Analytical solution for ground reaction curve of a tunnel excavated below groundwater table considering drained conditions

In this paper, an analytical solution for ground reaction curve is presented, under drained conditions considering pore pressure effect. A theoretical solution for seepage forces due to groundwater flow under steady-state flow is derived. Afterward, an elasto-plastic model based on a linear Mohr–Coulomb yield criterion is adopted and the ground reaction curve is evaluated based on plastic potential function for the non-associated flow rule. A simplified relation in elasto-plastic conditions is proposed for calculating the radius of plastic zone and a sensitivity analysis on the parameters affecting the ground reaction curve is also made. The ground reaction curve is in fact the relationship between the support pressure and the radial displacement of tunnel wall, which is used for an optimum design of the support system of the tunnels under ground water table influenced by seepage forces. A simplified and pervasive analytical solution to calculate stresses and displacements on the tunnel wall has not yet been suggested. In this paper, the approximation of calculating plastic radius is eliminated and a method to obtain its precise value, by a direct relation and a theoretical solution for the ground reaction curve, is proposed. Methodology and Approaches: In this paper, it is assumed that the seepage forces due to groundwater table are based on steady-state flow, the tunnel is circular, a plastic zone appears around the tunnel considering the Mohr-Coulomb failure criterion, and a drained condition is governed. Common stress-strain and pressure equations, and differential equations are used. The radial displacement for a circular tunnel can be calculated by means of the elasto-plastic theory. The plastic strains can be represented using the plastic flow rule. The strain in the plastic zone is the sum of elastic and plastic strains. Simplified relations, which the ground reaction curve would be expressed by these relations, are obtained. Results and Sensitivity analysis shows that by increasing the water height above tunnel crown, the ground reaction curve becomes more critical and the necessity of adequate tunnel wall will further be required. Moreover, by increasing the water height above tunnel crown, the influence of seepage increases as the effect of drainage is major in lower depths.