures, possible live load distributions over adjacent and alternate spans should be considered in estimating the local maximum for member forces. Very often, however, constmction·loads may be the most· severe that a building structure has to withstand. These include the weight of newly poured concrete and the fonnwork, and sometimes even the weight of the construction equipment and vehicles etc. There are two major types of horizontal loads: wind load and earthquake load. Wind. load depends on the height, size and shape ofthe building, as well as on its geographical location and the maximum design wind ·speed. Although wind loads are dynamic in character, they are often replaced by equivalent static loads to simplify the structural analysis. This is acceptable in most cases, but for exceptionally .taR slender or vibration-prone buildings, or for those of unusual shape or particular importance, dynamic analysis taking into account the gust characteristios and the frequency spectnlID. of the wind may be required. Earthquake load consists of the inertial forces of the building mass resulting from shaking of the building during an earthquake. Earthquake designs are normally based on the principles that buildings should: (a) resist minor earthquakes without .damage; (b) resist moderate earthquakes without structural damage, although accepting the probability ofnon-structural damage; (c) resist severe earthquakes with the probability of structural as· well as nonstructural damage, but without collapse. Wind and earthquake .loads have different frequency contents and dynamic characteristics.. A building structure that is sufficiently strong to resist wind does not necessarily imply that it can resist an earthquake inducing the same magnitude of equivalent static load. The nature frequency of the structure, damping capacity, any possible wind-structure interaction and soil-structure interaction etc will all affect the responses of the structure to wind and earthquake. For the particular

4

case of earthquake loading, the ductility and failure sequence of the structural components also play important roles.

Earthquake spectrum

Wind turbulence spectrum

0.001 I

1000

0.01 , 100

0.1

10

Frequency (Hz) «

I

10

0.1

Period (s) Fig. 1.1. Spectral densities of earthquakes and winds.

The ·above figure (Balendra 1993) depicts the frequency ranges of turbulent wind and earthquake. From this figure, it can be seen that strong winds have a dominant frequency range of approximately 0.01---0.1 Hz, while earthquake excitations have a dominant frequency range of approximately I-10Hz (note, however, that these are· very rough values and" the actual frequency compone~ts depend on many factors requiring both near-field and far-field analysis). The influence of these loads on buildings depends on the natural frequencies of the buildings. A stiff building with a period of 0.5s (natural frequency=2Hz) would only be slightly affected by the wind, but the effect of an earthquake on this bUilding could be serious. On the other hand, for a tall and flexible building of period 5s (natural frequency=O.2Hz), moderate earthquakes may have no serious effects, whereas winds could have a significant effect and would control the design. Thus, if both wind and earthquake loads need to be considered, it may be quite difficult to totally avoid resonant effects. Researchers are nowadays working hard on· the development of both active and passive damping devices for mitigating the resonant effects.

5

1.3 Structural forms A building structure generally consists of ( 1) a horizontal subsystem which provides flat surfaces for habitation and transfers all gravitational loads to the vertical subsystem; and (2) a vertical subsystem which cany all vertical and horizontal loads to the foundation and often at the same time forms external enclosure and .partitions for fimctionalpurposes. One can visualize the horizontal subsyste~ which comprises of beams and slabs, as 2-D wholes that act vertically to cany the floor or roof loads in bending,and horizontally as diaphragms and/or column connectors forming part of the frame structure. Similarly, the vertical subsyste~ which comprises of shear/core walls and columns, can be visualized as wholes that act to pick up loads from the horizontal subsystem and to .resist.lateral . loads as a vertical cantilever structure. Vertical subsystems are generally slender in one or both lateral directions (compared to the overall building height) and. cannot be very stable by themselves. They must be held in position by the ho~ontal subsystems which provide diaphragm (in-plane) actions to tie the vertical elements together. In fact, the horizontal subsystems may also provide coupling (out-of.plane) actions to improve the structural efficiency ofthe .vertical elements. Note that the design of the horizontal subsystem is related to the arrangement of the supporting vertical subsystem which determines the span lengths of the horizontal elements. On the other hand, the structural action of the vertical subsystem is affected by any COllpling actions of the horizontal subsystem. Hence, when actual designing, both subsystems must be considered more-or-Iess simultaneously. The structuralfonn depends largely on whether structural steel' or reinforced concrete are to be used for .the construction and on the type of buildings. Steel buildings are mostly frame or tress structures. Early day reinforced concrete buildings are basically· imitations of steel structures and are therefore generally frame structures. However, starting in· the 50s, engineers began . to realize that reinforced concrete frame structures without shear walls are not the most suitable for tall buildings (Fintel 1974). Since then, reinforced concrete buildings become normally incorporated with shear walls. Thus, most reinforced concrete buildings are some kind. of coupled. wall-frame structures.

6

Different structural systems have been evolved for residential and office buildings, which reflect their differing :functional requirements. The basic :functional requirement of a residential building is the provision of discrete dwelling units for groups of individuals. These have common requirements ofliving, sleeping, cooking and toilet areas, which must be separated by partition walls offering fire and acoustic insulation. Framed structures may be usefully employed for residential buildings, since the presence of permanent partitions allows the column layout to correspond to the architectural plan. However, these depend on the rigidity of the joints for their resistance to lateral forces, and tend to become uneconomic at heights above 2025 storeys. Since their introduction in the late 40s, shear walls, acting either independently or in the form of core assemblies, have been used extensively as additional stiffening elements for traditional frame structures. Typical planforms for tall buildings, whose load-resisting structure consists of interacting shear walls and frames are shown below.

•

•

: [ J:

•

•

(a)

(b)

(c)

(d)

7

In order to provide adequate fire· and acoustic insulation between dwelling units, infill panels of brickwork or blockwork are introduced into frames. However, although techniques exist for assessing the influence of these infill panels on the stiffuess and strength of the frame,·theyaregenerally assumed to be non-load-bearing, in view of the danger that they may be either removed or perforated for a change offunction at some future date, as well as the difficulty of achieving a tight fit between an infilled panel and. the surrounding frame. Consequently, later trends were to omit the infill panels and to utilize the walls which are required for space division in a structural context. This has led to the development ofthe shear wall system, in which structural walls are used to divide and enclose· space, while simultaneously resisting ·both vertical and horizontal loads. They tend to become economical as· soon as. lateral forces affect the design. However, they do possess the disadvantage of an inherent .lack of flexibility for future ·modifications.· Thus, they are more suitable for residential buildings in which each floor is permanently divided into dwelling units by partition walls. Typical planforms for shear wall buildings are shown below.

[

r

(a)

(b)

8

The essential functional requirement of an office building is the provision of areas unobstructed as far as possible by walls or columns to allow each occupant to design the partitioning most suitable for his particular business. The partition layout will generally alter when tenants change, and this necessitates flexibility in the distnoution of services to any floor area.. As a result, services tend to be carried vertically within one or more service cores, and a distnoution network run beneath the floor slab to the entire floor area. By judicious planning of the wall and column layout to maximize the open floor areas, shear wall-frame interactive structures may also be employed for office blocks, although the presence of internal walls and columns may make it difficult to achieve the desired planning flexibility. One simple method of creating open floor areas is to use a central shear wall core, which carries all essential services and which is designed to resist all lateral forces. The floor system spans between the central core and the exterior facade columns, and a large unobstructed floor area is created between the two .. vertical components. Another possibility is to provide a core at each end, especially if the building is long. However, in order to support the floor slabs in the interior, it is then necessary to provide a spine beam running between the cores, which may require additional supporting interior columns.

(a)

(b)

9

As, building become taller, the use of one or more cores acting independently to resist lateral forces will lead to unusually large cores, occupying too large a ratio of a given floor area, and leading to uneconomic solutions. A further increase in stiffuess and structural efficiency can. be achieved if the central core is tied to the exterior columns by deep (usually storey height) flexural members or truss at the top and possibly also at other intermediate levels. The objective of these connections is to integrate the central core and the exterior columns together so that the wind moment is resisted not only by bending of the core wall, but also by the couple formed of the axial forces in the exterior columns. This allows the axial stiffuess of the exterior columns to be mobilized to resist the wind moment. The resulting' .increase in the level arm of the base resisting forces leads to significant increase in stiffhess and structural efficiency.

---.... -;...... --..... --...... --.... .......-...

-----... .......

....... ........ --..... .......

The efficiency of the structure can also be increased substantially if the outer facade is' replaced by a, rigidly jointed framework" which can be used to resist lateral as well as vertical forces. The outer shell then acts effectively as a closed box-like structure whose faces are formed of rigidly jointed frame panels, or as a highly perforated tube whose cross-sectional shape is maintained by the

10

floor slabs acting as horizontal diaphragms. A combination of the framed-tube concept with the shear wall-frame interaction concept yields the structural form termed the tube-in-tube syste~ in which an exterior closely spaced column system is constrained by the floor slabs to act in collaboration with a very stiff shear core enclosing the central service area. The large level arm involved between opposite normal faces of the exterior tube gives rise .to an efficient moment-resisting structure, akin to an ordinary tubular structural component.

The closely spaced columns in a framed tube may pose problems in gaining access to the building at ground level, and some structural re-arrangement may be necessary in that region. Several columns may be run into one at regular.intervaIs, or a deep girder may be provided at first floor level to transfer column loads to more widely spaced ground level columns. The pure framed tube has the disadvantage that under bending action, a considerable degree of shear lag occurs in the faces normal to the wind, as a result of the flexibility of the spandrel beams. This has the effect of increasing the stresses in the comer columns, and of reducing those in the inner· columns of the normal panels, and results in loss of efficiency in the desired pure tubular action of the structure. Warping of the floor slabs, and consequently deformations . of

11

interior partitions and secondary structure will occur, which may become of importance in design.

True tubular cantilever stress

Actual stress due to shear lag

~---~ Actual stress True cantilever stress

t Wind load One technique which has been employed to help overcome this problem is to add substantial diagonal bracing members in the planes of the exterior frames. The eXterior columns may then be more widely spaced, and the diagonals, .aligned at some 45 to the vertica~ serve to tie together the exterior columns and spandrel beams to form.. facade trusses. Consequently, a very rigid cantilever tube is produced. As the diagonal bracing members may be subjected to both tension and compression forces, they are normally constructed of structural steel.

An alternative method ofreducing shear lag effects is to add infill panels at selected locations (normally in a cross diagonal pattern) so that the infill panels act together as diagonal bracing members tying the exterior columns and spandrel beams together in the same way as steel diagonals does. This would, however, require some window openings to be permanently blocked.

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1.4 Design assumptions An important first step in the analysis of a tall building is the selection of an appropriate idealized mode~ to include all the significant load-resisting elements and their dominant modes of behaviour. The design assumptions usually made are listed below:

(1)

The floor slabs are rigid in their own plane, so that each floor is subjected to a rigid body movement in plan. Consequently, the vertical elements at any floor level undergo the same horizontal and rotational components of displacement in the horizontal plane.

(2)

The out-of.plane flexural stiffuess of the floor slabs are negligible. As a result, the restraint of the floor slabs against the warping of horizontal sections of the building is ignored. However, in the case of shear walls coupled. by floor slabs only, it is· still necessary to take into account the contribution ofthe flexural stiffuess ofthe floor slabs towards the coupling of the shear walls, and this is normally done by treating" the coupling slabs as equivalent beams.

(3)

The vertical elements may be regarded as assemblies of inter-connected vertical planar sub-structures of which the out-of:.plane stiffiless are negligible. Thus for the case of shear walls, the out-of-plane flexural stiffuess of the shear walls are neglected. For the case of core walls, the core walls are treated as assemblies of inter-connected planar wall units which act individually as two dimensional membrane structures. This avoids the necessity to model the walls as shell elements. For the case of frames, the frames are treated as assemblies of two dimensional plane frames inter-connected at the common columns. The out-of.plane stiffiless

(4)

ofthe plane frames are neglected and as a result, the number of d.o.f/joint is reduced from 6 to 3. Shear deformation of the beams may be neglected. However, this assumption is not applicable to beams with small span/depth ratios.

(5)

Shear deformation of the walls may be neglected. There is, however, little advantage in neglecting the shear deformation of the walls. In fact, the negligence of the wall shear deformation may lead to numerical difficulties and significant errors in some cases. Thus the shear deformation of "the walls should be taken into account as far as possible.

13

(6)

(7)

(8)

(9)

(10) (11)

Local deformations at the beam-wall joints, which are often neglected but can in fact be quite significant in many cases, may be taken·into account by increasing the effective length of the beams or by including rotational springs at the beam-walljoints. The shear lag effects in the shear/core walls are negligible. Hence the shear/core walls may be analysed by applying the plane sections remain plane assumption to the individual planar wall units. The shear lag effects are normally more significant at the· wall bases. If required, the shear lag effects at the wall bases may be evaluated by analysing only the lower part ofthe structure with the finite element method. Alternatively, design charts maybe used. The rotations of the beam-wall joints may be taken as the rotations of the horizontal rigid arms which are incorporated to allow for the· finite width of the ·walls. This assumption is often made when the frame method is applied. However, this would lead to incompatibility between the beam and wall elements at their common interfaces. To ensure compatibility at the beam-wall joints, the joint rotations should be defined as the rotations of the beam-wall interfaces, i.e. the rotations of the vertical fibres at the joints. The cross-sectional shape of the shear/core walls would remain undistorted. Hence the rotations of all beam-wall joints can be related to the horizontal displacement ofthe building thereby reducing the number of unknowns to be solved. The P-D effects may be neglected. For dynamic analysis, the masses of the building may be lumped. at the floor levels so that the mass matrix becomes a diagonal matrix. Furthermore, it is also often assumed that the vertical. inertia. of the building may be neglected when only lateral vibration is considered.

14

2. Analysis of coupled planar walls

2.1 Methods of analysis Continuous connection method (EQuivalent continuum method) - treats the coupling beams as an equivalent continuous shear connection.medium - the walls are regarded as vertical cantilever beams Other assumptions made: (a) the properties are assumed to be constant with height (b) the coupling beams are assumed to be axially rigid (c) points of contraflexure occurs at mid-span of the beams a second order differential equation from which: the shear forces in the beams and consequently the forces and moments in the walls as well as the beams can be detennined =>

Due to assumption (a), the method is severely limited to relatively simple case.

\

15

Frame method (Eguivalent frame method) - treats the coupled walls as a plane frame structure - the walls are modelled as column elements residing at the centroidal axis of the walls - the coupling beams are taken as beam elements with rigid arms to take account of the [mite width of the walls The rigid arms may be incorporated by: - modifying the stiffness matrix of standard beam members; or - adding very stiff beam members as the rigid arms while using standard plane frame programs in which it is not practical to modify the stiffness matrix. In this method, the beams are assumed to be connected

rigidly to the axis of the walls. However, it has been· proved by some researchers that slight rotation exist at the connection due to local deformations at the beam-wall joints and that as a result, the effective stiffness of each beam is reduced. This effect of joint flexibility can be taken into account by extending the coupling beams by, say, half their depth, at each end.

16

Finite element method - the most powerful tool of analysis available - can be applied to the analysis and design of any type of buildings and for various support conditions The standard procedure for the method is to divide the frames, shear walls and coupling beams into many small elements, using: frame elements for the frames and coupling beams; and plane stress elements for the walls (out-of-plane bending actions of walls neglected). The number of nodes and elements required to model a building structure is usually very high and consequently the computer solution is very costly (not only the cost of computer time, but also the cost of preparing data and interpreting the computer results). There are, however, a number of problems with the use of the fmite element method for tall building analysis, apart from the high cost. These will be discussed in detail later on.

17

Finite striIJ method - the shear walls are modelled as assemblages of strip elements - the frame part of the structure is modelled either: (1) as a continuum having equivalent properties; or (2) in such a way that the columns are treated as line elements and the connecting beams as discrete beam members spanning between the strip or line elements In the frrst approach, it is necessary to assume that the locations of the points of contraflexure of the actual beams and columns are at the mid-points of the respective members. The true locations will of course be somewhat different, but the resulting errors are small. In the alternative approach, the stiffness properties of each beam are· formulated through the basic beam theory, and then subsequently transformed to the adj acent strip and line elements.

18

2.2 Continuous connection method

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2.3 Frame method

A plain shear wall acts principally as a vertical cantilever beam, and hence it is natural to model a shear wall as a vertical frame member residing at the centroidal axis of the wall, Fig. lea). Such frame analogy was extended in the 60s to deal with plane coupled shear walls (Clough et ale 1964, Candy 1964, MacLeod 1967) by adding beam elements to model the coupling beams and by incorporating horizontal rigid arms into the beam elements to take into account thefmite width of the shear walls, Fig. 1(b). This method can also be applied to walls as shown in Fig. I(c).

20

mu~tiple

coupled shear

U)

:J

c

&E o :J

CUO C

.«

0

E

~..----+--~----------r'\

CO

Q)

"'0

o

Z

21

APPENDIX I. BEAM STIFFNESS ~1ATRIX FOR WIDE COLUMN FRAME

Figure 12 sho\\'s a beam ah bet\veen t\VO shear walls (columns). Points A and B are on the centroidal axes of the columns at beam level and are the node points.

.-y

--'~.. /'--

FIG.

12.. Notation for deformation of wide column frame.

Normally the member stiffness for a beam element is written in terms of the deformations at its ends, Le.

6

Va

12 L3

6 L2

-12 L3

6

-L2

4

-L

-6

M"

Vb

-12 L3

-6

12

L2

L3

-6 L2

Mb

-L1-

6

-L

-6 . L2

-L

= EI

2

-L

L2

2

~a

L2

Va

4

Oa X ~b

Db

Vb

.~)

! A{tI( ~

~I

L

M"

(Since the axial terms are independent of the other terms they have been· omitted for conciseness.) I.e.

P"b

=

KM"tl

X

Ullb •

(1)

Since the equilibrium equations of the structure are written in terms ·of the· actions and deformations at the node points it is necessary to transform expression (1) so that the beam stiffness is also defined in terms of these vaiables. The deformations at the ends of the beam are related to the nodal deformations by the expression: ~a

Oa

-

..lb

I da 0

0

0

0

0

I

j.B

08

1

0 0 0

= TlJ..lll-

(2)

P. 111 = T'P",,_

(3)

U"b

I.e.

OA X

1 -bd

0 0

Ob

~A

By contragredience(l:!) Substituting (2) and (3) in (I) gives P. 11t = T'K,W""TU. 11J Thus TKM""T' = KJ"f.'ll is the beam stiffness matrix in terms of the nodal actions and deformations. The complete matrix \vith the axial terms and shear deformation included is quoted in Table 5.

TARLE

5.

BEAM STIFfNESS MATRIX.

r;x

r l

J.f. 1

N.A~(I

1.

rI) B

I ,fa

.1 (

L

)1 (

db

.1

~

,\f~

.~

..lX.1

Sy"'lmetric

o

=£x

0([. clu -CI' ~

+ t')

(f. tla"

+ 2t'. do + eI)

o

-I o (f. dh + ~) o

A

;: = T:

C

b

=

..lY.1

x

o

.~

(-.{.tla-f')

0

(.r. Ila. llh + ~(,Iu + db) + C') /0/3 - 2h) L •S .,

= 2/(1 .. f. +L:I') • ..·T

d =

2IC 113 + h) L.S '

[

()

<-f. db -

jYII e)

I

e

{I. (/h~ + 2eo. db + cI) 21

= L~ 5' f = I.~ S'

•.

= ElJulv~.lcnl she~,r area.

•

L'

{J.I

~XJl

•

1 S = (; •

= POisson s ralill.

+ 2h.

011

UDC 6:!
F·rame idealization for shear wall support systems

very careful not to nlake the difference between the rigidities of the nle.nbers too high since this can result in numerical instability in the solution. This instability is not easily identified and results can be obtained which appear to be correct but which are in fact appreciably in ~error. For this type of analysis axial defornlation nlust be considered for the colunlns but nlay be neglected for the beanls. Shear defornlation of·the bea.ns can have a significant eflect bUI it is not absolutely essential to include it.

I. A. MacLeod SSc (Eng), PhD. CEng, MIStructE, MICE

D. R. Green MS. SSc, PhD LC'C'fI'~' s. Dc~p.JlltJtc"t of Ci'liIElIg;neeriIJO. University of Glasgow

Synopsis The paper shows how a beanJ and colunJn support ,syste/n for a shear wall can be included in a plane franJe ifllalysis. This technique which is valid for both vertical cl/,d lateral load can 'be used 10 analyse tile wall and Its' supporl sysleln III one cO/nputer run. COluparlson willi finite elelJJelJ1 results illustrales I he accuracy. Introduction Many' shear wall structures are supported at first floor level on a frame system to allow large open spaces at ground floor level (Fig 1). This type of support arrangenlcnt nlay affect the behaviour of the wall under g"ravity and latcral load.

",A1EH~l LOAD 2 "'4/""

-1uN1f~~,","

"

,

OtS1RMkJt(C)I

,_." ..

\

i·D I .[1 I

"I:::

i:

i

I~

D. t /

r U"J

I :

.

t.

;

! :

i·

.tI

I

"

i

I :

.. ! .: ;

I : ;.

"

D

I. Ct.

A,

l:l stOREYS

Q

COHM:CIIfoG e(AAt

The (rarlle idealization Fig 2(0) illusl(at~s it typic<..tl tr(llue idc<.IIiLittion of (I shear wall supported on colun'\ns. The best way to deal with the rigid parts 01 the traole is to treat thelll as being fully rigid," i.e, to have an clcnlcnl in the COl1lputcr prOfJranl of the IYPl~ shown in Fig 2{b) (St~C Appendix 1). AltcnHltively. Ihe riUid parts can b(~ u;vc:n h;~Jh hut finite rigidity and (J stafl'd
:.::""

lea. Az

i - ----

Note: The continuous connection techniqueS has been applied 10 this type of problenl. This method has its attractions but the range of wall shapes which can be handled is Ii.nited and the calculations are such thut tho usc of a conlputor is dcsirablo. If a COlllputer is to be used then it is bettcr to adopt a Inethod which has Ill0re flexibility. Parameters affecting behaviour The following parameters have an inlportant effect on the behaviour of a wall supported as in Fig 1. 1. The position of the openings. With no openings or centrally located openings. vcrtical load is transnlHted to the supports by arching action. The support beam provides the tie for the arch and the wall (or connecting beams where central openings occur) transmits the compression at the crown of the arch. For off-centre openings the arching action is more complex since the beams do not coincide with the crown of the arch and arc thus recluired to transfer axial and shcar forces. Our hypothesis is that a secondary arch forms in the stiffer part of thcwall between the edge of the opcning and the support column. The relative n'lagnitude of this secondary arch depends on the relative stiffness of the support beam and the connecting beanls. 2. The axial and bending stiffness of the support beam. 3. The bending stiffness of the connecting beams. 4. The bending stiffness of the support colum::;.

,

,

t I

~,..l

\. SU"f'ORl IS£AM

COd",',."

\.0:

D£U\f.Ef. OF f'''''CCOQt.A-,

.,

-~l .tHtOtNfSS

- 0"'7::'..

\~

~"tMMt. fP!CA4.

.:

\.)rli.~If\.~

Fig 1. Wall shapes c,onsidered

: -

R'c;,O-'--~

·"-:-CON<\:ll:>NS

.

j-

(b) GENERAL!FORM

OF ELEMENT

Recent studies •• .:!,·7 haveshown that it is possible to use sin.. ple formulae estimate the force actions in support beanls of such shear walls under gravity load when no openings are present. When openings are pres'ent. especially if they are not synlnletrically placed. the development of forn,ula~ is inlpractical and sonle forr:ll of overall analysis is necessary. " This paper shows how a franle analysis can be used lo.analys·e relatively con'lplex shear wall configurations.

to'

~ . "T"h!

Srruclu(i\1

Enoinet!(IF~bu.il(Y 1973/No

2/Volume 51

r--l=~--1 ~StPPOAT 8t .... M

.

::

Co\'U,AN

(it) TYPICAL fRAME IDEALIZA TION

Fig 2. The {rjflue idealization 11

5. The finite width of the support colun'lns_ 6. The type of loading (vertical and lateral)_ The use of the frame idealization of Fig 2(a) is the simplest way to take accou.nt of all these parameters.

parameter aH is a measyre or this interilclion.-t aH is defined by tX

H Ib h b I

Comparison with finite elenlent results We use here finite element analysis 6 as a basis for defining the relative accuracy of the frame nlethod. The former method normally predicts stress and deformation to an accuracy equivalent to that of a perspex model with strain gauges, i.e. provided the nlesh size is chosen sensibly, solutions by finite element may be treated as being close to the 'exact' elastic solution. The basic wall shapes considered arc shown In Fig 2. The following were varied: (see Table 1). 1. Position of openings. Central and oft-centre cases are considered. 2. Horizontal and vertical load. The relative magnitudes of the vertical and lateral loadings are such that the vertical stress in the wall at the support beam level on the windward side will approach zero. 3. The stiffness of the connecting beams. This has an important effect on the degree of Interaction between the two wall sections. The dimensionless TABLE 1. Details of frames analysed

Az

Resulls

0·0875 0·015

I

At

Table 2 gives predictions of stress actions which are important in design, see also Figs 3 and 4. In all cases, except for the maximum tie force in the support bearn with off-centre openings, the frame solutions compare favourably with the finite element results. The reason for the discrepancy is that the frame analysis gives the tie force at the opening whereas the maxinlum tie force occurs below the stiffer wall section. A simple approximation Is now developed to inlprove on the frame results for tie force. First of all we postulate that when the openings are off-centre two arch systenlS form-a primary one and a secondary one as shown in Fig Sea). We therefore assume that the maximum tie force in the support beam T is approximated by

0-0875

0·00625

Elc

=

A(m 2 )

0·003125

hbJ

= height of wall

= moment of inertia of connecting beanl = storey height = clear width of openings = distance centre to centre of wall sections I,. = moment of Inertia of wall section A = area of wall section See Fig 1. Two values of aH were used. o:H 4 represents a wall where the connecting beams .are flexible enough to have an Important effect on the overall behaviour of the wall. With o:H = 15 the connecting bean,s arc stifl and there is a high degree of composite action b~twecn th(' wall sections.

-..-,---

--, -

J12". [~ ~- -~-- + -~-]-

=

0·3

T·- T"'I T"

(1)

TABLE 2. Comparison of force Actions by frame idealization and flnlte element Idealization

_____ (k_N_l

i ~c~~ ~r~~~~,~~I)llOrt

Finite Finite Finite F_ra_n_'c __I __c_lc_m_c_n_t_1_ _F_r_a_rn_e_ _ !__c_le_m_e_n_t_II__F_ra_m_e_ol__e_le_rn_e_n_t_:I__F_r_a_m_e_i.:

I__ I·

1----------. Maximum axial force "'0

co

o

..J

·f ~

t!)

In support beam

.3 ;;

.3~

+125·3

I

+104·6

+78·2

+63·9

+121·1

1·10-53

I

I 82·1

:

169·3

!-----i------I------II------I------I----- - - - - +152·8 + 127·1 + 104·6 + 178·8* +105·3 +203·9

1----------1------1-----1-----1-----111-----.1-----1-----!-----1

Shear force in support beam at opening

0

0

-100-6

-91-1

o

o

-155·3

-142·1

I-----

1------'----1·-----1-----'1-----1-----111-----·1-----1----:-- ----Shear In +1 I 0 0 -15·1 -64·5 0 0 -19·1 I --19·4 connecting

+2

o

o

-31·4

-31·5

0

0

-13·5

beams

+3 .

o

o

-11-9

-14·3

0

0

~I

±20·8

:1.19·9

:f:2·6

:! 3·9

Axial force in support beam at opening

~

-I- 125.3

Finite clement_

·!:2·6

Maximum t\xlal force In support beam

t

I·

:!:12-5

-·15·8

·-13·6

.I·S-1

·1

I

See note below

.----------!,-----I-----'I-----·I-----II-----·I-----II---_.-- - - - Shear force In support beam at openIng

:1;26·1! 19·8

Shear force

--4·1

-130·7

in connecting

-4-2

:+:33-3

I .~~~n~~

•

"-_3_

I-

1..~1~~

-1.60·0

±44·3

±79·5

±66·5

! 34

1:26-4

131·9

±12·S

:l14

:1-36-5

_t 27·5

:l:31·8

• .... ! 29·5

:'~~:~

__•

~ ~~.5 .•

I

__ • __

I

.1·17

1:11·6

.119.~ J ~.~~.~.. ..

!:106 I

12

.' 14·5 115·5

-:1 92·8

I-----

i

!

14·4

t

16-5

~

17·0

• By approximate formula (1) t L:ttcr"llOi\d on bC'i\n\ supported ShCClf w;\lIs dons produce axltll force in the support bean, which is generally Insignificant. In son'e f:a~I·~. hnwpv(!r. whC'"rt, the" :-:;ullport hl-:"" (:-:; ~(Hf In conlnnfi~on with Ih~ conn('ctinq h~:tn,~ tI,i~ r.flc"c:1 n';lV hC" a~ tllu("h :l~ 20 pC"~ ("{"'Ill of lho "Ialtll''''"1 a,rlal fuft:n clun to 11 •• IvItV louel. Thc" halllf;' hh-.,lIt"lion c:nnuul Ilrc·
"~i . ",,,

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....... Z

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SOO·

~

\

\'.... _.

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-(

v

;: soo

-' oC

\

v

CI.

w

;: -soo d

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\

PRIMARY ARCH

\'.

w

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1000

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I

r' !

-1000

"l",J

-11\)0

-.sao

fP-AM£

AHALYS. S

flNlTt

f.L(UE:4T ANAlYSIS

Fig 3. Vertical stress ilJwall. lale;al load case, a~H::: 15,.

(el) ARCHING ACTION

:..:

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L"

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r.:

+1~O

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-ISO

a Q.

:>

SECONDARY ARCH

+100-

or:(

x

.. ---

~~:\~ "19l

~

0'

va

-200

LEGEND

-A-.-A- FRAME ANA~YSIS --0---0-- o
ANALYSIS

= tie force due to primary arch =::: axial force in support beam from frame analysis

T. = tie force due to secondary arch ~.

shear in support beam for frame analysis

Justification The main assumption Is that the shear In the support beanlls equal to the vertical reaction of the secondary arch. We define this reaction as R and assume that the vertical load on the secondary arch is 2R (uniformly The Structural Engineer/February 1973/No

Fig 5. Arch action wilh off centre openings.

distributed). Taking a free body diagram atone half of the secondary arch (Fig 5(b» and assuming that the centre of compression of the arch is at a height L.12 above the tie beanl. (L li is the span of the secondary arch. Taking the centre of compression at L, /2 is normally conservative'.) Taking nloments about 0 in Fig 5(b)

L,

R 4" = T, L,/2

fig 4. Axial force in supporl beam. where Tp

(b) FREE BODY DIAGRAM OF SECONDARY ARCH

2/Vo'um~

51

. • T. = R _. •• 2 In view of the itnportance of this action and the degree of approximation used, it (s safer to take T. = R. Use of equation (1) gives satisfactorily conservative results for the cases given In Table 2. If a more accurate estimate Is requIred then a flolte element analysis can be used. However, nleasurements have Indicated stress levels In the tie steel much lower than calculations predict.2 73

Conclusion A shear wall with openings supported by a beam and eolunln system ean be satisfactorily analysed using a plane franle idealization wit~ rigid joints. This approach is much more efficient computationally than finitc elenlents and only marginally less accurate.

Appendix 1 Derivation of stiffness matrix for member with

(n)

rigid ends

For the member of Fig A 1(a) the relationship between thc end action :P.'u,: and the corresponding deformations : Viti': is

= [KM AIl ] (U... ,,~

{P At')

where [KM..clll = [T]T [H]T [KrJ [H] [T]

~(

rK,]

rclates the end actions of Fig A 1(b) ~Pr} (where . {P,: 7· :-: IN", M", M,,}) to the corresponding dcformations, Le.

AIL

rX,]

=~

E

8

0

0

0

Kn "

X""

0

X,,,,

K,,,,

21{1 ,. to)

= ...- -_..

[~

K""

:.-,c

0

1

0

l/L l/L

1

0

l/L

0

0

1/l

IT)""

~

l

1 0

0

Y1 0 0

Xl 0 0 0

0

1

0 0 1

0 0 C

0 0 0 1

0

Y2

L{28

H4

1/6)

~] 0 0

0 0 1 X2

institution notes

continued (ron, page 60

Environment, and by the Construction Industry Research and Information Association, as well as with projects of the Building Research Establishment. Application forms are obtainable from G. F. Goody, CIRIA. 6 Storey's Gate, London SW1P 3AU. and should be returned not later than 14 March.

Representation Th~

following members have nccel,tcd the invit;tClon of the Council to serve as Representatives in the capacities mentioned: , BSt Tecllnlcal Committee CEB9: Readynlixed concrete: Mr. J. H. Featherstone (F) replacing Mr. J. A. Derrington. aSI Technical Comnliltcc ISElt2: Steel for Bridqc~ f,nd General Building Cons(rClclion: Mr. J. E. Ti'ylor (M) rcplClcing Mr. S. A. C. Henry.

PYG

) Mb Pxb

(e)

Fig 1A. Metnber with rigid ends

2. Green. o. R., MacLeod. I. A., and Girnrdau. R. S •• 'Foree! action In shearwall support systems', pi\p~r pr('s('nled at Symposium on Response of Buildings Ie- Lntr,,,1 Forc('s. Bufialo, American Concr('l~ Inslilut<" 1971-

0 0 0 0 0

3. Macleod, I. A., 'Lftt('ral slinnes~ 01 sh("~, Willis with Ol)("n· Ings" Tall buildings (eds. Coull ilncl Stafford-Slnilh). Lonclon, Perg~mon, 1967.

4. Schwaighofer, J. and Microys, F. H .• 'AnC\lysis of sheeu Willi structures using standard computer pros:arcul'\S', JOl1rl1. Cone. Ins/, Vol. 66. No. 12. Dccen,bcr 1969, p. 1005.

A,,,.

1

References 1. Green, D. R.• 'The stress an;tlysis of shear walls'. PhD thesis, University of Glasgow, 1970.

Pyb

(t f p:;- ~--'~----i

[Tl'" relates :p. II ,: to :P",,; where :p.-t,,:7· :.: :PX..I, p,..." M,I, PXII,.P rll , Mil:

_

A

2/(1/3 ! B) •• - - - •• _- • . L(28 , 1/6)

[H1T' relates :P",.} to ~Pr: where :Pnl,lT = {Prn , P"tI, M", Prl" p.If'" M ,,:

r

l

(b)

equivalent shear area.

0

)Mb 8

r--··--1

28) = 1(1/3 .......- ...

K.",=.· K""

A=

A L2

IH] =

K"" ':::

He

5.. Rosman, R. t 'Approximate analysis of shear wnlls subjecllo lateral loads', Journ. Am. Conc. Inst, Vol. 61, No.6, June 1964, p. 717.

\ .

6. Zienciewicl. O. C. Tlte (illite clenu~nl ,net/.ocI in cng;lJC'cr;/JCI science, McGraw-Hili, 1971. . . 7. Green, D. R. 'The Interaction of solid shear Willis nnd their supporting structures,' Building Science. Vol. 7, 1972. p.239.

8S1 Code o( Practice Committee BLCP/84: Internal Partllions: Dr. W. W. L Chan (F). 8S1 Code o( Practice Committee BLCP/86: Flat Roofs: Mr. F. E. S. West (M). Organizing Commlltc~ (or Joint teE//SE Conference on Off-s/'ore Structures: Mr. J. A. Derrington (F) and Mr. W. J. Shirley (F).

CEI MacRoberl Award Evaluation Conlmiltee: Mr. E. C. Beck to represent ISEt ICE/IMunE replacing Dr. A. A. Fullon.

Forthcoming Examinations The next Institution Part 3 examination and Technican Test will be held on Friday 6 July 1973. For those non-grtlduate candidates who have completed or been exempted fronl the former Part 2 requirements of the Institution the July 1973 Part 3 examination is the last opportunity to complete the corporC\te membership requirements and obtCl;n registration as ft chtutered engineer by the end of 1973. Thereafter, it. will be recalled, none will be (ldnlitt~d

to membership of the Institution unless, prior to taking the Institution Pari 3, the CEI Part 2 examination has been ptlss~d or a qunlificC\tion cxr.,npling Ihercfro," (genernUy " UK engin~crino degree) has been obtained. Those non-graduates who have passed or were exempted from the Institution Pftr.2 prior to 1970 who wish to attempt the Part 3 examination after 1973 must first pass the 'cel 2 subiect test' details 01 which were given in the booklet AdnJission to tile Institution, a copy of which Is available on ftpp'ication to the Secretary. Entries for the July 1973 Pcut 3 ~JtClfnin.'­ lion nod Technician Test nllist be re· ceived at the Institution from home cnndid"tes not 1<'ler than 31 May 1973. The closing date (or overseas entries hC\s now passed. The ,,~xt eEl P;ut 1 ~nd ParI 2 cxtl.ninalions will b(~ ht~lci in M"y 1973: (he closing dilt!! for bolh hon1(' ilno ov{'rsnClS entries h~s

now (lClss("cL

3. Analysis of coupled nonplanar walls

3.1 Methods of analysis Coupled nonplanar walls maybe analysed by: (1) continuous connection method; (2) frame method; and (3) fmite element/strip method. The continuous connection method has the advantage that for uniform sections, differential equations can be established from which closed form solutions can be obtained. It is therefore the most convenient approach for simpler problems, especially where hand calculation is possible or design charts are available for the solution. The general application of the method is, however, difficult as it becomes rather complicated in dealing with non-uniform structures. On the other hand, the fmite element method, being the most powerful tool of analysis available, is applicable to any type of building structure, but the cost of data preparation and computation is often intractable.

22

It is therefore apparent that, relatively speaking, the frame method is the most generally useful approach. It is computationally more efficient than the [mite element method and yet sufficiently versatile to handle most building structures. It also has the advantage of allowing straightforward interpretation of the structural behaviour of the elements. In fact, it is the most popular method among design engineers. Herein, only the frame method is studied.

23

3.2 Frame method Two distinct frame methods have been developed to deal three dimensional coupled nonplanar wall structures. Heidebrecht and Swift (1971) and Taranath (197"5, 1976) developed special three dimensional nonplanar wall elements, with seven degrees of freedom per node to model the nonplanar wall units. They used Vlasov's open-section beam theory to account for warping of the nonplanar walls and to derive the stiffness matrices' of the nonplanar wall elements. As a results, their . methodology is severely limited by the intrinsic assumption made in Vlasov theory that the shear deformation of the walls is negligible. While this is not really a problem for open sections, it can become a serious problem for core walls closed by lintel beams, because the lintel beams could induce substantial Bredt shear flow round the cores. The method would thus lead to significant errors in the analysis of partially closed core walls, and in the case of nearly closed core walls, it becomes totally inapplicable (Kahn and Stafford Smith 1975, Rutenberg and Tso 1975). Moreover, since evaluation of the shear centres and sectorial coordinates of the nonplanar walls are required, the method is rather inconvenient to. use, especially for those engineers who are not familiar with warping theories.

24

In contrast, MacLeod (1973, 1976) and MacLeod and Hosny (1977) adopted a more fundamental approach of modelling the nonplanar shear walls as assemblies of inter-connected planar wall units, with the warping displacement of the nonplanar walls evaluated as an integral part of the solution. Determination of the shear centres and sectorial coordinates of the nonplanar walls is no longer required, and the Vlasov theory is dispensed with all together. Relatively, the frame method developed by MacLeod is much more practical than the frame method which relies on Vlasov theory. However, in the 90s, Kwan has found two problems with MacLeod'sframe method: - the practice of taking the beam end rotations as the rotations of the horizontal rigid arms is erroneous; this would lead to incompatibility between the beam and wall elements and consequently underestimation of beam stiffness; the vertical displacements of adjacent wall elements are not compatible and as a result, artificial flexure of the wall elements occur leading eventually errors in bending stresses. Herein, only the frame methods developed by MacLeod and Kwan are presented.

25

UDC 69.022.32:624.042

Structural analysis of wall systems I. A. Macleod,

SSe, PhD, CEng, MIStructE MICE

Professor 01 Civil Engine~ing" Paislt!y College 01 Technology

Introduction The term "shear wall" has the connotation of a high wall which

The $tructural mode of action of shear walls If a shear wall is considered over th3 height of one storey it looks like a deep beam (Fig 1 a) and if experience with frames is extrapolated to shear walls then there would be a temptation to treat the wall as a deep beam. This would normally be incorrect. For most shear walls the dominant deformation mode is bending which (unlike the shear mode deformation of an unbraced frame which is effectively uncoupled· between the storeys) demands that the wall be treated as a building height unit. Looked at as a building height unit a shear wall can normally be analysed without recourse to deep beam theory.

resists lateral 'oad but in fact aU load bearing waifs "are laterally stiff and will tend to be the principal lateral bracing elements whether designed to do so or no·t. In the past walls tended not to be designed for lateral load. This situation is changing in that firstfy we need to inject some science into the design process as we build higher, and secondly modern science is capable· of helping to tackle such problems. In this paper the recent developments in techniques for analysis of waifs are reviewed. These techniques are principally used for laterallaad but analysis under verticallaad can also be useful. Since this cannot be a comprehensive statement I intend to concentrate on what I believe to be the more practical techniques and make no apology far the obvious personal bias. The term"she'ar wall" is used to denote any load bearing wall under lateral load.

