The displacement capacity of reinforced concrete coupled walls

8/18/2019 The displacement capacity of reinforced concrete coupled walls

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Engineering Structures 24 (2002) 1165–1175

www.elsevier.com/locate/engstruct

The displacement capacity of reinforced concrete coupled walls

Tom Paulay ∗

Department of Civil Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

Received 20 December 2001; received in revised form 8 January 2002; accepted 8 April 2002

Abstract

With the identification of criteria of performance-based seismic design, the need to focus on estimations of displacement capacities

of ductile system emerges. This involves redefinitions of some properties of reinforced concrete structures. A system comprisingcomponents with very different characteristics, a coupled wall structure, is used to demonstrate how displacement and ductilitycapacities, satisfying specific performance criteria, can be predicted simply, even before the required seismic strength of the systemis established. An attractive feature of this approach is that the strengths of components, which contribute to the required seismicstrength of the system, may be freely chosen. The astute designer may advantageously exploit this freedom. © 2002 Elsevier ScienceLtd. All rights reserved.

Keywords: Displacements; Coupling beams; Ductility; Stiffness; Strength

1. Introduction

The prediction of displacement demands imposed on

structures by earthquake motions has been one of theimportant issues, challenging the earthquake engineeringresearch community. Relatively few studies addressedexplicitly the displacement capacity of reinforced con-crete ductile structures. A rational evaluation of displace-ment capacities, associated with both elastic and post-elastic response, satisfying specific performance criteria,should enable acceptable seismic displacement demands,relevant to local seismic scenarios, to be more convinc-ingly established.

To allow displacement capacities to be realisticallyestimated, some traditional definitions of structuralproperties, particularly those applicable to homogeneousmaterials, need to be redefined. Relevant principles arepresented first. Subsequently applications are illustratedusing a coupled wall example structure. It is postulatedthat the displacement capacity of such a system shouldbe controlled by that of its component with the smallestdisplacement capacity. Therefore, instead of commonlyspecified or judgement-based global displacement duc-tility factors, the deliberate evaluation of these for each

∗ Corresponding author. Tel.: 64-3-364 2249; fax: 64-3-364 2758.

E-mail address: [email protected] (T. Paulay).

0141-0296/02/$ - see front matter © 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 1 4 1 - 0 2 9 6 ( 0 2 ) 0 0 0 5 0 - 0

specific system is advocated. The approach relies thuson the hierarchy of the displacement ductility capacitiesof constituent components.

The procedure is claimed to be rational, realistic andsimple. It is design oriented. Redefined properties of components, as constructed, may then be used, to ana-lyze, if necessary, a structural system comprisingcomponents with different characteristics but knownstrengths.

In this presentation abstract definitions of quantitiesare, in general, immediately followed by their numericalevaluations relevant to a particular example structure.

2. The traditional treatment of coupled walls

Some 50 years ago the analysis of elastic coupledwalls structures was a challenging topic for researchersin several countries. With the arrival of computer tech-nology this pioneering work, based on innovative mode-ling [1–6], has become also accessible to the structuraldesign profession. Even though during significant seis-mic events, reinforced concrete structures are expectedto perform in the inelastic domain, the assignment tocomponents of lateral design strength is still widelybased on elastic structural response. However, in recog-nition of ductile behaviour, within specified limits, aredistribution between components of internal designactions, so derived, has been considered acceptable [7].


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1166 T. Paulay / Engineering Structures 24 (2002) 1165–1175

Nomenclature

Ae effective area of cracked concrete section

Asd area of diagonal reinforcement in one direction Awe aspect ratio of a wall in terms of he Awm aspect ratio of a wall in terms of hm Awi aspect ratio of a wall based on its full height

d b diameter of a bar Db overall depth of a beam

Dwi length (overall depth) of a wall E c modulus of elasticity of concrete

f y yield strength of reinforcing steelh total height of structure

he height where maximum storey drift occurs

hm height above base of center of accelerated mass

I e second moment of effective area of cracked reinforced concrete section I g second moment of an area of gross concrete section

l internal lever arm of coupling system l p length of equivalent plastic hinge

k i stiffness of component M overturning moment at a level

M ni nominal flexural strength of a section M o overturning moment at the base of the structure

M yi flexural yield strength of a section M 1 ,M 2 moments assigned to components (1) and (2)

s clear span of coupling beamT lateral force-induced axial load on coupled walls

V b total base shear for the structure

V nb nominal shear capacity of coupling beam

V ni nominal strength of a wall component in terms of its base sheara inclination of diagonal reinforcement b a moment ratiod p post-yield storey driftd u maximum acceptable storey driftd y storey drift at the nominal yield displacement of a walle y yield strain of reinforcing steelh coef ficient relevant to nominal yield curvatureqb beam chord rotationqby chord rotation of beam associated with nominal yield curvatureqw wall slope (storey drift)qwy wall slope associate with nominal yield curvature mb displacement ductility imposed on a beam mw displacement ductility relevant to a wall m system displacement ductilityx coef ficient defining the position of the neutral axis relative to the tension edge

fby nominal yield curvature in a beamf yi nominal yield curvature at the critical sectionfwyi nominal yield curvature of a wall sectiona anchorage deformation

by nominal yield displacement of coupling beamc diagonal shortening of coupling beam

e lateral displacement of elastic elementsT elongation of diagonal bars in tension

p post-yield displacementu maximum limit displacement

y nominal yield displacement of a ductile system

yi nominal yield displacement of a wall component


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1167T. Paulay / Engineering Structures 24 (2002) 1165 – 1175

In the design of coupled walls the fact that they are

simply cantilever structures, is often overlooked. As Fig.

1 shows, only the mode of the resistance to lateral force-

generated overturning moments is different in coupled

walls. The well established equilibrium requirement is

M o

M 1

M 2

l T (1)

where the components of flexural resistance are shownin Fig. 1(c). The axial force at any level, T, results from

the summation of the shear forces transferred by coup-ling beams above that level. The distance between the

centroidal axes of the two walls is usually taken as the

lever arm, l , on which the axial forces, T , operate. These

3 moment contributions are traditionally assigned pro-

portionally to component stiffness. The latter are based

on flexural rigidities, E c I e, of prismatic components,where E c is the modulus of elasticity of the concrete and

I e is the second moment of effective area of the cracked

reinforced concrete section. This is usually expressed in

terms of a fraction of the second moment of the gross

concrete sectional area, I g. Values of I e /I g, recommended

in some codes or used in publications [7,8] or design

practice, vary in a wide range of 0.2 to 1.0. While the

allocation of design strength to various components is

not sensitive to such assumptions, predicted displace-ments of elastic coupled walls may involve errors of the

order of several hundred percent. A particular disadvan-

tage of the use in seismic design of crudely estimated

values of I e, is the inability to predict realistic values of

yield deformations of both components and the system.

Fig. 2(a), showing 3 interconnected rectangular cantil-

ever walls, is used to summarize the force-displacementrelationship based on traditional bi-linear modeling. Therelative lengths, Dwi, of the rectangular walls with ident-

ical thickness are 1.00, 1.59 and 2.00, respectively.

Consequently the relative flexural rigidities of the wallsections, E c I e, being proportional to D

3wi, are 1, 4 and 8.

Fig. 1. Comparison of flexural resisting mechanisms in structural

walls.

Fig. 2. Bi-linear idealisation of the response of interacting cantil-

ever walls.

Lateral design strength to components are routinely

assigned in the same proportions. These stiffness-pro-

portional strengths, associated with a given displace-

ment, e, are shown in Fig. 2(b). It is then commonlyassumed that, having developed these strengths, compo-nents will simultaneously enter the inelastic domain of

response. This fallacy [9], relevant to ductile behaviour

shown in Fig. 2(b), is discussed in the next section.

Assumptions with respect to the stiffness of coupling

beams were considered [1,8,10,11] to affect both the

intensity and the variation with building height of the

shear forces generated in coupling beams. With minor

modifications [7,10,11] stiffness-dependent strengthshave been routinely adopted in conventional seismicdesign. The ratio

l T / M o b (2)

quantifies the degree of coupling. Figs 1(b) and (c) illus-trate examples of relatively high and low degrees of

coupling. This ratio has been the subject of differingviews in the relevant literature [8]. Some studies sug-

gested [12] that there is an optimal value for ß, whichpromises favourable dynamic seismic response. Others

held the view that large lateral force-induced axial

forces, T , would be dif ficult for the foundations toabsorb. However, it is not likely that separate foun-

dations for each coupled wall, i.e., a foundation structure

different from that required for a cantilever wall, shown


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1168 T. Paulay / Engineering Structures 24 (2002) 1165 – 1175

in Fig. 1(a), would be contemplated. For some 25 years

the use of squat coupling beams, if possible, was advo-

cated in New Zealand design practice [7,11]. It was per-

ceived that for ductile systems, a high degree of coupling

could be an ef ficient and in many cases the major sourceof energy dissipation and hence hysteretic damping. For

example a relevant code [10] specifies system displace-ment ductility capacities, m , in the range of 6 m5for values of 2 / 3 b1/3. A value of m 5 was con-sidered [10] applicable to appropriately detailed ductilecantilever walls.

3. Principles of displacement estimates for ductile

wall systems

Bi-linear modeling of force displacement relationships

for reinforced concrete components or systems, is gener-ally accepted as being adequate for purposes of seismic

design. Implications of a more realistic use of this simple

technique, studied recently [9,13–15], are briefly sum-marized here. Fig. 2 is used to complement this review.

3.1. Nominal yield curvature

Using first principles, it has been shown [13] that thenominal yield curvature at the critical section of a

reinforced concrete wall component i, associated with itsnominal flexural strength, M ni, can be very satisfactorilyapproximated by

f yi he y / Dwi (3)

where e y and Dwi are the yield strain of the reinforcingsteel used and the length (depth) of the wall, respect-

ively. The coef ficient h quantifies the combined effectsof the ratio of the nominal to yield flexural strength, M ni /M yi, and the distance, x Dwi, of the neutral axis of thesection from the extreme tension fiber, thus

h ( M ni / M yi) / x (4)

Typical values of these parameter are presented in Fig.

3. It has been found [15,16] that the ratio of flexuralreinforcement and the intensity of axial compression

loads, usually encountered to act on walls of multistorey

buildings, are responsible for only negligible variations

in eq. (4). When axial forces are significantly larger orsmaller than those anticipated to act on cantilever walls,

as in the case of coupled walls, acceptable estimates of

the corresponding changes of the relevant parameters,

listed in Fig. 3, can be readily made. Important features

of nominal yield curvature to be noted are, that it isinversely proportional to wall length, Dwi, and that, con-

trary to traditional usage, for design purposes, it is inde-

pendent of strength.

Fig. 3. Parameters affecting the nominal yield curvature of sections.

3.2. Nominal yield displacement

With the assumption that neutral axes at all levels of

a wall are located approximately as at the critical base

section, for a given pattern of moments, the displacementat any level can be readily obtained. The assumption

implies that the extent of cracking over the height of the

walls is similar and that shear and anchorage defor-

mations are neglected. When warranted, these additional

sources of displacements may, however, be included.

Under repeated reversing lateral displacements, effectsof tension stiffening may also be considered negligible.

Of particular interest are displacements of walls at

specific levels, such as that of the center of horizontallyaccelerated mass, hm, associated with the nominal yield

curvature at the base of a wall. This, when combinedwith the nominal strength of a wall, expressed in terms

of the base shear or moment sustained, enables compo-

nent stiffness to be defined. Because displacements are


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proportional to curvatures, the relative value of the nom-

inal yield displacement of a wall is

yif yih2me y Awmhm (5)

where Awm hm / Dwi is the effective aspect ratio of the

wall. Eq. (5) emphasizes the 3 important parameters to

be considered when attempting to estimate wall displace-ments.

3.3. Assignment of lateral strength

Because, as eq. (5) demonstrates, the nominal yielddisplacement is independent of strength, in contrast with

traditional usage, the latter can be assigned arbitrarily to

interacting components of a wall system. This freedom

in the choice of component contributions to total

required strength can be advantageously exploited by the

astute designer. Of course equilibrium requirements

must not be violated. With the knowledge of the nominalstrength of a component, V ni, its stiffness is uniquely

defined as

k i Vni / yi (6)

3.4. Stiffness and ductility relationships

The application of the above principles, controlling

the compatibility of the yield displacements of different

wall components, is illustrated here with the aid of a

simple example. The walls shown in Fig. 2(a) will be

considered again. As eq. (3) stated, nominal curvatures

at the base of these walls are inversely proportional totheir length, Dwi. Hence the relative yield curvatures of walls (1), (2) and (3) are 1.00, 1/1.59=0.63 and 1/2=0.50,

respectively. If, as one of the possibilities, the strength

allocation recorded in Fig. 2(b) is adopted, the bi-linear

force-displacement simulations, presented in Fig. 2(c),

are established. Therefore, the relative stiffness of all 3

components are determined. For example k 2

(4/13)/0.63 0.488.

As Fig. 2(c) shows, nominal strengths of componentsare attained at different displacements. The superposition

of the idealized component responses describes the non-

linear system response. However, in seismic design this

can also be modeled using a bi-linear relationship. The

equivalent nominal yield displacement of the system is

then

y V ni / k i (7)

In this specific example this corresponds to 0.557 dis-placement units.

The linear elastic response of components is an ideal-

isation, which again is considered to be acceptable inthe design for systems for ductile response. After the

attainment of the nominal yield displacement of the criti-

cal element, such as component (3) in Fig. 2, some

changes in the moment and shear patterns of the walls

may occur.

Fig. 2(d) illustrates similar relationships when compo-

nent strengths were chosen arbitrarily. In this example

wall strengths were made proportional to D2wi rather than D3wi, used in the previous examples. The appeal of this

choice is that it results in approximately identicalreinforcement ratios in all walls. A slight reduction of

system stiffness leads to a correspondingly small

increase of the nominal yield displacement of the sys-tem. The examples used demonstrate also the relation-

ship between the displacement ductility capacities of the

components and that of the system [9,13,15]. In this

example it was assumed that adequately detailed walls

have a displacement ductility capacity of 4. Wall (3)

being critical ( y3 0.5), the seismic displacementdemand on the ductile system must be limited to

max 4 × 0.5 2.0 displacement units. This corre-

sponds to system displacement ductility capacities of m 2.0/0.5573.6 or m 2.0/0.5783.5, respect-ively. In existing strength-based seismic design pro-

cedures these values will control the required design

strength of the system.

4. A 12 storey service core

To illustrate the application to a coupled wall structure

of the principles outlined in the previous sections, a spe-

cific example was chosen. While different aspects of dis-placement estimates are considered, as stated earlier, the

evaluation of relevant quantities will not only be givenin abstract terms, but will also be simultaneouslyexpressed in terms of the selected structural dimensions.

This should assist in the appreciation, particularly by

design practitioners, of the simplicity of the approach

employed.

The principal dimensions of a 12 storey service core,

comprising 2 channel shaped reinforced concrete

coupled walls, and its relevant details are shown in Figs

4(a) and (b). All dimensions are expressed in terms of

the total height, h, of the building. Because referencedisplacements are strength-independent, only the pattern

of the lateral design forces need to be known. In terms

of a unit base shear, chosen for convenience, these are

given in Fig. 5(b).

Therefore, the overturning moments and shear forces

at each level of the cantilever structure with fullyrestrained base, are readily determined. They are

presented in Figs 4(c) and 5.

4.1. Wall properties

The aspect ratio of the individual walls with respect

to the full height, h, is Awi 1/0.1357.4. As Wall (1),

shown in Figs 4(a) and (b), is expected to be subjected to


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Fig. 4. Principal dimensions and bending moments relevant to a

coupled wall structure.

significant axial tension, the location of its neutral axis,measured from the tension edge, is estimated as sug-

gested in Section 3.1, with the aid of the information

provided in row 9 of Fig. 3, i.e. with the parameterx 0.83 0.70. Considerations of Wall (2), subjectedto gravity and lateral force-induced compression (row 8in Fig. 3), lead to a similar value of x 0.83 0.94.With the assumption that the yield strain relevant to this

example structure is e y 0.002, we find from eq. (4)that the yield curvature factor is h1 h2 1.55.

The important property of the walls, the nominal yield

curvature at the base, is thus from eq. (3)

fwy1 fwy2 1.55e y / 0.135h 0.023/ h (8)

That nominal yield curvatures for two walls, although

subjected very different axial loads, are, in this rather

exceptional case, about the same.

Fig. 5. Bending moments and shear forces applicable to the walls of

a coupled wall structure.

4.2. Assignment of component strength

As stated in section 3.3, the assignment of seismic

strengths to components of the coupled wall structure

should be the designers’ experience-based choice. Forthe purpose of estimating wall actions, satisfying equi-

librium criteria, the axes of the walls are assumed tocoincide with the centroidal axes of the gross concretesections. As Fig. 4(b) shows, the distance between these

axes is l 0.233h. In this example it has been decided

that b 0.56 (eq. (2)), i.e. l T 0.56 M o at the base.Hence the lateral force-induced axial force in the walls

is T max (0.56 × 0.711hV b)/0.233h 1.709V b. Con-

trary to traditional procedures [7,8,11], identical

strengths are assigned to coupling beams at all levels,

i.e. 1.709V b / 12 0.142V b. The moment increment

introduced by the coupling beams at each level is M 0.233h × 0.142V b 0.033hV b (Figs 4

and 5(a)).

