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Journal of Earthquake Engineering, Vol. 10, No. 1 (2006) 73–96
c Imperial College Press
OPTIMAL PERFORMANCE-BASED SEISMIC DESIGN
USING MODAL PUSHOVER ANALYSIS
DONALD E. GRIERSON∗
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Department of Civil Engineering, University of Waterloo
Waterloo, Ontario, N2L 3G1, Canada
grierson@uwaterloo.ca
YANGLIN GONG
Department of Civil Engineering, Lakehead University
Thunder Bay, Ontario, P7B 5E1, Canada
LEI XU
Department of Civil Engineering, University of Waterloo
Waterloo, Ontario, N2L 3G1, Canada
Received 30 April 2004
Reviewed 13 June 2005
Accepted 20 June 2005
The paper presents a computer-automated performance-based design procedure for optimally proportioning a steel building framework to resist earthquakes. Modal pushover
analysis is employed to evaluate structural demands imposed by design earthquakes
representing a range of hazard levels (e.g. immediate-occupancy, life-safety and collapseprevention). First-order Taylor series and sensitivity analysis are employed to formulate
corresponding performance constraints explicitly in terms of member sizing variables
and target lateral displacements. A generalised optimality criteria algorithm is employed
to find the least-weight structure that experiences minimal damage (as characterised by
ductility demand) while simultaneously satisfying all performance constraints at all hazard levels. The design methodology is illustrated for a nine-storey planar steel building
framework.
Keywords: Earthquake engineering; steel building frameworks; performance-based
design; modal pushover analysis; sensitivity analysis; structural optimisation.
1. Introduction
Since the publication of FEMA-273 [1997], the performance-based seismic design
(PBSD) concept is gaining acceptance in professional practice. This is reflected
in subsequent documents regarding seismic rehabilitation of existing buildings
[FEMA-356, 2000] and seismic design of new building structures [FEMA-350,
2000]. Albeit, but few publications exist on the topic of design optimisation using
performance-based criteria. Foley et al. [2003] summarised the state of the art
∗ Corresponding
author. Tel.: +1-519-888-4567-x2412; fax: +1-519-888-6197
73
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D. E. Grierson, Y. Gong & L. Xu
in performance-based design optimisation. Gong et al. [2003] proposed a multicriteria performance-based design optimisation methodology using conventional
single-mode pushover analysis. Minimum earthquake damage was adopted as one
of two design criteria, the other being minimum weight. Ganzeri et al. [2000] integrated pushover analysis with the performance-based design concept for the design
optimisation of reinforced concrete portal frames. The design objective was the
minimisation of material cost. Only performance at one hazard level, immediate
occupancy, was considered in the design formulation, while performance at life
safety and collapse prevention hazard levels was checked by a final pushover analysis. Charney [2000] expressed the need for a broader set of basic limit states and
improved analysis procedures to implement performance-based design optimisation.
Beck et al. [1999] developed a general framework for optimal design suitable for
performance-based design of structural systems operating in an uncertain dynamic
environment. A preference function, which consisted of multiple criteria, was constructed as the objective function. However, the work was applicable only within
the range of elastic behaviour.
Four different analytical methods have been variously proposed for use in conjunction with the performance-based seismic design methodology: Linear static,
linear dynamic, nonlinear static (pushover) and nonlinear dynamic analysis procedures [FEMA-273, 1997]. The use of nonlinear dynamic analysis is often not justified
in a number of cases, but it can play a significant role in assessing a design once
completed. Pushover analysis is often seen to be a viable and attractive alternative to dynamic analysis. The essential feature of pushover analysis is that it is
a nonlinear procedure in which monotonically increasing lateral loads along with
constant gravity loads are applied to a framework until a control node (usually
referred to the building roof) sways to a predefined ‘target’ lateral displacement
corresponding to an earthquake hazard level. Structural deformation and member
forces are monitored continuously as the model is displaced laterally. The method
can oftentimes reasonably trace the sequence of yielding and failure at the member and system levels, respectively, such as to determine the inelastic drift distribution over the height of the building and the final collapse mechanism for the
structure.
In spite of the simplicity of pushover analysis, at times component and system deformation demands may not be evaluated with the precision required for
some structures. The prescribed lateral inertia load pattern for pushover analysis
is based on the premise that the response of the multi-degree-of-freedom structure
is mainly controlled by a single vibration mode, and that the shape of this mode
remains invariant throughout the time history. Generally, the fundamental vibration
mode for the structure is selected as the dominant response mode and the influence of the higher modes is ignored. For this reason, pushover analysis is usually
only used for predicting the seismic displacement response of low-rise to mid-rise
buildings.
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Optimal Performance-Based Seismic Design
75
Pushover analysis has been extensively applied to evaluate structural seismic
demand, including the works by Saiidi and Sozen [1981], Lawson et al. [1994],
Biddah and Naumoski [1995], Kilar and Fajfar [1997], Krawinkler and Seneviratna
[1998], Gupta and Krawinkler [1999], Kunnath and John [2000], and others. Interest in developing new pushover analysis techniques has increased among researchers
in recent years. Hasan et al. [2002] proposed a pushover analysis method by introducing ‘fictitious plastic-hinge connections’ at the ends of beam-column elements.