Analysis ofplane walls A single shear wall without openings therefore requires. only elementary analytical techniques. However, a fair proportion of walls do have at least one vertical row of openings and the analysis of these is less straightforward. A large number of papers have been published in recent years on the analysis of this type of problem. Fortunately this work is easily summarised. There is only one practical model for analysis of waifs with row ~penin~~ i.e. the frame J!!Q9~r.~f.:fi.g .1!?..~01~~~ ~ither by the ~Stiffness method' or the ·Continu~us.~9nnectipnmethod"; It s'lou'rd be 'noted that to· achieve accuracy equivalent to that from the frame model with plane stress finite elements an expensively large number af. elements is required and finite element solutions should normally be avoided for analysis of wall systems except possibly to examine detailed problems.

Synopsis Significant developments in techniques for analysing complex wallsystems have occurred over the past 15 years" These developments are reviewed and the relative importance of various parameters is discussed. The behaviour and analysis ofboth plane and 3-D wallsystems is considered.

Continuous Connection Bet.oJeen \o;a 11

Rigid

Sections.

Column

(Flexible)

Storeyl· Height

Unit

(a)

Plane Shear Wall Without Openings.

Fig 1. The frame

(b)

Frame Hodel for Wa 11 wi th Row 0 f

(c)

Continuous

Connection Method.

Openings.

model

The Structural Engineer/November 1977/No. 11/Volume 55

487

Continuous connection method This is normally described as a separate method but it is worthwhile to note that the basic model is 1hat of Fig 1b with the extra assumption that the connecting beams provide a continuous connection between the wall sections (Fig 1 c).

This is normally a good assumption (or high uniform walls where the wall sections are stiff compared with the connecting beams. Several solutions with varying numbers of rows of openings, different support conditions and variation of properties with height are available. "'- 's Reaaily available connection sc;>lutions will be more efficient than corresponding stiffness solutions but it is my experience that the versa tility of the latter type of solution far outweighs the fact that it requires a programme for a medium size computer.

.--n---.__--.; . .__-.i._x_Z_i

of c~~=­ ~;o,~.'; 11

I

:.f~,,:lcc~~~

. I

Hand analysis or computer analysis Hand solutions ~re possible with the continuous connection method although in many cases the use of a small computer is advisable. '6 It seems that there is a reluctance on the part of designers to use anything more than the simplest of analytical techniques. Many situations do occur where hand analysis is not realistic and some form of computer technique is essential.

line (lement wi th Benc:i,uJ and Shea,. ;:)e(o""" t i on A~ial.

l

~

Rigid Part

xz

j

12

r-. (b)

Shear deformation Terms which approximate the effect of shear deformation can easily be included in the frame element stiffness matrix 17 although Poisson·s ratio and an equivalent she~r area factor have to be added as data for each element. Shear deformation can also be included in continuous connection solutions. Fig 3 indicates the contrib~tion of shear def;~matiOn-to-to-p­ deflection of a cantilever under uniformly distributed load. Note that sheardeformatjon is much more important in flanged walls. Its effect on stress is however. much less significant than its effecton d~.fp!.f1J~Jion•..__.__ - _. . . __ It is worthwhile to include shear deformation in a computer frame analysis of wall structures but lack of this facility in the programme to be used should not normally cause results to be unacceptable.

Rotationa J Spring

COl~

Fig 2. 8 asic elements

The frame method (stiffness analysis) ',2,3 The basic difference between the frame method for walls and normal frame analysis is that the finite widths of the members and in particular those of the "columns' cannot be neglected. The term "wide column frame" is sometimes used to describe this method but f now find this description somewhat restricting and prefer to use the term "frame method" to denote a stiffness solution which takes account of the finite widths of the members as appropriate. The method requires the use of a plane frame programme with the non-standard elements shown in Fig 2 and is applica~re to a wide range of problems with no restriction on loading, variation with height or degree of foundation fixity.

1.. 25

X

The stiffness of the connecting beams There are two basic problems here i.e. : 1. How good is the assumption of full fixity between the beams and the walls 7 This problem has been studied'8,'9 and opinions vary. I tE~nd to ignore the effect of the ffexibility

ASF(H/D)2

100 + 6 BID

o

V')

o

<:

)(

+

en

a::l

<

80

_---1_

<:

c a C+6

f.=

top

b

I H

60

010

e .... lva

Q.I ' -

-Gol

c...o ~Ic..

40

'0

CJ

a.c:. f- V')

Plan

cO'\

-

Top Deflection Due·Shear Only .

c:

s..

+6

'-

Q,I

OU

Top Deflection Due Bending Only

20

1.0

I- l.I.I Q\

0.5 0.2 0.0

QJ

H2:

2

3

B/D

ASF

Area Shear Factor 5/6 when B = 0 • 1.0 when B i 0 E

5

4 H/O

Fig 3. Effect of sh~ar deformation

488

The Structural Engineer/November 1977/No. 11/Volume 55

~

____.__----,S

of this connectIon unlE:~s the re,nfofced detailing is such that a pin is likely to develop.~~C!,~~_ea~'i...toinct!J-.9~_ an estimate of this flexibility or to create a pin via the

~ (ic;~I spr~~~~~~~~~ ~ !~~~_~~t.;'~e.~~ ih~ ~·igl~~~=f!qj~_~,~~b.f~-

-p-~~lhe 1.!ame..eLement l FiY.)J.l1. 2. ~What is the contribution of the floor slabs to the connecting beam stiffness? Again this has been studied 20 , 21., 22 but we do not yet know the best assumption to make in all situations.

~,.,~

~~

be

o-'-c-.. . ~·lf_',I,.

,

[The nodeJ.!.~ed.i!.(!1,la_b~ Each node in a s((ucturaf idealisation is normally assi;)ned certain degrees of freedom. (In this context the degree of freedom is the position and direction which are common to a given force and its corresponding deformation) .. Some pro· gramming systems assign freedom numbers (or Code numbers"') to degrees ot freedom of the structure. Tr.e 'node freedom table' then defines the relationship between freedom numbers and "node numbers". For example. in the portal frame of Fig 4, nodes 1 and 2 have three degrees of freedo:n

The effect 01 cunlleclluns between IJrge panels

In large panel construction the connections between the units do affect the overall flexibility of the structure. This ffexibility can be modelled by introducing a 'column' between adjacent nodes on either side of the connection at each storey level. 23 The axial stiffness of the column then models the storey height stiffness of the vertical connection between waHs.24&.27 If would appear that this joint flexibility can be neglected in taU large panel buifdings 2J • 28 but it may have greater significance in lower buildings. Foundation fixity A small amount of rotational movernent at the base'

0

5

'\

t \

Freedom

I

Numbers

6 -----"4

~Ll

~

'Node Nur."ber

a taU

cantilever can have a most significant effect on its overall stiffness. The introduction of foundation springs in the 'Frame method' presents no difficulty but the values of the spring stiffnesses to be used are not at all easy to define.

Behaviour of three dimensional wall systems It is normal practice to treat a load bearing wall structure under latera' load as a system of parallel walls connected by floor slabs. Analysis is carried out for loading parallel to the two major axes of the structure. In generat out of plane bending of the walls is justifiably neglected as is bending of the floor slabs except as previously noted under 'Stiffness of the connecting beams'. The role of the floor slabs bending in .their own plane is discussed later. I n many cases the wans perpendicular to the direction of loading act as flanges to the walls parallel to the loading and the whole system behaves rather like a closed cell system provided the connections between the walls have sufficient strength and stiffness, As will be. shown, this three dimen· sic.;nal behaviour can now be mode!led in a quite straightforward way. For lateral load analysis,the best approach is to carry out a preliminary distribution of the lateral load to the waifs in proportion to their top stiffness (the calculation of which may require a computer analysis). This gives a feel for the importance of the various parts of the structure and it can then be decided whether a more complete analysis is required. Vertical load is transmitted by the connections between the openings. the effectiveness of which can be estimated. 29 Alternatively., a better solution will be obtained by applying vertical load to the frame model.

HOOE fREEDOH TABLE

XF

NODE

YF

THETAF

2

J

2

4

5

6

J

o

o

o

4

o

o

o

XF - Number of

X Direction Freedom YF - Number of Y Direction freedom THETAF - Number of Rotational Freedom o - Restrained freedom

Fig 4. The node freedom tab/~

.

f \5

-_1

r
II

.n FRAME

Analvsis of wall systams in three dimensions Again a frame model is the only practical approach although solution techniques tend to be more complex than for walls in a single plane. Variations on the continl;]ous connection theme have been devised J '· JJ but discrete element solutions have greater versatility) ...... '. I propose to discuss here ontya discrete element stiffness app:oach which I believe has the best versatility in combination with a simple scheme for data input. ~2 : .. A. three dimensional (rame model need not involve nod'e·s ~ ~ with six degrees of freedom and more efficient solutions can i be obtained using systems of interconnected plane frames. ~ -·There .are various ways by which interconnections can be • made. The ~~~.o···.~e~hodS) discussed here are both readily understood by the analyst and allow wide ffexibility in use. The Structural Engineer/November 1977 INc. 1 t /Volume 55

WAll

Hoo( fR((OC't rAel[

xf

;, s

Fig 5, No axial deform 3 cion of bei1ms

489

I \::' ,: -..----------~' -

__ 1.

4

fRAMES

SYSTEM

Dire c t i on

0

f

NODE

loading

FREEDOM TABLE

NODE

XF

YF

THETAF

1

1

2

3

2

1

4

5

3

1

6

-

~

8

1

4 ~

7 ,

9

_e1~

PLAN Fig 6. Para/ell walls connected (no torsion)

2( \s

2f \3 ----- 1

D

\..:/

CD

®

\6.

Loading

(c)

I Wa 11 1. I---Centroidal Axis-

1 Axis -

Wall 2. (b)

FRAMES

NODE FREEDOM TABLE

®

PLAN

~

I

SYSTEM

t

~.

I

Wa 11 1.

Direction

-3


leen troi da

of

f "\7

.J)

I

(a)

6

NODE

XF

YF

1

1

2

3

2

4

2

5

3

4

6

7

~

THETAF

et( .

Fig 7. Walls at right angles (no torsion) 490

The Structural Engineer/November 1977/No. 11/Vofume 55

4

21

1

\3

10

'\] \

--5

-1

1 \11

l~ "5 '-

.... ~ ,.~

-I

.......

J i .-.j

"7 • ..:.-!

_J 5

1. \

f

at

J

~1

SYSTEM

FRA:-l[S

9

f\ l-~

\8

12

ROOF SLAB

r'-'

•

"~

I

Direction of loading

9

NODE FREEDOM TAelE NO~E

XF

I!

YF

THETAF

1

I

2

3

1

4

6

7

1

PLAN

2

0

3

5

4

0

5

9

6

0

j j

i i

I

5

i i

10

I etr

9

I

8

,

11

12

Fig 8. Walls connected by floor slabs (with torsion.(S)

(normal fora plane frame). The relevant-node freedom' table is given in· Fig 4 .(freedom number 0 represents a restra int). Such tables are used to set up the structural stiffness matrix and would normally be established automatically by ths computer programme. However, if the table is read as data and freedoms at different nodes are given the same freedom number in the table. this wiUforce the deformations which correspond to these freedoms to be the same. Figs 5, 6, 7 and 8 show four examples of how this technique can be used to reduce the order of solution and to model three. di.~ensi~~_!1

floor Slab (Assumed R;9 id )

afl,\

@~ master-slavetec.'!!!iqu!] This technique was developed independently a few yearsago.c... using the terms "Definitive- and -Redunda"r- rather than •Master- and -Slave-. The Americans use the latter terminology which is somewhat more picturesque and is adopted here. The Structural Engineer/November 1977/No. 11/Volume 55

t 2/~·~er

11

Displaced

f]

.lition

w~--T_--f-=-~x

behaviour. Thus to create-i-three-· diniensfon3r system, the\ node freedom table is read as data by the programme bU~__\ otherwi~e ~~~~~~_.~s t~e.sa~e as "that for Plane.f~a"l~S.-:._~===, _" VirtuallY any system of frames and floors which are at fight : ~" angles to each other can be analysed using this technique. _" : ---rTh~;e--are'however: some practical limitations. First of one naturally wishes tf? n:'i.nimis~ ~he" cost of data prep~rati~~".J and com uter time. lThis is where the preliminary analysis can help to ens~r~·that only the important parts of the stru.c~u~_t: " are included in the three dimensional model.' Secondly, be ~ i careful with the numerical stability of the calculations.. Tall, \ flexible units interconnected with taU stiff units can give . trouble and it may be better to remove the flexible part from :" I; the analysis if it is of secondary importance. :·With respect to - ~~merical conditioning of a solution the more significant figures (i.e. the longer the word length) used in the solution routine the better. .

freedom

1_

Sa

.1

Sa

I

Fig 9. Plan of3-wall system

As mentioned before. it is normal to assign degrees of freedom to nodes but in some cases these freedoms are not independent but are directly related. As a simple example,consider the floor slab shown in Fig 9. If we wish to aSsume that the floor is rigid in its own plane then (neglecting deflection in the X direction) two of the three degrees of freedom shown are sufficient to define the in-plane movements. If freedoms 1 and 3 are defined as master freedoms, freedom 2 will be the slave freedom since the corresponding deformations are related bV

A2

i.e. where and

-

[0·50-5] {~:} .-

[Al{.1",} {.d.l} is the vector of slave deformations {.dill} is the vector of master deformations

.d.. -

TheA matrix is used to eliminate. the slave freedoms as variables in the stiffnesssolution 44• In general, any three 491

freedoms which are not co-linear can define tOe movement of a plane and all otter freedoms in that plane can be eliminated by thi~ technique (if the prane is rigid)_ The fonowing scheme for defining relationships between rigidly connected freedoms has been developed to simplify data input_ ,..

t

A method of establishing the A matrix from this information ;s given in Appendix L This technique can be used to analyse a wide variety of systems of connected plates assuming only membrane action. Th"e main applications of the master-slave technique in building structures are in modelling: 1. The action of floor slabs as fully rigid in their own plane. 2. The three dimensional behaviour of systems of walls which are not at right angles to each other_ Other uses are to connect frame elements to finite elements. simulate a rigid pile capl etc.

I.

Fig 10. Plan with wallnot at right angles As previously mentioned. a freedom defines a position and direction. A convenient way to define a freedom ;s therefore. bV two co-ordinate points. the first of which ;s the position of the freedom and the direction of the freedom is from the first point to the second po;nt. For example. for the system of walls shown in Fig 10 the master freedoms can be defined by

Xl

t Hode

f

X2

1

t

·'

Degrees of Freedom

Hode 2

1/ L

Freedom number

Position

Direction

1

0·0. 5-0 15·0.10-0 15·0. 0-0

0-0. 6-0 16-0.. 10-0 16-0. 0-0

Position

Direction

2 3 and the slave freedom by Freedom number

4

30-0, 0-0 The co-ordinate position quoted above are in order x. y.

Node 4~

/'

t

,.

X3

X4

25-0. 5-0

.,

Convention As in Fig.

z.

Fig 11. The:solid wall element

10

I hI -8

-------~.

D

o

FlWiES

NODE fREEDOM TABLE

SYSTEM

0r

zn

-"""tz.,....,..zJ;D

,....,e::z-.z--a

(0

®

P;W:i1

t

Direction

of loading

PLAN Fig 12. Web wall without opening-the solid wall element 492

NODE

XF

YF

1

1

2

3

Z

1

4

5

3

6

4

0

4

6

7

0

5

8

7

9

6

8

10

11

------- ------

etc.

THETAF

:...--

-

The Structural Engineer/November 1977{No. 11/Volume 55

5

4

7

4

(1 I --1

I

9 . __8

9

2

I 1

10 ,

~ :Z"

