optimal seism pbd of rc blgs using pushover anal -05

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Engineering Structures 27 (2005) 1289–1302

www.elsevier.com/locate/engstruct

Optimal seismic performance-based design of reinforced concretebuildings using nonlinear pushover analysis

X.-K. Zou, C.-M. Chan∗

Department of Civil Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

Received 29 April 2004; received in revised form 23 March 2005; accepted 4 April 2005

Available online 10 May 2005

Abstract

Performance-based design using nonlinear pushover analysis, which generally involves tedious and intensive computational effort, is a

highly iterative process needed to meet designer-specified and code requirements. This paper presents an effective computer-based technique

that incorporates pushover analysis together with numerical optimization procedures to automate the pushover drift performance design of

reinforced concrete (RC) buildings. Steel reinforcement, as compared with concrete materials, appears to be the more cost-effective material,

that can be effectively used to control drift beyond the occurrence of first yielding and to provide the required ductility of RC building

frameworks. In this study, steel reinforcement ratios are taken as design variables during the design optimization process. Using the principle

of virtual work, the nonlinear inelastic seismic drift responses generated by the pushover analysis can be explicitly expressed in terms of

element design variables. An optimality criteria technique is presented in this paper for solving the explicit performance-based seismic

design optimization problem for RC buildings. Two building frame examples are presented to illustrate the effectiveness and practicality of

the proposed optimal design method.

© 2005 Elsevier Ltd. All rights reserved.

Keywords: Performance-based design; Pushover analysis; Virtual work; Plastic hinge; Structural optimization design; Optimality criteria method

1. Introduction

Performance-based design appears to be the future

direction of seismic design codes [1–3]. In the newly

developed performance-based seismic design approach,

nonlinear analysis procedures become important in iden-

tifying the patterns and levels of damage for assessing a

structure’s inelastic behavior and for understanding the fail-

ure modes of the structure during severe seismic events.

Pushover analysis is a simplified, static, nonlinear procedurein which a predefined pattern of earthquake loads is applied

incrementally to framework structures until a plastic col-

lapse mechanism is reached. This analysis method generally

adopts a lumped-plasticity approach that tracks the spread of

inelasticity through the formation of nonlinear plastic hinges

∗ Corresponding author. Tel.: +852 2358 7173; fax: +852 2358 1534.E-mail address: [email protected] (C.-M. Chan).

0141-0296/$ - see front matter © 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2005.04.001

at the frame element’s ends during the incremental loading

process.

As graphically presented in Fig. 1, the nonlinear static

analysis procedure requires determination of three primary

elements: capacity, demand and performance. The capacity

spectrum can be obtained through the pushover analysis,

which is generally produced based on the first mode

response of the structure assuming that the fundamental

mode of vibration is the predominant response of the

structure. This pushover capacity curve approximates howa structure behaves beyond the elastic limit under seismic

loadings. The demand spectrum curve is normally estimated

by reducing the standard elastic 5% damped design spectrum

by the spectral reduction method. The intersection of the

pushover capacity and demand spectrum curves defines the

“performance point” as shown in Fig. 1. At the performance

point, the resulting responses of the building should then be

checked using certain acceptability criteria. The responses

can be checked against acceptability limits on both global

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1290 X.-K. Zou, C.-M. Chan / Engineering Structures 27 (2005) 1289–1302

Fig. 1. Nonlinear analysis procedure.

system levels (such as the lateral load stability and theinterstory drift) and local element levels (such as the

element strength and the sectional plastic rotation) [2].

When the responses of a structure do not meet the targeted

performance level, the structure needs to be resized and

the design process repeated until a solution for the desired

performance level is reached. In general, the determination

of the satisfactory performance response that fulfills both the

system level response and element level response requires a

highly iterative trial-and-error design procedure even with

the aid of today’s engineering computer software.

It has been recognized that the interstory drift perfor-

mance of a multistory building is an important measure of structural and non-structural damage of the building under

various levels of earthquake motion [4]. In performance-

based design, interstory drift performance has become a

principal design consideration [1,2]. The system perfor-

mance levels of a multistory building are evaluated on the

basis of the interstory drift values along the height of the

building under different levels of earthquake motion [5]. The

control of interstory drift can also be considered as a means

to provide uniform ductility over all stories of the building.

A large story drift may result in the occurrence of a weak

story that may cause catastrophic building collapse in a seis-

mic event. Therefore, a uniform story ductility over all sto-

ries for a multistory building is usually desired in seismic

design [6].

Although lateral drift performance is a principal concern

in the seismic design of structures, economically designing

elements of building structures for various levels of elastic

and inelastic lateral drift performance under multiple

levels of earthquake load is generally a rather difficult

and challenging task. Lateral drift design requires the

consideration of a proper distribution of the stiffness of all

structural elements and, in a severe seismic event, also the

occurrence and redistribution of plasticity in the structural

elements. Structural engineers are faced with the problem

of efficiently proportioning structural materials throughoutthe building to limit the inelastic seismic drift responses of

a structure. Due to the lack of an automated optimization

technique, performance-based seismic drift design is usually

carried out by trial-and-error methods based on intuition and

experience.

Numerous studies on structural optimization in the

seismic design of structures have been published in the past

two decades, including Cheng and Botkin [7], Feng et al.

