PDF interpolation technique for seismic fragility analysis of bridges

Engineering Structures 29 (2007) 1312–1322 www.elsevier.com/locate/engstruct

PDF interpolation technique for seismic fragility analysis of bridges Jin-Hak Yi a,∗ , Sang-Hoon Kim b , Shigeru Kushiyama c a Coastal Engineering Research Department, Korea Ocean Research and Development Institute (KORDI), Ansan, Gyeonggi 426-744, Republic of Korea b Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, USA c Department of Architectural Engineering, Hokkai Gakuen University, Sapporo, Japan

Received 19 January 2004; received in revised form 7 August 2006; accepted 7 August 2006 Available online 9 October 2006

Abstract This study presents a probability density function (PDF) interpolation technique for the evaluation of seismic fragility curves as a function of the return period. Seismic fragility curves have been developed in terms of various seismic intensities, such as the peak ground acceleration (PGA), peak ground velocity and pseudovelocity spectrum. However, of these seismic intensity measures, the return period of design earthquakes is more useful, as seismic hazard curves are generally represented with the return period of design earthquakes; seismic design codes also require consideration of the return period of the design earthquake spectrum for a specific site. Therefore, this paper focuses on the evaluation of seismic fragility curves as a function of the return period. Seismic fragility curves based on the return period are compared with those based on the PGA as an example bridge. The seismic fragility curves developed in this study make more intuitive sense for the design, retrofit and performance evaluation of bridges. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Probability density function; Seismic fragility analysis; Return period; Seismic damage

1. Introduction Performance-based seismic design methodologies have been improved for the design of new bridges and the retrofit of existing ones. There are two generally accepted recent performance-based design concepts: (1) the multilevel performance design concept introduced from SEAOC’s Vision 2000 projects [1] and BSSC’s NEHRP Guidelines for Seismic Rehabilitation of Buildings [2], and (2) the probabilistic concept for treating seismic events as random due to the uncertainties of earthquake events. Based on the abovementioned recent trends, the seismic fragility analysis, which evaluates damage exceeding the probability of a given structure’s response to varying intensities of seismic excitation, is most popular for the evaluation of the seismic performance of civil infrastructures. Seismic fragility analysis was originally conducted to evaluate the seismic safety of nuclear power plants, which has recently been accepted as a reliable method for the evaluation ∗ Corresponding author. Tel.: +82 31 400 7811; fax: +82 31 408 5823.

E-mail address: [email protected] (J.-H. Yi). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.08.019

of the seismic performance of civil infrastructures, such as bridges and buildings. In the early seismic fragility analysis, the seismic fragility curves were usually represented as functions of the peak ground acceleration (PGA) [3]. Hirata et al. [4, 5] tried to express the seismic fragility as a function of the peak ground velocity (PGV) as well as of the spectral pseudovelocity for base-isolated reactor structures. Dimova and Hirata [6] investigated several seismic intensity parameters, including PGA, PGV and the pseudovelocity spectrum (PVS) for a seismic fragility analysis of building structures. They demonstrated that the mean value of the PVS provided the best estimate of response for a given seismic intensity in a least-square sense. More recently, seismic fragility analyses were carried out for evaluation of the performance of network systems, including a transportation system [7–10]. Even though the PGA value has been most widely used as a measure of the magnitude of earthquakes in the seismic fragility analyses of bridges, the return period of a design earthquake, of the seismic intensity measures, including PGA and PGV, can also be very useful because seismic hazard curves or maps are generally represented using the return periods of design earthquakes; seismic design codes also require consideration of

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the return period of an earthquake for a specific site. In this study, seismic fragility curves are developed as a function of the return period by utilizing the probability density function (PDF) interpolation technique. A similar approach can be found in Wen and Wu’s study [11]. They interpolated the median values of the structural responses to the seismic events of many samples for comparing their proposed method with the results of the conventional method. This study can be considered as an extended version of one part of their research. In addition, a conventional seismic fragility analysis, based on PGA index and maximum likelihood estimation, is also briefly introduced. Finally, the results of the seismic fragility curves based on the return period are compared with those based on PGA using the same example bridge.

where, xi = 1 or 0, depending on whether the bridge sustains the jth damage state under ground motion, with PGA = ai , and N is the number of sample input ground motions. Two parameters c j and ζ j (or ζcommon for F jII ) in Eqs. (1) and (2), respectively, are computed by maximizing the likelihood function, as follows: ∂ ln L Ij ∂c j ∂ ln L IIj ∂c j

=

=

∂ ln L Ij ∂ζ j

= 0,

∂ ln L IIj ∂ζcommon

= 0,

j = 1, 2, . . . , Nstate j = 1, 2, . . . , Nstate .

