5 structural walls pdf

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5 structural walls

The usefulness of structural walls in the framing of buildings has long been recognized. When walls are situated in advantageous positions in a building, thcy can form an efficient lateral-force-resisting system, while simultaneously fulfilling other functional requirements. For buildings up to 20 stories the use of structural walls is often a matter of choice. For buildings over 30 stories, structural walls may become imperative from the point of vicw of cconorny and control of lateral deflection [B4].

Becausc a large fraction of, if not the entire, lateral force on the building and the horizontal shear force resulting from it is often assigned to such structural elements, they have been called shear walls. The name is unfortu- nate, for it implies that shear might control their behavior. This need not be so. It was postulated in previous chapters that with few exceptions, an attempt should be made to inhibit inelastic shear modes of deformations in reinforced concrete structures subjected to scisrnic forces. It is shown in subsequent sections how this can also be achieved readily in walled structures. To avoid this unjustified connotation of shear, the term structural walls will be used in preference to shear walls in this book.

The basic criteria that the designer will aim to satisfy are those discussed in Chapter 1 (i.e., stiffness, strength, and ductility). Structural walls provide a nearly optimum means of achieving these objectives. Buildings braced by structural walls are invariably stiffer than framed structures, reducing the possibility of excessive deformations under small earthquakes. It will thus oftcn bc unnecessary to separate the nonstructural components from the lateral-force-resisting structural system. The necessary strength to avoid structural damage under moderate earthquakes can be achieved by properly detailed longitudinal and transverse reinforcement, and provided that special detailing measures are adopted, dependable ductile response can be achieved under major earthquakes.

The view that structural walls are inherently brittle is still held in many countries as a consequence of shear failurc in poorly detailed walls. For this reason some codes require buildings with structural walls to be designed for lower ductility factors than frames. A major aim of this chapter is to show that the principles of the inelastic seismic bchavior of reinforced concrcte components developed for frames are generally also applicable to structural

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1

STRUCTURAL WALL SYSTEM 363

walls and that it is relatively easy to dissipate seismic energy in a stable manner [P3Y, P42). Naturally, because of the significant differences in geometric configurations in structural walls, both in elevation and sections, some modifications in the detailing of the reinforcement will be required.

In studying various features of inelastic response of structural walls and subsequently in developing a rational procedure for their dcsign, a number of fundamental assumptions are made:

1. In all cases studied in this chapter, structural walls will bc assumcd to possess adequate foundations that can transmit actions from the superstruc- ture into the ground without allowing the walls to rock. Elastic and inelastic deformations that may occur in the foundation structure or the supporting ground will not be considered in this chapter. Some scismic features of foundations, including rocking, are, however, revicwed in Chapters 6 and 9.

2. The foundation of one of several interacting structural walls docs not affect its own stiffness relative to the other walls.

3. Inertia forces at each floor are introduced to structural walls by diaphragm action of the floor system and by adequate connections to the diaphragm. In terms of in-plane forces, floor systems (diaphragms) arc assumed to remain elastic at all times.

4. The entire lateral force is resisted by structural walls. The interaction of frames with structural walls is, however, considered in Chapter 6.

5. Walls considered here are generally deemed to offer resistance inde- pendently with respect to the two major axes of the section only. It is to be recognized, however, that under skew earthquake attack, wall sections with flanges will be subjected to biaxial bending. Suitable analysis programs to evaluate the strength of articulated wall seclions subjected to biaxial bending and axial force, are available. They should be employed whcncvcr parts of articulated wall sections under biaxial seismic attack may be subjected t o significantly larger compression strains than during independent orthogonal actions.

5.2 STRUCTURAL WALL SYSTEM

To facilitate the separation of various problems that arise with the design of structural walls, it is convenient to establish a classification in terms of geometric configurations.

5.2.1 Strategies in the Location of Structural Walls

Individual walls may be subjected to axial, translational, and torsional displacements. The extent to which a wail will contribute to the rcsistance of overturning moments, story shear forccs, and story torsion dcpencls on ils

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364 STRUCTURAL WALLS

geometric configuration, orientaGon, and location within the plane of the building. The positions of the structural walls within a building are usually dictated by functional requirements. These may or may not suit structural planning. The purpose of a building and the consequent allocation of floor space may dictate arrangements of walls that can often be readily utilized for lateral force resistance. Building sites, architectural interests, or clients' desires may lead, on the other hand, to positions of walls that are undesirable from a structural point of view. In this context it should be appreciated that while it is relatively easy to accommodate any kind of wall arrangement to resist wind forces, it is much more difficult to ensure satisfactory overall building response to large earthquakes when wall locations deviatc considerably from thosc dictated by seismic considerations. The difference in concern arises from the fact that in the case of wind, a fully elastic response is expected, while during large earthquakc demands, inelastic dcformations will arise.

In collaborating with architects, however, structural designers will often be in the position to advise as to the most desirable locations for structural walls, in order to optimize seismic resistance. The major structural considerations for individual structural walls will be aspects of symmetry in stiffness, torsional stability, and available overturning capacity of the foundations. The key in the stratcgy of planning for structural walls is the desire that inelastic deformations be distributed reasonably uniformly over the whole plan of the building rather than being allowed to concentrate in only a few walls. The latter case leads to the underutilization of some walls, while others might be subjected to excessive ductility demands.

When a permanent and identical or similar subdivision of floor areas in all stories is required, as in the case of hotel construction or apartment buildings, numerous structural walls can be utilizcd not only for lateral force resistance but also to carry gravity loads. Typical arrangements of such walls are shown in Fig. 5.1. In the north-south direction the lateral force per wall will be small as a result of a large number of walls. Often, code-specified minimum levels of reinforcement in the walls will be adequate to ensure elastic response even to large earthquakes. Behavior in the east-west direo tion of the structure in Fig. 5.l(a) will be more critical, because of reduced wall area and the large number of doors to be provided.

Fig. 5.1 Typical wall arrangements in hotels and apartmcnt buildings.

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STRUCTURAL WALL SYSTEM 365

Fig. 5 2 Examplcs for the torsional stability of wall systems.

Numerous walls with small length, because of door openings, shown in Fig. 5.l(b), will supplement the large strength of the end walls during scismic attack in the north-south direction. Lateral forces in the east-west direction will be resisted by the two central walls which are connected to the end walls to form a T section [Fig. 5.l(b)]. The dominance of earthquake effects on walls can be conveniently expressed by the ratio of the sum of the sectional areas of all walls effective in one of the principal directions to the total floor area.

