14 / Finite element analysis in geotechnical engineering: TheoryIn terms of total stress, the equation reduces to: H,,,-4% ---Ywhere S,,is the undrained strength. Note: This solution is identical to the upper bound solution obtained assuming a planar sliding surface (see Section 1.9.3). The lower bound solution gives half the above value.1.9.2 Stress field solutionIn this approach the soil is assumed to be at the point of failure everywhere and a solution is obtained by combining the failure criterion with the equilibrium equations. For plane strain conditions and the Mohr-Coulomb failure criterion this gives the following: Equilibrium equations:+am,=ar, C)Mohr-Coulomb failure criterion (from Figure 1 .12):D; - g; = 2c%osqrSt1:+ (m,' + D;) sinp'(1.16)YNoting that:s = c'cot p'+ (D,' + m,') = c'cotq' + X(o,' + 4"')Figure 1. 12: Mohr's circle of S tressand substituting in Equation (1.16), gives the following alternative equations for the Mohr-Coulomb criterion:t= Ssin@(1.17)[!,((nxl-0;1)2+ z;.]~ '= [clcotp' + x ( n x 1 +a;')]sinp'(1.18)'The equilibrium Equations ( l . 15) and the failure criterion ( l . 18) provide three equations in terms ofthree unknowns. It is therefore theoretically possible to obtain a solution. Combining the above equations gives:
