Fragility analysis of reinforced concrete structures using a response surface approach

Student: M.IJ. Schotanus
Supervisor: Prof. P.E. Pinto



A general and sophisticated method for seismic fragility analysis of systems, originally proposed by Veneziano et al. (1983), is further developed in this work and applied to a reinforced concrete frame. The method is not limited to a specific class of structures or systems, or to a particular representation of the random input. Sophistication lies in the incorporation of state-of-the-art mechanical models; use of realistic models for the seismic action; assignment of probabilistic models for mechanical parameters and capacity; and in accounting for multiple failure modes and their correlation.

The basic proposal is to use a response surface to represent the capacity part in an analytical limit state function (g-function) as input for SORM analysis. Response surface techniques are used to replace the algorithmic g-function, or the capacity part of it, with an explicit functional relationship, fitting a second order polynomial. Such an explicit format highly reduces the number of expensive numerical analyses needed compared to classical methods that determine the failure domain. In order to keep the method competitive, only a small number (with a maximum of about 6) of parameters are chosen to enter in the function as explicit variables, whose effect is denominated fixed. The response surface methodology is then used as an iterative and evolutive procedure, to help distinguishing the important variables from the less important and to support the choices made. Those random variables that are not explicitly incorporated are accumulated in an error term together with errors originating from the lack-of-fit of the model. This error term is itself random and transfers the uncertainty to the output quantity. More specifically, the effect of a large number of implicit variables can be grouped together in a few additive random variables (random effects), improving the model's descriptive power at a moderate additional cost. The effect of earthquake loading on the response and the spatial fluctuation of the mechanical parameters are treated in this way.

The unknown model parameters (fixed effect coefficients and distribution of random effects) are then statistically estimated from numerical analyses, carefully planned in the experimental design. After the model is fitted, a SORM procedure is used to obtain the fragility curve for the system. Further statistical analysis gives insight in the accuracy of the results obtained.

The developed theory is then applied to a three bay, six-storey reinforced concrete frame, representing a system that is considered sufficiently complex to serve as a challenging test case. First only one random effect variable is introduced, describing dependence of the response on the earthquake, then a second one, introducing spatial variability of the concrete ultimate strain. In depth investigations are carried out to test the stability and accuracy of the suggested procedure, consisting of: treating the model coefficients both as fixed and as random parameters in the limit state function; testing the contribution to the accuracy of the model of the basic input variables chosen; observing the sensitivity of the results to the location of the centre of the experimental design; determining the effect of the choice of different earthquake records; comparing some of the results with exact results found by Monte Carlo simulation.

The results of the computations, allow a number of general observations to be made that are summarised in a final discussion.

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