Probabilistic damage control seismic design of bridges using structural reliability concept

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1 Probabilistic damage control seismic design of bridges using structural reliability concept A. Saini Doctoral Student, University of Nevada, Reno, NV, USA A. Vosooghi Bridge Engineer III, Ph.D., P.E., AECOM Transportation, Sacramento, CA, USA M. Saiid Saiidi Professor, Ph.D., P.E., University of Nevada, Reno, NV, USA ABSTRACT: The objective of this study was to develop a probabilistic damage control approach (PDCA) for seismic design of bridge columns with specified reliability level. This new approach was developed by incorporating damage index, structural reliability index, and earthquake intensities with different probability of exceedance. Damage index was defined as the ratio of plastic displacement demand to plastic displacement capacity. Different apparent damage states were defined, and a large database of large-scale column bents and bridges tested on shake tables was used to determine damage index associated with each damage state. To take into account the effect of scatter in data, distribution functions were applied to establish correlation between damage states and damage indices. Reliability analysis was performed to calculate reliability index for each damage state. For each bridge category, design value for damage index was proposed so that there is a uniform reliability index for a given damage state. Key words: bridge columns; damage index; damage states; earthquakes; return periods; reliability index. 1. INTRODUCTION Most existing bridge columns were designed based on deterministic approaches and does not take into account the non-deterministic features of response. Therefore, in order to get a reasonable response from the structure under seismic loading, it becomes necessary to analyze the structure using probabilistic approach by incorporating the reliability analysis, which takes into account the randomness of the involved parameters and consequently would be able to minimize the risk of failure of such structures. The concept of PDCA approach was proposed by Tourzani et al (2008), for designing and evaluating bridge column based on predefined or expected performance level. The performance based design was proposed by incorporating performance with hazard levels. The performance of bridge column was measured through application of damage index (DI). DI was defined as the ratio of plastic displacement demand to plastic displacement capacity. However the uncertainties in earthquake demands and bridge responses were not included in their study. The mathematical formulation of DI was presented by Equation 1. (1) Where, D d, D Y, and D u are the displacement demand, effective yield displacement, and ultimate displacement of the bent The objective of this study was to develop a probabilistic damage control approach (PDCA) for seismic design of bridge columns with specified reliability level. As part of a study funded by the California Department of Transportation (Caltrans), this innovative design methodology is developed by incorporating Damage Index (DI), structural reliability index (β), and different earthquake return periods. DI was correlated to each damage state (DS) of bridge columns. Furthermore, each DS and associated DI was assigned to each earthquake return period. Six apparent DS s defined in a previous study at University of Nevada Reno (Vosooghi and Saiidi 2010) was utilized and their correlation with DI was established. Vosooghi and Saiidi (2010) proposed the following six DS s for bridge columns: DS1 = flexure cracks, DS2 = minor spalling, DS3 = Extensive spalling, DS4 = visible lateral/longitudinal bars, DS5 = imminent core failure and DS6 = failure/fractured bars. The correlation between DI and DS was determined from a statistical analysis of measured data for 21 bridge column models subjected to seismic loads on shake table. To take into account the effect of scatter in data, distribution func- 1

2 tions were applied to establish correlation between DS s and DI s (Vosooghi and Saiidi 2012). These distribution functions serve as the resistance model in the reliability analysis. In general, reliability of a structure is measured in terms of reliability index, β. The choice of target reliability index for any structure is bit subjective. The AASHTO LRFD recommends target reliability index of 3.5 for gravity loads. Because gravity loads are deterministic and are always present on the structure. In this report it was decided to use β of 3 against failure for earthquake loading. Because earthquake loading is non-deterministic and not always present on a structure, therefore, the choice of β equal to 3 against failure was considered reasonable. Moreover, high β greater than 3 for earthquake loading involve high construction cost which may not be acceptable. To conduct reliability analysis, the bents were designed for DS3 under design earthquake with return period of 1000 years. Initially to design column bents, average DI of 0.35, which corresponds to 50% probability of exceeding DS3 was considered to obtain a target reliability index of 3 against bent failure. To perform reliability analysis First Order Second Moment reliability method (FOSM) was utilized for calculating reliability index. 