D UAlLS SYSTEH HODE FREEDOH TABLE

~~~""';"''-''';"".,L.~:;-L..dIIII:t1'"'

~ Bracing Element See Ref. 46.

YF

XF

WOE

THETAF

.1

1

2

3

Z

1

4

5

3

6

4

0

4

6

7

0

S

R

7

n

6

8

9

0

7

10

9

0

8

10

2

0

~L-----...:.tc.

PLAN carry out the necessary calculations. In other words, if suitable material laws can be defined then the technology is available to process them into an overall analysis. However, the cost of computer time for large scale dyna:r:ic non-linear analysis is such that one could not describe this yet as a practical design procedure except for very expensive structures. Therefore. the step from linear to non-linear analysis is not a simple progression and a better understanding of detailed behaviour is needed before accurate. predictions can be contemplated. ..!..!'eli~ve t.ha~ th~. bigge~t I?~!ential be')efi's from research in building aesign could come from observatio~ of . ~t"er~al behaviour of buildings rather than from developments' in analysis. .

Fig 13. Core system-the solid wall element

The solid wall element In some cases one needs to connect to a wall at both its edges. This' is quite easy if the wall has a row of openings but when the connecting wall has no openings the normal frame system requires modification since the column element of Fig 2 has only two nodes. The connection can· be achieved using the "master.;.slave· facility previously described but it is more efficient to use' a special element-the solid wall element shown in Fig 11 .6. Fig 12 shows the solid wall element used to. model a· web waflwith no openings and. Fig 13 shows how a core system can be treated.. G•

~ ,

--

--.-

Treatment ofthe In-plane action of the /Ioor slabs It is normal practice to assume that the floor slabs are fully rigid in their own planes although this assumption may· be somewhat inaccurate for long narrow structures J8 , .7, "S. Fig 8 shows how in-plane bending can· be introduced in the frame method and the fully rigid assumption can ·be implemented via the "master-slave- technique. The use of the fully rigid in-plane floor assumption has opposing effects. Firstly., it will improve the numerical conditioning since it removes high but finite stiffness from the structural stiffness matrix. Secondly, it will decrease the number of freedoms but effectively double the band width of the structural stiffness matrix resulting in increased computing costs.

r--------- -------..--

_..

.-.__ .

I

.(

• --.J-

Future developments in analysis of load bearing wall .tructu.... A more accurate model could include: 1. Prediction of post yield behaviour to provide more realistic estimates of ultimate strength. 2. Prediction of behaviour under earthquake conditions. 3. Prediction of long term behaviour including effect of creep. shrinkage and soil deformation. The earthquake problem is being given considerable attention at present. particularly in USA and New Zealand. It is clear from this work .that our knowledge of the detailed ~haviour of the materials and connections in building structures is considerably le~ well developed than· our ability to

The Structural Engineer/November 19TI/No. 11/Volume 55

Conclusions There is no doubt that the frame n,odel is a good representation of the equivalent· monolithic elastic system. One would normally expect differences b~tween results from this model and the real system due to : 1. Whatever values are adopted for the elastic constants to be used they will only give a very crude approximation to the real behaviour. For this reason estimates of deflection in particular will not correlate well with real behaviour except bvchance. 2. The difficulty of estimating the stiffness of the connectklg beams. 3. The effect of joints in large panel construction or in brickwork buildings. 4. The difficulty of defining the foundation fixity. With all these imponderables is the elastic frame model worthwhile 7 Is it any better than simple bending theory which does not take account· of the flexibility of the connecting beams 7 My answer to both these questions is yes. Use of the techniques described in this paper does promote a better understanding of . the· behaviour of a' load bearing wall structure and will lead to improved design if used with good judgement.. Appendix 1 In Fig 14 freedoms 1, 2 and 3 are master freedoms and S is a slave freedom. If all these freedoms move in a rigid X- Y plane then the force and deformation corresponding to freedom· S -are not independent of those corresponding to freedoms 1, 493

convention for M. Similarly P s is transformed to the" origin:

y

.... (2) a

where {Pols is the vector of resultant stave actions at the origin. and P s is the slave action.

s

Inverting (1) {P}m - (B]-' {Pa}", \lI;c:=:::-----------------~x

.... (3)

{PO}m is staticaUy equivalent to {Po}s i.e.

{PalM - {Pol,

.... (4)

Fig 14. Master andslavIJ freedoms

Substituting (2) and (4) in (3) gives

2 and 3. The equilibrium relationship between these freedoms can be defined as follows: P,. P 2 and P3 can be transformed to a system of force actions at the origin of the X .. Y system viz (Pi is the force correspondi og to freedom i) :

-lo ~~s:.l ~~S~2 ~~~3] (;:J {;;l MJ a, P3

i.e.

{po}m - [A]I p.

with the chosen

thus :Aj' - (8]-1 [C] is the tran"spose of the required A matrix"'~. If the master" or slave freedom is rotational then the corresponding column of (8J or (C] is (transposed) {a 0 1}. Further slave freedoms can be added to increase the number of columns of ~A]t. Situations with two or single master freedoms can be handfea similarly.

1. Candy, C. F. ·Analysis of shear-wall frames by computer.... New Zealand Engineering (Wellington).. Vol. 19.. No.9. Sept. 15. 1964. pp. 342-347. 2. Macleod. I A. 'lateral stiffness of shear walls with openings". Proceedings Symposium on" Tall Buildings with Panicufar Reference to ShearWaU Structures. University ofSouthampton" April. 1966. Pergamon Press. 1 967. pp. 223-252. 3. Macfeod. I. A. &. Green. D. R. ·Frame idealisation for shear wall support systems·.. Structural Engineer 1973. Vol. 51 .. No.2. Feb. pp. 71-74. 4. Beck.. H. 'Contribution to the analysis of coupled shear wafts'. American Concrete Institute Journ.1 Proceedings., Vol. 59" Aug. 1962. p. 1055. · 5. Rosman. R. 'Approximate analysis of shear walls subject to lateral 10ads'pAmerican Concrete InstiluteJournalProceedings, Vol. 61. June 1964.. p;. 717. 6. Rosman. R. Tables for the internal forces of pierced shear walls subject to lateral loads'" Bauingenieur.. Praxis.· Heft 66. W. t"rnst and Sohn. Bef.in. 1966. 7. Rosman. R. "Statik und dynamik der scheibensysteme des hochbaues"" Springer.. Verlag. Berlin. West Germany. 1968. a. Traum. E. E. "Multistorev pierc"ed shear walls of variable crosssection. Proceedings. Symposium on tall buildings with particular reference to shear wall structures. University of Southampton. April. 1966.PergamonPress. 1967. pp. 181 .. 206. 9. Coull. A. and Choudhury.. J. R. "Analysis of coupled shear walls'. American Concrete Institut" Journal Proceedings. Vol. 64. September. 1967. p. 587. .".:f O. Coull. A. 'Pierced shear walls. of stepwise variable thickness .. Journal of the Structural Division American Society of Civil Engineers" Vol. 100. No. S.T..5" May 1974. p. 1157. 11. Coull. A. and Chantaksinopas. B. "Design curves for coupled chear walls on flexible bases"" Proceedings Institution of Civil Engineers" Vol. 57. Part 2. December 1974. 12. Coull. A. and Subedi. N. K. 'Coupled shear walls with two and three bands of openings', Buildings Science" Vot 7, 1972. pp.81-86. 13. Rosman. R. "The effect of temperature on multi-storey structures'. Proceedings, Institution of Civil Engineers, Vol. II. 71-72.. No.8. February 1972. pp. 359-373. " 14. Tso. W. K. and Chan. P. C.I<. "Flexible foundation effect on coupled shear shear walls·" American Concrettf Institut~ Journ6/. Proceedings" Vol. 69. No. 65.. November 1972. pp. 678-683. Authors erasure" American concrete Institute Journal· Proceedings Vol. No.5. May 1973. pp. 372-373. 1 S. Rosman. R. 'Pierced walls subject to gravity loads"" Concrete (london). June. 1968. pp. 252-258. 16. Pearce. O. J .• and Mathews. D. D. "An appraisal of the design of shear walls in box-frame structures"" Report No. 0.690.. 12/1 Directorate of Civil Engineering Development.

Department of the Environment" Croydon. England. 1973. pp.41. '..17. Macleod.. I. A. "General frame element for shear waif analysis'. Proceedings" Institution of Civil Engineers" Part 2.. 1976.. Vol. 61, December. pp. 785-790. 18. Michael. D. the effect of local wall deformations on the elastic interaction of cross walls coupled by beams·. Proceed· ings. Symposium on Tall Buildings with Particular Reference to Shear Wall Structures. University of Southampton. Apr.1. 1966.. Pergamon Press. 1967. pp. 253·272. 19. Harrison. T.. Siddall. J. M. and Yeadon. R. E. "A modified beam stiffness matrix for interconnected shear walls'. BUilding Sci~nce, Vol. 10.. No. 2. July. 1975. pp. 89-94. 20 Barnard" P. R.. and Schwaighoffer. J. ,.he interaction of shear walls connected solely through slabs·" Proceedings.. Sympo· sium on TaU Buildings with Particular Reference to Shear Wall Structures.. University of Southampton" April. 1 966. Pergamon Press. 1967. pp. 157-180. 21. Quadeer. A." and Stafford Smith. B. ,.he bending stiffness of slabs· connecting shear walls". American Concrete Institute. June. 1969. :>p. 464-473. (Discussion: December. 1969. pp. 1021-1022). 22. coun. A•• and EI Hag.. A. A. ·Effective coupling of shear walls by floor slabs·, American Concrete Institute Journal. August" 1975. pp. 429-431. 23. Macleod. r. A. and Green.. D. R. 'Three dimensional analysis of shear wall buildings'. Bulletin International Association for Space & SheU Structures. No. 60. 1975. 24. Burnettr E. F" P.. and Re;endra. R. C. S.• 1972" ·'nfluence of joi nts in panefized structural systems.'. Americm Society of Civil Engineers Journal. 98 (ST9). pp. 1943-1955. 25. Pollner. E." Tso.. W. K... and Heidebrecht.. A. C. "Analysis of shear walls in large panel construction', Canadian Journal of Civil Engineering" Vol. 2. No.3.. pp. 357-367.. 1975. 26. Nayar. K. K. and Coull. A. ·Elastoplastic analysis of coupled shear walls'" Journal of the Structural Division" American Society of Civil Engineers" Vol. 102. No. ST9. Proceedings Paper. 12401 September 1976.. pp. 1845-1860. 27. Bluger" F. 'Oetermination of deformability characteristics of ve11ical shear joints in precast buildings". Building & Environ· ment. Vol. II. No.4 1976: pp. 217-232. 28. Bhatt." P. "Influence of Vertical joints on the behaviour of precast shear waUs·. Building Science. Vot 8. pp. 221·224. Perg~mon Press" 1973. ~ 29. MacLeod. I. A.." and Hosny, H. The distribution of vertical load in shear waif buildings". The Structural Enginel:r. Vol. 54. No. 2.. February 1976. pp. 67-71 .. ~. 30. Rosman. R. •Analysis of spatial concrete shear wall systems·.. Proceedings. Institution of Civil Engineers (London).. Supplement vi. Pa~( 7266 s. 1970. pp. 131-152 (Oi~ussion: : :~Supplement xvi. 1970. pp. 371-374).

62

"3

i.e:2{Po}", - [B]{P}m where

•.•. (1)

{Po}/ft is the vector of resultant master {P}/ft is the vector of master actions.

origin an

The sign of

8, must be chosen to

~ccord

actions at the

References

494

The Structural Engineer/November 1977/No. 11/Volume 55

31.

Petersson, H. 'Analysis of Loadbearing Walls in Multistorey BuildIngs', Chalmers Institute of Tp.chnology, Gothenborg, Sweden. 1974. '2 Coull. A.• and Irwin. A. W. 'Analysis of load distribution in . \,3 . multlstorey shear wall structures'. The Structural £ nglneer.. August. 1970, pp. 301.:306. . 33. _ Biswas, J. K., and Two. W. K. 'Three-Dimensional Analysis V of Shear Wall Buildings Subjected to Lateral loads', Journal the Structural Division. American Society of Civil £ngine.ers, Vol. 100. No. STS. May. 1974. PP. 1019·1036.- (Discussion: January, 1975, pp. 357 -358; Julv. 1975, p. 1616). 34. - Clough. R. W., King, I. P., and Wilson, E. L 'Structural a.n~lysi·s of multi-story buildings', -Journal of the Structural DIvIsion. American Society 01· Civil Engineers, Vol. 90, ST3, 1964.. pages 19-34. . 35. Weaver, William~ Jr., and Nelson. M. F. ahree-dlmenslonal analysis of tier buildings', Journal of the Structural Division, American Society 0/ Civil Engineers, Vol. 92, ST6.. December, 1966, pages 385-404. . ,36 Winokur. Arnold and Gluck, Jacob, 'lateral loads in asym· \J • metric multistorey structures'. Journal of the Structural Division, Arne/ican Society of C;vil Engineers. Vol. 94". ST3, March. 1968, pages 645-656. :'7. Webster. J. A. 'The static and dynamic analysis of orthogonal structures composed of shear walls and frames', Proceedings, Symposium on Tall Buildings with particular reference to Shear Wall Structures, UniverSItY of Southampton April 1966 Pe/gamon Pess, 1967 pages 377-399. 38. Goldberg. J. E. "Analysis of mult;storey buildings considering shear wall and floor deformations', Proceedings. Symposium on Tall Buildings with pan.cular reference' to Shear Wall Structures, University of Southampton, April. 1966, Pe/gamon Press. 1967. pp. 349-375. , 39.. Heidebrecht. A. C.• and Swift, R. D. 'Analysis of Asymmetrical \.. Coupled Shear Walls', Journal of the Structural Division, American Society of Civil Enginee/s, Vol. 97, No. SIS. May 1 971. pp. 1407 ·1422.

The Structural En~ineer/November 1977/No. t1/Volume 55

MacLeod. I. A. 'Analysis of shear wall buildings by the frame method·" Proceedings. Institution of Civil Engineers, Vol. 55. September. 1973. pp. 593-603. . 41. Taranath. B. S. ·Analysis of interconnected open section shear wall structures· Journal of the Structural Division. American Society 01 Civil Engineers. Vol. 101, No. ST11. November, 1975, pp. 2367 -2384. 42. MacLeod. I. A, ·Shear-Wall/l'. Genesys Subsystem Users Manual. Genesys Ltd., Loughborough, 1976. 43. Rubinstein 'Matrix compu(er analysis of structures·, Prentice Hall, 1966. 40.

44.

45.

46.

47.

48.

49.

50.

Macleod. I. A .• Wilson. W .• Bhatt. P.• and Green. O. R. 'Two· dimens'onal treatment of complex structures·, P/oceedings. Institution of Civil £ngine,ers (London), Vol. 53" part 2, December, 1972. pp. 589·596. Macleod. I. A. 'Analysis of Tall Buildings InCluding Torsion' Proceedings of the Conference on Tall Buildings. Hong Kong. \ September. 1976. MacLeod. I. A.• and Hosny, H. M. "Frame analysis of shear wall cores'.. Paper presented at American Society of C,Vil Engineers Specially Conference, Madison" Wisconsin, U.S.A., August, 1976. Dickson. M. G. T., and Nilson. A. H. 'Analysis of cellular buildings for. lateral loads'. American Concrete Institute Jou/nal. December, 1970.. pp. 963-966. Irwin" A. W. ·Analysis of tall shear wall buildings inclUding in-plane floor deformations', Build International, Vol. 8, No.1, January-February. 1975. pp. 43-55. _ Singh" G.• and Schwaighofer, J. 'A bibliography on shear walls'" Report Depa/tment a/Civil Engineering. University of Toronto, 1976. Ramakrishnan. V.• Vadivelu, S. K.• and Prasad, N. M_ Report SDSM&T-CNSF 7405. Department of Civil EngineerJng. South Dakota School of Mines &. Technology, Rapid City. South Dak.ota 57701.

495

j.

Reformulation of the frame method A. K. H. Kwan, BSc(Eng), PhD, MICE

Proc.lnstn Civ. Engrs Structs & Bldgs, 1992, 94, Feb., 103-116 Paper 9809

I

Am.ong design engineers, the frame method is popular for the analysis of coupled wall-frame buildings. However, there are several problems with it. Firstly, shear deformation of the walls is either totally neglected or inappropriately allowed for. Secondly, the rotational degrees of freedom at the beam-wall joints have been mistaken as the rotations of the horizontal rigid arms, leading to incompatibility between the beam and wall elements. Thirdly, there is also the problem of artificial flexure due to discrete modelling of the continuous connection between adjacent wall units. An attempt to solve these proble~s was made by the Author in a previous paper; and a further investigation is carried out here. It is found that, to overcome these problems, substantial reformulation of the frame method is necessary. The reformulation leads to a new solid wall element which is really a strain-based finite element with rotational degrees of freedom. Numerical examples are given for comparing with other theoretical and experimental results. Notation A b E G h

I

R t U,v (Xi

Pi

£x' £, £,.0

'1 1"

w

n1 n1 n 3 , n4 B E K K1 K'1

K1 K'1

K,3

sectional area breadth of element Young's modulus shear modulus height of element moment of inertia radius of curvature of frame axis thickness of element horizontal and vertical displacements coefficients in mixed displacement/strain function formulation coefficients in strain function formulation strain in x and y directions value of £, at the centroidal axis shear strain shear stress rotation of vertical fibre strain energy due to axial and bending strain strain energy due to shear strain energy lost at vertical joints between adjacent elements strain -displacement matrix elasticity matrix element stiffness matrix axial and bending stiffness matrix axial and bending stiffness matrix after reformulation shear stiffness matrix shear stiffness matrix after reformulation stiffness matrix for recovering energy lost at wall joints

Existing frame methods The frame method was used originally for the analysis of skeletal frame structures. It was adapted to the analysis of plane coupled shear wall structures by incorporating rigid arms in the beam elements to take the finite width of the shear walls into account. 1.2 2. The method had been extended to the analysis of three-dimensional coupled nonplanar shear/core wall structures in two different ways, namely the non-planar wall element approach and the planar wall element approach. 3. In the non-planar wall element approach, 3,4 the non-planar walls of the building structure are modelled by special threedimensional non-planar wall elements which have seven degrees of freedom at each end, among which the first six are the same as those of normal space frame elements, while the seventh is the warping displacement of the wall section. Vlasov theory is used to evaluate the warping effects and to derive the stiffness matrices of the non-planar wall elements. Calculation of the shear centres and sectorial coordinates of the non-planar walls is required; therefore, frankly, this method is not convenient to use, especially for those engineers who are not familiar with warping theories. 4. On the other hand, with the planar wall element approach,5.6 the non-planar walls of the building structure are treated as assemblies of two-dimensional planar wall units interconnected at their vertical joints. Each planar wall unit is modelled individually by planar wall elements, and, in so doing, the warping displacements of the non-planar walls are evaluated as an integral part of the solution. Hence reliance on Vlasov theory is not necessary, and determination of the shear centres and sectorial co-ordinates of the non-planar walls is no longer required. Two types of element-the solid wall element and the generalized column element (Fig. I)-were developed to model the planar wall units. The solid wall element is used to model solid walls connected to other wall units at both edges, while the generalized column element is used to model wall units con· neeted to coupling beams. In general cases, a mixture of the two types of element is needed.

Problems with existing frame methods 5. Both the above forms of frame method have been popular among design engineers for many years. However, there are still a number of problems with them.

Written discussion closes 15 April 1992

A. K. H. Kwan, Lecturer, Department of Civil and Structural Engineering, University o/Hong Kong

-

103

KWAN

:1

I I

I v,

t

-

i

u,

(b)

Fig. 1. Planar wall elements developed to model planar wall units :6 (a) solid wall element; (b) generalized column element

Fig. 2. Rotations o/beams when·the walls deflect in shear: (a) conventional frame method-the beams would rotate with the horizontal rigid arms; (b) in actual structure-the beams should rotate with the axis 0/ the wall Axis

-

(a)

104

(b)

6. The problems with the non-planar wall element approach arise mainly from their reliance on Vlasov theory which was intended originally for open sections only. In the Vlasov theory, the warping effects are evaluated on the basis of the assumptions that the shear centre is fixed and the shear deformation of the walls is negligible. In recent years, however, the validity of the concept of a fixed shear centre has been questioned by many researchers. In 1984, Stafford Smith and Abate 7 showed that the location of the shear centre is not really fixed; there could be a fairly large shift of the shear centre at regions of warping restraints. Moreover, since the shear deformation of the walls is totally neglected, the non-planar wall elements are not applicable to cases such as core walls subjected to torsion (or combined bending and torsion) and shear/core walls subjected to concentrated vertical loads, etc., where shear deformation can be significant. 7. For the above reasons, the planar wall element approach, which is more fundamental and does not rely on Vlasov theory, is generally preferred. However, the existing frame methods that adopt the planar wall element approach are not without problems. The major problem is that the effects of wall shear deformation have not been allowed for fully or appropriately. This is not at all obvious because the shear deformation of the walls has already been taken into account in the derivation of the stiffness matrices of the wall elements. What has been missing is the effect of wall shear deformation on the coupling beams, as explained later. In the current practice, the lintel beams are assumed to be rigidly connected to the horizontal rigid arms such that the rotations of the beams at the beam-wall joints are equal to those of the rigid arms. Consequently, when the walls deflect in shear with the rigid arms remaining horizontal, the beams would not rotate with the walls (Fig. 2(a». This is, however, incorrect because, when the walls deflect in shear, the beams should rotate as in Fig. 2(b). It is therefore seen that the beam rotations due to wall shear deformation have been omitted. On account of this omission, it has become possible for the walls to deform in shear without straining the coupling beams (Fig. 3(a», while in actual fact, the coupling beams should be strained (Fig. 3(b». This results in underestimation of the coupling effects, or, in other words, overestimation of the shear flexibility of the coupled wall structure. Therefore, the way that shear deformation of the walls is allowed for in the existing frame methods is not really appropriate. This problem was addressed in 1986 by Rutenberg et al.,8 who also demonstrated that partly because of this, the existing frame methods (planar wall element approach) grossly overestimate the torsional rotations of core walls closed by medium

REFORMULATION OF FRAME METHOD

to heavy lintel beams. 8. This problem is related closely to the Question of how the rotational degrees of freedom at the beam-wall joints should be defined. The intrinsic assumption in the frame methods that the rotations of the beams at the beam -wall joints are equal to those of the horizontal rigid arms is equivalent to defining the rotational degrees of freedom at the joints as the rotations of the horizontal fibres in the walls. However, this definition would lead to incompatibility between the beam and wall elements, as shown in Fig. 4(a). To ensure compatibility between the beam and wall elements, the rotational degrees of freedom at the joints must be defined as the rotations of the beam -wall boundaries: i.e. the rotations of the vertical fibres at the joints (Fig. 4(b», because this is the only way to guarantee conformity of the boundary displacements. It is noteworthy that the difference in rotations of the horizontal and vertical fibres is actually the shear strain, and that, because of possible difference in shear strain at the two sides of a beam-wall joint, the rotation of the horizontal fibre at the beam side and that at the wall side are not necessarily equal. So there is no unique horizontal fibre rotation at a beam-wall joint at all. Hence the existing definition of joint rotations as the rotations of the horizontal fibres is incorrect; the correct definition should be the rotations of the vertical fibres at the joints. 9. There is yet another problem with the planar wall element approach which is also associated with the shear stress or strain of the walls. Consider the interconnected planar wall units in Fig. 5{a). As a result of the interaction between adjacent wall units, developed shear stresses would be evident along the vertical edges of the wall units. In open sections, the vertical shear stresses are generally negligible. However, in core walls partially closed by lintel beams or even completely closed by themselves, the vertical shear stresses developed at the element edges could be quite significant owing to the circulatory Brecht shear flow round the cores. The vertical shear stresses at the edges should theoretically be distributed continuously along the height of the wall elements. However, as the continuous connection between adjacent wall units is actually modelled by discrete connections at the nodes, the continuous shear stresses can be treated only as equivalent to concentrated vertical forces acting at the nodes (Fig. 5(b». As a result, a solid wall element subjected to pure shear would, in effect, be subjected not only to shear stresses but also to bending moments at the top and bottom ends of the element. As these bending moments do not exist physically, they are considered parasitic. Such parasitic moments cause artificial flexure of the wall elements, as shown in Fig. 5(c), and eventually

(~

(~

Fig. 3. Deformation of the lintel beams when the walls deflect in shear: (a) conventional frame method-the beams remain unstrained; (b) in actual structure-the beams should be strained

overestimation of the shear flexibility of the wall assembly. This problem was noted early in 1979 by Stafford Smith and Girgis. 9 It can be reduced by dividing the building structure into more layers-say, two or more layers per storey-so that the discrete connection can simulate the behaviour of the continuous con· nection more closely but cannot be totally eliminated.

Resolving the problems with beam-wall incompatibility 10. An attempt to solve the above problems was made recently by the Author. 10 First, he pointed out that in order to ensure compatibility between the beam and wall elements, the rotational degrees of freedom at the beam-wall joints must be defined as the rotations of the vertical fibres at the joints, and that the conventional definition as the rotations of the horizontal rigid arms is erroneous. Then, distinguishing the horizontal fibre rotation and vertical fibre rotation in the derivation of the stiffness matrices, he found that the shear deformation of the walls should really be allowed for in the rigid arms rather than in the

Fig. 4. Rotational degrees offreedom at the beam-wall joints defined as: (a) rotations of horizontal fibres,. (b) rotations of vertical fibres Shear strain

Horizontal

Axis of

fibre

wall

Incompatibility

(a)

(b)

-

105

KWAN column elements which model the axial and bending behaviour of the wall units. Based on this finding, he developed a solid wall element with rotational degrees of freedom which are defined as the rotations of the vertical fibres (Fig. 6(a». Unlike other solid wall elements, this element can be used for both solid walls connected to other wall uni ts at each edge and walls coupled with beams. Therefore, the Author's solid wall element is generally more versatile and convenient to use ; and, more importantly, his element yields much better results than the conventional frame methods in the analysis of shear walls subjected to concentrated vertical loads, and core walls subjected to torsion where shear deformation is significant.

Vertical joint between adjacent wall units

~I~ Vertical

1J

rshear t stress

tJ

t~ tJ

tJ (a)

These two nodal forces induce parasitic. moment

Present study Nodal force due to vertical shear stress

t

I I

J-+H*---

I

Additional deflexion due to artificial flexure

I (b)

(c)

Fig. 5. Parasitic moments and artificial flexure in planar wall elements: (a) shear stresses along vertical joints,. (b) equivalent nodal forces and parasitic moments; (c) artificial flexure and additional deflexion

Fig. 6. Kwan's solid wall elements: (a)discrete member model (as developed in reference 10); (b) continuum model (adopted in this Paper) (Note: (JJ defined as - auf Oy: i.e. rotation of vertical fibre, displacement vector defined as {u 1 (JJ 1V 1V 2 U 2 (JJ 2 V 3 V 4}') ClJ2

V3!

1\

106

~ W, u,

-+

b

V3t

~ y,v

h

Column member

VII

-

tv.

II ~igid arm

h

u

W2

~

lV2

+-

Lx.u

tv. Thickness

I.-

=t

11. The Author's frame method, as presented in reference 10, resembles the conventional frame methods, in that his solid wall element is also modelled by discrete frame members (one column member and two rigid arms). It will be shown in this Paper that there are some problems with such a discrete member modelling method-such as discrete variation of shear strain, error in shear strain distribution and error in lateral deflexion, etc.-that can never be completely resolved. In order to overcome these problems, which are inherent in discrete member models,·the frame method is extensively reformulated by treating the wall element as an elastic continuum rather than as being·composed of discrete members. The reformulation leads to a new solid wall element which is really a strain-based finite element with rotational degrees of freedom. Hence the frame method is related to the finite-element method. This solid wall element is generally more efficient than other finite elements, which allow only uniform bending moment within the elements. However, the new element is still afflicted by artificial flexure. A simple method of reducing artificial flexure is proposed in the later part of the Paper.

Shear strain in the solid wall element Problems arising from discrete variation of shear strain 12. The Author's solid wall element, as developed in reference 10, is composed of three discrete members: one column member at the centroidal axis of the wall unit, and two horizontal rigid arms connecting the column member to the nodes (Fig. 6(a». The column member simulates the axial and bending behaviour of the wall element, while the two rigid arms, which are deformable in shear, allow for shear deformation of the element. In reference 10, the stiffness matrix of Kwan's solid wall element is given by

REFORMULATION OF FRAME METHOD (1)

where K 1 is the stiffness matrix of the column member (axial and bending stiffness matrix of the element) as given by equation (2), and K 2 is the stiffness matrix of the two rigid arms (shear stiffness matrix of the element) as given by equation (3), in which I and A are the moment of inertia and cross-sectional area· of the wall element respectively: i.e. 1= tb 3 /12 and A = tb. 13. This solid wall element has the peculiar characteristics that: shear deformation is allowed for in the rigid arms rather than in the column member; and shear strain is assumed to be uniform in each rigid arm, but the shear strain in one rigid arm may be different from that in the other rigid arm. As the top and bottom rigid arms represent the upper and lower halves of the element respectively, and the shear strains in the two rigid arms are independent of each other, the shear strain varies along the height as a discrete function. This can be demonstrated by analysing a cantilever wall subjected to a concentrated top load as shown in Fig. 7(a), using only one element. From the results plotted in Fig. 7(b), it can be seen that this solid wall element yields a discrete shear stress distribution which varies from zero at the upper half abruptly to two times the correct value at the lower half. As the correct answer should be a constant shear stress along the entire height, the shear stress distribution is erroneous. Apart from the error in shear stress distribution, there is also error in lateral deflexion as illustrated in Table 1, in which the exact solution obtained by the engineering theory of bending is also tabulated for comparison. It also shown in this table that the error in lateral deflexion arises solely from the overestimation of the shear deflexion of the cantilever wall which, in turn, is due to the error in shear strain at the base of the wall.

12EI

K1

=

14. The Author's way of accounting for the shear deformation of the walls by incorporating shear flexibility in the two rigid arms, which leads eventually to discrete variation of shear strain within the element, is a little awkward. It would appear more reasonable if the variation of shear strain within the element could be taken as a continuous function. Since the shear strain at the top of the element and that at the bottom are independent, and hence generally not equal, the shear strain within the element must vary at least as a linear function.

0

0

0

0

EA

EA

4h

4h

EA

EA

4h

4h

4EI

6EI

-

-J;2

h

0

0

0

0

GEl

12El

-y

J;2

GEl

2EI

-J;2

0

0

0

0

0

2EI h

0

0

EA

EA

4h

4h

EA

EA

4h

4h

0

0

0

0

EA 4h EA 4h

EA

6EI

-

J;2 0

0

0

0

12£/

GEl

0

y

J;2

0

0

6EI

4EI

J;2

h

EA 4h

EA

0

0

EA

EA

4h

4h

0

0

4h

0

0

0

0

0

0

0

0

0

0

0

Glh 2b

0

0

0

0

0

0

0

0

0

Gthb 2 Glh 2 Gth 2

Gth

Gth

Glh.

2

2

Gth

2

Gth 2b Gth 2b

0

0

-

2

Gth

-

2

Gth

0

=

0

0

Gth

0

K2

0

0

Gthb

0

0

0

0

0

0

0

0

0

0

0

2b

0

0

0

0

0

0

0

4h EA

4h

2

2

Gth

Glh 2b Glh

2b

Gth 2b

2b (3)

Table 1. Results of analysing the cantilever wall in Fig. 7 Bending deflexion:

Shear deflexion:

Total deflexion:

K

mm

mm

mm

% error in total deflexion

Exact solution

-

(1) (2)

Xl +K:z X 1 +K; X'l+K;

5·400 5·400 5·400 5·400

0·375 0·750 1·500 0·375

5·775 6·150 6·900 5·775

0 6.5 19·5 0

Method of analysis

(3)

~

10 m

1-

300 k

"Roof

(2)".

(1)

'"'. Thickness =0·3m

E o

C?

I VI (3)

'

)--.l(l) :(3)

v = 0·25

__ Ground

o (a)

Exact

solution

....

Midheight

E = 20 kNlmm2

Fig. 7. Analysis of a cantilever wall using the solid wall elements: (aJ cantilever wall analysed; (b) shear stress distribution

6EI

-J;2

0

h

0

12E/

-y

(2)

is

Reformulation of the element assuming linear variation of shear strain

6EI

-J;2

Y

:

~.~2) .

"

(1) K = K,

+ K2

(2) K = K,

+ K2 '

(3) K = K,'

+ K2 '

---'-_""""-----..Io_~ _

0·1 ' 0·2 0·3 0·4 Shear stress: N/mm 2 (b)

_

10

KWAN

Assuming a linear variation of shear strain

Bending strain in the solid wall element Shear-bending interaction 17. The loophole, after some painstaking studies, is found to be the omission of shearbending interaction in the formulation of the axial and bending stiffness matrix as explained below. 18. Ina conventional frame element, the longitudinal strain is taken as

where

and V4 -

V3

Y2 = -.-b- -

(6)

(,02

(9)

The shear strain energy is given by

TI 2 =

f ~ C·l d(vol)

(7)

Using the Principle of Virtual Work, as in the finite-element method, the shear stiffness matrix of the solid wall element is obtained as 0 0 0 0 K~=

0 0 . 0 0

0 Gthb

0 Gth

--

-

Gth 3 Gth 3 0 Gthb

Gth 3b Gth 3b 0 Gth 6 Gth 6b Gth 6b

3

-6

Gth 6 Gth

-

6

3

0 Gth 3 Gth 3b Gth 3b 0 Gth 6 Gth 6b Gth 6b

0 0 0 0 0 0 0

0

0 Gthb -6 Gth 6 Gth 6 0

Gthb -3 Gth 3 Gth 3

0 Gth 6 Gth 6b Gth 6b 0 Gtk

3 Gth 3b Gth 3b

0 Gth 6 Gth 6b Gtk 6b 0 Gth 3 Gth 3b Gth 3b

(8)

-

The overall stiffness matrix of the reformulated element is obtained simply by replacing K 2 in equation (1) with the above shear stiffness matrix: i.e. adding K 1 and K~ together. 15. The above reformulated element is used to analyse again the cantilever wall in Fig. 7(a). The corresponding results on the shear stresses along the height of the element (Fig. 7(b» show that the shear stress distribution is now continuous but still erroneous because it varies linearly from a negative value at the top to four times the correct value at the bottom, while the correct answer should be a constantshear stress throughout the element. As in the previous case, owing to the error in shear strain at the base of the elemen t, there are also errors in the lateral deflexion of the cantilever wall (Table 1). 16. Therefore, no matter if the shear strain variation is taken to be a discretestep function or a continuous linear function, the result on shear strain distribution is still erroneous; somewhere in the formulation process there must be a loophole.

108

where £yO is the axial strain at the centroidal axis due to axial load, R is the radius of curvature of the frame axis after bending, and x is the distance from the centroidal axis. For small displacement, the curvature (I/R) is given by 1

R=

-

02 U oy2

(10)

Hence £,

=

eyO

02 U oy2

-

X

(11)

From this it would seem that the longitudinal strain is independent of the shear strain. 19. If, however, the fundamental definition of £, is given at the start as

av

(12)

£=-

1

oy

then the longitudinal strain would turn out to be dependent on the shear strain. By definition, the shear strain is equal to

au ov oy ax

y=-+-

(13)

Rearranging

av oX

OU

-=Y--

oy

(14)

Integrating this equation on the assumption that plane sections remain plane, i.e. v varies linearly with x, gives

v

= vo

+(y -:}

(15)

where V o is the value of v at the centroidal axis. The longitudinal strain is thus equal to &7

=;

[v + (y - :}]

=

+

&70

o

G;-~;~)x

(16)

20. It can therefore be concluded that the bending strain (the second term o(the above equation) is not just a function of the curvature, but also a function of the rate of change of shear strain alongthe length of the element.

REFORMULATION OF FRAME METHOD

The omission of the term (iJyl iJy)x in the expression for the longitudinal strain is the loophole in the formulation which is causing the problems described in the previous section. Reformulation taking into account shear-bending interaction 21. The solid wall element is now reformulated, using equation (16) instead of equation (11) for the longitudinal strain. 22. The axial strain eyO' which is assumed uniformly distributed along the frame axis within the element, is given by e

=

yO

! (V 3+ V

4 _

h

VI

2

+ V2) 2

(17)

Assuming a linear variation of shear strain as in the previous section

12£1

Y 6EI

-hl

K'l=

6EI

-hl 3EI h

0

0

0

0

12£1

GEl

-y

hl

6E1

3El h

-hl 0

0

0

0

0

0

0

0

EA 3h EA 6h

EA 6h

EA

-

3h

12£/

6EI

6EI

hl

3EI h

0

0

0

0

- y -hl

0

0

12£1

6EI

hF

hl

0

0

6EI

3EI

h2

h

EA

EA 6h EA 3h

0

0

0

0

3h EA

6h

0

0

0

0

EA 3h EA 6h

EA 6h EA 3h

0

0

0

0

EA

EA

3h

6h

EA

EA

6h

3h

-

(22)

(18)

The term o2 u joy2 can be obtained by fitting a cubic polynomial for the lateral deflexion of the frame axis and differentiating the polynomial twice as Zienkiewicz did in reference 11. The result, which is identical to that obtained by the conventional beam theory, is

(19) Substituting into equation (16), the longitudinal strain is obtai~ed as B,

= ~ (V 3;

V 4

_

VI ;

V2)

The strain energy due to the axial and bending strain is given by fIt

=

f~ &:

devol)

(21)

Using the Principle of Virtual Work, the axial and bendin.g stiffness matrix of the solid wall element is obtained as given in equation (22). The overall stiffness matrix of the reformulated element is then obtained simply by adding K't and K~ together.

23. This new element is used to analyse again the cantilever wall in Fig. 7(a), whereupon the correct shear strain distribution and lateral deflexion are obtained (Fig. 7(b) and Table 1). This is achieved just by modifying the element stiffness matrix; no additional compu· tational effort is involved.

Equivalence to finite-element for~ulation

24. The major advancement resulting from the above reformulation is the change from the conventional frame method of modelling the solid walls by discrete frame members to the approach of modelling the wall elements as an elastic continuum, which is more realistic and logical. If the discrete member model is not replaced by the continuum model, the problems with discrete variatioD of shear strain, error in shear strain distribution, and error in lateral deflexion, etc., could never be completely resolved. Since similar methodology is adopted, the new frame method using the reformulated solid wall element resembles the finite-element method in many ways. In fact, the new solid wall element may also be regarded as a special kind of plane-stress finite element as depicted below. 25. The solid wall element in Fig. 6{a) will be considered again, but this time as an eight degrees of freedom plane-stress finite element (Fig. 6(b». Starting with the following displacement and strain functions (23) (24) (25)

it can be seen that these are the same as those assumed in the previous reformulation process. Integrating the two strain functions gives

-

IO!

,

KWAN v

= as + asY + [(a6 + a 7 y) - (a 2

+ 2a 3 y + 3a 4 y 2 )]x

(26)

in which as is an integration constant. The eight coefficients in equations (23) and (26) are determined by equating the nodal translations and rotations to the eight degrees of freedom and solving the equations thus obtained. The values of the coefficients are then substituted into the displacement functions and the displacement functions differentiated to obtain the strain-displacement matrix as follows

(27) Ut

e.x ey Y%y

0 -12xy

0 6xy

h3

h2

0

-h +2y 2h

0 -b +2x 2bh -h +2y 2bh

0 0 -b -2x 12xy h3 2bh h -2y 0 2bh

6xy

h2 -h -2y 2h

0 b -2x 2bh -h -2y 2bh

0

b

+ 2x

Wi

Vt

V2

2bh U2 h+2y! W z 2bh V 3

V4 The element stiffness matrix is evaluated by the standard expression

K

=

JB'EB d(vol)

(28)

where the matrix B is~the strain-displacement matrix given in equation (27), and the matrix E is the elasticity matrix given by

E=[~: ~]

(29)

It should be noted that in the matrix Eabove, the Poisson ratio effects have been neglected because the lateral stress (1.x is generally of no significance. The stiffness matrix so derived is identical to that obtained in the previous section, Le. identically equal to K't + K~. 26. Hence the new solid wall element is shown to be equivalent to a plane-stress finite element formulated on the basis of a mixed set of displacement and strain functions. This element behaves very like a frame element, and· is particularly suited for the analysis of shear walls which act basically as frame members.

Reform.ulation of the element as a strain-based finite element 27. The new solid wall element can also be formulated as a strain-based finite element starting with the following strain functions (30) Sy

=Pt + (P2+ P3Y)X

Yxy=P4+PSY

-

SJC is set eq ual to zero because the· horizon tal strains are neglected. The coefficient PI rep-

110

(31)

(32)

resents a constant axial strain along the centroidal axis, while the term (P2 + P3Y)X represents bending strains varying linearly with height. The last term (fJ4 + PsY) allows for linear variation of shear strain along the beam axis. It can be shown that the above strain functions satisfy the second-order differential equation governing internal compatibility. 12 28. Integrating the strain functions and adding the rigid body mode U=P6-PSY

(33)

V=P7+P S X

(34)

the displacement functions are obtained as U

= P6 ~ (/3s -

P4Jy ~ t
Ps)y2 -

iP 3y3 (35)

v = P7 + PlY

+ (Ps + PzY + tP 3y Z)x

(36)

Solving for the coefficients by equating the nodal translations and rotations to the eight degrees of freedom and substituting back into the strain functions, ~ the same straindisplacement matrix B as given in equation (27) is obtained. Using equation (28), the same stiffness matrix as before is derived. 29. It can therefore be concluded that the new solid wall element is really a strain-based finite element. Strain function formulation has proved very successful in many applications. 13 • 14 It has been applied to shear wall analysis by Sabir 1s and by Ha and Desbois. 16 Sabir developed two strain-based finite elements,one without rotational degrees of freedom and the other with rotational degrees of freedom defined as the mean of the rota tions of the horizontal and vertical fibres. Ha and Desbois, on the other hand, developed just one

REFORMULATION OF FRAME METHOD

strain-based element with no rotational degrees of freedom. These strain-based elements are not afflicted by parasitic shear under bending mode and are therefore most suitable for the analysis of structures whose principal mode of action is bending, such as shear wall structures. However, as discussed in § 8, Sabir's definition for the rotational degrees of freedom is not correct. Hence these strain-based elements have either no rotational degrees of freedom (without rotational degrees of freedom, coupling with beams would be difficult) or rotational degrees of freedom of incorrect definition. Compared with these elements, the new solid wall element is more advantageous: it has rotational degrees of freedom for easy coupling with beams and it would not lead to errors as a result of incorrect definition of rotational degrees of freedom. 30. It should be noted that in the strain function for the longitudinal strain (equation (3I», the strain state of pure bending can be exactly represented. Therefore, the new solid wall element is also free of parasitic shear. Furthermore, since linear variation of bending moment with height is allowed, it can model beam bending actions more closely than other finite elements in which only uniform bending moment within an element is allowed, as will be illustrated in the following examples.

Examples Example 1: coupled shear walls 31. A typical coupled shear wall structure is analysed as shown in Fig. 8(a). For comparison, the structure is first analysed by the proposed solid wall element and then by Sisodiya and Cheung's element. 17 Sisodiya and Cheung's element was originally developed for bridge analysis. It was adapted for building analysis by Cheung in reference 18. This element has 12

degrees of freed'om: two translational and one rotational at each of the four nodes. However, it can be simplified to have the same degrees of freedom as the proposed solid wall element by incorporating the assumption of ex = 0 without loss of accuracy, because in building structures, the lateral strain is generally negligible. The simplified Sisodiya and Cheung's element which has the same degrees of freedom as the proposed solid wall element is used in this example. Local deformation at the beam/wall joints is allowed for by increasing the beam span by half beam depth at both ends. Fig. 8(b) presents the results on lateral deflexion, from which it can be seen that the deflexion curve obtained using the proposed solid wall element, with one layer of elements per storey, coincides almost exactly with the corresponding curve by Sisodiya and Cheung's element, with two layers of elements per storey, and that Sisodiya and Cheung's element slightly underestimates the lateral deflexion if only one layer of elements is used per storey. The bending moment and shear forces in the shear walls at the lowest two stories are plotted in Fig. 9. It is shown that Sisodiya and Cheung's element. which allows only uniform bending moment within an element, yields a stepwise variation of bending moment with height. These bending moment results may be taken to be the average bending moment within the elements and assigned to be the bending moment values at the centres of the elements, but will not give the maximum

Fig. 8. Example 1 : analysis ofa coupled shear

wall structure: (a) coupled shear wall; (b) deflexion curve (Note: SW elementproposed solid wall element; SC elementSisodiya and Cheung's element)

--r-r----u----H-500 kN/storey-

32 SC element (1 layerlstorey)~

--u---r-r----u--

24 1.1.

" I.

~~d~=

~ 16

==t1~:

1 or 2 layers-[ of elements per storey _

I.

'0)

8

~=b==

==0== (a)

/.

::I:

==-0==

Thickness = 0·4 m, depth of beam E = 2x 10 10 ·N/m2 • v = 0·25

I.

E

1

SW element (1 layer/storey) .coincide or SC element almost exactly (2 layers/storey)

= 0·8 m

20 40 Deflexion: mm (b)

-

11:

KWAN

r

8 -------

8

"-

,

, \

2nd storey

~-

y. r

/

SC element (2 layers/storey)

-.. ~ ,/SC element (2 layers/storey)

Interpolated from SC element

I

I I

,

E

SWelement (1 layer/storey)

~ 4

SWelement I( 1 layer/storey)

- --==:

"0)

:r: 1st storey

L

I I

•

I

00

5

10

15

-0·5

Bending moment: MN-m

bending moment required for designing the structure. To obtain the maximum bending moment, it is necessary to useat least two layers of elements per storey and to extrapolate from the bending moment values at the centres of the elements, as shown in Fig. 9(a). In contrast, the proposed solid wall element requires only one layer of elements per storey to evaluate the bending moment variation within a storey. As revealed from Fig. 9(a), the .bending moment variation obtained by the solid wall element with only one layer of elements per storey agrees closely with the corresponding results obtained by Sisodiya and Cheung's element using two layers of elements per storey. The shear forces obtained using the two different elements are compared in Fig. 9(b), from \vhich it can be seen that Sisodiya and Cheung's element gives large fluctuation of shear forces wi thin an element, while the solid wall element yields a smoother shear force distribution which is more reasonable.

Fig. 10. Example 2: analysis of a coupled nonplanar wall structure: (a) model tested by Tso and Biswas; (b) dejlexion curve

48r Continuous connection method, Tso & Biswas

Example 2: coupled non-planar walls 32. The coupled non-planar wall structure studied by Tso and Biswas 19 is analysed using the new solid wall element as shown in Fig. lO(a). Only one layer of elements per storey is used in the analysis. The results on lateral deflexion are plotted in Fig. lO(b), where the theoretical and experimental results obtained by Tso and Biswas are also plotted for comparison. The figure shows that Tso and Bis\vas's theory, which neglects the shear deformation·of the walls, underestimates the lateral deflexion of the structure, while the proposed element agrees fairly closely with the experimental results.

ti>

1

3 in'"

4 in "L 3

36

in ~

.5

~24

'0)

:r:

• Experiment (Tso and Biswas)

12

oo'-----L-.--O......0-4----"---0~.5":":'"8---"---O-=~ Horizontal displacement:. in (a)

112

2·0

(b)

Fig. 9. Example 1 : bending moment and shear force at lowest two storeys (Note: SW element-proposed solid wall element; SC element~Sisodiya and Cheung's ele,ment)

C') (l')

1·0 Shear force: MN

(a)

.5

0

(b)

2

Example 3: partially closed core wall 33. The Perspex core wall model tested by Tso and Biswas 20 is analysed using the solid

REFORMULATION OF FRAME METHOD H

Torsion at top = 2001b-in

-

UmanSky-Benscoter

-0-244 in

)

3H/4

Bea~:/::----­

f

L

2-5 in

t1-S in ,

2·5 in

+

1:

-~ H/2

J:

Proposed solid wall element (shear energy lost at vertical joints recovered)

,:

~proposed

solid wall element

• Experiment (Tso and Biswas) H/4

Wall

Beam

Wall

(a)

wall element as shown in Fig. ll(a). This model has also been analysed by Rutenberg et al. 8 using several continuous connection methods. As suggested by Rutenberg et al., local deformation at the beam/wall joints need not be considered for this particular model because the roots of the beams are provided with fillets of radius 1/3 the depth of the beam, which has a stiffening effect that is probably commensurate with the loss of stiffness resulting from local rotation at the joints. Only one layer of elements per storey is used in the analysis. Fig. Il(b) shows the results of the analysis, together with the experimental results of Tso and Biswas and the theoretical results obtained by Rutenberg et ale using Vlasov theory and Umansky-Benscoter theory. As expected, Vlasov theory which ignores shear deformation in the walls yields over-stiffened results. The results of the proposed solid wall element agree very closely with those of the UmanskyBenscoter theory which, according to Rutenberg et al., is the most accurate and widely applicable method among the various continuous connection methods. The agreement with the experimental results is reasonable.

Example 4: closed core wall 34. A closed core wall of square shape subjected to torsion (Fig. 12(a» is analysed. The structure is divided into 20 storeys, each of height 5·0 ffi. Each storey is modelled by four solid wall elements interconnected to form a hollow section. The results on·torsional rotation, shear stress, and bending moment are shown in Figs 12(b), 13(a) and 13(b) respectively, where exact theoretical values (by Bredt-Batho theory) are also plotted for comparison. Fig. 12(b) shows that this frame

4 8 Rotation: x 10-3 rad (b)

12

Fig. 11. Example 3: analysis of a partially closed core wall: (a) core wall model tested by Tso and Biswas,o (b) comparison with Vlasov theory, Umansky-Benscoter theory and experimental results

method overestimates the twisting angles by about 10%. The results on shear stresses are, however, much better; they coincide exactly with the theoretical values eccept at the region where the thickness changes abruptly. The reason for the overestimation of the torsional shear deformation is given in Fig. I3(b), where it can be seen that the solid wall elements are subjected to parasitic bending moments in addition to the shear stresses due to torsion.

Artificial flexure of the solid wall element Incompatibility along vertical joints and shear energy lost 35. The results in Example 4 indicate that the new solid wall element is afflicted by parasitic bending moments and that, owing to the artificial flexure so caused, the proposed frame method tends to overestimate the shear deforrna tion of the walls. This artificial flexure problem actually occurs also in the conventional frame methods which adopt the planar wall element approach. Although the problem was identified early in 1979,9 it remains unresolved to date. 36. A closer look at the element reveals that the vertical displacement function of the element is not conforming along the vertical edges, as the vertical displacemen ts there vary

-

113

KWAN z

H

~",

Torsion = 100 tm

/ ", / ,,'

~'Proposed solid wall element ~, ,

Bredt-Batho theory

,,

3H/4

E o

%

to

Thickness = O·Sm

~,'

%'' ' !;f' ~,'

E

.~ H/2

:c

~,

E oll')

.

(shear energy lost at vertical joints recovered)

Proposed solid 'wall element

I

'f,' I

Thickness = 1·0m

H/4

~

Fig. 12. Example 4: analysis of a closed core wall: (a) variable thickness shear core; (b) variation of rotation with height

Fig. 13. Example 4: shear stresses and parasitic bending moments: (a) shear stress in each wall; (b) parasitic bending moment in each wall H

3H/4

Bredt-Batho theory

\

E

.~ H/2

1:

.~ H/2

:x:

:x:

\

Proposed solid wall element

H/4

0·5 Shear stress: Vm 2 (a)

114

1·0

,,

0·1

E = 105 Vm 2 • G = Q·5X10S tlm 2 (a)

3H/4

,W ,

H/4

o

- -.c:::;;....--'0------2"'-02O Bending moment: tm

L-.-.-_...L

(b)

0·2 0·3 Rotation: x 103 rad (b)

0·4

as quadratic functions while there are only two vertical degrees of freedom at each edge. Such incompatibility leads to relative slip along the vertical joints between adjacent elements and, eventually, to loss of shear energy at the vertical joints. It is this shear energy loss that results in the underestimation of the torsional shear stiffness of the core wall structure in Example 4. " 37. It should nevertheless be noted that such incompatibility permits linear variation of bending moment with height, and it is this factor that allows the solid wall element to model beam bending actions more closely than other conforming finite elements which allow only uniform bending moment within an element. 38. Whether or not the shear energy lost at the vertical joints is significant depends on the magnitude of shear stresses there. In open sections, the" shear stresses at the vertical joints are generally small; it is expected, therefore, that in the analysis of open section wall structures, such shear energy loss would not result in significant errors. In fact, Example 3 demonstrates that even in partially closed core walls subjected to torsional shear stresses, the errors involved are still acceptable. However, in closed core walls subjected to pure torsion where shear stresses dominate as in Example 4, the errors due to the shear energy.loss may be significant.

Recovering shear energy lost along the vertical joints 39. A simple method of mitigating the above problem by recovering the shear energy lost at the vertical wall joints is presented next.

REFORMULATION OF FRAME METHOD 40. The vertical displacement function of the element can be written as .

v=v(~-~)(~-~)+v(!+~)(~-~) 12 b 2 h 2 b 2 h 2

for element connected to adjacent element at one edge

(a)

K = K'I

+ K~ + ~ K)

(44)

(b) for element connected to adjacent elements

at both edges

K = K/1 + K; + K 3

(45)

where K) is given by 12E/

Y In the above displacement function, while the first four terms are conforming, the last term which has zero values at the nodes is nonconforming. This non-conforming term causes slip between adjacent elements. The slip at the edge x = -b12 is given by

51 = [:3

(U l -

U 2) -

:2

(WI

+(

2)J

GEl

-h2 0.

KJ =

0 12EI

-y GEl

-h2 (38)

GEl

-h2

0.

3EI 0. h 0. 0. 0 0. GEl 0.

h2

3El 0 h

0.

12El

-/;3

GEl

-h2

0.

0

0

0

0.

GEl

3EI

h2

h

0. 0.

0. 0. 12El

0

0.

0

0. GEl

0

0

0

0

0.

0.

0

/;3

h2

0.

6El

3EI

h2

h

0.

0.

0

0.

0.

0.

0 0

0.

0.

0.

0.

0.

0

0

(46)

0

while the slip at the edge x = bl2 is given by

52 = [:3

(U l -

U 2) -

:2

(WI

+

(2)J

Examples 42. The partially closed core wall in (39) Example 3 and the closed core wall in Example 4 are analysed again, using the above method of recovering the shear energy lost at the verti41. Alternatively, the shear stress in the cal wall joints. The results for Example 3 are element can be approximated by plotted in Fig. II(b) alongside the previous (40) results for comparison. In this example, since -r=U/A the artificial flexure is not significant in any where U is the lateral force on the element case, the change in recovering the shear energy given by lost at the wall joints is very small. Nevertheless, there is some slight improvement in accuracy. The effects ~f recovering the shear energy loss is more conspicuous in Example 4 because, Thus, if the element is connected to an adjacent in this case, the shear deformation of the walls element at the edge x = -bI2, shear energy will dominates. From the results on torsional rotation shown in Fig. 12(b), it can be seen that by be lost given by recovering the shear energy lost at the vertical wall joints, the error in torsional rotation is fi 3 t5 1 t dy reduced to only 5% which should be acceptable from the practical application point of view. 3E/ [ (u -u )-2(W h (42) There is practically no change in shear stress, =};3 2 1 +W 2 ) t but the magnitude of parasitic moments is decreased by exactly one half (the new results Similarly, if the element is connected to an on shear stress and parasitic moments are not adjacent element at the edge x = b/2, there will plotted in the figures to avoid confusing the be shear energy lost given by original results).

=~

f

J2

fi 4 =

~

f-

t5 2 t

dy

3EI [ (U -U )-2(W h =};3 2 t 1

Conclusions +W 2)

J2

(43)

Incorporating n J and/or fi 4 in the derivation of the stiffness matrix gives ..

43. The problems with the existing frame methods are identified and discussed. The Author's previous postulations that the rotational degrees of freedom of the wall elements should be defined as the rotations of the verti-

-

115

KWAN cal fibres, and that the conventional definition as the rotations of the horizontal rigid arms is erroneous are reaffirmed. However, it is found in this study that Kwan's solid wall element, which was developed by treating the element as composed of discrete members as in the conventional frame methods, has problems in the evaluation of shear stress distribution in certain cases. 44. To overcome these problems, the solid wall element is extensively reformulated by treating the solid wall element as an elastic continuum instead of as being composed of discrete members. The final outcome is a special kind of plane-stress finite element formulated on the basis of a mixed set of displacement and strain functions. This new solid wall element can also be formulated from a set of strain functions and is thus really a strain-based finite element. 45. The new element has been applied to a number of examples, and comparison with other theoretical and experimental results demonstrates that -the element is both accurate and versatile. It is nev~rtheless still subjected to artificial flexure owing to incompatibility of the vertical displacements at vertical joints between adjacent elements. A simple method of reducing artificial flexure by recovering the shear energy lost at the vertical joints is proposed. This method can suppress artificial flexure to a negligible degree in practically all shear wall structures, including completely closed core walls.

References 1. CANDY C. F. Analysis of shear wall-frames by computer. N. Z. Engng, 1964, 19, No.9, Sept., 342-347. 2.. MACLEOD I. A. Lateral stiffness of shear walls with openings. Proc. Symp. on Tall Buildings, University of Southampton, April 1966, Pergamon Press, New York, 1967,223-252. 3. HEIDEBRECHT A. C. and SWIFT R. D. Analysis of asymmetrical coupled shear walls. j. Struc!. Div. Am. Soc. Civ.Engrs, 1971, 97, No. ST5, May, 1407 -1422. 4. TARANATIi B. S. Analysis of interconnected open section shear wall structures. j. Struct. Div. Am. Soc. Civ. Engrs, 1975, 101, No. STl1, Nov., 23672384.

-

116

5. MACLEOD I. A. Analysis of shear wall buildings by the frame method. Proc. lnstn Civ. Engrs, Part 2, 1973, 55, Sept., 593-603. 6. MACLEOD I. A. and HosNY H. M. Frame analysis of shear wall cores. j. Struc!. Div. Am. Soc. Civ. Engrs, 1977, 103, No. STI0, Oct., 2037 -2047. 7. STAFFORD SMITH B. and ABATE A. The effects of shear deformations on the shear centre of opensection thin-walled beams. Proc. lnstn Civ. Engrs, Part 2,1984,77, Mar., 57 -66. 8. RUTENBERG A. et al. Torsional analysis methods for perforated cores. ].Struct. Div. Am. Soc. Civ. Engrs, 1986, 112, No.6, June, 1207 -1227. 9. STAFFORD SMITH B. and GIRGIS A. The torsional analysis of tall building cores partially closed by beams. Proc. Symp. on Behaviour of Building Systems and Building Components, Vanderbilt University, Nashville, Mar. 1979. 10. KWAN A. K. H. Analysis of coupled wall/frame structures by frame method with shear deformation allowed. Proc. lnstn Civ. Engrs, Part 2, 1991, 91, June, 273-297. 11. ZIENKIEWICZ O. C.The finite element method. McGraw-Hili, London, 1977, 3rd edn, ch. 2, 20-41. 12. TIMOSHENKO S. P. and GOODIER j. N. Theory of elasticity. McGraw-Hill, New York, 1988, Int. edn, ch. 3, 35 -64. 13. ASHWELL D. G. et al. Further studies in the application of curved finite elements to circular arches. Int.]. Mech. Sci., 1971, 13, 507. 14_ ASHWELL D. G. and SABIR A. B. A new cylindrical shell finite element based on simple independent strain functions. Int.]. Mech. Sci., 1972, 14, 17115. SABIR A. B. (CHEUNG Y.K. and LEE P. K. K. (eds». Strain based finite elements for the analysis of shear walls. Proc. 3rd Int. Coni. on Tall Buildings, Hong Kong and Guangzhou, Dec. 1984,447-

453. 16. HA K. H. and DESBOlS M. Finite elements for tall building analysis. Comput. & Structs, 1989, 33, No.1, 249-255. 17. SISODIYA R. G. and CHEUNG Y. K. (ROCKEY K. C. et ala (eds». A higher order in-plane parallelogram element and its application to skewed girder bridges. Developments in bridge design and construction. Crosby Lockwood, London, 1971, 304317. 18. CHEUNG Y. K. (KONG F. K. et ala (eds»). Tall buildings 2. Handbook 01 Structural concrete. Pitman Books, London, 1983,Ch.38. 19. Tso W. K. and BISWAS j. K. General analysis of non planar coupled shear walls.]. Struct. Div. Am. . Soc. Civ. Engrs, 1973, 99, No. ST3, Mar., 365-380. 20. Tso W. K. and BISWAS J. K. Analysis of core wall structures subjected to applied torque. Building Sci., 1973,8, 251-257.

, J

t

J

4. Coupling effects of beams and slabs

4.1 Types ofbeamlslab/walljoints Beam-slab-wall joints

I..inrcl hcam SJah

I~illtel

lJealll

Beam-wall joints

Slab-wall joints

Sial

I "illtcl l"'>Calll

26

4.2 Coupling effects of beams 4.2.1 Rotational d.o.f. at beam-wall joints: - .for compatibility between the beam and wall elements, the rotational d.o.f. at the beam-wall joints should be defmed as the rotations of the beam-wall interfaces - since the beam-wall interfaces are vertical, the rotational d.o.f. at the beam-wall joints should be taken as the rotations of the vertical fibres at the beam-wall interfaces 4.2.2 Shear deformation of beams: - if the beams are long (with span!depth ratios > 4), then the effects of shear deformation of the beams are small

- many coupling beams, however, havespanJdepth ratios smaller than 4, or even smaller than 1; in· such cases, the effects of shear deformation of the. beams should be taken into account

27

4.2.3 Local deformation at beam-wall joints: - stress concentration occurs at beam-wall joints, and as a result the bending stress distributions are not linear across the beam sections near the joint - the beam-wall joints translate and rotate relative to the remaining part of the wall as if the wall is an elastic support - Michael (1967) suggested to allow for the effects of j oint flexibility by extending the beams at each end by half the beam depth into the wall (equivalent length method) - the above method can be applied easily when the frame method is used for analysis by increasing the lengths of the beams and shortening the lengths of the horizontal rigid arms linking the beam ends to the wall axis, but is not applicable when the fmite element method is used for analysis because the fmite element method does not permit the lengths of the beams to be adjusted - when thefmite element method is used for analysis, it is better to allow for the local deformation at the beamwall joints by adding joint elements to model the joint behaviour

28

0045-7949/93 $6.00 + 0.00 (l) 1993 Pergamon Press Ltd

Computers &: Structures Vol. 48. No.4. pp. 615-625. 1993 Printed in Great Britain.

LOCAL DEFORMATIONS AND ROTATIONAL DEGREES OF FREEDOM AT BEAM-WALL JOINTS A. K. H.KwAN Department of Civil and Structural Engineering, University of Hong Kong, Pokfulam Road, Hong Kong (Received 9 April 1992)

Abstract-Local deformations· at beam-wall joints can significantly reduce the effective stiffness of coupling beams in shear/core wall structures. This phenomenon has been studied by many researchers and several methods of allowing for such effects have already been. developed. However, in the existing methods, the beam-wall joint rotations are often mistaken as the rotations of the horizontal rigid anns leading to incompatibility between the beam and wall elements. Moreover, many practical difficulties with the actual applications of these methods have been encountered. In this paper, it is proposed that in order to resolve the problem of incompatibility between the beam and wall elements, the definition of the joint rotations should be changed to the rotations of the beam-wall interfaces. A new method of using joint elements to model the joint deformations, which can overcome the problems with the existing methods, is proposed and two alternative beam elements with joint deformations taken·into account are developed. Finite element analysis is used· to evaluate the local deformations and determine the joint element properties.

NOTATIONS

A'

c d E

e G I

i L

/ M t V v

w a a' v

A. (JJ

equivalent shear area of beam cantilever span, i.e. distance from support to point of contraflexure depth of beam Young's modulus equivalent length of beam....wall joint shear modulus moment of inertia deflection factor equivalent t~tal length of beam beam span, i.e. physical length of beam bending moment at joint thickness of beam or·wall shear force at joint vertical deflection of joint width of wall shear deformation factor of the beam itself shear deformation factor of the equivalent beam Poisson's ratio joint flexibility coefficient rotation of joint

INTRODUCTION

It is well known that when the coupling beams in shear/core- wall structures deflect under load, stress concentrations and local deformations occur around the beam-wall joints. Cutting the coupling beams at their points of contraflexure, they can be regarded as cantilevers supported by elastic foundation (the walls), see Fig. 1. Local defonnations at the beam-wall joints produce additional deflections of the cantilevers and as a result reduce the effective stiffness of the coupling beams. The additional deflection of a cantilever due to elasticity of its support was first· studied by Weber [I] in 1949. He evaluated the local deformations at the

cantilever support by treating the foundation as a semi-infinite domain and assuming that the moment load at the built-in end of the cantilever was transmitted to the foundation by a linear distribution of normal stresses. The stress function method was used to determine the deflection of the elastic foundation and the support rotation was evaluated as the mean rotation of the beam-foundation interface. Weber's solution for the plane stress case is given by

(JJ

=

5.73(~), Etd

(1)

where OJ is the rotation of the support and M is the moment load transmitted to the foundation. In 1960, O'Donnell [2] tackled the same problem by assuming a cubic distribution of bending stresses, instead of a linear one, at the beam-foundation interface. The support defonnationswere evaluated by superimposing the surface displacements caused by·a large number of point loads acting on the foundation which was again treated as a semi-infinite domain. Apart from the rotation of the support due to moment load, the rotation due to shear load was also taken into account. O'Donnell's result for the support rotation is given by

where the second tenn on the right-hand side of the equation is ·the rotation due to shear load and. V is the shear load acting on the support. In 1967, Michael [3] used several different assumed bending and shear stress patterns at the beam-wall

A. K. H.

616

KWAN

t

---£]

w

c

---8 (a) coupled shear walls

(b) cantilever supported by elastic foundation

Fig. 1. Walls regarded as elastic foundations for cantilever beams.

interfaces to evaluate the local deformations at the joints. A similar method to that of O'Donnell's was used to determine the surface displacements at the beam-wall interfaces: As well as the rotations, the vertical deflections of the joints were also calculated. The results revealed that the joint rotations and deflections are more or less the same for the different stress patterns considered and that they can be given approximately by

(J)

v

= 6.00(E~2) + o.

76(;d)

=O.80(~)+ 1.74(~}

(3)

(4)

Based on these two equations, Michael proposed that the joint deformations can be accounted .for by extending the beam into the wall by a length of d/2. This method is called the equivalent length method. In the discussion on Michael's results, MacLeod [4] suggested that instead of increasing the effective beam length, the local deformation effects may also be allowed for in a simpler way by adding rotational springs with moment stiffness of Etd 2/6 at the ends of the beams. This is called the rotation spring method. However, since only the joint rotation due to moment load is considered, the joint rotation due to shear load and the joint deflection due to moment and shear loads would be effectively ignored. All these studies were carried out by treating the foundations (the walls) as semi-infinite domains. The advantage of doing this is that an analytical solution is possible. However, this method is applicable only when the walls are wide compared to the sizes of the beam-wall joints. Moreover, since the beams and walls are dealt with separately, it is not possible to take into account beam-wall interactions, and as a result, the interaction stresses have to be assumed rather than evaluated as an integral part of the solution. In order to resolve these problems, in the