Apart from the large number of walls, the suitability of the systems shown in Fig. 5.1 stems from the positions of the centers of mass and rigidity being close together or coincident [Section 1.2.3(b)]. This results in small static eccentricity. In assessing the torsional stability of wall systems, the arrangement of the walls, as well as the flexural and torsional stiffness of individual walls, needs to be considered. It is evident that while the stiffness of the interior walls shown in Fig. 5.1(6) is considerable for north-south seismic action, they are extremely flexible with respect to forces in the east-west direction. For this reason their contribution to the resistance of forccs acting in the east-west direction can be neglected.

The torsional stability of wall systems can be examined with the aid of Fig.

5.2. Many structural walls are open thin-walled sections with small torsional rigidities. Hence in seismic design it is customary to neglect the torsional resistance of individual walls. Tubular sections are exccptions. It is seen that torsional resistance of the wall arrangements of Fig. 5.2(u) (6) and ( c ) could only be achieved if the lateral force resistance of each wall with rcspect to its weak axis was significant. As this is not the case, these examples represent torsionally unstable systems. In thc case of the arrangement in Fig. 5.2(a) and (c), computations may show no eccentricity of inertia forces. However, these systems will not accommodate torsion, due to other causes described in Section 1.2.3(b) and quantified in Section 2.4.3(g), collectively refcrred to as accidental torsion.

Figurc 5.2(d) to (f) show torsionally stable configurations. Even in the case of the arrangement in Fig. 5.2(d), where significant eccentricity is

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Fig. 5.3 Torsional stability of inelastic wall systems.

present under east-west lateral force, torsional resistance can be efficiently provided by the actions induced in the plane of the short walls. However, eccentric systems, such as represented by Fig. 5.2(d) and (f ), are particular examples that should not be favored in ductile earthquake-resisting buildings unless additional lateral-force-resisting systems, such as ductile frames, are also present.

T o illustrate thc torsional stability of inelastic wall systems, thc arrangements shown in Fig. 5.3 may be examined. The horizontal force, H, in the long direction can be resisted efficiently in both systems. In the case of Fig.

5.3(a) the eccentricity, if any, will be small, and the elements in the short direction can provide the torsional resistance even though the flange of the T section may well be subject to inelastic strains duc to thc scismic shcar H.

Under earthquake attack E in the short direction, the structure in Fig.

5.3(a) is apparently stable, despite the significant eccentricity between the center of mass (CM) and center of rigidity (CR), defined in Section 1.2.3(b) and shown in Fig. 1.12. However, n o mattcr how carcfully the strengths of the two walls parallel to E are computed, it will be virtually impossible t o ensure that both walls reach yield simultaneously, because of inevitable uncertainties of mass and stiffness distributions. If one wall, say that a t B, reaches yield first, its incremental stiffness will rcducc to zero, causing excessive floor rotations as shown. There are n o walls in the direction transverse to E (i.e., the long direction) to offer resistance against this rotation, and hence the structure is torsionally unstable.

In contrast, if one of the two walls parallel to E in Fig. 5.3(6) yields first, as is again probable, the walls in the long dircction, which remain elastic under action E, stabilizc the tendency for uncontrolled rotation by devclop- ing in-plane shears, and the structure is hence torsionally stable.

Elevator shafts and stair wclls lcnd thcmselvcs to thc formation of a reinforccd concrete core. Traditionally, these have been used to provide the major component of lateral force resistance in multistory office buildings.

Additional resistance may bc derivcd, if necessary, from perimeter frames as shown in Fig. 5.4(a). Such a centrally positioned large corc may also provide sufficient torsional resistance.

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( 0 )

Fig. 5.4 Lateral force resislancc provided ^byreinforced concrclc cores.

When building sitcs a r c small, it is oftcn nccessary to accommodatc tlic core close to one of the boundaries. Howcvcr, eccentrically placcd scrvicc cores, such as seen in Fig. 5.4(b) lead t o gross torsional imbalance. It would be preferable to provide torsional balance with additional walls along the other three sides of the building. Note that providing onc wall only o n the long sidc oppositc tlic corc for torsional balancc 1s inadcqualc, lor rcasons discussed in relation to the wall arrangement in Fig. 5.3(a). If it is not possible to provide such torsional balance, it might be more prudent to eliminate concrete structural walls either functionally or physically and to rcly for latcral forcc rcsistancc on a torsionally balanced ductile framing system. In such cases the service shaft can be constructed with nonstructural materials, carefully separated from the frame so as to protect it against damage during inelastic response of the frame.

For better allocation of space or for visual cffccts, walls may be arranged in nonrectilinear, circular, elliptic, star-shaped, radiating, or curvilincar patterns [S14]. While the allocation of lateral forces to elcments of such a complex system of structural walls may require special treatment, the undcr- lying principles of the seismic design strategy, particularly thosc relevant t o torsional balance, remain the same as those outlined above for the simple rectilinear example wall systems.

In choosing suitable locations for lateral-force-resisting structural walls, three additional aspects shoutd be considercd:

1. For the best torsional resistance, as many of the walls as possiblc should be located a t the periphery of the building. Such an example is shown in Fig. 5.50(a). The walls on cach side may be individual cantilevers or they may be coupled to cach othcr.

2. The more gravity load can be routed to the foundations via a structural wall, the less will be the demand for flexural reinforcement in that wall and the morc readily can foundations be provided to absorb the overturning moments generated in that wall.

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\

368 STRUCTURAL WALLS

Fig. 5.5 Common scctions of structural walls.

3. In multisto~y buildings situated in high-seismic-risk areas, a concentra- tion of the total lateral force resistance in only one or two structural walls is likely to introduce very large forces to the foundation structure, so that special enlarged foundations may be required.

5.2.2 Sectional Shapes

Individual structural walls of a group may havc dilferent scctions. Some typical shapes are shown in Fig. 5.5. The thickness of such walls is often determined by code requirements for minima to ensure workability of wet concrete or to satisfy fire ratings. When earthquake forces are significant, shear strength and stability requirements, to be examined subsequently in detail, may necessitate an increase in thickness.

Boundary elements, such as shown in Fig. 5.5(6) to ( d l , are often present to allow effective anchorage of transverse beams. Even without beams, they are often provided t o accommodate the principal flexural reinforcement, to provide sPability againsl laleral buckling of a thin-walled section and, if necessary, to enable more effective confinement of the compressed concrete in potential plastic hinges.