2. RELIABILITY ANALYSIS CONCEPT 2.1 Limit state function The concept of limit state is generally used to define a failure in the context of structural reliability analyses. A limit state is the boundary between the desired and undesired response of a structure. The limit state in reliability analysis is composed of two components; resistance component and loading component. Limit state function can be used to define structural performance and generally expressed in terms of mathematical equation. In general, let Q be the load parameter in a structural member and R be the resistance parameter in the structural member, then, the limit state function g can be expressed as: (2) If the load effect exceeds the resistance, then the component will fail and the probability of failure, P F, can be represented by ( ) (3) Both resistance and load effect are functions of multiple parameters. These parameters vary randomly. Because both the magnitude of the load effect and the resistance are subjected to statistical variation, therefore, by knowing the statistical distribution of the load effect and resistance, the probability of failure can be computed. In the present study the reliability model based on DI was developed for bridge columns. Both R and Q are continuous random variables and each has a probability density function (PDF) as shown in Figure 1 (Nowak and Collins 2000). Furthermore R-Q is also a random variable with its own PDF. In general limit state function can be function of many variables (load components, resistance parameters, dimensions, material properties, analysis factors etc.), therefore, direct calculation of probability of failure may be very difficult. Due to these reasons it is convenient to measure structural safety in terms of reliability index, β. Figure 1. Probability distribution function of load, resistance, and safety margin (Nowak & Collins 2000). 2.2 Reliability Index definition and probability of failure Reliability index or safety index is the distance of the mean of limit state function from failure surface measured in standard deviations. The graphical representation of β (Cornell 1969) is shown in Figure 2. Figure 2. Graphical representation of reliability index and probability of failure (Cornell 1969). β is directly related to the probability of failure P f as (Nowak and Collins 2000): Or ( ) (4) ( ) (5) 2

3 P ( DI DS) Where Ф -1 is the inverse standard normal distribution function. Table 1 show the probability of failure corresponding to different reliability indices. AASHTO LRFD Code recommends target β of 3.5 for bridges under gravity loading and does not consider the affect of earthquake loading. Therefore, it is reasonable to consider a target reliability index of 3 against failure for bridges under seismic loading. Table 1. Reliability index and their probability of exceedance. Reliability index, β Probability of exceedance Reliability analysis method There are various procedures available for calculation of β. In this study First Order Second Moment method (FOSM) was utilized for calculating reliability indices. First order implies that this method considers only linear limit state function, while second moment refers to the fact that the first two moments of a random variable, the mean value and the standard deviation, are considered. If both R and Q are independent normally distributed random variables, then β can be calculated as (Ayyub and McCuen 2003): (6) Where μ R and μ Q are the means of R and Q, respectively, σ R and σ Q are the standard deviations of R and Q, respectively, and δ is the coefficient of variation. If R and Q are log-normally distributed random variables, then, β can be calculated as (Ayyub and McCuen 2003): ( ) [( )( )] Where, ; ; (7) 3. COLUMN BENTS RESISTANCE MODEL Damage index (DI) is the most important parameter in seismic design of bridge columns. DI is a measure of the lost/ reserve capacity in the column at certain damage state. Therefore, in this study, DI is focused on for investigation. The resistance or capacity component of bridge column mostly determined from material properties and dimensions. To consider resistance model, the fragility curves developed by Vosooghi and Saiidi (2010) for damage index corresponding to each damage state was utilized. It was found that both normal and lognormal distributions are valid for these fragility curves. The normal and lognormal distribution of resistance model was justified using Kolmogorov Smirnov test with 10% significance level (90% confidence limits). Fragility curves used for resistance or capacity model in reliability analysis are shown in Figure 3.. DS6 is corresponds to failure with DI equal to one, therefore, even though in some cases where DI was greater than one, it was considered equal to one and consequently, no scatter in the data was considered and no fragility curve was developed for DS6. Also for reliability analysis mean and standard deviation of DS6 for resistance model was considered one and zero, respectively. 