Cloud analysis considering global dynamic instability

Cloud Analysis is based on simple regression in the logarithmic space of structural response versus seismic intensity for a set of registered records. This method is particularly efficient since it involves non-linear analysis of the structure subjected to a set of un-scaled ground motion time-histories. The simplicity of its underlying formulation makes it a quick and efficient analysis procedure for fragility assessment and/or performance-based safety-checking. Nevertheless, the Cloud Analysis has some limitations; such as, the assumption of a constant conditional standard deviation for probability distribution of the structural response given intensity. Arguably, with the increasing levels of intensity, the conditional dispersion in displacement-based response parameters given intensity may increase. Another complication arises when the structure becomes dynamically unstable (or when the analysis software encounters non-convergence problems, or the Collapse of the building due to large demands takes place) by subjecting to some of the ground motion records used for Cloud Analysis. In such cases, the assumption that structural response given intensity is described by a Log Normal probability distribution with constant standard deviation (one of the underlying assumptions of the Cloud Analysis) no longer holds. However, the method can still be applied to the portion of Cloud response that does not include cases of dynamic instability. Thus, the probability of exceeding a specific structural response value given ground motion intensity can be expressed, using the Total Probability Theorem, as the sum of two probability terms. These terms correspond to the two mutually exclusive and collectively exhaustive portions of the Cloud response; namely, without cases of dynamic instability and with cases of dynamic instability. In such formulation, the probability of dynamic instability given seismic intensity can be calculated by using a generalized regression model (e.g., logistic regression). The transverse frame of a seven-story existing building in Van Nuys, CA, which is modeled in OpenSees considering the flexural-shear-axial interactions, is employed in order to demonstrate this procedure. The critical demand to capacity ratio, corresponding to the component or mechanism that leads the structure closest to the onset of limit state (e.g., near collapse), is adopted as the structural response parameter. This structural response parameter, that is equal to unity at the onset of limit state, can encompass both ductile and fragile failure mechanisms. Moreover, it can register a possible shift in the governing failure mechanism with increasing intensity. The results for probabilistic demand assessment based on the Cloud Analysis considering the cases of dynamic instability are compared with those obtained based on non-linear dynamic analysis methods such as Incremental Dynamic Analysis and Multiple Stripe Analysis. It is demonstrated that, with a careful selection of ground motion records, this method leads to reasonably accurate results with considerably reduced analysis effort.