Walls meeting each other at right angles will give rise t o flanged sections.

Such walls are normally required to resist earthquake forces in both principal directions of the building. They often possess great potential strength. It will be shown subsequently that when flanges are in compression, walls can exhibit large ductility, but that T- and Lsection walls, such as shown in Fig.

5 3 e ) and ( g ) , may have only limited ductility when the flange is in tcnsion.

Some flanges consist of long transverse walls, such as shown in Fig. 5.5(h) and ^{( j ) .}T h e designer must then decide how much of the width of such wide flangcs should be considercd to be effective. Code provisions [All for the effective width of conlpression flanges of T and L beams may be considered to be relevant for the determination of dependable strength, with thc span of the equivalent beam being taken as twice the height of thc cantilever wall.

As in the case of beams of ductile multistory frames, it will also be necessary to determine the flexural ovcrstrcngth of the critical scction of ductile structural walls. In flanged walls this ovcrstrength will bc governed primarily by the amount of tension reinforcement that will be mobilized

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\ S T R U F U R A L WALL SYSTEM 369

,Fig. 5.6 Estimation of effective flangc widths in structural walls.

during a large inelastic seismic displacement. Thus some judgment is required in evaluating the effective width of the tension flange. The width assumed for the compression flange will havc ncgligible effect on the estimate of flexural overstrength.

A suggested approximation for the effective width in wide-flanged structural walls is shown in Fig. 5.6. This is based on the assumption that vertical forces due to shcar stresses introduced by the web of the wall into the tension flange, spread out at a slope of 1 : 2 (i.e., 26.6" ). Accordingly, with the notation of Fig. 5.6, the effective width of the tension flange is

For the purpose of the estimation of flexural overstrength, the assumption above is still likely to be unconservative. Tcsts on T-section masonry walls [P29] showed that tension bars within a spread of as much as 45" were mobilized.

As stated earlier, the flexural strength of wall sections with the flange in compression is insensitive with respect to the assumed effective width. It should be noted, however, that after significant tension yield excursion in the flange, the compression contact area becomes rather small after load rever- sal, with outer bars toward the tips of the flange still in tensile strain. It may be assumed that the effective width in compression is

b,, = 0.3hW

+

b, I b ( 5 . l b )

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\

3'10 STRUCTURAL WALLS

The approximations abwe represent a compromise, for it is not possible to determine uniquely the effective width of wide flanges in the inelastic state.

The larger the rotations in the plastic hinge region of the flanged wall, the largcr the width that will bc mobilized to develop tension. The foundation system must be examined to ensure that the flange forces assumed can, in fact, be transmitted at the wall base.

I

5.2.3 Variations in Elevation

In medium-sized buildings, particularly apartment blocks, thc cross section of a wall, such as shown in Fig. 5.5, will not change with height. This will be the case of simple prismatic walls. The strenglh demand due to lateral forces reduces in upper stories of tall buildings, however. Hence wall sizes, particularly wall thickness, may then be correspondingly reduced.

More often than not, walls will have openings either in the web or the flange part of the section. Some judgment is required to assess whether such openings are small, so that they can be neglected in design computations, or large enough to affect either shear or flexural strength. In the latter case due allowance must be made in both strength evaluation and detailing of the reinforcement. It is convenient to examine separately solid cantilever structural walls and those that are pierced with openings in some pattern.

(a) Cantilever Walls Without Openings Most cantilever walls, such as shown in Fig. 5.7(a), can be treated as otdinary reinforced concrete bcam-columns.

Lateral forces are introduced by means of a series of point loads through the floors acting as diaphragms. The floor slab will also stabilize the wall against lateral buckling, and this allows relatively thin wall sections, such as shown in

-

-C

- -

-C

S

*

_C

-

C

*

( a ) ( 6 1

Fig. 5.7 Cantilever structural walls.

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\ STRUCTUKAL WALL SYS'I'EM 371

Fig. 5.5, to be used. In such walls it is relatively easy to ensure that when required, a plastic hinge at thc basc can dcvclop with adequate plastic rotational capacity.

In low-rise buildings or in the lower stories of medium- to high-rise buildings, walls of the type shown in Fig. 5.7(b) may be used. These are characterized by a small height-to-length ratio, hw/l,. The potential flcxural strength of such walls may be very large in wmparison with the lateral forces, even when wde-specified minimum amounts of vertical reinforcement are used. Because of the small height, relatively large shearing forces must be generated to develop the flcxural strcngth at the base. Therefore, the inelastic behavior of such walls is often strongly affected by effects of shear.

In Section 5.7 it will be shown that it is possible to ensure inelastic flexural response. Energy dissipation, however, may be diminished by effects of shear.

Therefore, it is advisablc to design such squat walls for larger lateral forcc resistance in order to reduce ductility demands.

To allow for the effects of squatness, it has been suggested [X3] that the lateral design force specified for ordinary structural walls bc increased by the factor Z,, where

It is seen that this is applicable when the ratio h w / l w < 3. In most situations it is found that this requirement does not represent a penalty because of the great inherent flexural strength of such walls.

While the length of wall section and the width of the flanges are typically constant over the height of the building, the thickness of the wall [Fig.

5.8(a)], including sometimes both the web and the flanges [Fig. 5.8( f

11,

may be reduced in the u m e r stories. The reduction of stiffness needs to be taken A A

into account when the interaction of several such walls, to be discussed in Section 5.3.2(a), is being evaluated. More drasiic changes in stiffness occur when the length of cantilever walls is changed, either stepwise or gradually, as seen in Fig. 5.8(b)'to ( e ) . Tapered walls, such as shown in Fig. 5.8(d), are

Fig. 5.8 Nonprismatic cantilever walls.

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structurally efficient. However, care must be taken in identifying the locations and lengths of potential plastic hinge regions, as these will critically affect the nature of detailing that has to be provided. The inefficiency of the tapered wall of Fig. 5.8(e), sometimes favored as an architectural expression of form, is obvious. If a plastic hinge would need to be developed at the base of this wall, its length would bc critically restricted. Therefore, for a given displacement ductility demand, excessive curvature ductility would develop (Section 3.5.4). Such walls may be used in combination with ductile frames, in which case it may be advantageous to develop the wall base into a real hinge.

(b) Stnrctumf W& with Openings In many structural walls a rcgular pattern of openings will be required to accolnmodate windows or doors or both.

