A view of seismic robustness based on uncertainty

A simplified view of robustness and redundancy is presented for the seismic assessment and design of structures. It is argued that the topological simplicity of the seismic load, i.e., its highly correlated nature of application for practically every component in all but the ultra-long structures, means that simpler formulations can be devised compared to blast, wave or wind hazards. In that sense, robustness may be considered to express the influence of structural uncertainties on the seismic performance of the structure, in essence showing the available margin of safety subject to material variability. Quantitatively, a pertinent robustness/redundancy index is defined as the ratio of two different estimates of the mean annual frequency (MAF) of exceeding a limit-state of interest, such as global collapse: On the denominator lies the MAF estimate that incorporates all sources of variability while on the nominator is the estimate that neglects structural uncertainties. It is shown that a simple closed form solution is available that directly relates robustness to the dispersion of response due to model parameter uncertainty. As an example, a steel frame where only beams are allowed to yield is shown to be more robust compared to another version where columns become the sacrificial element. Structural robustness and redundancy are highly desirable structural properties that are understood to be related to structural safety, especially against collapse. Qualitatively, one may think of robustness as the property of the system that prevents isolated local failures from progressing unchecked to become extensive or even global. According to EN1990, robustness in design is enforced by mandating that “a structure should be designed and executed in such a way that it will not be damaged to an extent disproportionate to the original cause”. Quantitatively, this is a difficult concept to capture, as it pertains to the level and magnitude of loading, structural typology, member properties and even the consequences of failure (e.g. loss, time-to-repair etc.). Various robustness indices have been proposed at times, each carrying its own combination of factors that contribute to increased structural safety. In general, the focus has been on assessing either the consequences or the safety associated with a locally damaged structure. For example, Baker et al (2008), suggest separating the consequences (e.g. losses) due damage into (a) the direct losses of the damaged components and (b) the indirect ones due to cascading failures of other components or loss of functionality. Then the ratio of direct over the total losses defines a robustness index. Another approach sees the comparison of the global safety of the damaged structure versus the intact, where a wealth of different indices is available to supply quantification; see for example Sorensen (2011), Mondal and Tesfamariam (2013), Starossek and Haberland (2011) and references therein for a more comprehensive review. Redundancy is a concept quite similar to robustness. For example, Frangopol and Curley (1987) define it as the absence of critical components whose failure would cause global collapse. It is also generally taken to be associated to the presence of alternate load paths within the structure, and thus the presence of multiple elements to resist a given load. It has also received a fair share of attention, especially in seismic engineering, due to the inclusion of an explicit redundancy factor in US codes (ICBO 1997) that favors some structural configurations 12 International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2 over others, an issue that has received considerable criticism (Wen and Song, 2002, Liao et al 2007). In the end, as elaborately argued by Kanno and Ben-Heim (2011), redundancy and robustness can be thought as two faces of the same coin, conveying the ability of the system to transfer loads with safety in face of the uncertainty in the structure itself and its environment. 1. CONCEPTUAL DEVELOPMENT Existing approaches to determine robustness and redundancy need to be comprehensive, and thus often undeniably complex, mainly due to the need to examine the potential failure of many different components due to localized loads. In this particular case, though, the focus is on structures under seismic loads. This is a highly specific situation where topology is not necessarily important. It is quite different in comparison to, e.g., a lifeline network under seismic loads. There, having multiple alternate routes connecting a given node often confers an advantage, as each may experience different loading levels due to the spatial variability of the ground motion. It would also be different for the blast assessment of a building where the highly localized effects of an explosion become important. Then, having a large number of widely-spaced structural elements working in unison improves the survivability. Topology becomes important for such cases because the damaging loads have a very distinct spatial distribution. A single structure subject to an earthquake, instead, generally experiences the same (or a highly correlated) ground motion in all of its components (with the probable exception of long bridges). Based on the above, it can be reliably stipulated that for any single structure we can define a seismic robustness index that does not need to account for topology per se. Then, structural robustness becomes a simpler issue of quantifying how the safety margin of capacity versus demand is influenced by the uncertainty arising from the structure itself. In that sense, robustness is not far from the notion of a reliability index that pertains to the influence of epistemic and aleatory uncertainties inherent to the structure, rather than the method of analysis or the seismic hazard. Following this mode of thinking, a simple definition for a seismic robustness index is proposed to characterize the “resistance” of structural systems to uncertainty. Specifically, the uncertainty robustness index (URI) is defined as the ratio of two estimates of the mean annual frequency (MAF) of exceeding a limit-sate of interest. At the nominator is the MAF that includes only (aleatory) record-to-record randomness, λR, while the denominator is the MAF estimate that also incorporates (aleatory or epistemic) structural uncertainties, λRU: