# Damage-based design earthquake loads for SDOF inelastic structures

**ABSTRACT** This paper develops a new framework for modeling design earthquake loads for inelastic structures. Limited information on strong ground motions is assumed to be only available at the given site. The design earthquake acceleration is expressed as a Fourier series, with unknown amplitude and phase angle, modulated by an envelope function. The design earthquake is estimated by solving an inverse dynamic problem, using nonlinear programming techniques, such that the structure performance is minimized. At the same time, the design earthquake is constrained to the available information on past recorded ground motions. New measures of the structure performance that are based on energy concepts and damage indices are introduced in this paper. Specifically, the structural performance is quantified in terms of Park and Ang damage indices. Damage indices imply that the structure is damaged by a combination of repeated stress reversals and high stress excursions. Furthermore, the use of damage indices provides a measure on the structure damage level and thus a decision on necessary repair is possible. The material stress-strain relationship is modeled as either bilinear or elastic-plastic. The formulation is demonstrated by deriving the design earthquake loads for inelastic frame structures at a firm soil site. The damage spectra for the site are also established, which provide upper bounds of damage under possible future earthquakes.

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**ABSTRACT:**This paper deals with damage assessment of adjacent colliding buildings under strong ground motion. In previous studies, the structure input-response pair is used to examine pounding effects on adjacent buildings under seismic loads. In this paper, pounding of adjacent buildings is assessed using input energy, dissipated energy and damage indices. Damage indices (DI) are computed by comparing the structure’s responses demanded by earthquakes and the associated structural capacities. Damage indices provide quantitative estimates of structural damage level, and thus, a decision on necessary repair can be taken. Adjacent buildings with fixed-base and isolated-base are considered. The nonlinear viscoelastic model is used for capturing the induced pounding forces. Influences of the separation distance between buildings, buildings properties, such as, base-condition (fixed or isolated), and yield strength on damage of adjacent buildings are investigated. The set of input ground motions includes short-, moderate- and long-duration accelerograms measured at near-fault and far-fault regions with different soil types. Earthquake records with different characteristics are considered to study damage of adjacent buildings under seismic loads. Numerical illustrations on damage of fixed-base and isolated-base adjacent buildings with elastic–plastic force–deformation relation are provided.Engineering Structures 03/2014; 61:153–165. · 1.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Multiple acceleration sequences of earthquake ground motions have been observed in many regions of the world. Such ground motions can cause large damage to the structures due to accumulation of inelastic deformation from the repeated sequences. The dynamic analysis of inelastic structures under repeated acceleration sequences generated from simulated and recorded accelerograms without sequences has been recently studied. However, the characteristics of recorded earthquake ground motions of multiple sequences have not been studied yet. This paper investigates the gross characteristics of earthquake records of multiple sequences from an engineering perspective. The definition of the effective number of acceleration sequences of the ground shaking is introduced. The implication of the acceleration sequences on the structural response and damage of inelastic structures is also studied. A set of sixty accelerograms is used to demonstrate the general properties of repeated acceleration sequences and to investigate the associated structural inelastic response.Earthquakes and Structures 01/2012; 3(5). · 1.38 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In recent decades, it has been realized that increasing the lateral stiffness of structure subjected to lateral loads is not the only parameter enhancing safety or reducing damage. Factors such as ductility and damping govern the structural response due to lateral loads. Despite the significant contribution of damping in resisting lateral loads, especially at resonance, there is no accurate mathematical representation for it. The main objective of this study is to develop a damping identification procedure for linear systems based on a mixed numerical-experimental approach, assuming viscous damping. The proposed procedure has been applied to a laboratory experiment associated with a numerical model, where a hollow rectangular steel cantilever column, having three lumped masses, has been fixed on a shaking table subjected to different exciting waves. The modal damping ratio has been identified; in addition, the effect of adding filling material to the hollow specimen has been studied in relation to damping enhancement. The results have revealed that the numerically computed response based on the identified damping is in a good fitting with the measured response. Moreover, the filling material has a significant effect in increasing the modal damping.Earthquakes and Structures 01/2013; 4(2). · 1.38 Impact Factor

Page 1

Damage-Based Design Earthquake Loads for

Single-Degree-Of-Freedom Inelastic Structures

Abbas Moustafa1

Abstract: This paper develops a new framework for modeling design earthquake loads for inelastic structures. Limited information on

strong ground motions is assumed to be available only at the given site. The design earthquake acceleration is expressed as a Fourier series,

with unknown amplitude and phase angle, modulated by an envelope function. The design ground acceleration is estimated by solving an

inverse dynamic problem, using nonlinear programming techniques, so that the structure performance is minimized. At the same time, the

design earthquake is constrained to the available information on past recorded ground motions. New measures of the structure performance

based on energy concepts and damage indexes are introduced in this paper. Specifically, the structural performance is quantified in terms of

Park and Ang damage indexes. Damage indexes imply that the structure is damaged by a combination of repeated stress reversals and

high-stress excursions. Furthermore, the use of damage indexes provides a measure on the structure damage level, and making a decision

on necessary repair possible. The material stress-strain relationship is modeled as either bilinear or elastic-plastic. The formulation is

demonstrated by deriving the design earthquake loads for inelastic frame structures at a firm soil site. The damage spectra for the site

are also established, to provide upper bounds of damage under possible future earthquakes. DOI: 10.1061/(ASCE)ST.1943-541X

.0000074. © 2011 American Society of Civil Engineers.

CE Database subject headings: Ductility; Damage; Optimization; Earthquake loads; Seismic design; Inelasticity; Earthquake resistant

structures.

Author keywords: Design earthquake loads; Input energy; Inelastic structures; Ductility ratio; Hysteretic energy; Damage indexes;

Damage spectra; Nonlinear optimization.

Introduction

The assessment of seismic performance of structures under future

earthquakes is an important problem in earthquake engineering.