When arranging openings, it is essential to ensure that a rational structure results, the behavior of which can bc predicted by bare inspection [PI]. The designer must ensure that the intcgrity of the structure in terms of flexural strength is not jeopardized by gross reduction of wall area near the extreme fibers of the section. Similarly, the shear strength of the waH, in both the horizontal and vertical directions, should remain feasiblc and adequate to ensure that its flexural strength can be fully developed.

Windows in stairwells are sometimes arranged in such a way that an extremely weak shear fiber results where inner edges of the openings line up, as shown in Fig. 5.9(a). It is difficult to make such connections sufficiently ductile and to avoid early damage in earthquakes, and hence it is preferable to avoid this arrangement. A larger space between the staggered openings would, however, allow an effective diagonal compression and tension field to develop after the formation of diagonal cracks [Fig. 5.9(h)]. When suitabIy reinforced, perhaps using diagonal reinforcemcnt, distress of regions between openings due to shear can be prevented, and a ductile cantilever response due to flexural yielding at the base only can be readily enforced.

Overall planning may sometimes require that cantilever walls be discontin- ued at level 2 to allow a large uninterruptcd space to be utilized between levels 1 and 2. A structure based on irrational concepts, as seen in Fig.

5.10(a), may result, in which the most critical region is deliberately weak- ened. Shear transfcr from the wall to foundation level will involve a soft-story

Fig. 5.9 Shear strcngth of walls as aIfcctcd by opcn-

ings. ( a l

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t 4 l 4

f a 1 f b ) Fig. 5.10 Structural walls supportcd on columns.

sway mcchanisrn with a high probability of excessive ductility demands o n the columns. T h e overturning moment is likely to impose simultaneously vcry large axial forces on one of the supporting columns. This system must be avoided! However, often it is possible to transfer the total seismic shear above the opening by means of a rigid diaphragm connection, for example, to other structural walls, thus preventing swaying of the columns (props). This is shown in Fig. 5.10(b). Because of the potential for lateral buckling of props under the action of reversed cyclic axial forces, involving yielding over their full length, they should preferably be designed to remain elastic.

Extremely efficient structural systems, particularly suited for ductile response with very good energy-dissipation characteristics, can be conceived when openings are arranged in a regular and rational pattern. Examplcs are shown in Fig. 5.11, where a number of walls are interconnected or coupled to each other by beams. For this reason they are generally referred to as coupled structural walls. The impIication of this terminology is that the connecting beams, which may be relatively short and decp, are substantially weaker than the walls. T h e walls, which behavc predominantly as cantilcvcrs, can then impose sufficient rotations on these connecting beams to make them yield. If suitably detailed, the beams are capable of dissipating energy over the entire height of the structure. Two ideniical walls [Fig. 5.11(a)] o r two walls of differing stiffnesses [Fig. 5.11(b)] may b e couplcd by a single line of

. .

r c i i d

i

Fig. 5.11 Typcs of couplcd structural walls.

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Fig. 5.12 Undcsrrable p~erccd walls for earthquake resistance.

beams. In other cases a series of walls may be interconnected by lines of beanis between them, as seen in Fig. 5.11(c). The coupling beams may be identical at all floors or they may have different depths or widths. In service cores, coupled walls may extend above the roof level, where lift machine rooms or space for other services are to be provided. In such cases walls may be considered to be interconnected by an infinitely rigid diaphragm at the top, as shown in Fig. 5.11(d). Because of their importancc in carthquake rcsislancc, a detailed examination of the analysis and design of coupled structural walls is given in Section 5.3.2(c).

From the point of view of seismic resistance, an undesirable structural system may occur in medium to high-rise buildings when openings are arranged in such a way that the connccting bcams arc strongcr than the walls. As shown in Fig. 5.12, a story mechanism is likely to develop in such a systcm because a series of piers in a particular story may be overloaded, whilc nonc of thc dccp beams would become inelastic. Because of the squatness of such conventionally reinforced piers, shear failure with restricted ductility and poor energy dissipation will characterize the response to large earthquakes. Even if a capacity design approach ensures that the shear strength of thc picrs cxccedcd thcir flcxural strength, a soft-story sway mechanism would result, with excessive duct~lity demands o n the hinging piers. A more detailed examination of this issue is given in Section 7.2.l(b).

Such a wall system should be avoided, or if it is to be used, much larger lateral design force should be used to ensure that only reduced ductility demand, if any, will arise.

Designers sometimes face the dilemma, particularly when considering shear strength, as to whether thcy should trcat couplcd walls, such as shown in Fig. 5.11(a) or (d), as two walls interconnected or as one wall with a series of openings. The issue may be resolved if one considers the behavior and rncchanisms of resistance of a cantilever wall and comparcs these with those of coupled walls. Thesc aspects are shown qualitatively in Fig. 5.13, which compares the mode of flexural resistance of coupled walls with different strength coupling beams with that of a simple cantilever wall.

It is seen that the total overturning moment, Mot, is resisted at the base of the cantilever [Fig. 5.13(a)] in the traditional form by flexural stresses, while in the coupled walls axial forccs as wcll as moments are being resisted. These

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Fig. 5.13

satisfy the following simple equilibrium statement:

for the coupled walls to carry thc same moment as the cantilever wall.

The magnitude of the axial forcc, being Lhe sum of the shear forces of all the coupling bcams at uppcr lcvcls, will dcpcnd on thc stiffness and strength of those beams. The derivation of axial forces on walls follows the same principles which apply to columns of multistory frames, presented in Section 4.6.6. For example, in a structure with strong coupling beams, shown in Fig.

5.13(b), the contribution of thc axial forcc to thc total flexural resistance, as expressed by the parameter

will b e significant. Hence this structure might behave in much the same way as the cantilever of Fig. 5.13(a) would. Therefore, one could treat the entire structure as one wall.

When the coupling is relatively weak, as is often the case in apartment buildings, where, because of headroom limitation, coupling by slabs only is possible, as shown in Fig. 5.13(c), the major portion of the moment resistance is by moment components M , and M,. In this case the value of A [Eq. (5.411 is small. One should then consider each wall in isolation with a rclatively small axial load induced by earthquake actions. An example of the interplay of actions in coupled walls is given in Section 5.3.2(b) and Fig. 5.22.

In recognition of the significant contribution of appropriately detailed beams to energy dissipation a t each floor in walls with strong coupling EFig.