Non-dimensional ‘plasticity-factors’ were introduced to monitor the progressive
plastification (stiffness degradation) of building members under increasing lateral
loads. Bracci et al. [1997] and Gupta and Kunnath [2000] have proposed adaptive
pushover procedures which account for the effect of higher modes and time-varying
structural stiffness. Kunnath [2004] investigated modal combination schemes to
account for higher mode effects. Elnashai and his co-workers [Mwafy and Elnashai,
2001; Elnashai, 2002] have conducted extensive comparisons between dynamic and
pushover analyses in order to identify the domain where a pushover analysis is
valid. Chopra and Goel [2002] introduced the Modal Pushover Analysis (MPA) concepts to overcome the limitation of the conventional single-mode pushover analysis
method which ignores the contributions of other modes. The MPA method evaluates
seismic demand in two stages: (i) Multiple single-mode pushover analyses are first
carried out for different vibration modes (e.g. modes 1, 2, 3, etc.) to determine the
corresponding modal responses at target displacement levels; (ii) the total structural
response is then estimated by combining the multiple mode responses according to
an appropriate modal combination rule. In their paper, Chopra and Goel [2002]
proved that modal pushover analysis was equivalent to standard response spectrum analysis in the elastic range. Through comparison with the results of nonlinear dynamic analysis, they demonstrated that the MPA method provides good
estimations for floor displacements, interstory drifts, and location of plastic hinges,
but that it does not seem to predict plastic hinge rotations satisfactorily. They concluded that the MPA method is accurate enough for practical application in building
evaluation and design. The application of modal pushover analysis in the context of
performance-based seismic design requires considerable computational effort since
the iterative synthesis process involves many re-analyses of the structure. For practical use in engineering offices, the optimal performance-based seismic design of steel
building frameworks requires a computational algorithm that efficiently integrates
the modal pushover analysis together with a design optimisation methodology.
2. Load- and Displacement-Control Pushover Analysis
Most studies in the literature define earthquake hazard levels by displacement limits. For example, a framework that undergoes a roof drift of 0.7%, 2.5% or 5% of
the building height is taken to be at the Immediate Occupancy (IO), Life Safety
(LS) or Collapse Prevention (CP) hazard level, respectively [FEMA-273, 1997]. A
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D. E. Grierson, Y. Gong & L. Xu
displacement-control pushover analysis procedure is thus required to evaluate seismic demand at the corresponding displacement limit. However, such an analysis
procedure is not applicable for general seismic design, since structural designs are
not based on displacements but on loads specified by governing codes and standards.
For example, the magnitude of the base shear due to an earthquake is determined
from a corresponding acceleration spectrum specified by a design code or standard.
To evaluate the seismic demand imposed on a structure by the earthquake-induced
base shear, a load-control pushover analysis procedure can be applied. That is,
instead of terminating the pushover analysis when a maximum specified displacement limit is reached at a target node (e.g. the roof), the analysis is instead terminated when a maximum specified design load (e.g. base shear) limit is reached.
Since the lateral stiffness of a structure at its point of incipient plastic mechanism formation is so small that trivial increase in lateral forces will cause significant increase in displacements, many researchers believe that seismic demand
evaluations should be based on displacement rather than force criteria. In other
words, displacement-control pushover analysis is often thought to be more rational
than load-control pushover analysis. However, Gupta and Kunnath [2000] demonstrated that the load-control procedure (called spectrum-based analysis in the literature) can reasonably predict seismic demands that earthquakes impose on buildings
when the acceleration response spectrum is based on site-specific ground motions.
Gong [2003] illustrated that a new building structure can be first designed using
a load-control pushover analysis, and then its performance can be checked using a
displacement-control pushover analysis.
If a gradient-based optimisation technique is used to conduct the performancebased design, a load-control pushover analysis procedure is more desirable for the
design of new structures, since it allows the direct evaluation of the sensitivities of
design base shears [Gong et al., 2005]. On the contrary, a displacement-control procedure is likely to be more suitable for the pushover analysis of existing structures
to evaluate their need for seismic rehabilitation, as specified by FEMA-273 [1997].
For this study, which concerns the design of new structures, design earthquakes are
represented by acceleration spectra, which translates to the design being directly
governed by specified loads and, hence, a load-control modal pushover analysis procedure is adopted to evaluate seismic demands for the performance-based design
optimisation process.
3. Modal Pushover Analysis
3.1. Single-mode pushover analysis
Upon introducing ‘potential plastic-hinges’ at both ends of a general beam-column
member (e.g. see Fig. 1 for a planar member), the corresponding member stiffness
matrix K can be found as [Xu, 2001; Hasan et al., 2002]:
K = K e Ce + K g Cg ,
(1)
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2
1
5
p1
E, I, A
@
p2
77
6
@
4
3
L
@ represent a plastic hinge
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Fig. 1.
Planar beam-column element.
where Ke and Kg are the standard elastic and geometric stiffness matrix respectively, and Ce and Cg are the corresponding correction matrices expressed in terms
of plasticity-factors p that characterise the degradation of flexural stiffness due to
post-elastic deformation. The plasticity-factor p at each member end is defined as
p=
1
,
1 + 3EI/Rp L
(2)
where Rp is the instantaneous rotational stiffness of the plastic hinge section, and
L and EI are the length and elastic flexural stiffness of the member, respectively.
For a fully elastic section Rp = ∞ and p = 1, for a fully plastic section Rp = 0 and
p = 0, while for a partially plastic section 0 < p < 1.
3.1.1. Single stress yield condition
The moment-rotation (M –φ) relation that characterises the nonlinear variation in
the post-elastic rotational stiffness Rp of a plastic-hinge section under increasing
moment (see Fig. 2) is taken to be [Hasan et al., 2002]:
M (φ) = My + (Mp − My )2 − [(Mp − My )(φp − φ)/φp ]2 ,
(3)
where My = Sσye and Mp = Zσye are the first-yield and fully-plastic moment
capacities of the beam section, respectively (S and Z are the elastic and plastic
Fig. 2.
Post-elastic moment-rotation relation.
