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Engineering Structures 29 (2007) 21722182

www.elsevier.com/locate/engstruct

Fragility relationships for torsionally-imbalanced buildings usingthree-dimensional damage characterization

Seong-Hoon Jeonga,, Amr S. Elnashaib,1

aDepartment of Architectural Engineering, Inha University, Incheon, Republic of KoreabDepartment of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, United States

Received 17 June 2006; received in revised form 13 November 2006; accepted 13 November 2006

Available online 21 December 2006

Abstract

In this paper, a methodology for the derivation of fragility relationships for three-dimensional (3D) structures with plan irregularities is

developed. To illustrate the procedure, fragility curves are derived for an irregular reinforced concrete (RC) building under bi-directional

earthquake loadings. In order to represent the damage state of irregular structures, a spatial (3D) damage index is employed as the salient response

parameter. The feasibility of using a lognormal distribution for the bounded response variables, as in the case of structural fragility analysis, is

investigated. Through the comparison between the fragility curves derived using the spatial and the previously-existing damage indices, it is shown

that the proposed method provides realistic results and is therefore recommended for fragility analysis of buildings with significant torsional and

bi-directional responses.c 2006 Elsevier Ltd. All rights reserved.

Keywords: Fragility curves; Earthquake response; Irregularity; Torsion; Bi-directional; Damage index

1. Introduction

Fragility curves, used for the assessment of seismic

losses, are in increasing demand, both for pre-earthquake

disaster planning and post-earthquake recovery and retrofitting

programs. This is due to the difficulties associated with

analyzing individual structures and the importance of obtaining

a global view of anticipated damage or effects of intervention,

before and after an earthquake, respectively. Analytically-

derived, mechanics-based fragility relationships result in

reduced bias and increased reliability of assessments compared

to the fragilities based on post-earthquake observations [1] or

on expert opinion (e.g. HAZUS [2]). Since analytical methods

are based on statistical damage measures from analyses of

structural models under increasing earthquake loads, employing

Corresponding address: Department of Architectural Engineering, InhaUniversity, 253 Yonghyun-dong, Nam-gu, Incheon, 402-751, Republic ofKorea. Tel.: +82 32 860 7580; fax: +82 32 866 4624.

E-mail addresses: [email protected] (S.-H. Jeong), [email protected](A.S. Elnashai).

1 2129e Newmark CE Lab. 205 North Mathews Ave, Urbana, Illinois, 61801,USA. Tel.: +1 217 265 5497; fax: +1 217 265 8040.

an appropriate damage assessment method is central to deriving

fragility curves.

For the seismic assessment of structures with planar

irregularities, a damage measure should be able to reflect 3D

structural response features such as torsion and bi-directional

response. In this study, a 3D damage characterization is

utilized to represent the damage states of buildings with

plan irregularities. The latter method accounts for the

multi-directionality of earthquake motions as well as the

asymmetry of the structure. It therefore captures the true

three-dimensional inelastic effects that govern the response

of structures. The adoption of such a damage measure opensthe door to the derivation of spatial fragility relationships of

irregular structures which have 3D responses, bi-directional

deformation and torsion. In deriving fragility curves with

the proposed damage measure, the validity of the statistical

manipulation methods is carefully investigated. A systematic

methodology to exclude unrealistic analyses results from the

statistical treatment of response variables is proposed, and

the feasibility of using lognormal distributions for bounded

response variables, such as in the case of fragility derivation,

is investigated.

0141-0296/$ - see front matter c

2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2006.11.010
http://www.elsevier.com/locate/engstructmailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.engstruct.2006.11.010http://dx.doi.org/10.1016/j.engstruct.2006.11.010mailto:[email protected]:[email protected]://www.elsevier.com/locate/engstruct

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S.-H. Jeong, A.S. Elnashai / Engineering Structures 29 (2007) 21722182 2173

Fig. 1. Plan of an irregular building and frames used in planar decomposition.

2. Damage assessment method for spatially-responding

buildings

The seismic assessment of buildings with irregular plans

requires special attention, while regular structures can be

readily idealized and assessed using the conventional 2Ddamage measures. Plan irregularities cause non-uniform

damage levels among the members within a story and thus

story-level damage indicators are inadequate in such cases. For

instance, interstory drift cannot capture the localized variation

in demand because the drift of columns varies according to their

positions in their plane, due to torsion. In order to overcome

the limitations of conventional damage measures, a 3D damage

assessment method for torsionally imbalanced buildings is

proposed, as described in subsequent sections of this paper.

2.1. Planar decomposition and local damage measure

To account for the torsional effects, a 3D structure is

decomposed into planar frames that are considered to be the

basic elements of lateral resistance, as shown in Fig. 1. Planar

decomposition (Fig. 1) is not a method that physically separates

structural components, but rather an approach that conceptually

limits the response monitoring scope to a basic component

(planar frame) in an integrated 3D structure. Therefore, while

the geometry of a planar frame is defined in 2D, the response

of the frame is not constrained to two-dimensional space. A

planar frame may respond out-of-plane and be subjected to

forces from other members orthogonally connected to it. Thus,

the damage measure for planar frames (local damage measure)

should be sensitive to these out-of-plane responses.The comparison of the response of an RC column under

unidirectional and bi-directional static loading is depicted in

Fig. 2. Curves A and B are obtained from pushover analyses

on the RC column subjected to unidirectional and bi-directional

loadings, respectively. It is shown that the out-of-plane response

(Curve B: bi-directional loading) leads to a strength reduction

compared to the in-plane response (Curve A: unidirectional

loading). Since the backbone envelope curve is obtained by

a 2D pushover analysis, the differences from the latter curve

mean that there exist additional damage-inducing factors other

than in-plane monotonic deformation, which is the only source

of damage featured in Curve A or the backbone curve.

Fig. 2. Comparison of responses with and without out-of-plane loadings.

Thus the strength reduction below the latter curve can be a

measure of additional damage due to the out-of-plane response

(bi-directional loading). In cases of cyclic loading, strength

reduction may also be caused by the effect of load reversals.

Therefore, at a given deformation value, the strength reduction

from the backbone envelope curve reflects the combined effects

of out-of-plane actions and cyclic loading.

Based on the above discussion, the damage level (D) of a

planar frame is defined as a combination of the damage due

to in-plane monotonic displacement and the strength reduction

from the backbone envelope curve, as given in Eq. (1).

D =

p

u+

1 pu

F0 Fp

F0 Fffor p u

p

u

for p > u.

(1)

The parameters used in Eq. (1) are explained in Fig. 3,

where a typical forcedisplacement relationship of an RC frame

under bi-directional loading and its backbone envelope curve

are presented.p andu are the displacement at peak response

and the ultimate displacement, respectively. The peak response

point (p) is not necessarily the maximum displacement.

Instead, the damage level (D) needs to be monitored at several

candidate peak response points (p) that may lead to the

maximum damage level. Since the maximum value of D

represents the maximum damage level of a planar frame during

its response history, it is defined as the damage index (Di ) of

the planar frame i .

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2174 S.-H. Jeong, A.S. Elnashai / Engineering Structures 29 (2007) 21722182

Fig. 3. Parameters to define damage due to strength degradation.

At a given displacement (p), the term F0 Fp representsstrength degradation due to bi-directional and cyclic loading

effects while Ff is the corresponding failure strength. The

failure strength is assumed to be a linear function ofthe displacement which connects the origin and ultimate

displacement (u ). The latter definition has been used as an

upper-bound for the failure strength of RC members [3,4]. It

is conceptually similar to the stiffness at ultimate state used in

secant stiffness damage indices [5,6].

In the study of Park [7], the post-peak displacement

corresponding to a small reduction in load (about 15%) is one

of the most realistic definitions of the ultimate displacement

(u ) for a ductile system. For non-ductile frames, using

displacements corresponding to brittle failure for predicting the

ultimate displacement is recommended. For the cases examined

in this paper, the ultimate displacement is defined as the post-

peak displacement corresponding to 85% of the peak strength,which is a common practice for systems without significant

capacity reduction after their peak strengths [7].

Structural components acting in 3D may have various

forcedisplacement curves according to the response mon-

itoring directions. However, for a planar frame, the re-

sponse monitoring direction is predetermined, and a single

forcedisplacement relationship in the latter direction is mean-

ingful to its damage assessment. For instance, the Frame y1 in

Fig. 1 is considered as the basic element of the lateral load re-

sistance in the y direction. Therefore, the damage assessment of

the Frame y1 is performed based on the forcedisplacement

curve in the y direction (the plot of y-directional force vs. they-directional displacement) only. Its interdependence on the

forcedisplacement curve in the other direction is represented

by the strength drop-off as shown in the formulation (Eq. (1)),

i.e., at a given displacement, the strength is lower than the back-

bone envelope due to the interdependence.

2.2. System damage measure

The local damage measures for planar frames are combined

to formulate a system-wide global damage index. For a given

direction, all planar frames in the direction of consideration

participate in the unidirectional global damage index. After

obtaining a global damage index in one direction, the process

is repeated for the other direction, and the overall damage

state of a story is determined by the damage index of the

critical direction. The proposed combination method adopts

two important assumptions that have been generally used in

formulating weighted average damage indices [8,4]. These

assumptions involve (i) placing emphasis on severely damaged

local elements, and (ii) using the gravity loads supportedby each local component for its weighting factor. The latter

assumptions are justified on the grounds of the criticality of

members that are supporting high levels of axial force to the

gravity load stability of the frame.

Based on the above assumptions, the weighting (Wi ) of local

damage is defined as:

Wi = wi ACi (Di ) (2)where, wi is the gravity load on the contribution area (ACi ). The

contribution area is defined as a function of the local damage

level (Di ) and determined by inspecting the floor area affected

by the local damage, as illustrated in Fig. 4. In the latter figure,

D1 and D2 are the damage levels of Frame 1 and Frame 2,

respectively. The influence area (Ai ) of each frame changes

from Ai,min to Ai,max according to the damage level of the

corresponding planar frame. Definitions of the various areas

used in the proposed combination method are given in Table 1.

The first row of the influence area matrix in Fig. 4 represents

the influence areas of Frame 1. The influence area in each

element of the first row is determined as follows:

(a) Case A: When the damage level of Frame 1 is minor (Dm),

the local contribution of a planar frame is determined by its

own tributary gravity load. In this case, the influence area

of the local damage is termed the tributary area (Ai,min).

(b) Case B: The influence area (A1) is determined by

interpolating between the two cases (Case A and Case C).

The value of A1 is between the tributary area (Case A) and

the failure consequence area (Case C). It is expressed as:

A1 = A1,min + (A1,max A1,min) D1 (3)where, A1,min and A1,max are the tributary area and failure

consequence area, respectively. D1 is the damage index of

Frame 1. In the latter formulation, the minor damage level

in Case A is assumed as Dm = 0.(c) Case C: If Frame 1 fails (D1 1.0), then the consequence

is not limited to the tributary area but extends to the

neighboring planar frame. In this case, the whole areashared by the two frames is used for the influence area of

Frame 1. The area is referred to as the failure consequence

area (Ai,max).

The second row represents the influence area of Frame 2 and

the calculation method is the same as that of Frame 1, which

is presented above. Finally, the third row of the influence area

matrix (Fig. 4) represents local contribution areas (ACi ). They

are determined based on the influence areas (Ai ) of individual

frames, which are tentatively used for deriving the contribution

area (ACi ). In order that the sum of all contribution areas

is equal to the total area, any overlapping areas should be

subtracted. The overlapping influence areas of Frame 1 and

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Fig. 4. Influence area matrix.

Table 1

Description of areas used in the proposed combination method

Areas Description Symbols

Influence area The area affected by local damage AiTributary area The minimum influence area when the local damage is minor Ai,minFailure consequence area The maximum influence area when the local component has failed Ai,maxCommon influence area The area shared by the influence areas of two neighboring local components Acm,i jContribution area The area for determining the local damage contribution to the overall damage level ACi

When Di < D j

ACi = Ai Acm,i j , AC j = AjCommon failure consequence area The area shared by the failure consequence areas of two neighboring local components; maximum common area AC F,i j

Frame 2 is termed common influence area (Acm,12), expressedin Eq. (4).

Acm,12 = (A1,max A1,min) D1 + (A2,max A2,min) D2.(4)The common influence area is regarded as part of the

influence area of the more damaged local component, i.e., the

common influence area is governed by the damage level of the

critical component. If Frame 1 is less damaged than Frame 2(D1 < D2), their contribution areas (ACi ) can be calculated as

follows:

AC1 = A1 Acm,12 = A1,min (A2,max A2,min) D2

= A1,min AC F,12

2 D2 (5)

AC2 = A2 = A2,min + (A2,max A2,min) D2= A2,min +

AC F,12

2 D2 (6)

where AC F,12 is the common region between the failure

consequence areas of the two neighboring planar frames(A1,max, A2,max). It is referred to as the common failure

consequence area. In Case A (the first column of Fig. 4),

Frame 2 has failed and its tributary region will be the failure

consequence area (the whole area between two frames), and

there is no contribution area for Frame 1. Case C is the opposite

of the latter case.

Using the local contribution (Wi ) in Eq. (2), the global

damage index (Dg) of the example structure in Fig. 4 is written

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Fig. 5. Plan of the example structure for the damage assessment comparison.

as:

Dg =W1 D1 + W2 D2

Wtotal= w1 AC1 D1 + w2 AC2 D2

Wtotal(7)

where w1 and w2 are the gravity loads on contribution areas

(AC1, AC2). Assuming that the gravity loads are uniformly

distributed, Eq. (7) may be rewritten as

Dg =AC1 D1 + AC2 D2

Atotal. (8)

The latter relationship gives general formulations of the

global damage index (Dg) for the case of uniformly distributedgravity loads, as shown below:

Dg =

DiAi,min

Atotal

+i=j

AC F,i j

2 Atotal Max(Di , Dj ) |Di Dj |

(9)

where the subscripts i and j identify the individual local

components. Di and Ai,min are the local damage index and the

tributary area of the local component i , respectively. AC F,i jis the common region between the failure consequence areas

of two neighboring planar frames and is termed the common

failure consequence area.

2.3. Limit state

The assessment results obtained using the spatial damage

index are compared with those from previous studies. For

the comparison, the Park and Ang [8] and interstory drift

damage indices [9] are selected, because they are the two most

extensively used damage measures. The damage assessment

was performed on the single-story RC frame shown in Fig. 5.

Columns C2 and C4 have larger sections than the other

columns, and their strong axes are along the global x and

y directions, respectively. All other columns have the same

(a) Unidirectional loading, 2D.

(b) Unidirectional loading, 3D.

Fig. 6. Comparison of damage assessment results by the proposed damage

index, Park and Angs damage index, and the interstory drift-damage index.

section sizes. The masses are uniformly distributed over the

plan, and the center of rigidity changes according to the

distribution of inelasticity. The distance between the center of

mass and the center of rigidity varies between 0.6 m and 1.0 mwith an average of 0.8 m, under an earthquake with a PGA of

0.3g.

Damage assessment comparisons for two different cases:

(i) unidirectional loading and 2D responses (i.e. out-of-plane

responses restrained) and (ii) unidirectional loading and 3D

responses are represented in Fig. 6(a) and (b), respectively.

Since the spatial and Park and Ang damage indices are

very similar in cases of unidirectional earthquake loadings,

the damage scale proposed in Park et al. [8] is considered

to be viable for the spatial damage index. The suggested

classification of the limit states is as follows.

D < 0.25 no damage or Minor damage light

cracking0.25 D < 0.4 moderate damage severe cracking,

localized spalling0.4 D < 1.0 severe damage crushing of concrete,

reinforcement exposedD 1.0 collapsed.