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t (

5.13(b)], it has been suggested [X8] that they be treated as ductile concrete frames. Accordingly, the force reduction factor, R, for ductility, given in Table 2.4, may be taken as an intermediate value belween the limits recom- mended for slender cantilever walls and ductilc framcs, depending on the efficiency of coupling defined by Eq. (5.4) thus:

when

Various aspects of coupling by beams are examined in Section 5.3.2(b).

A rational approach to the design of walls with significant irregular openings is discussed in Section 5.7.8(d).

5.3.1 Modeling Assumptions

(a) Member Stifiess To obtain reasonable estimates of fundamental period, displacements and distribution of lateral forces between walls, the stiffness properties of all clcmcnts of rcinforccd concrete wall structures should include an allowance for the effects of cracking. Aspects of stiffness estimates were reviewed in Section 1.1.2(a) and shown in Fig. 1.8. Displace- ment (0.75AJ and lateral force resistance (0.75S,), relevant to wall stiffness estimate in Fig. 1.8, are close to those that develop at first yield of the distributed longitudinal reinforcement.

1. The stiffness of cantilever walls subjected predominantly to flexural deformations may be based on the equivalent moment of inertia I, of the cross section at first yield in the extreme fiber, which may be related to the moment of inertia I, of the uncracked gross concrete section by the following expression [P26]:

where P, is the axial load considered to act on the wall during an earthquake taken positive when causing compression and ^fy is in MPa. (The coefiient 100 becomes 14.5 when ^fyis in ksi.)

In ductile earthquake-resisting wall systems, significant inelastic deformations are expected. Consequently, the allocation of internal design actions in accordance with an elastic analysis should be considered only as one of several acceptable solutions that satisfy the unviolable requirements of inter-

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\

ANALYSIS PROCEDURES 37'1

nal and external equilibrium. As will be seen subsequently, redistribution of design actions from the elastic solutions are not only possible but may also be desirable.

Dcformations of the foundation structure and the supporting ground, such as tilting or siiding,.will not be considered in this study, as these produce only rigid-body displacement of cantilever walls. Such deformations should, however, be taken into account when the period of the structure is being evaluated or when the deformation of a structural wall is related to that of adjacent frames or walls which are supported on independent foundations.

Elastic structural walls are vely sensitive to foundation deformations [P37].

2. For the estimation of the stiffness of diagonally reinforced coupling beams [PX] with depth h and clear span I, (Section 5.43,

For conventionally reinforced coupling beams [P22, P23] or coupling slabs,

In the expressions above, the subscripts e and g refer to the equivalent and gross properties, respectively.

3. For the estimation of the stiffness of slabs connecting adjacent structural walls, as shown in Fig. 5.13(c), the equivalent width of slab to compute I, may be taken as the width of the wall b, plus the width of the opening between the walls or eight times the thickness of the slab, whichever is less..

The value is supported by tests with reinforced concrete slabs, subjected to cyclic loading [P24]. When flanged walls such as shown in Fig. 5.6 are used, the width of the wall bw should be replaced by the width of the flange b.

4. Shear deformations in cantilever walls with aspect ratios, hw/lw, larger than 4 may be neglected. When a combination of "slender" and "squat"

structural walls provide the seismic resistance, the latter may be allocated an excessive proportion of the total lateral force if shcar distortions are not accounted for. For such cases (i.e., when h , / l ,

<

4) it may be assumed that

where

In Eq. (5.9) some allowance has also been made for deflections due to anchorage (pull-out) deformations at the base of a wall.

Deflections due to code-specified lateral static forccs may be determined with the use of the equivalent sectional properties above. However, for

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consideration of separation of nonstructural components and the checking of drift limitations, the appropriate amplification factors that make allowance for additional inelastic drift, given in codes, must be used.

(b) Geometric Modeling For cantilever walls it will be sufficient to assume that Lhe sectional properties are concentrated in the vertical centerline of the wall (Fig. 5.11). This should be taken to pass through the centroidal axis of the wall section, consisting of the gross concrete area only. When cantilever walls are interconnected at each floor by a slab, it is normally sufficient t o assume that the floor will act as a rigid diaphragm. ElTecls of horizontal diaphragm flexibility are discussed briefly in Scction 6.5.3. By neglecting wall shear deformations and those due to torsion and the effects of restrained warping of an open wall section on stiffness, the lateral force analysis can be rcduccd to that of a set of cantilevers in which flexural distortions only will control the compatibility of deformations. Such analysis, based on first principles, can allow for the approximate contribution of each wall when it is subjected to deformations due t o floor translations and torsion, as shown in Section 5.3.2(a). 11 IS to be remembered that such an elaslic analysis, howcvcr approximate it might be, will satisfy the requirements of static equilibrium, and hence it should lead t o a satisfactory distribution also of internal actions among the walls of an inelastic structure.

When two or more walls in the same plane are interconnected by beams, as is the case in coupled walls shown in Figs. 5.11 and 5.12, in the estimation of stiffnesses, it will be necessary to account for more rigid end zones where beams frame into walls. Such structures are usually modeled as shown in Fig.

5.14. Standard programs writtcn for framc analyscs may then be used. It is emphasized again that the accuracy of geometric stiffness modeling may vary considerably. This is particularly true for deep membered structures, such as shown in Fig. 5.14. In coupled walls, for example, axial deformations may be significant, and these affect the efficiency of shear transfer across the coupling system. It is difficult to model accurately axial deformations in deep members afler lhe onset of flexural cracking.

Figurc 5.15 illustrates the difficulties that arise. Structural properties are conventionally concentrated at the reference axls of the wall, and hence

wall frames. l b l

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ANALYSIS PROCEDURES 379

Fig. 5.15 Ellcct of cuwaturc on uncrackcd crackcd wall scctions.

and

under the action of flexure only, rotation about the centroid of the gross concrete section is predicted, as shown in Fig. 5.15, by linc 1. Aftcr flcxural cracking, the same rotation may occur about the neutral axis of the crackcd section, as shown by line 2, and this will result in elongation A, measured at the reference axis. This deformation may affect accuracy, particularly when the dynamic response of the structure is evaluated. However, its significance in tcrms of inelastic rcsponse is likcly to bc small. It is evidcnt that if onc were to attcmpt a more accuratc modcling by using thc neutral axis of thc cracked section as a reference axis for the model (Fig. 5.141, additional complications would arise. The position of this axis would have to change with the height of the frame due to moment variations, as well as with the direction of lateral forces, which in turn might control the sense of the axial force o n the walls. These difficulties may be overcome by employing finite elcmcnt [P25] analysis techniques. However, in design for earthquake resistance involving inelastic response, this computational effort would seldom be justified.