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D. E. Grierson, Y. Gong & L. Xu
section moduli, respectively, and σye is the expected yield stress of the material),
and φ is the extent of post-elastic rotation occurring somewhere between first yielding (φ = 0) and full plastification (φ = φp ) of the cross-section. From Eq. (3), the
post-elastic moment varies in the range My ≤ M (φ) ≤ Mp as the plastic rotation
varies in the range 0 ≤ φ ≤ φp . Upon differentiating Eq. (3) with respect to φ, the
post-elastic rotational stiffness of the plastic hinge is found as
Rp =
(Mp − My )2 (φp − φ)
dM (φ)
= 2
dφ
φP (Mp − My )2 − [(Mp − My )(φp − φ)/φp ]2
(0 < φ ≤ φp ).
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(4)
Rewriting Eq. (3) as,

φ = φp 1 −
1−
M/My − 1
ς −1
2


(0 < φ ≤ φp ),
(5)
where M = M (φ) and ς = Mp /My = Z/S is the section shape factor, and substituting for φ from Eq. (5) into Eq. (4), the post-elastic rotational stiffness is given
by,
2
ς −1
(ς
−
1)M
y
p
− 1 (0 < φ ≤ φp ).
(6)
R =
φp
M/My − 1
3.1.2. Combined stress yield condition
Though the foregoing developments were made through reference to the pure bending case, they are readily extended to account for combined stress states. For example, consider the case of combined bending moment M and axial force N for members of planar frameworks. The reduction in the moment capacity of a member
cross-section due to the presence of axial force can be accounted for through the
following interaction constraint equation having lower and upper bounds that correspond to first-yield and fully-plastic behaviour, respectively [Hasan et al., 2002]:
M
1
≤
+
ς
Mp
N
Np
a
≤ 1,
(7)
where the shape factor ς depends on the cross-section type (e.g. ς = 1.12 ∼ 1.16
for wide-flange steel beam sections); Np = Aσye is the fully-plastic axial force
capacity (where A is the cross-section area); and the exponent a depends on the
cross-section shape (e.g. a = 2 for a rectangular section, while a = 1.2 ∼ 1.5 for a
wide-flange section [see e.g. Duan and Chen, 1990]).
The two bounds of Eq. (7) can be viewed as defining the ‘plasticity domain’
shown in Fig. 3. Assuming that the ratio M/N remains constant in the post-elastic
response range for the combined stress case, identical satisfaction of the lower bound
of Eq. (7) at a generic point Oy in Fig. 3 corresponds to first-yield behaviour
occurring at the reduced yield moment level Myr = Mp /ξς (where ξ > 1), while
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Optimal Performance-Based Seismic Design
Fig. 3.
79
Idealised plasticity under combined bending moment and axial force.
identical satisfaction of the upper bound of Eq. (7) at related point Op corresponds
to fully-plastic behaviour occurring at the reduced plastic moment Mpr = Mp /ξ =
ςMyr . Upon replacing My and Mp with the reduced capacities Myr and Mpr , Eqs. (3)
and (4) then respectively define post-elastic moment-rotation and flexural-stiffness
relations that account for the influence of axial force on bending moment capacity
of member sections, and the post-elastic analysis can proceed exactly as for the
pure bending case [Hasan et al., 2002; Gong, 2003].
3.1.3. Basic analysis concepts
The pushover analysis uses centreline dimensions to model members. Rigid beamto-column connections are assumed. Panel zone deformation is neglected. Stability
effects are accounted for in Eq. (1) by the full geometric stiffness matrix Kg for
beam-column members. The incremental load-step method (also called the Euler
method, see McGuire et al. [2000]) is adopted to conduct the analysis. The first
loading step is taken arbitrarily small so that the structure will remain elastic. The
structure global stiffness matrix is initially formed from member stiffness matrices
defined by Eq. (1) by setting all plasticity-factors equal to unity (p = 1) and all
element axial forces equal to zero. After each load increment, the structure tangent
stiffness matrix is updated for member stiffness matrices modified by plasticityfactors p and axial forces found from the previous loading step. A detailed description of the single-mode pushover analysis procedure is given by Hasan et al. [2002].
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3.2. Modal pushover analysis
In this section, a load-control (or so called spectrum-based) multi-mode pushover
analysis procedure is described that retains the conceptual simplicity and computational attractiveness of the single-mode procedure. The effective modal mass Mk
for the kth vibration mode is defined as [Chopra, 2001]:
Mk =
(φTk M I)2
,
φTk Mφk
(8)
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where φk is the mode shape; M is the lump-mass matrix; and I is the unity vector.
The lateral load profile vector for the kth mode is determined as
Ck = Mφk
(9)
and is assumed to be invariant over the loading history, which is equivalent to
assuming that φk is invariant. The total structural response is obtained by applying
a modal combination rule, such as the square-root-of-the-sum-of-squares (SRSS)
rule [Chopra, 2001], i.e.
1/2
nm
uc =
2
(uk )
,
(10)
k=1
where uc is the combined structural response; uk is the structural response associated with the kth mode; and nm is the number of vibration modes under consideration. The design earthquake base shear at hazard level i for the kth mode is
calculated from
i
i
= Mk Sa,k
Vb,k
(i = hazard level),
(11)
i
is the spectral acceleration response corresponding to the earthquake
where Sa,k
i
value depends on the
associated with hazard level i for the kth mode. Each Sa,k
site conditions, damping, hazard level, and elastic period Tk of the kth mode and,
for this study, is defined by the following design response spectrum [FEMA-273,
1997],
 i
Ss (0.4 + 3Tk /T0i )
0 < Tk ≤ 0.2T0i



 i
Ss
0.2T0i < Tk ≤ T0i
i
=
(i = hazard level),
(12)
Sa,k
i


S
 1

Tk > T0i
Tk
where T0i is the period at which the constant acceleration and constant velocity
regions of the response spectrum intersect for the design earthquake associated
with hazard level i, and Ssi and S1i are corresponding short-period and one-secondperiod response acceleration parameters, respectively. The elastic period of the kth
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Optimal Performance-Based Seismic Design
81
mode Tk is found as [IBC, 2000]:
ns
(ms vs2 )
Tk = 2π · nss=1
s=1 (V1 Ck,s vs )
1/2
,
(13)
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where ms is the seismic mass of storey s; ns is the number of building storeys; and
Ck,s are the entities of vector Ck defined by Eq. (9). V1 is a base shear force taken
to be sufficiently small to ensure that the resulting lateral displacement vs of storey
level s corresponds to elastic behaviour of the structure.