(4a) (4b)

3. PDF interpolation technique for seismic fragility analysis

2. Seismic fragility analysis 2.1. Revisit to seismic fragility analysis in terms of peak ground acceleration Shinozuka et al. expressed the seismic fragility curves using two-parameter lognormal distribution functions, and estimated the two parameters (median and log-standard deviation) with the aid of the maximum likelihood estimation method [12]. A brief description is introduced, as follows: The fragility curve for the jth damage state, F jI (a), takes the following form   ln(a/c j ) (1) F jI (a; c j , ζ j ) = Φ ζj where a is a PGA value of the ground motion, Φ(·) the standard normal distribution function, and c j and ζ j the median and log-standard deviation values of the fragility curve for jth damage state, respectively. Even though the fragility curves can be expressed as Eq. (1), the fragility curves for different damage states can be observed to intersect due to the different log-standard deviations. In such a case, the damage exceeding probability for more severely damage states are estimated as being higher than for less severe damage. To avoid such an intersection, Kim and Shinozuka [13] proposed a common logstandard deviation, using all sample data for every damage states, for the maximum likelihood estimation. In this study, the following seismic fragility curves, with a common log-standard deviation (noted as F jII (a)), are also investigated.   ln(a/c j ) II F j (a; c j , ζcommon ) = Φ , ζcommon j = 1, 2, . . . , Nstate

(2)

where ζcommon denotes the common log-standard deviation. The likelihood function for the jth damage state, L Ij and L IIj , can then be introduced as: L Ij =

N h Y

ixi h i1−xi F jI (ai ) · 1 − F jI (ai )

for F jI (a)

(3a)

i=1

L IIj =

N h Y i=1

ixi h i1−xi F jII (ai ) · 1 − F jII (ai )

for F jII (a)

(3b)

3.1. Seismic hazard curve Seismic fragility curves can be evaluated by calculating the probability of exceeding predetermined performance limit states for several return periods, if the PDF of the maximum response of a given structure is known under the ground motions with an arbitrary return period. However, it is impractical to obtain the PDF of maximum structural responses under an earthquake with an arbitrary return period. One solution can be achieved by interpolating the characteristics of the PDF of maximum structural responses at existing probability levels (e.g. 50, 10 and 2% in 50 years), and a reliable basis function is essential to interpolate the PDFs. For this purpose, the mathematical form of a seismic hazard curve is utilized as a basic function for interpolating the characteristics of PDFs of maximum structural responses, because earthquake related parameters, such as seismic damages, are reasonably represented by a seismic hazard curve. In addition, the seismic hazard curve can be obtained by estimating two parameters from a few sampling points at the available return periods. A similar idea for obtaining the structural responses with respect to the return periods can be found in the study of Wen and Wu [11]. They interpolated the probability of exceedance of the column drift ratio in a 50 year period, and obtained the column drift demand at various return periods. In this study, the following two assumptions were considered to establish the seismic fragility curves based on return period: (1) structural responses under the earthquake ground motions at the same exceedance probability level follow a lognormal distribution (see Eq. (6)); and (2) the characteristics of the PDF of seismic responses follow the seismic hazard curve proposed by Cornell [14] as: P E 1 = P[θmax > θ] = H (θ ) = K 0 θ −K 1

(5)

where P E 1 is an annual exceedance probability, θmax the maximum structural response, and K 0 and K 1 are the two parameters representing the seismic hazard curve. Graphically, K 0 and K 1 are given as the intercept and slope of a regression line of the logarithmic plot of a seismic hazard curve.

