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Applications of Statistics and Probability in Civil Engineering – Kanda, Takada & Furuta (eds)© 2007 Taylor & Francis Group, London, ISBN 978-0-415-45134-5

Seismic risk assessment and expected damage evaluationof railway viaduct

H. YoshikawaMusashi Institute of Technology, Tokyo, Setagaya, Japan

T. Ohtaki, H. Hattori, Y. Maeda, A. Noguchi & H. OkadaTokyu Construction Co., Ltd., Tokyo, Shibuya, Japan

ABSTRACT: This paper shows an analytical procedure of seismic risk estimation, which consists of fourphases; Calculation of seismic hazard curves at a construction site, Evaluation of structural performance of aviaduct with pushover analysis and equal energy principle, Vulnerability and damage evaluation of the structure,and Seismic risk assessment based on the information in the previous phases. Numerical simulations on tworeinforced concrete railway viaducts were carried out and the seismic risk were assessed. The results providedfragility curves, damage functions, expected damage loss, and risk curve. The annual expected loss of the viaductdesigned with the current design code reduced by 37% and 81%, for transverse and longitudinal response ofthe structure, respectively, as compared with those of the structure designed with the former provisions. Theproposed system is quite useful for estimating the seismic risk of reinforced concrete structures and could beapplicable for various types of structures.

1 INTRODUCTION

Railway system is an important infrastructure forurban traffic that requires daily mass transportation.Needless to say that securing safety of life is essentialfor the system in case of disaster such as earthquakes,and that the damage on the owners’ and users’ con-venience due to depression or stop of the function ofthe system is serious as well. The disadvantage for theowners is evaluated from the operating loss during theinterrupted period of the transportation and the costof the restoration, repair and retrofitting. It is veryimportant for the owners to know how to reduce theloss described above and the procedure to minimizethe total expenses for maintenance from the viewpointof life cycle cost of the structure. Seismic risk assess-ment is one of the procedures which provide usefulinformation on the judgment of such cost and benefitevaluation.

The structural damage due to earthquakes couldbe computed from the deterministic parameters suchas intensity of the excitation at the construction siteand the seismic performance of the structure. How-ever, in order to qualitatively assess the expecteddamage loss due to such uncertain attack in a giventime scale, probabilistic approach is required. Inthis study, seismic risk evaluation system, basedon reliability theory, for reinforced concrete railway

Phase 2 Structural Performance

Phase 1 Seismic Hazard Curves

Earthquake Structure

Phase 3 Vulnerability Evaluation • Fragility Curve • Seismic Loss Function

Phase 4Seismic Risk Assessment • Expected Loss Evaluation • Seismic Risk Curve

Figure 1. Flow diagram for seismic risk assessment.

viaducts was proposed and numerical simulations weredemonstrated.

2 RISK ASSESSMENT PROCEDURE

The procedure of the seismic risk assessment is shownin Figure 1 (Hattori et al. 2006). Phase3 and 4 are appli-cable not only for monetary loss but also for service

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interruption date (Ohtaki et al. 2006). The each phaseis described as follows:

2.1 Phase1 seismic hazard curves

Seismic hazard curve exhibits probability of annualexceedance of seismic intensity at a specific con-struction site. Probability density function of annualoccurrence is obtained as a tangent of the probabil-ity of annual exceedance. In this study, a hazard curvefor the maximum acceleration on the rock surface at aconstruction site of the target structure was calculatedand used for the risk assessment.

2.2 Phase2 structural performance

Nonlinear pushover analysis is employed for the eval-uation of seismic performance of the target struc-ture. The structural damage events obtained from thepushover analysis correspond to the member dam-age events. Hence, the structural damage level will beevaluated based on the member damage level. The def-initions of the member damage event and the memberdamage level are shown in Figure 2.

In order to estimate the structural response fromseismic excitation, an empirical equation was applied.According to Kanda et al. (Kanda et al. 1998) the rela-tionship between average response acceleration andthe bedrock acceleration is given by Equation 1. Then,the inelastic deformation of the structure can be cal-culated assuming equal energy principle as given byEquation 2.

where αE = structural response acceleration; α =acceleration on bedrock.

where δresp = inelastic overall structural response;αY = yield acceleration; δy = yield displacement ofstructural system.

2.3 Phase3 Vulnerability evaluation

In this phase, fragility curves, defined as probabil-ity of exceedance of certain damage state of thestructure, and expected damage loss corresponding toseismic acceleration are calculated based on event treeanalysis.