The load-control MPA procedure is then carried out as follows:
(i) Determine the mode shapes φk under consideration (k = 1, 2, . . . , nm ).
(ii) Evaluate effective modal masses Mk (k = 1, 2, . . . , nm ) by Eq. (8).
(iii) Determine modal lateral load distribution vector Cv,k (k = 1, . . . , nm ) by
Eq. (9).
(iv) Set modal index k = 1 (the gravitational loads are only applied for the first
mode).
(v) Calculate the kth modal period Tk by Eq. (13).
(vi) Compute the kth modal response acceleration by Eq. (12).
(vii) Calculate the kth modal design base shears by Eq. (11).
(viii) Perform single-mode pushover analysis for the kth mode.
(ix) Set k = k + 1; if k > nm , go to step (x), otherwise go to step (v).
(x) Evaluate combination of modal responses according to Eq. (10).
In step (iv) of the foregoing procedure, gravity loads are included with the first mode
analysis alone [Chopra and Goel, 2002] to ensure that gravity effects are accounted
for only once when multiple modes are considered through the combination rule, Eq.
(10). In step (x), the SRSS rule is used to compute combined responses for bending
moments, Mc , axial forces, Nc , and displacements. Thereafter, the combined plastic
state of the structure is determined through Eqs. (7), (6) and (2), for M and N
replaced by Mc and Nc , and the ratio Mc /Nc assumed to remain constant in the
post-elastic response range (see Fig. 3).
4. Design Problem Formulation
4.1. Objective functions
Minimum structural weight is adopted as one of two design objectives, and is formulated as
f1 (x) =
1
Wmax
n
ρLj Aj ,
(14)
j=1
where x is a generic representation of design variables; ρ is the material mass
density; n is the number of members; and Lj and Aj are the length and crosssection area of the jth member, respectively. Section areas are design variables,
while beam spans and storey heights remain constant. The weight function f1 is
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D. E. Grierson, Y. Gong & L. Xu
normalised by the maximum possible weight of the frame Wmax = ρLj AU
j , where
AU
j is the upper-bound cross section area for member j (e.g. the maximum area
of commercially available standard steel sections applicable for member j [see, e.g.
AISC, 1994]).
In addition to minimising the structure cost, minimising the damage of structures under earthquake loading is another desirable objective. To this end, it is necessary to quantitatively express damage in terms of structural response. Interstorey
drift is the primary parameter in evaluating structural performance [FEMA-350,
2000], and is widely regarded as a major parameter characterising the extent of
plastic deformation of a structure. Therefore, it is possible to quantify structural
damage in terms of interstorey drift. This study quantifies structural damage in
terms of the post-elastic interstorey drift distribution of a building framework at
the extreme collapse prevention (CP) hazard level. It has been observed in many
collapsed structures that plastic deformation concentration occurs at a soft (or
weak) storey under severe earthquake loading. It is reasonable to assume that the
structure will undergo less damage if this deformation concentration can be mitigated; i.e. that less damage will occur if the structure exhibits a uniform interstorey
drift distribution, since minimum deformation concentration occurs when all the
storeys of a structure have the same interstorey drift. Thus, the damage-mitigating
objective can be stated as pursuing a uniform interstorey drift distribution in the
post-elastic response range since this is equivalent to achieving a uniform ductility
demand in all storeys. Since a linear storey drift distribution is equivalent to a uniform interstorey drift distribution, the damage function f2 to be minimised can be
defined as
2 1/2
n −1 CP
vs (x)/Hs
1 s
−1
,
(15)
f2 (x) =
ns s=1 ∆CP (x)/H
where ns is the number of building storeys; vsCP and ∆CP are the drift of storey s
and the roof drift at the CP hazard level, respectively; Hs is the vertical distance
from the base of the building to storey s; and H is the height of the building. In
effect, f2 defines the coefficient of variation of the lateral deflection distribution,
since vsCP/Hs and ∆CP/H represent storey-drift ratio and mean-drift ratio, respectively. By definition, the value of f2 is not less than zero, and it is only for the
extreme case of a perfectly uniform interstorey drift distribution that f2 = 0 (in
fact, while striving to achieve a linear interstorey drift distribution provides useful
results, identically achieving it is not practical in most cases).
Damage of a structural member also depends on its cross-section depth, in that
the extent of damage caused by a given lateral drift will differ depending on the
section depth. Since the design methodology presented by this study adopts predetermined nominal depths for structural members that remain virtually constant
during the optimisation process (see Secs. 4.2 and 6), the section depth is not
included as a parameter in the damage function f2 .
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4.2. Design variables
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The cross section sizes of the members are taken to be the design variables for this
study. The four cross-sectional properties of area A, moment of inertia I, elastic
modulus S and plastic modulus Z are required for pushover analysis [see Eqs. (2),
(3) and (7)]. For specified type and nominal depth of commercially available standard steel sections, these four properties can be related together through functional
relationships as follows:
I = C1 A2 + C2 A + C3 ,
(16)
S = C4 A + C5 ,
(17)
Z = ςS,
(18)
where C1 to C5 are constants determined by regression analysis [Gong, 2003]. Having such relationships, the cross section area A can alone be taken as the design
sizing variable, thereby reducing the number of design variables significantly (see
Sec. 6). Pushover analysis results are very sensitive to the sectional properties I, S
and Z, and Eqs. (16) to (18) ensure that these properties are estimated as accurately
as possible.