As reported in previous studies [8,10], the variation in

the value of a damage index for a given damage condition

is significant. Therefore, it is not feasible to estimate the

damage condition using deterministic limit states. The limit

state uncertainty can be calculated, for example, from the

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Fig. 7. Probabilistic damage scale.

statistical data in Park et al. [11], for which the total coefficient

of variation (COV) is 0.6 and the COV reflecting only the

inherent randomness in the damage index is 0.5. Based on the

latter data, Singhal and Kiremidjian [10] calculated the COV

due to uncertainty (u ) as follows:

u =

2T 2r =

0.62 0.52 = 0.33 (10)

where, T and r are the total COV and the COV due to inherent

randomness, respectively.

Based on the above, in this study, a probabilistic index

versus limit state relationship is used to account for the

uncertainties in definition of the limit state (Fig. 7). In deriving

the latter relationship, it is assumed that for a given limit state,

the associated damage index follows a lognormal probability

distribution with the mean value of the above limit states and

a COV of 0.33. Additional details on the three-dimensional

damage index are given in Jeong and Elnashai [12].

3. Derivation of fragility curves

The proposed approach of constructing fragility curves

for buildings with plan irregularities is presented through an

example derivation in the following sub-sections. The main

differences between the proposed and the conventional fragility

assessment procedures involve: (i) applying bi-directional

earthquake loading to the structure, (ii) using the spatial

index to capture the damage state of a structure with an

asymmetric plan, and (iii) filtering out of the unrealistic analysis

results during the statistical manipulation. Additionally, thefeasibility of using the lognormal distribution, intrinsically

relevant to unbounded problems, for the damage index, a

bounded response variable, is investigated.

3.1. Example structure and input ground motions

A three-story, 2 2 bay RC frame with an asymmetricplan is used for the example derivation of fragility curves. The

structure was designed for a full-scale pseudo-dynamic test at

the Joint Research Center, Ispra, Italy under the auspices of

the European Union project Seismic Performance Assessment

and Rehabilitation (SPEAR) [13]. The layout of the structure

is represented in Fig. 8. The large column (C6) in Fig. 8

(a) 3D view of the test structure.

(b) Plan of the test structure.

Fig. 8. Overview and plan of the test structure.

(b) contributes significantly higher stiffness and strength in

the y direction than in the x direction. The thickness of the

slab is 150 mm and the total beam depth is 500 mm. The

sectional dimension of C6 is 750 250 mm, whereas all othercolumns are 250 250 mm. The structure is a strong-beamweak-column system, and the second story is a soft story. The

finite element analysis program Zeus NL [14] was utilized to

perform the analyses necessary for the assessment. A detailed

description on the structure and its analytical modeling are

given in Jeong and Elnashai [15]. The application of the planar

decomposition to the SPEAR test structure, its capacity curves

and the calculation of ultimate displacements of individual

planar frames are described in Jeong and Elnashai [16].

Since the aim of this study is to develop an approach

for the probabilistic fragility analysis of structures with plan

irregularities, issues of seismic hazard are not considered in the

example derivations using the spatial damage index. Therefore

a set of randomly selected records are used for the simulation.

The list of the earthquake records used is given in Table 2.

Each of the records consists of two orthogonal components

(Longitudinal and Transverse) of horizontal accelerations, and

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(a) Components applied in the x direction.

(b) Components applied in the y direction.

Fig. 9. Elastic response spectra of the applied records (5% damping).

Table 2

List of earthquake records with two orthogonal components

No. Earthquake Station

1 Montenegro 1979 Ulcinj2 Montenegro 1979 Herceg Novi

3 Friuli 1976 Tolmezzo

4 Imperial Valley 1940 El Centro Array #9

5 Kalamata 1986 Prefecture

6 Loma Prieta 1989 Capitola

7 Imperial Valley 1979 Bonds Corner

8 Athens 1999 Metro

9 Parkfield 1969 Temblor

10 Duzce 1999 Duzce

is modified from the natural record to be compatible to a smoothcode spectrum (Eurocode 8, Type 1, Soil C, 5% damped design

spectrum) [17]. The response spectra of the applied records are

shown in Fig. 9. The three main structural response periods arealso shown in the figure.