As the lateral force-induced axial load on the walls,T , increases (stepwise) linearly to its maximum at the

base, the corresponding (stepped) wall moments

( M 1 M 2), are derived. These are shown by theshaded area in Fig. 4(c). With V b 1.00, the sum of

the base wall moments is thus

M 1 M 2 (0.711 0.233 × 1.709)h 0.313h (9)

i.e. 44% of the total overturning moment (eq. (1)). The

presentation in Fig. 4(c) of these moments is informative

because it shows clearly the effects of the chosen beam

strengths on the wall moment patterns. It is evident that


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at approximately he 0.57h above the base, wall

moments became negligibly small. This enables critical

wall deformations to be readily estimated. The same wall

moments are presented in the more conventional form

in Fig. 5(a).The final stage of strength assignment, not affecting

deformation estimates for the walls, involves the distri-bution of the required wall flexural strength, M1

M2, between components (1) and (2). This too can be

done arbitrarily. As wall (2) will be subjected to signifi-cant axial compression, it will be able to develop sig-

nificant flexural strength with only a modest quantity of tension reinforcement in the vicinity of the door open-

ings. For example the designer may choose a strength

ratio of M 2 / M 1 7/3. Therefore, the total shear force

to be assigned to the walls should be V1 0.3Vb andV20.7Vb, respectively. This is shown in Fig. 5(b). To

inhibit the interference of possible shear mechanisms

with the intended ductile response of walls, the nominal

shear strength of the walls, as constructed, needs to be

well in excess of that satisfying static equilibrium [7].

Fig. 5(b) also shows the chosen distribution over the

height of lateral static forces. In this case 92% of the

unit base shear was distributed in the traditional pattern

of an inverted triangle, while 8% of the base shear was

added to the lateral force at level 13. Modal shapes will

affect lateral force patterns relevant to elastic systems,the displacements of which, as in the cases studied here,

are controlled by full height walls. Once walls entered

the inelastic domain of response, higher modes of

vibrations will have negligible effect on overall system

displacements, such a shown in Fig. 8. Therefore, anytype of commonly used lateral design force pattern, lead-ing to displacements consistent with elastic first moderesponse, should be considered to be adequate for the

purpose of displacement estimates.

4.3. Wall deformations

The typical moment pattern, applicable to the walls

and shown in Figs 4(c) and 5(a), suggests that for the

purpose of displacement estimates, linear variation over

the height he 0.57h may be considered. This is

shown by the dashed line in Fig. 5(a). Hence the nominalyield deflection of the walls (eq. (5)) at that height maybe estimated by

y1 y2 fwyi h2e / 3 (10)

(0.023/ h)(0.57h)2 / 3 0.0025h

The slope of the walls, i.e. the drift in the 8th storey, is

qwy fwyihe / 2 (0.023/ h)(0.57h) / 2 (11)

0.0066 rad

These are two important quantities which enable dis-

placement limits for the ductile system subsequently to

be established.

4.4. Beam deformations

4.4.1. Conventionally reinforced coupling beams

A convenient form of expressing beam deformationsis by defining the chord rotation at the development of nominal yield curvatures, shown in Fig. 6(a) as qby

by / s, where by is the relative vertical displacement

of the ends of the beam with clear span s. The nominal

yield curvature for such a beam, with depth Db

0.018h, is estimated as

fby he y / Db 1.7 × 0.002/(0.018h) (12)

0.189/ h

The corresponding transverse beam displacement is

by

fby

s2 / 6 (0.189/ h)(0.045h)2 / 6 (13a)

0.064 × 103h

However, due to steel strain penetrations at the beam bar

anchorages, particularly after a few elastic displacement

reversals, additional beam displacements must beexpected. It is assumed that this anchorage deformation,

a, is in the order of yield strain over 8 times the diam-

eter, d b, of bars in tension [7].

In the example structure d b0.55 × 103h, and

hence a

8 × 0.002 × 0.55 × 103h 9 × 106h. The cor-

responding beam deflection is

by (s / Db) a

(0.045/0.018)9 × 106h (13b)

Fig. 6. Sources of nominal yield displacements in reinforced concrete

coupling beams.