For this study, the optimal design for a steel framework is to be found using
commercially available section shapes for the members. To this end, the design
process is divided into two phases. During Phase I, the design optimisation treats
the sizing variables as being continuous valued. Phase II then utilises a dynamic
rounding-off strategy to size the members using discrete commercial sections [Gong,
2003].
4.3. Design formulation
The design optimisation problem is formulated as,
Minimise: f (x) = f1 (x) + f2 (x)
ω1
Wmax
=
+ ω2
n
ρLj Aj
j=1
1
ns
Subject to: δsi (x) ≤ δ̄ i
¯i
∆i (x) ≤ ∆
AL
j
≤ Aj ≤
AU
j
ns −1 CP
vs (x) /Hs
∆CP (x) /H
s=1
2 1/2
−1
.
(19)
(i = 1, . . . , nh ; s = 1, . . . , ns ),
(20)
(i = 1, . . . , nh ),
(21)
(j = 1, 2, . . . , n) [Phase I],
(22)
or
Aj ∈ aj
(j = 1, 2, . . . , n) [Phase II],
(23)
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where nh is the number of earthquake hazard levels of concern to the design; superscript i refers to the ith hazard level; f is a combined (cost + damage) objective
function; ω1 and ω2 are combination factors for the objectives f1 and f2 from
Eqs. (14) and (15), respectively, which turn the design problem into a single criterion problem. (Note that f1 and f2 are two different entities, and the appropriate
selection for the values of ω1 and ω2 requires numerical experiments that account
not only for how f1 and f2 are defined and normalised but also for how much
the gradients of the two objectives conflict with each other [Gong, 2003]). Equations (20) are performance constraints on interstorey drift constraints, where δsi is
the interstorey drift of storey s and δ̄ i is the allowable interstorey drift at hazard level i. Equations (21) are performance constraints on roof drift, where ∆i is
¯ i is the allowable roof drift at hazard level i. Equations (22)
the roof drift and ∆
are fabrication constraints specifying lower and upper bounds on the design sizing
U
variables for the Phase I continuous design, where AL
j and Aj are the lower- and
upper-bound cross section areas for design variable j, respectively. Equations (23)
are fabrication constraints for the Phase II discrete design, where aj is the set of
discrete commercial-section areas possible for member j.
In Eqs. (19) to (21), the f1 (x) term of Eq. (19) is alone an explicit function of
the section-area design variables Aj , while the f2 (x) term of Eq. (19) and all drift
constraints are implicit functions of the design variables. To facilitate computer
solution of the design optimisation problem, it is necessary to use an approximation technique to formulate the objective function f2 (x) and all drift constraints
explicitly in terms of design variables. High quality approximations can be obtained
by adopting the reciprocal sizing variables [Schmit and Farshi, 1974]:
xj = 1/Aj
(24)
and then employing first-order Taylor series to reformulate Eqs. (19) to (23) as the
explicit design optimisation problem:
n
Minimize: f (x) =
ω1
1
ρLj
Wmax j=1
xj

n
0
+ ω2 (f2 (x)) +
j=1
n
0
Subject to: δsi (x) +
j=1
dδsi (x)
dxj
0
df2 (x)
dxj
0
CP
xj − x0j  .
(25)
xj − x0j ≤ δ̄ i
(i = 1, . . . , nh ; s = 1, . . . , ns ), (26)
0
∆i (x) +
n
j=1
U
xL
j ≤ xj ≤ xj
i
d∆ (x)
dxj
0
¯i
xj − x0j ≤ ∆
(j = 1, 2, . . . , n) [Phase I],
(i = 1, . . . , nh ), (27)
(28)
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85
or
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xj ∈ Xj
(j = 1, 2, . . . , n) [Phase II],
(29)
U
where superscript ‘0’ represents values for the current design; xL
j and xj are lower
U
and upper bounds on the reciprocal area xj , respectively, where xL
j = 1/Aj and
U
L
xj = 1/Aj ; Xj is the set of reciprocal discrete commercial-section areas for variable j; and the derivatives dδ/dxj , d∆/dxj , df2 /dxj are interstorey drift, roof drift
and ductility-demand sensitivity coefficients, respectively. These sensitivity coefficients take on different forms depending on whether the pushover analysis underlying the design has concern for single-mode or multi-mode structural responses.
Gong et al. [2005] have presented detailed calculations of sensitivity coefficients for
single-mode response, which are extended in the following to account for multi-mode
responses.
5. Multi-Mode Sensitivity Analysis
To obtain sensitivity coefficients for multi-mode response, it is necessary to differentiate Eq. (10) with respect to the design variable xj according to the Chain Rule
[Kaplan, 1973] to get,
n
1/2
nm
nm
m
d
duc duk
uk duk
duc
2
=
uk
=
=
,
(30)
dxj
dxj
duk dxj
uc dxj
k=1
k=1
k=1
where the derivative duc /duk is obtained from Eq. (10) by differentiating the
combined multi-mode response uc with respect to the individual single-mode
responses uk . Equation (30) indicates that the sensitivity of uc to change in the
design variable xj is equal to the weighted combination of the corresponding sensitivities duk /dxj of the k individual modes, where the weighting factors are equal
to the corresponding displacement ratios uk /uc . (See Gong et al. [2005] for detailed
calculation of the sensitivity coefficients duk /dxj for single-mode responses.) Having
Eq. (30), the sensitivity analysis for multi-mode response proceeds as follows:
(i)
(ii)
(iii)
(iv)
Set vibration mode index k = 1 (i.e. the first mode).