100% 80% 60% 40% 20% 0% DS-1 DS-2 DS-3 DS DI Figure 3. Fragility curves of damage index for different damage states (Vosooghi & Saiidi 2010). 4. COLUMN BENTS LOAD MODEL DS-5 Demand damage index (DI d ) was considered as load component for reliability analysis. To develop a load model, non-linear dynamic analysis was performed on various bridge columns utilizing SAP In the present study, each bent was designed for a target DI of 0.35 for design earthquake of 1000 year return period. DI of 0.35 is corresponds to a 50% probability of exceedance of DS3. Xtract software and design spectrum were used to design bents. All bents were analyzed under 10 far-field and 15 near-field ground motions selected from PEER 3

4 Spectral Acceleration, Sa (g) strong ground motion data base. DI d was calculated for each bent for each ground motion. To consider uncertainties in the load model, various parameters are considered. These parameters are divided into two categories; bridge site class and column bent properties. 4.1 Bridge site class In the present study bridge site class was divided into two categories; site B/C and site D. In various codes (AASHTO and ASCE) these site classes are defined based on soil type and shear wave velocity (V S30 ). According to the definition given in AASH- TO 2010, site B, site C, and site D is correspond to the rock, soft rock, and stiff soil, respectively. Because site B and C both represents rock, therefore, for sake of simplicity they are lumped together as B/C. Caltrans ARS online (Caltrans 2012) and USGS de-aggregation beta website (USGS 2008 Interactive Deaggregations Beta) were utilized to determine design spectrum for site class B/C and D, respectively. For design spectrum shear wave velocity (V S30 ) of 760 m/s and 270 m/s was used for site B/C and site D, respectively. 760 m/s represents the median of V S30 of site class B and C, whereas, 270 m/s represents the average V S30 of site class D. The earthquake design spectrum used for site B/C and site D with 5% damping is shown in Figure Site B/C Site D Period, secs Figure 4. Design spectrum for site class B/C and D. 4.2 Selection and scaling of ground motions Un-scaled ground motions were selected from PEER strong ground motion database website (PEER 2011). The selection criteria of ground motions were based on the V S30, distance to the fault (R jb ), and scaling factors (SF). Ground motions selected for site B/C are based on 500 m/s< V S30 < 1500m/s. V S30 between 500 to 1500 m/s and 200 to 360 m/s were considered to select ground motions for site class B/D and D, respectively. R jb between 0 to 15 km and 15 to 30 km was selected for near-field and far-field ground motions, respectively. Ground motions are selected so that the scale factor calculated based on spectral acceleration associated with period of one second is not greater than near-field and 10 far-field ground motions were selected for each site class. 4.3 Column bent properties and analytical analysis The expected material properties as specified in SDC Caltrans 2010 were used to design column bents. Grade 60 steel was used for longitudinal and spiral reinforcement. The expected concrete compressive concrete strength and steel yield strength were used as 5 ksi and 68 ksi, respectively. To develop load model for reliability analysis, non-linear dynamic analysis was utilized. To capture the randomness in load parameter (DI d ), various column bent properties like, column height to diameter ratio (H/D), longitudinal steel ratio (ρ l ), column support conditions, and number of columns in each bent were used in the analyses. Circular column section of six feet diameter was used for both SCB s and TCB s. Square cap beam section of seven feet was used in all TCB s. Table 2 shows the column bent properties used in the analyses. Fiber plastic hinge model available in SAP 2000 was utilized for both single column and two column bents (SCB and TCB). Axial load index of 10% was used for both single column and two columns bent. Each bent was analyzed under 10 far-field and 15 nearfield ground motions. Ground motions scaled at the spectral acceleration associated with the fundamental period of the bent. The peak relative displacement was determined and consequently, DI d associated with peak displacement was calculated. To take into account the scatter in DI d normal and lognormal distribution functions were utilized as shown in Figure 6 and 7. These functions are load distribution functions in the reliability analysis. It was observed that DI d of single column bents follow the lognormal distribution while DI d of TCB s follow the normal distribution. 