[8], Bhatti and Pister [9], Balling et al. [10], Cheng and

Truman [11], Arora [12]. However, most of these previous

research efforts were concerned with optimization through

prescriptive-based design concepts. Recently, Beck and hisassociates [13] developed an optimization methodology for

performance-based design of structural systems operating

in an uncertain dynamic environment. Ganzerli et al. [14]

presented an optimal performance-based design method for

a reinforced concrete (RC) frame, in which performance-

based constraints were implemented in terms of plastic

rotations of the beams and columns of the frame. Foley [15]

provided a review of current state-of-the-art seismic

performance-based design procedures and presented the

vision for the development of performance-based design

optimization. It has been recognized that there is a pressing

need for developing optimized performance-based design

procedures for seismic engineering of structures [15,16].

Chan [17,18] developed an efficient computer-based

optimization technique for lateral stiffness design of tall

buildings. The optimization technique, developed on the

basis of a rigorously derived Optimality Criteria (OC)

approach, is capable of optimizing large-scale tall steel

and/or reinforced concrete buildings subject to multiple

static wind drift and dynamic wind-induced vibration

design constraints. The effectiveness of the state-of-the-art

optimization technique has been demonstrated through its

actual design application to a number of the tall buildings

in Hong Kong [18]. It should be noted that the research has



X.-K. Zou, C.-M. Chan / Engineering Structures 27 (2005) 1289–1302 1291

been primarily focused on the wind-induced serviceability

performance design of tall buildings. Much effort is needed

to extend the current optimization technique to inelastic

seismic design of multistory buildings.

This paper presents an effective optimization technique

for the inelastic drift performance design of RC building

frames under pushover loading. Attempts have been madeto automate the performance-based seismic design of RC

buildings using an optimization procedure. The quantities of

steel reinforcement, the only effective material that provides

ductility to RC building frameworks, are considered as

design variables in the inelastic seismic drift optimization.

With careful tracking of the location and extent of plastic

hinge occurrence, the pushover drift can be explicitly

expressed in terms of the steel reinforcement sizing variables

using the principle of virtual work. The optimization

methodology for the solution of the nonlinear seismic drift

design of buildings is fundamentally based on an OC

approach. Two RC building frame examples are presented

in this paper. A simple, one-story, one-bay frame is firstpresented to verify the inelastic drift formulation. A ten-

story, two-bay planar frame building is then presented to

illustrate the details of the OC optimization method for

inelastic seismic drift performance-based design.

2. Optimal inelastic seismic design problem

2.1. Implicit design optimization problem

In seismic design, it is commonly assumed that a

building behaves linear-elastically under minor earthquakes

and may respond nonlinear-inelastically when subjectedto moderate and severe earthquakes. Under such an

assumption, the entire design optimization process can

therefore be decomposed into two phases [19–21]. In the

first phase, the structural concrete cost is minimized subject

to elastic drift responses under minor earthquake loading

using elastic response spectrum analysis. In this phase,

concrete member sizes are considered as the only design

variables since the concrete material plays a more dominant

role in improving the elastic drift performance of the

building. Once the optimal structural member sizes are

determined at the end of the first phase of the optimization,

the steel reinforcement quantities can then be consideredas design variables in the second phase. In controlling the

inelastic drift responses, steel reinforcement is the only

effective material that provides ductility to an RC building

structure beyond first yielding. In this second design phase,

the member sizes are kept unchanged and the cost of the steel

reinforcement is minimized subject to design constraints on

inelastic interstory drift produced by the nonlinear pushover

analysis. The emphasis of this paper is on the second phase

of the design optimization, the inelastic seismic drift design

optimization. The details of the first phase of the design

optimization (i.e., elastic seismic drift design optimization)

can be found in the work of Zou [21,22].

For an RC building having i = 1, 2, 3, . . . , N i members

and 2N i plastic hinges (assuming one hinge at each end of

a member), the tension steel reinforcement ratio, ρi , and the

compression steel reinforcement ratio, ρ

i, for a rectangular

cross section are taken as design variables in the design

optimization, whereas the member sizes, Bi (width) and Di

(depth), are fixed. If the topology of a building’s structuralsystem is predefined, the steel reinforcement cost of the RC

framework is minimized as

Minimize: steel cost =

N ii=1

wsi (Lsi ρi + Lsi ρ

i) (1)

where wsi is the cost coefficient for steel reinforcements;

and Lsi and L

si are respectively the lengths of the tension

and compression steel reinforcements for member i , which

can be predefined on the basis of code requirements. Only

the longitudinal flexural reinforcement of member sections

is considered as a design variable in this study, while the

transverse shear reinforcement is considered invariant underthe assumption that adequate shear capacity strength is

provided for each member.

In the performance-based design, it is necessary to

check the “capacity” of a structure against the “demand”

of an earthquake at the performance point. In this study,

the interstory drift responses of a building, generated

by a specified earthquake demand, are checked against

appropriate limits corresponding to a given performance

level. That is, for a multistory building structure, the

interstory drift ratio caused by pushover loading should

comply with the following requirement:

u j

h j=

u j − u j −

1h j

≤ d U j (2)

where u j is the interstory drift of the j th story; u j

and u j −1 are the respective story displacement of the two

adjacent j and j − 1 floor levels; h j is the j th story

height; and d U j is the specified interstory drift ratio limit

representing the damage threshold for the j th story.

Besides checking the interstory drift responses discussed

above, local response quantities (i.e., sectional plastic

rotation and strength of all members) at the performance

point also must not exceed appropriate response limits.

Therefore, the plastic rotation, θph , at the hth end of member

i (where the subscript h represents one end of a member andh = 1, 2) should be constrained in the optimization by

θph ≤ θU p (3)

where θU p is the rotation limit of member i for a specific

performance level. Once the designer determines the per-

formance levels of the structure (e.g., immediate occupancy,

life safety, collapse prevention), the corresponding limiting

value of θU p is then determined. Unless specific design crite-

ria are stated, ATC-40 [2] and FEMA-273 [3] provide guide-

lines for estimating the limiting values of plastic rotation

of the flexural members for various levels of performance

criteria of an RC frame. In practical multistory building




structures, an excessive number of design constraints may

create enormous increases in computational effort. In order

to reduce the practical building design problem to a manage-

able size, the strength design of each member is not consid-

ered explicitly as a design constraint; rather, the strength-

based steel reinforcement ratios in accordance with code

specifications are first calculated and these values are thentaken as the lower size bound for each member in the inelas-

tic seismic drift design optimization [21,22].

In addition to the design performance constraints on the

system-level story drift and element-level sectional plastic

rotation, the steel reinforcement variables are constrained

within the minimum and maximum steel reinforcement

ratios as

ρLi ≤ ρi ≤ ρU

i ; (4a)

ρLi ≤ ρ

i ≤ ρU i (4b)

where the superscripts L and U denote the minimum and

maximum limits of the design variables, ρi and ρ

i .In order to facilitate a numerical solution of the drift

design problem, it is necessary that the implicit story

drift constraint (Eq. (2)) and the plastic rotation constraint

(Eq. (3)) be expressed explicitly in terms of the design

variables, ρi and ρi .

2.2. Explicit drift formulation

Before the drift formulation is discussed, three assump-

tions must be made. The first is that all the inelastic defor-

mation is assumed to occur at the plastic hinges, which are

located at the ends of each frame member and members arefully elastic between the plastic hinges. Secondly, the plastic

hinges are assumed to be frictionless and have zero length.

The third assumption is that beam–column joints are much

stronger than any adjacent framing components so that the

joint region may be modelled as a stiff or rigid zone.

On the basis of the internal element forces and moments

of the structure obtained from the pushover analysis at

the performance point, the principle of virtual work can

be employed to express the pushover displacement. The

pushover story displacement, u j , at the performance point

is equal to the sum of the virtual work, u j,memb, produced

by the structural members and the virtual work, u j,hinge,generated by the plastic hinges. That is,

u j = u j,memb + u j,hinge (5)

in which

u j,memb =

N ii=1

Li

o

F X f X j

E A X

+F Y f Y j

G AY

+F Z f Z j

G A Z

+M X m X j

G I X

+M Y mY j

E I Y

+M Z m Z j

E I Z

dx

i

(6)

u j,hinge =

N i

i

=1

2

h

=1

m0p j hθph

i

. (7)

Fig. 2. Definition of local coordinate system.

In Eq. (6), Li is the length of member i ; E and G are the

axial and shear elastic material moduli; A X , AY and A Z are

the axial and shear areas for the cross-section; I X , I Y and

I Z are the torsional and flexural moments of inertia for the

cross-section; F X , F Y , F Z , M X , M Y and M Z are the element

internal forces and moments, as depicted in Fig. 2; f X j , f Y j ,f Z j , m X j , mY j and m Z j are the virtual element forces and

moments due to a unit virtual load applied to the building

at the location corresponding to the story displacement, u j .

Considering rectangular concrete elements with width (Bi)

and depth (Di ) and expressing the cross sectional properties

in terms of Bi and Di , the displacement, u j,memb, in Eq. (6)

can be simplified in terms of Bi and Di [18,21]. During

the inelastic drift design optimization process, u j,memb is

kept unchanged since Bi and Di of each member section

are fixed. That is, the displacement, u j,hinge, caused by the

formation of the plastic hinges alone varies.

In Eq. (7), m0

pj his the virtual end moment at the location

of the hth hinge of a member; θph is the actual plastic

rotation at the location of the hth plastic hinge of a member,

which is equal to zero when no plastic hinge is found.

As shown in Fig. 3, the behavior of a plastic hinge is

modeled as a bilinear curve: the elastic segment, AB, and the

hardening segment, BC, where Point A corresponds to the

unloaded condition, Point B is the first yield moment point,

Point C is the ultimate moment capacity, which generally

corresponds to the structural stability performance level in

ATC-40 [2]. On the basis of the line segments A–B–C, the

plastic rotation, θp, can be given as follows:

θp = M − M yM u − M y

θU p ≤ θU

p (8)

where θU p is the ultimate plastic rotation which can be

established on the basis of experimental tests or can be

obtained directly from design guidelines such as the ATC-

40 [2]; M is the applied moment at the location of the plastic

hinge; M y is the bending moment at the first yielding of the

tensile steel; and M u is the ultimate moment of resistance.