The fragility curves Fi(α) are calculated from Equa-tion 3 (Endo & Yoshikawa 2003) with mean value ofthe structural response δresp and the displacement δicorresponding to the structural damage event obtained

Figure 2. Definition of member damage event and memberdamage level.

Figure 3. Relationship between structural response andfragility curves.

from the pushover analysis. As the overall responseis a function of the acceleration, the probability ofthe occurrence of each damage state is described asa conditional occurrence probability of seismic accel-eration Fi(α). Thus, the probability of occurrence ofeach structural damage level is obtained as Equation 4.Figure 3 shows the relationship between the structuralresponse and the fragility curves.

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Table 1. Damage event tree.

Probability of Monetary DamageEvent occurrence loss state

Earthquake P(c1|α) c1 1α (gal) P(c2|α) c2 2

. . . . . . . . .

P(ci|α) ci i. . . . . . . . .

P(cn+1|α) cn n + 1

�(ci|α) = 1

where ζ2x = ln{(1 + ν2

i )(1 + ν2R)}, νi = cov of δi,

νR = cov of δresp.

where i = number of structural damage event(i = 1, 2, . . . , n + 1).

Table 1 shows a damage event tree for a rein-forced concrete frame structure. From the probabilityof occurrence and corresponding damage loss ci for agiven α, expected damage loss cm, denoted as cNEL:normal expected loss, and its variance σc can becalculated from Equations 7 and 8.

The seismic loss function is then obtained as therelationship between the input acceleration and theexpected damage loss.

2.4 Phase4 Seismic risk assessment

Seismic risk is assessed with annual expected loss andrisk curve. The annual expected loss demonstrates thestructural vulnerability at the construction site, namelyseismic hazard, by means of repair cost. Seismic riskcurve is given as relationship between damage lossand its probability of annual exceedance. The mag-nitude and the shape of the risk curve exhibit thecharacteristics of the seismic risk.

Annual expected loss to specific bedrock accelera-tion can be evaluated with the seismic hazard curvesand the seismic loss function. Figure 4 shows the

Figure 4. Calculation procedure of annual expected loss.

procedure of the calculation. The probability densityfunction of annual exceedance of bedrock accelerationis given by

where PA(α) = probability of annual exceedance ofbedrock acceleration obtained from seismic hazardcurve.

Then, the annual risk of the structure is evaluatedas expected loss density el(α) and integrated loss ELas given by Equations 10 and 11, respectively.

Introducing β distribution with mean and varianceof cm and σc, probability density function of monetaryloss is given by

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Figure 5. Seismic hazard curve at Tokyo Shibuya region.

Figure 6. Dimensions of viaduct 1 (1992 code).

where cmax = maximum expected loss.Then, the probability of exceedance of the loss c

is given by Equation 15, and the seismic risk curve isobtained as an integration of the product of the proba-bility density function of the acceleration pA(α) and the

Figure 7. Dimensions of viaduct 2 (2004 code).

conditional probability of the loss R(c|α), as definedby the Equation 16.

3 NUMERICAL SIMULATION

3.1 Seismic hazard curve (Phase1)

Taking Tokyo Shibuya region as a construction site forthe simulation, the seismic hazard curve is obtained asan exponential approximation of Hazen plot (Hazen1930) of the peak bedrock acceleration calculatedwith Equation 17 (Fukushima 1996). The parametersrequired for the equation are given by the historicalearthquake data (NAOJ 2003). The result is shown inFigure 5.

where M = magnitude; r = hypocentral distance.

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0

2000

4000

6000

8000

0 50 100 150 200 250Displacement (mm)

Lat

eral

for

ce (

kN)

Longitudinal

1

2 20

12

0

1000

2000

3000

4000

5000

0 05 100 150 200 250Displacement (mm)

Lat

eral

for

ce (

kN)

Transverse

12

3 4 5 6

Figure 8. Force-displacement response of viaduct 1.

3.2 Seismic performance (Phase2)

Two different types of reinforced concrete railwayviaducts were analyzed and compared. The dimen-sions of the viaducts are shown in Figures 6 and 7.Both viaducts are rigid frame structures with two-column bent and five continuous spans, designed with1992 (RTRI 1992) and 2004 (RTRI 2004) provisions,respectively. The foundations of the viaduct 1 arespread-footings isolated each other and have no piles,while the columns of the viaduct 2 were connectedrigidly with underground beams and have piles fromthe column base straight down into the ground.