Perform pushover analysis for the single mode k (see Sec. 3.1)
Conduct sensitivity analysis for the single mode k [see Gong et al., 2005].
Set k = k+1 and go to step (v) if k > nm (where nm is the number of vibration
modes under consideration), otherwise go to step (ii).
(v) Calculate sensitivity coefficients for multi-mode response through Eq. (30).
The sensitivity coefficients from step (v) are substituted into Eqs. (25) to (29) to
obtain the desired formulation of the seismic design optimisation problem having
account for the influence of multiple vibration modes for the structure. A generalised optimality criteria algorithm, called the Dual method [Fleury, 1979; Gong,
2003], is applied by this study to solve the optimisation problem of Eqs. (25) to (29)
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to find member sizes for which structure weight and damage are both minimised.
The design algorithm can start from any design point. Since the design formulation
imposes only drift constraints, and since various combinations of column and beam
sections can result in an identical lateral stiffness distribution, the selection of optimal sections for the individual members is not unique. (See Gong [2003] and Xu
et al. [2005] for full details of the design process.) An illustrative design example is
presented in the following from a variety of viewpoints.
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6. Design Example
Consider the nine-storey steel moment frame shown in Fig. 4 (also studied by Hasan
et al. [2002]). All five bays have the same span of 9.15 m (centerline dimension), and
all stories are 4.0 m high except the first storey which is 5.5 m high. The frame has
rigid moment connections. All columns are wide-flange sections having yield stress
of 400 MPa, while all girders are wide-flange sections having yield stress of 340 MPa.
The lumped seismic masses for the frame are 505 Mg for the first floor level, 495 Mg
for each of the second to eighth floor levels, and 531 Mg for the roof level (i.e. the
total seismic mass for the frame is 4501 Mg). The constant uniformly distributed
gravity loads on the girders for the first to eighth floor levels are 32.2 kN/m, and
28.8 kN/m for the roof-level girders.
Fig. 4.
Nine-storey steel moment frame.
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Optimal Performance-Based Seismic Design
87
The framework consists of 99 members. Through member linking, the number
of section-area design variables is reduced to 18, designated as A1 to A18 in Fig. 4.
Design values for the column variables A1 through A9 are chosen from among
available W14 sections [AISC, 1994], while design values for the girder variables
A10 to A13 , A14 , A15 to A16 and A17 to A18 are chosen from among W36, W33,
W30 and W24 sections, respectively. For each section type, Eqs. (16) to (18) are
employed to express the corresponding inertia and moduli parameters Ij , Sj and
Zj in terms of area Aj . While providing considerable a priori knowledge to the
design automation algorithm, the predetermination of the type and nominal depth
of the set of sections from which a member is to be selected is entirely consistent
with conventional design practice where a designer generally knows which type of
section and approximate nominal depth is used for a certain type of frame member.
For instance, it is well known that W14 sections are often used for columns of
low- to mid-rise steel frameworks, while W24 to W36 sections are often used for
girder members. If a reasonable section is not able to be chosen from among the
predetermined sections to meet the design requirements, the predetermined sections
are upgraded to a larger available nominal depth and the member design process
is repeated.
The design optimisation simultaneously considers the three hazard levels of
Immediate Occupancy (IO), Life Safety (LS) and Collapse Prevention (CP),
[FEMA-273, 1997]; i.e. Eqs. (26) and (27) are formulated as constraints on performance at the nh = 3 hazard levels i = IO, LS and CP. Three corresponding
design acceleration spectra, representing three different earthquake intensity levels
having 20%, 10% and 2% probability of exceedance in a 50-year period, are taken
to be as indicated in Table 1. The modal pushover analysis considers the first three
vibration mode shapes 1, 2, and 3 shown in Fig. 5 to evaluate the earthquake loading for the framework, [Gong, 2003]. The effective modal masses for modes 1, 2,
and 3 are 3787, 547 and 167 Mg, respectively (i.e. the summation of modal masses
is 4501 Mg, which is equal to the total seismic mass of the frame). The allowable
interstorey drift ratios are taken to be 0.01, 0.02 and 0.055 for the IO, LS and CP
hazard levels, respectively, while the allowable roof drift ratios for the three hazard
levels are taken to be 0.007, 0.025 and 0.05, respectively. The two combination factors in Eq. (19) are assigned values ω1 = 0.95 and ω2 = 0.05. A detailed description
on how to select ω1 and ω2 values is given by Gong [2003].
The modal shapes in Fig. 5 are assumed to remain unchanged during the optimisation process. As such, the modal masses calculated through Eq. (8) remain
Table 1.
Acceleration spectra data for design earthquake hazard levels.
Hazard level
Immediate occupancy
Life safety
Collapse prevention
Earthquake intensity
(Exceedance probability/years)
Ss
(g)
S1
(g)
T0
(sec.)
20%/50
10%/50
2%/50
0.29
0.40
1.15
0.14
0.19
0.65
0.49
0.48
0.56
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88
00233
D. E. Grierson, Y. Gong & L. Xu
Fig. 5.
Vibration mode shapes 1, 2 and 3.
unchanged as well. This assumption is consistent with current practice of the engineering community. For example, the static force procedure [SEAOC, 1999] distributes the design base shear over the height of the structure according to the
period of the structure, where the period is approximated by an empirical formula without explicit reference to the actual structure. These assumptions obviate the need for dynamic analysis, as herein. As pointed out by Elnashai [2002],
though pushover analysis cannot replace dynamic analysis for all structures, the
domain where it is applicable is continually expanding with the results of ongoing
research.
