Hello everyone. In this sequence, we shall assess the fragility of structure through seismic fragility curves. You will see how these curves are generated through various concepts from the previous sequences. Let's introduce the concept of seismic fragility curves. One of the major indicators in the framework of safety assessment is the probability of failure. In our case, the probability of failure can be given by the integral in the load space ES, of the conditional probability of failure knowing the load multiplied by the probability of occurrence of this load. In our context, the PGA is often used to represent the seismic load. The conditional probability is also called seismic fragility. Plotting the evolution of the seismic fragility with respect to the load, we get the so-called seismic fragility curve. Taking into account the uncertainties in the process allows us to produce several curves each one corresponding to a different quantile. In this plot, for example, the brown curve is the 50 percent quantile. You can also see in red the fifth percent, and in blue, the 95 percent quantile. As the seismic fragility reflects the probability of failure of a structure or component to a given event with respect to its seismic intensity, we easily understand that its determination raises the needs for seismic hazard assessment, including soil response and site effects, soil structure interaction analysis, structural response estimation to get the floor spectrum, for instance. Then we'll begin the study of the seismic response of the structure or component. The most common method to produce the fragility curve is probably the one proposed by the Electric Power Research Institute, EPRI. In this method, the seismic capacity of the structure or component is given by a reference acceleration. This acceleration is often the result of a deterministic mechanical study and includes safety margins. So, the real seismic acceleration capacity, A, of the structural component can be found by multiplying the reference acceleration and the safety margin factor F. This global factor can be determined by decomposition in the structural response margin factor, the equipment response margin factor, the elastic resistance margin factor, and the ductility or inelastic dissipation margin factor. To evaluate them, each factor is also decomposed in unit factors considered as three parameters lognormal random variables. The central limit theorem allows us to write that the global factor is also a lognormal variable with three parameters: the median, FMEF, and 2 overs parameters: BETA_R describing the random variability and BETA_U, the epistemic uncertainty. If F is lognormal, then the structural or component acceleration capacity, A, is also lognormal with a median AM and the same BETA. Then we can plot the probability density function of A in which we can see the reference acceleration, the margin factor, F, and the median acceleration, A_m. The BETA parameters drives the PDF variability. Back to the plot, we have already seen the fragility curve is the cumulative distribution function of the lognormal variable A and its expression is given with respect to the previous parameters, where PHI is the standard normal cumulative distribution function and Q, the quantile. In that plot, you can read some useful indicators corresponding to a failure probability of 0.5 the median acceleration AM and associated to the probability of 0.05 in the 95 percent quantile curve, the high confidence low probability of failure acceleration, A_HCLPF. Another way to build fragility curves uses simulations and uncertainty propagation methods. Back to the deterministic chain from seismic hazard to the structure or competent response, the second way is based on the comparison between the stress generated by the quake and the mechanical resistance capacity of the structure or components. It's important to notice at that point that the comparison have to be done for every failure mode. The stress and the resistance capacity are probabilistic variables. To find their distribution, we can use simulations to propagate uncertainties in every step of the deterministic chain , the seismic hazard, the soil structure interaction, the structural response to reach finally the probabilistic floor spectrum, and the physical excitation indicators corresponding to the failure mode. The resistance capacity is also probabilistic and maybe compared to the excitation. Graphically, we can plot at a given PGA the probability density function of the excitation as well as that of the resistance capacity. The common area of the two distributions represents the conditional failure probability corresponding to the PGA and defines one point of the fragility curve. As a result of the simulation process, following a design of experiments, we can obtain this type of scatter plot. Each point is the result of the deterministic chain for one input data value set of the DOE. The scatter have to be compared to the red resistance threshold deterministic in this example to build the fragility curve. To do so, numerous method, more or less sophisticated, exist. The easiest method often used to have the first approximation considering intervals in the PGA domain. For instance, 0.5 G For each interval, we estimate one point of the cumulative distribution function representing the fragility curve by the rate of points overpassing the threshold. We built point by point, the fragility curve. A more sophisticated method consist in estimating the relation between the scatter of deterministic calculations and the PGA. That step requires to transform the dataset in order to get a linear regression with respect to the homoscedasticity property and allowing to know or approximate the distribution of the residuals. The same transformation has to be applied to the threshold. At that point, we have the dataset, the threshold, the transformation, and the residual distribution that is enough to calculate for each point of the dataset, the corresponding conditional probability of failure given the PGA at any quantile. In other words, to plot the fragility curves. To sum up the sequence, we have seen that the fragility curves gives the probability of failure of a structure or component for a given seismic load. We introduced two different methods to build them: one with a strong hypothesis on the distribution law of the fragility and another based on simulation and uncertainty propagation through deterministic mechanical response models and different statistical post-treatments of the results.