The basic objective of the structural engineer is to design structures

that are safe against possible future earthquakes and economical at

the same time. To achieve this goal, the following criteria should be

fulfilled: (1) robust definition of earthquake ground motions for the

site, (2) accurate mathematical model for the material behavior, and

(3) reliable structural damage descriptors that accurately describe

possible structural damage under seismic loads.

The earthquake-resistant design of structures has been an active

area of research for many decades. Early works have dealt with

specifying earthquake loads in terms of the elastic and inelastic

design response spectra for the site (e.g., Mahin and Bertero

1981; Newmark and Hall 1982; Riddell 1995), specifying the time

history of the ground acceleration (e.g., Bommer and Acevedo

2004), or using the theory of random vibrations (e.g., Kiureghian

and Crempien 1989; Conte and Peng 1997). Hazard response spec-

tra have also been established by many researchers (e.g., Reiter

1990 and McGuire 1995). The development of mathematical

models to describe the hysteretic nonlinear behavior of the structure

during earthquakes has been carried out by several researchers

(e.g., Takeda et al. 1970; Otani 1981; Akiyama 1985). However,

the inadequate performance of structures during recent earthquakes

has motivated researchers to revise existing methods and to develop

new methods for seismic-resistant design. This includes new design

concepts, such as, energy-based design (e.g., Akiyama 1985; Goel

1997; Decanini and Mollaiodi 2001; Wong and Yang 2002),

performance-based design (e.g., Park et al. 1985; Fajfar 1992;

Fajfar and Krawinkler 1997; SEAOC 2000; Bozorgnia and Bertero

2004; Choi and Kim 2006) and optimum damper placement for

seismic-resistant design (e.g., Nakashima et al. 1996; Yamaguchi

and El-Abd 2003).

This paper develops a new framework for specifying robust

design earthquake loads for seismic-resistant design of structures

using the method of critical excitations. This method relies on

the high uncertainty associated with the occurrence of earthquakes

and their characteristics (e.g., time, location, magnitude, duration,

frequency content, and amplitude), and also on the safety require-

ments of important and lifeline structures (e.g., nuclear plants, stor-

age tanks, and industrial installations). A limited material on

modeling critical earthquakes for nonlinear structures is available

in the literature (e.g., Moustafa 2002; Abbas 2006; Takewaki 2002,

2007). Drenick and Iyengar provided early research thoughts on

this subject. Iyengar (1972) computed critical seismic inputs for

nonlinear Duffing oscillators by constraining the input energy.

Drenick (1977) derived critical excitations for nonlinear systems

in terms of the impulse response of the linearized system.

Philippacopoulos and Wang (1984) expressed the critical input

as a linear summation of recorded accelerograms and established

critical inelastic response spectra for the site. This series representa-

tion, however, is questionable. Westermo (1985) used calculus of

1Department of Civil Engineering, School of Engineering, Nagasaki

Univ., Nagasaki 852-8521, Japan, and Dept. of Civil Engineering,

Faculty of Engineering, Minia Univ., Minia 61111, Egypt (corresponding

author). E-mail: abbas.moustafa@eng.miniauniv.edu.eg; abbas.moustafa@

yahoo.com

Note. This manuscript was submitted on June 23, 2008; approved on

March 30, 2009; published online on February 15, 2011. Discussion period

open until August 1, 2011; separate discussions must be submitted for

individual papers. This paper is part of the Journal of Structural Engineer-

ing, Vol. 137, No. 3, March 1, 2011. ©ASCE, ISSN 0733-9445/2011/3-

456–467/$25.00.

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Page 2

variation to show that the critical inputs for elastic-plastic systems

are not harmonic. Recently, Takewaki (2001) used the equivalent

linearization method to estimate critical probabilistic earthquakes

for elastic-plastic structures that maximize the interstory drift.

Abbas (2006) derived critical seismic loads for inelastic structures

by maximizing the ductility ratio. Similarly, probabilistic critical

earthquakes were computed for inelastic and parametrically excited

structures by using first-order reliability method (FORM) and re-

sponse surface approximations (Sarkar 2003; Abbas and Manohar

2005, 2007).

From the above discussion it can be seen that most of the afore-

mentioned research work is either conceptual, uses approximations

in representing the ground motion or in calculating the structure

response or is based on maximization of single response parameter.

This paper avoids these approximations and introduces modern

measures of structural damage to develop robust earthquake loads

on inelastic structures. The structure performance is quantified

using Park and Ang damage indexes and thus a quantitative mea-

sure of the structure damage and necessary repair are possible.

Response and Damage Characterization of Inelastic

Structures under Earthquake Loads

This section demonstrates briefly the seismic-response analysis and

energy quantification for single-degree-of-freedom (SDOF) inelas-

tic structures under earthquake loads. Subsequently, the use of

inelastic response parameters and energy absorbed by the structure

in developing damage indexes is explained.

Dynamic Analysis of Inelastic Structures

The equation of motion for a nonlinear SDOF structure under a

single component of earthquake acceleration € xgðtÞ is given as

m€ xðtÞ þ c_ xðtÞ þ fsðtÞ ¼ ?m€ xgðtÞ

where m and c = mass and damping coefficient of the structure,

respectively; fsðtÞ = spring hysteretic restoring force; xðtÞ =

structure displacement and dot indicates differentiation with

respect to time. Fig. 1 depicts the relationship between the inelastic

deformation and the spring hysteretic force for bilinear and elastic-

plastic materials. Eq. (1) can be rewritten as

€ x þ 2η0ω0_ x þ ω2

Here η0and ω0= preyield damping ratio and natural frequency;

xy= yield displacement; and?fsðtÞ = normalized hysteretic force.

Eq. (2) can be further recast as

ð1Þ

0xy?fsðtÞ ¼ ?€ xgðtÞð2Þ

€ μðtÞ þ 2η0ω0_ μðtÞ þ ω2

0?fsðtÞ ¼ ?ω2

0

€ xgðtÞ

ay

ð3Þ

where μðtÞ ¼ xðtÞ=xy= ductility ratio; and ay¼ fy=m = constant

that can be interpreted as the acceleration of the mass necessary

to produce the yield force. The response of inelastic SDOF struc-

tures can be computed by solving the incremental form of Eqs. (1)

and (3) using numerical integration at discrete points of time. In this

study, we use the Newmark β-method. For bilinear behavior, an

iterative procedure is adopted to correct for approximations of

the secant stiffness used from previous time step. The next subsec-

tion demonstrates the quantification of the earthquake input energy

and associated energy dissipated by the structure.

Earthquake Input Energy and Energy Dissipated

by Inelastic Structures

The energy balance for the SDOF inelastic structure can be

obtained by multiplying Eq. (1) by the relative velocity _ xðτÞ and

integrating, thus (Zahrah and Hall 1984; Akiyama 1985; Uang

and Bertero 1990; Takewaki 2004; Kalkan and Kunnath 2008)

Z

¼ ?

0

t

0

m€ xðτÞ_ xðτÞdτ þ

Z

Z

t

0

c_ x2ðτÞdτ þ

Z

t

0

fsðτÞ_ xðτÞdτ

t

m€ xgðτÞ_ xðτÞdτ

ð4aÞ

EKðtÞ þ EDðtÞ þ ESðtÞ ¼ EIðtÞð4bÞ

Eqs. (4a) and (4b) represent the relative energy terms (see, e.g.,

Uang and Bertero 1990; Kalkan and Kunnath 2008). Here EIðtÞ

is the earthquake relative input energy to the structure, because

the ground shakes until it comes to rest. EKðtÞ is the relative kinetic

energy ½EKðtÞ ¼ m_ x2ðtÞ=2? and EDðtÞ is the energy absorbed by

damping. The energy ESðtÞ represents the total relative energy

absorbed by the spring and is composed of the recoverable elastic

energy and the hysteretic cumulative plastic energy EHðtÞ.

At the end of the earthquake duration the kinetic and elastic

strain energies diminish. Thus, the earthquake input energy to

the structure is dissipated by the hysteretic and the damping

energies. The next section demonstrates the use of the response

parameters and plastic energy in developing damage indexes.

Safety Assessment of Inelastic Structures Using

Damage Indexes

The literature on damage measures of structures under ground

motions is vast (e.g., Cosenza et al. 1993; Ghobarah et al. 1999).

Damage indexes are based on either a single or combination of

structural response parameters. Table 1 summarizes several

damage measures that are based on a single response parameter

(Powell and Allahabadi 1988; Cosenza et al. 1993). The first

measure indicates the ultimate ductility produced during theground

shaking. Clearly, this measure does not incorporate any information

on how the earthquake input energy is imparted on the structure

nor how this energy is dissipated. Earthquake damage occurs

not only due to the maximum deformation or ductility but is asso-

ciated with the hysteretic energy dissipated by the structure as well.

The definition of structural damage, in terms of the ductility factor,

is inadequate. The last three measures indicate the rate of the earth-

quake input energy to the structure (i.e., how fast EIis imparted by

the earthquake and how fast it gets dissipated). Damage indexes can

be estimated by comparing the response parameters demanded by

the earthquakewith the structural capacities. Powell and Allahabadi

(1988) proposed a damage index in terms of the ultimate ductility

(capacity)μuand the maximum ductility attained during ground

shaking μmax:

(a) (b)

fs(t)

x(t)

fy

xy

k0

kp=α k0

fs(t)

x(t)

fy

xy

k0

kp=0

xmax

xmax

Fig. 1. Force-displacement relation for nonlinear materials: (a) bilinear

model; (b) elastic-plastic model

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Page 3

DIμ¼xmax? xy

xu? xy

¼μmax? 1

μu? 1

ð5Þ

However, DIμdoes not include effects from hysteretic energy

dissipation. Cosenza et al. (1993) and Fajfar (1992) proposed a

damage index based on the structure hysteretic energy EH:

DIH¼EH=ðfyxyÞ

μu? 1

ð6Þ

A robust damage measure should include not only the maximum

response but the effect of repeated cyclic loading as well. Park and

coworkers developed a simple damage index, given as (Park et al.

1985; Park and Ang 1985; Park et al. 1987)

DIPA¼xmax

xu

þ βEH

fyxu

¼μmax

μu

þ β

EH

fyxyμu

ð7Þ

Here xmaxand EH= maximum displacement and dissipated hyster-

etic energy (excluding elastic energy) under the earthquake. Note

that xmaxis the maximum absolute value of the displacement

response under ground motion, xuis the ultimate deformation

capacity under monotonic loading and β is a positive constant that

weights the effect of cyclic loading on structural damage. If β ¼ 0,

the contribution to DIPAfrom cyclic loading is omitted.

The state of the structure damage is defined as: (a) repairable

damage, when DIPA< 0:40, (b) damaged beyond repair, when

0:40 ≤ DIPA< 1:0, and (c) total or complete collapse, when

DIPA≥ 1:0. These criteria are based on calibration of DIPAagainst

experimental results and field observations in earthquakes (Park

et al. 1987). The Park and Ang damage index reveals that both

maximum ductility and hysteretic energy dissipation contribute

to the structure resistance during ground motions. In Eq. (7)

damage is expressed as a linear combination of the damage caused

by excessive deformation and that contributed by repeated cyclic

loading effect. Also, the quantities xmax, EHdepend on the loading

history while the quantities β, xu, fyare independent of the loading

history and are determined from experimental tests. In this paper

we adopt Park and Ang damage index in deriving the design earth-

quakes. The next section develops this formulation.

Damage-Based Design Earthquake Loads

for Inelastic Structures

The derivation of critical earthquake loads for SDOF inelastic

structures is developed in this section. The ground acceleration

is represented as a product of a Fourier series and an envelope

function:

€ xgðtÞ ¼ eðtÞ

X

Nf

i¼1

Ricosðωit ? φiÞ

¼ A0½expð?α1tÞ ? expð?α2tÞ?

X

Nf

i¼1

Ricosðωit ? φiÞð8Þ

Here A0= scaling constant, and the parameters α1, α2impart the

transient nature to € xgðtÞ. Next, Riand φiare 2Nfunknown ampli-

tudes and phase angles, respectively and ωi, i ¼ 1;2;…;Nfare the

frequencies presented in the ground acceleration that are selected to

span satisfactory the frequency range of € xgðtÞ. In constructing criti-

cal seismic inputs, the envelope function is taken to be fully known.

The information on energy E, peak ground acceleration (PGA) M1,

peak ground velocity (PGV) M2, peak ground displacement (PGD)

M3, upper bound Fourier amplitude spectra (UBFAS) M4ðωÞ,

and lower bound Fourier amplitude spectra (LBFAS) M5ðωÞ are

also taken to be available, which enables defining the following

constraints (Abbas and Manohar 2002; Abbas 2006):

?Z

maxmax

∞

0

€ x2

gðtÞdt

?1=2

≤ E max

0<t<∞j€ xgðtÞj ≤ M1

0<t<∞jxgðtÞj ≤ M3

0<t<∞j_ xgðtÞj ≤ M2

M5ðωÞ ≤ jXgðωÞj ≤ M4ðωÞ

Here XgðωÞ = Fourier transform of € xgðtÞ. The constraint on the

earthquake energy is related to the Arias intensity (Arias 1970).

The UBFAS and LBFAS constraints aim to replicate the frequency

content and amplitude observed in past recorded accelerograms on

the design earthquake. The ground velocity and displacement are

obtained from Eq. (8) as follows:

ð9Þ

_ xgðtÞ ¼

X

X

Nf

i¼1

Nf

Z

Z

t

0

RieðτÞcosðωiτ ? φiÞdτ þ C1;

xgðtÞ ¼

i¼1

t

0

RieðτÞðt ? τÞcosðωiτ ? φiÞdτ þ C1t þ C2

ð10Þ

Making use of the conditions xgð0Þ ¼ 0 and limt→∞_ xgðtÞ → 0

(Shinozuka and Henry 1965), the constants in the above Eq. can

be shown to be given as (Abbas and Manohar 2002; Abbas 2006):

C2¼ 0;

C1¼ ?

X

Nf

i¼1

Z

∞

0

RieðτÞcosðωiτ ? φiÞdτ

ð11Þ

The constraints of Eq. (9) can be expressed in terms of the variables

Ri, φi, i ¼ 1;2;…;Nfas

Table 1. Response Descriptors for Inelastic Buildings under Earthquake Ground Motion

S. no.Response parameterDefinition

1

2

3

4

5

6

7

8

9

10

11

Maximum ductility

Number of yield reversals

Maximum normalized plastic deformation range

Normalized cumulative ductility

Residual (permanent) ductility

Normalized earthquake input energy

Normalized total hysteretic energy dissipated

Ratio of total hysteretic energy to input energy

Maximum rate of normalized input energy

Maximum rate of normalized damping energy

Maximum rate of normalized hysteretic energy

μmax¼ max0≤t≤tfjxðtÞ=xyj

Number of times velocity changes sign

Δ? xp;i¼ max0≤t≤tfjΔxp;i=xyj

μac:¼PN

?EI¼ 1=ðfyxyÞRtf

rE¼?EH=?EI

PI;max¼ max0≤t≤tf½dEIðtÞ=dt?=ðfyxyÞ

PD;max¼ max0≤t≤tf½dEDðtÞ=dt?=ðfyxyÞ

PH;max¼ max0≤t≤tf½dEHðtÞ=dt?=ðfyxyÞ

i¼1jΔxp;ij=xyþ 1

μres¼ jxðtfÞ=xyj

?EH¼ 1=ðfyxyÞRtf

0EIðtÞdt

0EHðtÞdt

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Page 4

?

A2

0

X

Nf

m¼1

X

????A0½expð?α1tÞ ? expð?α2tÞ?

0<t<∞

n¼1

????A0

? A0t

n¼1

????A0

Nf

n¼1

RmRn

Z

∞

0

½expð?α1tÞ ? expð?α2tÞ?2cosðωmt ? φmÞcosðωnt ? φnÞdt

?1=2

≤ E

max

0<t<∞

X

Nf

n¼1

Rncosðωnt ? φnÞ

????≤ M1

max

????A0

X

X

Z

Nf

Z

Z

t

0

Rn½expð?α1τÞ ? expð?α2τÞ?cosðωnτ ? φnÞdτ ? A0

X

Nf

n¼1

Z

∞

0

Rn½expð?α1τÞ ? expð?α2τÞ?cosðωnτ ? φnÞdτ

????≤ M2

max

0<t<∞

Nf

n¼1

t

0

Rn½expð?α1τÞ ? expð?α2τÞ?ðt ? τÞcosðωnτ ? φnÞdτ

X

Nf

∞

0

Rn½expð?α1τÞ ? expð?α2τÞ?cosðωnτ ? φnÞdτ

Z

????≤ M3

M5ðωÞ ≤

X

Nf

n¼1

∞

0

Rnfexp½?α1τ? ? exp½?α2τ?gcosðωnτ ? φnÞexp½?iωτ?dτ

????≤ M4ðωÞð12Þ

Here i ¼

M3, M4ðωÞ, and M5ðωÞ it is assumed that a set of Nrearthquake

records denoted by € vgiðtÞ, i ¼ 1;2;…;Nrare available for the site

under consideration or from other sites with similar geological soil

conditions. The values of energy, PGA, PGVand PGD are obtained

for each of these records. The highest of these values across all

records define E, M1, M2, and M3. The available records are further

normalized such that the Arias intensity of each record is set to

unity (i.e.,½R∞

M4ðωÞ ¼ E max

ffiffiffiffiffiffiffi

?1

p

. To quantify the constraints quantities E, M1, M2,

0€ v2

giðtÞdt?1=2¼ 1, Arias 1970), and are denoted by

i¼1. The bounds M4ðωÞ and M5ðωÞ are obtained as

M5ðωÞ ¼ E min

f€ ? vgigNr

1≤i≤Nrj?VgiðωÞj;

1≤i≤Nrj?VgiðωÞjð13Þ

Here?VgiðωÞ, i ¼ 1;2;…;Nrdenotes the Fourier transform of the

ith normalized accelerogram € vgiðtÞ. The bound M4ðωÞ has been

considered earlier (Shinozuka 1970; Takewaki 2001; 2002). The

lower bound was considered by Moustafa (2002) and Abbas

and Manohar (2002).