late 1960s, researchers changed to the use of the finite element method to study the local deformation effects. Early in 1969, Hall [5] applied the finite element method to analyse the local deformations around beam-wall joints. In his study, the beams and the walls were analysed as integral structures and hence any interactions between them were automatically taken into account. Both L- and T-joints were studied. Several element meshes of varying degrees of fineness were used for the analysis, but even the finest mesh was rather coarse. The use of the equivalent length method to account for the joint deformations was recommended. From the finite element analysis results, Hall detennined the equivalent lengths as equal to O.28d and 0.42d for T- and L-joints, respectively, when the walls are wide and half the wall width when the walls are narrow. In 1973, Bhatt [6] carried out another study on the problem using a slightly finer element mesh than those used by Hall. Three alternative methods of allowing for the joint deformations were proposed. The first . method, which may be called the deflection factor method, is to use a deflection factor, i, to increase the flexibility of the coupling beam so as to account for the joint deformations. Actually this i-factor also allows for the shear deformations of the beam. It was presented as a function of the wall width/beam depth and cantilever span/beam depth ratios in the form of a small-scale graph. The second method is to increase the effective length of the beam by a factor of 0.5 x J(i) at each end. This equivalent length allows for both local deformations at the joints and shear deformations of the beam. In other words, the shear deformations of the beam are not allowed for by incorporating a shear deformation factor in the stiffness matrix but rather by extending the length of the beam beyond that required for taking into account joint deformation effects. The third method is to add rotational springs at the ends of the beam, as MacLeod suggested.

L9cal deformations at beam-wall joints

More recently, Cheung [7] used quadratic elements to analyse the joint deformations. Details of the finite element analysis were not given, but heoretically quadratic elements should be able to give more accurate results than lower order elements. Cheung proposed to increase the length, I, of the beam to PI, where P is given in the form of a table, to allow for the joint deformations. His method is actually identical to that of Bhatt's equivalent length method. And as in Bhatt's method the equivalent length Pi allows for both local deformations at the joints and shear deformations of the beam. Although several methods have already been developed to allow for the local deformation effects, and they have been in use for a long time, there are still problems with them. Firstly, many practical .difficulties with their· actual applications have been encountered; for instance, the methodology of extending the beam a certain length into the wall to allow for joint deformations is simply not applicable if the shear/core wall structure is to be analysed by the finite element. method which

617

requires the nodes to be fixed right at the physical locations of the beam- wall joints. Secondly, and in fact more importantly, the rotational DOF at the beam-wall joints have often been mistaken as the rotations of the horizontal rigid arms. The author [8, 9] has found in some recent studies that the practice of taking the joint rotations as the rotations of the horizontal rigid arms will cause incompatibility between the beam and wall elements. Such an incompatibility will in turn lead to errors in the joint rotations and underestimation of the effective stiffness of the coupling beams.. An attempt to resolve these problems is made in this paper. It is postulated that in order to ensure compatibility between the beam and wall elements, the joint rotations should be defined as the rotations of the beam-wall interface-s. Modifications necessary for the existing methods of allowing for joint deformations after changing the definition of the joint rotations to the rotations of the beam-wall interfaces are studied. It is found that they all have shortcomings and that a better method is to use joint elements to model the joint deformations. A parametric study of the local deformations using

vertical fibre

horizontal fibre -- - -

'----~p---

-

beam

(a) joint rotations defined as rotations of horizontal fibres

vertical fibre

horizontal fibre

(b)

joint rotations defined as rotations of vertical fibres

Fig. 2. Rotations of beam-walljoints. CAS 48!4-E

A. K. H.

618

finite element analysis is carried out to determine the joint element properties. Based on the finite element results, two alternative beam elements with joint deformations taken into account by incorporating joint elements at the ends are developed. ROTATIONAL DOF AT BEAM-WALL JOINTS

Many different definitions have been used for the beam-wall joint rotations in the existing methods of analysis. In the continuous connection method [10], they are taken to be the rotations of the vertical axis of the walls. This is equivalent to defining the rotations of the joints as the rotations of the vertical fibres there. In the frame method [II], they are assumed to be equal to the rotations of the horizontal rigid arms which are actually horizontal fibres in the walls. With regard to the finite element method, different researchers used different definitions. Abu-Ghazaleh [12] defined the rotational OOF at the joints as the mean of the rotations of the horizontal and vertical fibres. MacLeod [13] defined them as the rotations of the horizontal or vertical fibres at alternate nodes. On the other hand, in Sisodiya and Cheung's element, which was originally developed for bridge analysis [14] and subsequently adapted for building analysis [7], the joint rotations are taken as the rotations of the vertical fibres. Mohr [15, 16] initially defined the nodal rotations as the sum of the rotations of the horizontal and vertical fibres and later followed Abu-Ghazaleh's definition. The definitions described above are not equivalent. Figure 2 illustrates the different effects of adopting two common definitions for the joint rotations. In Fig. 2(a), the joint rotations are defined as the rotations of the horizontal fibres, while in Fig. 2(b), they are taken as those of the vertical fibres. The differences in deformed shapes of the beams clearly show that the bending and shear stresses and hence the effective stiffness of the coupling beams are dependent on how the joint rotations are defined. From Fig. 2(a), it can be seen that incompatibility between the beam and wall elements at their joints would arise if the joint rotations are taken as the rotations of the horizontal fibres. Figure 2(b), on the other hand, shows that to ensure compatibility between the beams and the walls, the rotations of the vertical fibres must be kept continuous across the beam-wall interfaces. Note that the difference in rotations of the horizontal and vertical fibres is actually the shear strain and that because of possible difference in shear strain at the two sides of a beam-wall joint, the rotation of the horizontal fibre at the beam side and that at the wall side are not necessarily equal. So there is no unique horizontal fibre rotation at a beam-wall joint at all and therefore it would be a mistake to define joint rotations as horizontal fibre rotations or any mathematical function of them. It is thus clear that the joint rotations should be defined as the vertical fibre

KWAN

rotations, i.e. the rotations of the beam-wall interfaces, otherwise the beam and wall elements would be incompatible and the effective stiffness of the beams in error. It is noteworthy that when Weber [I], O'Donnell [2] and Michael [3] evaluated the joint rotations, they determined the joint rotations as the mean rotations of the beam-wall interfaces which are really the vertical fibre rotations at the joints. Thus, in eqns (1)-(4), the rotations (jJ are in fact vertical fibre rotations. It was only during the subsequent development of the various methods of allowing for the local deformation effects that the joint rotations became mistaken as something else. EXISTING METHODS OF ALLOWING FOR JOINT DEFORMATIONS

As described in th.e Introductiop., several methods, namely' the deflection factor· method, the equivalent length method and the rotational spring method, have already been··developed. A detailed discussion on their accuracy, versatility, and the modifications required to change the joint rotations to the vertical fibre rotations is presented below. Deflection factor methods The deflection factor method was proposed -by Bhatt [6]. Basically, a deflection factor, i, is used to increase the flexibility of the beam to allow for the joint deformations. This factor also allows for the shear deformations of the beam itself. Since the i-factor is quite sensitive to the cantilever span/beam depth ratio, it requires the location of the point of contraflexure to be known before it can be determined (note .that the cantilever span is the length from the beam-wall joint to the point of contraflexure; it is not the same as the beam span). If the location of the point of contraflexure is indeterminate, such as in beams connected to walls at one end and frames at the other end, this method would fail. Apart from the above, there is another problem which is even more serious. It arises from the use of the i-factor to allow for the shear deformations of the beam, as explained below. The stiffness matrix equation of a standard beam element (Fig. 3), with shear deformations taken into account is given by l

Fig. 3. A standard beam element.

Local deformations at beam-wall joints

6 /2

-[3

4+ex I

-p.

12

VI

[3 6

Mt

=

EI/(l

+ ex)

12

-p.

6

2-ex

r

VI

[3

[i

-[3

6

4 I

-[2

Elli

=

12

6

12

-12

6

M2

12

6

-[3

[3 6

2

-p.

I

fi

Comparing the above two stiffness matrix equations, it can be seen that the two equations would not give the same results unless COl = (iJ2. Hence the use of an i-factor to account for beam shear defonnations is applicable only when COt = ()J2' or in other words, only when the point of contraflexure is known to be at the centre of the beam. If (01 and CO 2 can be different, then this method should not be used. Nevertheless, provided the above condition of COl = (02 is satisfied, no modification to the method is required after changing the definition of the joint rotations to vertical· fibre rotations.

(5J

6

-p.

V2

4+a I

CO 2

--

There are two different versions of equivalent length method. The first version was developed by Michael [3] and Hall [5]. They ,proposed to extend the end of the beam by an equivalent length equal to a

6

r

COt

I

Equivalent length method

12

Mt V2

6

-p.

VI

2-ex --

[3

I

where a is the shear deformation factor defined by a = (12EI/1 3) x (I/GA '). The corresponding stiffness matrix equation of the same beam element with shear deformations allowed for by an i-factor is given as follows:

6

f

12

6

-[3

M2

6

--

fi

V2

12

619

6

r

VI

2

-

COl

I

(6)

6

-/2

V2

4 I

(JJ2

-

certain fraction of the depth of the beam into the wall to account for the joint deformations. The second version was developed by Bhatt [6] and Cheung [7]. In this latter version, the length of the beam is extended into the walls to allow for both local deformations at the joints and shear deformations of the beam. Hence, the equivalent lengths derived by Bhatt and Cheung are longer than those by Michael and Hall. Whichever version is used, as the beam is extended into the walls, the beam-wall joints are effectively shifted into the walls (Fig. 4). This requires the horizontal rigid arms, which are there to allow for

A

~

v

y

flexible portion of beam element

I

e

I

1

-tc/ /////~ ////

/A

rigid" arm \ I

I.

wall

/

- ,

I

·

~+

- ---

beam

\

I

e

beam-wall joint shifted into wall

v/////.

rigid arm

wall

A

A 'f

V

Fig. 4. Equivalent length method.

·

I·

A. K. H.

620

the finite widths of the walls, to be shortened at the same time. Thus, this method, in its current form, is applicable only if the shear/core wall structure is to be analysed by the frame method in which the lengths of the beams can be increased by reducing the lengths of the rigid arms. If the finite element method is used to analyse the structure, then since the nodes have to be placed right at the physical joints and cannot be shifted, this methodology of extending the beams into the walls would not be applicable. After changing the definition of the beam-wall joint rotations to the vertical fibre rotations, as the joint rotations and the rigid arm rotations would then be unequal, two rotational DOF, namely the horizontal fibre rotation and the vertical fibre rotation, would be required at each end of a beam element with rigid arms. For connection to the beam elements, the column elements which model the walls need also to have two rotational DOF at each end. Extensive reformulation of the frame method is needed. However, there is a possibility of locating the nodes right at the physical positions of the joints and incorporat- . ing the horizontal rigid arms into the wall elements instead of in the beam elements [8, 9]. In that case, only one rotational DOF, the vertical fibre rotation, would be required at each node.. But then, since the nodes have to be fixed at the joints, the equivalent length method would become inapplicable. A solution to this problem is proposed later in this paper. Rotational spring method The advantage ·of the rotational spring method, which was first suggested by MacLeod [4], is that it is simple to use and easy to understand. The nodes can be located immediately on the beam-wall joints so there is no problem with the shifting of the joint locations. However, the joint ·rotations due to shear and joint deflections caused by both moment and shear would be ignored. This could cause significant errors in accounting for the joint deformation effects when the span/depth ratio of the beam is small. No modification to the method is required after changing the definition of the joint rotations to the vertical

KWAN

c

fixed

fixed Fig. 5. Finite element analysis of local deformations around beam-wall joints.

fibre rotations. It should, nevertheless, be noted that after changing the joint rotations to the vertical fibre rotations, the beam elements should no longer be incorporated with horizontal rigid arms because two rotational DOF at each end of the beam element would then be required. For this reason, the beam element developed by Bhatt [6] with both rotational springs and horizontal rigid arms incorporated should not be used after the definition of the rotational DOF is changed. FINITE ELEMENT ANALYSIS

A parametric study of the local deformations around the beam-wall joints is carried out by finite element analysis using a very fine mesh of rectangular bilinear elements as shown in Fig. 5. The structural parameters studied are the half wall width/beam depth ratio, (w/2d), and the cantilever span/beam depth ratio, (c/d). In the investigation, the problem is analysed for (w /2d) = 4, 3, 2, 1 and 0.5, and (c /d) = 4, 3, 2, 1, and 0.5. All together, 25 combinations of the two parameters· are studied. The Poisson ratio is

contour of max. principal stress as multiple of Vc/td 2

Fig. 6. Local deformations and stress concentrations around abeam-wall joint.

Local deformations at beam-wall joints

taken to be 0.25 throughout. Figure 6 illustrates the defo~ed shape of a typical beam-wall joint and the stress concentrations around the joint. The joint rotations and deflections due to local deformations are determined from the following equations

v

v

=

(c

621

+ e)3(c + e) 3EI

c3 c GA' = 3EI + GA'

+

joint deflection by finite element analysis - joint deflection with local deformations ignored

(7)

+ COJ = tip

deflection by finite element analysis - tip deflection with local deformations ignored

in which the joint and tip deflections with the local deformations ignored are evaluated by hand calculation with the walls and beams treated as·frame members connected by rigid joints. The results for the· joint rotations and deflections are summarized by the following equations

where· the dimensionless joint flexibility coefficients At ,A2 and A3 are as given in Table 1. Note that the A-values vary with the half wall width/beam depth ratio but are independent of the cantilever span/beam depth ratio. Compared with the results obtained by O'Donnell [2] and Michael [3] using analytical methods with the finite widths of the walls and the beam-wall interactions ~gnored, the joint deformationsobtained by the finite element analysis are significantly smaller. The joint flexibilities as given by eqns (9)'and (10) can be converted into equivalent lengths which give the same tip deflections by. the following equation

(8)

It should be noted that the equivalent length, e, evaluated above only allows for the joint deformations; the shear defonnations of the beam are to be separately allowed for as in Michaers [3]- and Hall's [5] methods. Table 2 presents the results so obtained for the equivalent lengths. It can be seen from the table that the equivalent length of a joint varies with both the half wall width to beam depth ratio and the cantilever span to beam depth ratio. Generally, the equivalent length is longer when both the wall width and cantilever span are large, and shorter when the wall width and cantilever span are small. The dependence of the equivalent length on the cantilever span creates an inherent difficulty with the equivalent length method; it requires the location of the point of contraflexure to be known before the equivalent length can be determined. Fortunately, unlike the equivalent lengths in Bhatt's [6] and Cheung's [7] methods, the equivalent lengths tabulated in Table 2 vary only slightly with the cantilever span/beam depth ratio. In order to make the method more practicable, it is proposed to ignore the· effect of the cantilever span on the equivalent length and account for the effect of the wall width approximately by the following equation

e

. .

= mInImum

{O.29d

0.22).~'.

(12)

Table 1. Joint flexibility coefficients

11

12

4.53 4.49 4.41 4.18 3.83

0.58 0.55 0.51 0.42 0.30

The errors in the· effective stiffness of the coupling beam due to the above approximation are given in Table 3 from which it can be seen that with the equivalent length taken to be O.29d when the wall is wide and 0.22w when the wall is narrow, the error in the effective stiffness of the coupling beam is at most ± 100/0 within the range of c/d and w/2d studied. From an engineering application point of

Table 2.. Equivalent length of joint expressed as a fraction of beam depth (e /d) Cantilever span to beam depth ratio (c /d)

Half wall width to beam depth ratio (w /2d)

4.0

3.0

2.0

1.0

0.5

4.0 3.0 2.0 1.0 0.5

0.366 0.362 0.354 0.334 0.304

0.362 0.358 0.350 0.329 0.299

0.355 0.350 0.341 0.319 0.287

0.338 0.329 0.317 0.291 0.254

0.315 0.301 0.284 0.248 0.201

622

A. K. H. KWAN Table 3. Percentage errors in beam stiffness, if e is taken to be 0.29d or 0.22w, whichever is smaller Cantilever span to beam depth ratio (c/d)

Half wall width to beam depth ratio (w /2d) 4.0 3.0

2.0 l.0 0.5

4.0

3.0

2.0

LO

0.5

5.3 5.0 4.4 3.0 5.9

6.4 6.1 5.3 3.4 7.2

8.0 7.4

9.1 7.4 5.1 0.2 6.6

6.2 2.7 -1.4 -9.8 -4.7

6.3 3.5 8.5

view, such errors should be acceptable and It IS therefore recomtpended to use eqn (12) instead of Table 2.

In general, the joint deformations can be evaluated by

BEAM ELEMENTS WITH JOINT DEFORMATIONS ALLOWED FOR

A new method of using joint elements to model the deformations at the beam-wall joints is developed in the following. Consider the walls and the coupling beam in Fig. 7, where the nodes 1 and 2 are the nodes at the wall sides of the beam-wall joints and the nodes I' and 2' are the nodes at the beam sides of the joints. The flexibilities of the joints at each end of the coupling beam are to be modelled by joint elements. Briefly, a joint element is one that has two nodes with the same coordinates but different translational and rotational OOF. For instance, the joint element at the left-hand side of the beam consists of two nodes, nodes .1 and 1', which have the sam~ nodal coordinates. Likewise, the joint element at the right-hand side of the beam also consists of two nodes, nodes 2 and 2', which are located at the same position but on different sides of the joint. VI

12

6

13

f2

-13

M.

p.

6

4+a --

-/2

V2

EI =-l+a

l

V

wall

6

-12

-[2

6

2-a

r

t 2_P:

I

6

12

13 6

-p.

6

P2-a -I

6

-p.

VI'

WI'

(15) v 2'

4+a

-I

W2'

Two alternative beam elements with the joint deformations taken into account by incorporating joint elements at the ends are developed in the following sections.

V

be_a_m

12

I

12

M2

-t :~f_l/

where VI' v2 , (lJI and (J)2 are the translational and rotational OOF at the wall sides of the joints and VI" V 2" ro l , and {J)2' are the corresponding DOF at the beam sides of the joints. To ensure compatibility between the beam and wall elements, the rotational DOF are defined as vertical fibre rotations. Thus, if the lateral strains in the walls are negligible, the rotations (J). and 002 would be equal to the rotations of the vertical axis of the respective walls. On the other hand, the rotations 00 1, and 0)2' are the rotations of the beam-wall interfaces at the two ends of the beam, respectively. The coupling beam can be modeled by a standard beam element with shear defonnations taken into account. The stiffness matrix equation of the beam element is given by .

Flexibility coefficient method

wall

From eqns (9) and (10), the flexibility matrices of the two joint elements are obtained as

(16) Fig. 7.A beam element with joint deformations allowed for.

Local deformations at beam-wall joints

623

in which L = I +e.

(17)

ct.

where AI' A2 and A) are the joint flexibility coefficients as given by Table 1.. and the items in parentheses with subscripts 1 and 2 are the corresponding values at nodes I and 2, respectively. Appending these two joint elements to the ends of the beam element, an element with four nodes., two internal nodes l' and 2' and two external nodes I and 2, is formed. Substituting the above equations for the flexibility matrices into eqns (13) and (14), and eliminating the DOF at the internal nodes from eqns (13)-{15).. the stiffness matrix equation of the beam element with joint deformations allowed for is obtained as follows. VI

MI

V2 M2

=(I+K{~I

:J)-I

and

COl

. (18)

Let the equivalent lengths of the beam-wall joints at the left and right-hand sides of the coupling beams be e J and e2 ~ respectively. The joint flexibility matrices are as follows:

F1 =

F2 =

(d)

(e2EIf )

(;~)

(3EI e~ e2 ) + GA' -

(d)

2El .

2EI

-

(19)

(e~ ) 2EI (20)

(il)

Substituting into eqns (13) and (14) and eliminating the DOF at the internal nodes, the following stiffness matrix equation for the beam element with joint deformations taken into account by means of equivalent lengths is obtained 12 VI

MI

V2

M2

L

3

EI =T'·-1 +a'

12 - L3

6

L

2

I

0

0

0

0

0 1 e'! 0

(24)

,

VI

2-a

,

L2 12 - L3

6 -L 2

12

L3

6 - L2

6

2-a' :L

6 - L2

--

L

0

0

0

6

6 - L2

2

-e 1 0

[2

4+a -L

6

(23)

Since the end nodes are placed at the physical positions of the beam-wall joints.. both the above elements can be applied to the finite element analysis of coupled shear/core wall structures in which the nodes are required to be located right at the joints. Moreover, when applied to the frame method.. as the nodes can be located immediately on the beam-\vall joints, they allow the joint. deformations to be taken into account without shifting the beam-wall joints into the walls. This is important because when the frame method is reformulated to have the joint rotations redefined as the vertical fibre rotations.. the nodes must be moved from the centroidalaxis of the walls to the beam-wall joints or else the use of two rotational DOF at each node, which is rather cumbersome.. would be required.. Although the rotational spring method can allow joint deformations to be taken into account without shifting the joints, it only accounts for the joint rotations due to moment loads and is therefore less accurate. Each of the ·above beam elements has its own advantages and disadvantages. Generally speaking, the beam element based on the flexibility coefficient ;~ . method is more versatile; it can be applied even to those cases in which the walls and the beams are of different materials or thickness. Theoretically.. it can also give more accurate results. However, it requires

Equivalent length method

el )

~)

L) )(GA'

Applications

V2

There is no simple algebraic expression for the stiffness matrix of this beam element. However, it can be obtained numerically with no particular difficulties as part of the computer analysis.

3£/+ GA'

(22)

It is interesting to note that the stiffness platrix in eqn (21) is very similar to that of a standard beam element with horizontal rigid arms incorporated; the only differences between them are just the signs and values of the rigid arm lengths. Hence~ existing computer programs need only be modified slightly with the rigid ann lengths changed.

(JJ2

(d

2EI

I

T=

VI

Kh

'=C

+ e2

L

4+a' L

WI

T

(21) V2

W2

.

624

A. K. H. KWAN

either the flexibility coefficients to be input or the the beam elements. This would, however, require the whole of Table I to be stored in the computer program nodes to be moved from the centroidal axis of the and the actual coefficients interpolated according walls to the physical positions of the beam-wall to the wall width. Moreover, as there is no simple joints. And, as the joints cannot then be shifted, the algebraic expression for the stiffness matrix, it takes joint deformations would have to be allowed for by extra computer time to obtain the stiffness matrix some means without shifting the locations of the numerically. On the other hand, the beam element beam-wall joints. based on the equivalent length method is much A new method of allowing for the joint deformsimpler to apply. In fact, its stiffness matrix resembles ation effects by using joint elements to model the joint that of a standard beam element with horizontal rigid deformations is proposed. This method can allow arms so closely that existing computer programs need the joint deformations to be accounted for without only be slightly modified with the rigid arm lengths shifting the locations of the joints and can therefore changed. There will be some errors in the effective be applied to both finite element analysis and frame stiffness of the coupling beams due to the negligence method analysis of the structures. A parametric study of the effect of cantilever span on the equivalent of the joint deformations using finite element analysis length but the errors are at most only 10% which is carried out to detennine the joint element properties. should be acceptable in normal engineering applica- The results of the finite element analysis are presented tions. Unfortunately, however, eqn (21) would not be . in both the forms of flexibility coefficients and equivapplicable if the walls and the beams are of different alent lengths. Two alternative beam elements, one materials or thickness. Summing up, it is suggested with the joint deformations allowed for by means that if highest generality and accuracy are desired, of flexibility coefficients and the other by means of tl1en the beam element based on the flexibility equivalent lengths, are developed. The first element is coefficient method should be used. But, for most more general and can give more accurate results, but engineering applications in which the walls and the is also more involved in computer implementation. beams are of the same material and thickness, the Relatively, the second element is easier to use. In fact, beam element based on the equivalent length method this element can be implemented just by modifying the values of rigid arm lengths in existing computer is recommended. programs. However, it is restricted only to those cases in which the walls and beams are of the same material CONCLUSIONS and thickness and is slightly less accurate than the The existing methods of allowing for the effects first element. of joint deformations are reviewed. It is found that despite being used for many years, there are still a REFERENCES number of problems with them. The major problem 1. C. Weber, The deflection of loaded gears and the effect is with the definition of the joint rotations which have on their load carrying capacity. Department of Scientific often been mistaken as the rotations of the horizontal and Industrial Research, Germany, Report No.3, Part 1, rigid arms thereby leading to incompatibility between England (1949). 2. W. J. O'Donnell, The addi~ional deflection of a cantithe beam and wall elements. Moreover, many practical lever due to elasticity of the support. J. Appl. Mech., difficulties with their actual applications have been enASME 461-464 (1960). countered. For instance, the deflection factor method 3. D. Michael, The effect of local wall deformations on is, strictly speaking, applicable only when the point the elastic interaction of cross walls coupled by beams. Proceedings, Tall Building Symposium, University of of contraftexure is at the centre of the beam. The Southampton, pp. 253-270. Pergamon Press (1967). equivalent length method, on the other hand, suffers 4. I. A. MacLeod, Discussions on ref. [3]. Proceedings, from the shortfall of requiring the beam-wall joints Tall Buildings Symposiuln, University of Southampton, to be shifted as the lengths of the beams are increased pp. 271-272. Pergamon Press (1967). to allow for the joint deformations. The rotational 5. A. S. Hall, Joint deformations in building frames. Civil Engng Trans, InSf. Engrs, Australia 60-62 (1969). spring method does not require the position of the 6. P. Bhatt, Effect of beam-shea rwa II junction defonnations joints to be shifted, but it is not capable of taking into on the flexibility of the connecting beams. Building Sci. account the joint rotations due to shear loads and 8, 149-151 (1973). the joint deflections due to moment and shear loads. 7. Y. K. Cheung, Chapter 38: Tall Buildings 2. Handbook of Structural Concrete (Edited F. K. Kong et af.). It is postulated that in order to restore compatibility Pitman, London (1983). between the beam and wall elements, the beam-wall 8. A. K. H. Kwan, Analysis of ocupled wall/frame joint rotations should be redefined as the rotations structures by frame method with shear deformation of the beam-wall interfaces, i.e. the vertical fibre allowed. Proc./nsl. Civ. Engrs, ParI 291,273-297 (1991). rotations. As the joint rotations and the rigid arm 9. A. K. H. Kwan, Re-fonnulation of the frame method. Proc. Insf. Civ. Engrs J. Struct. Bldgs 94, 103-116 (1992). rotations would then be unequal, each end of a beam element with horizontal rigid arms incorporated must 10. J. K. Biswas and W. K. Tso, Three dimensional analysis of shear wall buildings to lateral load. J. Struct. Div., have two rotational DOF. The use of two rotational ASCE 100, 1019-1036 (1974). DOF per node can be avoided by incorporating II. I. A. MacLeod, Structural analysis of wall systems. Structural Engineer 55, 487-495 (1977). the rigid arms· into the wall elements rather than in

r.

Local deformations at beam-wall joints 12. B. N.Abu-Ghazaleh, Analysis of plate-type prismatic structures. Ph.D. thesis, University of California, Berkeley (1965). 13. I. A. MacLeod, New rectangular finite element for shear wall analysis. J. SlrUCI. Div., ASCE 95, 399-409 (1969). 14. R. G. ·Sisodiya and Y. K. Cheung, A higher order in-plane parallellogram element and its applications to skewed girder bridges. In Developments in Bridge

~25

Design and Construction (Edited by K. C. Rockey et al.), pp. 304-317. Crosby Lockwood, London (1971). 15. G. A. Mohr, A simple rectangular membrane element including the drilling freedom. Comput. Struct. 13, 483-487 (1981). 16. G. A. Mohr, Finite element formulation by nested interpolations: application to the drilling freedom problem. Comput. Struct. 15, 185-190 (1982).

4.3 Coupling effects of slabs 4.3.1 Rotational d.o.f. at slab-wall joints: - if the slabs are modelled as thin plates, then the slab-

wall interfaces are horizontal - hence, the rotational d.o.f at the slab-wall joints should be defmed as the rotations of the horizontal fibres in the walls 4.3.2 Shear deformation of slabs: - since the span!depth ratios of the slabs are normally greater than 4, the effects of shear deformation of the slabs are negligible 4.3.3 Stress concentration around slab-wall joints: - bending of slab is maximum near the wall and decreases with distance from the plane of the wall - effective bending stiffness of slab normally expressed in terms of the effective width of the slab - there is also local deformation of the wall which will affect coupling stiffness of the slab

29

4.3.4 Effective width of couIJling slabs Results obtained by Coull and Wong (1981): 0-8

0-6

~

>........

u

>-

i

'0-4

=~1_-_

0-2

0

----' (a)

y.

~ = l-~(!:-) -1 5 Y

Y

0-8

•

1

~

__ o=--o==-o 0=-0=0-

YL ~ ... ~

,~o-=-

o~<:J

.

O~\

,/'

. if' · ~

0-6

0/ "'"

Ye

•

o

Generalized design centre

.

./

0-2

.

L

YY

Normal section

2 L

L

SY'

Y

Reciprocal section

o

2

3

U·y (bl Effective slab width for p(anewall configuration: (a) generalized design curve; (b) empirical curve

30

4.4 Coupling effects of beams and slabs together

- the beam and slab act compositely like aT-beam - coupling effect of the T-beam is highly dependent on the effective flange width of the beam - effective flange width depends on two factors: the outof-plane bending action of the slab and the in-plane membrane action of the slab - both the out-of-plane bending and in-plane membrane actions exhibit stress concentration characteristics - moreover, there is also lqcal deformation in the wall which will also affect the coupling action of the composite beam-slab element - behaviour very complicated, no defmite solution at the moment, no design guidance available, more research needed

31

5. Finite element analysis of tall buildings

5.1 Difficulties with the finite element method Application of the [mite element method to shear/core wall analysis can be dated back to the 60s'. Theoretically, the [mite element method, being the most powerful tool of analysis available, can be applied to any type of building structures. However, due to relatively low efficiency and high computing cost, full [mite element analysis of coupled shear/ core walls hav,e never been popular. There are all together three causes for the low efficiency of the method.

32

·The fust cause is the inefficiency of using 2-d plane stress elements to model the coupling beams. During the earliest applications of the fmite element method to coupled wall structures, both 2-d plane stress elements (Girijavallabhan 1969) and I-d beams elements (MacLeod 1969) had been used to model the coupling beams. In 1975, a comparative study (Al-Mahaidi and Nilson 1975) revealed that it is much more accurate and efficient to model the coupling beams by I-d beam elements. Since then, it became widely recognized that the coupling beams should be modelled as I-d elements and, in order to allow direct connection with the beam elements, the walls should be modelled by plane stress elements with rotational d.o.f. The incorporation of rotational d.o.f. into plane stress elements had been found a formidable task (Mohr 1982). Nevertheless, various forms of plane stress elements with rotational d.o.f: have been developed. A review of these elements regarding their suitability for tall building analysis was given by the Author in a previous paper (Kwan 1992). When incorporating rotational d.o.f into plane stress elements, it should be borne in mind that the rotations of the vertical fibre and that of the horizontal fibre are not the same (they differ by an amount equal to the shear strain) and that for beam-wall compatibility, the rotational d.o.f at the beam-wall joints should be defmed as the rotations of the vertical fibres at the beam-wall interface.

33

The second cause is the occurrence of stress concentratio,ns and local deformations around the beam-wall joints. In order to deal with high stress gradients near the joints, the element mesh needs to be refmed at least locally there. Numerical experience (Hall 1969, Bhatt 1973) indicated that very fme meshes around the joints are required to give acceptable accuracy. It is thus not practical to allow for local joint deformations by refming the element mesh near every joint in a full scale analysis (not only because of the high computing cost, but also because of the difficulties with the renumbering of the nodes due to mesh refmement). A more practical solution is to account for the local deformations by adding rotational springs or both rotational and shear springs at the beam-wall joints (Harrison et al. 1975).

34

The third cause is the presence of parasitic shears in many of the lower order elements which render the elements too stiff under bending unless the elements are very small. Since shear/core walls are subjected principally to bending actions, this is a very serious problem indeed. The source of the trouble is the inability of the elements to curve themselves to follow the deformed shape of the structure when subjected to bending. Various techniques of overcoming this problem had been developed (Cook 1975). However, it is felt that the best method of dealing with parasitic shears is to avoid them by using elements which can exactly represent the strain state of pure bending.

35

In order to improve the computational efficiency of the f~te element method, fmite strip elements (Cheung and Swaddiwudhipong 1978) and higher order elements (Chan and Cheung 1979) were developed to model the walls. The fmite strip elements extend the whole height of the building while the higher order elements span at least several stories high. Thus the number of unknowns to be solved is generally smaller than those of other fmite element methods. These elements are not subjected to parasitic shears. However, as . they span at least several stories, they are afflicted by an excess continuity problem which arises from the fact that the elements can only yield continuous stress distributions within an element but due to the diaphragm actions of the floor slabs and the concentrated moments from the coupling beams, there could be discrete changes in shear force and bending moment across a floor level.

36

5.2 Suitable finite elements for tall building analysis

It became clear at this stage that a good element for modelling the walls should be: (1) one that has rotational d.o.f for direct connection to the beams; (2) can represent the strain state of pure bending so as to avoid parasitic shears; and (3) spans at most only one story so that there will be no excess continuity problem. In 1983, Cheung adapted the so called beam-type element, which was originally developed for bridge deck analysis (Sisodiya and Cheung 1971), to apply to buildings (Cheung 1983). This beam-type element is a rectangular four node element with three d.o.f. at each node. The nodal d.o.f. include a horizontal translation, a vertical translation and an in-plane rotation defmed as the rotation of the vertical fibre at the node. This element satisfies all the above criteria for a good wall element.

5.3 Beam-type element for shear/core wall analysis See attached paper.

37

Analysis of coupled shear/core walls using a beam-type finite element A. K. H. Kwan and Y. K. Cheung Department of Civil and Structural Engineering, University oj Hong Kong,Hong Kong (Received July 1992,. revised version accepted March 1993)

The Sisodiya and Cheung so-called beam-type element, which was originally developed for bridge analysis, had been successfully adapted by Cheung to apply to coupled shear/core wall structures. This element is not afflicted by shear Jocking and has rotational qegrees-of-freedom for direct connection with the coupling beams. Moreover, the definition adopted for each of its rotations is the vertical fibre rotation which, as Kwan recentl,y found, is the only definition that can ensure compatibility between the beam and wall elements. In this paper, further studies on the application of this element are carried out and improvements, which can lead; firstly, to higher computational efficiency of the method; secondly, enable .more reasonable shear stress results to be obtained, and finally, allow the maximum bending stress to be determined for practical design purpose without using a fine mesh, are proposed. Numerical examples are given to demonstrate the improvements achieved. Keywords: shear/core wall structures, FEM

Existing methods for coupled shear/core· wall analysis may be categorized into the continuous connection method 1 • 2 , the frame method 3 • 4 and the finite element methodS. 6. Among them, the finite. element method, being the most powerful tool of analysis available, is the most versatile; it can be applied to any form of structures with various support conditions and subjected· to any type of loadings. Many different types of finite elements are now available. However, not all of them are suitable for coupled shear/core wall analysis. For instance, some of the lowerorder elements, such as the bilinear element, are found to be afflicted by the shear locking problem which renders the elements overstiff under bending actions. Since shear/ core walls are subjected principally to bending actions, such shear locking can severely reduce the convergence rate and thus the efficiency of the elements. Hence for shear/core wall analysis, elements not afflicted by shear locking are preferred. Moreover, as the coupling beams are normally modelled by beam elements with end rotations, the elements which model the walls need to have inplane rotations so that they can be connected directly with the beams. A number of plane stress elements with rotationaldegrees-of-freedom (DOF) have been developed and many different definitions for the rotations adopted 7- 1o • However, Kwan 11. 12 has recently found that the various definitions adopted are not equivalent to each other and that if the rotational DOF are not defined

0141-0296/94/020111-08

©

1994 Butterworth- Heinemann Ltd

as the rotations of the beam-wall interfaces, i.e. the rotations of the vertical fibres at the beam-wall joints, the beam and wall elements would be incompatible.. Therefore for modelling shear/core walls, elements with rotational DOF. defined as the vertical fibre rotations are more appropriate. In 1983, Cheung 13 adapted the Sisodiya and Cheung!4 so-called beam-type element!, which was originally developed for bridge analysis, to apply to coupled shear/ core wall structures. This element (Figure 1) has rotational DOF defined as vertical fibre rotations for direct connection with the beam elements and is not afflicted by shear locking. Hence it is a suitable element for coupled shear/core wall analysis. It is called a beam-type element because its lateral displacement function is similar to that of a beam element. This beam-type element can generally· produce very accurate results for the lateral deflections. However, like most other elements formulated by the displacement method, it is less accurate in the stress evaluations. Firstly, as its shear strain function is quadratic along the vertical direction, it should theoretically yield reasonably accurate shear stress results, but in reality it tends to produce unrealistically large fluctuations in shear stresses. Secondly, since the element allows only a constant bending moment within an element, it yields a stepwise variation of bending moment with height. The bending moment results may be taken as the average

Engng Struct. 1 994, Volume .16, Number 2

111

Analysis of coupled shear/core walls: A. K. H. Kwan and Y. K. Cheung

tion while v is linear in both directions, i.e.

+ 'X 2 ,., + ':t 3 17 2 + ':1. 4 17 3 ) + ~(a9 + alo'l + £1 11 11 2 + Ct. 12 113 ) v = (as + ct6~ + ':t,fl + ~8~YJ) which ~ = x/a and 'I = y/b. The close U

b

v

~(~2

.. _ul

a

_ u2

a

Sisodiya and Cheung beam-type element

bending moments within the elements and attributed as the bending moment values at the centres of the elements but would not give directly the maximum bending moments which usually occur at either the top or bottom of the elements and are needed·for structural design. In this paper, the above problems are studied and methods of overcoming them are proposed. Additionally, the computational efficiency of the method is also improved by neglecting the lateral strains in the walls which are generally of little significance.

Notation E Young's modulus G

Ii

u v t (Xi

ex e, 1X1 (j % (j,

7:

%,

co

shear modulus shape functions horizontal displacement vertical displacement thickness of wall coefficients in displacement functions horizontal axial strain vertical axial strain shear strain horizontal axial stress vertical axial stress shear stress inplane rotation defined as -ou/oy

The Sisodiya and Cheung beam-type element This is a rectangular four-noded element with three DOF at each node (Figure 1). The nodal OOF are the horizontal displacement u, the vertical displacement v, and the inplane rotation co which is defined as -(au/oy)., i.e. t.he vertical fibre rotation. It was formulated by starting with displacement functions which are such that u is linear in the horizontal direction and cubic in the vertical direc-

112

(2)

(3)

~~f----} Figure 1

(})

V

1

t_~- :.W_l_

I-I

(£1 1

in resemblance of the above displacement functions to the actual deformation modes of a beam renders this element very beamlike. It is this beam-like characteristic that makes it particularly suited for the analysis of shear/core walls ·which act principally as vertical cantilever beams. Solving the a coefficients in terms of the nodal DOF, the displacement field is obtained as

y,v

.b

=

Engng Struct. 1994, Volume 16, Number 2

where

*

f 1 i = (1 + ~ ei )( 1 + '1 r/i) f 2i = t(2 + 11Y/i - 11 2 )f 1£ f 3i = t b11l1 - 11 2 )fli

(4)

(5)

(6)

e

ei

and and 11i are simply the values of and '1 at node i. The strain functions and the stiffness matrix of the element can be derived by following the standard procedure of finite element formulation. The above element has been successfully applied to various kinds of coupled shear/core wall structures 13 and implemented in some tall building analysis software. However, its computational efficiency can actually be further improved by-neglecting the lateral strains in the walls which are generally insignificant. With the lateral strains neglected, the horizontal displacement u of the element becomes independent of and the number of DOF of the element can be reduced from twelve to eight as shown in Figure 2. The resulting element, hereafter called the simplified beam-type element, is computationally much more efficient than the original element. The idea of making the assumption that the lateral strains in the walls are negligible so as to improve the computational efficiency of the analysis method is not new. Although often not explicitly stated, this assumption has been incorporated in the frame method 3 • 4 . Cheung l3 and Cheung and Swaddiwudhipong lS have applied ·this idea to the finite strip method. K wan 12 has also incorporated such an assumption in his strain-based element and had in fact suggested previously that this idea could be applied also to the beam-type e.lement. Neglecting the lateral strains, the displacement functions of the element become

e

=

(!Xl

v=

(!X s

U

+ ':1. 2 '1 + ':1. 3 11 2 + ~4113) + ':1.6~ + ':1. i '1 + !X8~'1)

(7)

(8)

Solving the (X coefficients in terms of the nodal parameters shown in Figure 2{ a), the displacen1ent field of the element is obtained as

+ '1 3 )U 1 +!b( -1 + 11 + 11:! - '1 3 )W 1 l(2 + 3'1 - ,,3 )U 2 + ib( I + 'I - ,,2 - 11 3 )w 2 (9) v = !( 1 - ~)( I - 'l)V 1 + !( 1 + ~)( I - 11)r 1 +!( I - ~)( 1 + '1)v 3 + i( I + ~)( 1 + ,,}t'4 (10)

Ll =

*(2 -

311

Analysis of coupled shear/core walls: A. K. H. Kwan and Y. K. Cheung

M v

V

4

3 --.f--.

t

V

).

3

1

b

2

V

U

4

2

1

1

V

V

thickness t

=

b

v

v

1

t

~

W

.

u

1

1

2

t

t

J.. M

1

a Figure2

a

t 1

1

b

a

Simplified beam-type element. (a), nodal displacements; (b), nodal forces

It should be noted that the horizontal displacement function u is identical to the lateral displacement function of a beam element· while the vertical displacement function v is the same as that of a standard bilinear element. Although this element resembles a bilinear element in the v function, it is not afflicted by shear locking because its u function allows it to bend like a beam when subjected to bending. The rest of the formulation follows the standard finite element procedure.

Shear stress in the element Differentiating the displacement functions given in eq uations (9) and (10), the shear strain in the element is obtained as "txy

U

2

= ~ (V2~ VI

-

W1

+KV4 ~ V3 -W 2

)(1 -

)(1 +

1])

1])

+~U2 ~ U l + W l + W2 )(1 - 1]2)

(11)

It is thus seen that the shear strain in the element varies as a quadratic function in the vertical direction. So theoretically, this element should be able to give reasonably accurate shear stress· results. However, experience from actual applications of the element revealed that this is not really the case. The problems with the shear stress evaluation are best illustrated by studying Example 1 given in Figures 3 and 4. In Example 1, a typical 8-storey coupled shear wall structure subjected to lateral loads, Figure 3a, is analysed using two layers of beam-type elements per storey. For comparison, the structure is also analysed by the frame method. The deflection results as obtained by the two methods of analysis are plotted in Figure 3 (h) from which it can be seen that the two methods give more or less the same lateral deflections. However, the shear

stress results obtained by the two methods are quite different from each other. From Figure 4{ a), in which the shear force distribution at the lowest two storeys are plotted, it can be seen that while the frame method yields exact shear force results, the finite element method produces a fairly large fluctuation of shear force with height which is not reasonable from the physical point of view. The large and erratic fluctuation of shear stresses in the finite element results is rather annoying. When a structural design engineer comes across such shear stress results, he/she is often lost as to what shear force values should be used for the design because in most practical cases, it is difficult to guess the correct shear forces in the walls. Fortunately, however, the remedy to this situation is actually quite simple. There are two possible ways of overcoming this problem. The first way is to take the shear stresses everywhere within an element as always equal to that at the centre of the element. In displacement formulated finite elements, the stress and strain results are usually more accurate at the centroids of the elements and so it is thought that the same should also apply to this element. Putting 11 = 0 in equation (11), the shear strain at the centre of the element is obtain as "txy

= i(U 2~ U l )

1(

+4

WI

+ (J)2 +

V2

+ v4

-. Vl -

a

V3 )

(12)

from which the shear force acting on the element can be determined as

U

= 2Gtay.~). 3Q =Gt( 2b (11 2

+!(V2 + V4

a

ud + 2 (w l + W2)

-

-

Vl -

»)

V3

Engng Struct. 1994, Volume 16, Number 2

(13)

113

Analysis of coupled shear/core walls: A. K. H. Kwan and Y. K. Cheung

32

'I

V

finite element method

v

1

rt

,,'1,1

24 rt

frame method

'I

E

o

'dt @)

16

(J)

QJ ....-4

.... o

~

[f)

00

8

0'-------...1.0.-----...1.-----. .. 60 40 20 o thickness = O.4m beam depth = O.8m E = 20GPa. v = 0.25

deflection (mm)

b

a Figure 3

Example 1, coupled shear walls. (a), structure analysed; (b), deflection curve

.

finite element method using

- - -- -- L -. - - - - - - --Eqn. (15)

r

8

....

. "

8

frame method or finite element method using Eqn. (18)

2nd storey

-..r-:: ' )

"j ~

___ - - ......:-----_-_ _

4

,.

DO

(coincide with exact results)

Y f l n i t e element method using

"1

,

a Figure 4

114

extrapolated from finite element results

I

frame method 4

I

O---...r.------a.--"'--~--

o

2

shear force

Eqn. (19)

..c:

Eqn. ( 17 )

L -----a.-----&--__..I_...c....L..&.--1 a

Q)

fInite element method using

P

~<

-o.S

-

t

.

- - -....

1st storey

,_yr

.....

..c:

(MN)

b

5

10

lS

bending moment (MNm)

Example 1, shear force and bending moment results. (a), shear force in walls; (b), bending moment in walls

Engng Struct. 1994, Volume 16, Number 2

Analysis of coupled shear/core walls: A. K. H. Kwan and Y. K. Cheung

Taking this shear force as the shear force acting at all horizontal sections of the element, a stepwise variation of shear force as shown in Figure 4{a) is obtained. This results in a less fluctuating and apparently more reasonable shear force distribution, but compared to that evaluated by the frame method, it is still less accurate. The second way is to determine the shear force in the element from the horizontal nodal forces acting at the top or bottom of the element, Figure 2(b), which can be obtained simply by multiply the stiffness matrix with the Flodal displacements, i.e.

u=

U2

= G{~: (u 2 +teV2

+ V4 -

u1)

+; + (w 2

Vi - V3

Wi)

»)

(14)

The shear force results so obtained are plotted in Figure 4{a) alongside the other results. It can be seen from the figure that the shear force results now coincide with the exact values. Thus this method is more accurate than the previous one. This is because whilst the strains evaluated from the strain-displacement relation are generally one order less accurate than the displacements, the nodal forces which are always in equilibrium with the external loads are generally as accurate as and sometimes even more accurate than the displacements especially if the structure is close to a statical determinate one. As the amount of computational work for the two methods are the same, this latter method, which is more accurate, is recommended.

Figure 4(b). To determine the maximum bending moment for structural design, first attribute the bending moment results as the bending moment values at midheight of the finite elements and then estimate the bending moment values at the top and bottom of the elements by linear extrapolation as shown in the figure. The bending moments at the top and bottom of the elements give the maximum bending moment values needed. As can be seen from the results plotted, the maximum bending moments so obtained by extrapolation agree closely with those evaluated by the frame method. It should be noted that .such extrapolation would not be possible if only one layer of element is used per storey. Hence this beam-type element needs to be used at a rate of at least two layers per storey. On the other hand, since in the physical structure, the variation of bending moment with height is only linear (to be exact, the variation of bending moment with height should be piecewise linear but within a storey, it is linear), a linear extrapolation within each storey should suffice and therefore the use of two layers of elements per storey should be sufficient. A more general method of axial and bending stress evaluation based on the above methodology is developed below. As discussed above, the beam-type element should be used at a rate of two layers per storey. Consider each pair of beam-type elements comprised of one element stacked above the other (see Figure 5), as a composite element. Using linear extrapolation, the axial strain at the top of the composite element is obtained as By =

+

Axial and bending stresses in the element Differentiating the vertical displacement function given

in equation (10), the axial strain in the element is obtained as By =

i(V

3

~ V1 )(1 - e) + i(V4 ~. V2)(1 + e)

cvs -

This gives .a linear variation of axial strain across a horizontal section. However, the axial strain is constant in the vertical direction. Therefore, this finite element method always produces a stepwise variation of bending moment with height. The bending moment results may be taken as the average bending moments within the elements but would not give directly the maximum bending moments in the walls which are needed for structural ·design purposes (it is always the maximum bending moment that is more critical than the average bending moment). If the bending moment results produced by the finite element method are used directly as the maximum bending moments for the structural design, there will be errors which are not on the safe side. Depending on how rapidly the bending moments vary with height,·a very fine element mesh is required to reduce such errors to within an acceptable level. Nevertheless, the maximum bending moment may be accurately determined wi.f.hout using a fine element mesh by·extrapolation as depicted below. Consider the bendingmoment results for. the lowest t\VO storeys of the structure analysed· in Example 1 which are plotted in

V6

-

:~4 + V2)(1 + e)

(16)

Likewise the·axial strain at the bottom of the composite element is derived as By

(15)

C

:~3 + V 1 )(1 - e)

=

(-vs + :~3

+(

-V6

-

+ :~4

)(l -

3V 1

-

e)

)(1 + e)

3V 2

(17)

The maximum axial force and bending moment acting on the walls can then be evaluated by integrating the axial stresses so determined.

Numerical examples Example 2: coupled nonplanar .walls

To illustrate the application of the beam-type element, the coupled nonplanar wall structure studied by Tso and Biswas 16, Figure 6{ a), is analysed. This structure can be treated as composed of four planar wall units and a row of coupling beams. The four planar wall units are modelled by two layers of beam-type elements per storey while the coupling beams are modelled by one-dimensional beam elements. For comparison, the structure is also analysed by the frame method. The results on the lateral deflections are plotted in Figure 6 (b) where the theoretical and experimental results obtained by Tso and

Engng Struct. 1994, Volume 16, Number 2

115

Analysis of coupled shear/core walls: A. K. H. Kwan and Y. K. Cheung

from the results that the frame method yields inconsistent vertical stresses for the waH units at the vertical wall joints. This is due to the presence of parasitic moments in the solid wall elements which arise from the incompatibility of vertical displacements along the vertical wall joints 11. The finite element method, on the other hand, does not have such a problem. So relatively speaking, the results of the finite element analysis are more realistic.

Example 3: partially closed core wall In this example, the core wall model tested by Tso and Biswas 17 , Figure 7(a), is analysed. The model is a 20storey symmetric core wall structure formed of two channel-shaped nonplanar wall units coupled together by lintel beams at each floor. In the finite element analysis, each channel-shaped wall unit is treated as an assembly of three planar wall units, i.e. one web and two flanges, and each planar wall unit is modelled by two layers of beam-type elements per storey similar to Example 2. For comparison, the model is also analysed by the frame method.. The torsional rotation results obtained by the two methods agree fairly closely with each other.. However, the two different methods again yield different stress results. The vertical stresses at the ground level as obtained by the two methods are plotted in Figure 7 (b) . In the figure, the results obtained by the frame method, the finite element method using one layer of beam-type elements per storey, and the finite element method using two layers of beam-type elements per storey together with linear extrapolation as per equation (17) are given as cases I, II and III, respectively. As in the previous example, the frame method yields inconsistent stresses at the vertical wall joints.. The results by the finite element method are more consistent but comparing the results of cases II and III with those of case I, it is apparent that the finite element method yields accurate results only when at least two layers of elements are used in the analysis..

one

storey

Figure 5

Treating each pair of beam-type elements as a composite

element

Biswas are also plotted for checking.. It is shown that both the finite element method and the frame method agree quite closely with the experimental results.. The results for the shear forces acting on the walls obtained by the finite element method using equation (14) and those obtained by the frame method agree almost exactly with each other but the results on the axial stresses obtained by the two methods differ significantly.. Figure 6 ( c) shows the axial stresses at the base· of the structure evaluated by the two methods.. It can be seen

/

//

48

load

~/

..c:

60

f/e

.5 ~

.

I

~

36

~"

~

f in1 te element

{,,--method

GJ

..c:

,

{"

.

120

f Incon- .......... slstent

frame method

6S

193

frame method

j,

24

/I

1,' I'

~I

12

.

1S experiment, 1so & Blswas

r' I

1SO "'----+---,..01;.-.

b

a

0'-------'------'------'---.08 .06 .04 .02 0

deflection (in) Figure 6

116

c

S2 150

flnite element method

Example 2, coupled non planar walls. (a), model tested by Tso and Biswas; (b), deflection curve; (c), axial stress at base (psi)

Engng Struct. 1994, Volume 16, Number 2

Analysis of coupled shear/core walls: A. K. H. Kwan and Y. K." Cheung 104

76

....-

1:: : : r'---

_-1-----...

_ I

frame method

)

S"

torsion at top = ZOO Ib-in - --I I _- -/- ..-.-_-----'

r..-

109

92

beam

i--z-.~5,-,----,f-l-·1-.-5-"

--z.-5-"- } -

-#-of

no. of stories

20

storey height

2.45"

beam depth

II

III

finite element method (1 layer/storey)

finite element method (2 layers/storey)

= 0.375"

b

a Figure 7

49~"""--""

47..--'!1I----....

Example 3, partially closed core wall. (a) model tested byTso and Biswas; (b), axial stress at base (psi)

Conclusions The Sisodiya and Cheung beam-type element is found to be particularly suitable for the analysis of coupled shear/ core wall structures. However, it is not without problems. Firstly, when connected with coupling beams, it yields large fluctuations of shear stresses which are not realistic. Secondly, it gives only the average bending moments within the elements but would not give directly the maximum bending moments needed for structural design. Thirdly, the finite element method is computationally less efficient than many other methods. These problems have been studied and the following remedies are proposed. To resolve the problem with shear stress evaluation, it is suggested that the shear stresses in the element should be determined from the horizontal nodal forces acting on the element instead of from the strain-displacement relation of the element. This can eliminate all the unrealistic fluctuation of shear stresses and produce shear stress results which are always in equilibrium with the external loads. To resolve the problem with bending stress evaluation, it is proposed to use the element in pairs in the form of a composite element and apply linear extrapolation to determine the maximum axial and bending stresses. Finally, in order to improve the computational efficiency of the n1ethod, the number of unknowns to be solved is reduced by neglecting the lateral strains in the walls which are generally insignificant. After thesemodifications, it is believed that'the improved beam-type element n1ethod is a better method than most others for the analysis of coupled shear/core wall structures.

Acknowledgments The work presented herein is part· of a research project aiming at the development of suitable methods for the computer aided design or tall building structures. Financial support from theU.P.G.C. Research Grants Council is gratefully acknowledged.

References

2

3 4 5

6 7 8 9

10

Rosman, R., 'Analysis of spatial concrete shear wall systems', Proc. Instn. Civ. Engrs., 1970, Suppl. 131-152 Biswas, J. K. and Tso, W. K. 'Three-dimensional analysis of shear wall buildings to lateral load', J. Struct. Div. ASCE 1974, 100, (5), 1019-1036 MacLeod, I. A. 'Analysis of shear .wall buildings by the frame method', Proc. Instn. Civ. Engrs., Part 2, 1973, 55, 593-603 MacLeod, I. A. 'Structural analysis of wall systems', Struct. Eng., 1977, 55 (11) 487-495 Zienkiewicz.. O. C., Parekh, C. 1. and Teply, B. 'Three dimensional analysis of buildings composed of floor and wall panels" Proc. Illstn. Civ. Engrs., Part 2, 1971,49, 319-332 Ha. K. H. and Desbois, M. Finite elements for tall building analysis', Comput. & Struct., 1989,33, (1) 249-255 MacLeod, I. A., 'New rectangular finite element for shear wall analysis·. J. Struct. Div. ASCE.. 1969, 95, (3) 399-409 ivfohr, G. A., · A simple rectangular membrane element including the drilling freedom', C0I11pUt. & StruC:l, 1981. 13, 483-487 Mohr, G. A., · Finite element formulation by nested interpolations: application to the drilling freedom problem". Conlput. &. Struct. 1982.. 15, 185- t 90 Allnlan, D. J., •A compatible triangular element including vertex rotations for plane elasticity analysis', Conlput. & Struc/, 1984, 19, 1-8 &

Engng Struct.1994, Volume 16, Number 2

117

Analysis of coupled shear/core walls: A. K. H. Kwan and Y. K. Cheung It 12

13 14

118

Kwan, A. K. H. 'Analysis of coupled wall/frame structures by frame method with shear deformation allowed', Proc. Instl1. Civ. Engrs, Part 2, 1991,91, 273-297 K wan, A. K. H. 'Analysis of buildings using a strain-based element with rotational d.oJ.', J. Saucr. Engng. ASCE, 1992, 118 (5), 1191-1212 Cheung, Y. K., 'Chapter 38: Tall buildings 2', Handbook o.f Structural Concrete, F. K.Kong et af (eds). Pitman, London, 1983 Sisodiya, R. G. and Cheung, Y. K. 'A higher order in-plane parallelogram element and its application to skewed girder

Engng Struct.1994, Volume 16, Number 2

bridges', in K. C. Rockey et al. (eds) De,;elopn1enls in Bridge Design and Construction, Crosby Lockwood, London, 1971, 304-317 15 Cheung, Y. K. and Swaddiwudhipong, S. ' Analysis of frame shear wall structures using finae strip elements', Proc. In:an. Civ. Engrs., Part 2, 1978,65, 517-535 16 Tso, W. K. and Biswas, 1. K. 'General analysis of non planar coupled shear walls', J. Scruct. Diu. ASC£, 1973, 99, (ST3), 365-380 17 Tso, W. K. and Biswas, J. K., ~ Analysis of core wall structures subjected to applied torque', Building Sci., 1973, 8, 251-257

6 Wall-frame interaction In the design of composite shear (core) wall/frame building structures, it is-often assumed that all horizontal loads

are

carried

by

the

shear

(core)

walls

only;

the

contribution of the frame part of the building is neglected. Effect of the assumption: the load carried by the walls and hence

the

resulting

deflection

will be overestimated. Although. the

contribution of

the

frame

portion

is

neglected, it may still be necessary to check and allow for the effects of the lateral deflection on the frame. If

the frame

necessary

horizontal

portion is relatively stiff,

to determine forces

the

between

elements.

38

distribution the

wall

and

it

of

is the

frame

2.

H

€.JcT~ .~

~,""JId

~"j

D~~~~

'rt.t.A~(~~~ '«Alrt'o ~ ~:;

et

2.6,

~

.,."..1-.....---......-.......... -----~--~ .-t----w-----tot---+t

{1{~?/1 e,i\~ ~ ~IS~~

~':J

'J

~ A-ftYLA ~

~

3 Analysis of Yall/Frame Interaction

If similar

~llowed

to deform independently under the action of

horizontal

loads,

the

adopt different configurations,

cantilever

wall

and

as shown in the following

figure.

Interaction forces I

t

--....

--... --.....

---

-,.....

w

--.....

---... --... -.... --..... -.-

--....

(a)

Figure 8-1

(b)

frame

(c)

Intera:ction between frame and shear wall constrained to act together by floor system

Consequently, when the two components are constrained to deflected together by the floor slabs, they will have to be

'pulled together' base.

at the top,

A redistribution of horizontal

place throughout the height, the

and 'pushed apart'

top

and

bottom.

load must

near the then take

wi th heavy interactions near

The

frame

will

thus

resist

proportionately more load at the top, whilst the wall will tend to' pick up a larger proportion in the lower regions of the bui lding. An approximate continuum based method of analysis may

again be derived to yield simple formulae and design curves for both preliminary and final design calculations [1].

Ref.

ell: Heidebrecht A C and

St~ford

Smith B, "Approximate

analysis of tall wall-frame structures", J. Struct Div, ASCE, Vol 99, No.ST2, Feb 1973, pp199-221.

The

composite

wall-frame

structure

is

assumed

to

consist of a combination of a vertical flexural beam and a

shear cantilever. The

equations

governing

the

behaviour

of

the

two

components are:

4

E I

shear wall:

(in flexure)

d Yb dx

4

= wb(x)

(1)

w (x)

(2)

2

d y

frame (in shear)

- GA

dx 2

s

=

s

in which:

. the subscripts tb' and's' refer to the wall (in bending) and frame (in shear) respectively, w is the distributed lateral load intensity, and El and GAare the equivalent flexural and shear p~gidities

of the wall and frame respectively.

~

E1 ~ d,Xj'1- -

---?>

-?"--

Wb

-7 ..-;L

d.-f1

-r

cl~

=-7

tls

....,.

----4X

-7

, -II

,

-......

x

I I

/~/

I

-J 1

/ " .,

tJ\

= .... s ---...... -wb

7 The two components are assumed linked by floor slabs so that they have identical horizontal deflections throughout the height.

It

the discrete set

is then assumed that

links may be replaced by an equivalent .pin-ended connecting medium.

so

only

that

hor~zontal

Consequently,

transmitted between the two.

induced a set of distributed horizontal

at.

the

connecting

medium

to

of

continuous forces

are

there wi 11 be

interaction forces

maintain

horizontal

compatibility of the wall and frame. Let

y =y

b

= y

(horizontal

s

w (x) + w (x) = w(x)

Since

b

co~patibility)

(horizontal equilibrium)

s

therefore: -

- a.

2

=

w

(3)

E I

in which: W

= w

(

b

+

distribu~ed

)

W

s

-

load. and a

ratio, g1 venby Equation

closed-form constants of

(3)

may

solution

is the intensity of the applied

(X

2

int~gration

is the relative stiffness

= GA/EI.

readily

for

2

any

be

integrated

applied

loading.

to

give

The

a

four

which arise in the solution may be

determined from the known boundary condl t ions at the base

and the top of the structure.

8 In the particular case of a structure which is free at the top and rigidly built in at the base) and subjected to a

uniformly dlstriputed

load

of

intensi ty

w.

complete

the

solution becomes, in non-dimensional form:

y =

~

wH

8

EI

-

= -

1

4

~ ex

sinh

(X

wH

_tl __ s_i_nh __ a_+_l_

{

4

(cosh a z - 1)

cosh ex.

.

(X

Z

+ ex 2 [ z - -1- z 2 2

4

(4)

- - F (z,a.)

8

EI

The solution is dependenc~

01

1

in this form to emphasize the

express~d

the

horizontal

deflection

on

dimensional parameters only, t.he height ratio z the relative stiffness « Since

lateral

deflection

at

cantilever of flexural

non-

(= x/H) and'

a H).

(=

1 8

the term

two

4

w H /EI

the

top

represents the maximum

of

rigidity EI,

a

uniformly

the function F

loaded 1

is a

measure of the stiffening influence of the frame component. If F

1

1s large, of the order of unity, the stiffness of the

building Is derived almost entirely from the wall component,

whereas if F

1

is small, the frame component provides a major .

part of the lateral stiffness of the building.

Once

the

deflect ion

is

known.

the

other

force

components follow from the force-displacement relationships. The

val ues of the bending moment

flexural

cantIlever,

and shear force

on the

expressed in design form in terms of

the maximum applied moment and shear force at the base of the structure, then become:

(5)

(6)

.

in which the moment and shear functions are:

F

F

2

3

=

2 -2

{a sinh

cosh

(X

=

1 ~

ex. + 1

{

cosh a z - a sinh a z - 1 }

(7)

(X

a. + 1 sinh az } .acosh az - asinh cosh

(8)

(X

These functions indicate graphically the

influence of

the f'rame component on the force distribution in the wall element.

The corresponding bending moment and shear force on the

frame will be given by: M = _1_ w H2 ( 1 - z ) 2 - M s 2 b

(9)

S

(lG)

s

=wH(l-z)-S

b

Cl

6 4 3 2 1.5 1 0.50 1.0 ,....--..........-."....-....,.......-..-......--~-".-"..-.,...-...........-...-..--

0.8 t\I

o

0.6

+J

co L-

......,

..c. en

0.4

Q)

I

0.2

o Figure 8-2

0.. 2

0.8 Deflection function, F, 0.4

0.6

1.0

Variation of deflection function

1.0 , . - - - - . . . . . - - . . . , , . . . - - - - , - - - - , . - - . . . - - - - . . - - - r - _ -

-

_

0.8 I\,J

....

o 0.6

~

ro

'-

+-'

..c

.~ 0.4

:r:

0.2

Q

0.2

0.4

0.6

0.8

1.0

Moment function,F 2

Figure 8-3

Variation of wall bending moment function

(2..

0.8 N

o 0.6 +J

co

~ ......, ..r:: .Cn

0.4

Q)

I

0.2

-0.2

o

0.2

0.4

0.6

0.8

1.0

Shear force function, F3

Figure 8-4

Variation of wall shear force function

(~ Evaluatio~

of Equivalent Shearing Rigidity of Frame

Due to the racking action which

oc~urs

over each storey

height, plane frames in tall buildings behave essentially as shear components, since the overall mode of deformation 1s more akin to a shearing than a frame

component may then be replaced for

'shear

equivalent rigidity,

the

bending act Ion.

GA,

cantilever')

with

beam deflects

only

~LD8.~~fJt!WThe overall

in

analysis by an effective

such that when subjected to

substitute

The tall shear

lateral forces,

shear.