Given the quantity of the steel reinforcement used in a

concrete section, the values of M y and M u can then be

determined. For simplicity, M u can be approximately related

to M y as M u = 1.1M y [2]. As a result, the plastic rotation,




Fig. 3. Moment–rotation curve.

θp , given in Eq. (8), can be rewritten as

θp =θU

p

0.1

M

M y− 1

. (9)

For the explicit problem formulation, it is necessary that

the plastic rotation, θp , be explicitly expressed in terms

of the design variables (i.e., ρ and ρ

). Furthermore, agood formulation should reflect accurately the change in the

plastic rotation, θp, due to a change in the design variables

during the optimization resizing process. In other words, any

change in the design variables, ρ and ρ, during the inelastic

optimization process requires a corresponding update on the

values of M and M y .

In pushover analysis, pure moment hinges as well as

axial-moment hinges are widely used and are generally

assigned to the two ends of each beam or column. In fact,

the inelastic displacement, u j,hinge, in Eq. (7) includes the

displacement generated by moment hinges (usually in the

beams) and that generated by axial-moment hinges (usually

in the columns). By the force equilibrium shown in Fig. 4,where f c is the stress at the extreme compression concrete

fiber, f s is the stress in the compression steel, f y is the yield

strength of the tension steel, and d is the effective depth,

which is equal to the distance from the extreme compression

fiber to the centroid of the tension steel, M y for a moment

hinge (where there is no coexisting axial force) can be

expressed in terms of design variables, ρ and ρ, as

M y = 0.5 f c Bk d

kd

3− d

+ f y Bd (d − d )ρ (10)

where k is the neutral axis depth factor at the first yield and

it is given as

k =

(ρ + ρ)2n2

sc + 2

ρ + ρ

d

d

nsc − (ρ + ρ)nsc

(11)

in which nsc =E sE c

where E c and E s are the moduli of

elasticity of the concrete and the steel, respectively.

Similarly, M y for an axial-moment hinge (wherein there

is coexisting axial force) can be expressed in terms of design

variables, ρ and ρ, as

M y =f y Bd

2n

sc

· D

2

−kd

3 ·

k 2

1− k

Fig. 4. Doubly reinforced member section at first yield.

+f y Bd (d − d )2

2d ·

ρ

1 − k (12)

where k is given as

k = −

2ρ +

F X

f y Bd

nsc

+

2ρ +

F X

f y Bd

2

n2sc + 2nsc

ρ

D

d +

F X

f y Bd

. (13)

To take into account the change in θp due to a change

in ρ and ρ while maintaining an instantaneously fixed

value of M , a second-order Taylor series approximation for

evaluating the value of θp is given as

θp(ρ) = θp

ρ=ρ0 +

∂θp

∂ρ

ρ=ρ0

(ρ − ρ0)

+1

2

∂2θp

∂ρ2

ρ=ρ0

(ρ − ρ0)2 (14)

where the tension steel ratio, ρ, is considered as the major

design variable affecting θp; for simplicity, the compression

steel ratio, ρ, is assumed to be linearly related to ρ forbeams and to be the same as ρ for columns such that the two

steel ratios can be reduced to one design variable for each

member. Given the explicit expression of M y as a function of

ρ from Eqs. (10) and (12), the gradient,∂θ p

∂ρ, and the second-

order term,∂2θp

∂ρ2 , can be analytically calculated from Eq. (9).

By substituting the explicit plastic rotation, θp(ρ), given

in Eq. (14) into Eq. (7), the pushover displacement, u j , in

Eq. (5) can be explicitly expressed in terms of the design

variable, ρi , as

u j (ρi ) = u jρi =ρ0

i +

N ii=1

∂u j

∂ρi

ρi =ρ0

i

(ρi − ρ0i )

+1

2

N ii=1

∂2u j

∂ρ2i

ρi =ρ0

i

(ρi − ρ0i )

2. (15)

2.3. Plastic rotation constraint and sizing constraint

In this design optimization, when the plastic rotation,

θp , is to be modified with changes in the design variable,

ρi , it is necessary to make sure that θp does not exceed

the specified threshold of plastic rotation, θU p , for each




specified performance level. Moreover, in order to prevent

drastic changes in the internal element force and moment

distribution due to the changes in the design variables and

the resulting fluctuation of solution convergence during the

pushover reanalysis and design optimization processes, it

is necessary that each plastic hinge remain plastic once it

appears during the resizing iteration of the design variables(i.e., no local unloading). Furthermore, to maintain the

accuracy of the Taylor approximation of the pushover

displacement in Eq. (15), it is necessary to ensure that

the variation of ρi for the members with plastic hinges

be restricted within a sufficiently small range described as

follows.

It is found from Fig. 3 that, in order to ensure that 0 ≤

θp ≤ θU p , the internal moment, M , leading to the occurrence

of a plastic hinge must satisfy the following condition:

M y ≤ M ≤ M u . (16)

On the one hand, in order to maintain the plasticity of

the plastic hinge, an increase of the value of ρi causinga corresponding increase in M y must be limited by the

moment, M , acting at the location of the plastic hinge.

Therefore, by setting M y = M where M y can be found

from Eq. (10) for a moment hinge and Eq. (12) for an axial-

moment hinge, the corresponding value of ρi can be solved

for, and this value can then be taken as the instantaneous

upper bound value of ρi such that ρU i

= ρi .