Finally, the problem of deriving critical earthquake loads for

inelastic structures can be posed as determining the optimization

variables y ¼ fR1;R2;…;RNf;φ1;φ2;φ3…;φNfgtsuch that DIPA

is maximized subjected to the constraints of Eq. (12). The solution

to this nonlinear constrained optimization problem is tackled by

using the sequential quadratic programming method (Arora

2004). The following convergence criteria are adopted:

jfj? fj?1j ≤ ε1;

jyi;j? yi;j?1j ≤ ε2

ð14Þ

Here fj= objective function at the jth iteration; yi;j= ith optimiza-

tion variable at the jth iteration; and ε1, ε2= small quantities to be

specified. The structure inelastic deformation is estimated using the

Newmark β-method which is built as a subroutine inside the opti-

mization program. The details of the procedure involved in the

computation of the optimal earthquake and the associated damage

index are shown in Fig. 2.

The quantities μðtÞ and EHðtÞ do not reach their respective

maxima at the same time. Therefore, the optimization is

performed at discrete points of time and the optimal solution y?¼

½R?

mum DIPAacross all time points. The critical earthquake loads are

characterized in terms of the critical accelerations and associated

damage indexes, inelastic deformations and energy dissipated by

1;R?

2;…;R?

Nf;φ?

1;φ?

2;…;φ?

Nf?tis the one that produces the maxi-

the structure. The next section provides numerical illustrations

for the formulation developed in this section.

In the numerical analysis, the constraints quantities E, M1, M2,

M3, M4ðωÞ, and M5ðωÞ are estimated using past recorded earth-

quake data. These quantities are taken as the extreme values of

the associated parameters across the set of past recorded ground

motions. These parameters define the energy, PGA, PGV, PGD,

and upper and lower bounds on the Fourier amplitude spectra of

past recorded ground motions at the site under consideration or

other sites with similar geological soil conditions. This approach

is considered to be consistent with the aspirations of the ground-

motion models that are commonly used by engineers, that basically

aim to replicate some of the gross features of recorded motions,

such as amplitude, frequency content, nonstationarity trend, local

soil amplification effects, and duration. Predictive or physical

models for ground motions that take into account several details,

such as fault dimension, fault orientation, rupture velocity, magni-

tude of earthquake, attenuation, stress drop, density of the interven-

ing medium, local soil condition, and epicentral distance, have also

been developed in the existing literature, mainly by seismologists

(see, e.g., Brune 1970; Hanks and McGuire 1981; Boore 1983;

Queck et al. 1990). In using these models, one needs to input values

for a host of parameters and the success of the model depends upon

how realistically this is done. It is possible to formulate the optimal

earthquake models based on the latter class of models where one

can aim to optimize the parameters of the model to realize the least

favorable conditions. It is important to consider that the class of

admissible functions, in the determination of critical excitations,

in this case, becomes further constrained by the choice that one

makes for the physical model. The approach adopted in this study,

in this sense, is nonparametric in nature. A comparison of results

based on this approach with those from “model-based” approaches

is of interest; however, these questions are not considered in the

present study.

Numerical Illustrations and Discussions

Bilinear Inelastic Frame Structure

We consider a SDOF building frame with mass 9 × 103kg, initial

stiffness k0¼ 1:49 × 105N=m and viscous damping of 0.03 damp-

ing ratio. The initial natural frequency was computed as 4:07 rad=s

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and the strain hardening ratio is taken as 0.05. These parameters are

changed later to study their influence on the estimated optimal

earthquake loads and corresponding inelastic deformations. The

yield displacement is taken as 0.10 m and the structure is taken

to start from rest. The objective function is adopted as the Park

and Ang damage index DIPAgiven by Eq. (7). The parameters

of the Newmark β-method are taken as δ ¼ 1=2; α ¼ 1=6;

and Δt ¼ 0:005 s.

Earthquake Data at the Site and Quantification of

Constraints

A set of 20 earthquake ground motions is used to quantify the con-

straint bounds E, M1, M2, M3, M4ðωÞ, and M5ðωÞ [Consortium of

Organizations for Strong Motion Observation Systems (COSMOS)

2005]. Table 2 providesinformation onthese records (Abbas 2006).

Based on numerical analyses of these records the constraints were

computed as E ¼ 4:17 m=s1:5, M1¼ 4:63 m=s2(0.47 g), M2¼

0:60 m=s and M3¼ 0:15 m. The average dominant frequency of

the ground accelerations was about 1.65 Hz. The envelope param-

eters were taken as A0¼ 2:17, α1¼ 0:13, and α2¼ 0:50. The con-

vergence limits ε1, ε2were taken as 10?6and the convergence

criterion on the secant stiffness is taken as 10?3N=m.

The frequency content for € xgðtÞ is taken as (0.1–25) Hz. Addi-

tionally, in distributing the frequencies ωi, i ¼ 1;2;…;Nfin the

interval (0.1,25), it was found advantageous to select some of these

ωito coincidewith the natural frequencyof the elastic structure, and

also to place relatively more points within the modal half-power

bandwidth.

The constraint scenarios considered in deriving the optimal

earthquake inputs are listed in Table 3. The constrained nonlinear

optimization problem is tackled using the sequential quadratic

optimization algorithm “fmincon” of the Matlab optimization tool-

box (Caleman et al. 1999). In the numerical calculations, alternative

initial starting solutions, within the feasible region, were examined

and were found to lead to the same optimal solution. To select the

number of frequency terms Nfa parametric study was carried out

and Nf¼ 51 was found to give satisfactory results. Fig. 3 depicts

the influence of the frequency terms Nfon the convergence of the

objective function for constraints scenarios 1 and 4 (see Table 3).