10% level of significance was used for both single column and two columns bent. Figure 6 show that, 18% of the calculated data is falling outside the limit curves. It was observed that, the normal fragility curves for TCB s are very sensitive to the number of failures (DI = 1). In TCB s six failures were observed which cause this data to fall outside the limit curves. Excluding these six data points, only 5% of the total data falls outside the limit curves. 4

5 P (DI d ) P (DI d ) Table 2. Analytical single column and two columns bent model. Bent Site Column H/D ρ l Column Yield Demand Ultimate DI configuration class end period displacement displacement displacement Conditions. % secs in in in in/in Single column bent B Cantilever Single column bent D Cantilever Single column bent B Fix-Fix Single column bent D Fix-Fix Two column bent B Fix-Pin Two column bent D Fix-Pin Two column bent B Fix-Fix Two column bent D Fix-Fix Single column bent D Cantilever Single column bent D Fix-Fix Two column bent D Fix-Pin Two column bent D Fix-Fix % 100% 80% 80% 60% 60% 40% 20% Fragility Curve (Lognormal) Limit Curves SAP Calculated Data 40% 20% Fragility Curve (Normal) Limit Curves SAP Calculated Data 0% DI Figure 5. Lognormal fragility curve of DI d for single column bents. 0% DI Figure 6. Normal fragility curve of DI d for two column bents. 5

6 Reliability index, β DS Reliability index, β DS 5 RELIABILITY INDEX CALCULATIONS AND RESULTS The probability of failure of bent is related to the return period or exceedance probability of the design earthquake. In the present paper all bents were designed for 1000 year return period year return period is corresponds to the 7% probability of exceednace in 75 years (AASHTO 2010). 75 years corresponds to the life of a structure. Therefore, it is important to determine the probability of bent failure considering the probability of earthquake exceedance during the life time of a structure. Considering annual earthquake events are independent, the probability of earthquake exceedance during lifetime of a structure can be calculated as (Yen 1970): ( ) (8) Where, t and T is the life of a structure and return period, respectively. The P EQ was calculated utilizing Equation 8 and is equal to The probability of bent failure (P BF ) and earthquake exceedance during the life time of a structure can be calculated using conditional probability as: ( ) ( ) (9) ( ) (10) To determine P BF P EQ, the bents were designed for target DI of 0.35 and analyzed using non-linear dynamic analysis by utilizing SAP 2000 as described in section 4.4 of this paper. The target β was calculated for SCB s and TCB s utilizing Equation 6 and 7, whichever was applicable. After determining β, P BF P EQ was back calculated utilizing Equation 10. β for SCB s and TCB s was calculated for DS3 using Equation 9. Based on the distribution of DI d, Equation 7 and 9 and 6 and 9 was utilized to determine β for single column and two columns bent, respectively. β was also calculated for DS4, DS5, and DS6 to check the reserve capacity of bents against these damage states. Figures 7 to 14 represent the reliability index for SCB s and TCB s under various longitudinal steel ratio, site class, support conditions, and H/D ratio. The cumulative reliability index of all SCB s and TCB s against failure (DS6) is 2.9 and 4.4, respectively, shown in Figure 15. Figure 15 shows that β for TCB s is higher compared to SCB s. There are various parameters which causing higher reliability index in TCB s such as, redundancy effect, higher yield displacement, and cap beam flexibility. These parameters cause the mean DI d for TCB s lower than SCB s and consequently, results in higher reliability. Because β for SCB s is 2.9, which is close to 3, therefore, the choice of design DI of 0.35 for SCB s is reasonable. Considering yield displacement and displacement demand is independent of volumetric spiral ratio for a given longitudinal steel ratio, design DI was back calculated utilizing Equation 6. Design DI for TCB s was calculated as 0.5 to obtain a reliability index of 3 against failure. 1% Long. Reinf. 2% Long. Reinf. 3% Long. Reinf. Single Col. Bents, Site B/C Single Col. Bents, Site D Figure 7. Reliability index for single column bents under different longitudinal steel ratio. Figure 8. Reliability index for single column bents for various site classes. 6