On the other hand, a reduction in the value of ρi leads to

a corresponding decrease in the ultimate moment capacity,

M u , but this reduced M u must be greater than the applied

moment, M , in order to satisfy the design requirement that

M ≤ M u . On setting M u = M and assuming M u = 1.1M y ,the corresponding value of ρi can then be found such that

the instantaneous lower bound value ρLi = ρi . As a result,

on the basis of Eq. (16), the lower and upper bounds of ρi for

each plastic hinge can be instantaneously established during

the OC iterative resizing process.

2.4. Explicit design problem formulation

Upon establishing the explicit formulation of the inelastic

drift, Eq. (15), the optimization problem of minimizing the

steel construction cost of a multistory RC building can be

explicitly written in terms of the design variable, ρi , as

Minimize: F (ρi) =

N ii=1

wsi ρi (17)

subject to

g j (ρi ) =1

h j

u j

ρi =ρ0

i+

N ii=1

α1i(ρi − ρ0i )

+1

2

N ii=1

α2i(ρi − ρ0i )

2

≤ d U

j ( j = 1, 2, 3, . . . , N j )

(18)

ρLi ≤ ρi ≤ ρU

i (i = 1, 2, 3, . . . , N i ) (19)

where

α1i =∂u j

∂ρi

ρi =ρ0

i

=

2

h=1

m0j h

∂θph

∂ρ

ρi =ρ0

i

i

(20a)

α2i

=∂2u j

∂ρ2i

ρi =ρ0

i

= 2h=1

m0

j h

∂2θph

∂ρ2ρi =ρ0

i

i

. (20b)

In Eq. (17), w

si is the cost coefficient for the steel

reinforcement, ρi . Eq. (18) defines the set of seismic

interstory drift performance constraints under specified

earthquake ground motions. Each performance level

is viewed as corresponding to an earthquake of a

specified intensity (spectral acceleration) and performance

requirement (drift limitation). For example, if there are

three performance levels of concern, such optimization

formulation involves three sets of interstory drift constraints.

Eq. (19) defines the sizing constraints for the steel

reinforcement, where ρLi and ρ

U i correspond to the lower

and upper size bounds specified for the tensile steel

reinforcement variable, ρi , and they should be updated after

each nonlinear pushover analysis. Note that the compressive

steel reinforcement, ρ

i , has been assumed to be linearly

related to ρi and, therefore, it is not included in the explicit

optimization problem Eqs. (17)–(19), (20a) and (20b).

3. Optimality criteria algorithm and design procedure

3.1. Optimality criterion algorithm

Upon formulating the explicit design optimization

problem, Eqs. (17)–(19), an OC approach is developed to

solve the problem. In this approach, a set of necessary

optimality conditions for the optimal design is first

derived and a recursive algorithm is then applied to

achieve indirectly the optimum by satisfying the optimality

conditions.

For convenience of discussion, xi (i = 1, 2, . . . , N i ) is

defined as a design variable vector for member i . From

Eqs. (17)–(19), the optimal design problem with N jdisplacement constraints and N i sizing constraints can be

generally defined in terms of N i design variables as

Minimize the objective function F = F (xi ) (21)

subject to g j (xi ) ≤ d U j ( j = 1, 2, . . . , N j ) (22)

xLi ≤ xi ≤ x

U i (i = 1, 2, . . . , N i ). (23)

In classical optimization theory [23], the necessary

optimality criteria for the constrained optimization problem,

Eqs. (21)–(23), can be obtained indirectly by first converting

the constrained problem into an unconstrained Lagrangian

function and then solving for the stationary condition of

the Lagrangian function, L. By temporarily omitting the

sizing constraints in Eq. (23), the unconstrained Lagrangian




Fig. 5. E xample 1: One-story, one-bay frame.

9. Apply the recursive OC optimization algorithm using

Eqs. (26) and (27) to resize all steel reinforcement

design variables and to identify the active inelastic drift

constraints.

10. Check convergence of the steel cost and the inelastic

drift performance of the structure. Terminate with the

optimum design if the solution convergence is found;

otherwise, return to Step 6.

4. Illustrative examples

4.1. Example 1: A one-story, one-bay frame

A one-story, one-bay planar frame is used to verify the

pushover displacement formulation (Eqs. (5)–(7)), including

the element elastic strain energy generated by the membersand the plastic hinge strain energy generated by the plastic

hinges. The structural geometry of the frame is given in

Fig. 5. Concrete with the cylinder strength of 20 MPa and

steel reinforcement with the yield strength, f y , of 335 MPa

are used for all members. A uniform distributed gravity

load of 4 kN/m is applied to the beam. The lateral load

denoted as P is applied at the top of the structure as shown

in Fig. 5. Flexural moment hinges are assigned to the ends

of the beam, while the axial-moment hinges are assigned to

the ends of each column. The plastic rotation limit, θU p , is

assumed to be 0.02 rad for moment hinges and 0.015 rad

for axial-moment hinges [2,3]. The framework is analyzedusing the pushover method with member sizes of 300 mm ×

300 mm for the two columns and 200 mm × 450 mm for the

beam [25].