Results and Discussions

The numerical results obtained are presented in Figs. 4–9 and

Table 4. Fig. 4 shows results for constraint scenario 1 and similar

results for case 4 are shown in Fig. 5. Each of these figures shows

the Fourier amplitude spectrum of the optimal ground acceleration,

the inelastic deformation, the hysteretic force and the energy dis-

sipated by the structure. Fig. 6 shows the time history of the ground

acceleration and velocity for cases 1 and 4. Based on extensive

analyses of the numerical results, the following observations

are made:

1. The frequency content and Fourier amplitude of the design

earthquake are strongly dependent on the constraints imposed

(see Table 3). If available information on earthquake data is

limited to the total energy and PGA, the design input is narrow

band (highly resonant) and the structure deformation is conser-

vative (see Fig. 4 and Table 4). Furthermore, most of the power

of the Fourier amplitude is concentrated at a frequency close to

the natural frequency of the elastic structure. This amplitude

gets shifted away from the natural frequency toward a higher

frequency when the strain hardening ratio increases. The

Fourier amplitudes at other frequencies are low and uniformly

distributed. The results of this constraint scenario match well

with earlier work reported by this writer for elastic-plastic

structures (Abbas 2006) and also by Takewaki (2001) on prob-

abilistic earthquake inputs for elastic-plastic structures. This

result, however, is substantially different from that for the elas-

tic structure where all power of the acceleration amplitude is

concentrated around ω0with no amplitude at other frequencies

(Abbas and Manohar 2002). Additional constraints on the

Fourier amplitude spectra (see Table 3) force the Fourier

amplitude of the optimal acceleration to get distributed across

other frequencies. The critical acceleration possesses a domi-

nant frequency that is close to the average dominant frequency

observed in past records (see Fig. 5). The realism of the earth-

quake input is also evident from the maximum damage index it

produces. For instance, the damage index for case 4 is 0.37

which is substantially smaller than 1.15 for case 1 (Table 4).

The constraints on PGV and PGD were not found to be sig-

nificant in producing realistic critical inputs compared to the

constraints on UBFAS and LBFAS. Also, the realism of the

Fig. 2. Flowchart for deriving optimal earthquake loads

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Page 6

optimal acceleration for case 4 can be examined by comparing

the Fourier amplitude spectra and frequency content of the

design acceleration (Figs. 4 and 5) with the Fourier amplitude

spectra of past recorded earthquakes (Fig. 7). It may be empha-

sized that while the constraint scenario 1 leads to pulselike

ground motion, such scenario was in deed observed during

some of the recent earthquakes (e.g., 1971 San Fernando,

1985 Mexico, and 1995 Hyogoken-Nanbu earthquakes).

Resonant or pulselike earthquakes are also observable in

near-field ground motion with directivity focusing, known

as forward- and backward-directivity ground motion, which

resemble fault-parallel and fault-normal components (Housner

and Hudson 1958; Kalkan and Kunnath 2006; He and Agrawal

2008; Moustafa 2008). The realism of optimal earthquake

loads can be also examined by comparing maximum response

from optimal accelerations with those from past recorded

ground motions. Thus, the maximum ductility factor of the

structure from the design earthquake is about 3.9 (case 1)

and 2.6 (case 4) times that from the Hyogoken-Nanbu earth-

quake and is 2.7 (case 1) and 1.5 (case 4) times that from the

San Fernando earthquake.

2. To examine the effect of the strain hardening ratio on the

design earthquake acceleration computed, limited studies were

carried out. The value of α was changed and the critical input

was determined by solving a new optimization problem.

Namely, α was taken as 0.20, 0.10, 0.05, and 0.01. The strain

hardening ratio was not seen to significantly influence the

Table 3. Nomenclature of Constraint Scenarios Considered

Case Constraints imposed

1

2

3

4

Energy and PGA

Energy, PGA, PGV, and PGD

Energy, PGA, and UBFAS

Energy, PGA, UBFAS, and LBFAS

(b)

Ductility factor

4

0

102030 40 50

Nf

60

70

80

90

100

0

0.5

1

1.5

2

2.5

3

3.5

Ductility factor

05 101520 25

Nf

303540 45 50

0

(a)

1

2

3

4

5

6

Fig. 3. Convergence of objective function in terms of frequency terms Nf: (a) Case 1; (b) Case 4

Table 2. Information on Past Ground-Motion Records for Firm Soil Site

Earthquake dateMagnitude Epic. dist. (km) ComponentPGA (m=s2) PGV (m=s) PGD (m) Energya(m=s1:5) Site

Mamoth Lakes

05.25.1980

Loma Prieta

10.18.1989

Morgan Hill

04.24.1984

San Fernando

02.09.1971

Parkfield

12.20.1994

Caolinga

05.02.1983

Northridge

01.17.1994

Cape Mendocino

04.25.1992

Westmorland

04.26.1981

Imperial Valley

10.15.1979

aE ¼ ½R∞

6.21.5W

S

W

S

4.02

3.92

3.91

4.63

3.06

1.53

3.09

2.66

2.88

3.80

2.83

2.20

3.81

3.43

3.25

2.89

4.35

3.54

2.68

1.98

0.21

0.23

0.31

0.36

0.40

0.30

0.17

0.28

0.44

0.10

0.26

0.26

0.60

0.34

0.45

0.24

0.33

0.44

0.22

0.19

0.05

0.05

0.07

0.11

0.07

0.02

0.04

0.10

0.01

0.01

0.10

0.10

0.12

0.09

0.15

0.08

0.11

0.15

0.10

0.15

3.73

4.01

3.82

2.61

2.33

1.64

2.07

2.47

1.33

1.74

2.67

2.14

4.17

3.50

2.44

2.31

3.26

3.25

2.30

2.14

Convict Greek

7.09.7 Capitola

6.1 4.5S60E

S30W

N69W

N21E

W

S

W

N

S74E

S16W

W

S

E

S

S45W

N45W

Halls Valley

6.6 27.6Castaic Old Ridge

5.09.1 Parkfield fault

6.530.1 Cantua Creek

6.7 5.9Canoga Park

7.0 5.4Petrolia general

5.0 6.6Westmorland fire

6.4 17.4Calexico fire

0€ v2

gðtÞdt?1=2(Arias 1970).