(c) Analysis of Wall Sections T h e computation of deformations, stresses, or strength of a wall section may be based o n the traditional concepts of equilibrium and strain compatibility, consistent with the plane section hy- pothesis. Because of the variability of wall scction shapes, design aids, such as standard axial load-moment interaction charts for rectangular column sections, cannot often be used. Frequently, the designer will have to resort to the working out of the required flexural reinforcement from first principles [Section 3.3.l(c)]. Programs to carry out the seetion analysis can readily be developed for minicomputers. Alternatively, hand analyses involving succes- sive approximations for trial sections may be used such as shown in design examples at the end of this chapter. With a little expcricncc, convergence can be fast.

T h e increased computational effort that arises in the section analysis for flexural strength, with or without axial load, stems from thc multilayered arrangement of reinforcement and the frequent complexity of section shape.

A very simple example of such a wall section is shown in Fig. 5.16. It represents one wall of a typical coupled wall structure, such as shown in Fig.

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Fig. 5.16 Axial load-moment intcraction curves for unsymmetrically reinforced rectangular wall section.

5.11. The four sections are intended to resist the design actions at four different critical levels of the structure. When the bending moment (assumed t o be positive) causes tension a t the more heavily reinforced right-hand edge of the section, net axial tension is expected to act on the wall. On the other hand, when flexural tension is induced at the left-hand edge of the section by (negative) moments, axial compression is induced in that wall. Example calculations are given in Sections 5.5.2 and 5.6.2.

The moments are expressed as the product of the axial load and the eccentricity, measured from the reference axis of the section, which, as stated earlier, is conveniently taken through the centroid of the gross concrete area rather than through that of the composite o r cracked transformed section. It is expedient to use the same reference axis also for the analysis of the cross section. It is evident that the plastic centroids in tension or compression d o not coincide with the axis of the wall section. Consequently, the maximum tension o r compression strength oE the section, involving uniform strain across the entire wall section, will result in axial forces that act eccentrically with respect to the reference axis oE the wall. These points a r e shown in Fig.

5.16 by the peak values at the top and bottom meeting points of the four sets of curves. This representation enables the direct use of moments and forces, which have been derived from the analysis of the structural system, because in both analyses the same reference axis has been used.

Similar axial load-moment interaction relationships can be constructed for different shapes of wall cross sections. An example for a channel-shaped section is shown in Fig. 5.17. It is convenient to record in the analysis the neutral-axis positions for various combinations of moments and axial forces, because these give direct indication of the curvature ductilities involved in developing the appropriate strengths, a n aspect examined in Section 5.4.3(b).

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ANALYSIS PROCEDURES 381

Fig. 5.17 Axial load-moment interaction relationships for a channel-shaped wall section.

Because axial load

P,

will vary between much smaller limits than shown in Fig. 5.17, in design office practice only a small part of the relationships shown need be produced. For walls subjected to small axial compression or axial tension, linear interpolations will often suffice.

53.2 Analysis for Equivalent Lateral Static Forces

The choice of lateral design force level, based on site seismicity, structural configurations and materials, and building functions has been considered in detail in Chapter 2. Using the appropriate model, described in preceding sections, the analysis to determine all intcrnal design actions may then bc carried out. The outline of analysis for two typical structural wall systems is given in the following sections.

(a) Zntemcting CMtilever W& The approximate elastic analysis for a series of interacting prismatic cantilever walls, such as shown in Fig. 5.18, is based on the assumption that the walls are linked at each floor by an infinitely rigid diaphragm, which, however, has no flexural stiffness. There- fore, the three walls shown and so linked are assumed to be displaced by identical amounts at each floor. Each wall will thus share in the resistance of

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382 SrRUCTURAL WALLS

Fig. 5.18 Model of interacting cantilever wails.

a story force, F, or story shear, V, or overturning moment, M, in proportion to its own stiffness thus:

where the stiffness of the walls is proportional to the cquivalent moment of inertia of the wall section as discussed in Section 5.3.l(a).

The stiffness of rectangular walls with respect to their weak axis, relative to those of othcr walls, is so small that in general it may be ignored. It may thus be assumed that as for wall 1 in Fig. 5.19, no lateral forces are introduced to such walls in the relevant direction. A typical arrangement of walls within the total floor plan is shown in Fig. 5.19. The analysis of this wall systcm is based on the concepts summarized in Appendix A. The shear force, V, applied in any story and assumed to act at the point labeled CV in Fig.

5.19, may be resolved for convenience into components

C:

and V,. Uniform deflection of all the walls would occur only if these component story shear forces acted at the center of rigidity (CR), the chosen center of the coordi- nate system for which, by analogy to the derivalions of Eq. (A.20), the following conditions are satisfied:

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I

ANALYSIS PROCEDURES 383 where I,,, Zi, = equivalent moment of inertia of wall section about the x and

y axis of that scction, respectively

xi, y, = coordinates of the wall with respect to the shear centers of the wall sections labeled 1,2,. .

.

, i and measured from the center of rigidity (CR)

Hence for the general case, shown in Fig. 5.19, the shear force for each wall at a given story can be found from .

where (V'e,

-

Vyex) is the torsional moment of V about CR,

C

( ~ ~ 1 , ~

+

yiZIiy) is the rotational stiffness of the wall system, and ex and e, are eccentricities mcasurcd from thc center of rigidity (CR) to thc ccntcr of s t o ~ y shcar (CV).

Note that the value of e, in Fig. 5.19 is negative. With substitution of I, for Dl, the meaning of the exprcssions above are identical to those given in Sections (f) and (g) of Appendix A.

The approximations above are also applicable to walls with variable thickness provided that all wall thicknesses reduce in the same proportions at the same level, so that the wall stiffnesses relative to each other do not change. When radical changes in stiffnesses (I,, and I,,) occur in some walls, the foregoing approach may lead to gross errors, and some engineering judgment will be required to compensate for this. Alternatively, a more accurate analysis may be carried out using established computer techniques.

The validity of the assumption that the interaction of lateral force resisting components is controlled by an infinitely rigid connection of the floor diaphragm is less certain in thc casc of structural walls than in the case of structures braced by interacting ductile frames. The in-plane stiffness of walls and floor slabs, especially in buildings with fewer than five stories, may be comparable. Thus diaphragm deformations in the process of horizontal force transfer can be significant, particularly when precast floor systems arc uscd.