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3.2. Seismic fragility curve When the maximum structural responses are distributed following a lognormal distribution function, the PDF of the maximum structural responses, θmax , is expressed as: "   # 1 1 ln θ − λ 2 pθmax (θ) = √ (6) exp − 2 η 2π ηθ

Fig. 1. Elevation of sample bridge.

where λ and η are log-mean and log-standard deviation values, respectively, for the PDF of θmax . In order to determine the PDF at an arbitrary exceeding probability level, two seismic hazard curves for median (θmedian ) and deviated values (defined as θdeviated ≡ eλ−η ) have to be drawn. From Eq. (5), the above two seismic hazard curves are expressed as follows: θ

P E 1median = K o,median θ −K 1,median

(7)

θ P E 1deviated

(8)

= K 0,deviated θ −K 1,deviated

where K 0,median and K 1,median are two constants of a seismic hazard curve for the median value (θmedian ), and K 0,deviated and K 1,deviated are those for the deviated value (θdeviated ). The median and deviated values can be obtained as follows: θmedian = eλ

(9)

θdeviated = eλ−η .

(10)

After the two constants (K 0 and K 1 ) have been obtained using linear regression of the logarithmic plot of Eqs. (7) and (8), the log-mean value, λi and log-standard deviation, ηi at θ an arbitrary return period, Ti (Ti = 1/P E 1,i , P E 1,imedian = θ

P E 1,ideviated = P E 1,i ), can be calculated as follows:   P E 1,i 1 λi = − log K 1,median K 0,median   P E 1,i 1 log ηi = λi + . K 1,deviated K 0,deviated

(11) (12)

The PDF for the maximum structural responses under an earthquake with an annual exceedance probability, P E 1,i (or return period Ti ), can be expressed by a lognormal distribution function, as follows: "   # 1 ln θ − λi 2 1 exp − . (13) pθmax ,i (θ ) = √ 2 ηi 2π ηi θ Finally, the seismic fragility curves at an arbitrary return period, Ti , are evaluated by calculating the probability of exceeding the predefined performance limit state (θ j ’s), as follows: Z θj F j (Ti ) = P[θmax > θ j | T = Ti ] = 1 − pθmax ,i (θ )dθ.

Fig. 2. Nonlinearity in bridge model.

superstructure consists of a longitudinally reinforced concrete deck slab, and is supported by two pairs of columns. Each pair has three columns with a circular cross-section diameter of 0.8 m. The bridge is modelled to experience nonlinear behaviour at the columns as a form of plastic hinge. A column is modelled as an elastic zone, with a pair of plastic zones at each column end, considering double bending behaviour. Each plastic zone is then modeled to consist of a nonlinear rotational spring and a rigid element, as depicted in Fig. 2. The plastic hinge formed in the bridge column is assumed to have bilinear hysteretic characteristics. The column ductility program, developed by the authors based on the work by Priestley et al. [15], is used to obtain the moment–curvature relationship of the plastic hinges for the columns. From the moment–curvature curve, the moment–rotation relationship is calculated using the plastic hinge length (which is calculated from the sectional analysis [15]). The critical parameter used to describe the nonlinear structural response in this study is the rotational ductility demand at the plastic hinges. The ductility demand is defined as θ/θ y , where θ is the rotation of a bridge column in its plastic hinge and θ y the corresponding rotation at the yield point. 4.2. Response analysis of bridge

−∞

(14) 4. Example study 4.1. Description of a sample bridge Fig. 1 shows a sample bridge for an example analysis. This bridge has an overall length of 34.0 m, with three spans. The

The SAP2000/Nonlinear finite element computer code [16] is utilized for extensive two-dimensional response analyses of the bridge for sixty (60) Los Angeles earthquake time histories. The earthquake data were developed for the FEMA SAC (SEAOC-ATC-CUREe) project, as listed in Table 1 [17]. These acceleration time histories were derived from historical records and physical simulations, with some linear adjustments, and

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Table 1 FEMA-SAC LA earthquakes PGA (cm/s2 )

SAC name

Record

Earthquake magnitude

Scale factor

LA01 LA02 LA03 LA04 LA05 LA06 LA07 LA08 LA09 LA10 LA11 LA12 LA13 LA14 LA15 LA16 LA17 LA18 LA19 LA20

Imperial Valley, 1940, El Centro Imperial Valley, 1940, El Centro Imperial Valley, 1979, Array #05 Imperial Valley, 1979, Array #05 Imperial Valley, 1979, Array #06 Imperial Valley, 1979, Array #06 Landers, 1992, Barstow Landers, 1992, Barstow Landers, 1992, Yermo Landers, 1992, Yermo Loma Prieta, 1989, Gilroy Loma Prieta, 1989, Gilroy Northridge, 1994, Newhall Northridge, 1994, Newhall Northridge, 1994, Rinaldi RS Northridge, 1994, Rinaldi RS Northridge, 1994, Sylmar Northridge, 1994, Sylmar North Palm Springs, 1986 North Palm Springs, 1986