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Page 7

0510

Frequency (Hz)

15 20 25

0

0.1

0.2

0.3

0.4

0.5

Fourier amplitude (m/s)

Critical input

M4(ω)

M5(ω)

(a)

05 1015 2025 30

-3

-2

-1

0

1

2

3

Time (s)

Ductility factor

(b)

-4-3 -2 -10123

-4

-2

0

2

4x 10

4

Ductility factor

Restoring force (N)

(c)

05 10 1520 25 30

0

5000

10000

15000

Time (s)

Dissipated energy (N m)

EY

ED

EK + ES

(d)

Fig. 5. Optimal earthquake input and associated structural responses for case 4: (a) Fourier amplitude of the ground acceleration; (b) inelastic

deformation; (c) hysteretic restoring force; (d) dissipated energy

(a)(b)

(c) (d)

0510

Frequency (Hz)

152025

0

0.2

0.4

0.6

0.8

1

Fourier amplitude (m/s)

051015 2025 30

-6

-4

-2

0

2

4

Time (s)

Ductility factor

-10-505 10

-5

0

5x 10

4

Ductility factor

Restoring force (N)

05 1015 202530

0

1

2

3

4x 10

4

Time (s)

Dissipated energy (N m)

EY

ED

EK + ES

Fig. 4. Optimal earthquake input and associated structural responses for case 1: (a) Fourier amplitude of the ground acceleration; (b) inelastic

deformation; (c) hysteretic restoring force; (d) dissipated energy

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Page 8

frequency content of the critical earthquake input. It was

observed, however, that the inelastic structure with lower

values of α yields more frequently compared to the same struc-

turewith higher α values. Accordingly, the cumulativehystere-

tic energy dissipated was observed to decrease for higher

values of α [Fig. 8(a)]. This feature is particularly remarkable

at the end of the earthquake duration. It was also observed that

the results on critical earthquake accelerations for bilinear

inelastic structure with α ¼ 0:01 are close to those for the

elastic-plastic structure (Abbas 2006).

3. To investigate the influence of the damping ratio on the com-

puted design earthquake load, limited studies were carried out.

The damping ratio was changed (namely, 0.01, 0.03, and 0.05)

while all other parameters were kept unchanged. The critical

earthquake is computed by solving a new optimization pro-

blem for each case. The effect of the change in η0was seen

to be similar to that attributable to α. In other words, the value

of the damping ratio was not seen to significantly influence the

frequency content of the earthquake acceleration. It was

observed, however, that the ductility ratio and the maximum

inelastic deformation for the structure decrease for higher

dampingratios. Thus, the ductility ratio decreases to 2.43 when

the damping ratio is taken as 0.05 while the ductility ratio

increases to 2.89 when the damping ratio reduces to 0.01.

051015 202530

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.5

051015202530

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

05101520 2530

-0.5

Acceleration / g

Time (s) Time (s)

1

10 1202468 14 16 1820

-1

(a)(b)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Velocity (m / s)

Time (s) Time (s)

Fig. 6. Optimal earthquake acceleration and velocity: (a) Case 1; (b) Case 4

0510

Frequency (Hz)

15 2025

0

0.01

(a)(b)

0.02

0.03

0.04

0.05

0.06

Fourier amplitude (m/s)

05 10

Frequency (Hz)

152025

0

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 7. Fourier spectra of recorded earthquakes: (a) San Fernando 1971; (b) Hyogoken-Nanbu 1995

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It was also observed that the inelastic structure with higher

damping ratio dissipates more energy through damping com-

pared with the same structure with lower damping ratio [see

Fig. 8(b)]. The damage index also reduces when the damping

ratio increases.

4. To assess the structure safety, Eq. (7) was used to estimate the

damage index of the structure subjected to the critical earth-

quake load. We first examine the effect of the parameter

β on the damage index. Based on experimental tests, it was

reported that β ranges between 0.05 and 0.20 with an average

value of 0.15 as suggested by Park et al. (1987). Fig. 9(a)

shows the influence of β on the damage index. To study the

effect of the initial natural frequency of the structure on the

damage index, the structure stiffness was varied while keeping

all other parameters unchanged and the critical earthquake was

computed for each case. Subsequently, the value of DIPA

was calculated for each case. In the numerical calculations

β was taken as 0.15 and xmax, μmaxare taken as 0.10 m

and 2.64, respectively. The value of μuwas taken as 6 in

Fig. 9(a) and 8 in Fig. 9(b). It was found that the damage index

for the structurewith initial natural frequency smaller than 1.65

is higher than 0.40 and thus either total collapse or damage

beyond repair of the structure is expected. The value of

DIPAfor the structure with ω0greater than about 1.70 Hz is

less than 0.40 and thus the structure does not experience total

damage but repairable damage. This observation is consistent

since the site dominant frequency is around 1.65 Hz and since

the Fourier amplitude of the ground acceleration is seen to be

located in the stiff side of the initial frequency of the inelastic

structure.

The numeric illustrations of the formulation developed in this

paper were demonstrated for simple structures modeled as

0

(a) (b)

5 101520 25 30

0

5000

10000

15000

Time (s)

EH (N m)

α = 0.20

α = 0.10

α = 0.05

α = 0.01

0510 15202530

0

2000

4000

6000

8000

10000

Time (s)

ED (N m)

η0 = 0.05

η0 = 0.03

η0 = 0.01

Fig. 8. (a) Effect of strain hardening ratio on dissipated yield energy; (b) effect of damping on dissipated damping energy

00.050.1 0.15

β

0.2 0.250.3

0

(a)(b)

0.2

0.4

0.6

0.8

1

DIPA

01234

0

0.5

1

1.5

2

2.5

3

Frequency (Hz)

DIPA

η0 = 0.03

η0 = 0.05

Fig. 9. (a) Effect of the value of β on the damage index; (b) damage spectra for SDOF inelastic structures

Table 4. Response Parameters for Alternative Constraint Scenarios (α ¼ 0:05, ζ ¼ 0:03)

Casexmax(m)

μmax

xp(m)Nrv

a

DIPA

Damage status

1

2

3

4

aNrv= number of yield reversals (see Table 1).

