Updating the Seismic Design Process of Bridges Using Bridge Specific Fragility Analysis
Jazalyn Dukes, Jamie E. Padgett, and Reginald DesRoches
ABSTRACT
Bridge fragility analysis has come to be prevalent in research and practice as a tool used to determine the seismic risk of a bridge structure. Fragility analysis is used in many capacities, such as providing performance information of the bridge that can be used in retrofit decisions, emergency response planning, and seismic risk assessment of a transportation network. Noticeably, most of the applications of bridge fragility analysis are intended for analysis of an existing bridge or transportation network. This paper will introduce a bridgespecific fragility methodology intended for the application of fragility analysis into the seismic design process for new bridge designs. Bridge specific fragility analysis allows the current seismic design process to use a performancebased approach as opposed to just a prescriptive approach. The method introduced in this paper can be used during the design process to estimate the fragility of a bridge without extensive modeling or simulations. The method also allows the design engineer to modify certain design details of the bridge and compare the resulting fragility curves to determine the effect of design decisions on the performance of the bridge. This application of fragility analysis in the seismic design process could ultimately lead to safer as well as more efficient bridge designs. This paper will present this method in the context of a concrete box girder bridge class commonly found in California.
_____________
Jazalyn Dukes, Presenting and Corresponding Author, Graduate Student Researcher, Georgia Tech, 500 10^{th} St NW, Atlanta, GA 30332
Jamie E. Padgett, PhD, Assistant Professor, Rice University, 6100 Main St, MS318, Houston, Texas 77005
Reginald DesRoches, PhD, Karen and John Huff School Chair and Professor, Georgia Tech, 500 10^{th} St NW, Atlanta, GA 30332
INTRODUCTION
Fragility analysis has become useful in many areas of research and industry applications. Specifically, seismic fragility analysis of bridges has developed over the past years and has been used to provide support to engineers and owners in the areas of seismic risk assessment in a region, retrofit planning, emergency response planning, among other postevent applications (Mackie and Stojadinović 2005; Basoz and Mander 1999; Padgett and DesRoches 2008). However, fragility analysis has not been used extensively in the seismic design of bridges. Integrating fragility analysis into the design process can provide many benefits to design engineers, in that it could create a performance based method of design as compared to a prescriptive approach.
California is a state with a high seismic hazard and a history of damaging earthquakes. Because of this, California has been in the forefront of seismic design in the United States. A brief history of seismic design in California is including in this paper. Recently, as part of the need to quantify and mitigate earthquake damage, the California Department of Transportation (Caltrans) has embraced the use of bridge fragility models, particularly in postevent response programs (Lin and Wald 2008). As part of the authors’ project with Caltrans, fragility analysis is being explored in a possible capacity in the seismic design process.
The California Department of Transportation (Caltrans) seismic bridge design process specifies the design engineer to meet minimum requirements resulting in a bridge that should avoid collapse in the event of a Design Seismic Hazard (Caltrans 2010). By following the guidelines set forth by the SDC, a design engineer should produce a bridge design that meets that performance requirement. However, the procedure set forth in the SDC does not provide quantitative information on how a bridge will perform under the given hazard. Therefore, there is a need for a methodology that will provide performance information on bridge performance for the given hazard level, and also for demands other than the design hazard level. There is also a need for design engineers to have an understanding of the effects of variations in design parameters on the performance of a bridge.
In this paper, the authors will present a tool developed for the California Department of Transportation that allows design engineers to produce fragility curves specific to their bridge design. This tool will be used as a design check for compliance with seismic design criteria. This tool is useful in providing the engineer with a probabilistic fragility analysis of the performance of their bridge in the event of an earthquake with a specific hazard. It can be used to provide more efficient bridge designs in that it allows the engineer to compare the effects of changing certain design aspects of the bridge on the fragility and performance of the bridge. It also gives the engineer insight into how the responses of different components of the bridge contribute to the overall fragility. The tool and method is presented here for a bridge type common in California, the single frame integral concrete box girder bridge. An example of the use of this tool as a final design check in the context of the seismic design code in California is given.
SEISMIC BRIDGE DESIGN IN CALIFORNIA
Early Seismic Code Provisions for Bridge Design in Caltrans