For such buildings, variations of the order of 20 to 40%, depending on slab flexibility, have been predicted in the distribulion of horizontal forces among elastically responding walls [U3]. This aspect is of importance when assessing the adequacy of the connection of precast floor panels to the structural walls.

Diaphragm flexibility is examined further in Section 6.5.3.

The adequacy of the connections between the floor system, expected to function a s a diaphragm, and structural walls must be studied at an early stage of the design process. Large openings for services are often required immediately adjacent to structural walls. These may reduce the effective

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s =clear span of coupling beams

SPECIFIED SUBSTITUTE

FDRtEs mCEs

Fig. 5.20 Modcling of the lateral forces and the structure for the laminar analysis coupled walls.

stiffness and strength of the diaphragm and hence also the effectiveness of the poorly connected wall. In buildings with irregular plans, such as an L shape, reentrant corners in the floor system may invite early cracking and consequent loss of stiffness [P37] [Section 1.2.3(a)].

The larger the expected inelastic response of the cantilever wall system, the less sensitive it becomes to approximations in the elastic analysis. For this reason the designer may utilize the concepts of inelastic force redistribution to produce more advantageous solutions, and this is discussed in Section 5.3.2(c).

(b) Coupled W a k Some of the advantages that coupled wall structures offer in seismic design were discussed in Section 5.2.3(b). Analysis of such struclures may be carried out using frame modcls, such as shown in Fig. 5.14, or that of a continuous connecting medium. The latter, also referred to as laminar analysis, reduces the problems of a highly stalically indeterminate structure to the solution of a single differential equation. Figure 5.20 shows the technique used in the modeling with which discrete lateral forces or member prope.rties are replaced by equivalent continuous quantities. This analysis, rather popular some 25 years ago, has been covered extensively in the technical literature [ B l , C6, PI, R2, R31 and is not studied further here.

It has bccn employed to derive the quantities given in Fig. 5.22, which are used here to illustrate trends in the behavior of coupled walls.

To study the effect of relative stiffnesses on the elastic behavior of coupled walls, the results of a parametric study of an example service core structure, with constant sectional dimensions as shown in Fig. 5.21, will be examined.

The depth for the two rectangular, 300-mm-thick coupling beams at each floor will be varied bchvcen 1500 and 250 mm. Beam stiffnesses are based on Eq. (5.8~) when the depth is 400 mm or more, and on Eq. (5.86) when the

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\ 1 , 'ANALYSIS PROCEDURES 38s

Fig. 521 Dimensions core structure.

example service

depth is less than 400 mm. AJso, coupling by a 150-mm-thick slab with 1200-, GOO-, and 350-mm effective widths is considered. Figure 5.22 summarizes the response of this structure to lateral forces which, in terms of patterns in Fig.

5.20, were of the following magnitudes: F, ⁼2000 kN, F, = 700 kN, and F3 = 300 kN.

Figure 5.22(a) compares the bending moments in the two walls which would have been developed with both walls remaining uncracked, with those obtained if some allowance for cracking in thc tcnsion wall 1 only is made.

For both cases, beams with 1000 mm depth were considered. It is seen that the redistribution of moments due to cracking of wall 1 is significant.

Figure 5.22(c) compares the variation of laminar shear forces, q (force per unit height), over the full height of the structure as the depth of the beams is varied. It is seen that with deep beams the shear forces are large in thc lowcr third of the structure and reduce rather rapidly toward the top. This is due to the fact that under the increased axial load on the walls, axial deformations in the top stories become more significant and these relieve the load on the coupling beams. On the other hand, the shear, q, in shallower beams is largely controlled by the general slope of the wall; hence a more uniform distribution of its intensity over the height results. The outermost curve shows the distribution of vertical shear that would result in a cantilever wall (i.e., with infinitely rigid coupling). This curve is proportional to the shear force diagram that would be obtained from the combined forces I;,, F,, and F3 on the structure.

Within a wide range of beam depths, the shear force and hence the axial force T, being the sum of the beam shear forces, does not change significantly in the walls, particularly in the upper stories. This is seen in Fig.

5.22(d). There is a threshold of beam stiffness (depth), however, below which the axial force intensity begins to reduce rapidly. This is an important feature of bchavior, and designers may make use of it when deciding how efficient coupled walls should be.

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1

386 S'I'KUC'I'URAL WALLS

Fig. 5.22 Response of an example coupled wall service core. (1 mm = 0.394 in., 1 MNm = 735 kip-ft, 1 kN/m = 0.0686 kip/ft, 1 kN = 0.225 kip).

The interplay of internal wall forces at any level is expressed by Eq. (5.3):

and this is shown in Fig. 5.22(b). It is seen again that in this example with beams 500 mm o r deeper, very efficient coupling is obtained because the 1T component of moment resistance is large. Significant increase of beam stiffness does not result in corresponding increase of this component. How- ever, when beams shallower than 250 mm and particularly when 150-mm slabs, irrespective of effective widths, are used, the moment demand on the

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\

ANALYSIS PROCEDURES 381

walls (M,

+

M,) increases rapidly. With the degeneration of the coupling system, the structure reverts to two cantilevers. The shaded range of the IT component of moment resistance shows the limits that should be considered in tcrms of potential energy dissipation in accordance with Eq. (5.6). Finally, Fig. 5.22(e) compares the deflected shapes of the elastic structure. It shows the dramatic effects of efficient coupling and one of the significant benefits in terms of seismic design, namely drift control, which is obtained.

(c) Lateral F m e Redistribution Between Walls The ~ r i n c i ~ l e s of the redistribution of design actions in ductile frames, estimated by elastic analyses, were discussed in Section 1.4.4(iv) and in considerable detail in Section 4.3. Those principles are equally applicable to wall structures studied in this chapter because with specific detailing they will possess ample ductility capacity [S2].

In Section 5.3.2(a) the elastic analysis of intcrwnnected cantilever walls, such as shown in Fig. 5.18, was presented. During a large earthquake plastic hinges at the base of each of the three walls in Fig. 5.18 are to be expected.

However, the base moments developed need not be proportional to those of the elastic analysis. Bending moments, and correspondingly lateral forces, may be redistributed during the design from one wall to another when the process leads to a more advantageous solution. For example, wall 3 in Fig.

5.18 might carry considerably larger gravity loads than the other two walls.

Therefore, larger design moments may be assigned to this wall without having to provide proportionally increased flexural tension reinforcement.

Moreover, it will be easier to transmit larger base moments to the foundation of wall 3 than to those of the other two walls.

