The displacement capacity of reinforced concrete coupled walls
-
Upload
crimsonposh -
Category
Documents
-
view
213 -
download
0
Transcript of The displacement capacity of reinforced concrete coupled walls
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
1/11
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].
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
2/11
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
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
3/11
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
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
4/11
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
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
5/11
1169T. Paulay / Engineering Structures 24 (2002) 1165 – 1175
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
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
6/11
1170 T. Paulay / Engineering Structures 24 (2002) 1165 – 1175
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
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
7/11
1171T. Paulay / Engineering Structures 24 (2002) 1165 – 1175
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.
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
8/11
1172 T. Paulay / Engineering Structures 24 (2002) 1165 – 1175
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
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
9/11
1173T. Paulay / Engineering Structures 24 (2002) 1165 – 1175
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
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
10/11
1174 T. Paulay / Engineering Structures 24 (2002) 1165 – 1175
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.
-
8/18/2019 The displacement capacity of reinforced concrete coupled walls
11/11
1175T. Paulay / Engineering Structures 24 (2002) 1165 – 1175
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.
References
[1] Chitty L. On the cantilever composed of a number of parallel
beams interconnected by cross bars. The London, Edinburgh and
Dublin Philosophical Magazine and Journal of Science
1947;38:685–99.
[2] Beck H. Ein neues Berechnungsverfahren für gegliederter
Scheiben dargestellt am Beispiel der Vierendelträgers. Der Bauin-genieur 1956;31(12):436–43.
[3] Albiges M, Goulet J. Contreventment des batiments. Annales de
l’Institute Technique du Batimants et des Travaux Public
1960;13(149):473–500.
[4] Rosman R. Beitrag zur statischen Berechnung waagrecht belas-
teter Querwände bei Hochbauten. Der Bauingenieur
1960;35(4):133–136,1962;37(1) :24–26,(8);303–308.
[5] Beck H. Contribution to the analysis of coupled shear walls. Pro-
ceedings ACI Journal 1962;59(8):1055–70.
[6] Coull A, Choudhury JR. Analysis of coupled shear walls. Pro-
ceedings ACI Journal 1967;64(9):587–93.
[7] Paulay T, Priestley MJN. Seismic Design of Reinforced Concrete
and Masonry Buildings, p. 767, Wiley.
[8] Harries AK. Ductility and deformability of coupling beams in
reinforced concrete coupled walls. Earthquake Spectra2001;17(3):457–78.
[9] Paulay T. A simple displacement compatibility-based design
strategy for reinforced concrete buildings. Proceedings of the
12th World Conference on Earthquake Engineering, Auckland,
New Zealand, 2000. Paper No.0062
[10] NZS 3101:1995, Standards New Zealand. Concrete Structures
Standard Part 1 — The design of concrete structures, p. 256, Part
2 — Commentary on the design of concrete structures, p. 264
[11] Park R, Paulay T. Reinforced Concrete Structures, p. 786.
Wiley, 1975.
[12] Saatcioglu M, Derecho AT, Corley WG. Parametric study of
earthquake-resistant coupled walls. ASCE Journal of the Struc-
tural Division 1987;113(1):141–57.
[13] Paulay T. A re-definition of of stiffness of reinforced concrete
elements and its implications in seismic design. Structural Engin-eering International 2001;11(1):36–41.
[14] Paulay T. Some seismic design principles relevant to torsional
phenomena in ductile buildings. Journal of Earthquake Engineer-
ing 2001;5(3):273–308.
[15] Paulay T. Seismic response of structural walls: recent develop-
ments. Canadian Journal of Civil Engineering 2002;28:in.
[16] Priestley MJN, Kowalsky MJ. Aspects of drift and ductility
capacity of rectangular walls. Bulletin of the New Zealand
Society for Earthquake Engineering 1998;31(2):73–85.
[17] Paulay T. Diagonally reinforced coupling beams of shear walls.
ACI Special Publication SP-42 1972;1:579–98.
[18] Paulay T, Santhakumar AR. Ductile behaviour of coupled shear
walls. Journal of the Structural Division, ASCE 1976;102:ST1.
[19] Paulay T, Spurr DD. Simulated seismic loading on reinforced
concrete frame-shear wall structures. Sixth World Conference onEarthquake Engineering, New Delhi 1977;3:221–6.
[20] Galano L, Vignoli A. Seismic behaviour of short coupling beams
with different reinforcement layouts. ACI Structural Journal
2000;97(6):876–85.
[21] Priestley MJN. Myths and fallacies in earthquake engineering —
Conflicts between design and reality. American Concrete Insti-
tute, (SP-157) Recent developments in lateral force transfer in
buildings, 231-257