The pushover analysis of the structure is first carried out

by the SAP2000 software [24] and the top displacement

response of the structure caused by pushover loading is also

explicitly calculated by the virtual work method as stated

in Eqs. (5)–(7). Table 1 lists the step-by-step displacements

calculated by both the software and the virtual work

method and the percentage of difference between them. It

is found that the difference ranges between 0.0% and 0.5%,

indicating that the explicit displacement values calculated by

Fig. 6. E xample 1: Top displacement contributed by members and hinges.

the virtual work method match very well at each pushover

step with those obtained from the pushover analysis by the

software.

Table 1

Example 1: Comparison of pushover lateral displacements

Pushover steps SAP2000

displacement

(mm)

Virtual work

displacement

(mm)

Percentage

of difference

(%)

Push 0 0.005 0.005 0.000

Push 1 5.176 5.198 0.416

Push 2 5.757 5.787 0.522

Push 3 6.282 6.313 0.491

Push 4 10.872 10.900 0.258

Push 5 28.971 28.921 −0.170

Push 6 40.971 40.920 −0.125

Push 7 49.981 49.929 −0.105

Push 8 49.982 49.867 −0.230

Push 9 50.154 50.038 −0.232

Push 10 50.535 50.419 −0.230

Table 2 presents the portion of the top displacement

contributed by the elastic virtual strain energy of the

members, the portion of displacement contributed by the

plastic virtual strain energy of the plastic hinges and the ratio

of each portion to the total displacement. It is observed that

with the increase in the pushover analysis steps, the lateraldisplacement contributed by the plastic hinges becomes

increasingly large, i.e., goes from 0% to 90.3%, while

that contributed by members becomes increasingly small,

i.e., goes from 100.0% to 9.7%. That is to say, with

the occurrence of plastic hinges and an increase in the

hinge rotation, the contribution of the elastic energy of

the members to the total lateral displacement becomes less

dominant than the contribution of the inelastic energy of the

plastic hinges. This point also can be observed in Fig. 6,

which shows the comparison in each pushover step between

the lateral displacements contributed by both the elastic

members and the plastic hinges.




Table 2

Example 1: Contribution of members and plastic hinges to total pushover

displacement

Pushover steps Virtual work displacement Percentage distribution

Members Plastic hinges Members Plastic hinges

(mm) (mm) (%) (%)

Push 0 0.005 0.000 100.000 0.000Push 1 5.198 0.000 100.000 0.000

Push 2 5.565 0.222 96.171 3.829

Push 3 5.699 0.613 90.287 9.713

Push 4 6.183 4.716 56.729 43.271

Push 5 6.387 22.534 22.085 77.915

Push 6 6.504 34.416 15.893 84.107

Push 7 6.591 43.338 13.201 86.799

Push 8 4.673 45.194 9.371 90.629

Push 9 4.775 45.263 9.543 90.457

Push 10 4.870 45.549 9.659 90.341

4.2. Example 2: A ten-story, two-bay frame

A ten-story, two-bay planar frame is used to illustrate

the proposed optimal design method. The geometry of the

example is given in Fig. 7. Concrete with the cylinder

strength of 20 MPa and steel reinforcement with the yield

strength, f y , of 335 MPa are used for all members. The

loads considered in the pushover analysis are lateral seismic

loads and vertical gravity loads. While the lateral loads

are incrementally applied, the gravity loads are maintained

unchanged during the nonlinear pushover analysis process.

To illustrate the effectiveness of the optimal design

technique, two cases are conducted in this example. Case A

considers a uniformly distributed gravity load of 30 kN/m

to be applied to the beams of each story, while in Case Bthe distributed gravity load is intentionally reduced from 30

to 10 kN/m. It is intended that the different gravity loadings

will cause different axial effects on the columns that may

result in different pushover performance and capacity of the

framework example.

Initial member sizes of the framework used to start the

inelastic design optimization are shown in Table 3 [21,22].

Initial steel reinforcement ratios are first calculated on the

basis of the strength requirements of the RC members in

accordance with the Chinese seismic design code [25]. Such

strength-based reinforcement ratios are taken initially as the

lower bounds for the inelastic design optimization. AlthoughCase B is applied with lighter gravity loads, the initial steel

reinforcement design of Case B is deliberately assumed to

be the same as that of Case A for simplicity and comparison

reasons. The upper size bounds of the steel reinforcement

ratios are assumed to be 6.0% for columns and 4.0% for

beams. For simplicity, symmetrical arrangement of the steel

reinforcement of each member is assumed such that ρi =

ρi . Flexural moment hinges and axial-moment hinges are

assigned to the end locations of the beams and columns,

respectively. The ultimate plastic hinge rotation, θU p , is

assumed to be 0.02 rad for the moment hinges on the beams

and 0.015 rad for the axial-moment hinges on the columns.

Fig. 7. Example 2: Ten-story, two-bay frame.

The 5% damped design spectrum with an initial peak

acceleration of 1.4g according to the Chinese seismic design

code [25] is modified by the spectral reduction method [2] inthe pushover analysis of this example. The pushover analysis

of the structure is conductedwith the SAP2000 software [24]

on the basis of the assumption that the fundamental mode

of vibration is the predominant response of the structure.

The initial lateral loads applied in the pushover analysis are

proportional to the product of the story mass and the first

mode shape of the structure.

