(FOSM analysis for functions of many variables) PROBABILISTIC ANALYSIS OF FOUNDATION SETTLEMENT Mechanical Model

Consider a footing resting on a compressible soil stratum of depth z0. Below z0, the soil may be considered incompressible (e.g. there is bedrock or the stresses induced by the foundation are small enough that soil deformation may be neglected). An important problem is to evaluate the amount of settlement. The total settlement D may be found by adding the vertical strain of the soil elements along the vertical line through the center of the footing. This gives zo D = ∫ ∆σ(z) dz (1)M ( z) 0 where ∆σ(z) is the increment in the vertical stress at depth z due to the foundation load and M(z) is the elastic modulus of the soil at that depth (a deformability parameter of the soil). For simplicity, we divide the compressible soil stratum into n layers of equal thickness (z0/n) and replace the integral in Eq. 1 with the discrete summation n D = zo ∑ ∆σi (2)n i=1 Mi where ∆σi = ∆σ(zi), Mi = M(zi), and zi = (i- 0.5) z0/n is the average depth of layer i. If the footing is circular of radius R and the load per unit area transmitted by the footing is