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0.023 × 103h

Therefore, the nominal yield chord rotation of a conven-

tionally reinforced coupling beam will be of the order of

qby (by by) / s (0.064 (14)

0.023)103h / (0.045h) 0.0019 rad

4.4.2. Diagonally reinforced coupling beams

Early studies [17,18] indicated that to avoid sliding

shear failures in the inelastic regions of conventionally

reinforced coupling beams, which are subjected to highshear demands, diagonal reinforcement could be used.

Typical details are shown in Fig. 6(b). Such beams have

been first used in New Zealand and subsequently inmany other countries. Their design and behaviour has

been extensively reported [7,11].Occasionally claims are made [8] that, because of the

reduced inclination of the diagonal reinforcement inslender beams, shear resistance becomes inef ficient.Such claims fail to recognize the simple equilibrium-dic-

tated fact that, irrespective of the inclination, a , (Fig.6(b)) diagonal steel forces can resist simultaneously thetotal moment and shear generated by earthquake-

imposed chord rotations. Test beams with bar incli-

nations as small as a 6, exhibited [19], as expected,Ramberg-Osgood type of hysteretic response with mod-

erate stiffness degradation and displacement ductility

capacities in the order of 14.

The properties of such beams [7,15,17] are:The nomi-

nal strength, in terms of the shear force sustained by

diagonal bars with area Asd , as shown in Fig. 6(b), is

V bn 2 Asd f ysina (15)

where f y is the yield strength of the steel used.

The elongation of the diagonal bars in tension is

T (s / cosa 16d b)e y (16a)

where, as in section 4.4.1, allowance was also made foranchorage deformations. For the beams of the structure

shown in Fig. 4(a), a 18 and d b0.55 × 103h.

Therefore,

T (0.045/ cos18

16 × 0.55 × 103)0.002h (16b)

0.112 × 103h

The shortening of the diagonal compression chord,C , depends on the ratio of diagonal reinforcement used.

An approximation, acceptable for seismic design pur-poses and in agreement with observed magnitudes [18],

results in C 0.3T .

The relative vertical displacement at the ends of the

this beam is

by 1.3T / (2sin a ) (17a)

The nominal yield chord rotation of the beam, as

shown in Fig. 6(b), is thus

qby by / s (17b)

which is found for the example structure to be

qby

1.3 × 0.112 × 103h / (0.045h × 2 × sin 18°) (17c)

0.00524 rad

If there is any effective horizontal reinforcement present,

for example in a flange formed by a floor slab (Fig. 6(a)),the flexural resistance of the coupling beam will corre-spondingly increase at one end only. The contribution

of such reinforcement, subjected to tension only, can be

readily determined [7]. Strength enhancement will how-ever, diminish during hysteretic response of the beam.

Such horizontal reinforcement will increase beam

strength only when the imposed ductility demand is

larger than any previously imposed one. The partici-

pation in strength development of such bars is similar to

those placed in tension flanges of beams in frames.In some experimental studies [20] it has been found

that, when the elongation of coupling beam test speci-

mens is prevented during cyclic and reversing loadingby artificial restrainers, other forms of diagonalreinforcement are likely to result in better ductile

response. In real structures full restraint of beam elonga-

tions does not exist. Moreover, in axially restrained

beams, the contribution of shear forces by means of a

diagonal concrete compression field is significant. Under

reversing inelastic displacements the deterioration of thecompressed concrete eventually leads to drastic loss of beam resistance. As the model in Fig. 6(b) suggests, in

the elastic range of response, diagonal forces associated

with shear transfer can be sustained predominantly, and

in many cases entirely, by the reinforcement without any

reliance on concrete compression strength.

4.5. Relationships between beam and wall

deformations

In this section the estimated displacements of the

walls and the critical pair of coupling beams, associated

with 3 distinct limit states, are compared. These states

refer to (i) the elastic limit of wall response, (ii) accept-

able maximum storey drift and (iii) the displacement

ductility capacity of the walls.

