PSI - Issue 64

Edward Steeves et al. / Procedia Structural Integrity 64 (2024) 1975–1982 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

1978

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Risk-based: robustness index (Baker et al. 2008): = + (3) is the direct risk associated with a given damage scenario, and is the indirect risk associated with said damage (Baker et al. 2008). General requirements for robustness measures have been published, including properties such as expressiveness, simplicity, and calculability (Starossek and Haberland 2011). As in the case of Eq. (1), deterministic measures are often composed of simple comparisons between the performance of the structure in an intact versus damaged state. However, their simplicity and calculability can often compromise their expressiveness. Unlike deterministic, probabilistic and risk-based robustness measures, as in Eqs. (2) and (3), account for the aleatoric uncertainties associated with loads and resistances. Although arguably more expressive in this sense, these measures can be computationally onerous, thus limiting their calculability. Furthermore, risk-based measures do not align with the definition of structural robustness used in this research: in this work robustness is treated as an intrinsic structural property meaning its quantification is conditioned on a local damage, while Eq. (3), for example, accounts for the probability of exposure that could lead to damage within its formulation. The three measures defined above are bound between zero and one, allowing simple comparisons to be made with different damage states and even other structures, but there is a clear lack of distinction between robustness and redundancy. Moreover, none of the existing measures account for the ductility of the system when the collapse load is reached. In short, these measures do not explicitly account for the performance of all elements, the redundancy of the damaged state with respect to the intact performance, and system ductility. In response, Steeves and Oudah (2024) formulated a holistic structural robustness index and associated structural redundancy index specifically for truss bridges. The indices are deterministic, bound between zero and one, and provide a balance between simplicity, calculability, and expressiveness to maximize their utilitarian value for practicing structural engineers. They are simple since they use concepts familiar to structural engineers, and expressive because they account for the performance of all elements in a quantifiable manner. The indices are quantified following a user-friendly framework of analysis that allows for the incorporation of different material properties, connection capacities, and damage states (Steeves and Oudah 2024). The framework consists of developing a finite element (FE) model of the bridge, followed by an incremental nonlinear static pushover or pushdown analysis, depending on the direction of loading. Hinges with nonlinear material properties are assigned to all necessary elements within the model, where an element refers to a specific limit state of a structural member or connection. Damage is included by reducing the capacity of the hinges and/or reducing the stiffness of the members in the FE model (Steeves and Oudah 2024). A brief overview of the holistic structural robustness index and the associated structural redundancy index is provided in the following subsections, while the detailed formulations are included in Steeves and Oudah (2024). (4) The first term relates to the external loads, , of the damaged system with respect to the intact version, the second term accounts for utilization ratios, , for each element of the structure in a similar manner, and the third term accounts for system ductility, ; utilization ratios refer to the ratio of unfactored load to nominal resistance. Stemming from the definitions highlighted in the previous section, Eq. (4) measures the performance of the damaged system compared with the intact state by incorporating redundancy and system ductility in the formulation. 3.2. Holistic structural robustness index from Steeves and Oudah (2024) , =[ ∑ , == 1 −∑ , == 1 ∑ , | == 1 ] ∙ [ ∑ , == 1 −∑ , == 1 ∑ , | == 1 ] ∙ [ (( ),1)]