A typical unit construction cost of the steel reinforcement

(including the costs of both steel material and labor) of

US$950/ton is assumed. Interstory drift constraints are

considered with an assumed allowable interstory drift ratio

limit of 1%. The initial preprocessor, in which analyticaloptimization with the top displacement constraint alone is

considered, is applied before the multiple inelastic interstory

drift constrained optimization is invoked. The design process

is deemed to converge when the difference in the structure

costs for two successive design cycles is within 0.5% and

when the difference between the active interstory drift value

and its allowable limit at the performance point is within

0.5%.

The optimal design histories of the two cases are

presented in Fig. 8. The initial preprocessing is first

completed and the multiple inelastic interstory drift

constrained optimization is then commenced. It is found




Table 3

Example 2: Initial and final steel reinforcement ratios

Element type Story level Member group Initial member sizes Case A Case B

Steel ratios Steel ratios

Width (mm) Depth (mm) Initial (%) Optimal (%) Initial (%) Optimal (%)

9th–10th C1, C3 350 350 0.990 1.096 0.990 1.506

C2 350 350 1.386 1.386 1.386 1.5788th C1, C3 350 350 0.849 0.860 0.849 1.524

C2 350 475 0.958 0.958 0.958 0.958

Column 7th C1, C3 350 350 1.163 1.163 1.163 1.922

C2 350 475 1.092 1.092 1.092 1.092

6th C1, C3 350 400 0.854 0.864 0.854 1.200

C2 350 575 0.831 0.831 0.831 0.874

5th C1, C3 350 400 0.969 1.004 0.969 1.428

C2 350 575 1.002 1.002 1.002 1.002

4th C1, C3 350 450 0.833 0.873 0.833 1.356

C2 350 600 0.739 0.739 0.739 0.739

3rd C1, C3 350 450 0.825 0.876 0.825 1.536

C2 350 600 0.857 0.857 0.857 0.857

2nd C1, C3 350 450 1.044 1.124 1.044 1.644

C2 350 650 1.225 1.225 1.225 1.225

1st C1, C3 350 450 1.514 1.514 1.514 1.514C2 350 650 1.844 1.844 1.844 1.844

9th–10th B1, B2 200 400 0.800 0.800 0.800 0.800

8th B1, B2 200 450 0.838 0.941 0.838 0.838

Beam 7th B1, B2 200 450 0.915 1.320 0.915 1.053

6th B1, B2 250 450 0.897 1.310 0.897 1.355

5th B1, B2 250 450 0.958 1.493 0.958 1.383

4th B1, B2 300 450 0.915 1.457 0.915 1.430

3rd B1, B2 300 450 0.958 1.484 0.958 1.440

2nd B1, B2 300 450 0.982 1.512 0.982 1.404

1st B1, B2 300 450 0.843 1.006 0.843 1.119

Fig. 8. Example 2: Design history of steel reinforcement costs.

from Case A that there is a relatively large increase of

14% in the steel cost from the initial US$1693 to US$1931

after the initial preprocessing with only the top displacement

constraint. Therefore, the Case A optimal design process

with the multiple drift constraint converges slowly but

steadily after 11 design cycles, with only a slight difference

of 2% in the steel cost from US$1931 to the final US$1978.

As shown in Fig. 8, Case B is found to converge to a higher

value of optimal steel cost of US$2225 in 18 design cycles.

Relatively slow, but steady, convergence is found for bothcases due to the need for maintaining a small change in

the steel reinforcement ratios during the inelastic design

optimization process.

It should be noted that Case B subjected to lighter

gravity loading has actually resulted in a higher value

of the final steel cost than Case A with heavier gravity

loading in order to satisfy the same inelastic pushover drift

performance requirements. A close scrutiny of the internal

force distribution of the Case B structure indicates that the

lateral pushover loading at the performance point of the

structure has caused net tension in the left side columns

due to the overturning action of the lateral loading. Thenet tension in the columns has led to a reduction in the

first yielding moment, M y , thereby resulting in a weaker

structure that requires an increase in the quantity of steel

reinforcement compared to Case A to fulfill the same

inelastic interstory drift design criteria.

Table 3 presents the initial and optimal steel reinforce-

ment ratios for the two cases. Initially, the starting design

with strength-based steel reinforcement is found to be in-

feasible in terms of the assumed allowable interstory drift

limit. After the optimization, the steel reinforcement ratios

of the beams greatly increase particularly in the lower levels

of the structure (e.g. from 0.982% to 1.512% for Case A and




(a) Case A.

(b) Case B.

Fig. 9. Example 2: Comparison of performance points.

from 0.982% to 1.404% for Case B at story level 2), while

those of the side columns are found with little changes for

Case A (e.g. from 1.044% to 1.124% for columns C1 and

C2 at story level 2), but with a rather significant increasefor Case B (e.g. from 1.044% to 1.644% for columns C1

and C2 at story level 2). Although the overturning action of

the lateral loading has caused an uplifting tension on the left

side columns (with negative weakening effect on moment

resistance) and down-pressing compression on the right side

columns (with positive stiffening effect on moment resis-

tance), the steel reinforcement of these side columns must

be sized the same to take into account the reversal seismic

loading effects. When comparing the results of the steel re-

inforcement of the columns for the two cases, the columns

of Case A are found to be mainly controlled by the minimum

strength requirements with the final steel ratios being set to

(a) Case A.

(b) Case B.