~~~

shear stiffness will depend

on the individual member stiffness, the frame configuration, and the rigidity of the joints. Assumption: points of contraflexure occurs at the mid-height positions of all

columns and at

the

mid-span

positions of all beams. The forces on a typical interior frame segJUent

J

bounded

by the assumed points of contraflexure and subjected to a

horizontal shear force QJ are shown in the following figure, in which the appropriatoe boundary condi t ions are indicated.

(4 Q .....

Q

h -rrrT-

I,·J

'I

1'1

II

II

I

\~~t= ~ .~ ~ ~~t" T\J~j

f)~

tC.P+t~'~

bj

~~

of

The horizontal displacement, 6, equivalent shear cantilever,

the segment of the

of storey he ighth,

subjected

to the same shear force Q 1s given by:

=(

~

(11)

Q / GA ) h

The horizontal deflection of the frame segment may be

and on equating the

calculated,

two

displacements,

the

effective shearing rigidity, GA, established. If the relative size of the

joint

1s small and the

rigid arms are 'omitted, GA is given by:

1

(12)

G A

where I d

'

I

h

d

h

= second moments of area of beams and column;

= bay width; and = storey height.

The finite size of the joints may 'be allowed for by

incorporating

short

deformation of the (Refer to

II

~tiff

memb~rs

arms

at

the

joints.

Shear

and joints may also be included.

Hand,book of Structural Concrete" edi ted by F. K.

Kong etal) Chapter 37 for the formulae).

I~ Corresponding expressions for an exterior column may readily be deduced, by the simple expedience of omitting the contribution 01 one of the beams and its associated stiff

arm at the Joints.

I

\

L 7 Framed-tube structures or ~

A

:framed-tube

is

essentially

a

per:forated

boxA

comprising four orthogonal frame panels of closely spaced

columns connected by spandrel beams around the perimeter at IAS~

each floor level. The outer tube is ~ designed to resist al
r-

Figure3-8

-,-

Planform of tube-in-tube structure

39

In a preliminary approximate analysis. assumed that

th~

it is sometimes

side frames parallel to the wind carryall

lateral loads, so that a plane frame analysis can be used.

However,

the

frames

normal

t,o

wind

the

directions

are

constrained by the floors to deform as flanges in the same

mode as the side frames, and can playa significant part in re~isti~g

wind loads (or other lateral loads). In this case,

these normal :frames are subjected mainly to axial forces and the

side

primary

spandrel

frames

Q,et"~o\t'\

axt ion

beams

are

subjected

1s complicated

allowing

stresses in the corner

a

to by

shearing the

shear lag

column~t

inner columns of" the normal

actions.

The

flexi bil i ty of

the

which

the

increase

and reduces those in the

rram::J The

major interactions

between the 'two types of frame are the vertical shear f'orces at the corners.!

True tubular cantilever stress

Actual stress due to shear lag

~--;i>] Actu aI stress True cantilever stress

\

\

\ -1

t Wind load

3 By

recognizing the

panels, behaviour

a

more

may

dominant

accurate again

be

modes

of

assessment; of achieved

using

action of the a

the

structural plane

frame

analysis. Methods of Analysis: (A) Bending action only:

The side and norma.! frames may be considered to lie in the same plane,

and are connected in series by ficti tious

linking members whose stiffnesses are appropriately chosen to allow only verticalforqes

frames.

This

may

be

to

be transmitted between the

performed

most

conveniently

incorporating a hinge release in the fictitious members.

~ ~ to~~\S")\

111/,/1' g~~,

~

N

ll-S"

by

(b) Bending and twisting:

/

~ ~~ oX~-f·

If a symmetric framed-tube structure is subjected to bending

and

separately.

twisting

forces,

the

two

may

In the case of pure torsion.

be

considered

the stiff floors

ensure that the cross-sect ion shape is maintained at each level. so that the applied torque is resisted primarily by the

shearing

periphery.

resistance

The main

of

each

interactive

plane forces

frame

around' the

between the frame

panels are again the vertical shear forces at the corners. Since the rotation of each frame relative to the centre of the building is the same. 'the shear deformation in the plane of each frame is equal to the product 01 the rotation and the distance from the is not

centre).

necessari ly

centre

(note

~troportion~l·

Knowing the shear

to

that

the shear force

the distance from the

stiffness

of each frame.

and

using overall condit!ons of compatibility and equilibrium, the

horizontal

forces

on

each

frame

and

the

vertical

interaction forces at the 90rners can be established.

€lA

A~~\MIt*iGM~.

CD

~~

1~

Cb

~ ~s

~ ~

c5l

r

~~

= A/H ~

oX ~cl-t"T.

