Definition of fragility curves through nonlinear static analyses: procedure and application to a mixed masonry-RC building stock

Abstract

Seismic risk analyses at large scale represents a fundamental support to effective mitigation policies. Evaluating fragility curves able to capture the huge variety of existing buildings is one key point of this analysis. Within this context, this paper proposes a procedure for the evaluation of the fragility curves that aims to limit the computational effort without losing the reliability of the achieved results. This is reached through the execution of a limited number of nonlinear static procedures based on the use of the sensitivity analysis carried out according to the simplified star design with central point approach. The main strength of the procedure is the ability to explicitly quantify the various contributions of uncertainty to the dispersion, associated to those on the structural capacity (taking into account both aleatory and epistemic sources) and on the seismic input. As known, the adoption of a nonlinear static approach for the seismic assessment implies various assumptions, such as the load pattern applied, the criteria adopted to compare the capacity and the demand, and the definition of the damage levels. All these issues potentially affect the reliability of the final fragility curves, which are defined through a proper combination of such various options or they can be selected based on the ones more representative of the expected behaviour of the class. To improve this aspect, the evidences from nonlinear dynamic analyses are used. The feasibility and effectiveness of the procedure is duly demonstrated in this paper through its application to a building stock typology, consisting of existing mixed masonry-reinforced concrete structures, representative of one of the largest portions of the existing residential buildings in Lisbon. The attention is focused only to the global in-plane behaviour by adopting as modelling approach the equivalent frame method, that has been proven particularly efficient and accurate enough in representing the nonlinear behaviour of the examined structures.

Appendix

In reference to Sect. 2.2, the procedure to compute $\beta_{c}$ is graphically illustrated as a cube (Fig. 18), by supposing the three factors, x₁, x₂ and x₃, each at two levels, are of interest. Using the notation, i.e. “+ 1” and “− 1” to represent the “max” and “min” levels of each factor (input variable), the six runs could be listed in 2 × 3 design in the tabular format shown in Fig. 18. Usually, this is called the design matrix, which collects the row vectors from all the “n” experiments. In detail, the procedure was carried out in the following way: the first run was performed by adopting the median values of all the factors considered, whereas each following run considered the “max” or “min” level of one factor (independent variable) of the rational range defined, keeping all other factors at median values, represented here as “0”. Thus, seven runs in total will be defined in this case and presented in the design matrix, i.e. the aforementioned matrix Z. Then, the value of the contribution $\beta_{c}$ can be derived. The method presented here may be generalized to the case of 2 × N sensitivity parametric design, that is a design with “N” factors, each at two levels. In the case presented, N was equal to 11. It should be mentioned that the response spectra $S_{a1,16} (S_{d} )$ and $S_{a1,84} (S_{d} )$ were defined and used for $\beta_{D}$, while the spectrum of $S_{a1,50} (S_{d} )$ was considered for $\beta_{c}$.

Fig. 18

Definition of the contribution $\beta_{C}$