Component Capacity or Limit State Models

Seismic fragility involves the convolution of the demand and capacity models. The formulation of the demand models was explained in the previous section. Definition of the component capacities or limit states is not a trivial task and is a crucial step in the fragility formulation. The individual limit states are characterized by representative values for the median, S , and dispersion, β , (see equation (5.1)) for the component

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damage states distributions which are also assumed to be lognormal akin to the PSDMs. Discrete damage states are defined for each component corresponding to significant changes in its response and consequences to its own performance and performance of the bridge at the global or system level. Although the damage state definitions are discrete, the assumption is that a continuous range of damage exists between the discrete states to enable the closed-form computation of the component fragility curves. It is essential that the limit state definitions use the same metric as the EDP for the respective bridge components. Table 5.3 listed the bridge component EDPs that are used to monitor the response of specific components and assess their performance.

A significant contribution in the present study is that the damage state definitions for the components are derived in such a way that they align with the Caltrans design and operational experience. This will facilitate the evaluation of repair-related decision variables, repair cost and repair time, which are the end products in a typical risk assessment procedure. The major challenge lies in being able to group components that have similar consequences at the system level in terms of functionality and repair consequences. A common question that could arise is: “Do the complete collapse of columns have the same effect on bridge functionality as the complete damage to a shear key or tearing of an elastomeric bearing pad?” In order to be able to address the aforementioned concerns, two classes of components are proposed viz., primary and secondary. Primary components are defined as those that affect the vertical stability and load carrying capacity of the bridge. Extensive or complete damage to these components might lead to closure of the bridge. Columns and abutment seat belong to this category with regards to the bridge classes considered in this research. When looking at bridges with in-span hinges, which is out of the scope of the present study, the internal hinge is also considered as a primary components as excessive hinge opening (values exceeding the support seat length) could lead to unseating of the superstructure.

Secondary components may be defined as the ones that do not affect the vertical stability of the bridge. Failure of these components will not force closure of the bridge but might lead to restrictions on the travel speed and traffic conditions on the bridge. Table 5.6 lists the primary and secondary components in the bridge classes considered in this study for both diaphragm and seat abutments.

Table 5.6: List of primary and secondary components in the bridge classes considered in this

study

Seat Abutments Diaphragm Abutments

Primary components

Columns Columns

Abutment seat

Secondary Components

Joint seal Maximum deck displacement

Elastomeric bearing pads Bent foundation translation

Restrainers Bent foundation rotation

Maximum deck displacement Abutment passive displacement Bent foundation translation Abutment active displacement Bent foundation rotation Abutment transverse displacement Abutment passive displacement Joint seal*

Abutment active displacement Elastomeric bearing pads* Abutment transverse displacement Restrainers*

Shear key displacement Shear key displacement*

*These components are only present in the case of MSCC-IG bridges with diaphragm abutments

Tables 5.7 and 5.8 show the general description of the bridge system level damage states (BSST) and the component damage thresholds (CDT) for primary components, respectively. The bridge system level damage state descriptions, BSST-0 through BSST-3 are defined in Table 5.7 and are aimed at operational consequences in the aftermath of an earthquake. The CDT of primary components map directly to the BSSTs since the loss of a primary component affects the load carrying capacity and overall stability of the bridge system. In the case of secondary components, only two broad CDTs are defined, CDT-0 and CDT-1 and these map directly into BSST-0 and BSST-1, respectively. The damage state descriptions for CDT-0 and CDT-1 in the case of

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Damage Thresholds (CDT) of primary and secondary components, detailed in Table 5.7, are aimed at achieving similar consequences in terms of bridge operations (repair and traffic implications) in the aftermath of an earthquake. As described in Table 5.7, the primary components: columns and abutment seat (the latter only in the case of seat abutments) directly map into the BSSTs and equally contribute to the vulnerability across all damage states. On the other hand, the secondary components (detailed in Table 5.6) map into BSST-0 and BSST-1 since there complete failure will not have a similar consequence as that of the primary components. Both these tables are developed in close collaboration with Caltrans (Caltrans, 2010-2012) to ensure that the component mapping is in alignment with the inspection/maintenance closure decisions and the training guides for post-earthquake inspections (Sahs et al., 2008). The CDTs may be broadly defined as below:

• CDT–0 (Aesthetic damage) is a performance parameter threshold beyond which aesthetic damage of the component occurs. The associated repair is primarily aimed at restoring the aesthetics

• CDT–1 (Repairable minor functional damage) is a performance parameter threshold beyond which significant repairs are required to restore component functionality

• CDT–2 (Repairable major functional damage) is a performance parameter threshold beyond which extensive repairs are required to restore component functionality

• CDT–3 (Component replacement) is a performance parameter threshold beyond which component replacement is likely to be the most cost-effective means to restore component functionality

The CDT values can be described using a prescriptive (physics-based) approach, descriptive (judgmental-based) approach or by incorporating both (Padgett et al., 2007) using Bayesian updating principles. The prescriptive approach is based on the mechanics of the problem where a functional level is associated with component damage such as

spalling of cover concrete in a column, buckling or rupture of the longitudinal column reinforcement etc. The descriptive approach is based on the functionality level of the components post disaster and is usually in terms of repair cost and downtime. In this study a combination of both techniques are used to define the threshold value.

Having broadly defined the CDTs for various components, the threshold values are determined based on experimental studies from the literature and based on extensive input from the Caltrans design and bridge maintenance groups. The subsequent sections provide these median values, SC, for the CDTs along with visible signs of associated

damage and repair strategies. As mentioned before, the capacity distributions are assumed to be lognormal similar to the demand distributions. The uncertainty associated with the median values of the CDTs is prescribed in the form of a logarithmic standard deviation or dispersion, βC. The assignment of dispersion is done in a subjective manner

due to lack of enough information to quantify it and a dispersion value of 0.35 is adopted across the components and the respective damage states. This value is particularly a good estimate for columns and is consistent with the test results documented in the PEER column structural performance database (Berry and Eberhard, 2004).

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Table 5.7: General description of bridge system level damage states along with component damage thresholds Bridge system damage states BSST-0

MINOR