In order to obtain response variables as functions of

earthquake intensities, inelastic response history analyses were

performed with the records in Table 2 by scaling their PGAsfrom 0.05 to 0.4g with a step of 0.05g. It is emphasized here

that this is a reference implementation of 3D fragility analysis;

hence strong motion selection and scaling are not the focus of

the work presented.

3.2. Exclusion of unrealistic analytical results

In constructing analytical fragility curves, the damage state

of a structure is estimated from simulation results. If the

demand is far larger than the capacity of the structure, analytical

simulations often lead to erroneous response estimates due to

numerical instability. The latter situation yields unrealistically

large response values. Such results should be excluded

from the statistical data set of structural responses, because

they distort the statistical parameters (mean and variation).

Therefore, it is proposed that response parameters largerthan a certain threshold value should be filtered out in

calculating the statistical parameters, which cannot therefore

be fully automated. The filtering threshold is determined as

a spatial damage index of 2.0, since this value corresponds

to deterministic structural failure (with more than 99%

confidence, from Fig. 7). The values of the damage indices

obtained from the example simulation in Section 3.1 and the

filtering threshold used (DI = 2.0) are shown in Fig. 10(a).The means of the post-filtering values (DI < 2.0) are

represented as a function of earthquake intensities; this is a

3rd order polynomial equation between 0.05 to 0.4g PGA. The

probability of failure calculated by Eq. (11) is represented in

Fig. 10(b).For a given earthquake intensity, the procedure of calculating

a probability of damage index exceeding a given limit state (LS)

is described as follows:

(1) The probability of failure is separately calculated using theresponses larger than the filtering threshold value (DI >2.0).

PFailure

= number of analyses where the structure is assumed to have failedtotal number of analyses

.

(11)

(2) For values of the damage index between 0 and 2.0, the

conditional probability of exceeding a limit state (LS)

is calculated by Eq. (12), which is the product of the

probability of non-failure and the limit state probability

under this condition.

P((DLS D < 1)|s) = (1 PFailure) (1 F(DLS)|s )(12)

where, F(DLS)|s is the cumulative probability of obtainingthe damage index (D) between 0 and DLS when the

earthquake intensity is s. DLS is a threshold response

quantity (threshold damage index) for a limit state. The

statistical parameters (mean and variation) to obtain thecumulative probability F(DLS)|s are calculated for damageindices less than 2.0.

(3) Finally, the probability of the maximum response exceeding

a predetermined limit state is calculated by the total

probability theorem, combining the probability of failure

and the probability of structures suffering a certain level of

damage (calculated with the damage index values between

0 and 2.0), and is given by:

P((DLS D)|s) = PFailure+(1 PFailure) P((DLS D < 2)|s)

= PFailure + (1 PFailure) (1 F(DLS)|s ) (13)

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Fig. 10. Response distribution and mean damage index as functions of earthquake intensity and probability of failure: (a) Filtering threshold and mean of post

filtering responses; (b) Probability of failure.

Fig. 11. Viability of using the lognormal probability distribution: (a) Comparison of fragility curves using Beta and lognormal distributions; (b) Lognormal

probability paper.

where, s is a given earthquake intensity. The fragility curve

for a limit state is obtained by plotting the total probability

from Eq. (13) over a range of earthquake intensities.

3.3. Probability distribution

The response variable used in this study is the damage

index, which is bounded between 0 and 2.0. Therefore, strictly

speaking, the Beta distribution is suitable for its probabilistic

treatment. However, the use of the lognormal probability

distribution is preferred, because it is simple in formulation and

convenient to deal with in terms of implementing uncertainties.

The question does not arise when using drift as the assessment

measure, since it is, at least theoretically, unbounded.