6.9 6.9 6.5 6.5 6.5 6.5 7.3 7.3 7.3 7.3 7 7 6.7 6.7 6.7 6.7 6.7 6.7 6 6

2.01 2.01 1.01 1.01 0.84 0.84 3.2 3.2 2.17 2.17 1.79 1.79 1.03 1.03 0.79 0.79 0.99 0.99 2.97 2.97

452.03 662.88 386.04 478.65 295.69 230.08 412.98 417.49 509.70 353.35 652.49 950.93 664.93 644.49 523.30 568.58 558.43 801.44 999.43 967.61

LA21 LA22 LA23 LA24 LA25 LA26 LA27 LA28 LA29 LA30 LA31 LA32 LA33 LA34 LA35 LA36 LA37 LA38 LA39 LA40

1995 Kobe 1995 Kobe 1989 Loma Prieta 1989 Loma Prieta 1994 Northridge 1994 Northridge 1994 Northridge 1994 Northridge 1974 Tabas 1974 Tabas Elysian Park (simulated) Elysian Park (simulated) Elysian Park (simulated) Elysian Park (simulated) Elysian Park (simulated) Elysian Park (simulated) Palos Verdes (simulated) Palos Verdes (simulated) Palos Verdes (simulated) Palos Verdes (simulated)

6.9 6.9 7 7 6.7 6.7 6.7 6.7 7.4 7.4 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1 7.1

1.15 1.15 0.82 0.82 1.29 1.29 1.61 1.61 1.08 1.08 1.43 1.43 0.97 0.97 1.1 1.1 0.9 0.9 0.88 0.88

1258.00 902.75 409.95 463.76 851.62 925.29 908.70 1304.10 793.45 972.58 1271.20 1163.50 767.26 667.59 973.16 1079.30 697.84 761.31 490.58 613.28

LA41 LA42 LA43 LA44 LA45 LA46 LA47 LA48 LA49 LA50 LA51 LA52 LA53 LA54 LA55 LA56 LA57 LA58 LA59 LA60

Coyote Lake, 1979 Coyote Lake, 1979 Imperial Valley, 1979 Imperial Valley, 1979 Kern, 1952 Kern, 1952 Landers, 1992 Landers, 1992 Morgan Hill, 1984 Morgan Hill, 1984 Parkfield, 1966, Cholame 5W Parkfield, 1966, Cholame 5W Parkfield, 1966, Cholame 8W Parkfield, 1966, Cholame 8W North Palm Springs, 1986 North Palm Springs, 1986 San Fernando, 1971 San Fernando, 1971 Whittier, 1987 Whittier, 1987

5.7 5.7 6.5 6.5 7.7 7.7 7.3 7.3 6.2 6.2 6.1 6.1 6.1 6.1 6 6 6.5 6.5 6 6

2.28 2.28 0.4 0.4 2.92 2.92 2.63 2.63 2.35 2.35 1.81 1.81 2.92 2.92 2.75 2.75 1.3 1.3 3.62 3.62

578.34 326.81 140.67 109.45 141.49 156.02 331.22 301.74 312.41 535.88 765.65 619.36 680.01 775.05 507.58 371.66 248.14 226.54 753.70 469.07

Probability of exceedance

10% in 50 years

2% in 50 years

50% in 50 years

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(a) Example of 50 in 50 Set (LA56).

Fig. 4. Moment–rotation curve for plastic hinge of column 1. Table 2 Description of damaged states

(b) Example of 10 in 50 Set (LA17).