0.47

0.45

0.41

0.26

4.65

4.53

4.14

2.64

0.07

0.06

0.07

0.05

60

54

49

44

1.15

0.97

0.72

0.37

Total collapse

Damaged beyond repair

Damaged beyond repair

Repairable damage

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SDOF

deformation laws. The application of the proposed method to

multi-degree-of-freedom (MDOF) structures and the use of

more detailed degradation models (e.g., trilinear degradation,

Takeda and Clough models) need to be investigated. Addition-

ally, in this paper Park and Ang damage index has been used to

assess the structure performance. This damage index has some

limitations, which have been discussed by Mehanny and

Deierlein (2000) and Bozorgenia and Bertero (2003). Among

these drawbacks are: (1) its weak cumulative component for

practical cases giventhe typical dominance of the peak displace-

ment term over the accumulated energy term, (2) its format

using a linear combination of deformation and energy in spite

of the obvious nonlinearity of the problem and the interdepend-

ence of the two quantities, and (3) its lack of considering the

loading sequence effect in the cumulative energy term. Further-

more, when EH¼ 0 (elastic behavior), the value of DIPAshould

be zero. However, the value of DIPAcomputed from Eq. (7) will

be greater than zero. Similarly, when the system reaches its

maximum monotonic deformation, while DIPAshould be 1.0,

however, Eq. (7) leads to DIPAgreater than 1.0. Chai et al.

(1995) proposed modification to DIPAto correct for the second

drawback only. The study, also, examined experimentally the

implication of the energy-based linear damage model of DIPA.

Despite the drawbacks of DIPA, it has been extensively used by

many researchers, mainly due to its simplicity and the extensive

calibration against experimentally observed seismic structural

damage during earthquakes (mainly for reinforced concrete

structures). Bozorgenia and Bertero (2003) proposed two

improved damage indexes that overcome some of the draw-

backs associated with DIPA.

In this paper, the optimal earthquakes that maximize the

structural damage were obtained using deterministic methods.

The design earthquake loads can be formulated based on hazard

analysis using probability of occurrence, which provides a power-

ful alternative to the methodology developed in this paper.

systemswithbilinear andelastic-plastic force-

Concluding Remarks

This paper developed a methodology for specifying earthquake

ground motions as design inputs for inelastic structures at sites

having limited earthquake data. New damage descriptors are

introduced in deriving optimal earthquake loads. Specifically,

the structural damage is quantified in terms of Park and Ang dam-

age indexes. Damage indexes are mathematical models for quanti-

tative description of the damage state of the structure and they

correlate well with actual damage displayed during earthquakes.

It is believed that the quantification of structural damage in terms

of damage indexes is of substantial importance in deriving critical

earthquake loads for inelastic structures. This is because damage

indexes imply that the structure is damaged by a combination of

repeated stress reversals and high-stress excursions. The quantifi-

cation of the structure damage in terms of damage indexes makes it

possible to assess the safety of the structure and provides an idea on

necessary repair.

The design earthquake is estimated on the basis of available

information using inverse dynamic analysis and nonlinear optimi-

zation methods. It was seen that if available information is limited

to the energy and PGA, the resulting earthquake is highly resonant

and produces conservative deformation. However, if extra informa-

tion on the Fourier amplitude spectra is available, more realistic

earthquakes are obtained, in terms of frequency content, amplitude,

inelastic deformations and damage indexes they produce. The

influences of the strain hardening and damping ratios on the esti-

mated design loads were also studied. Critical damage spectra for

the site were also established. These spectra provide an upper

bound on structural damage and necessary repair under possible

future earthquakes. The formulation developed in this paper was

demonstrated for frame structures modeled as SDOF systems.

The application of the proposed method to MDOF structures is

currently under investigation. In this case, the global damage index

of the structure is defined in terms of a weighted function of the

damage indexes for the individual structural members.

Notation

The following symbols are used in this paper:

ay= ratio of yield force to mass of the structure;

C1, C2= quantities;

c = damping coefficient;

DI = damage index;

DIPA= Ang and Park damage index;

E = Arias intensity (energy) of ground

acceleration;

ED= energy dissipated by damping;

EH= hysteretic energy;

EI= earthquake input relative energy;

Ek= kinetic energy;

ES= strain energy

eðtÞ = envelope function;

fsðtÞ = hysteretic restoring force;

fy, xy= yield force and yield displacement,

respectively;

k0, kp= preyield and postyield stiffness;

M1, M2, M3= peak ground acceleration, velocity and

displacement, respectively;

M4ðωÞ, M5ðωÞ = lower and upper Fourier amplitude spectra,

respectively;

m = mass;

Nf= number of frequencies;

Ri, φi= ith amplitude and phase angle of the ground

acceleration;

t = time;

€ x, _ x, x = acceleration, velocity and displacement

responses of SDOF system, respectively;

€ xg, _ xg, xg= ground acceleration, velocity and

displacement, respectively;

xmax= maximum displacement of inelastic structure;

xu, μu= ultimate displacement and ductility under

monotonic load;

A0, α1, α2, β = positive quantities;

α = strain hardening ratio;

ε1, ε2= small positive quantities;

η0, ω0= damping ratio and natural frequency of elastic

structure;

μ = ductility ratio; and

τ = dummy variable indicating time.

Acknowledgments

The author thanks three anonymous reviewers for their careful

reading of the paper and insightful comments they made. This

research work is partly supported by research funds from the

Japanese Society for the Promotion of Science. The support is

gratefully acknowledged.

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Page 11

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