Fig. 10. Example 2: Initial and final interstory drift ratios.

the initial strength-based limits, whilst the columns of Case

B are governed by the inelastic interstory drift performance

requirements with the final steel ratios being larger than the

initial strength-based limits.Figs. 9(a) and (b) present the initial and final performance

points respectively for Cases A and B. In Case A, the

performance point “P1” of the initial structure has a spectral

acceleration of 0.068g and a spectral displacement of

0.262 m. The optimized structure corresponding to the

final performance point “P2” has a spectral acceleration of

0.086g and a spectral displacement of 0.211 m. Such a

result of shifting the spectral displacement from 0.262 m to

0.211 m indicates that, for the optimized inelastic frame, the

inelastic lateral load resistance has been enhanced through

optimal resizing of the steel reinforcement by the OC

procedure. Also, the shifting from the initial performance




(a) Number of plastic hinges.

(b) Case A: Initial design. (c) Case A: Final design. (d) Case B: Initial design. (e) Case B: Final design.

Fig. 11. Example 2: Initial and final plastic hinge distribution.

point to the final performance point has resulted in an

increase in the spectral acceleration values from 0.068g to

0.086g, indicating that there has been an increase in the

seismic loading action for the optimized structure. The OC

procedure developed is found to be able to automatically

drive from any initial performance point to the final

performance point resulting in the minimum cost design.

Similar trends of results have also been observed in Case B.

In fact, the performance points of the optimized structures

of the two cases are found to be relatively close to eachother.

The initial and final interstory drift ratios are shown

in Fig. 10(a) for Case A and Fig. 10(b) for Case B. The

initial interstory drift constraints at the second through

the eighth floors are found to violate substantially the

allowable interstory drift ratio limit of 1% for both cases,

resulting in the occurrence of weak stories at these floor

levels of the building. However, these pushover interstory

drift constraints are found to be close to and within the

allowable value ranges after the optimization, indicating that

a rather uniform interstory drift distribution over the height

of the building has been achieved and the occurrence of a

weak story has been prevented at the optimum performance

point.

Fig. 11(a) includes a table showing the number of

plastic hinges at various ranges of performance states, i.e.,

Immediate Occupancy (IO), Life Safety (LS) and Collapse

Prevention (CP). In this figure, the values of plastic rotation

limits corresponding to the performance states of IO, LS

and CP are defined to be 0.005 rad, 0.01 rad and 0.02 rad

for all beams, and 0.005 rad, 0.01 rad and 0.015 rad for

all columns, respectively, in accordance with the guidelinesprovided by ATC-40 [2]. It is found that Case B with

lighter gravity loading has resulted in tension forces in

the columns under the lateral pushover loading so that a

larger number of plastic hinges have been found to appear

than in Case A. Figs. 11(b)–(e) show the initial and final

plastic hinge distributions under the pushover loading at

the performance point of the structure for the two cases.

No plastic hinge rotation is found to exceed the specified

threshold of plastic rotation, θU p . As shown in Fig. 11(b)

for Case A and Fig. 11(d) for Case B, the rotations of

twenty plastic hinges of the initial design are found to be

located between the LS and CP states. However, after the




optimization, most of the plastic hinges are found to have

smaller values of plastic rotations within the ranges of the

B–IO and IO–LS states and only one hinge is found to be in

the more critical range between the LS and CP states, as can

be observed from the optimized framework in Figs. 11(c)

and (e). Furthermore, the interstory drifts along the height of

the building are also found to be almost all fully constrainedat the optimum, resulting in a rather linear deflected profile

of the inelastic design. Such a result further indicates

that the optimization method developed can automatically

resize the steel reinforcements of all members to attain a

uniform ductility demand over the height of the multistory

building.

5. Conclusions

It has been demonstrated that steel reinforcement plays

a significant role in controlling the lateral drift beyond

first yielding and in providing ductility to an RC building

framework. Using the principle of virtual work and the

Taylor series approximation, the inelastic performance-

based seismic design problem has been explicitly expressed

in terms of the steel reinforcement design variables. Axial-

moment hinges and moment hinges should be considered

in the nonlinear pushover analysis of a frame structure so

that the behavior of columns and beams can be effectively

modelled. It is important that uplifting tension induced by

seismic loading should be prevented in an RC building as the

net tension in columns tends to weaken the lateral resistance

of such a structure and to result in a less economical design.

It is demonstrated that the OC design method is able to

improve automatically and gradually a performance-basedinterstory drift design to attain optimal performance. Also,

this OC design method developed is able to automatically

shift any initial performance point to achieve the final

optimal performance point. However, a restrictive move

limit imposed on the steel reinforcement design variables

is necessary to ensure a smooth and steady convergence

of the inelastic drift design process. At optimum, a

uniform lateral drift or ductility demand over all stories

of the building with the minimum cost is achieved. It is

also believed that this optimization methodology provides

a powerful computer-based technique for performance-

based design of multistory RC building structures. Theproposed optimization methodology provides a good basis

for more comprehensive performance-based optimization of

structures as more accurate nonlinear pushover procedures

taking into account the higher mode effects are developed

and multiple levels of performance criteria and design

objectives are to be simultaneously considered.

Acknowledgment

The authors are grateful for the financial support provided

by the Research Grants Council of Hong Kong under Project

No. HKUST6249/00E.

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