Damage state

Description

Drift limits

Almost no damage Slight damage Moderate damage Extensive damage Complete damage

First yield Cracking and spalling Loss of anchorage Incipient column collapse Column collapse

0.005 0.007 0.015 0.025 0.050

Table 3 Peak rotational ductility demand of columns of a sample bridge

(c) Example of 2 in 50 Set (LA22). Fig. 3. Acceleration time histories generated for Los Angeles area.

consist of three groups (each consisting of 20 time histories) with probabilities of exceedance of 50%, 10% and 2% in 50 years at the Los Angeles site. These sets of ground motions are referred to as 50 in 50 Set, 10 in 50 Set, and 2 in 50 Set. It should be noted that these three sets should only be used as a set, and not individually or as small subsets, in order to correctly represent the probability levels specified, which means that the median spectral acceleration of the set might reasonably match the target value at any particular period, and the spectral acceleration for an individual record might be quite different from the expected target spectral acceleration. Typical ground acceleration time histories for each group are plotted on the same scale in Fig. 3 for comparison of the intensities of the acceleration. Typical ground acceleration data are selected taking into consideration the average PGA values for each data set. The average PGA values were measured as 0.426, 0.588 and 0.881g for the 50 in 50 Set, 10 in 50 Set and 2 in 50 Set. The nonlinear response characteristics associated with the bridge, based on a moment–curvature curve analysis, are incorporated taking into consideration the axial loads and confinement effects. The moment–rotation relationship is obtained using the plastic hinge length (L p = 0.58 m), and is modelled using a bilinear rotational spring element without any stiffness degradation; its parameters are established according to the equations in Priestley et al. [15]. The moment–rotation curves were obtained, as shown in Fig. 4, and typical nonlinear

Damage state

Column 1

Column 2

Almost no Slight Moderate

1.0000 1.3263 2.6315

1.0000 1.3263 2.6316

responses at the column bottom end of column 1 are plotted in Fig. 5. 4.3. Damage states While the seismic damage states can be classified in several ways, Dutta’s recommendation is now being widely used as an indication of reasonable damage states [18–20] as the guideline was conducted using real-scaled bridge column tests. A set of five different damage states was introduced, as shown in Table 2, which provides a description of the five damage states and the corresponding drift limits for a tested column. In this study, the drift limit was transformed to the rotational ductility demand of the plastic hinge at the columns, as the structural analysis programme used in this study utilized the nonlinear rotational spring element; therefore, it is easier to read the rotational displacements than the drift [15]. A similar approach can be found in the other paper by Choi et al. [20], where the curvature ductility demand was used instead of drift limits. Table 3 shows the rotational ductility demand limits for the sample bridge used in this study. It is noticeable that two peak rotational ductility demands for two columns are virtually identical because the values were directly obtained from the sectional analyses of the same sections of the two columns. Small discrepancies occurred due to the difference in the axial forces on the sections.

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(a) Example of 50 in 50 Set (under LA56).

(b) Example of 10 in 50 Set (under LA17).

(c) Example of 2 in 50 Set (under LA22). Fig. 5. Responses at column 1.

The corresponding return periods were easily obtained, as 72, 475 and 2475 years, by taking the reciprocals of the annual exceedance probabilities, as follows: T =

Fig. 6. Responses at column end of column 1.

4.4. Seismic fragility curve by PDF interpolation technique Fig. 6 shows the results of nonlinear time history analyses in terms of the ductility factors for the sample bridge employing the three data sets. The annual exceedance probability, P E 1 , corresponding to 50 for 50 Set, 10 for 50 Set and 2 for 50 Set were calculated to be 0.0138, 0.0021, and 0.0004, respectively, using the following equation: P E 1 = 1 − (1 − P E 50 )1/50 .

(15)

1 . P E1

(16)

Table 4 lists the two parameters for the lognormal distributions (λ and η), and the median and deviated values (eλ and eλ−η ) for the three prespecified return periods. Table 5 lists the constants for two seismic hazard curves obtained by interpolating the characteristics (i.e. the median value θmedian and the deviated value θdeviated in Eqs. (7) and (8)) for the PDF of maximum structural responses for the three different return periods, those being, 72, 475 and 2475 years. Fig. 7 shows the seismic hazard curves obtained for maximum structural responses (rotational ductility factors) for columns 1 and 2, where the circle and square marks depict the median and deviated values obtained at the sampling points, respectively. From Tables 4 and 5, the probabilistic characteristics are observed to be quite different for columns 1 and 2, even though the ductility demands and sectional properties are almost the same, as shown in Table 3. This is due to the difference in the column lengths. As shown in Fig. 1, the length of

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Table 4 Log-mean and log-standard deviation for three data sets Return period