Seismic design in the US has evolved significantly over the past 100 years, with observably a lot of the innovation in design coming after large earthquake events. In the United States, seismic design codes started to form after the 1906 San Francisco earthquake (FEMA, 2006). The first seismic design provision in California for bridges was developed in 1940. The design criteria stated that bridges should be designed for a seismic force placed horizontally at the center of mass in any direction. The force was a percentage of the dead load which was determined by the design engineer. In 1943, the design criteria became more specific. It stated that the seismic force applied to the center of gravity of the weight of the structure should be between 2% and 6% of the dead load of the structure, depending on the type of foundation. In 1965, the criteria incorporated more characteristics of the bridge into the calculation of the seismic force. The seismic force equaled two coefficients, K and C, modifying the dead load of the structure. The coefficient K represented the energy absorption of the structure, and is determined based on the bent system (wall, versus single and multicolumn piers). The coefficient C represented the structure’s stiffness, and is based on the natural period of vibration. The minimum force was 2% of the dead load of the structure, and the engineer was instructed to give special consideration to structures founded on soft soils, and structures with massive piers (Moehle, et al. 1995).
The 1971 San Fernando earthquake prompted major changes in the seismic bridge design code. For bridges in construction, lateral design forces were increased by a factor of 2 or 2.5. Design for new bridges then had to account for many new factors, including fault proximity, site conditions, dynamic response and ductile design for reinforced concrete structures. These changes were included in the 1974 seismic code for Caltrans (Sahs, et al. 2008). Practice in design continued to evolve to improve the reinforcement details of columns and to design for plastic shear in the column. From this era, the criteria for design provided more details for the proportions of the components that would lead to ductile response in the columns, and elastic response in other parts of the bridge (Moehle, et al. 1995). After the 1989 Loma Prieta earthquake, Caltrans asked the Applied Technology Council (ATC) to review and revise their design criteria (Sahs, et al. 2008).
Once the ATC completed its ATC32 report for Caltrans, Caltrans incorporated nearly all of the recommendations made therein into its design code in 1996 after the 1994 Northridge earthquake. Figure shows how the seismic design spectra have changed throughout the years based on the code provisions (Moehle, et al. 1995). The new recommendations included a capacity design approach to ensure flexural failure in the column, which would be made possible by carefully designing the joints, column geometry, footing connection, among other things (Sahs, et al. 2008).
Figure : Caltrans design spectrum for a certain type of bridge (Moehle, et al. 1995).
Current Seismic Design Practice in CT SDC
The current seismic design code available for bridges is the Caltrans Seismic Design Criteria (SDC) version 1.6 released in 2010 (Caltrans 2010). The SDC specifies the minimum requirements for seismic design of bridges that correspond to the performance goals for ordinary bridges. This document goes through the requirements for determining the demands and capacities of structural components, compares the demand versus capacity, lists appropriate analysis methods of the structure, specifies how to assess the seismicity of a site and the foundation performance, and details specifying design requirements to be met. Of particular interest to this project is the section dedicated to the design of the bridge. It has the requirements for frame design, superstructure, bent caps, joint design, bearings, columns and pier walls, foundations and abutments. The first requirements are that the frame is balanced in terms of stiffness, mass and geometry. The SDC gives recommendations to follow to ensure a balanced frame, which is intended to increase the chance of the structure responding in the fundamental mode of vibration. Ensuring this type of response is intended to reduce the chance of producing a nonlinear response. Balancing the fundamental periods between frames is also meant to reduce the relative displacements due to outofphase movements (Caltrans 2010)
The Caltrans SDC lists many design specifications to ensure proper performance of all bridge components. The minimum seat width of the deck at abutments is 24 inches to prevent unseating of the deck. For bent caps, a section describes requirements for integral and nonintegral bent caps. A section for superstructure joint design gives equations to ensure proper performance and proportioning of joints, and different requirements for tjoints and knee joints, as well as proper detailing for bent caps and joints. For the design of columns, the SDC specifies the analytical plastic hinge length for different column types. Detailing for column flares mainly states that care should be taken to avoid a flare design that would increase the seismic shear demand on the column. Other components addressed are bearings, foundation and pile performance, and abutment design.
As was demonstrated, current seismic design process gives many directives and requirements for the capacity of many bridge components. Bridges in California designed today not only have to meet general bridge design requirements, but also have to be designed to withstand an expected earthquake load. The following section describes the steps that need to be taken to ensure a proper seismic design of a new bridge. The steps detailed in this paper are used by Caltrans design engineers to check the design of each bridge and ensure compliance with the SDC (Setberg, 2011). Each design check should be considered during the design process and after the design is complete.
Design Checks from the SDC for a New Bridge Design
This section will illustrate the design checks that are derived from requirements in the SDC, and used by Caltrans design engineers to ensure a proper seismic design (Setberg 2011). As was mentioned, the current seismic design process is very thorough in terms of setting requirements for checking the capacity and demand of many bridge components. Each design check should be considered during the design process and afterwards. The design checks mostly deal with the relative stiffness of the structure, ductility of columns, and the structure displacement demand.
The following describes the process of checking the bridge design for compliance with the Caltrans the SDC. The first design check is that of balanced stiffness. This check requires that the stiffness of the bents, k_{i}, of the bridge system are close enough so there would not be an issue of unbalanced responses due to an earthquake load, which could lead to complex nonlinear response, increased damage in the stiffer bents, and possible column torsional response (Caltrans 2010). The design check specifies that adjacent columns within a bent need to have stiffness within 75% of the other, shown in Eqn. (1). More details on balanced stiffness requirements are found in Section 7.1.1 of the SDC.
(1)
The second design check is that of the local member ductility capacity, μ_{c}, quantitatively shown in Equation (2). This design check is referenced in the SDC in section 3.1.4.1 as the Minimum Local Displacement Ductility Capacity for each ductile member in the bridge. This requirement is to ensure adequate rotational capacity in the plastic hinge regions of the ductile member (Caltrans 2010). The ductility capacity has to be greater than 3. Eqn (2) shows the parameters that must be calculated from the properties of the bridge, including the yield displacement of column at the formation of plastic hinge (Δ_{y}), and the displacement capacity (Δ_{c}).
(2)
The next design check is that of the displacement ductility demand, μ_{d}, shown in Equation (3). This requirement is described in Section 2.2.4 of the SDC. The displacement ductility demand describes the postelastic bending of the ductile member (Caltrans 2010), and is defined as the ratio of the displacement demand (Δ_{d}) and the yield displacement.
(3)
The next design check is that of the global displacement criteria, shown in Equation (4), where the demand should be determined to be less than the capacity of the system. The global displacement criteria is given in Section 4.1.1 of the SDC, listed as a Performance Criteria for the bridge design. The SDC mentions that care should be taken to compare the two values, the global displacement demand and the displacement capacity, as calculated along the same local axis to ensure the proper comparison.
(4)
The final design check is that of the loaddisplacement, or PΔ, effect. This requirement is found in the SDC in Section 4.2. This design check is to determine if the lateral displacements caused by the axial load on the ductile member, or column, can be ignored, along with further nonlinear analysis (Caltrans 2010). To check this requirement, Equation (5) is used, where the dead load on the column (P_{dl}), the displacement demand (Δ_{dg}), and the plastic moment (M_{p}) is used in this design check.
(5)
This section described the past and present bridge seismic design process in California along with important design checks to be employed during the design of the bridge. This process, however, does not provide the designer with critical information about specific performance of the bridge at the design seismic hazard. It does not account for the uncertainty inherent in the capacity of the structure against collapse for a design event. Neither does the process produce the effects on that performance given a change in any design detail. Fragility analysis determines the probability of a structure or system experiencing a seismic demand exceeding the structural capacity defined by a limit state (Hwang, Liu, & Chiu, 2001). Fragility curves graphically show the performance of a bridge or bridge component at different ground shaking levels and at different damage levels. Thus, fragility analysis and fragility curves can be used to fill the gap of quantitative performance evaluation in the seismic design process. Forthcoming sections will describe how fragility can be used in the design process that will enable performancebased design decisions.
BRIDGE SPECIFIC FRAGILITY METHOD
Bridge seismic fragility curves are statistical functions that give the probability of exceeding a certain damage level or damage state as a function of a ground motion intensity measure. The fragility function can be written as P[DS_{i}  IM=y], where IM=y stands for a ground motion intensity measure taking a particular value, and DS_{i} is the exceedance of the damage state in question. Today, more research has been done to create additional fragility methodologies and analysis types and to analyze many different structures, including bridges. Applications of fragility curves include aiding in emergency response optimization, design support for performancebased engineering, planning support for seismic events, and policy support (Mackie and Stojadinović 2005; Basoz and Mander 1999; Padgett and DesRoches 2008).
To develop fragility analysis for applicability to specific bridge designs in consideration, a fragility method was needed that could produce fragility curves that are specific to the design bridge, as opposed to fragility curves developed for a general class of bridges. The fragility methodology highlighted in this paper is a type of analytical fragility process. Analytical bridge models were analyzed with time history analyses using the PEER ground motion suite (Baker, et al. 2011), a suite of ground motions that is applicable to a range of sites and structural properties. Once the analyses were done on the bridge models using the suite of ground motions, the responses of each bridge component were collected, and analysis on that data was performed. This is the general framework of analytical fragility analysis, however, the method presented here differs in several ways from past methods.
There are certain components of the bridgespecific fragility method (BSFM) developed by the authors that differentiate from other fragility methods that develop curves for bridge classes (Nielson 2005; Ramanathan 2012). One of the components is the set of design parameters of the bridge design that affect the response of the bridge. These design parameters are used in the fragility method to define the demand model and also in the fragility definition. The design parameters used in this paper are the following: longitudinal steel ratio in the column, transverse steel ratio of the column, span length to column height ratio, superstructure depth to column dimension ratio, and column height to column dimension ratio. More information about these design parameters can be found in the paper by Dukes, et. al (2012).
The next component of the BSFM is the multiparameter probabilistic seismic demand model (PSDM). Cornell, et. al (2002) proposed a relationship that involved the ground motion characteristic used to predict the component response. For the BSFM, the authors expanded on this commonly used relationship and a parameterized approach proposed by Ghosh, et. al (2012) to create a multiparameter PSDM, as shown in Eqn. (6), where DP_{i} is the design parameter, Y is the component response, IM is the ground motion intensity measure, and the β_{i}‘s are the regression coefficients of the demand model. This PSDM includes the design parameters mentioned earlier and allows the creation of bridgespecific fragility curves.
(6)
Finally, logistic regression is used to convolve the demand and capacity models. Many techniques in past research have been used to create the fragility estimation, such as Monte Carlo simulations and twoparameter lognormal distributions (Nielson and DesRoches 2007; Mackie and Stojadinovic 2007; Shinozuka, Banerjee and Kim 2007). Logistic regression analysis was used here to convolve the demand and capacity models in order to allow the design parameters to be used in the estimation of the fragility; thus, the probability of failure can be conditioned on the ground motion characteristic as well as the design parameters. This creates the bridgespecific fragility estimation. The logistic regression equation used in the method is shown in Eqn. (7), where the coefficients α_{i}’s are the results of the logistic regression analysis. Analysis is first done at the component level. The bridge system level fragility estimates are determined using a series approach, where the bridge is said to fail if at least one component fails. Logistic regression analysis is then performed to find the bridge system fragility.
(7)
BRIDGE SPECIFIC DESIGN SUPPORT TOOL
One of the main outcomes of the Caltrans research project, for which this research was conducted, is the implementation of a design support tool that utilizes bridgespecific fragility analysis described earlier. This tool is meant to be used by design engineers as a final design check and an exploration on the effects of design parameters on the response of the bridge. With this tool, an engineer can determine if their design meets the criteria set by the design code and other criteria established for their particular project based on analytical fragility analysis. The design support tool can be used as the final design check to determine the performance of the bridge, its compliance with the requirements of the SDC, and the performance of the bridge in other limit states.
Figure 2 shows a snapshot of the input section of the design support tool which was developed in an Excel spreadsheet. As is shown, the design engineer inputs certain aspects of the design into appropriate cells, such as longitudinal steel ratio and other geometric properties of the bridge. These design aspects correspond to the design parameters mentioned earlier. The output of the tool includes fragility curves and specific fragility points at specified hazard levels. System fragility curves as well as individual component fragility curves can be developed within the tool based on bridgespecific fragility analysis. Specific probabilities of failure can be displayed for any value of the ground motion intensity level specified by the designer. The user of the tool can also enter upper and lower bound values for each design parameter in order to produce fragility curves based on the original design parameters and on the upper and lower bounds of the design parameters. This allows the user to see the effect of design decisions on the performance of the bridge. The bridge specific design support tool is presented in a Microsoft Macroenabled Excel worksheet. The spreadsheet utilizes Visual Basic Macros in order to produce the fragility curves. The design tool includes hidden and protected sheets in which the logistic regression equations for the fragility curves are placed in order to ensure the integrity of the analyses. The previous processes described earlier, generation of the PSDM and the logistic regression to obtain fragility information, were completed and verified before incorporating the results into the design tool. Separate worksheets would be provided for different ground motion intensity measures, such as peak ground acceleration and spectral acceleration at one second, when available. The user only needs to input the design parameters into the sheets and click the appropriate buttons to generate the fragility curves of choice.
EXAMPLE OF APPLICATION OF FRAGILITY ANALYSIS IN SEISMIC DESIGN PROCESS
As described earlier, the current seismic design process in Caltrans is a prescriptive approach designed to ensure a nocollapse state after a Design Seismic Hazard event. The procedure set forth in the SDC does not provide for any details on the anticipated performance of the bridge or its components at the Design Seismic Hazard or any other hazard level. The bridge specific design tool developed for this project does not aim to supersede the SDC, but to supplement it as a design check. After the bridge design is finalized using the SDC, and the bridge design passes design checks described earlier, then this tool can be used to check that the design meets the criteria of nocollapse for a Design Earthquake set forth in the SDC. The design tool can also be used by a designer to quantify the effect of modifying aspects of the design on the response of the bridge. With the design tool, a design engineer can input key design aspects of their bridge, and a fragility curve will be instantly created that is specific to that bridge.
Figure . Snapshot of the input section of the design support tool.
TABLE : GEOMETRIC PROPERTIES OF EXAMPLE BRIDGE.
GEOMETRIC PROPERTY

UNITS

VALUE

Span Length

(ft)

122

Column Height

(ft)

24.6

Deck Depth

(in)

59.0

Column Diameter

(in)

60

Longitudinal Steel Ratio

(%)

1.17

Transverse Steel Ratio

(%)

1.02

The following is an example detailing the use of the tool within the context of the seismic design process as a design check after the design of a bridge is complete. The bridge analyzed here is a bridge type common in California: a single column 2 span concrete box girder bridge. Table I shows the geometric properties of the example bridge that will be analyzed. These properties were randomly generated based on the distribution of the parameters determined during a survey of California bridge plans. The bridge was checked with the SDC design checks to ensure compliance with the code. For the sake of brevity, those results are not included here.
The design parameter values were then put in the design support tool to determine the fragility of the bridge. The results of the fragility estimation are included in Table II, and bridge system level fragility curves are shown in Figure 3. The bridge was analyzed at a design hazard level of 0.3g peak ground acceleration (PGA). The system level and the component level fragility estimates are included in Table II. As is shown, the probability of exceeding the threshold for the collapse limit state (LS3) is 1.1%. Depending on the performance requirements of the particular design, this probability may be acceptable. With this information, the design engineer can get performance information of their bridge design and quantify the risk of meeting the collapse state of the bridge, as well as learn the performance of the bridge at other damage states. With this tool, it would also be possible to determine the effects of changes in the design parameters in the performance of the bridge design.
Figure . Bridge system level fragility curves for design example bridge.
TABLE : FRAGILITY ESTIMATES FOR BRIDGE AND COMPONENTS AT 0.3g PGA.
SPECIFIC FRAGILITY ESTIMATES:

LS0

LS1

LS2

LS3

System

100.0%

81.3%

8.6%

1.1%

Column Component

100.0%

76.6%

8.6%

1.1%

Gap at Abutment Component

97.6%

11.7%

0.0%

0.0%

Bearing Component

97.6%

3.0%

0.0%

0.0%

Joint Seal Component

97.8%

0.0%

0.0%

0.0%

CONCLUSION
The seismic design process of bridges in California has performance criteria for a new bridge design specifying that the bridge should not collapse during a Design Seismic Hazard event. While the design code gives many detailing specifications and analysis requirements, it does not provide the designer a way to quantitatively determine if the design passes the nocollapse performance criteria. Here, fragility analysis is introduced as a complement to the seismic design process to give the user performance information on their bridge design. As a design check, the use of the fragility analysis method proposed in this paper can help the user determine if performance criteria have been met, and also give the user information on potential uncertainty of the performance of the design.
The bridgespecific fragility method is a fragility analysis approach that allows for tailoring fragility curves to a specific bridge design without having to complete extensive computer simulations and analysis. This method was developed to apply fragility analysis into the seismic design process of bridges in California. The key components of bridgespecific fragility analysis that differentiate it from other fragility approaches that produce fragility curves for bridge classes include the inclusion of design parameters in the fragility estimation, a multiparameter PSDM, and logistic regression to estimate the fragility.
The vehicle by which this method is presented to the design engineer is the design support tool. The design support tool houses the results of the fragility method in order to facilitate the use of bridgespecific fragility in the design process. This paper highlighted the SDC and the design checks that design engineers use after the design to ensure a proper design. The purpose of using the design support tool as a design check after a bridge design was to add a performancebased design approach to the seismic design process described by the SDC. The tool gives the engineer performance and behavior information of the bridge design in order to ensure that the design meets the design criteria of nocollapse at the Design Seismic Hazard level, as well as other hazard levels. The tool can also be used to help the user make more informed design decisions by allowing the user to determine the effects of the design parameters values on the performance of the bridge. This could lead to safer as well as more efficient designs.
ACKNOWLEDGEMENTS
This project was funded by the California Department of Transportation. It involved an extensive investigation into improving fragility relationships of California bridges as well as developing ways to integrate fragility analysis into the design process. The CoPrincipal Investigators of this project were Dr. Reginald DesRoches of Georgia Tech and Dr. Jamie Padgett of Rice University.
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