For nonlinear pushover analyses, the viaducts weremodeled as two-dimensional frames in transverse andlongitudinal direction, respectively. The structural ele-ments such as columns and beams were modeled as abeam element with nonlinear components with char-acteristics as shown in Figure 2 at plastic hinge region.The foundation springs for the spread-footings and thepiles were appropriately modeled with lateral, verti-cal and rotational stiffness calculated based on eachdesign code.

Under the initial stress state due to dead load ofthe structure, seismic lateral force was applied tothe model monotonically in order to obtain force

0

2000

4000

6000

8000

10000

12000

0 100 200 300 400 500Displacement (mm)

Lat

eral

for

ce (

kN)

Longitudinal

1

7

18

8

0

2000

4000

6000

8000

10000

12000

0 100 200 300 400 500

Displacement (mm)

Lat

eral

for

ce (

kN)

Transverse

12

65

14

10

Figure 9. Force-displacement response of viaduct 2.

deformation relationship. Then, the structural damagestate can be assessed with the structural response.

The results obtained from the pushover analyses areshown in Figures 8 and 9 demonstrating the relation-ship between the lateral force and the displacementin transverse and longitudinal direction, respectively,for the viaduct 1 and viaduct 2. The numberings inthe figures indicate the consecutive structural dam-age events associated with the member damage eventsshown in Figure 2. As the deformation increases, thestructural capacity degrades with the members reach-ing their damage state such as yield, maximum andultimate at the possible plastic hinge regions in beamsand columns. Finally, overall structural mechanism isattained.

Table 2 shows the member damage levels and thedefinition with the basic repair procedures. Thesedamages should be repaired after earthquakes and therepair cost can be estimated. Thus, the monetary lossdue to earthquakes can be calculated by summing upthe repair cost required for members appropriate forthe damage level (Maeda et al. 2006).

Tables 3 and 4 summarize the structural damageevent number and the member damage levels with theestimated total cost for repairs in longitudinal direction

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Table 2. Member damage levels and required repair procedure.

DamageMember level Definition Repair required

Beams 1 Slight cracking None

2 Yield of longitudinal Temporary scaffoldreinforcement Crack injectionFlexural cracking or shearcracking

3 Spalling off of cover concrete Rail removalBuckling of longitudinal Temporary scaffoldreinforcement Crack injection

Adjustment of reinforcementPatch-up cover concreteSlab waterproofingRail restoration

4 Damage on core concrete Slab underpinningFracture of longitudinal Rail removalreinforcement Temporary scaffoldFracture of lateral Concrete removalreinforcement Replace reinforcement

Concrete castingSlab waterproofingRail restoration

Columns 1 Slight cracking None

2 Yield of longitudinal Temporary scaffoldreinforcement Crack injectionFlexural cracking or shearcracking

3 Spalling off of cover concrete Temporary scaffoldBuckling of longitudinal Crack injectionreinforcement Adjustment of reinforcement

Patch-up cover concrete

4 Damage on core concrete Slab underpinningFracture of longitudinal Temporary scaffoldreinforcement Concrete removalFracture of lateral Replace reinforcementreinforcement Concrete casting

for the viaduct 1 and 2, respectively, showing that thelarger the deformation, the lager the monetary loss.

3.3 Vulnerability evaluation (Phase3)

3.3.1 Fragility curvesThe probability that the overall structural responseδresp exceeds the displacement of the structural damageevent δi is given by Equation 3 as fragility curves. Thedisplacements corresponding to the structural damageevents were obtained from the pushover analysis andthe structural response due to earthquake is evaluatedwith bedrock acceleration using Equations 1 and 2.

Assuming the coefficients of variance for δi andδresp are 0.3, the fragility curves are plotted as shown inFigures 10 and 11 in longitudinal and transverse direc-tions for viaduct 1 and 2, respectively. One fragility

curve corresponds to one damage event of one mem-ber. The structural characteristics such as location andthe sequence of occurrence of the plastic hinges arereflected on the shape and the distribution of thefragility curves, depicting relationship between theinput excitation and the damage level that the structuremight experience.

The probability of exceedance of damage events ofViaduct 1 is lager than that of Viaduct 2 at the sameacceleration, indicating insufficient performance ofthe structure designed with 1992 code.

3.3.2 Seismic loss functionThe various monetary loss, ci , shown inTable 2 are cal-culated as cumulative repair cost for damaged plastichinges taking symmetric damages in the frame modelunder cyclic loading of earthquakes into consideration.

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Table 3. Structural damage event number and member damage level of viaduct 1.