72 475 2475

Column 1

Column 2

λ

η

eλ

−0.2914 0.4123 1.2514

0.6616 0.5593 0.7019

0.7472 1.5103 3.4952

eλ−η

λ

η

eλ

eλ−η

0.3856 0.8633 1.7324

−0.1893 0.5312 1.3596

0.6675 0.5604 0.6924

0.8275 1.7010 3.8946

0.4245 0.9712 1.9488

Table 5 Constants for seismic hazard curves for columns 1 and 2 Column 1 K 0,median

K 1,median

K 0,deviated

K 1,deviated

Column 2 K 0,median

K 1,median

K 0,deviated

K 1,deviated

0.0064

2.2771

0.0015

2.3481

0.0082

2.2712

0.0019

2.3140

(a) For column 1.

(b) For column 2. Fig. 7. Seismic hazard curves for the example bridge.

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J.-H. Yi et al. / Engineering Structures 29 (2007) 1312–1322 Table 6 Probability characteristics obtained from seismic hazard curves Return period

100 1000 10000

Column 1

Column 2

eλ

eλ−η

λ

η

eλ

eλ−η

λ

η

0.8240 2.2650 6.2264

0.4427 1.1803 3.1467

−0.1936 0.8176 1.8288

0.6212 0.6518 0.6824

0.9180 2.5300 6.9730

0.4899 1.3251 3.5842

−0.0856 0.9282 1.9421

0.6280 0.6468 0.6655

Fig. 9. Seismic fragility curves utilized by PDF interpolation technique. Fig. 8. Probability density function for earthquakes with 1000 year return period.

column 1 is shorter than that of column 2, which makes the applied moments at the two sections different. This difference is transferred to the probabilistic characteristic values. The PDF of the maximum structural responses (ductility factors) under earthquakes with an arbitrary return period can be obtained using the seismic hazard curves, as shown in Fig. 7, which makes it possible to evaluate the seismic fragility curves from the calculation, using Eq. (14), of the exceedance probability of pre-defined damage states. Table 6 lists the probability characteristics obtained from the seismic hazard curve with return periods of 100, 1000 and 10 000 years. The median values of the structural responses under the earthquakes with return periods of 100, 1000, and 10 000 years were calculated to be 0.8240, 2.2650 and 6.2264, respectively. This implies, for example, that half of the maximum responses under the earthquakes with a 1,000 year return period might have a ductility factor over 2.2650. Fig. 8 shows the PDF of the maximum structural response under earthquakes with a 1,000 year return period and predefined damage state. Fig. 9 shows the seismic fragility curves represented by the return periods. From the result shown in Fig. 9, the probability of exceeding the damage states for an earthquake with an arbitrary return period can be obtained easily. The probability of exceeding a complete damage state, for example, is almost 40% under earthquakes with a 10 000 year return period; whether the structure should be upgraded to resist the earthquakes for a specific probability of a damage state can also be easily determined. Furthermore, if the fragility curves are periodically updated, based on real measurements, the structures can be more reliably maintained [21].

5. Discussion Seismic fragility curves for the same sample bridge are evaluated using the maximum likelihood estimation in terms of a PGA index. Fig. 10(a) and (b) show the analytical seismic fragility curves obtained using Eqs. (1) and (2), respectively. From Fig. 10(a), the fragility curves for almost no and slight damages are observed to intersect around 0.22g, and the damage exceeding probabilities for slight damage were evaluated to be slightly larger than those for almost no damage; in the range of 0–0.22g, due to the different log-standard deviation. In Fig. 10(b), the intersection of two fragility curves have been removed by adopting the common log-standard deviation. However, it can also be observed that the overall shapes are severely altered. The damage exceedance probabilities of the 50 in 50 set, 10 in 50 set and 2 in 50 set for the five damage states are compared with those obtained by a previous approach. Since the two indices are different (i.e. return period and PGA indices), the equivalent measure should be considered in their comparison. In this respect, the equivalent damage exceedance probabilities are calculated from the fragility curves indexed, using PGA values, for several return periods. We recall that the damage exceedance probability function can be expressed as follows:   ln(a/c j ) I (17a) F j (a; c j , ζ j ) = Φ ζj   ln(a/c j ) F jII (a; c j , ζ j ) = Φ . (17b) ζcommon The first equivalent probability of exceeding the jth damage state ( F˜ jI,1 and F˜ jII,1 ) can be evaluated by reading the fragility

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(a) F jI (a).

