Seismic risk assessment and expected damage evaluation of railway viaduct

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1 Applications of Statistics and Probability in Civil Engineering Kanda, Takada & Furuta (eds) 2007 Taylor & Francis Group, London, ISBN Seismic risk assessment and expected damage evaluation of railway viaduct H. Yoshikawa Musashi Institute of Technology, Tokyo, Setagaya, Japan T. Ohtaki, H. Hattori, Y. Maeda, A. Noguchi & H. Okada Tokyu Construction Co., Ltd., Tokyo, Shibuya, Japan ABSTRACT: This paper shows an analytical procedure of seismic risk estimation, which consists of four phases; Calculation of seismic hazard curves at a construction site, Evaluation of structural performance of a viaduct 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 two reinforced concrete railway viaducts were carried out and the seismic risk were assessed. The results provided fragility curves, damage functions, expected damage loss, and risk curve. The annual expected loss of the viaduct designed with the current design code reduced by 37% and 81%, for transverse and longitudinal response of the structure, respectively, as compared with those of the structure designed with the former provisions. The proposed system is quite useful for estimating the seismic risk of reinforced concrete structures and could be applicable for various types of structures. 1 INTRODUCTION Railway system is an important infrastructure for urban traffic that requires daily mass transportation. Needless to say that securing safety of life is essential for the system in case of disaster such as earthquakes, and that the damage on the owners and users convenience due to depression or stop of the function of the system is serious as well. The disadvantage for the owners is evaluated from the operating loss during the interrupted period of the transportation and the cost of the restoration, repair and retrofitting. It is very important for the owners to know how to reduce the loss described above and the procedure to minimize the total expenses for maintenance from the viewpoint of life cycle cost of the structure. Seismic risk assessment is one of the procedures which provide useful information on the judgment of such cost and benefit evaluation. The structural damage due to earthquakes could be computed from the deterministic parameters such as intensity of the excitation at the construction site and the seismic performance of the structure. However, in order to qualitatively assess the expected damage loss due to such uncertain attack in a given time scale, probabilistic approach is required. In this study, seismic risk evaluation system, based on reliability theory, for reinforced concrete railway Earthquake Phase 1 Seismic Hazard Curves Phase 4 Seismic Risk Assessment Expected Loss Evaluation Seismic Risk Curve Figure 1. Structure Phase 2 Structural Performance Phase 3 Vulnerability Evaluation Fragility Curve Seismic Loss Function Flow diagram for seismic risk assessment. viaducts was proposed and numerical simulations were demonstrated. 2 RISK ASSESSMENT PROCEDURE The procedure of the seismic risk assessment is shown in Figure 1 (Hattori et al. 2006). Phase3 and 4 are applicable not only for monetary loss but also for service 1

2 interruption date (Ohtaki et al. 2006). The each phase is described as follows: 2.1 Phase1 seismic hazard curves Seismic hazard curve exhibits probability of annual exceedance of seismic intensity at a specific construction site. Probability density function of annual occurrence is obtained as a tangent of the probability of annual exceedance. In this study, a hazard curve for the maximum acceleration on the rock surface at a construction site of the target structure was calculated and used for the risk assessment. 2.2 Phase2 structural performance Nonlinear pushover analysis is employed for the evaluation of seismic performance of the target structure. The structural damage events obtained from the pushover analysis correspond to the member damage events. Hence, the structural damage level will be evaluated based on the member damage level. The definitions of the member damage event and the member damage level are shown in Figure 2. In order to estimate the structural response from seismic excitation, an empirical equation was applied. According to Kanda et al. (Kanda et al. 1998) the relationship between average response acceleration and the bedrock acceleration is given by Equation 1. Then, the inelastic deformation of the structure can be calculated assuming equal energy principle as given by Equation 2. Figure 2. Definition of member damage event and member damage level. where α E = structural response acceleration; α = acceleration on bedrock. where δ resp = inelastic overall structural response; α Y = yield acceleration; δ y = yield displacement of structural system. 2.3 Phase3 Vulnerability evaluation In this phase, fragility curves, defined as probability of exceedance of certain damage state of the structure, and expected damage loss corresponding to seismic acceleration are calculated based on event tree analysis. The fragility curves F i (α) are calculated from Equation 3 (Endo & Yoshikawa 2003) with mean value of the structural response δ resp and the displacement δ i corresponding to the structural damage event obtained Figure 3. Relationship between structural response and fragility curves. from the pushover analysis. As the overall response is a function of the acceleration, the probability of the occurrence of each damage state is described as a conditional occurrence probability of seismic acceleration F i (α). Thus, the probability of occurrence of each structural damage level is obtained as Equation 4. Figure 3 shows the relationship between the structural response and the fragility curves. 2

3 Table 1. Damage event tree. Probability of Monetary Damage Event occurrence loss state Earthquake P(c 1 α) c 1 1 α (gal) P(c 2 α) c P(c i α) c i i P(c n+1 α) c n n + 1 (c i α) = 1 where ζ 2 x = ln{(1 + ν2 i )(1 + ν2 R )}, ν i = cov of δ i, ν R = cov of δ resp. where i = number of structural damage event (i = 1, 2,..., n + 1). Table 1 shows a damage event tree for a reinforced concrete frame structure. From the probability of occurrence and corresponding damage loss c i for a given α, expected damage loss c m, denoted as c NEL : normal expected loss, and its variance σ c can be calculated from Equations 7 and 8. Figure 4. Calculation procedure of annual expected loss. procedure of the calculation. The probability density function of annual exceedance of bedrock acceleration is given by The seismic loss function is then obtained as the relationship between the input acceleration and the expected damage loss. 2.4 Phase4 Seismic risk assessment Seismic risk is assessed with annual expected loss and risk curve. The annual expected loss demonstrates the structural vulnerability at the construction site, namely seismic hazard, by means of repair cost. Seismic risk curve is given as relationship between damage loss and its probability of annual exceedance. The magnitude and the shape of the risk curve exhibit the characteristics of the seismic risk. Annual expected loss to specific bedrock acceleration can be evaluated with the seismic hazard curves and the seismic loss function. Figure 4 shows the where P A (α) = probability of annual exceedance of bedrock acceleration obtained from seismic hazard curve. Then, the annual risk of the structure is evaluated as expected loss density el(α) and integrated loss EL as given by Equations 10 and 11, respectively. Introducing β distribution with mean and variance of c m and σ c, probability density function of monetary loss is given by 3

4 Mori CH077.tex 5/6/ : 3 Page 4 Figure 5. Seismic hazard curve at Tokyo Shibuya region. Figure 7. Dimensions of viaduct 2 (2004 code). conditional probability of the loss R(c α), as defined by the Equation Figure 6. Dimensions of viaduct 1 (1992 code). NUMERICAL SIMULATION 3.1 Seismic hazard curve (Phase1) Taking Tokyo Shibuya region as a construction site for the simulation, the seismic hazard curve is obtained as an exponential approximation of Hazen plot (Hazen 1930) of the peak bedrock acceleration calculated with Equation 17 (Fukushima 1996). The parameters required for the equation are given by the historical earthquake data (NAOJ 2003). The result is shown in Figure 5. where cmax = maximum expected loss. Then, the probability of exceedance of the loss c is given by Equation 15, and the seismic risk curve is obtained as an integration of the product of the probability density function of the acceleration pa (α) and the where M = magnitude; r = hypocentral distance. 4

5 Lateral force (kn) Longitudinal Lateral force (kn) Longitudinal Displacement (mm) Displacement (mm) Lateral force (kn) Transverse Lateral force (kn) Transverse Displacement (mm) Displacement (mm) Figure 8. Force-displacement response of viaduct 1. Figure 9. Force-displacement response of viaduct Seismic performance (Phase2) Two different types of reinforced concrete railway viaducts were analyzed and compared. The dimensions of the viaducts are shown in Figures 6 and 7. Both viaducts are rigid frame structures with twocolumn bent and five continuous spans, designed with 1992 (RTRI 1992) and 2004 (RTRI 2004) provisions, respectively. The foundations of the viaduct 1 are spread-footings isolated each other and have no piles, while the columns of the viaduct 2 were connected rigidly with underground beams and have piles from the column base straight down into the ground. For nonlinear pushover analyses, the viaducts were modeled as two-dimensional frames in transverse and longitudinal direction, respectively. The structural elements such as columns and beams were modeled as a beam element with nonlinear components with characteristics as shown in Figure 2 at plastic hinge region. The foundation springs for the spread-footings and the piles were appropriately modeled with lateral, vertical and rotational stiffness calculated based on each design code. Under the initial stress state due to dead load of the structure, seismic lateral force was applied to the model monotonically in order to obtain force deformation relationship. Then, the structural damage state can be assessed with the structural response. The results obtained from the pushover analyses are shown in Figures 8 and 9 demonstrating the relationship between the lateral force and the displacement in transverse and longitudinal direction, respectively, for the viaduct 1 and viaduct 2. The numberings in the figures indicate the consecutive structural damage events associated with the member damage events shown in Figure 2. As the deformation increases, the structural capacity degrades with the members reaching their damage state such as yield, maximum and ultimate at the possible plastic hinge regions in beams and columns. Finally, overall structural mechanism is attained. Table 2 shows the member damage levels and the definition with the basic repair procedures. These damages should be repaired after earthquakes and the repair cost can be estimated. Thus, the monetary loss due to earthquakes can be calculated by summing up the repair cost required for members appropriate for the damage level (Maeda et al. 2006). Tables 3 and 4 summarize the structural damage event number and the member damage levels with the estimated total cost for repairs in longitudinal direction 5

6 Table 2. Member damage levels and required repair procedure. Damage Member level Definition Repair required Beams 1 Slight cracking None 2 Yield of longitudinal Temporary scaffold Crack injection Flexural cracking or shear cracking 3 Spalling off of cover concrete Rail removal Buckling of longitudinal Temporary scaffold Crack injection Adjustment of Patch-up cover concrete Slab waterproofing Rail restoration 4 Damage on core concrete Slab underpinning Fracture of longitudinal Rail removal Temporary scaffold Fracture of lateral Concrete removal Replace Concrete casting Slab waterproofing Rail restoration Columns 1 Slight cracking None 2 Yield of longitudinal Temporary scaffold Crack injection Flexural cracking or shear cracking 3 Spalling off of cover concrete Temporary scaffold Buckling of longitudinal Crack injection Adjustment of Patch-up cover concrete 4 Damage on core concrete Slab underpinning Fracture of longitudinal Temporary scaffold Concrete removal Fracture of lateral Replace Concrete casting for the viaduct 1 and 2, respectively, showing that the larger the deformation, the lager the monetary loss. 3.3 Vulnerability evaluation (Phase3) Fragility curves The probability that the overall structural response δ resp exceeds the displacement of the structural damage event δ i is given by Equation 3 as fragility curves. The displacements corresponding to the structural damage events were obtained from the pushover analysis and the structural response due to earthquake is evaluated with bedrock acceleration using Equations 1 and 2. Assuming the coefficients of variance for δ i and δ resp are 0.3, the fragility curves are plotted as shown in Figures 10 and 11 in longitudinal and transverse directions for viaduct 1 and 2, respectively. One fragility curve corresponds to one damage event of one member. The structural characteristics such as location and the sequence of occurrence of the plastic hinges are reflected on the shape and the distribution of the fragility curves, depicting relationship between the input excitation and the damage level that the structure might experience. The probability of exceedance of damage events of Viaduct 1 is lager than that of Viaduct 2 at the same acceleration, indicating insufficient performance of the structure designed with 1992 code Seismic loss function The various monetary loss, c i, shown intable 2 are calculated as cumulative repair cost for damaged plastic hinges taking symmetric damages in the frame model under cyclic loading of earthquakes into consideration. 6

7 Table 3. Structural damage event number and member damage level of viaduct 1. Member damage level Stractural Monetary damage Column bottom Column top Beams Deformation loss event Element δ i c i number number 54,84 60,78 66,72 52,82 58,76 64,70 2,49 9,42 12,39 19,32 22,29 mm yen Table 4. Structural damage event number and member damage level of viaduct 2. Stractural Underground Monetary damage Column bottom Column top Beams beams Pile top Deformation loss event Element δ i c i number 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 yen

8 Probabilty of exceedance Longituinal F 1 F2 F 19 F Peak bedrock acceleration (gal) Probabilty of exceedance Transverse F 1 F 2 F 3 F Peak bedrock acceleration (gal) Figure 10. Fragility curves of viaduct 1. Figure 11. Fragility curves of viaduct 2. The seismic loss function can be computed with Equation 7, denoted as normal expected loss, c NEL, in which the conditional probability of damage occurrence to the input acceleration is considered. The results are shown in Figures 12 and 13 for viaduct 1 and 2, respectively. The individual loss c i is expressed as a step function of the bedrock acceleration, while the seismic loss function c NEL resulted into the smoothing curve. The seismic loss function for viaduct 1 exhibits the larger loss for transverse direction than that of longitudinal direction, while viaduct 2 shows almost equivalent value in both directions up to bedrock acceleration of 600 gal. It can be seen that the total expected loss of viaduct 2 was significantly reduced compared to viaduct 1 as a result of revision of the design code. Figure 12. Seismic loss function of viaduct Seismic risk assessment (Phase4) Annual Expected Loss Based on the procedure described in Phase4, the annual expected loss density function el(α) and the annual expected loss EL can be computed from Equations 10 and 11, respectively. The results are shown in Table 5 and Figure 14. As a result, the annual expected loss of viaduct 2 in transverse direction drastically reduced to about 19% of that for viaduct 1 and about 63% in longitudinal direction. It should be noted that the repair cost is thoroughly authors estimation based on the unit construction cost in 2006 and could be changed due to the conditions requisite for the repair work. 8

9 Figure 13. Seismic loss function of viaduct 2. Figure 15. Seismic risk curve of railway viaducts. Table 5. Annual expected loss EL for viaduct. 4 CONCLUDING REMARKS Transverse Longitudinal Yen Yen Code Viaduct Viaduct / % 62.7% Figure 14. Annual expected loss density of railway viaducts Seismic risk curve Figure 15 shows the seismic risk curve obtained from Equation 16. The differences in the risk curves are not significant except for the curve for viaduct 1 in transverse direction, exhibiting about four times loss of other cases at 0.2% probability of annual exceedance. Thus, seismic risk curve gives quantitative information on the amount of loss and its probability and could be quite useful for strategic maintenance planning provided that the importance of the structure is also taken into account. The procedure for the seismic risk evaluation was presented in this paper, and numerical simulations were made for the different types of railway viaducts. The proposed procedure integrated the four Phases; each of which is basically dealt with on the analytical bases and the well known techniques. The particular features in the proposed method are summarized as follows: The structural analysis in Phase2 was made by the inelastic pushover analysis to identify the structural damage levels. Here, classification of the member damages, their repair methods and the cost of repairs were carefully examined from the experiences of the past major earthquakes in Japan. The vulnerability evaluation on the Phase3 was done to analytically obtain fragility curves and seismic damage loss functions. As the final step, Phase4 was carried out for the seismic risk assessment, which provided the amount of annual expected damage and the seismic risk curves. It should be noted that the density function of expected seismic loss given in relation of peak bedrock acceleration helps engineers to develop the retrofit strategy. Moreover, the proposed risk analysis may lead to the new seismic design framework to take the place of the present performance design philosophy. REFERENCES Endo, A.,Yoshikawa, H., Application of Seismic RiskAssessment to Single Reinforced Concrete Pier, J. of Structural Engineering, Vol.49A, (in Japanese) Fukushima Y., Derivation and Revision of Attenuation Relation for Peak Horizontal Acceleration Applicable to the Near Source Region, Technical Report, Vol.63, Shimizu Corp (in Japanese) 9

10 Hattori, H. et al., Expected Damage and Seismic Risk Curve for Railway Structures. The 12th Japan Earthquake Engineering Symposium, (in Japanese) Hazen,A., Flood Flows, a Study in Frequency and Magnitude, J. Wiley and Sons, New York, Kanda, J. et al., Seismic Hazard Analysis Considering Active Faults and Application to Optimum Reliability for Structural Safety, The 10th Japan Earthquake Engineering Symposium, (in Japanese) Maeda,Y. et al., Damage event analysis and structural damage evaluation for a RC railway viaduct under seismic loading, J. of Structural Engineering, Vol.53A, (in Japanese) National Astronomical Observatory of Japan, Chronological scientific, (in Japanese) Ohtaki, T. et al., Seismic Risk Assessment and Expected Damage for a Railway Viaduct, Technical Reports, Vol.32. Tokyu Construction Co., Ltd (in Japanese) Railway Technical Research Institute, Design Provisions for Railway Structures, Concrete Structures, (Japanese) Railway Technical Research Institute, Design Provisions for Railway Structures, Concrete Structures, (Japanese) 10