Member damage levelStractural Monetarydamage Column bottom Column top Beams Deformation lossevent Element δi cinumber number 54,84 60,78 66,72 52,82 58,76 64,70 2,49 9,42 12,39 19,32 22,29 mm ∗10000 yen

0 1 1 1 1 1 1 1 1 1 1 1 0 01 1 1 1 1 1 1 1 2 1 1 1 46 462 1 1 1 1 2 1 1 2 1 1 1 73 1053 1 1 1 1 2 2 1 2 1 1 1 74 1644 1 2 2 1 2 2 1 2 1 1 1 79 1965 1 2 2 1 2 2 2 2 1 1 1 80 2426 2 2 2 1 2 2 2 2 1 1 1 83 2577 2 2 2 1 2 2 2 2 1 2 1 95 3038 2 2 2 2 2 2 2 2 1 2 1 101 3629 2 2 2 2 2 2 2 2 1 2 2 108 408

10 2 2 2 2 3 2 2 2 1 2 2 130 42211 2 2 2 2 3 3 2 2 1 2 2 134 43612 2 3 3 2 3 3 2 2 1 2 2 138 46313 3 3 3 2 3 3 2 2 1 2 2 149 47614 3 4 3 2 3 3 2 2 1 2 2 167 167315 3 4 3 2 4 3 2 2 1 2 2 168 171316 3 4 3 2 4 4 2 2 1 2 2 169 175417 3 4 4 2 4 4 2 2 1 2 2 172 179418 3 4 4 3 4 4 2 2 1 2 2 177 180819 4 4 4 3 4 4 2 2 1 2 2 184 186220 4 4 4 4 4 4 2 2 1 2 2 229 1902

Table 4. Structural damage event number and member damage level of viaduct 2.

Stractural Underground Monetarydamage Column bottom Column top Beams beams Pile top Deformation lossevent Element δi cinumber number 1,6 2,5 3,4 1,6 2,5 3,4 1,5 1,5 others 1,5 other 1,6 2,5 3,4 mm ∗10000 yen

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 01 1 2 1 1 1 1 1 1 1 1 1 1 1 1 46 252 1 2 2 1 1 1 1 1 1 1 1 1 1 1 52 503 1 2 2 1 2 1 1 1 1 1 1 1 1 1 55 1224 1 2 2 1 2 2 1 1 1 1 1 1 1 1 58 1945 1 2 2 1 2 2 2 1 1 2 1 1 1 1 76 5636 2 2 2 1 2 2 2 1 1 2 1 1 1 1 82 5887 2 2 2 2 2 2 2 1 1 2 1 1 1 1 85 7058 2 3 2 2 2 2 2 2 1 2 1 1 1 1 271 7369 2 3 3 2 2 2 2 2 1 2 1 1 1 1 277 766

10 2 3 3 2 3 2 2 2 1 2 1 1 1 1 286 79611 2 3 3 2 3 3 2 2 1 2 1 1 1 1 292 82712 3 3 3 2 3 3 2 2 1 2 1 1 1 1 307 85713 3 3 3 3 3 3 2 2 1 2 1 1 1 1 325 88814 3 4 3 3 3 3 2 2 1 2 1 1 1 1 340 223415 3 4 4 3 3 3 2 2 1 2 1 1 1 1 349 227416 3 4 4 3 4 3 2 2 1 2 1 1 1 1 355 231517 3 4 4 3 4 4 2 2 1 2 1 1 1 1 364 235618 4 4 4 3 4 4 2 2 1 2 1 1 1 1 388 2396

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0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 300 600 900 1200 1500 1800

Peak bedrock acceleration (gal)

Prob

abilt

y of

exc

eeda

nce

Longituinal

F1

F2

F20

F19

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 300 600 900 1200 1500 1800

Peak bedrock acceleration (gal)

Prob

abilt

y of

exc

eeda

nce

Transverse

F1F2

FF3

6

Figure 10. Fragility curves of viaduct 1.

The seismic loss function can be computed with Equa-tion 7, denoted as normal expected loss, cNEL, in whichthe conditional probability of damage occurrence tothe input acceleration is considered.

The results are shown in Figures 12 and 13 forviaduct 1 and 2, respectively. The individual loss ciis expressed as a step function of the bedrock accel-eration, while the seismic loss function cNEL resultedinto the smoothing curve.