It is therefore suggested [X3] that if desirable in ductile cantilever wall systems, the design'lateral force on any wall may be reduced by up to 30%.

This force must then be redistributed to other walls of the svstem. there being no limit to the amount by which the force on any one wail could be increased.

The advantages of the redistribution of design action can be utilized to an even greater degree in coupled walls, such as shown in Figs. 5.21 and 5.23(d).

The desired' full energy-dissipating mechanism in coupled walls will be similar to that in multistory frames with strong columns and weak beams, as shown in Fig. 1.19(a). This involves the plastification of all the coupling beams and the development of a plastic hinge at the base of each of the walls, as seen in Fig. 5.23(d) with no inelastic dcformation anywhere else along the height of the walls. This is because the walls are usually very much stronger than the connecting beams.

The elastic analysis for such a structure (Figs. 5.21 and 5.22) may havc resulted in bending moments M, and M, for the tension and compression walls, respectively, as shown by full lines in Fig. 5.23(a) and ( b ) . In this analysis it is assumed that the elastic redistribution of moments due to the effects of cracking, as outlined in Section 5.3.l(a), has already been considered. In spite of MI being smaller than M,, more tension reinforcement is

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1

-

= Redstribuled moments

Fig. 5.23 Ductile response of an example coupled wall service core.

likely to be required in wall 1 because it will be subjected to large latcral force induced axial tension [Fig. 5.22(d)]. The flexural strength of wall 2, on the other hand, will be enhanced by the increased axial compression. It is thercfore suggested that if desirable and practical, the moments in the tension wall be reduced by up to 30% and that these moments be redistributed to the compression wall. This range of maximum redistributable moments is shown in Fig. 5.23(a) and (b). The limit of 30% is considered to be a prudent measure to protect walls against excessive cracking during moderate earthquakes. Moment redistribution from one wall to another also implies redistribution of wall shear forces of approximately the samc order.

Similar considerations lead to the intcntional redistribution of vertical shear forces in the coupling system. It has been shown [PI] that considerable ductility capacity can be provided in the coupling beams. Hence they will need to be designed and detailed for very large plastic dcformations. This is considcrcd in Section 5.4.5. A typical elastic distribution of shear forces in coupling beams, in terms of laminar shcar, q, is shown in Fig. 5.23(c).

Coupling beam reinforcement should not be varied continuously with the height, but changed in as small a number of levels as possible. Shear and hence moment redistribution vertically among coupling beams can be utilized, and the application of this is shown by the stepped shaded lines in Fig.

5.23(c). It is suggested that the reduction of dcsign shear in any coupling beam should not exceed 20% of the shear predicted for this beam by the elastic analysis. It is seen that with this technique a large number of beams can be made identical over the height of the building.

When shcar is redistributed in the coupling system, it is important to ensure that no shear is "lost." That is, the total axial load introduced in the walls, supplying the IT component of the moment resistance, as seen in Fig.

5.22(b), should not be reduced. Thcrcfore, the area under the stepped and shaded lines of Fig. 5.23(c) should not be allowed to be less than the area

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( I DESIGN Ok NALL ELEMENTS FOR STRENGTH AND DUCTILITY 389

under the curve giving the theoretical elastic laminar shear, q. Neither should the strength of the coupling system significantly a c e e d the demand, shown by the continuous curve, because this may unnecessarily increase the overturning capacity of the structure, thus overloading foundations. It will be shown in the complete example design of a coupled wall structure in Section 5.6 how this can be readily checked.

While satisfying thc moment cquilibrium requirements of Eq. (5.31, it is also possible to redistribute moments between the ( M ,

+

M,) and 1T components, involving a change in the axial force, T, and hence in shear forces in the coupling system. However, this is hardly warranted because with the two procedures above only, as illustrated in Fig. 5.23, usually a practical and economical allocation of strength throughout the coupled structural wall system can readily be achieved.

5.4 DESIGN OF WALL ELEMENTS FOR STRENGTH AND DUCTILITY 5.4.1 Failure Modes in Structural Walls

A prerequisite in the design of ductile slruclural walls is that flexural yiclding in clearly defined plastic hinge zones should control the strength, inelastic deformation, and hence energy dissipation in the entire structural system [B14]. As a corollary to this fundamental requiremcnt, brittle failurc mccha- nisms or even those with limited ductility should not be permitted to occur.

As stated earlier, this is achieved by establishing a desirable hierarchy in the failure mechanics using capacity design procedures and by appropriate detailing of the potential plastic regions.

The principal source of energy dissipation in a laterally loaded cantilever wall (Fig. 5.24) must be the yielding of the flexural reinforcement in the plastic hinge regions, normally a t the basc of thc wall, a s shown in Fig.

5.24(b) and (e). Failure modes to b e prevented a r e those due to diagonal instability of ment, sliding shear along construction joints, shown in Fig. 5.24(d), and shear or bond

( a ) I b ) ic) Id l fr 1

Fig. 5.24 Failure modes in cantilcvcr walls.

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I

3 ~ 0 STRUCTURAL WALLS

Deflection lmml

1,

Fig. 5.25 Hysteretic responsc of a structural wall controlled by shear strength.

failure along lapped splices or anchorages. An example of the undesirable shcar-dominated response of a structural wall to reversed cyclic loading is shown in Fig. 5.25. Particularly severe is the steady reduction of strength and ability to dissipate energy.

In contrast, carefully detailed walls designed for flexural ductility and protected against a shear failure by capacity design principles exhibit greatly improved response, as seen in Fig. 5.26, which shows a one-third full size cantilever structural wall with rectangular cross section. The test unit simu-

Fig. 5.26 Stable hystcrctic rcsponsc of a ductilc wall structure [GI].

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, \

DESIGN OF WALL ELEMENTS FOR STRENGTll AND DUCTILITY 391

lates one wall of a coupled wall structure that was subjected to variable axial compression between the limits shown. It is seen that a displacemcnt ductility of approximately 4 has been attained in a very stable manner [GI, P44].

Failure due t o inelastic instability, to be examined subsequently, occurred only after two cycles to a displacement ductility of 6, when the lateral deflection was 3.0% of the height of the model wall [P44].

The hysteretic response shown in Fig. 5.26 also dcmonstratcs that the flexural overstrength devclopcd depends on the imposed ductility. The ideal flexural strengths shown were based on measured yield strength of the vertical bars which was 18% in excess of the specified yield strength that woul