~ers

~~~~4~~ ~y\h-

~

,/J»'\r-~

~. uw+;cM ~~ ~ 3~

t~t~~~~ ~s

,

~ ~ ~

~ . .h.. d

-,f-----;f-

6~

~J.

~

\,3

= ~

----.

6l

r."

( 2. E

a) 12- E

c6" (X

(Xh. . .!:L do

-.r~

T

D.J

h1

\L.e-I.~

\

~

G~

.........

~ 6l

ci

[

t+

!h.·~1 \, ~~

7

"\

Il

~~Uo"''''''

OtJ"'·

~

........ ...,............. ra

~

.I::J' J.

_r1.......-...............

,.......xas

----_ _.. ...

Journal of the

Proceedings of tlle American Society of Civil Engineers •~

gF7JJliBLL.....-........,...........

lj

ratelr the struclural beh:l\'lor of the systenl itl order to produce an e{[lcicnl d~sl~n.

S1"\RUCrUI~AL DIVISION --....

t)T

AUl!usl, 1971

2098

nn-:"'f..4: " - '.•- , _ - .-

C1":'U' (.. ,.-

{c {,'"

~.o;"''''&

tn this paper. a slnlpUCled nlethod 1s presented lor_the analysts of {ranled .. tube struc.turessubJected to bending due to lateralloadlnc. By recognizing the d(\:~ .. lnant nlo.:\e oCbeha\'lor or the structure, it \s possible to redu'ce the anal}'" sl$ to that of an equh·alent plane franle, \vllh a consequent large reduction In tha 4't1\Ount oCconlputalion required in a convent\onallhrec-dlmenslonnl ana \y. sis. The accurac):of the method Is tested by conlpar lng results wlth those from A.r,-n.~

t

A

'j'Oil

CJ

FRAMED-TUBE STRUCTURES FOR HIGH-RISE BUILDINGS

CJ

~

Cl

c.~

C3

8

By Alexander Coull,IM. ASeE and Nutan Kumar Subcdl,2 A.M. ASCE .~ • t ....

....

INTRODUCTION

.

Recent developments In high-rise bulldings have produced a number of new structural concepts whlch are erflclent and economlc in the use of materials. One such concept Is the framed-tube system, a natural evolution or the rigidly jointed frame, which has proved to'beecono1nic over·a wide range or building heights (1,2). The framed-tube structure conslsts of rour orthogonalrlg1d frame panels formlnga tube 1n plan, as indicated In Fig. 1. .:r!l~_!~3:..m.~s a~e ,formed by the perlm.~,~~.:!:. C;.91umns whlc~ are cQnnec~ed by spandrel beams at each story. level. 111.J!l_~ny. s.tructures, the ~xterlor tube Is designed to resist the ~nq.r.~ wind loading. Both steel and concrete have been used In the construcUon or such

Sfructures. .

.

The system has the advantage that it Is conlpatlble \vlth the traditlbnal architectural arrangelnents {or \vlndo\vs,and It cah be used for both conlnlerclal . and residential requlrelnents. WhUe the structure has a tube-llke appearance. the behavior Is much more complex than ~hat or a plain tube, and the sUffness is considerably reduced. In addition to the cantUevered tube action, whlch tends to produce tensile and compressive ror~ces in the colunlns on opposlte sides of the structure (AB and DC in Fig. 1), the frames parallel to the lateral load (AD and Be) undergo the usual shearlng action associated with an independent rigid frame. This basic a.ction is complicated by the fact that the flexibllityof the spandrel beams pro- . duces a[§hear lagl which has the effect of Incr~aslng the stresses In the corner columns, and reducing those In the Inner columns. This latter effect will pro- . duce \varplng or the rloor slabs, and thererore delornlatlons or inter lor parli- . · Hons and seconda.rystr)Jctures. Consequently, II Is essential to predict accuNotc.-Discu~slonopen unW Januar)' 1. 1972. To extend Ule closing date one month, a written request must bo CUed with the Executive Director, ASeE. 'I111s paper Is p3rt or the copyrIghted JOlU"nal oC the Structural Division, Proceedings or the American 50cletyoC Civil Enrjneera, Vol. 97. ST8. August, 1971. }Ylan\\scrlpt was subn\1tlcd fol' reYltnV for possible publication on Scptclnber 22, 1970. lProf. of Slructural EngTg., Unlv. of Strathclyde. Glasgow,Scotland. lResearch Student.Unh'. of Sn:athclyde, Glasgow, ScoU:md. ?n~'7

l.,:'( I...

,. I

1

71 In.

.

FIG. I.-PLAN VIEW OF FRAl\lED-TUD'E l\10DEL

Crom acomplete three-dlmenslonalsolutlon achieved with a commercially available standard computer program:

a modelln\'estlgatlonand

METHOD OF ANALYSIS In a framed-tube structure of the form sho\\'1l in Fig. 1, the lateral load is resls.ted malnl)' by the {ollo\vlng actlons: (1) Th~x:!g.~?..!:t.::J~lJ1~~.d. r!,,~I1)e .. a~.tlo.n. .!?U!!! she!!.:!.~~~~!~.~g .. p~tleJs parallel Lo. the ~lrectio~ or. the load(Ap a,n4 Be). and (2) the axial d.eformatlons of the rraf1le_{1~!l.~l~_~9r_mal to lhe diI;ccllon of j~e load .
beams.

Secondary out-of -plane actions wlll occur, but these will tend to be reslr lcled by lhe high In-plane sUrfness of the floor slabs, and can generally be assunled insignificant In relation to the primary acllons. By re:cognlzlng the tv/a dominant modes of action in the.orthogonalpancls, thre~·dlmenslonal frame may be reduced to the equlvalent plane !ranle shown In Fig. 2. For simplicity, It has been assumed as is generally the case. lhat the slructure Is symmetrical about both cenler Hnes, 50 that only onc-

the

ST Ii

FRAMED-TUDE STRUCTURES

2099

quarter need be considered In the analy·sls .. Appropriate joint condltltuis nre Indicated to give the required constraints at tho axes of syJnnlcl1'y. As the flOOr slabs are very stU! In their o"/n plane, it can be assunled f or the purposes or analysis th(~t tho lateral forces acUng on the norlnal face of the bUilding can be applled In the plane ot the parallel Caces, as lndicnted In Fig; 2. In Fig. 2 orthogonal pan\~ls AD and DC are shown joined at each floor level by !lCllllou~ ( \tt.o. -h' (A,'\ J'\u.. t' fN". !.c.r'1 )

alon~ junctloil D. l.e., along Hnes l\1~'l and LL In Fig. 2). The transfer or vertical she:\f Illay be achlc\'ed shuply by luaklng the approprlale clcnlenls in the ~tlffne~Snl3.trl~ for the ,"ertlc3.1 shear transfer 1l1tHllber a laree quantlly com" p3.l'ed to the elements In the ~tUCI\ess mall'lccs Cor the real members. The stiffness nlatrL~ lor a typical plane franle nlenlbcl' oriented along a horlzontal'coordlnate a.xis has the ,\"ell-kno\vn fOl'1l1 (e,g" s~e Ref. 3)

4y,.4....

•

M

I

)o/M

£A

/

~.

I, I

Panel DC

.:-M.

0

0

0

12 El --L3

IT

0

.[;2

EA

0

L

6E1

12£ 1

I

--rr

v

I

0

0

T

Fictitious memb.eu L

~

V

15th slol')'

4 £1

K ='

L

I

L ~

I

=P:

I

I

6El

~g

BEl

L

(1)

6 El

IT

L:r

Symmetric

I , ..... , ..

0

12 EI

I En 14th'

'1

51' 8

Augusl, 1971

21(\0

'4£1

L 13;th

using standard notations, in \\'hich the colunlns refer, respectively, to axial displacements, nornlal (vertical) displacements, and notations, or the corresponding forces. For slmpliclty, and to compare directly wlth the vertical shear transfer stiffness matrlx K l ' only the slnlple slUfness matriX K, referring to a horizontal member, is shown. · Uslng the same notation, the sUrfness matrix for a horizontal vertical shear transfer member can be written as

K

~ P-O.532Ib

~.

I I

M

J'i:1"

I

I

L

FIG. 2.-QUARTRR ~LEVATION OF EQUTVALENT PLANE FRA1\'tl:: STnUCTURE

horizontal members indicated by heavy lines. These beams lnay be ter'Jl1ed lhe vertical shear transfer members, whose sole purpose Is to transfer vertlcal shear forces between the two frames. By this. mechanism, only vertical lnter· acllon Corces are induced into the normal panel DC. The propcl'tle~ of the fletltlous members must be such thal the two panels rentAln conlpatlblC' VC'l·tlrnlh·

z:

0

'"~,, \:(~.(

0'

0

0

0

0

0

Q

0

0

10

0

0

0

0

0'

0

0

0

0

0

0

o

.-Q

0

0

Q

0

0

0

0

0

0

0

f ,:

I~,

(I' (, "

.. :. ,.

- Q 0 t

••••••••••

"

•••

"

•••••

•

•

I

,

(2)

In which Q = somerelatlvely large number" The analysis may then be carrled out on the modi fled plane frame by a conventional stUfness method, using a standard program, In the prest:nt work, a standard plane Craine pro~ram was modified to allow variations of lhe value of parameter Q in matr lx K 1. The modified progran1 picks out the largest element In Ule sllflncss matrices {or c.lthcr the beams or columns and mulUplles It by an arbitrary large number to gl~e Q. Various nunlbers were used, rt:.nging from 102 to 10 8 ~ all gave similar results, and the value Unally chosen 'was 10·. The results qUCll(;d in succec:dlng sectlons were achieved by usi'ng a value of Q = 10~ limes the largc6t clement In the other sliffneSR malrlces, The rnalr lx, K I ' ensur es thal only vertical Rhp.n r forr.('~ ~ r f\ trn n~f,..r T(\d

:f'HAMED.. 1.'UDE STRUCTURES

ST H

2101

across the JuncUons of the orthogonal frames, and thalverUcal conlpatlbllHy is achieved.

Au~ust.

2102

1971

£,)pctl'lcal restst:\nce slraln gages, and deflections ~ep.\ l"ate frallH~,

Model Details, -The 15-slory nlodel was constructed of Pcrspex, the columns being cut from a 3/1 a-ln. thick sheet and the Cloor slabs Crom a liS-in, thick sheet. Th~ nlodel had eight colulnns along one edge and five along the other, each column being 1/2 In. wide. \vith plan dimensions as sho,lm In Fig. I, The corner columns were glued together to form an angle member. The story height was 2-1/8 In., I.e., 2-ln, clear height betweenfloor slabs. The petlnleter

uy dial gages mounted on a

.

AGnEE~IENT DET\VEEN TllEOlt Y

COll1PARISON \'lI'l'H ItESUL1'S FR01\'fJvIODEL TESTS

8T 8

ANl)

EXPBnl~'I ENT

Fig. 3 s.ho,,"s lhe theoretlci\l and experinH?nlal stress dist'ributions at the third floer le,'el. Slnl11ar distributions v.:ere oblained at other levels. For simplicity, the lheoreltcal values' have been presented as contlnuous curves be .. cause a prototype (ralned-tube structure will contain a large n\.llnber or closely

t::

\:>am

~

cUII'{ \'~;~S

200

8.

::: ~

\1

"'!

§ i]

~

8.8. "0

:::

~'~l

o

+

I

I

.

I ·

I I

t::

~

en~i

.::

r--·--------l o

,_. ._._._._.+ ~ I +~

~

100 .5.. .c: ~ ~ ~.-

I

I

f

~.

::: ~

+

.LJ J .

.'

.

II

I I

I

(b) PLAN VIEW

I

--J

Cross-section o Experimental - - Theoretical

- :-...

-

FIG. 3.-11iEORETICAL AND MEASURED STRESS DISTRIBUTIONS AT THIRD FLOOH LEVEL OF l\10DEL STRUCTURE

colunlns were connected by floor slabs at each Cloor level. the colunlns being glued Into slots nlachlned in the slabs. (As the nlodcl was to be used for subsequent tests on a hull-core type oCstructural systell1, central holes were cut In the floor slabs to receive a box core ala later stage). At the base, th~ columns \\'erc glued Inlo slots passing through the entire depth of a l-ln.th~ck Perspcx base plate. The nlodel was cantilevered horizontally byclamplng the base plate to a test {ranle, using rectangular hollow steel sections passing as ncar Lo the structure as possible In an alt~lllpt to achieve a rigid foundation condltlon. Lateral loads were applied to lhe nlodel by lhe slnlplc expedient or hanging dead weights at each slory level. Strains were nlcasured 'at several levels b}'

-

C<,I',,-,n I

2

~rtl

H

,~

H

3

~

'-1loa

1-

I-.-.

..

3' 5 6 -;er ,~ ~r ~,r ~ r H H H H H

(e) (QUIVALE.'~T PLANE

II

~20 11+20 1\../.2010II~ fl

1~1 · I

I IL

•

FRAMe

rJri. 4.-t'HAl\1ED-TUUf·; :::;THlJc.;1'Un~~ FOB EXA~tPL~: PllOIJLEM

•••• p .......... ""

L)

r HAf\'H~U- 'l'U 13,(:; :)'fHUl:'l'Ull£;S

1 u

2103

I"

.)'1' ti

purpose, the G-story structure 6ho" ;0 in Fl~. 4 wns chosen, lhls being the I

spaced colulnns, and the dlslribution or-axial stresses will apprOx.inlate lo a ·continuous curve. In general, good. agreement Is oblained between theoretical and cxperhllcn(al results, p:\rUcularly bearing tn Inhad the difficulties or fab.. rlcation involved In a nlodel oC this nature, \vhlch makes it very dlCllcult 10 achieve a unlCornl structure. The stresses In the columns at the outer edges of the normal panels tend, generally, to be lo\ver than the theoretical values. This may be due to the influence of the increased sUrCness of the Cloor slab In the corner, where It Is restrained by the orthogonal edge colulnns in lhe side panels. . In order to evaluate the effective stiffness of the Cloor slabs Cor use In the theoretical calculations, separate tests were performed on a representative sectlon ot floor slab and columns. The curves Illustrate the severe shear lag effects whlch can exlst in structures of this forln. In the present case, the stresses at the center of the panels whlch lie normal to the directlon of the applied load are only of the order ot 26 % of the stresses at the edges. COMPARISON BETWEEN SIMPLE METHOD AND COl\1PLETE THREE-DIMENSIONAL ANALYSIS The proposed method \vas also tested against the accuracy achieved by a analysis using a standard space-frame program. For this

three-dlmension~l

TABLE

Spacclramc analysts deflection, in inches

(1)

(2)

1 2 3 4 6 6

0.363 0.854 1.272 1.589 1.803 1.918

SlmpllIled me thod deflection, in inches

(3) L'

0.365 0.860 1.281 1.599

1.814 1.929

TABLE 2.-EXJUfPLE PRODLE~t-AX1AL FOnCES AND FmsT STORY COLUhiNS SPACE FRAl\lE ANALYSJS Axial force, in tons

(1)

(2)

1

0 +0.5G9 -5.348

2 3 3 .(

5

G

-0.0'11 +0.001 -0.000

CONCLUSIONS

1.-EXA~fPLE PROB.LE~{-DEJ""LECTIONS

Story

Colwnn

Inl'(tcst confl~:uratlont \\rlthln the gLven series of constraints, which could be t'oh'ed by the a,·nllable progralll. In each case. because or syn1mclr}'. only one .. qU:\l't('r ot the; slructure need be conslcierecl. The sectlonal properties oC the colulnns and spn.ndl'el beanlS were ns rollow~ lor colulllns-S In. x Sin. )( 58 Ib; Ix A. = 227.3 in"' ; 1\'}' = 74,9 lno(; aren :: 17.0Gsq In.jandJ = 3.3ilno(.Forbeallls-5in. x 121n.·x 31,Slb;lxx = 215.8 In"; I yy = 9.5 1no(; area = 9.26 sq l112 j and J = 0.92 In
h·lomcnlR I in ton-lJ\ches Lower end Upper end (:1 ) (4) -115.3 -233.5 -205.0 -0.10 -0.088 -0.014 -0.006

-70.0 -145.0 -8~.5

-0.10 -0.142 -0.007 -0.005

BJ-~NDINO

l\IOl\-lENTS IN

A slmpl1!led method has been presented lor Ute analysis of framed-tube structures subje'cted to lateral loads. The method can be used with standard plane frame programs. and should be partlcularly useful where only a small computer Is available. As the sUllness matrlces are much smaller than those required {or a space frame analysis. the technique enables larger structures to be treated on a given computer.

The method has been lested under wldely dlllerent conditions-against model tests, using a structure with relatively stlC! spandre 1 beams, and against an exact solutlon, using a structure wlth weak spandrel beams. In each case, the degree or accurac)' was very satisfactory. By using conditlons of equUlbrlum and compatlbllLty at different levels on the stru~ture, the method could be developed to treat more complex slructures such as the hull-core or tube-In-tube systems for high-rise buildings,

SI~tPLIFIED ~IETHOD

Axial force. in tons

ACKNOWLEDGMENT

htoments, in ton-Inches

The work described In this report was aided by a Grant [rom the Sclence Research Counc 11. .

Lower cnd Upper end (5 )

(6)

(7)

0 +0.502 -2.69 -2.69 -0.0'14 .. 0.00·1 -0.000

-116.2 -235.1 -206.7 -0.08 -0.074 -0.005 -0.003

-70.6 -146.3 -89.0 -0.15 -0.131 -0.001 -0,00:1 _

APPENDIX I.-REFJ:;HENCES

I.Kh~n. F.

R., "Current Trend_ in Cunerelc IliSh·Rhe nuildincs,"

/'ft)Crt'cJill/;,I. S)'1I1PIlSiulll

on

lJ £

'"

,It'HAi\1EO- TU BE S'fnUC'l'UnJ::s

2105

rail DUlllJinJ;~, Univer:aily of SuullI;IOl·plon. P~r~amon Pn:s~. Lond\ln. t=:ntland. Coull und Star. lonJ Smilh. cds.• 1967, Pil. 571 590. •. 1. Khnn. F. R•• "Colun'n-Frcc Rox.r)·Il~ Franlin~ \Vilh und \\'ilhuUI Corc," Prclimin:lr)' f'uulica. lion. Eighth Con1H~u of the Internatiunal I\uociutiun fur Hrid~e .lnd Suutlurul engineering, New Ynrk. Sept.• 1968, pp. 261··273. 3. \\'(aver• .\V •• CtJII,pllltr PrugrQlI1S fur S""CllIr&lI.·'lIal.rS;J. VUH ~oslrand COn1IUln)', In..:,. Prince. ton. N.J .. 1967.

APPENDIX II.-NOTATION

The !ollowlng synlbols are used In this paper: 'A E

= cross-sectional area of member; modulus of elasticity;

I = moment of inertia of member; torslonal constant lor cross section; J K = sUfeness matrlxoC typical member; .;. K 1 = stlflness matrix tor horizontal vertl~al shear transfer member;

=

L = length or member;

and

Q = elenlcnt In matrix K l'

SIMPLE METHOD FOR ApPROXIMATE ANALYSIS OF FRAMED TUBE STRUCTURES By A. K. H. Kwan ' ABSTRACT: Framed tube structures are particularly suitable for tall buildings. They act primarily like cantilevered box beams and since they generally have much larger lateral dimensions than the internal shear wall cores, they arc more effective in resisting the overturning moments of the lateral loads. However. due to nexural and shear flexibilities of the frame members, the basic beam bending actions of the framed tubes are complicated by the occurrence of shear lag. which could significantly affect the stress distributions in the frame panels and reduce the lateral stiffnesses of the structures. In this paper, a simple hand-calculation method is proposed for approximate analysis of framed tube structures with the shear lag effects taken into account. This method is suitable for quick evaluations during the preliminary design stage and can provide a better understanding of the effects of various parameters on the overall structural behavior. Numerical examples are given to demonstrate the case of application and accuracy of the proposed method.

INTRODUCTION

Framed Tube System

The framed tube system is widely accepted as an economic solution for tall building structures over a wide range of building heights (Khan 1967. 1985; Wonget al. 1981; Spires and Arora 1990). In its basic form, the system consists of closely spaced perimeter columns tied at each floor level by deep spandrel beams to form a tubular structure, Fig. 1. It is compatible with the traditional architectural arrangements for windows and has the advantage that as the perimeter configuration is used to form the structure, the whole width of the building is utilized to resist the overturning moment due to lateral load. Under lateral load, a framed tube acts primarily like a cantilevered box beam. The overturning moment of the lateral load is resisted by axial stresses in the columns of the four frame panels, whereas the shear from the lateral load is resisted by in-plane bending of the beams and columns of the two side frames. If the frame members are very rigid, then the axial stresses in the columns due to the overturning moment may be determined by the normal "plane sections remain plane" assumption, as shown in Fig. 2. However, because of requirements for window provisions, there are practical limits to the sizes and hence rigidities of the frame members. As a result of the flexural and shear flexibilities of the frame members, the basic beam bending action of the framed tube is complicated by the Hshear lag" phenomenon which has the effects of increasing the axial stresses in the corner columns and decreasing those in the inner columns as illustrated in Fig. 2, and reducing the lateral stiffness of the structure. Shear lag could also produce warping of the floor slabs and consequently deformations of the secondary structures. ILect. Dept. of Civ. and Struct. Engrg., Univ. of Hong Kong, Pokfulum Road. Hong Kong. Note. Discussion open until September 1, 1994. To" extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on November 30, 1992. This paper is part of the Journal 01 Structural Engineering, Vol. 120, No.4, April, 1994. COASCE, ISSN 0733-9445/94/0004-1221/$2.00 + $.25 per page. Paper No. 5196. 1221

actual ·stress due axial stress if to shear lag beams are rigid

web

panel web panel

(1)

flange panel

::J

"0 U) U)

bO

...., ....."' (1) ~

U)

.....

~

td

4)

nS.c ::J en

....o

0

cd .....

~ .r-4 U) U)

bO

td

QJ ..... ~

..... U)

flange panel

~

"' 4)

........c:: cd

en

X

0

.r-4

lateral load

nS C

FIG. 2.

FIG. 1.

Typical Framed Tube Structure

Existing Analysis Methods Framed tubes, which are essentially three.. dimensional space.. frame structures, can be analyzed by most existing space-frame analysis programs. I-Iowever, as the out-of-plane deformations of the frame panels are insignificant and the interactions between the web and flange panels consist rnainly of vertical shearforces, a simpler and in fact better alternative is to analyze the three dimensional system as an equivalent plane frame by neglecting the out-of-plane actions and using various forms of fictitious frame elements to effect the vertical shear transfer at the panel junctions (Coull and Subedi 1971; Khan and Amin 1973). This could substantially reduce the amounts of data and computation r-equired. While computer programs can be used to numerically analyze the framed tube structures, they cannot substitute theoretical analysis methods which may offer better understanding of the structural system. Furthermore, in view of the wide applications of such system, there is an obvious need for a simplified analysis method that can be used during the preliminary design stage to give an initial assessment of the.structural behavior and in the final design stage for manual checking of the computer analysis results. A number of simplified analysis methods have been developed. Khan and Amin (1973) suggested that for very preliminary design purposes, the shear lag effects may be approximately allowed for by treating the framed tube structure as a pair of equivalent channels each with an effective flange 1222

Distribution of Axial Stresses In Framed Tube Structure

width of not more than half the width of the web panel or more than 10% of the building height. Chan et al. (1974) proposed to' evaluate the shear lag effects incnntilevered box structures with solid shear walls as \veb panels and rigidly jointed beam-column frames as flange panels by assuming the distribution of axial displacements across the width of the flange panels to be of either parabolic or hyperbolic cosine shape. Although the structures studied by them were not really franled tube structures, their nl~thodology of ullo\ving for sh~ur lag in the flange panels should also be applicable to framed tube structures. Coull and Bose (1975, 1976) and Coull anG Ahmed (1978) developed un orthotropic membrane analogy of transforming the framework panels into equivalent orthotropic membranes each with elastic properties so chosen to represent the axiaJ and shear behavior of the actual framework. They analyzed the equivalent membrane tubes by assuming the bending stress distributions to be cubic and parabolic in the web and flange panels respectively and using energy formulation to derive the governing differential equations. Khan and Stafford Smith (1976) have also developed an orthotropic membrane analogy for simplified analysis of framework panels by using finite element analysis to determine the equivalent elastic properties of the membranes. Although their membrane analogy was applied only to plane frames, it is actually also applicable to framed tube structures. . Subsequently, Ha et al. (1978) further developed the orthotropic membrane analogy to include the shear deformations of the framemen1bers and the deformations of the beam-column joints in the derivation of the equivalent elastic properties. Their membrane analogy is more refined than the others' and should thus be more accurate. 1223

Present Study The methodology of modeling the framework panels as equivalent orthotropic membranes so that the framed tubes can be analyzed as continuous structures is followed in the present study. There are two factors that can affect the accuracy of solutions based on this membrane analogy: (1) The equivalent elastic properties for the membranes; and (2) the method of analyzing the equivalent membrane structures. The primary concern in this investigation is to develop an analysis method for the membrane tubes that is simple to use and yet reasonably accurate. The analysis. method proposed herein has the characteristic that unlike previous methods, independent distributions of axial displacements are used for the web and flange panels. Thus the shear lag in each panel is individually allowed for; this is more reasonable because the shear lag in one panel is obviously more related to the properties of that particular panel rather than those of other panels and is therefore not necessarily so much dependent on the shear lag in the other panels. It will be seen that this can, in fact, also lead to simpler formulas for the evaluation of the shear lag effects.

z

I I I I

, I

I I

flange

I

E

I

r

PROPOSED METHOD OF ANALYSIS

Structural Modeling The framed tube structure shown in Fig. 1 can be considered to be composed of: (1) Two web panels parallel to the direction of the lateralload; (2) two flange panels normal to the direction of the lateral load; and (3) four discrete columns at the corners. These structural components are interconnected to each other along the panel joints and connected to the floor slabs at each floor level. The high in-plane stiffness of the floor slabs will restrict any tendency for the panels to deform out-of-plane and it may there-fore be assumed that the out-of-plane actions are insignificant compared to the primary in-plane actions. If the sizes and spacings of the frame members are assumed uniform, as is usually the case in practice, then each framework panel may be replaced by an equivalent uniform orthotropic membrane. Methods for determining the equivalent membrane properties have been given in the following references: Coull and Bose (1975). Khan and Stafford Smith (1976), and Ha et a1. (1978). Appendix I presents the method being used by the writer, which is actually an abridged version of Ha et al. 's method. This method is less sophisticated than the original Ha et al. 's method and is thus simpler to apply. On the other hand, since the shear deformations of the frame members are taken into account, it is more accurate than CouJl and Bose's method. Mathematical Formulation Shear lag occurs in both the web and flange panels and as a result, the distributions of axial stresses are no longer linear in the web panels or uniform in the flange panels. To take into account the shear lag effects in the flange panels, Chan et a1. (1974) allowed variations of the axial displacements across the width of the flange panels in the forms of either parabolic or hyperbolic cosine distributions. Coull and Bose (1975, 1976), on the other hand, took into account not only the shear lag effects in the flange panels but also the shear lag effects in the web panels. In their analysis method, the distributions of the axial stresses are assumed to be cubic in the web panels and parabolic in the flange panels. However, the cubic 1224

G

t

• I I

I

E

w

I

r

I I

r

I

G

w

t

I

Y ,--I

w

I

I

.,. ..J.....

FIG. 3.

web

I

I

I

_x

Orthotroplc Membrane Tube Analogy

distribution of axial stresses in the web panels is dependent on the parabolic distribution in the flange panels. They had tried to use an independent cubic distribution for the axial stresses in the web panels and found that this would lead to more accurate results but eventually decided not to pursue any further because this would make the governing differential equations rather laborious to solve (Coull and Bose 1977). Herein, an attempt to use independent distributions for the axial dis.. placements in the web and flange panels is made. The axial displacemen t distributions are assumed to be cubic in the web panels and parabolic in the flange panels, and the principle of minimum total potential energy is employed for the formulation. Consider the analogous membrane tube structure in Fig. 3. Due to shear lag, plane sections will no longer remain plane after the structure is loaded. Let the axial displacements in the web and flange panels, denoted respectively by wand w', be approximated by the following equations: 1225

a

a

I

aw'

e; = a;

f

(4)

Similarly, the shear strains in the web and flange panels are given respectively, by t

C'I.,pa

ow

au

'YXl - az + ax

(5)

(l-«);a iJw'

"I...: = ay

From these axial and shear strain expressions, the strain energy of the framed tube can be evaluated as

tangent to curve at centre assumed displacement curve

fit'. =

lH fa

(a) distribution of axial displacement in web I

I

b

b

(6)

o

-Q

/w(EwE~ +

I

+ Gw"Y;z) dx dz

f fb H

o

-b

l,(E,F.;2 + G,'Y;z) dy dz

. lH 2E",A

+ () .

k

E~

dz

(7)

On the other hand, the potential energy of the applied lateral load is given by the following equations, in which u(z) is the lateral displacement of the structure. Load case l~point load of magnitude P at top tl'a

I1p = - Pu(H)

(8)

Load case 2-uniformly distributed load of intensity U per unit height

(l-I3)c/>a

L H

rIp (b) distribution of axial displacement 1n flange FIG. 4.

=

w' =

TIp = -

~a [(1 - a) ~ + a (~r]

(1)

~a [(1 - (3) + (3 (~r]

(2)

which give cubic and parabolic distributions of axial displacements in the web and flange panels, respectively. "These assumed distributions of axial displacements are illustrated in Fig. 4. Note that is the rotation of the plane section joining the four corners of the tubular structure which initially lie on the same horizontal plane, and a and (3 are dimensionless shear lag coefficients representing the degrees of shear lag in the web and flange panels, respectively. The axial strains in the web and flange panels are given, respectively) by the following expressions:

aw

£z

= az 1226

Uu(z) dz

(9)

Load case 3-triangularly distributed load of intensity T per unit height at top and intensity zero at base

Assumed Distributions of Axial Displacements

w

=-

(3)

fH T H z u(z) dz

Jo

(10)

The total potential energy is just the sum of the potential energy of the applied force and the strain energy of the structure. Having obtained the expression for the total potential energy, the governing differential equations can then be derived by minimizing the total potential energy with respect to the unknown displacement functions and u and the unknown coefficients a and ~ using the calculus of variations. However, the set of governing equations so derived, which consists of four simultaneous firstor second-order partial differential equations, is rather difficult to solve. To make the solution more tractable, the following SiOlplifications are intro-

duced. Approximate Solution Method Minimization of the total potential energy with respect to

, which· may be interpreted as the moment equilibrium equation and expressed in the following form: £1 o = M dz 1227

(11 )

where El = effective bending stiffness of the tubular structure and M is the overturning moment of the lateral load. This equation is not easy to solve because £1 varies with height and is dependent on other unknowns. Nevertheless, if the effect of the variation of EI with height on the bending rotation is assumed negligible, then

l

z

=S

(13)

in which S = shear of the lateral load. This is actually the horizontal shear equilibrium equation. From this equation, u can be determined by direct integration as follows:

u

= (z ( _ S -

Jo

4G w twQ

<1»

dz

(14)

~

and

(X

l3

(2)

(3)

Puint luatl.at

= , 1.17nl". +

l.OU

QI

=

+

1.00

(X"

Q,

lOp

-

Uniform distributed load

= 111;

+ 2.67m", + 0.57

~~

= HI} + 11.20n" + Hl.OR

2.57m no + 1.12 = Ill;'. + 2.94/11". + 0.04

~.

=

+ 2.6711'". + U.S7

O.29In", m~.

O.03m". + 1.12 ex" = - - - - - - - - - Triangular distributed load

(X

-

m;. + 2.94m", + 0.64

,

+ 1.09 = m~. 2.22m". + 2.86,nt\' + 9.62

OJ

J.SOn" + 12.60 + tl.:!()I1'r + to.OX

~I

111 ~,.

= m;'.

O.10m", + 1.09 + 2.86111", + 0.62

O.RSln,. + 11.00

nl i,

7.721n, + 14.15 15111, + II ..l -' + I'- ... O.ORnl,. + 14.15 II ._ 3' 1.

~~ = rtl;. + I'_.."5 In, + J3. =

J3

6.67/111' + 13.71 . nlj + 12.01111{ + 10.97 7 __0_.2_9_11...:..'1_+_13_._ In} + 11.01",{ + 10.97 _1_

~

=

_ G w H2 -~ E w a2

(17)

G H2 m / / - E b'1

( 18)

m H'

Substituting the preceding values of

Q

(1 )

0

(~~ +
Formulas for

Load case

(12)

M dz

Likewise, minimization of the total potential energy with respect to u yields the following governing equation for u: 4G w t,.tl

TABLE 1.

/

It should be noted from the formulas given in the table that the shear lag coefficient of a frame panel is dependent only on the elastic properties of that particular panel, not on those of any other panels. EFFECTS OF VARIOUS PARAMETERS ON SHEAR LAG

where a .. (X2' f3 .. and f32 = unknown coefficients to be solved. Note that al and f31 are actually the values of (X and f3 at the base, while (X2 and f32 are the corresponding values at the top. Substituting the foregoing equations into the expression for the total potential energy, minimizing with respect to a. (X2' f3., and f32' and solving the set of algebraic equations thus obtained, the shear lag coefficients are determined for each loading case. The results are given in Table 1 in which the relative shear stiffness parameters m w and In, are defined by

The shear lag coefficients are plotted against the relative shear stiffness parameters of the frame panels for each loading case in Fig. 5. Fronl this figure, the effects of the various parameters on the degrees of shear lag are readily revealed. Firstly, it can be seen that in all cases, a. > a2 and f3. > f32 and therefore, the shear lag effects are generally greater at the base of the structu re than at higher levels. Secondly, it is apparent that the degrees of shear lag are dependent on the distribution of lateral loads. At the base of the structure, the degrees of shear lag increase with the following order of load cases: point load at top, triangularly distributed load and uniformly distributed load. However, at the top of the structure, the degrees of shear lag decrease wi th the same order of load cases. Thirdly, since the shear lag coefficients decrease with the relative shear stiffness parameters of the panels, the shear lag effects may be reduced by increasing the frame member sizes. Moreover, as the shear lag effects are largest when the relative shear stiffness parameters, which are proportional to the square of the height/width ratios of the frame panels, are small, the shear lag effects are generally more significant in low-rise buildings than in high-rise buildings. Lastly, it is identified that the single most important structural parameter

1228

1229

(1 - ~Y + [2 ~ - (~y] ~ = ~I (1 - ~r ~ ~2 [2~ - (~y]

a = a.

a2

t

(15)

(16)

flange panels can be evaluated by first differentiating the axial displacements with respect toz as per (3)-(4), and then multiplying the axial strains so determined bytheir respectively Young's moduli. The expressions thus obtained for the axial stresses generally consist of two terms, one proportional to d4>/dz and the other proportional to 4>. At the base of the structure, where the axial stresses are most critical, cP is equal to zero and the expressions for the axial stresses would be reduced to

load case

-.

---. ___ •

-- --.--j}

«1

2

5

4

3 m

w

1.0

load case

N

QJ. L-

0

-.-

0.8

1 2

ct:J.....

3

0.6

- - }~I - --.. .

0.4

0.2

0

0

} f3 2

1

4

3

5

~

that determines the degree of shear lag in a frame panel is the relative shear stiffness parameter of the panel as· defined by (17) or (18). The effects of changing the sizes of the frame members can be evaluated simply by calculating the new values of relative shear stiffness parameters and using the design chart given in Fig. 5 to determine the corresponding values of shcur I,lg coefficients. This nlethod is, therefore, particularly suitable for preliminary evaluations of the main structural element sizes during the early stages of design. STRESSES AND DEFLECTIONS

Axial Stresses

Having determined the shear lag coefficients ex and f3 and hence the distributions of the axial displacements, the axial stresses in the web and 1230

(20)

I:n 2t CTzX dx w

+

fb

2tf CT;a dy

+ 4A k CT k a

(21 )

= M

d/dzis determined as dcf>/dz = MIEI, where EI is given by

= ~ Ewtwa)

(1 - ~ ex) + 4E

f lf

a 2b

(1 - ~ f3) + 4E",A a k

2

(22)

2

r

Values of a and

(19)

At levels above the base, however,

EI

m FIG. 5.

= Ew ~~ a f

)«

2

[(1 - ex); + ex (;rJ CT; = E ~~ a [(1 - ~) + ~ (~r]

az

Putting the value of d4>/dz so obtained back into (19)- (20). the axial stresses can be expressed directly in terms of the overturning nlonlent. Lateral Deflections The lateral deflection u of the structure can be evaluated by first substituting the value of EI into (12) to solve for thus obtained into (14) to solve for u. Since the value of £1 varies with height, the resulting expressions for

(1

u= -P - H Z 2 EI 2

-

1) + -4G- -

-

6

1231

zJ

P

w l wQ

z

(23)

Load case 2-uniform load

(1

c:

1

1) + - -

u = -U - H2 z2 - - HzJ + £1 4

6

24

Z4

U

4G w tw a

1)

( Hz - 2

Z2

.2

U

.!

(24)

~

c

~ C .2

Load case 3- triangular load u = -T

(1 H2

EI 6

Z2 -

(1

1 H z3 + - 1 -ZS) + -T- - Hz - -1 -Z3) 12 120 H 40 w t..,a 2 6H

-

-g

o o....

(1)"8

(25)

8..£:

o·~

0.&

~

Example 1 A high-rise 40-story reinforced concrete framed tube structure, as shown in Fig. 6(a), is analyzed. All the beam and column members are of sizes 0.8 m x 0.8 m. The height of each story is 3.0 m and the center-to-center spacings of the columns are 2.5 m. The Young's and shear moduli of the material are 20 GPa and 8.0 OPa, respectively. A uniformly distributed lateral load of 120 kN/m is applied to the structure. The equivalent elastic properties of the analogous orthotropic membrane tube, as evaluated by the method given in Appendix I, are as follows:

!l

=~

III 61>' L. Q

+J

....

(

' ....

u

oII)

......... "

~

=

0, = 1.441 GPa

= 0.256 m

t w = I,

Ak

=0

(ij

·x

< ~

-0 OJ ~

c:

< OJ :;

(I)

~----,

U;~ ~

&1-4

eu eu &.. I:: \to4 eu

..0 )

(27)

+J

.£:

,- 7

."

I

N

(29)

.c:I

"0

..-4

GJ

~ ns

1-4

~

en Q)

GJ

.!.H

2

QJ

:r.

bO

~

U

.D

'/

..-4

From these elastic properties, the relative shear stiffness parameters pf the web and flange panels are worked out by (17)-(18) as

en

."

~

."

....

~

..0

-tJ

& ~

'"

:0

GJ

"0

o

0

o...c: 0 .....

... en

~

Q)

m,

= 3.388

(30a,b)

.0

o.e

1::lJ.I-J.L...J

...OJ

t>:::s

bO C co -e

GJ (/1"0

~ :s

co

&.. GJ

m.., = 4.611;

... en

N

(26)

(28)

en en Q)

n;

GJ

OK'

u

~o 0

~

Q)

Ew = £, = 20.0 GPa

Q)

-t ~

"0

~

......

~ns

~, Jl=

en C

QJ

' ... ""

.£:

.0 .t:

c ~

I.. GJ

EXAMPLES AND COMPARISONS WITH COMPUTER ANALYSIS

:;

e

&

....:::J

~

"C C1)

The shear lag coefficients are then determined by the formulas given in

E

co

Table 1 as

u: a, = 0.366;

Ct2

= 0.624;

~2

~l

= 0.035 = 0.223

OJ

(31a,b)

C/)

(32a,b)

Having determined the shear lag coefficients, the bending stresses at the base and at mid-height of the structure are evaluated by (19)-(20) and plotted in Fig. 6(b). The deflections of the structure are then calculated by (24) and plotted in Fig. 6(c). For comparison, the corresponding results obtained by using a standard space-frame analysis program are also plotted alongside the foregoing results. It is revealed that the proposed method underestimates the maximum axial stress at the base by 14% and overestimates the maximum lateral deflection by 13%.

_ ~

a:,

C'l .t:

i: '0 C/)

'iii >

'i r:

/

r

/

7

w Oll

< .,..I C1)

0..

E CO

)(

Example 2 In this example, a low-rise IS-story framed tube structure constructed of structural steel, Fig. 7(a), is analyzed. All the beam and column members arc 610 x 305 x 238 kg/n, Universal Benrns (I = 207,571 cnl"; A = 303.R C1l1 1 ; A. = 117.7 cm 2). The height of each story is 3.2 m and the center to

UJ

cD

l!o

f

1232

C1)

· :::Jc:

C)

1233

c

center spacings of the columns are 2.8 ffi. The Young's and shear moduli of the material are 200 OPa anti 80 OPa, respectively. A triangularly distributed lateral load of intensity 150 kN/m at the top and zero intensity at the bottom is applied to the structure. The equivalent elastic properties of the analogous orthotropic membrane tube are determined according to Appendix I as follows:

o

n Q)

~

o

..-.. ~

0

M

....

C .2 '5 .a .t:

! ~

~

U GJ

.... ....

(33)

U)

Ak = 0

(36)

ca ~

From these equivalent properties, the relative shear stiffness parameters are calculated as

1;)

d 0

-e

""

= 200 OPa G w = G, = 5.345 GPa I ... = If = 0.0109 m EM' = E1

C en CI)

...

..... ..... QJ

(34) (35)

Q)

"C

e

mK' = 0.218;

"C

m/

= 0.314

(37a,b)

Q)

-~

...

~

and the corresponding shear lag coefficients are obtained as:

coc

~

<

...

)

.tJ

..... .t:

QJ

GJ

..0

"0

-e

GJ

tl()

C

ns

S o6.J

ns

-~-:

~

"C GJ

...

CX2

= 0.861

(38a,b)

(31

1.065;

132

= 0.930

(39a ,b)

RS

-..0

~-g

Figs. 7(b and c) illustrate the final results for the axial stresses and lateral deflections. The corresponding frame analysis results are also plotted in the above figures "to demonstrate the accuracy of the proposed method. I t is seen from the comparison that the proposed method underestimates the maximum axial stress at the base by 5% and overestimates the maximum lateral deflection by 8%. These errors are quite acceptable for preliminary design purposes. However, since the values of a and ~ at the base of the structure are both greater than 1.0, the proposed method yields axial stresses of the wrong sign near the centroidal axis of the frame panels at the base of the structure. Fortunately, as the axial stresses there are actually very small, such errors are not of much practical importance.

en

..-.. .!,

...:JQ)

o...c

1j

c.e

..... en

...:J

Q+J L. GJ

~

Jl:J .... "C (1)

E ~ u. ~

CONCLUSIONS

a:

-nt

~ o

...I

....o

.u;U) ~

"i

c

<

I

C'I

r

1.129;

:J C'3

I

~

1j en

.t::

RS

:J

:&

t:IO

-e GJ

/

al

(1)

.tJ 4J

I

w 8~

I

Go)

0. E n) ><

w r-=

.

Q)

~

CJ :::s

i!o

1234

A simple hand calculation method for approximate analysis of framed tube structures with the shear lag effects taken into account was proposed . In the proposed method, independent distributions of axial displacCIl1Cnts are used for the web and flange panels and thus the shear lag in each panel is individually allowed for. Closed-form solutions are obtained, from which the effects of various parameters on the overall structural behavior can be readily evaluated. The single most important parameter that determines the degree of shear lag in a frame,panel is the relative shear stiffness parameter of the panel as defined by (17) or (18). Since the shear lag effects of each frame panel can be separately evaluated and the degree of shear lag in a frame panel is dependent only on the properties of that particular panel, the shear lag effects may be calculated by means of only a small set of design charts (Fig. 5). The proposed method is easy to use and yet reasonably accurate and is thus most suitable for preliminary design calculations. Numerical examples demonstrated that the method is applicable to framed tube structures over a wide range of building heights. 1235

Ii

APPENDIX I. ELASTIC PROPERTIES OF eQUIVALENT ORTHOTROPIC MEMBRANES

A typical frame segment bounded by the centers of the adjacent frame members, Fig. 8, constitute a basic unit of the frame and may be modeled 'as a solid membrane spanning the same area (shown by dotted lines in Fig. 8) provided the elastic properties of the membrane are so chosen to represent the axial and shear behavior of the actual framework. The method for evaluating the equivalent properties of the membrane is presented in the following. This method is applicable to both the web and flange panels and thus in the following, they are not distinguished from each other. When it is necessary, the corresponding properties may be denoted with a subscript w to signify their belonging to a web panel, or with a subscript f to signify their belonging to a flange panel.

h./2

h/2

Axial Stiffness Under the action of vertical axial forces, the load-deformation relationships for both the frame unit and the equivalent membrane will be equal if

Est = EmA e

(40)

where E = equivalent elastic modulus of the membrane; t= thickness of the membrane; Em = elastic modulus of the construction material; and A e = sectional area of the column. It is normal practice to fix the value of t such that the area of the membrane is equal to the sectional area of the column (Le. st = A(") and so that the axial stress in the column and that in the membrane are equal. In such a case t

= Ae

(41)

s

centre of column

r

---.,

I

column

h

l

centre of beam beam

I

I I

I

~

boundary of

I

menlbrane

equivalent

L_

.J

~ c

FIG. 8.

Membrane Analogy for Basic Frame Unit 1236

and

E

= Em

(42)

Shear Stiffness Consider now the case of the frame unit subject to a lateral force Q, Fig. 9. The lateral deflection may be computed as the sum of that due to bending 6 h and due to shear 6.s • The bending deflection Ah is given by

Q

I

dbI

Basic Frame Unit under Lateral Shear Force

6. h

I

I I

FIG. 9.

= (It

- db )3 12Em I("

+ (~) 2 (s - dr)'~} s

12E",lb

(43)

where Ib and 1(" = moments of inertia of the beam and column respectively. On the other hand, the shear deflection As is given by ., D. s db) (s - de) (44) Q = OmAs(" + ; G",A sb

(h -

(h) -

in which A sb and A s(" = effective shear areas of the beam and column respectively; and Om = shear modulus of the material. Equating the total lateral deflection of the frame unit to the shear deflection of the membrane, the following equation is obtained: h

Q OS!

= Ab + 6.s

(45)

where G = equivalent shear modulus of the membrane. From this equation, the value of G is derived as 1237

h G

=

st db

mw,m,

(46)

As

-+Q Q

t

Wt

I,

U W

w' a

in which /lJQand AJQ are as given by (43) and (44), respectively. APPENDIX II.

S

f3

REFERENCES

'Yxz

Chan, P. C. K., Tso, W. K., and Heidebrecht, A. C. (1974). "Effect of normal frames on shear walls." Building Sci., Vol. 9, 197-209. Cheung, Y. K. (1983). "Chapter 38: Tall buildings 2." Handbook of structural concrete, F. K. Kong et aI., eds., Pitman Books Ltd., London, England. Coull. A .• and Ahmed, A. A. (1978). "Deflections of frame-tube structures." J. Strllct. Div., ASCE, 104(5),857-862. Coull, A., and Bose, B. (1975). "Simplified analysis of frame-tube structures. J. Struct. Div., ASCE, 101(1 1),2223-2240. Coull. A" and Bose, B. (1976). "Torsion of frame-tube structures." J. Struct. Div., ASCE. 102(12),2366-2370. Coull, A., and Bose, B. (1977). "Discussion of 'Simplified analysis of frame-.tube structures. tt, J. Siruct. Div., ASCE, 103(1),297-299. Coull, A., and Subedi, N. K. (1971). "Framed-tube structures for high-rise buildings." J. Struct. Div., ASCE, 97(8), 2097-2105. Ha, K. H., Fazio, p.. and Moselhi, O. (1978). "Orthotropic membrane for tall building analysis." J. Struct. Div., ASCE, 104(9), 1495-1505. Khan, A. H., and Stafford Smith, B. (1976). "A simple method of analysis :or deflection and stresses in wall-frame structures." Building and En vir. , Vol. 11, tI

1yz

£k Ez E'z

ak az a'1
relative shear stiffness parameter's defined by (17) and (18);

= shear of lateral load;

equivalent thickness of web and flange panels, respectively; lateral deflection of structure; = axial displacement in web panel; axial displacement in flange panel; = shear lag coefficient of web panel; shear lag coefficient of flange panel; = shear strain in· web panel; shear strain in flange panel; = axial strain in corner column; axial strain in web panel; axial strain in flange panel; axial stress in corner column; axial stress in web panel; axial stress in flange panel; and rotation of plane section joining corners of tube.

=

69-78.

Khan, F. R. (1967). "Current trends in concrete high-rise buildings." Proc.. Symp. on Tall Buildings, Coull and Stafford Smith, eds., Pergamon Press, Oxford, England, 571-590. Khan, F. R. (1985). "Tubular structures for tall buildings. Handbook of concrete engineering, M. Fintel, ed., Van Nostrand Reinhold, New York, N.Y., 399-410. Khan, F. R., and Amin, N. R. (1973). "Analysis and design of frame tube struct,u:--es for tall concrete buildings." Struct. Eng., 51(3), 85-92. Spires. D., and Arora, J. S. (1990). "Optimal design of tall RC-framed tube building;." J. Strllct. Engrg., ASCE, 116(4), 877-897. Wong, C. H., EI Nimeiri, M. M., and Tang, J. W. (1981). "Preliminary unul}ksis and member sizing of tall tubular steel buildings. ,. Engrg. J., (Second Quarter), 33-47. It

APPENDIX III.

NOTATION

The following symbols are used in tfzispaper.·

Ak

sectional area of corner column; half width of web panel; half width of flange panel; Em = Young's modulus of material; equivalent Young's moduli of web and flange panels, respec£'1" £, tively; bending stiffness of structure; £1 shear modulus of material; Gm equivalent shear moduli of web and flange panels, respective:y; G"" G, overturning moment of lateral load; M

a b

1238

1?~A

8 Outrigger braced structures

40

......

l~

'j

SP

L

63-21

,j

Behavior of Multi-Outrigger Braced Tall Bui.lding Structures By B. Stafford Smith and I. O. Nwaka

.I:i.~ .j'

{

l I

Synopsis: A study iR made of the forces nnd displacements in multi-outrigger tnll huilding structures! SifTlplified r,eneral equations are developed for the restraining moment of the outriggers, the reduction in drift nnd the optimum location of the outriggers for maximum drift reduction. The efficiencies of various optimum and evenly spaced outriAger systems are presented. The assumptions used make the method of analysis suitable only for preliminary design guidance; however, some v,lluable general conclusions relating to the number and location of outriggers are

'j::; :11,.\

Ii:Ii I

:\:

.11; 'I~

III

1;; ,ill

T

i:i

drawn.

1'\

.,\\

m 11 1

Keywords: bending moments;brac;ng; columns (supports); highrise buildings; lateral pressure; loads (forces); mathematical models; structural analysis. I

~

itl

\:~

.:;~

jll

'It: ",i,

.).J'II

~

JI!;,1.,' ,II

. ~ll ,I'

"'d~1

j~

JI ,\I,'j

1'1 ! I

,I,

":If ~

I

,I

I

1

I

A

r.:1r.:

JIU

t,J La I • V

I U

.., • • ••

•

.....

MU1{I·UUlrlt;O~r ~ll U\...lUI \";~

Bryan Stafford Smith is a Professor of Civil Engineering at McGill University, Montreal. Previously he was a Professor of Civil Engineering at the University of Surrey in England. He has researched over a period of many years on various problems relating to tall building analysis.

acting about a common neutral axis and contributing to the total res~sting moment an amount

fc

1 1

{

,

,i INTRODUCTION Outrigger bracing is an efficient means of reducing the drift and forces in a tall building structure. In its simplest form the system consists of a reinforced concrete or braced steel core to which horizontal cantilevers - outriggers - are rigidly attached at one or mora levels, (Fig. 1). The ends of the outriggers connect to columns which, when the building is subjected to sideways loading, resist the rotation of the outriggers and core. In modifying the free deformation of the core, the total drift of the building and the moments in the core are reduced, (Fig. 2). If the principal restraining columns are on the face of the building it is sometimes expedient to mobili~e the axial stiffness of additional perimeter columns by running a deep, very stiff spandrel girder around the structure at the outrigger levels. If this is placed at the top, the system is sometimes called "hat" or "top-hat" br.acing, and if located at int ermed iate levels, "belt-bracing". Notable recent examples of belt-braced buildings include the 42 - storey First ~~isconsin Centre in 1.1ilwaukee, (Fig. 3), by Skidmore, Owings and Merrill of Chicago, and the 43 - storey Yasuda Insurance Company Building in Tokyn, (Fig. 4) designed by the Structural Department of the Tasei Corporation.

A

Taranath (1) who showed that the level of the outrigger for minimum drift is close to the mid-height of the building. ~tc~abb and Muvdi have also studied the problem, confirminA Taranath's results for a single outrigger (2) and extending their considerations to two outriggers (3). Further reductions in the total drift and the core bending moments can be achieved by increasing the size Rnd therefore nxi:ll Atiffnf'fHl of the' Q(ll\1mn~. ilnd hv lldd ln~ ou t r 19ge rH n t more leve 1M. The lmprovcmcn t d 1m1n 1 ::;hl"fi , however, for each additional level. Taken to the limit. c.)utrigp,ers plnccd at nn infinite number of levels, (FiA. 5) ,,,...\\lld r '" \l ~ l' the r 01 \I mn H t (\ he hnve f \l 11yenmr 0 ~ 1r l' 1~' \\' l t h t \ h" \- \"1 r l' •

(

c

EI + AE

c

d 2/2

.) . N

total

( 1)

in which EI is the flexural rigidity of the core, ARc is the axial rigidity of the columns and d/2 is the distance of the columns from the common neutral axis.

This concept, of the fully composite structure, will be used as a standard of efficiency when comparing different outrigger systems later in this paper. It is usually convenient. as well as necessary from strength considerations, to make the outrigger at least a full storey depth. Their bulk, and obstructive configuration, often makes it appropriate for them to share the plant room levels. This and aesthetic architectural considerations, as well as structural factors, will all have to be taken into account in arriving at the eventual number and location of the outriggers. For the structural engineer's contribution to the discussion it would be valuable if he could easily assess the relative performance of systems with different numbers of outriggers at different locations. He could then make known the structural and cost penalties incurred by departures from the structural optimum.

t

-l ~

i 1

The belt-hrnced structure hns been studied previously hy

2 d /2

AE M

Onyemaechi Nwaka graduated with an ~{.Eng. from McGill University in ·1977. This paper is based on the results of a research project which contributed to that degree. He now works in his home country, Nigeria.

...., .. "

1 ItA

In this paper, multi-outrigger structures are studied to obtain guidelines for the relative performance of different arrangements. Generalized formulae are developed for estimating the core moments, the column forces and the total drift in structures with any number of outriggers at any levels. General formulae are also derived which allow the optimum location of the outriggers to be determined. The formulae are used to give numerical. comparisons of efficiency for a range of structures with alternative arrangements· of single and multi-outrigger bracing. Some general conclusions are then 'drawn 'relating to the useful number and locations of outrigger bracing. It Rhould be restated that the Assumptions u5cd in developing the formulae in this paper, in particular the vertical uniformity of the structure, ~he flexural rigidity of the outr~ggers nnd the uniform distribution of the londing) re~tricts their use to the preliminary stages of design.

~ ",.

ThC' mnin vnl\1(' of tlH' work IH thcr('rore 1n thn p,cncrlll guldHIlCl! It pruvLul'H In numbcrlnR nnd locnt1.ng the oLltriggers for

their greatest effect.

'~

':~gt

"r-518 :

Stafford Smi.th and Nwaka

Multi-Outrigger Structures

519

j:

f.

ANALYSIS OF A SINGLE-BELT STRUCTURE

c) The outrigger action induceR only axial forces in the columns. The neglect of any bending in the columns due to rigidity o( their connections to the outriggers will be small and conservative.

The main purpose of this study is to develop general equations for the analysis of multi-belt structures. It is useful. however, to refer initially to a study of single-belt structures made by Tarannth (1). He developed equations for the outrigger restraining moment) drift and optimum location as follows.

d) The core is riRidly attached to the foundation. This will be ensured in the design o( the core to foundation connection. e) The sectional properties of the core and columns, and the distribution of horizontal loading) are uniform through the height. In a tall building this assumption will almost certainly be not valid. This reinforces the restriction that the results of this study are to be used for preliminary design guidance only.

Restraining moment (H 3 _ X3 )

w

M 1

;-;[ir + AE~d2J' wH

Drift

~

1

(H - Xl)

Analysis of Core Moments

M ~H2 _ X2)

4

= -BEl

1

1

2EI

in which w is the uniform horizontal distributed loading. the height of the structure and Xl is the distance of the rigger from the top of the building. The optimum location Xl of the outrigger for minimum drift is given by the solution of

4X~

+

3X~H

3 - U • 0

(4)

These equations will be used in conjunction with equations to be developed for two and three outrigger structures to establish general forms of the equations for restraining moments) drift and optimum outrigger locations. ANALYSIS OF A

DOUBLE-OUTRIGGE~

"

STRUCTURE

Assumptions These are The structure behaves linear elastically. This should be reasonably valid in both steel and reinforced concrete structures up to the design wind loading. In ~ases of reinforced concrete columns, a check should be made to assess the possibility of net tensile strt·sses in the columns. If thl8 occurH, nnypreoLctlolls for resisting moment and drift reduction based on ~ross sectional areas may be excessive.

~

"c~l.1l111

1I11.lInllll"

l,,·whlc·h I hc' oul rl)·aJ..'rli. "llh.""J.11 11"xllllllly

very Hllrf. :lrl' not rl",ld. Thfllr II .. xJhllltl(,:H will r .. d,u'p lit .. effectiveness o( the bracIng system. Thls important factor ls, t h l"' r ft f n r (', rl t Sf' \l H r; e cl R e pn rat e 1 v, 1 n t e r i nth e pnrc r .

Taking the restraining moments ~l and H2 as the redundant actions to be solved, the analysis can proceed using the conditions of rotational compatibility. Considering in Fig. 6a the rotation of the structure at the upper outrigger. the compatibility equation is H

e

a)

b) The outriggers are rigidly connected to the core and are thcmselveR flexurally rigi.d. Thi.R iR C1 groRR Rimplifi.cntinn of the

When a core stands as n free cantilever without outri~gerst it is statically determinate. The addition of a single outrigger system makes the structure once redundant. For each additional outrigger it becomes one degree more redundant. Referring now to the double cantilever structure shown in Fig. 6a) the structure is twice redundant. In addition to the "free l l bending moment distribution applied to the structure by the sideways force) Fig. 2b. each outrigger will impose on ~he core R restrAining moment which extends all the way froM that outrigger to the base, as shown in Figs. 6b and 6c.

1

2H l (H-X 1 )

-- +

c

Xl

2

(5)

w(HJ_X J )

HI (H-X l ) + H2 (H-X 2)

I

I

in which

'1

-

c

2) -12-.J: X (wx~ _ ~I )dx I _ -l.(",.~ 2 _ Ml-~12)dx"'() EI 2 I x E1 -

in which the first two terms represent the rotation of the upper outrigger and the last two represent the rotation of the core at the upper outrigger level. Equation 5 then reduces to

I

1.

£d 2 (H-X

I

2]

s= - + - - -2 TiT i\I\.c1 •

[

1

(6)

6EIS (7)

Considering now the rotation of the lower outrigger, it can he shown similarly that

i:~~

!'~!~I'I"" ~~ ':~f' j'

I

:1~

:..-.

Stafford Smith and Nwaka

520

3

+ M (l1-X )

M (H-X ) 1 2

2

2

MUlti-Uutrlgger

3

w
wH 6

(8)

6ElS

2

4

222

=--

(12 )

'3

M1

=

4X 1 ,+ 3X

(9a)

Xl

w 2 2 and (9b) M2 = 6EIS (H + HX 2 - X2X1 - Xl) \. Hl and H2 can b~ subtracted from the "free" bending moment diagram to give the iesulting bending moment distribution in the core, Fig. 6d. The forces in the columns just below the top outrigger are then! M1/d and, in the columns below the lower outrigger, ! (M1 + ~12)/d. The maximum moments in the outriggers will then be the product of the column force and the free length of the outrigger. Note that these will be less than the half moments Ml /2, M2/2 which are the moments of the column forces at the centroid of the core.

L[ IXl o

2

giving

6

wH

4

dx +

· x

I

2 (w~ 2-M ) .x.dx ' i

Xl

1

2

= 8El - ZEI [M l (H -

I

0

c

2 y 2 3 _ - 3X l X2 + 3•• " H - H - 0

( 13)

The procedures for deriving equations foi bending moments, drift and optimum locations have been given for a double-outrigger structure. Similar procedures can be used for structures with three or more outriggers to give equations corresponding to Eqn. 9 for the redundant moments, Eqn. 11 for drift and Eqn. 13 for optimum outrigger locations. For a triple-outrigger structure, these are Outrigger Moments

H

X

2

-wx

3

X - X 2 2

with respect to Xl'

ANALYSIS OF A TRIPLE-OUTRIGGER STRUCTURE

The moment area method is a simple means of determining the top deflection of the outrigger structure

= EI

3

2

1

~irst

The simultaneous solution of these gives the optimum values of Xl and X2 as O.3122H nnd O.6355H, respectively. These can he substituted into Eqns. 9 and 11 to determine the corresponding restraining moments ~nd minimum top deflection.

Analysis of Drift

tJ

222

l2(El) S

which, when partially differentiated, then to X2t yields

The simultaneous solution of Eqns. 6 and 8 leads to expressions for the redundant moments 2 2 6EIS (X 2 + X2X1 + Xl)

2

V_I

w[ (X +X X +X ) (H -X1)+(H +HX -X X -X ) (H -X )) 2 2 1 l 2 2 l 1 2 -2-

BEl

w

~lrU\;lUr~~

+

(w~ 2-~li-M2) .x.dx]

x2

(10)

2 2 2 Xl ) + MZ(H - X2 )J

(11)

-t

M)

Optimum Location of Outriggers The primary purpose of any bracing system is to resist drift. It is useful, therefore, to know at which levels the outriggers should he plRf:cO to nr.hirvp minimum rlri ft. T1H"HP """ he' fnul1c1 r Ir:al. In :
then to X2 . Therefore, substituting for M and H from Eq. 2 i into Eq. 11 gives

C)

w

= 6ElS

2

(X 2

+ X2Xi +

2

(14 a)

~~i)

__w_ 2 2 H2 - 6EIS (X 3 + XJ X2 - XZX l - Xl)

~. .,.

Hence, for a structure in which the dimensions, sectional properties and horizontal load intensity are specified, the actions in the structure can be derived from Eqns. 9, and the drift by substituting the obtained values of H and HZ into Eqn. 11. 1

by mlnlmll.lnJl, lilt, dC'lll'c:lloll f!'lunl Inn wlt.h r'·lIp.'c'l

Hi

w

2

= 6EIS (H + HX J

. (14b)

2

(14c)

- X1X2 - X2)

Drift

I

1

6

= wH

4

8J:.:l

_ _l_x [ l\{

2EI' 1

1

+

2 N (H 2- X )

2

2

+

2 2 N (H - X )

3

3

1

Optimum Outrigger Locations 'I'll,'"'' "rl' glvPIl hv XI' X'J. nnd X.\ frnm lhe' nnl"t'Inn nr l~qU:Il. IOIl~"

4X

~

(H 2_X2)

J 2 J + 1X X2 - X2 = 0 t t

( 15 )

522

~tanora ~mltn

ana

X3 _ 3X x2 + 3X 2X _ x3 1 1 2 233

3

22 X32 - 3X· 2X3 +03X 3 H

H

I~Wdl\.d

, l' I U • ". _. '"" .'-41 " . •

4X

o and the remaining

=

0

00""' · - .... - - - -- - --

32'3

1

~-l

+ 3X X2 - X 1 2

=

0

equations of the form

(16) ·

3

·22

Xj _ 1 - 3X j _ 1Xj + JXjX j +1

GENERALIZED EQUATIONS FOR MULTI-OUTRIGGER STRUCTURES Equations are now available to solve the redundant moments 2, g and l4}, drift (3, 11 nnd 15) and optimum outrigger locations (4, 11 and 16) in single, double and triple-outrigger structu~es. An inspection of these equations indicates sequences which allow corresponding generalized equations to be written for structures with any number of outriggers, as follows.

3

X'+l J

o

(20)

where j is equation number.

(Eqns.

Outrigger Moments

1

In a structure with N outriggers, the restraining moment due to the uppermost outrigger is given by

In the equation for a single-outrigger structlJre and in the for an N outrigger structure X2 = Hand XN+1 = H, respectlvely.

~th equ~tion

The formulation and simultaneous solution of the set of equations for a structure with N outriggers will determine their optimum locations for minimum drift.

THE

PE~FOR}~NCE

OF OUTRIGGER STRUCTURES

:1

I:

l l;. ~

"~

2 2 6ElS (X 2 + X2X1 + Xl) w

If the outrigger is at the top of the structure then Xl = O.

In an attempt to develop a feeling for the relative performance of optimum and certain non-optimum outrigger structures, the derived equations will be used to ohtain valueR of optimum locations. restraining moments and drift reductions.

In all other outriggers, i to N

Structures with Optimum Location of Outriggers

M1

=

(17)

w 2 Mi = 6ElS (X i +l + Xi+lX i - XiX i _ l ~+l

taking, for the lowest outrigger,

I

2

x.1- 1)

(18)

j

= H.

,I~

iI~I'ro 'tl.l ~~: ~ .i,

'f;· '~t

01 ~,~

:j." The results for optimum locations) restraining moments and drift reduction for structures with up to four levels of outriggers are given in Table 1. The presentation and implications of the results are discussed below.

:t,,·

l,:~

"i':·rl t.~

"

Drift The total drift for a structure with N outriggers is given

~

by

Ii

wH 4 BEl

___ _

1

N

r

2EI ~i=l

M (H 2

· 1

2

.(19)

- Xi )

The first term represents the drift of a free cantilever under the action of a distributed horizont~l load wand the second represents the reduction in drift due to the o\ltriA~cr syRtem. Optimum Location of Outriggers '1' II U

U

II L J IIIlJ III LtJ t: II L LUll tJ. X1

llJ

XN'

t)'

l

II C

(I 1I

l

r f ~ ~ l~ .. Ii

J It

•, "

N

1

Outrigger locations Values which are obtained from the simultaneous solution of sets of equations. as per Eq. 20, are optimized to produce maximum drift reduction. Although the drift reduction will also be influenced by the axial stiffness of the column system -the larger their stiffness the greater the reduction - it is evident from Eq. 20 that the optimum locations are independent of both the core and column stiffness properties. The resulting optimum locations are noteworthy in two respects. First. that the top of the structure is not nn optimum locntion for an outrigger in any of the cnsey. Second, thnt in nll cases the optimum locations are very close to the equal interval points in the height of the structure. For example, the optimum level of the outrigger in n HinKle outrigger structure is almost exactly at mid-height whilst those in a three-outrigger system nrc cl o~e to the quarter, hal f an1d three-quarter heights. Although th~ primary function of l1 bracing system is the reduction of drift, an associated merit of the o\ltrlp.Rcr bracing Rystem is the reduction in core moment. I': q. 2. r() r 1\ H In ~\1 C 0 \l t r 1RRc r H t r \l C t \1 r c. ah ow 8 t hn t the low crt he 1{t!9training moments

outrigger structure can be obtained from the simultaneous solution of n set of N equations including a first equation.

J1

.. :~

il~I'n':11 ••

~~I

:

IW:·l ,!\

:1

I~;:

q;~ 1i;:~

j·:X

11'~1

ii:

:!: "

~~lLll

.\il~,~ IIJ~ . !;~

!

I'

d 'I

.- ...

,---

.- .Stafford Smith and Nwaka

'-524

Multi::Outrigger Structures

outrigger the greater the resisting moment it provides. The benefit of placing it very low on the structure to give a high resisting moment is offset by the fact that it would not be as effective in reducing qrift nor would the reduction in moment, from the outrigger to the base, extend over much of the height of the core. Indeed the moment in the core above a very low outrigger might even be larger than below. The values of resisting moment in Table 1 were obtained from Eq. 17 and 18, and correspond to outriggers at the optimum drift locations. The resisting moments are expressed non-dimensionally as a percentage of the fully composite moment, Mfc (Eq. 1) .. The values are, therefore, absolute measures of the moment resisting efficiency of the system, independent of the column and core sectional properties. The justification for this form of pres entation is as follows 2 AE d /2

from Eq. 1 1-1 f

c

•

EI + AE d 2 /2 c

Mfc

= EIS

I

1

The results in Table 1 show that the larger the number of outriggers, the greater the'total moment reductton in the core. However, for each additional outrigger there is a diminishing return in the extra reduction of moment. The largest resisting moment of an individual outrigger occurs in a single-outrigger structure. In multi-outrigger structures the lowest outrigger carries the highest resisting moment and each successively higher outrigger carries less. Reductions in drift These are also given in Table 1 in a non-dimensional form, as a percentage of the drift reduction which would be achieved by the same system acting fully compositely. By simllar reasoning to that for the restraining moments, the fully composite reduction in moment due to the outriggers and columns would be

i

2 ( 21)

2

2 wH · '-2-

1

or

wH

c

j

?

AE d-/2

J

dLi

fc

c

• -

wH 2

EI + AE'd /2 c

wH

(22)

4 ( 26)

BEl

4 (27)

8(EI)2 S Taking Mi' the actual resisting moment of outrigger i from Eq. 18, and expressing it as a percentage of Mfc gives

*

Mi

Mi

= M:

fc

~ 100

100 w

2

= 6ElS (X i+l

2

+ Xi+1X i - XiX i _ l - Xi - l )

2EIS

·----2 wH

(23)

hence

*i =

JJ.J

M

(X~+l

+ Xi+1X i - XiX i _ 1 2 H

X~~l)

(24)

I i

Taking the second term of Eq. 19 as the actual reduction in drift, d6, and substituting in it for Mi from Eqs. 17 and 18. it can be expressed as a percentage of the fully composite reduction

by

~

d/\

Mt

'M~

Mi

= 100

•

wH2 ET~

where S is defined by Eq. 7.

dl\

'If

di\rt'

:< 100

~ 1(

Therefore, although the actUAl resisting moments of the outriggers are dependent on the size and spacing d, of the columns, the efficiency of the system, based on the fully composite hehnviour of the columns and core as heine 100% efficient, is ShO~l hy Eq. 24 to be dependent only on the number and locations of the outriggers. Values of in Table 1 can be converted to Actual moments ~11 in n pnr t iculnr struc t ure hy us inA

525

dl\

I

(:? '))

1 ':'

N + E 2

8fEI)2S wH 4

w

100 x ~~-' ·

(X~+l

[(X

l2(El) S

2 2

2 2 + X2X1 + Xl) (H

+ Xi+1X i - XiX i _1 - X:_ 1){H .

r'" * - 66.7 --,--II

2

2·

2

r (X " + X" X, + XI) (11

I

•

2

•

'2

- XtX i _1 - Xi_1)(H

2

- Xi)]

2

2

2

- X I)

-

X~)]

2 X) 1

(28)

~ 2 + f. (X, 1 1 + X 1 I. \ X1 :~

(29)

:>..-rr-~

-

.:)IdTTOfO :7IT1lln ano

-

J~Kd--

0-

O_---J~/

Generally, outrigger flexibility will shift the optimum locations of the outriggers, however many, up the structure from the posit ion s Cl u0 ted for r -i g i d 0 U t rig gerR. The ext en t 0 f the shift will depend on the rcductlon in stlffncHs or th~ outri~n~~r and also on the stiffness of the column system. The more flexibl~ the outriggers, the greater the shift. On the other hand, the stiffer the column system the greater will be the effect of any change in outriAgcr flexihility.

Equation 29 demonstrates that the efficiency of the system in reducing drift, taking the .fully composite system as 100% efficient, is dependent only on.the number and location of the outriggers. The efficiency~of the outrigger system in reducing drift is so high, 88%foro~e outrigger and 96% for two, that the extra return for additional outriggers diminishes even more than 1n the case of restraining moments. The extremely high, 98.6%, efficiency of a four-outri~gcr system implies that nny more than four is not worthwhile.

As an illustration of the influence of outrigger flexibility, Table 5 shows the change in the resisting moment and drift reduction efficiencies for structures with three values of core··to . . outrigger stiffness ratios, and three core-to·column stiffness ratios.~. Clearly,the influence of softening the outrigger is greater for structures with the high column stiffness Q = 1 than with the low-column stiffness, C1 = 10.0. Fig. 11 shows the shift in the optimum location of a single outrigger as it becomes more flexible. for hypothetical structures with axially 'rigid' columns. Two points are plotted in Fig. 11 for existing structures with known core-to-outrigger stiffness ratios, which show that even in the extreme case of assumed rigid columns, the shift in the optimum positions for these structures is not large.

Structures with Outriggers at Even Spacing

a,

To allow integration of the outriggers within the normal floor intervals. as well as for architec~ura1 and plant considerations,it may be more convenient to lo~~te the outriggers at equally spaced positions rather than at the optimum locations. Therefore Tables 2. 3 and 4 present the resisting moment and drift reduction efficiencies for equi-spaced outriggers. It is useful to compare their performances with the optimum. Table 2 compares the efficiencies of systems with a single outrigger located either at the top, or at mid-height, orat the optimum location, Xl = 0.455 H. Whilst the mid-height and optimum outriggers are Jndistinguishable in performance, a single, top outrigger behaves relatively poorly in providing only 60% of the moment resistance and 75% of the drift reduction of the same outrigger with an optimum location. Figures 7 and 8 compare the bending moment diagrams and deflected shape for top and mid-height single outrigger structures with those fqr a free cantilever. The core to column stiffness ratio a (where a = 2EI/ oAE c d 2) for this case is 1.0. The fully c~mposite, that is maximum possible, reduction which could have been achieved in these cases for moment and deflection is 0.5. Tables 3 and 4 are for· equi-spaced, multi-outrigger structures, the first including one at the top and the second without. It is evident that for systems with the same number of equi-5paced outriggers,those with an outrigger at the top are relatively inefficient. Indeed, the total resisting moment of any system which includes a top outrigger can b~ almost equalled by a system with one less outrigger, omitting the one at the top. Figures 9 and 10 compare the bending m~ment and deflection diagrams for a triple-belt structure with those for a free cnntllever core.

CONCLUSIO~S

The followinggeneial conclusions apply:

Allthc dcrlvutlons and results have safar assumed the outriggers to beflexurally rigid. In reality, their flexibility will reduce the overr.111 Rtiffnp.RR of thf' n\ltrtRRl"r HyHI.(\1ll /llul

J.I

Outrigger bracing, in single or multi-outrigger arrangements, is an efficient means of reducing drift and core bendinR moments in tall building structures.

2.

In single-outrigger structures, it is more efficient to locate the outrigger in the mid-height region than at the top of the structure.

3.

Multi-outrigger structures can be more effective than singl~­ outrigger systcmsj however with each Cldditional olltrigger the increment of additional stiffness and moment reduction diminishes. The possible reductions in drift for more than four outriggers are insignificant.

4.

If, in multi-outrigger structures, the outri~gers ore located

I

~:

fa t (! q \I I cl {H t nn t III Lg h t {11 t (' r v f\ 1H r r n III t 11 e J~ r 0 lJ n d. h 1I t nmit t 1n g une Ht the top of lhu structure, u close to optimum rcuuctlun in drift will be achieved. 1

'1

An investigation is proceeding into the detailed effacts of outrigger flexibility, but some general comments based on preliminary results ~an he mArtp..

1. !

INFLUENCE OF OUTRIGGER BENDING STIFFNESS

'."11ke it lell see fee tive in controlling the d r H t and core moment II

NlOl(f.UUtrrggeT-~1.rUClU-r~~

I

~~

Multi-Outrigger Structures

Stafford Smith and Nwaka

528

529

REFERENCES
1.

~

Taranath, B.S., "Optimum Belt Truss Location for High-Rise Structures", Structural Engineer, No. 8,V53, August, 1975.

c

o

or+oJ

U

U

'f-

~<)

2.

3.

McNabb, J.~., and Muvdi, B.B., "Drift Reduction Factors for Belted High-Rise Structures", Engineering Journal, American Institute of Steel Construction, Third quarter, 1975. McNabb, J.W., and Muvdi, B.B., Discussion of Reference (2) Engineering Journal, American Institute of Steel Construction First quarter, 1977.

"C"O

I

C1I

J,.'f+J

area of column horizontal distance between columns

d

a:-

\C

ID

.......

en

\G'i

~ f'.: .......

0)

0'\,

VI

x

CIt,

0i: '" Oft'

I

I

~

T

h:

~

'7

Ln ~

\0

It

:F

.....

~

::')

fU

o

:F

c:::

Vl UJ

'0:

.. rr:

...

t··

,Jrdl

N

;1-'1'"

intensity of uniformly distributed horizontal loading

~

~

co

(\.

0IIl:t

of[

\0

M

f

::')

Q)

....

0:

-4: r-4

VI

0:

LLJ

~

~

C' C

distance of outrigger from top of structure

c.!:l

..

"Z

C?

f',

:;'

'"

Ui)

-:e.-

,.rJ...

N

~

o

co

U).

I:::J

0')

oI

V\

E c

elastic modulus of columns

H

total height of structure

I

moment of inertia of core

ex

M fc

resisting moment of outrigger i

Mi expressed as a percentage

N

s

a parameter

VI + AE:d 2]

~ ..,..

.-:r ..

LIt

~

~

-

horizontal drift at top of building

d6

reduction in drift

0-

• v

~J ~

6

J

.

Cully composite reduction in drift

I

centage of dli fc

":'!:

~

l.L.

::::

0: Lt.J

0..

~g

\U

o

~ ~

Ln

a

M 4l:t' IIIl:1"

o

t---.

~

o

U'l ll;:I"

Q.

N

t'""

('\,

M

~.

C""!

o

C')

~ Q

C'.:

r",

q-

4'

C" L 0' GJo,... .J:J'-

t: ...,

;=?

R

Lt.J

...J

co

.... U')

,II

I

c(

V\

d6 expressed as n per-

.... 0..

o

it- L

d6*

:~l~-

~

'

g ,.,:.' ~

o

o

C"'-

C":~'

~ ~"i.I

i·i

1/

.. \

l.L.

+oJ

l5 L

I

w

W

'+-

i~

o

:::t:

5 1+;-

"'~I ~'1

";1'

•• ~ 1:1:

;\;1

~ ~

it )of(-

dL\fc

(;)

"i

;~~r , .\

.......J :::J 2::

'

V\ .,...

of H

fc total number of outriggers

core to outrigger relative stiffness parameter EI . d (EI)o H

=-

fully composite resisting moment 6

N*

core to column relative stiffness parameter = 2EI 2 AE d c

~

B

i

c-

:1

::, ;~~.

0:

.~

GI

N

Q

eo

co

o

elastic modulus of core

E

0-

'to- ,;-

HOTATION A

o

~l'

:·t

III I

'/

..

~

CJ ~

c:

,

Resisting moment as % of Mfc

location of outriggers

Drift reduction as % of d~fc

d~ (J ,...

~

Optimum X1/H = 0.4554

55.4

...=

o

88.0

c

CI

3

~1id-Ueight

58.3

X1/H = 0.5000

~

::J

87.5

~

::s

c

Top-Hat X1/H ::: fJ

..

~

f

I t

'!umbp.r of C'.Itriq
--

* Xl

, •

X*z

~

~ ~ ~

~

Distance from toP :

I

:2

66.6

PERFORMA~iCE OF SINGLE-OUTRIGGER STRUCTURES

TABLE 2

....

33.2

! ~

*

:<3

....1------ ..

Po

I i

*

XII

r

•

R~sistinn

\1 *

1

:

·.1?*

.--~::.;,:=,'=~'"

. ..

noment as

*

"'1

~

till·

...

.

of Mfc

'*

f

;

Mil

I.~~

Drift reduction dd as .~ of dt\fc

::;

I

i I

f

I

1

~

-

-

-

33.2-

2

()

0.5

-

-

8·3

t

..

.~

-

-

JJ.2

66.6

-

-

5S.,"3

91.6

oc ,.....

!

,i I t

50.0

:!.

QQ

I

(JQ

I

1 I

,

3

0

0.33

0.67

4

~

.Q.25

f).

-

8.6

23.1

4-3.9

-

70.6

96.3

0.75

2.1

12.. c;

2.5.0

,7.5

77.1

97.8

I

50

..

_--J

TABl E 3

PERFORMA:JCE. OF EQU I- SPACE') I)UTR I ~GC:R ST~UCTURES ~'IITH TOP 11UTR IGfiER

... r.r. ,.... ...c. !1 c... m to

I

iI

~

C

(I)

o

u.

~fflit£tiii'iii-_fl.I1_~d4t

• •Ii-WEg~~¥~::·:~·······

c.n

~I • I

f

Distancp.

'tumber of outriggers

i

Xl*

top : H

X,*

X*2

-

0.5

1

fro~

Resistino X* 4

t.4

I

'1

-.,!*

as

~

*

I

&.1 * 4

:

'~3

;

~fc

of

1:"1*

Drift reduction d6 as ~ of d6 fc

til ,..... ~

-

-

*

mo~nt

5S.3

-

-

-

87.5

5~.3

~

o

-t

C. 2

0:33

0.67

-

-

2'5.9

~4.

-

-

70.3

95.5

3

0.25

0.50

0.75

-

14.~

25.0

37.5

-

77.1

97.8

4

0.20

~.4

0.6

0.9

9.3

16.0

24.!)

32.0

81.3

98.5

.,,1

til

3 ;::;:

::r ~

::J C.

Z

~

,

. TABLE 4

}.. - - -

-t ,

PERFOR~A!lCE

10

EXCLUOI~G

.:.-

~

;::-

TOP OUTRIGGER

~

.-......--

.'.~'='.~

- ._.. - -._ .. _-- .. - ..... --------------------_.---Drift reduction d6 as % of d6 fc Resisting moment as % of Mfc ...

....

o

..- - _.._ - - - . .

hr·

.....

N

OF EQUI-SPACED OUTRIGGER STRUCTURES

. ..- ..... -..

1.0

I I

2.0

1.0

10.0

2.0

10.0

66.6

....~ oc•

33.2

33.2

33.2

66.6

66.6

32.0

32.4

33.0

64.0

64.8

66.0

51.9

61.6

23.6

a

C

~

,..... :!.

26.1

47.0

30.8

= Core: column stiffness ratio

O"Q CJQ

....

(D

,.....

CIl

2 AE d

-I

~

t: ~

2EI

C

-I

B = Core:

outr;~qer stiffness ratio

E1

• d

{fI}o

IT

(t)

en

TABLE 5 INFLUE~CE OF SI~GLE OUTRIGGER FLEXIBILITY O~ PERFORMANCE (JJ

W VJ

CJ1

-+

OUTRIGGERS

~.

--...

I

-.

HAT GIRDER

I

-+

c.n ,... ~

COLUMNS

---h. ......

o ., c.

8.M.

,', . :i:


II

I

I

Fig. 2a--"Free"

Fig. 2b--B.M .

cantilever

diagram for a

_.

3,...,.

free cantilever

CORE

I

CIl

:r ~

:::J

C.

Z ~

~

~ ~

Fig. 2d--B.M. diagram for outrigger braced

Fig.2c--Outrigger braced cantilever

Fig. I--Jutrigger braced structure

cantilever 1..---

__..

~.,

1201t

fO 1 404

I

41~

I

•

•

z..-::'l...r:·......·

_'" ,.

tJ. *",lSj';O_.,.

,r.

OUTRIGGERS 2 - STOREYS

..

..

OUTRtGGER BENTS

DEEP

r+ A

FLOOR

....

+-

o (\J

~

c 2! •

•

+-

oc

'f-

o

...

.....
o

~

......

:::!•

eJ r
C1Q

()Q

<.0

4A 5

e

40 ft

11:

200 ft

I

... ,.... ...enc Q. c...

to

tt>

U')

Fig. 3b--Typical plan of first Wisconsin Centre Fig. 3a--Section AA of first Wisconsin Centre

~:.~;}~:~~Y:---=

(J1

~

. ~__·:.}/:;:~~::;~~~1!!~ri-~:.:

- - - - -

-

. - '.'Stafford Smith and Nwaka

536

HAT GIRD

---

,.-

43 rd FLOOR

;,0

-. OLUMNS

.

4

_.... _21· ..

""V~

...

'. L....:.-

I

I

~

'" ~

~-

.,

I A

L

--+--+-

,

I

1 --,

\

I

11

8

--+ --+ ~

...!L

.: "~

537

CORE

35

BELT GIROE

.- - Multi-Outrigger - - Structures

-

INFINITE NUMBER OF OUTRIGGERS

.A.C. WALLS

}~.,~=

FOUNDATION

~

;

-.

,(

A

.

•

Fig. 5a--Fully composite outrigger braced structure

Fig. 4a--Typ;cal section of Yasuda Building

FREE CANTILEVER

B. M. DIAGRAM

OUTRIGGERS ....,..----.

:/

I

~

E

;t

~

J

• • ~~l

I~

E Q

. ~"l~_' - -_ _~1_m_.

Fig. 4b--Plon of Y4sudo Uu11d1ny

.-,ty..m~

B

B.M. IN CORE B.M. CARRIED BY COLUMNS

B.M.

FlU. ~Jb--J\11()CilLI()1l of hendlnu lI1o\li(!nl 'n fully composlLe structure

JI

':',

":\J~\t~!\: :,1~!11 ' .

j,",t'.:

."

..

~

_

,._ . . . . . . . . . v .

~~-

-'~-

.~, ~~.~~~~~~~;~~:~:~~~~;~~~ _:·:Z-:i... _•. _-- ...._ ... ~.~:.:: .-:_":w:.:• •-..·.,..~ . '::~7~: ..

l

en ."

r t .-.

-.os co c· n

!

~.

~

~-. _~.~:::::~:~.-:-:-=-' ._~.: --.;~_:

~r 20

c-t" 0'\

c:s:u -.os.

-t ~

I Z

~

o I o

fllGl

C

(i)~

t+

-

J.

..

I

-1-0

(,Q (,Q

\0

rn

CD

-s

en c..

•• :::c (1)

-'" c

r-+

c:» ::s

~

CJ

3:

",,.~ ,." NI ~

I

X

L

:r:

:0

·le

N

:s:

~

...0..

o

en

~

r+ ....... ."

o:::::s ....... ....... u:l co ::J3 0...0 m

.

3:

-30"

-s c:»


....... :::::s

-

~

:r

\0

Q..

3:

:::s n

......

:=:

S

3

CIl

3

I

N

I

:::c:»

L

NI~N

NI~N

~

CO

-l> -i o

CDI

-r'1

(JJ ~

C PI

.~

c+::J

a.

o

Z

~

PI

0...

~

~

.- ....... m

N

:J C.

010

Ul I C-SI

0

'"1

r+CI::O

(1)

-s3C'D

....... c.n c.n \0 r+

co

c...-.os

-.os

CD-

3: N

C'DCPI VI

::s

I

I

0..-_..

••

\

~.- ------6

2

tat

_

TOP ,

\

-rR..

TOP

0.1661---0 \

..

O. Loading w/tnt : 0.1 ~ Height _

0.1

\0.2 \

C

0.2L

~.3

O5 '

0.3

FREE CANTLEVER

\

A

~

C

0.4

:x:

::x: +

0..

0

t-

~j ~

0 0:

LL.

OQ

0.8

--'I

cr. ,.... --t

C

A

0.8.

a

0.9

0.9

BASE 1.0

CD

0.7

Ij

0.7

(f)

I

o

,

,

02

,

!

0.4

t

!

0.6

>

A.

0.8

!

>t

LO

2

BENDING ·MOMENT IN C.ORE + wH /2

Fig. 7--Bending moment diagram for single-outrigger structure

:~~~;;,.:.;~:::-:-.

--'I QQ.

0.6

~

~

o

E

0.5

+

SINGLE OUTRIGGER MID-HEIGHT

0-

g

S oc• ,.....

BASE 1.0

c--.r

2EI

d..

to

AE d 2 = 1

(I)

c

I

o

I

!

02

I

I

0.4

!

!

0.6

!

•

,

08

I

LO

4

HOOZONTAL DISPLACEMENT + wH /8 El

Fig. 8--Deflection diagram for single-outrigger structure

CJ V.

c.c

.~~~~~il.I~~~:5='~Jjf~::'~~~'~i~~;~;~::~~

. ...._.:-::. . _".- _.:.. ~

~ _.-.-~'~. ~~~

~

-=_r~" •

..

.

~ ~

:~.,j.:,I~~"''''' .,,,

~

~

•

..

>'. ••

#

._

-,...

'"

.....

.r , -.

•

.. •

. #

'"