The seismic loss function for viaduct 1 exhibitsthe larger loss for transverse direction than that oflongitudinal direction, while viaduct 2 shows almostequivalent value in both directions up to bedrockacceleration of 600 gal. It can be seen that the totalexpected loss of viaduct 2 was significantly reducedcompared to viaduct 1 as a result of revision of thedesign code.

3.4 Seismic risk assessment (Phase4)

3.4.1 Annual Expected LossBased on the procedure described in Phase4, theannual expected loss density function el(α) and theannual expected loss EL can be computed from Equa-tions 10 and 11, respectively. The results are shown inTable 5 and Figure 14. As a result, the annual expected

Figure 11. Fragility curves of viaduct 2.

Figure 12. Seismic loss function of viaduct 1.

loss of viaduct 2 in transverse direction drasticallyreduced to about 19% of that for viaduct 1 and about63% in longitudinal direction.

It should be noted that the repair cost is thoroughlyauthors’estimation based on the unit construction costin 2006 and could be changed due to the conditionsrequisite for the repair work.

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Figure 13. Seismic loss function of viaduct 2.

Table 5. Annual expected loss EL for viaduct.

Transverse LongitudinalYen Yen Code

Viaduct 1 64500 16600 1992Viaduct 2 12100 10400 20042004/1992 18.8% 62.7% –

Figure 14. Annual expected loss density of railway viaducts.

3.4.2 Seismic risk curveFigure 15 shows the seismic risk curve obtained fromEquation 16. The differences in the risk curves arenot significant except for the curve for viaduct 1 intransverse direction, exhibiting about four times loss ofother cases at 0.2% probability of annual exceedance.Thus, seismic risk curve gives quantitative informationon the amount of loss and its probability and could bequite useful for strategic maintenance planning pro-vided that the importance of the structure is also takeninto account.

Figure 15. Seismic risk curve of railway viaducts.

4 CONCLUDING REMARKS

The procedure for the seismic risk evaluation was pre-sented in this paper, and numerical simulations weremade for the different types of railway viaducts. Theproposed procedure integrated the four Phases; eachof which is basically dealt with on the analytical basesand the well known techniques.

The particular features in the proposed method aresummarized as follows:

• The structural analysis in Phase2 was made by theinelastic pushover analysis to identify the structuraldamage levels. Here, classification of the memberdamages, their repair methods and the cost of repairswere carefully examined from the experiences of thepast major earthquakes in Japan.

• The vulnerability evaluation on the Phase3 was doneto analytically obtain fragility curves and seismicdamage loss functions.As the final step, Phase4 wascarried out for the seismic risk assessment, whichprovided the amount of annual expected damage andthe seismic risk curves.

• It should be noted that the density function ofexpected seismic loss given in relation of peakbedrock acceleration helps engineers to develop theretrofit strategy. Moreover, the proposed risk anal-ysis may lead to the new seismic design frameworkto take the place of the present performance designphilosophy.

REFERENCES

Endo,A.,Yoshikawa, H.,Application of Seismic RiskAssess-ment to Single Reinforced Concrete Pier, J. of StructuralEngineering, Vol.49A, 2003. (in Japanese)

Fukushima Y., Derivation and Revision of Attenuation Rela-tion for Peak Horizontal Acceleration Applicable to theNear Source Region, Technical Report, Vol.63, ShimizuCorp. 1996. (in Japanese)

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Hattori, H. et al., Expected Damage and Seismic RiskCurve for Railway Structures. The 12th Japan EarthquakeEngineering Symposium, 2006. (in Japanese)

Hazen,A., Flood Flows, a Study in Frequency and Magnitude,J. Wiley and Sons, New York, 1930.

Kanda, J. et al., Seismic Hazard Analysis Considering ActiveFaults and Application to Optimum Reliability for Struc-tural Safety, The 10th Japan Earthquake EngineeringSymposium, 1998. (in Japanese)

Maeda,Y. et al., Damage event analysis and structural damageevaluation for a RC railway viaduct under seismic loading,J. of Structural Engineering, Vol.53A, 2006. (in Japanese)

National Astronomical Observatory of Japan, Chronologicalscientific, 2003. (in Japanese)

Ohtaki, T. et al., Seismic Risk Assessment and ExpectedDamage for a Railway Viaduct, Technical Reports, Vol.32.Tokyu Construction Co., Ltd. 2006. (in Japanese)

Railway Technical Research Institute, Design Provisions forRailway Structures, Concrete Structures, 1992. (Japanese)

Railway Technical Research Institute, Design Provisions forRailway Structures, Concrete Structures, 2004. (Japanese)

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