Below, the feasibility of using lognormal distributions for the

probability distributions of damage indices after filtering (0

0.2, (ii) damage index (DI) > 0.4, and (iii) damage index (DI)

> 0.6.

In Fig. 11(b), two sets of sample data from the example

simulation are plotted on the lognormal probability paper with

the vertical axis as the logarithm of the damage index. The

horizontal axis is the standard normal variate, which is the

inverse of the standard normal cumulative probability. The

cumulative probability of the mth value among the N data

points (x1, x2, . . . ,xN, arranged in increasing order) of the

logarithm of the damage index is determined by m/(N+1) andits basis is discussed in Gumbel [18]. The linear relationship

between the vertical and horizontal ordinates guarantees thatthe vertical axis can be used as a random variable with a normal

distribution [19]. Therefore, the linearity of the sample data sets

in Fig. 11(b) shows that the logarithm of the damage index

is normally distributed. Based on the above discussion, it is

concluded that assuming lognormal probability distribution is

feasible for damage indices between 0 and 2.0.

3.4. Effect of randomness in material properties on fragility

curves

In order to estimate the effects of material randomness on

the fragility analysis results, dynamic response history analyses

were performed on 30 analytical models that nominally

represent the example frame (Fig. 5) with 30 sets of random

material properties using the bi-directional earthquake records

in Table 2. Each set of the material properties are randomly

sampled from normal distributions described as the mean of

concrete compressive strength ( fc) and steel yield strength ( fy )of 26.4 MPa and 474 MPa, respectively. The yield stress of

steel is assumed to be normally distributed with a coefficient

of variation (COV) of 5.2% based on the investigation by Pipa

and Carvalho [20]. The Youngs modulus of steel is assumed as

deterministic because only a small variability of this parameter

has been observed in the latter reference. For concrete, Mirza

et al. [21] suggested a COV for the normal distribution of

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(a) Mean of responses.

(b) Variance of responses.

Fig. 12. Comparison of full and reduced simulations.

compressive strength equal to 12%, 15% and 18% for precast,

ready-mix and in situ concrete, respectively. In the present

simulation, 18% is adopted for the COV of the concretes

strength distribution.

The mean and variation of the response from the latteranalyses are compared with those from the example structures

with deterministic mean material properties ( fc = 26.4 MPa;fy = 474 MPa), as shown in Fig. 12. For comparison, thespatial damage index is employed as the response variable. The

statistical parameters based on the mean material properties

show very good agreement with those of the frames with the

30 random material strength values. This indicates that the

fragility curves, at least under the current set of conditions,

are little affected by material randomness, and thus they can

be derived considering only the randomness in ground motions.

This observation was also reported in Kwon and Elnashai [22].

The observation is attributed to the fact that the mean capacity

of a structure is marginally affected by the random material

properties, which is a very slightly skewed normal distribution

around the mean properties. Moreover, the effect of the material

randomness on the response variation is overshadowed by the

randomness in the earthquake record set. Based on the above

discussion, material randomness is not accounted for hereafter,

and earthquake ground motion is considered as the only random

aspect.

3.5. Fragility curves

Fragility curves for the three-story example frame (Fig. 8)

subjected to bi-directional earthquake loadings (records in

Table 2) are derived. Based on the assumption that a critical

story governs the overall damage state of the building, the 3D

damage measure of a critical story (often the 2nd story, and less

frequently the 1st story) is employed as the response variable.

The amplitudes of acceleration are from 0.05 to 0.4g. The

probability of exceedance of a limit state at a given earthquake

intensity is calculated from Eq. (13), which can be rewritten asfollows.

Prob(DL S D) = PFailure + (1 PFailure)

1

ln(DLS)

2R+ 2LS

(14)

where, () represents the cumulative standard normal

distribution and = ln(/

1 + 2). is the mean of responsevariables and is the coefficient of variation, equal to the ratio

of the standard deviation to the mean value of the damage

index at an earthquake intensity, i.e. /. R and LS are

response variabilities due to inherent randomness and limitstate uncertainty, respectively. The latter parameters can be

calculated as follows:

R =

ln(1 + 2), LS =

ln(1 + 0.332). (15)As discussed in Section 3.4, the effect of material

randomness is negligible and thus, only the response variation

caused by randomness in the earthquake record set is used for

R .