~~~~~~~~~~~~~~0~,~~~~~~~~~~~~~~~~~~h

TOP

0

TOP

I

2EI

do. = 0.1

\

TOP OUTRIGGER

8.M. DIAGRAM FOR

\

0.4 ..

0.6

~u

07

'"

,, ,

g

~-~ ,

if.X

3

(f)o-

'" 0.9',

I

!

o

,

Q2

,

~ 0.75 H

,

"" , ,

-

,

0.6

04

,

,

08

'"

I

BASE 1.0

LO 2

AE d 2

I

,

,

,

I

1

I

I

~

= ,

,

I

o 0.2 04 0.6 08 LO HORIZONTAL DISPLACEMENT e;. wH4;e EI

Fig. lO--Deflection diagram for triple-outrigger structure

for triple-outrigger structure

0.

d-.

c

Fig. 9--Bending moment diagram

......

2EI

I

BENDING MOMENT + w:H /2

....

&.1 ~

0.9

: "

.=1

HeicJht ::.

BOTTOM OUTRIGGER ...

11°·8

CANT1LE~lR

-

U>oding w IUit

0.7

:r Z

aM. DIAGRAM FREE

"

0.6

o

fE

::;:

FREE CANTILEVER

~

sanOM OUTRIGGER

0.8',

BASE 1.0

31

0.5

a..

~

~

-t

c. +

"

t·

I

....

::I: '

...

•

..

.. I~

n -n

0-

--'\0

c: •

5.Con

0 tv 0

.-

••

0

-0

C'+

3 c: 3

0

::r ro

0 W (J1

0

0

0

(J1

~.

h -

.I

YI!;.. ()

-.

\0

~

--i ~o C ::0 ..

::r

C

» -~ r G> (J1

C'+

0 -t)

--•

::;-

0

O(Jl

-t)

~O

-..II

~

--

).

CT

--"

CD

C ,...,.

::Oc C-f 0::0

C1Q

CG)

tn

::!.

(JQ

-f-

0

c

rt

-e til -e

::oG)

:l

,..

fTlfT1

to to CD

Ul~

t:

""1

--.-

n

0 ""1

1"'+

-.

t» X

~

--..

'<

-s

0:> 0

c:

GZ1

~

t'D

en

t .O

a-

~U

0 0

,,:',::::'.,-

I

I g:~ I

0.4

\

~

~

0.3

X2 t X3

MIDDLE OUTRIGGER X2 - O.SH

,.__ .1

+ 0.. o....

\ \

0.5

YII :. x·: It:

-_-

TRIPLE OUTRIGGER STRUCTURE -.~.,

\ \

lit

w J1Jnit Height

~\

\\

O.2~ ~m

r· Loodi"9

0.25 H

~

° 0.1

c

XI: .J ~

:I:

AE d 2 -

a

02"

0.3

_

I

CJ1

~ -4

d:-c!-=~·:'~:~:': '~~;";'~:~7~~::~~"'~]:;-' ;:;~.;;'.i~~~~~~~~~~.~::~~~~:~?-'~4-S't~~!!fii/fr~~-::t~?~~!.4#f3:~~~~f!l~~~~¥f:::'-;~"~

I

9 Shear lag effects in buildings 9.1 The shear lag phenomenon Shear lag is the lagging behind of bending stresses relative to that induced by pure bending action (as derived by the plane section remain plane assumption). It occurs whenever there is shear stress and is generally more significant for structures with relatively small height/width or length/width ratios. Shear lag leads to stress concentrations around the comers of the structures and additional deflection. Asa result, the overal structural efficiency of the structure would be reduced. Common structures whose structural efficiencies are affected by shear lag include framed-tube structures, shear/core wall structures and bridge deck structures. Shear lag occurs· in both the web and flange panels, but since the shear lag phenomenon is generally more significant in the flange panels, most researchers take into account only the shear lag effects in the flange.

41

9.2 Shear lag in framed-tube structures See attached paper.

9.3 Shear lag in shear/core wall structures See attached paper.

42

SHEAR LAG IN SHEAR/CORE WALLS

By A. K. H. Kwan l ABSTRACT: Shear lag occurs not only in bridge decks and framed tubes, but also in shear/core walls. However, there have been relatively few studies on shear lag in wall structures. Moreover, most existing theories neglect shear lag in the webs and, although they are acceptable for bridge decks that normally have flanges wider than webs, they may not be applicable to shear/core walls whose webs can be much wider than flanges. To study the shear lag phenomenon in wall structures, a parametric study using finite-element analysis is carried out. Unlike previous studies that neglected shear lag in the webs, many layers of elements are used for both the webs and flanges so that shear lag in the webs can also be taken into account The results indicate that the shape of the longitudinal stress distribution in an individual web or flange panel is quite independent of the dimensions of the other panels. Based on this observation, design charts and empirical formulas for estimating the shear lag effects are developed for practical applications.

INTRODUCTION

Shear Lag Phenomenon The Bernoulli-Euler assumption that plane sections remain plane after bending is often used for the analysis of beam structures. According to this assumption, the longitudinal stresses in the webs and flanges should be linearly and uniformly distributed, as in Fig. 1(a). However, this assumption is approximate and strictly applicable only when there is no shear force or when the structure has infinite shear stiffness. In actuality, when the structure is subjected to shear forces, a shear flow would be developed between the web and flange panels and, owing to shear deformations of the panels, the longitudinal displacements in the parts of the webs and flanges remote from the web-flange junctions would lag behind those at the junctions. As a result, the longitudinal stresses in the webs and flanges would become distributed as shown in Fig. l(b). Such "shear lag" phenomenon reduces the effectiveness of the webs and flanges and may significantly increase the longitudinal stresses at the web-flange junctions and the lateral deflections of the structure. Shear lag is most pronounced in beam structures with relatively wide webs and/or flanges such as bridge. decks and shear/core walls, and in beam structures with low shear stiffness such as framed tubes. However, although there have been a number of studies on the shear lag phenomenon in bridge decks and framed tubes, there were relatively few studies on shear lag in shear/core walls.

Existing Analysis Methods Existing methods for shear lag· analysis include (1) the folded-plate method; (2) the harmonic analysis method; (3) the finite stringer method; (4). the finite-element method; and (5) semiempirical methods. In the folded-plate method (DeFries-Skene and Scordelis 1964; Kristek 1979, 1983; Kristek and Skaloud 1991), the structure is treated as an assembly of plates interconnected at their longitudinal joints. The displacements and forces along the longitudinal joints are expressed as Fourier series of harmonic functions and by considering the plate bending and membrane actions of each individual plate, each tenn in the series for the joint forces is related to the corresponding term 1 Sr. Lect., Dept. of Civ.. and Struct. Engrg.y Univ. of Hong Kong, Pokfulum Rd., Hong Kong. Note. Associate Editor: ~c M. Lui. Discussion open until February -1, 1997. To extend· the clOSing date one month, a written request must be filed with the ASCE M~ager of Journals. The manuscript for this paper was submitted. ~or review and possible publication on November 22, 1994. This paper IS partor the Journal of Structural Engineering, Vol. 122, No.9, September, 1996. ~ASCE, ISSN 0733-9445/96/00091097-1104/$4.00 + $.50 per page. Paper No. 9648.

in the series for the joint. displacements by a stiffness matrix. Then, by also representing the external loads in the form of Fourier series, the entire analysis can be conducted separately for each term of the series and the final results can be obtained by summing the partial results. As for the folded-plate method, the hannonic analysis method (Abdel-Sayed 1969; Kristek 1983; Song and Scordelis 1990a,b; Kristek and Skaloud 1991) also represents the external loads as Fourier series. Unlike the former method, however, this method simplifies the analysis by. neglecting the outof-plane bending action of the individual plates and treating the web plates as simple bending elements so that the analysis can' be confined to the flange plates only. The finite-stringer method (Evans and Taherian 1977, 1980; Taherian and Evans 1977; Connor and Pouangare 1991) also treats the web plates as simple bending elements so that the analysis can be confined to the flange plates. Instead of using Fourier series solution method, it models the axial action of the flange plates by a finite number of stringers welded onto the plates, and the shearing action of the flange plates by the plates themselves that are assumed to take no axial loads. Thus, the axial and shearing actions of the flange plates are separated and the governing equations become easier to solve. Using only three stringers to model a flange plate, a simplified version, called the three-bar method, has been developed by Evans and Taherian (1977, 1980) for practical applications. The finite-element method, being the most powerful and versatile numerical method, can also be used to evaluate the shear-lag effects. Moffatt and Dowling (1975) had, by using the finite-element method, carried out a comprehensive para-

flange

web

Ca)

flange

web

(b)

FIG. 1. Axial Stress Distribution in Beam Structure: (a) with No Shear Lag; (b) with Shear Lag JOURNAL OF STRUCTURAL ENGINEER1NG./ SEPTEMBER 1996/1097

~,.-

metric study on shear lag in bridge decks.. It was found necessary to use fine mesh divisions over the width and length of the flange plates.. However, the web plates were assumed to behave in accordance with the elementary theory of ben~ing and, thus, each web plate was modeled by one layer of elements only. From the finite-element results, Moffatt and Dowling (1975) had produced a set of design values for the estimation of shear lag in bridge deck structures.. Apart from the previous method, there are also some semiempirical methods based on energy formulation (Coull and Bose 1975; Coull and Abu EI Magd 1980; Kwan 1994).. In these methods, various simplifying assumptions regarding the longitudinal stress distributions in the web and flange plates are made to render the analysis more tractable, and solutions are effected by minimizing either the strain energy or total potential energy of the structure.. They vary in their accuracy and are generally not as accurate as the rigorous methods.. Although shear lag can be accurately analyzed by many of the existing methods, such accurate analysis is generally very time consuming.. For practical applications, there is still the need for simple methods, which can allow quick estimation of the shear-lag effects without the use of computers, particularly during the preliminary design stage. There are, however, very few simplified methods for shear/core wall structures. Moreover, in most of the existing methods, the webs are assumed to act as simple bending elements and, as a result, any possible shear lag in the webs is effectively neglected. Although methods that neglected shear lag in webs are acceptable for bridge decks that nonnally have flanges wider than webs, they may not be applicable to shear/core walls whose webs can be much wider than flanges. It is, therefore, necessary to have a new method that is easy to use and yet capable of taking into account shear lag in both the webs and flanges for application to wall structures.

Present Study Herein, a parametric study on the shear-lag phenomenon in shear/core wall structures is carried out by using finite-element analysis. Unlike previous studies that neglected shear lag in the webs, many layers of elements are used for both the webs and flanges so that any shear lag in the webs can also be taken into account. Apart from point loads and unifonnly distributed loads, triangularly distributed loads, which are quite common in tall building structures, are also considered. The ultimate aim is to develop a simple method for estimating the effects of shear lag in both the webs and flanges of shear/core wall structures.

PARAMETRIC STUDY USING FINITE ELEMENT ANALYSIS

Finite-Element Analysis To study the shear lag phenomenon in wall structures, a parametric study is carried out by analyzing a number of core wall models using the finite-element method. The models are shown in Fig.. 2(a). Due to symmetry, only half of the core wall model is analyzed. Very fine element meshes of isoparametric eight-noded quadratic serendipity elements are used for both the web and flange panels, as illustrated in Fig. 2(b). Twenty layers of elements are used in each web or flange panel (since only half of each flange is included in the analysis for taking advantage of symmetry, 10 layers of elem~nts are used in each half flange). Along the height, there are 60 elements in each layer which are distributed in such a way that 40 of them are evenly distributed within the lower half while the other 20 are distributed within the upper half. The structural parameters studied are the flange width/web width ratio, the t

1098/ JOURNAL OF STRUCTURAL ENGINEERING / SEPTEMBER 1996

height/web width ratio, and the height/flange width ratio.. Altogether, 15 models are analyzed.. They are numbered from 1 to 15, and their (web width:flange width:height) ratios are given in the second column of Tables 1-3.. The flange width! web width ratios of the models range from 0.. 33 to 3.. 0, while the height/half web width and height/half flange width ratios both range from 3.. 33 to 40.. 0 . On the other hand, the web and flange panels are assumed to have the same and constant thickness along the height of the structure, and the Poisson ratio is taken to be 0.. 25 throughout. Three loading cases, namely point load at top, uniformly distributed load, and triangularly distributed load, are considered. The loads are applied laterally

20 ale.ents 1n upper

halt H

40 elements 1n lover half

Ca)

FIG. 2. Core Wall Model: (a) General Layout; (b) Half Model Analyzed by Finite-Element Method TABLE 1.

Finite-Element Results (Point Load at Top) Shear-lag Shear-lag coefficient coefficient

Model number (1)

2a:2b:H (2)

(3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3:1:5 3:1:10 3:1:20 2:1:5 2:1:10 2:1:20 1:1:5 1:1:10 1:1:20 1:2:5 1:2:10 1:2:20 1:3:5 1:3:10 1:3:20

0.386 0..209 0.. 102 0.301 0.151 0.. 065 0.. 175 0.075 0.035 0..202 0.094 0.. 042 0.222 0.. 107 0.049

TABLE 2. Model number

(1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


{4)

Stress factor at fixed end (5)

Deflection factor at free end (6)

0.. 313 0.163 0.073 0.290 0.150 0.069 0.267 0.139 0.064 0.427 0.253 0.. 128 0.. 538 0..345 0.. 191

1.299 1..139 1..060 1..255 1.116 1.049 1..221 1..101 1.044 1..432 1..215 1.098 1.657 1..331 1.. 159

1..017 1.006 1.002 1.015 1.005 1.. 002 1.014 1.004 1.002 1.071 1.018 1.004 1.. 162 1.043 1.010

~

Finite-Element Results (Uniformly Distributed Load) Stress Shear-lag Shear-lag coefficient coefficient factor at a. fixed end 2a:2b:H 13 (4) (5) (3) (2)

3:1:5 3:1:10 3:1:20 2:1:5 2:1:10 2:1:20 1:1:5 1:1:10 1:1:20 1:2:5 1:2:10 1:2:20 1:3:5 1:3:10 1:3:20

0.. 533 0..307 0.. 146 0..425 0.220 0..090 0.. 248 0.. 110 0.. 048 0.. 273 0.128 0.. 057 0.. 297 0.144 0.. 066

0.431 0.233 0.104· 0..415 0.. 216 0.100 0.384 0.202 0.096 0..588 0.370 0.. 191 0.707 0..489 0.. 281

1.470 1..217 1..089 1..412 1.179 1..072 1.355 1.156 1.068 1.. 724 1..353 1..155 2.. 100 1.552 1.256

Deflection factor at free end (6)

1..022 1.. 008 1.003 1.020 1..006 1..003 1..018 1..005 1.003 1.. 080 1.021 1..005 1.. 185 1.046 1.012

TABLE 3.

Finite-Element Results (Triangularly Distributed

Load) Model number (1 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Deflection factor at free end (6)

Stress Shear-lag She~r-Iag coefficient coefficient factor at 2a:2b:H fixed end a ~ (2) (5) (4) (3) 1.399 3:1:5 0.385 0.478 1.183 3:1:10 0.203 0.267 1.077 3:1:20 0.093 0.126 1.343 2:1:5 0.362 0.378 1.152 2:1:10 0.188 0.193 1.062 2:1:20 0.089 0.075 1.299 1:1:5 0.339 0.218 1.131 1:1:10 0.175 0.094 1.059 1:1:20 0.085 0.042 1.608 1:2:5 0.531 0.247 1.295 1:2:10 0.325 0.114 1.131 1:2:20 0.166 0.050 1.912 1:3:5 0.650 0.267 1.466 1:3:10 0.437 0.130 1.214 1:3:20 0.245 0.059

1.020 1.007 1.003 1.018 1.006 1.002 1.0.16 1.005 1.002 1.077 1.020 1.005 1.177 1.045 1.011

to the web panels in their in-plane directions. Twisting of the channel-shaped structure (half of the .core wall model) is prevented by restraining the free edges of the half flanges (line of· symmetry of the flange panels) from moving horizontally in the in-plane direction. The base of the model is assumed to be perfectly fixed.

Numerical Results The axial-stress distributions in the web and flange panels of a typical model, model 7, which has a flange width/web width ratio of 1.0 and height/half web width and height/half flange width ratios equal to 10.0, are shown in Fig. 3, from y 18014 Eqn. (2)

flange

-----....-x

flange

web

• r Inl te

FIG. 3.

elellant rewl t

Axial Stress Distribution at Base of Model 7

which it can be seen that significant shear lag occurs in both the web and flange panels. The degree of shear lag varies among the models and is dependent on the loading case. To allow detailed study of the shear lag phenomenon, it is proposed to measure shear lag in the web and flange panels in terms of the dimensionless shear lag coefficients, Ctand (3, whose definitions are depicted in Figs. 4 and 5, respectively. These shear-lag coefficients were first proposed by the writer and had been applied to shear lag analysis of framed tube structures in an earlier paper (Kwan 1994). From Fig. 4, which shows the distribution of axial stress across the web panel, it can be seen that when the degree of shear lag is small, the distribution of axial stress across the web panel is approximately linear..-;..-but when the degree of shear lag is large, the axial stress near the centroidal axis of the panel would significantly lag behind that given by a linear distribution leading to a reduction in the gradient of the stress distribution curve at the center in such a way that the larger the shear lag, the greater the reduction. Hence, the reduction in gradient of the stress-distribution curve at the center of the panel may be taken as a measure of the degree of shear lag in the web. The proposed shear-lag coefficient a is defined mathematically as the fractional reduction in· the gradient of the axial stress distribution curve at the center of the web compared to that of a straight line with zero stress at the center of the web and the same maximum stress at the web-flange junction (note that the straight line with zero stress at the center of the web and the same maximum stress at the web-flange junction is not the same as the stress distribution obtained without considering the shear-lag effects because negligence of shear lag would lead to a different value of maximum stress). Similarly, from Fig. 5 which shows the distribution of axial stress across the flange panel, it can be seen that when the degree of shear lag is small, the distribution of axial stress across the flange panel is approximately uniform-but when the degree of shear lag is large, the axial stress near the centroidal axis uf lhe:: panel would lag behind that given by a unifonn distribution leading to a significant reduction in stress at the·. center in such a way that the larger· the shear lag, the greater the reduction. Hence, the reduction in stress at the center of the panel may be taken as a measure of the degree of shear lag in the flange. The proposed shear-lag coefficient ~ is defined mathematically as the fractional reduction in the axial stress at the center of the flange compared to the maximum stress at the weh-flange junction (note that this maximum stress is the maximum stress obtained with shear-lag effects allowed for, not the maximum stress obtained without considering the shear-lag effects). These two shear lag coefficients are dimensionless and are therefore not dependent on the units used. When shear lag is small, the shear-lag coefficients would

e

webflange junction

e

webflange junction

I

I

I

I

I

}-

.

ct (0"

Iv

m

gradient reduced due to shear lag

_I~ } a ~ a

-,lI---L

,

Ca}

FIG. 4. a

(b)

~

a

/

a)

!

,

a

,

}-

~

/

L

'l

(e)

Axial Stress Distributions In Web Panels Illustrating Definition of a: (a) Small Shear Lag; (b) Large Shear Lag; (c) Definition of

JOURNAL OF STRUCTURAL ENGINEERING / SEPTEMBER 1996/1099

.GDiP?i

,

web~

I

flange Junction

b

b

~

I

~ I·

I·

~

stress reduced due to shear lag

J

b

Ca)

FIG. 5. afp

Junction

I

I

D:t::sJ I· J )

\leb-

flange

b

j

~

b

)

b

~

(e)

(b)

Axial Stress Distributions in Flange Panels Illustrating Definition of~: (a) Small Shear Lag; (b) Large Shear Lag; (c) Definition

be close to zero, and when shear lag is large, they would be close to unity. The finite-element results show that the values of Ct and f3 vary along the height of the wall structures and are generally greatest at the bases, Le., the fixed ends, of .the structures. Therefore, the shear lag effects are most critical at the bases. The values of a and 13 at the bases of the models, as determined from the finite-element results, are given in the third and fourth columns of Tables 1-3. From these results, it can be seen that ex is dependent mainly on the heightlhalf web width ratio, while 13 is dependent mainly on the heightlhalf flange width ratio. When the height-ta-width ratios of the panels are small, the values of a and f3 can be as large as or even larger than 0.5 and 0.7, respectively. Comparing the shear-lag coefficients under different load cases, it can also be seen that the importance of shear lag increases in the order of point load case, triangularly distributed load case, and uniformly. distributed load case. Regarding the axial-stress distributions in the web panels, most researchers neglected the shear lag in the webs and assumed a linear distribution of axial stresses across the width of the webs, except Coull and Bose (1975) and Kwan (1994), who considered shear lag in the webs and assumed that the deviation of the axial stresses in the webs from linear distributions can be expressed as third-order-polynomial functions. The present finite-element results, however, reveal that the deviation of the axial stresses in the webs from linear distributions may be more accurately represented by fifth-order polynomials. In Fig. 3, a fifth-order-polynomial curve, whose equation is given by

is plotted on the web panel alongside the finite-element results to demonstrate the close agreement between the axial stress distribution in the web panel and the fifth-order polynomial. Regarding the axial stress distributions in the flanges, the stress distributions across the width of the flanges were approximated as second-order-polynomial curves by Coull and Bose (1975), Coull and Abu EI Magd (1980), and Kwan (1994); as third-order-polynomial curves by Evans and Taherian (1977); and as fourth-order polynomial curves by Moffatt and Dowling (1975). Hence, there is no agreement between the different researchers on the stress distribution in the flange panels. The present finite-element results indicate that the stress distributions in the flange panels lie somewhere between a third-order polynomial and a fowth-order polynomial, but are generally closer to a fourth-order polynomial. A fourth-order polynomial curve, whose equation is given by

CT/(Y) =

el..

[(l - /3) + /3 (~r]

(2)

is plotted on the flange panel alongside the finite-element results in Fig. 3 to demonstrate how close the axial-stress distribution in the flange panel is to a fourth-order polynomial. Shear lag increases the axial stresses at the web-flange junctions and the lateral deflections of the structure. Such effects may be quantified in terms of a stress factor As and a deflection factor Ad as follows: A .r -

axial stress at web-flange junction with shear lag axial stress at web-flange junction without shear lag lateral deflection with shear lag

Ad

= lateral deflection without shear lag

(3)

4 ( )

Since the axial stresses are largest at the fixed ends while the lateral deflections are largest at the free ends, the stress factors at the fixed ends and the deflection factors at· the free ends are more important than those at other locations, and they are tabulated in the last two columns of Tables 1-3. From the tabulated results, it can be. seen that in the worst case of a short and wide wall structure subjected to uniformly distributed lateral loads, the stress factor at the fixed end can be larger than 2.0. The stress factor qecreases as the height of the structure increases, and when the height/half web width and heightJhalf flange width ratios are both greater than 40.0, the increase in bending stress due to shear lag becomes insignificant. On the other hand, the effects of shear lag on lateral deflections are generally much smaller. The increase in lateral deflection due to shear lag would become negligible when the heightlhalf flange width ratio is greater than 10.0.

Comparison with Others' Results The stress factors obtained from the present parametric study are compared to those obtained by Moffatt and Dowling's method (1975) and by Evans and ):aherian's method (1980) in Tables 4 and 5 for the load cases of point load at top and uniformly distributed load, respectively. There are no existing results for triangularly distributed loads and, thus, no similar comparison can be made for this load case. Moffatt and Dowling's (1975) results were obtained by finiteelement analysis, while Evans and Taherian's (1980) results were obtained by the three-bar method. In both Moffatt and Dowling's (1975) results and Evans and Taherian's (1980) results, the effects of any shear lag in the webs were neglected. Comparison of the present results with these" existing results reveals that when the flange width/web width ratio is equal to or greater than 2, there is very close agreement in the stress

1100 I JOURNAL OF STRUCTURAL ENGINEERING I SEPTEMBER 1996

.:~ .... ::,-..~ -'$J~~ .~ ~" ... 1....'/"

:- .•. ~:

.'P

·

TABLE 4. Top)

Comparison with Others' Results (Point Load at Stress Factor at Fixed End

(3)

Present analysis (4)

Moffatt and Dowling (1975) (5)

Evans and Taherian (1980) (6)

1/3 1/3 1/3 1/2 1/2 1/2 1 1 1 2 2 2 3 3 3

1.299 1.139 1.060 1.255 1.116 1.049 1.221 1.101 1.044 1.432 1.215 1.098 1.657 1.331 1.159

1.111 1.064 1.036 1.136 1.078 1.044 1.176 1.099 1.055 1.394 1.207 1.110 1.631 1.321 1.168

1.056 1.028 1.014 1.075 1.038 1.019 1.119 1.059 1.030 1.359 1.179 1.090 1.675 1.338 1.169

Model number (1)

2a:2b:H (2)

bla

1 2 3 4 5 6 7 8 9 10 11 12 13 . 14 15

3:1:5 3:1:10 3:1:20 2:1:5 2:1:10 2:1:20 1:1:5 1:1:10 1:1:20 1:2:5 1:2:10 1:2:20 1:3:5 1:3:10 1:3:20

TABLE 5. Comparison with Others' Results (Uniformly Distributed Load) Stress Factor at Fixed End Model number (1 )

2a:2b:H (2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3:1:5 3:1:10 3:1:20 2:1:5 2:1:10 2:1:20 1:1:5 1:1:10 1:1:20 1:2:5 1:2:10 1:2:20 1:3:5 1:3:10 1:3:20

bla (3)

Present analysis (4)

Moffat and Dowling (1975) (5)

Evans and Taherian (1980) (6)

1/3 1/3 1/3 1/2 1/2 1/2 1 1 1 2 2 2 3 3 3

1.470 1.217 1.089 1.412 1.179 1.072 1.355 1.156 1.068 1.724 1.353 1.155 2.100 1.552 1.256

1.190 1.099 1.058 1.238 1.121 1.071 1.316 1.156 1.090 1.699 1.378 1.182 2.092 1.562 1.290

1.106 1.054 1.028 1.143 1.073 1.037 1.228 1.116 1.059 1.674 1.348 1.177 2.249 1.650 1.331

factors. When the flange width/web width ratio is less than 2, however, the stress factors obtained by the present finite-elementanalysis are significantly higher than those obtained by Moffatt and Dowling's (1975) method or by Evans and Taherian's (1980) method. Such discrepancies may be attributed to· shear lag in the webs. From the differences in stress factors when the flange width/web width ratio is small, it can be seen that shear lag in the webs can contribute to a further increase in maximum bending stress of more than 20% when the webs· are short and wide.

SIMPLIFIED METHOD FOR PRACTICAL DESIGN APPLICATIONS Estimation of Shear Lag Coefficients

The .shear-jag .coefficients, a and 13, are plotted against Hla and Rib, respectively, for. the three loading cases in Figs. 68. From the plotted results, it can be clearly seen that a. is dependent mainly on Hla, while 13 is dependent mainly on HI b.The ratio bla does affect the values of ex and ~, but its effects are gener~lly. mi?or. As et and J3 define the shapes of the axial-stress. dls~butlons, it may be said that the shape of the shear lagged axIal stress distribution in an individual web

.._" .... _- .. ..... :-

....::...~.:_

.. -.

0.& , . . - - - - - - - - - - - - - - - - - - - - - -

•

a

bI.-IIJ

TABLE 6. Fixed End

0.6

Empirical Formulas for Shear Lag Coefficients at Shear lag coefficient

Load

• bfa-I

004

case

ex

(1 )

(2)

Point load at top

•

-

ex=

Prq:loI:cd

1DnnuIa

8

Uniformly distributed load Triangularly distributed load

0.2

Shear lag coefficient J) (3)

1.50 1.00

+ 0.76

1.25 (Hla)

1.59 ex= 1.00 + 0.54 (Hla) 1.56 ex= 1.00 + 0.62 (Hla)

J)

= 1.00 + 0.37

(Hlb)

1.31 J) = 1.00 + 0.24 (Hlb) 1.29 J) = 1.00 + 0.28 (Hlb)

O-+-------+-------+--------li---------f

o

10

20

30

40

Hla

0.1.,.-----------------------. P

•

bla-ll3

•

blplJ2

•

blpl

Point Load at Top

0.6

Model number (1)

a blP2 o bla=a3

0.4

-Prq:loI:cd formula

0.2

0-+-------+-------+----~1--------4

o

20

10

30

40

BIb

FIG. 8.

Values of a and

IJ for Triangularly Distributed Load

or flange panel is quite independent of the dimensions of the other panels. To simplify the estimation of the shear-lag coefficients, it is proposed to neglect the influence of bla on a. and ~, and take the values of ex and ~ as those corresponding to the case of bla = 1.0. Hence, the curves fitting the finite-element results for the case of bla = 1.0 may be taken as design charts for the estimation of ex and ~. Furthermore, to allow quick evaluation using hand-held calculators, formulas for these design charts are derived by empirically matching the finite-element results with different forms of equations. Good matching is found to be achieved with equations of the following forms: CI

a=---C2 + Hla

TABLE 7. Comparison of Stress Factors at Fixed End Obtained by Proposed Formulas with Those by Finite-Element Analysis

Finiteelement analysis (2)

1 2 3 4

1.299 1.139 1.060

5

6 7 8 9 10 11 12 13 14

1.116 1.049 1.221 1.101 1.044 1.432 1.215 1.098 1.657 1.331

15

1.159

1.2S5

Uniformly Distributed Load

Proposed formulas (3)

Finiteelement analysis (4)

Proposed formulas (5)

1.295 1.149 1.075 1.249 1.125 1.063 1.226 1.114 1.057 1.460 1.235 1.119 1.704 1.361 1.183

1.470 1.217 1.089 1.412 1.179 1.072 1.355 1.156 1.068 1.724 1.353 1.155 2.100 1.552 1.256

1.462 1.233 1.117 1.394 1.198 1.099 1.363 1.183 1.092 1.749 1.379 1.191 2.166 1.589 1.296

Triangularly Distributed Load Finiteelement analysis (6)

Proposed formulas (7)

1.399 1.183 1.077 1.343 1.152 1.062 1.299 1.131 1.059 1.608 1.295 1.131 1.912 1.466

1.214

1.390 1.197 1.099 1.332 1.167 1.084 1.306 1.154 1.078 1.630 1.319 1.161 1.971 1.494 1.250

for simplicity, that (j w and CTf can be approximated by (1) and (2). Substitution into (7) and then integration yields 0'",[(4/3)t",a2 (1 - 4na.)

+

4tf ab(1 - 4/5~)] = M

(8)

From this equation, the maximum bending stress is obtained as Ma

(9)

where I w and I, are given by I w = (4/3)tw a 3

(5)

(10)

(11) (6)

where Cl to C4 = unknown coefficients to be determined. The empirical formulas so derived are tabulated in Table 6, and plotted in Figs. 6-8 to demonstrate their close agreement with the finite-element results.

and are actually the moment of inertia of the webs and flanges when there is no shear lag. The effects of shear lag are now clear. Shear lag in the webs causes reduction of the effective moment of inertia of the webs to 1 - O.57a. times the original value, while shear lag in the flanges causes reduction of the effective moment of inertia of the flanges by a factor of 1 0.80(3. From (9), the stress factor is evaluated as

Estimation of Stress Factors The bending stresses are governed by the following moment-equilibrium equation:

f:a

2tw CTw X dx

+

f:'

2r,CT,a dy = M

(7)

in which t w and ~ = thickness of web and flange panels; and M = bending moment acting on section. It may be assumed, 1102/ JOURNAL OF STRUCTURAL ENGINEERING / SEPTEMBER 1996

As

[w + I, =-------~---[w(1 - 0.57a) + [,(I - 0.80~)

(12)

The stress factors so evaluated by using the empirical formulas given in Table 6 and the prior equation are compared with those obtained by finite-element analysis in Table 7. It is seen that within the range of parameters studied, the stress factors evaluated by the proposed fonnulas differ by, at most, a few percent from those by finite-element analysis. Hence, the pro-

-

;- ;;

~_

.

.

20

posed formulas are considered sufficiently accurate for practical applications.

fa

tf

10

ANALYSES

III

It

5

n

III

180 JcH/1i

n

-load

(b) II

3.62 HPa

i

I

1

90 leN/,.

3.62 HPa

\leb {lanalt

Ca)

(e)

FIG. 9. Example: (a) Core Wall SUbjected to Wind Load; (b) Cross Section; (e) Axial Stress at Base As Obtained by Finite-Element Analysis

high accuracy, the variation of thickness with height may be neglected in the evaluation of the degree of shear lag at the base. The value of Poisson ratio used in the analysis would also affect the numerical results for the shear-lag effects. There is, however, the problem of what Poisson ratio should be used in the analysis. Concrete has a Poisson ratio under static load of 0.15-0.20, but an average Poisson ratio under dynamic load of about 0.24 (Neville 1981). The presence of reinforcement affects the Poisson ratio, too. Although reinforced concrete is not homogeneous, it is often treated as a homogeneous material with equivalent properties in order to simplify the structural analysis. This can be done by smearing the reinforcement across the concrete section and taking the effective Young's modulus E of the reinforced concrete as E = (1 - p)Ec

+

pE.r

(13)

where Ec and E.r = Young's moduli of concrete and steel, respectively; and p = reinforcement ratio. However, since reinforced concrete walls can be deformed in shear without straining the horizontal and vertical reinforcement, the effective shear modulus G should be unaffected and remain the same as that of plain' concrete. For compatibility, the effective Pois.. son ratio of the reinforced concrete needs to be taken as that evaluated' by 2(1+ v) = EIG

Limits of Applications In the present study, homogeneous, isotropic walls of constant thickness and a Poisson ratio of 0.25 have been assumed. Because of the assumption made regarding homogeneity, the proposed method should not be applied to precast concretepanel structures whose joints, as planes of weakness, could render the structural behavior quite different from that of homogeneous walls. Theoretically, anisotropy in reinforced concrete walls due to differences in horizontal and vertical reinforcement could affect the stress distributions. But, as the lateral stresses in the walis are normally quite small compared to the longitudinal stresses, it is unlikely that the Young's modulus in the lateral direction would significantly affect the overall. structural behavior. Thus, the effects of anisotropy may be neglected and the present results should be applicable to reinforced concrete walls albeit they may not be entirely isotropic. Strictly speaking, variation of wall thickness with height may affect the shear-lag .phenomenon. Nevertheless, since shear lag is basically a local-stress-concentration .problem at the lower part of the wall structure, it· is anticipated' that the wall thickness at. the upper part of the structure would not significantly affect the shear lag at the base where it is most critical. Therefore, for preliminary designs that do not require

:: thickness ft or vall. :: - 0.3 III

ft Of

If

50

In actual practice, the applied lateral loads are never as simple as the three loading cases considered. Nevertheless, for the purpose of estimating the shear-lag coefficients, the actual loading case may be approximated, by exercising engineering judgment, as one of the three loading cases studied. An example of a core wall subjected to wind loads is given in Fig. 9. In this case, the applied loads consist of several uniformly distributed loads of increasingly larger intensity at greater height. Although the loads are not distributed as in any of the three loading cases studied, the load distribution is treated, for the purpose of estimating a. and ~, as a triangular distribution. Using the proposed formulas for the triangularly distributed load case, the values of a. and (3 at the base of the core wall are evaluated as 0.380 and 0.339, respectively. The bending moment acting at the base of the core wall is then calculated from the actual load distribution as 264.4 MNm. Finally, the maximum bending stress at the base is obtained by using (9) as 3.52 MPa. For comparison, finite-element analysis of the core wall using the actual load distribution is carried out and the bending stress results are shown in Fig. 9(c). The maximum bending stress obtained by finite-element analysis is 3.62 MPa, which is very close to that evaluated previously by hand calculation. Moffatt and Dowling (1975) suggested that for bridges, if the applied loads are type HA or HB vehicle loads, as defined in British Standard BS5400 ("Steel" 1978), for the .purpose of estimating the reduced effective widths of the flanges due to shear lag, a uniformly distributed load case may be assumed. For buildings, a' similar assumption may be made. If the applied loads are wind loads whose distribution is similar to that in the previous example, then a triangularly distributed load case may be assumed in the evaluation of the shear-lag coefficients. When the load distribution lies between unifonnand triangular distributions, then the mean value of the shearlag coefficients for uniformly and triangularly distributed load cases may be used.

III

U- l o a d

5IU

270 kN/1ll

Load Case

20

I

(14)

from which a higher Poisson ratio than that of plain .concrete would be obtained. For typical walls cast of concrete having a Poisson ratio of 0.15-0.20 and with 0.3-1.2% reinforcement provided, the effective Poisson ratio ranges from 0.18 to 0.30. Under dynamic load, the Poisson ratio would be slightly higher. In actual engineering practice, however, it is not really necessary to be so precise in the evaluation of the Poisson ratio. For most applications, it should be sufficiently accurate to just use an average value of 0.25 regardless of the amount of reinforcement provided, whether the load is static or dynamic.

CONCLUSIONS A parametric study of the shear-lag phenomenon in shearl core wall structures has been carried out by analyzing a number of core wall models with.the finite.;.element method. In the study, shear lag in allthe web and flange panels is taken into account, and the load cases considered include point. load at top, unifonnly distributed load, and triangularly distributed loads. The numerical results showed that (1) the degree of shear lag in a cantilevered wall structure varies along the height and is generally greatest at the fixed end; (2) the. axial stress disJOURNAL OF STRUCTURAL ENGINEERING I SEPTEMBER 1996/1103

tributions across the widths of the web and flange panels can be described approximately by polynomials of fifth and fourth order, respectively; (3) the importance of shear lag increases in the order of point load case, triangularly distributed load case, and uniformly distributed load case; and (4) the effects of shear lag in the web panels can be quite significant when the web panels are relatively short and wide, and hence, it should be prudent to also take into account the effects of any shear lag in the webs. Detailed analysis of the shear-lag phenomenon revealed that the degree of shear lag in an individual web or flange panel, measured in tenns of the dimensionless shear-lag coefficients a or J3, is dependent mainly on the height/width ratio of the panel. Plotting the shear-lag c;oefficients against the heightJ width ratios of the panels and matching the numerical results with empirical equations of different forms, design charts and empirical formulas for estimating the shear-lag coefficients are produced. A simple equation for evaluating the increase in maximum-bending stress due to shear lag in both the web and flange panels is also derived. Comparison with the finite-element. results confirmed that the proposed formulas are sufficiently accurate for practical applications. A numerical example has also been presented to demonstrate the ease of application of the proposed formulas.

APPENDIX I.

REFERENCES

Abdel-Sayed, G. (1969). ""Effective width of steel deck-plate in bridges." J. Struct. Div., ASCE, 95(7), 1459-1474. Connor, J. J., and Pouangare, C. C. (1991). ""Simple model for design of framed-tube structures." J. Struct. Engrg., ASCE, 117(12), 3623-

3644. Coull, A., and Abu EI Magd, S. A. (1980). Analysis of wide-flanged shearwall structures." Reinforced concrete structures subjected to wind and earthquake forces, ACI Spec. Publ. 63, Paper No. SP63-23, Concrete Institute, Detroit, Mich., 575 -607. Coull, A., and Bose, B. (1975). "'Simplified analysis of frame-tube structures." J. Struct. Div., ASCE, 101(11), 2223-2240. DeFries-Skene, A., and Scordelis, A. C. (1964). "'Direct stiffness solution for folded plates." J. Struct. Div., ASCE, Vol. 90(4), 15-48. Evans, H. R., and Taherian, A. R. (1977). '''The prediction of shear lag effect in box girder." froc., Ins tn. Civ. Engrs., Part 2, 63, Thomas Telford Services Ltd., London, 69-92. Evans, H. R., and Taherian, A. R. (1980). ""A design aid for shear lag calculations." Proc., Instn. Civ. Engrs., Part 2, 69, Thomas Telford Services Ltd., London, 403 -424. Kristek, V. (1979). ""Folded plate approach to analysis of shear wall systems and frame structures." Proc., Instn. Civ. Engrs., Part 2, 67, Thomas Telford Services Ltd., London, 1065 -1075. Kristek, V. (1983). "Chapter 6: Shear lag in box girders." Plated U

1104 / JOURNAL OF STRUCTURAL ENGINEERING / SEPTEMBER 1996

structures-stability and strength, R. Narayanan, ed., Applied Science Publishers, London and New York, N.Y., 165-194. Kristek. V. and Skaloud, M. (1991). Advanced analysis and design of plated structures-Developments in Civil Engineering 32. Elsevier Science Publishing Co. Inc., New York., N.Y. Kwan, A. K. H. (1994). "Simple method for approximate analysis of framed tube structures. u J. Struct. Engrg., ASCE, 120(4), 1221-1239. Moffa~ K. R., and Dowling. P. J. (1975). "Shear lag in steel box girder bridges." Struet. Engrg., 53(10), 439-448. Neville, A. M. (1981). "Chapter 6: Elasticity. shrinkage and creep." Properties of concrete. Longman Scientific and Tech. Ser., Longman Group U.K. Ltd.• Essex, England, 359-432. Song, Q., and Scordelis, A. C. (1990a). hFonnulas for shear-lag effect of T-, 1- and box beams." J. Struct. Engrg., ASCE, 116(5), 13061318. Song. Q., and Scordelis A. C. (1990b). "'Shear lag analysis of T-, 1- and box beams:' J. Struct. Engrg., 116(5), 1290-1305. "Steel, Concrete and Composite Bridges." (1978). BS5400, British Standards Instn., London. Taherian, A. R., and Evans, H. R. (1977). uThe bar simulation method for the calculation of shear lag in multi-cell and continuous box girders." Proc., lnsm. Civ. Engrs., Part 2, 63 Thomas Telford Services Ltd., London, 881-897. 9

9

9

APPENDIX II.

NOTATION

The following symbols are used in this paper: a = half width of web panel; b = half width of flange panel; C, = coefficients; E = effective Young's modulus; Ec = Young's modulus of plain concrete; E s = Young's .modulus of steel reinforcement; G = shear modulus of concrete; H = height of wall; IJ = moment of inertia of flange; I w = moment of inertia of web; M = bending moment; ~ = thickness of flange; tw = thickness of web; a = shear lag coefficient of web panel; ~ = shear lag coefficient of flange panel; Ad = deflection factor as defined by (4); AI = stress factor as defined by (3); v = Poisson's ratio; p = reinforcement ratio; a', = axial stress in flange panel; am = axial stress at web-flange junction (maximum stress in section); and a'w = axial stress in web panel.