The problem of the curse of dimensionality is frequently encountered in practical applications. It can be defined as the significant increase of the number of mechanical model calls with the number of uncertain parameters. To overcome this difficulty, a two-steps stochastic approach has been developed in this work. The first step of this approach consists in calculating the stochastic effective dimension by the means of Morris screening method. Once the most significant uncertain parameters on the variability of the mechanical responses are identified, they are modeled as random variables and the remaining parameters are fixed to their respective mean values. This allows us to reduce significantly the stochastic dimension of the problem in the second step of the approach where the decomposition method is used to estimate the statistical characteristics of the mechanical responses. The efficiency and the accuracy of this approach are evaluated through an academic problem dealing with the assessment of the integrity of a three-span five-story frame structure subjected to horizontal loads. We have demonstrate that we can reduce about 90% of the computation time required by the classical stochastic methods. Then, the proposed approach is used to the analysis of the integrity of timber roofs under seismic loading. The behaviour of this structure is described through a finite element model where the timber joints are modeled by anisotropic hysteresis law, and the seismic action is represented by eight real earthquake ground motion records. These accelerograms provided by the French institution involved in geosciences BRGM allow us to take into account different soil types according to the classification provided by the europeen design code dealing with seismic events Eurocode 8. The failure of timber roofs is reached when the damage levels in the timbers joints localized on the buckling and bracing members reach the critical value. It is shown, through a deterministic analysis, that the structure resists the seismic hazard representing the city of Le Moule in Guadeloupe. The stochastic analysis has shown that, among the 134 random variables representing the uncertainty in the nonlinear behaviour of the timber joints, only 15 have a significant effect on the variability of the structural response, which allow us to reduce the stochastic dimension in the computation of the statistical moments. According to the estimates of the mean and the standard deviation, we have shown that the variability of bracing members damage is greater than the variability of buckling members damage. Moreover, the variability of the bracing members damage is more significant for the earthquake ground motion records having the lowest collapse PGA.