Since the number of ground motions at each earthquake

intensity level is relatively small, the uncertainty due to the

use of a limited number of samples is accounted for. This

uncertainty is represented by the confidence interval of themean values. The confidence interval is determined by the

t-distribution [19] assuming that the 10 earthquake records are a

randomly selected set of samples from all possible earthquakes

at a given intensity. Fig. 13(a) shows the lower and upper bound

of mean values with a 90% confidence level and their regression

functions. Fragility curves based on the latter mean estimation

(Fig. 13(b)) shows that the effect of uncertainty due to the

limited sample size is significant for the collapse limit state.

In Fig. 14, fragility curves derived by the proposed damage

assessment method are compared with those obtained using

the Park and Ang damage index. For a clearer presentation,

the confidence intervals in Fig. 13(b) are omitted. The latter

damage index is utilized with planar decomposition, i.e., thePark and Ang damage index is applied to individual planar

frames and the local damage measures are combined by

the following method [8]: the overall damage of a building

is calculated by averaging the local damage indices (Di ),

weighted by the ratio of local energy absorptions to the global

energy absorption (Ei /Ei ), as shown in Eq. (16).

Dg =

DiEiEi

. (16)

While the Park and Ang damage index with planar

decomposition is sensitive to torsion, it is not sensitive to out-

of-plane response [12]. Therefore, fragility curves using the

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Fig. 13. Mean values and fragility curves with a 90% confidence interval: (a) Mean values and regression functions; (b) Fragility curves for various limit states.

Fig. 14. Comparison of fragility curves derived by thespatial and Park & Angs

damage indices.

spatial damage index represent higher risk than those using

the Park and Ang damage index, as shown in Fig. 14. The

difference between the two sets of fragility curves becomeslarger as the limit state represents more significant damage

levels.

4. Conclusions

In the derivation of analytical fragility curves, the response

of a structure is represented by a single local, or more

often global, quantity to enable the statistical treatment of

simulation results. Examples of response quantities are steel

strain, curvature ductility, interstory drift, roof drift or some

damage index. However, the absence of a single quantity to

measure the three-dimensional response taking into account

torsion and bi-directional responses has hitherto hampered thereliable fragility assessment of irregular structures under bi-

directional loadings. The spatial damage measure presented

in this study broadens the application of analytical fragility

curves to the important domain of 3D structures with significant

torsional and bi-directional effects. Below, the methodological

considerations needed to incorporate this spatial damageindex into the framework for deriving fragility curves are

summarized:

Comparing fragility curves obtained using the lognormaland Beta distributions confirmed the feasibility of using

the lognormal distribution to represent bounded response

measures.

The limit state probability is calculated by combiningthe conventional method of calculating the probability of

exceeding a certain damage level and the probability of

failure. The latter probability is separately calculated by

dividing the number of damage indices larger than the

threshold value by the total number of response points at agiven earthquake intensity.

In addition to the variability of the response due toinherent randomness, uncertainties due to errors in defining

limit states and using a limited number of samples are

incorporated into the study.

Through the comparisons between the fragility curves

derived by the new spatial damage index and damage indices

used in the published literature, it has been shown that the

proposed method provides realistic results. It is further noted

that using the conventional damage index is less conservative

for structures responding in 3D. Therefore, the proposed

method of deriving fragility curves is recommended for theprobabilistic seismic assessment of buildings with significant

torsional and bi-directional responses.

Acknowledgements

The work presented above was undertaken as part of

the Mid-America Earthquake (MAE) Center research project

CM-4: Structural Retrofit Strategies, which is under the

Consequence Minimization Thrust Area. The MAE Center

is a National Science Foundation Engineering Research

Center (ERC), funded through contract reference NSF Award

No. EEC-9701785. This paper was supported by the Inha

University.

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