4.5.1. At the attainment of the nominal yield

displacement of walls

Eq. (11) estimated the maximum wall rotation, i.e. sto-

rey drift, associated with the nominal yield curvature atthe wall base. The relationship between traditionally

evaluated [7] rotations of two identical rectangular walls,

based on E c I e, and the coupling beam chord rotations


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Fig. 7. Relationships between wall and coupling beam rotations.

are shown in Fig. 7(a). To estimate the relative vertical

displacements of the walls, based on the traditional

definition of axial stiffness [7,11], E c Ae, allowance isalso made for axial deformations of the walls, shown as1 and 2 in Fig. 7(a). The simulation implies that, under

the lateral force-induced axial compression load wall (2)

Fig. 8. Wall deformations associated with three limit states.

shortens by 2. However, this contradicts the fact that

after cracking walls expand vertically. Therefore, a more

realistic estimate of the differential axial deformations

of walls can be made if rotations are related to positions

of the neutral axes of the cracked elastic walls. In thedetermination of these, the simultaneous actions of

moment and axial force need be taken into account. Withthis simulation, shown in Fig. 7(b) the ratio of the beam

chord rotation, qb, and the wall rotation, qw, at a givenlevel is

qb / qw ( Dw c1 c2) / s w (18)

where the relevant dimensions are defined in Fig. 7(b).This magnification factor, w , affects dramatically dis-placement demands on coupling beams. For the channel

shaped walls of the example structure it was estimatedthat c1 c2 0.023. In this case the rotation magnifi-cation relevant to the beams is simply

w Dw / s 0.135/0.045 3 (19)

Hence with the known nominal yield rotation of the

walls, given by eq. (11), the beam chord rotation at level8 is estimated as

qb wqwy 3 × 0.0066 0.02 rad (20)

This is significantly larger than the nominal yield chordrotation of conventionally reinforced coupling beams,

given by eq. (14). The displacement ductility imposed

on the critical coupling beam at this elastic limit stage

of the walls is, therefore, of the order of

mb

qb / q

by 0.02/0.00191 10.5 (21)

a magnitude which would be dif ficult to sustain withoutsignificant loss of beam strength.

However, if diagonally reinforced beams are used,

from eq. (17c) it is found that

mb 0.02/0.00524 3.8 (22)

It may be readily shown that at the development of

nominal yield curvatures at the wall bases, all diagonally

reinforced coupling beams would have yielded. There-

fore, the development at this stage of all strength compo-nents, M 1, M 2 and l T , shown in Fig. 1(c), can be

expected.

4.5.2. At the attainment of the limiting storey drift

It is assumed that the adopted performance criterion

restricted the maximum storey drift to d u 1.5%. Eq.(11) established that at the nominal yield of the walls

the critical drifts was qwy d y 0.66%. Theadditional drift, requires plastic hinge rotations at the

wall base. The recommended [7] effective length of a

plastic hinge of the wall is

l p 0.2 Dw 0.044he (0.2 × 0.135 (23)

0.044 × 0.57)h 0.052h


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i.e. 40% of the length of the walls. The necessary lateral

post-yield displacement at the level of maximum drift

needs to be

p (he 0.5 l p)d p

(0.57 0.5 × 0.052)0.0084h 0.0046h(24)

The total displacement at level he, using eq. (10), is thus

u y1 p (2.5 4.6)103h (25)

0.0071h

implying a displacement ductility demand on the walls

of

mw u / y1 7.1/2.5 2.8 (26)

The corresponding chord rotation of the diagonally

reinforced coupling beams at level 8 will be of the order

of qbm wd u 3 × 0.015 0.045 rad. From eq.(17c) the displacement ductility demand on this pair of

beams is thus

mb qbm / qby 0.045/0.005248.6 (27)

which, with eq. (17a), translates into a steel tensile strain

ductility of the order of m 8.6/1.3 6.6. Themaximum steel tensile strain is thus esmax 1.3%.

4.5.3. At the attainment of the displacement ductility

capacity of the walls

Assuming that the displacement ductility capacity of

adequately detailed walls [7] is 5, the additional inelasticdisplacement of (5-2.8) y1 2.2 × 0.0025h

0.0055h of the walls would be acceptable. However,the associated lateral displacement near level 8 of max 5 × 0.0025h 0.0125h (Fig. 8) would

increase the maximum storey drift to d max 2.5%. Thedisplacement ductility on the critical coupling beam andthe maximum steel strain would increase to mb

14.3 and esmax 2.2%, respectively.

4.6. System response

The deformed shapes of the walls and lateral displace-ments just below level 8 and associated with the pre-

viously defined 3 limit states, are shown in Fig. 8.For the purposes of seismic design, the bi-linear

modeling of the ductile response of this example struc-

ture, as shown in Fig. 9, was claimed to be entirely

adequate. Lateral displacements of the walls shownrelate to the level of accelerated mass at hm 0.71h

above the base. These displacements can be readilyextrapolated from those previously evaluated at a lower

level, i.e. at he,. Strength increase with post-yield defor-

mations, having negligible influence on the response of the system, have not been considered. Displacement duc-

tilities, associated with the 3 selected limit states, are

also recorded in Fig. 9. This simple modeling, based on

Fig. 9. Bi-linear modeling of the ductile behaviour of a coupled

wall system.

realistic displacement estimates, for a single mass sys-

tem, may well replace popular pushover analyses tech-

niques.

As the data in Fig. 3 suggest, displacements during

the first elastic response of the structure can be predictedby bilinear modeling only if strength demands on the

walls do not exceed approximately 80% of their nominal

strength. Under the same circumstances the onset of yielding in some coupling beams can be expected at less

than 50% of the nominal strength of the structure.

The choice of the contribution to the total flexuralstrength of the system by the coupling beams, that is,

the l T component seen in Fig. 1(c), determines the height

at which the maximum storey drift can be expected. If

the l T / M o ratio would have been chosen 0.75, instead of 0.56, the maximum drift should have been expected inthe 5 storey, i.e., at he 0.38h. The corresponding

moments to be resisted by the walls are shown in Fig. 4

by the dashed stepped lines. This choice, requiring 34%

increase of beam strengths, would have led to 33%

reduction of the critical nominal yield drift. The inelastic

contribution of the walls at the 1.5% drift limit,

expressed in terms of their displacement ductility, mw,would have increased from 2.8 to 4.3. This alternative

illustrates how more ef ficient utilization of energy dissi-pation and hysteretic damping could be achieved by

deliberate increase of the contribution of the coupling

system to the resistance of overturning moments.

It is re-emphasized that, as eq. (5) has shown, dis-

placement limitations are strongly influenced by theyield strength of the reinforcement used. For example if steel with 25% larger strength, i.e. e y 0.0025, was

to be used, nominal yield displacements would corre-spondingly increase. At a drift limit of 1.5% the ductility

demand on the walls would reduce from 2.8 to 2.2. In

current force-based seismic design methods [21], thecorresponding design base shear for the system would

be increased, negating partly the economic advantages

which the use of higher strength steel would offer.


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5. Concluding remarks

To satisfy the intents of performance-based seismic

structural design, the importance of more realistic pre-

dictions of target displacement capacities should be more

widely recognized. For reinforced concrete structures,

addressed here, such displacement limits can be readilyand realistically predicted in a rather simple way withoutthe knowledge of the eventual seismic strength required.

Therefore, displacement estimates made during the pre-

liminary stage of the design, can immediately expose

undesirable features of the contemplated structural sys-

tem.

The use of a number of simple principles, often over-

looked or ignored in seismic design, was demonstrated.

These include: (a) The stiffness of a reinforced concrete

component depends on the strength eventually assigned

to it. Therefore, element or system stiffness cannot be a

priory assumed. (b) The nominal yield curvature of a

reinforced concrete section, and all displacements of acomponent associated with it, are insensitive to the

flexural strength of the section. (c) Because deformationlimits, applicable to components of ductile system, are

independent of the strength, the latter can be arbitrarily

assigned to them. This enables the astute designer to dis-

tribute the required seismic strength among components

so that more economical and practical solutions are

obtained.The estimation of displacement capacities of compo-

nents of a system, such as a coupled wall structure,

enables the critical component to be identified. Hence,

instead of assuming global ductility factors for structuralsystems, their displacement and hence ductility capacity

should be made dependent on that of the critical compo-nent. Such relationships can be established before

strengths are assigned to components.The approach, illustrated with the aid of an example

coupled wall structure, can be readily incorporated into

existing strength-based seismic design methods. Its

major appeal relates, however, to displacement-based

design strategies.

Coupled wall structures offer distinct advantages such

as: (i) very good displacement control, (ii) a strong coup-

ling system allows the use of slender walls without jeop-

ardizing drift limits, (iii) displacement limits during duc-tile response are not affected by higher mode dynamic

effects, (iv) with appropriate detailing of the reinforce-

ment, they can be expected to deliver larger hysteretic

damping than any other conventionally constructed

reinforced concrete system.

Acknowledgements

The contribution of Rolando Castillo to some of the

data presented, using moment-curvature analyses, is

gratefully acknowledged.

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