(b) F jII (a). Fig. 10. Fragility curves for the same sample bridge evaluated by Shinozuka et al.

PN value at the averaged PGA value ( i=1 ai /N ) for each data set using the following equation: F˜ jI,1 (a1 , a2 , . . . , a N ; c j , ζ j )    N    1 P cj  ai  ln N   i=1 = Φ    ζj F˜ jII,1 (a1 , a2 , . . . , a N ; c j , ζcommon )    N    1 P cj  ai  ln N   i=1 = Φ .   ζcommon

(18a)

(18b)

The second equivalent probability of exceeding the jth damage state ( F˜ jI,2 and F˜ jII,2 ) can also be obtained by averaging the fragility values under each ground motion in a data set, as follows: N   ln(a /c )  1 X i j I,2 ˜ F j (a1 , a2 , . . . , a N ; c j , ζ j ) = Φ N i=1 ζj (19) N   ln(a /c )  1 X i j II,2 ˜ F j (a1 , a2 , . . . , a N ; c j , ζcommon ) = Φ . N i=1 ζcommon

In Eqs. (18) and (19), a1 , a2 , . . . , a N are the PGA values for a certain earthquake group having the same return period, for example, 20 data in the 50 in 50 set. The sample means are used as the third equivalent damage exceeding probability, as follows: N 1 X F˜ j3 (a1 , a2 , . . . , a N ) = xi j N i=1

(20)

where xi j = 1 if the jth damage state occurs in a bridge subjected to a PGA value a = ai , or xi j = 0 in all other cases. For example, if the structures experience almost no damage for all individual earthquakes in the 10 in 50 set, the obtained equivalent damage exceedance probability will be 1.0. Fig. 11 shows a comparison of the results of the estimated damage exceeding probabilities from the present study with

those from the previous approaches, as well as the sample means. In this Figure, F˜ jI,1 ( F˜ jII,1 ), F˜ jI,2 ( F˜ jII,2 ) and F˜ j3 are referred to as ‘Previous 1’, ‘Previous 2’ and ‘Sample’. From these results, those from the present approach are closer to the sample means than those from the previous approaches. The damage probabilities are observed to be over estimated in the case of the previous results for the 50 in 50 set, and the damage probabilities are over- and under-estimated in the case of the 10 in 50 set. Also, no considerable discrepancies were found between the results from the present study and those from the previous methods, especially for the case with the common log-standard deviation (Fig. 11(b)). The common logstandard deviation derived for preventing intersection between the fragility curves with different damage states may have contributed to this discrepancy. This shortcoming is shown in Fig. 6, for the ductility factors of the sample bridge for the 60 Los Angeles earthquakes, and Fig. 10 for the seismic fragility curves. For the complete damage state, only one case is over the damage state limit, which can make the maximum likelihood estimation difficult and unreliable. From Fig. 6, the damage probabilities are observed to be underestimated in the case of almost no and slight damage states by incorporating the common log-standard deviations, while the damage probabilities are over estimated in the case of severe and complete damage states. This observation may explain why there are more discrepancies between the sample means and the damage probabilities with the previous approaches. The proposed method can be concluded to give unbiased information about the damage exceeding probabilities, whereas the previous approaches can over- or under-estimate the damage exceeding probabilities. 6. Conclusions The probability density function (PDF) interpolation technique is proposed for the evaluation of seismic fragility curves as a function of the return period. The seismic hazard curves for maximum structural responses are determined from the structural responses under FEMA SAC earthquake ground motions at a Los Angeles site. The following two assumptions were employed: (1) the maximum structural responses follow the lognormal

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(a) With individual logstandard deviation.

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(b) With common logstandard deviation.

Fig. 11. Comparisons between the results from present study and previous study.

distribution; and (2) the PDF can be interpolated from the form of seismic hazard curve proposed by Cornell. Under these assumptions, the log-mean, λ, and log-standard deviation, η, are easily obtained from two seismic hazard curves. Using the seismic hazard curves, the seismic fragility curves, as a function of return period, are then evaluated from the numerical integration of PDF at each exceeding probability level. Seismic fragility curves have been successfully constructed for a sample bridge, utilizing the proposed method, where the damage exceeding probability for a certain return period can be very easily read. The results obtained using the present method are compared with those obtained by the conventional method represented by PGA, it was found that the results indexed on PGA and the maximum likelihood estimation generally overor under-estimated the seismic fragilities. Acknowledgements This study was carried out with financial support from the Smart Infra-Structure Technology Centre (SISTeC), which was

established at the Korean Advanced Institute of Science and Technology (KAIST) under sponsorship from the Ministry of Science and Technology (MOST) and the Korean Foundation of Science and Technology (KOSEF). References [1] SEAOC. Vision 2000 — A frame work for performance based design, volumes I, II, III. Technical report. Sacramento (California): Structural Engineers Association of California, Vision 2000 Committee; 1995. [2] BSSC. NEHRP Guidelines for the seismic regulation of existing buildings and other structures. Technical report. USA: FEMA 273. Building Seismic Safety Council; 1997. [3] Kennedy RP, Ravindra MK. Seismic fragilities for nuclear power plant risk studies. Nuclear Engineering and Design 1984;79:47–68. [4] Hirata K, Kobayashi Y, Kameda H, Shiojiri H. Fragility of seismically isolated FBR structure. Nuclear Engineering and Design 1991;128: 227–36. [5] Hirata K, Ootori Y, Somaki T. Seismic fragility analysis for base-isolated structure. Journal of Structural and Constructive Engineering 1993;452: 11–9. Architectural Institute of Japan.

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J.-H. Yi et al. / Engineering Structures 29 (2007) 1312–1322

[6] Dimova SL, Hirata K. Simplified seismic fragility analysis of structures with two types of friction devices. Earthquake Engineering and Structural Dynamics 2000;29:1153–75. [7] Shinozuka M, Feng MQ. Nonlinear static procedure for fragility curve development. Journal of Engineering Mechanics, ASCE 1995;126(12): 1287–95. [8] Dong X, Shinozuka M. Performance analysis and visualization of electric power systems. In: Proceedings of the SPIE’s 10 international symposium on smart systems and NDE for infrastructures. 2003. [9] Zhou Y, Murachi Y, Kim SH, Shinozuka M. Seismic risk assessment of retrofitted transportation systems. In: Proceeding of the 13th world conference on earthquake engineering. 2004. Paper no. 1128. [10] Shinozuka M, Zhou YW, Banerjee S. Recent studies in fragility information on highway bridges. In: Proceedings of the 3rd US-PRC workshop on seismic behavior and design of special highway bridges. 2004. [11] Wen YK, Wu CL. Uniform hazard ground motions for Mid-America cities. Earthquake Spectra 2001;17(2):359–84. [12] Shinozuka M, Feng MQ, Kim H, Uzawa T, Ueda T. Statistical analysis of fragility curves. Technical report. Buffalo (NY): MCEER, State Univ. of New York; 2000. [13] Kim SH, Shinozuka M. Development of fragility curves of bridges

[14]

[15] [16] [17] [18]

[19]

[20] [21]

retrofitted by column jacketing. Probabilistic Engineering Mechanics 2004;19(1–2):105–12. Cornell CA. Calculating building seismic performance reliability: a basis for multi-level design norms. In: Proceedings of the 11th world conference on earthquake engineering. 1996. Priestley MJN, Seible F, Calbi GM. Seismic design and retrofit of bridges. John Wiley & Sons, Inc. p. 270–73. Computer and structures, Inc. SAP2000/Non-linear users manual version 8. Berkeley (CA, USA); 2002. FEMA-SAC. Develop suites of time histories project task: 5.4.1. Draft report of joint venture steel project phase 2. 1997. Dutta A, Mander JB. Seismic fragility analysis of highway bridges. In: Proceedings of the center-to-center project workshop on earthquake engineering in transportation systems. 1999. Dutta A. On energy based seismic analysis and design of highway bridges. Ph.D. dissertation. Buffalo (NY): Department of Civil, Structural and Environmental Engineering, State University of New York; 1999. Choi ES, DesRoches R, Nielson B. Seismic fragility of typical bridges in moderate seismic zones. Engineering Structures 2004;26(2):187–99. Yi JH, Kim SH. Probabilistic seismic safety assessment of bridges utilizing measured response data. Journal of Korea Society of Civil Engineers 2004;24(5A):1093–102.