Summarised in column 2 of Table 2 (Design 1) are the optimal discrete-section
design results found for the steel framework upon solving the design optimisation
problem of Eqs. (25) to (29) using MPA to account for the first three vibration
modes of the structure. For this design, the periods of the first three modes are
1.98, 0.87 and 0.47 seconds, respectively. The modal pushover curves are shown
in Fig. 6, where it is seen that the first mode undergoes large displacements into
the plastic response range, the second mode experiences slight plastification, and
the third mode is still in the elastic range. The plastic states of the optimal design
at the CP hazard level are found to be as shown in Fig. 7, where ovals represent
plastic hinges and the digit inside an oval is the corresponding percentage extent
of sectional plastification [expressed as 100 (1 − p), where p is the plasticity-factor
defined by Eq. (2)]. From the last row of Table 2, it is observed that the damage
index value (coefficient of variation) of 0.033 indicates that the optimal design
has a near-uniform ductility demand (i.e. little or no excessive damage due to
concentrated soft-storey lateral deformations).
For purposes of comparison with Design 1, given in Columns 3 and 4 of Table
2 (Designs 2 and 3) are corresponding optimal discrete-section design results found
using a single vibration mode (Mode 1 in Fig. 5) as the basis for the pushover
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Table 2.
Optimal design results.
Design 1
Design 2
Design 3
Using MPA
with modes
1, 2, 3
W-Section
Aj (mm2 )
First-mode
analysis with
total mass
W-Section
Aj (mm2 )
First-mode
analysis with
first-mode mass
W-Section
Aj (mm2 )
1
W14 × 311
58970
W14 × 426
80640
W14 × 311
58970
2
W14 × 233
44190
W14 × 311
58970
W14 × 233
44190
3
W14 × 233
44190
W14 × 257
48770
W14 × 176
33420
4
W14 × 211
40000
W14×257
48770
W14 × 193
36640
5
W14 × 605
114840
W14 × 808
152900
W14×605
114840
6
W14 × 426
80640
W14 × 550
104510
W14 × 426
80640
7
W14 × 370
70320
W14 × 500
94840
W14 × 370
70320
8
W14 × 257
48770
W14 × 342
65160
W14 × 283
53740
9
W14 × 211
40000
W14 × 159
29940
W14 × 120
22770
10
W36 × 256
48640
W36 × 328
62190
W36 × 245
46510
11
W36 × 210
39870
W36 × 260
49350
W36 × 194
36770
12
W36 × 194
36770
W36 × 256
48640
W36 × 194
36770
13
W36 × 182
34580
W36 × 245
46510
W36 × 182
34580
14
W33 × 169
31930
W33 × 221
41930
W33 × 152
28840
15
W30 × 148
28060
W30 × 211
40000
W30 × 173
32770
16
W30 × 108
20450
W30 × 148
28060
W30 × 108
20450
17
W24 × 68
12970
W24 × 104
19740
W24 × 76
14450
18
W24 × 68
12970
W24 × 62
11740
W24 × 55
10450
0.84
[CP]
0.80
[CP]
0.85
[CP]
0.90
[CP, 4]
0.84
[CP, 2]
0.99
[CP, 4]
Design variable index
(j)
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89
Most critical roof drift response ratio
[at performance level i]
Most critical interstorey drift response ratio
[at performance level i, storey s]
Weight f1 · (Wmax ) (kN)
1995
2562
1934
Damage function f2
0.033
0.052
0.066
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D. E. Grierson, Y. Gong & L. Xu
Fig. 6.
Modal pushover curves for Design 1.
analysis. The single-mode seismic mass for Design 2 was taken to be the total
seismic mass of the frame, 4501 Mg, while for Design 3 it was taken to be the
first mode mass, 3787 Mg. Adopting total seismic mass as in Design 2 is consistent
with current engineering practice, such as the static force procedure adopted by
the Structural Engineers Association of California with the view to provide an
upper-bound value on the combined mode base shear [SEAOC, 1999]. For Design 2,
the periods of the first three modes are 1.66, 0.79 and 0.45 seconds, respectively;
while for Design 3 they are 1.98, 0.92 and 0.53 seconds, respectively. The plastic
states of Designs 2 and 3 at the CP hazard level are found to be that as shown in
Figs. 8 and 9, respectively. From the last row of Table 2, the damage index values
of 0.052 and 0.066 for Designs 2 and 3 indicate that these two designs also have
a near-uniform ductility demand like Design 1. At the same time, however, from
the second-last row of Table 2, the weight values of 1995 and 2562 for Designs 1
and 2, respectively, indicate that the design solution found using MPA with total
seismic mass is about 22% lighter than that found using single-mode pushover
analysis with total seismic mass. The main reason for the significant differences in
the two design solutions is that when computing the design base shear for mode
1, Design 1 uses only the first-mode seismic mass while Design 2 uses the total
seismic mass. As a result, Design 2 significantly overestimates the base shear and
displacement contributions from the first mode. On the other hand, from Table 2,
because Design 1 accounts for the combined seismic mass of three modes it is
found to be about 3% heavier than Design 3 which accounts only for the firstmode seismic mass. (Note that the small difference in the weights of Designs 1
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Optimal Performance-Based Seismic Design
r = degree of plastification = 100 (1-p)
r
100
100
100
100
88
100
100
100
100
100
100
100
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98
96
95
100
98
93
98
73
98
93
99
92
96
98
98
98
99
98
99
98
99
84
84
90
93
98
95
95
98
98
98
85
79
96
95
98
98
74
98
96
98
98
95
98
92
98
95
95
98
98
96
96
98
93
98
98
71
88
92
93
98
98
100
89
98
98
92
92
98
88
98
98
100
96
93
74
88
98
98
100
98
98
74
88
84
100
90
98
100
98
100
93
98
100
91
98
90
98
100
98
98
93
90
98
99
96
100
84
100
100
100
71
98
100
98
98
100
91
98
98
98
98
96
100
100
100
88
84
100
98
100
100
71
91
91
100
100
84
100
100
88
71
84
100
100
100
88
71
100
91
97
97
95
94
94
94
94
97
100
100
100
100
100
100
Fig. 7.
Structural plastification at the collapse prevention level for Design 1.
and 3 indicates the contribution of the higher modes to lateral displacements is
somewhat insignificant.)
The plastic states of the optimal designs shown in Figs. 7 to 9 are somewhat
uniform over the structure, which is consistent with the small damage index values
for the three designs in Table 2. For the same amount of lateral displacement at
the CP hazard level, the girder members in the upper storeys of Fig. 7 have slightly
more plastification than those members in the corresponding stories of Figs. 8 and 9.
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D. E. Grierson, Y. Gong & L. Xu
r
r = degree of plastification = 100 (1-p)
97
95
94
97
78
97
96
81
97
97
94
95
94
95
96
77
97
95
95
90
96
98
98
96
97
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98
99
70
97
85
86
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95
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94
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95
20
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98
95
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92
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84
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90
36
97
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91
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36
97
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28
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90
94
95
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94
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94
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90
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93
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94
80
92
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98
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98
98
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96
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98
Fig. 8.
Structural plastification at the collapse prevention level for Design 2.
This is essentially because, with little exception, the Fig. 7 design has lighter girder
members in the upper stories than the Figs. 8 and 9 designs. From Table 2, the
exterior column members in the upper storeys have less weight in Design 3 than
Design 1, which, in turn, has less corresponding weight than Design 2. At the same
time, the interior column members in the upper storeys have less weight in Design 3
than Design 2, which, in turn, has less corresponding weight than Design 1. Due in
part to the splicing of interior columns at the second floor level where the column
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r = degree of plastification = 100 (1-p)
95
92
97
95
97
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97
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50
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80
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91
89
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96
96
96
97
Fig. 9.
Structural plastification at the collapse prevention level for Design 3.
sections are reduced abruptly (see Fig. 4 and Table 2), it can be noticed in Fig. 7
that at the CP hazard level the first and second storeys are essentially weak or ‘soft’
stories. This phenomenon does not conflict with the design objective to minimise
excessive local damage, because the structure is intended to be at the point of
incipient collapse at the CP hazard level.
Finally note from Table 2 that the most critical response ratios for the three
optimal designs occur at the CP hazard level and that, with the exception of interstory drift for Design 3, they are all somewhat less than unity. This indicates that
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D. E. Grierson, Y. Gong & L. Xu
there are essentially no displacement constraints “active” for each final design. As
displacements at the CP hazard level are very sensitive to change of member sizes,
this is primarily due to the rounding-off procedure applied to convert continuousvalued design results to discrete sections [Gong, 2003].
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7. Summary
This paper described a performance-based seismic design optimisation method based
on a load-control multi-modal pushover analysis procedure modelled after a singlemode pushover procedure recently developed by Hasan et al. [2002]. The MPA procedure retains the conceptual simplicity and computational efficiency of the single-mode
pushover analysis while providing improved estimates of demands imposed by design
earthquake events. The optimisation model adopted both minimum structural weight
and minimum earthquake damage as design objective criteria, as well as both interstorey and roof drift performance constraints at multiple hazard levels corresponding
to different earthquake intensities. The applicability of the proposed design methodology was illustrated for a nine-storey steel moment frame, including comparisons
with optimal designs found using single-mode pushover analysis. It was found that
the optimal design found using MPA was lighter than that found by current engineering practice, i.e. single-mode pushover analysis accounting for total seismic mass,
primarily because the latter analysis overestimates the lateral displacements resulting
from the actual inertial earthquake loading. At the same time, regardless of whether
MPA or single-mode pushover analysis was used, all optimal designs were found to
have a near-uniform ductility demand (i.e. having little or no excessive damage due
to concentrated soft-storey lateral deformations). While the reported results are only
applicable for the given example and will vary for other structures, it is quite possible
that the foregoing comments concerning structural weight and ductility demand hold
true for a range of conventional steel building frameworks. Finally, it is important to
remember that designs based on single-mode or multi-mode pushover analysis rely on
approximate estimates of response behaviour and ultimately should be assessed by a
nonlinear dynamic analysis in many instances, particularly when higher mode effects
are important.
Acknowledgements
The authors wish to thank NSERC (Canada) for the financial support of this work.
The second author would like to acknowledge the graduate scholarship provided
through the Ontario Graduate Scholarship Program by the Ministry of Training,
Colleges and Universities of Ontario. The authors are indebted to the reviewers for
their valuable comments and suggestions to improve the paper.
References
AISC [1994] Manual of Steel Construction, Load and Resistance Factor Design (American
Institute of Steel Construction, Chicago, IL).
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Optimal Performance-Based Seismic Design
95
Beck, J. L., Chan, E., Irfanoglu, A. and Papadimitriou, C. [1999] “Multi-criteria optimal
structural design under uncertainty,” Earthquake Engineering and Structural Dynamics 28, 741–761.
Biddah, A. C. and Naumoski, N. [1995] “Use of pushover test to evaluate damage of
reinforced concrete frame structures subjected to strong seismic ground motions,”
Proc. of 7th Canadian Conference on Earthquake Engineering, Montreal.
Bracci, J. M., Kunnath, S. K. and Reinhorn, A. M. [1997] “Seismic performance and
retrofit evaluation of reinforced concrete structures,” Journal of Structural Engineering, ASCE 123, 3–10.
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