7 Reliability index, β DS Reliability index, β DS Reliability index, β DS Reliability index, β DS Reliability index, β DS Reliability index, β DS Single Col. Bents, Cantilever Single Col. Bents, Fix-Fix Single Col. Bents, H/D=5 Single Col. Bents, H/D=10 Figure 9. Reliability index for single column bents under different support conditions. Figure 10. Reliability index for single column bents under different column heights to diameter ratio (H/D) % Long. Reinf. 2% Long. Reinf. 3% Long. Reinf Two Col. Bents, Site B/C Two Col. Bents, Site D Figure 11. Reliability index for two columns bent under different longitudinal steel ratio. Figure 12. Reliability index for two columns bent for various site classes. Two Col. Bents, Cantilever Two Col. Bents, Fix-Fix Two Col. Bents, H/D=5 Two Col. Bents, H/D=10 Figure 13. Reliability index for single column bents under different support conditions. Figure 14. Reliability index for single column bents under various column heights to diameter ratio (H/D). 7

8 Reliability index, β DS Figure 15. Cumulative reliability index for single column and two columns bent. 6. CONCLUSION Single Col. Bents Two Col. Bents A new approach for seismic design of bridges is presented in this paper by incorporating damage index with earthquake return period to obtain a uniform reliability index against failure. The results presented in this study show that, the choice of design damage index of 0.35 for single column bents is reasonable and can be used for design purposes. Results also shows that reliability index against failure in twocolumn bents is much higher than reliability index against failure in single-column bents indicating a much higher safeguard against failure, when the columns are designed to be at DS-3 under 1000-year earthquake. Therefore, two column bents can be designed for higher probability of exceeding DS-3 without causing a concern for failure. The new approach presented in this paper provides sufficient means to design a bridge column to reach a given damage state with a specified reliability under an earthquake with a given return period. ACKNOWLEDGEMENTS The research presented herein was sponsored by California Department of Transportation (Caltrans) under Grant No. 65A0419. Mr. Abbas Tourzani, Dr. Mark Mahan, Dr. Amir Malek, and Mr. Sam Ataya, of Caltrans are thanked for their feedback and interest in different aspects of the project. Special thanks due to Mr. Peter Lee, the Caltrans Research Program manager, for his support and advice. Ayyub, B. & McCuen, R. H Probability, Statistics, and Reliability for Engineers and Scientists, CRC Press LLC, Boca Raton, Florida. Caltrans 2012, ARS Online, California Department of Transportation, Sacramento, CA. Caltrans Seismic Design Criteria (SDC), version 1.6, California Department of Transportation, Sacramento, CA. Cornell, C A. (1969) A Probability Based Structural Code, Journal of American Concrete Institute 66(12): Fernandez, B. & Salas, J.D Return Period and Risk of Hydrologic Events I: Mathematical Formulation, Journal of Hydrologic Engineering, October Massey, F. J The Kolmogorov-Smirnov Test for Goodness of Fit, Journal of the American Statistical Association, V. 46, No. 253, March, pp Naeim, F., Hagie, S., Alimoradi, A., and Miranda, E. (2005). Automated Post-Earthquake Damage Assessment and Safety Evaluation of Instrumented Buildings, Report No , John A. Martin & Associates, Inc., Los Angeles, CA Nowak, A. S. & Collins, K. R Reliability of Structures, McGraw-Hills Companies Nowak, A. S. & Grouni, H. N., 1988, "Serviceability Considerations for Guide ways and Bridges," Canadian Journal of Civil Engineering,Vol. 15, No. 4, August, pp Nowak, A. S. & Zhou, J. H., 1985, "Reliability Models for Bridge Analysis," Report No. UMCE85-9, University of Michigan, March. PEER 2011, Users Manual for the PEER Ground Motion Database Web Application, Beta Version, November, Tourzani,A.M, Malek,A.M, Ataya, S, and Mahan, M 2008, Probabilistic Damage Control Approach (PDCA) and Performance Based Design of Bridges, Sixth National Seismic Conference on Bridges and Highways: Seismic Technologies for Extreme Loads, Multidisciplinary Center for Earthquake Engineering Research. USGS NSHMP PSHA Interactive Deaggregation, Geologic Hazard Science Center. Vosooghi, A. & Saiidi, M "Seismic Damage States and Response Parameters for Bridge Columns," ACI Special Publications, SP271-02, Structural Concrete in Performance-Based Seismic Design of Bridges, V. 271, May 24, pp Vosooghi, A. & Saiidi, M "Experimental Fragility Curves for Seismic Response of Reinforced Concrete Bridge Columns," ACI Structural Journal, V. 109, No. 6, November-December, pp Yen, B. C Risks in Hydrologic Design of Engineering Projects, Journal. Hydrologic. Division, ASCE, 96(4), REFERENCES AASHTO, 2010, "AASHTO LRFD Bridge Design Specification," American Association of State Highway and Transportation Officials, 5 th Edition, Washington, D. C. 8