# Stress and Displacement Analysis in Tunnel

I am simulating in flac3D a simple model of a circular excavation (radius = 2 m) in an elastic rock medium with a square shape as outer limit (40 m) and with a very small depth in the axial axis (Y axis) of 0.2 (m). The model is initially constrained normally on all faces except the ‘East’ face (this face is free). Additionally, a stress is applied on the free face (‘East’ face) in a Cartesian way (sigma xx, sigma yy, sigma zz) with k = 2 (vertical stress = 2*horizontal stress). Also mention that gravity is activated. The displacements are reset to zero once the stresses have been loaded, to obtain the induced displacements. The model consists of two stages, stage 1 where the complete model is found with the loaded stresses and displacements restarted as I mentioned, and stage two, where the circular excavation of 2 m radius is carried out. The results obtained show that the magnitude of the total displacements are greater in the ceiling of the excavation while the highest stresses are found in the wall of the excavation. It makes sense to me that the stresses are concentrated on the wall, but I don’t understand why the displacements and therefore the nodal velocities are greater on the roof. Just as the highest stresses are in the wall, shouldn’t the highest displacements also be here? I enclose images of the direction of the vectors of the maximum stresses (in the image they appear as minimums because they are compression stresses) and of the displacements at the end of the excavation.

Hello Cristian,

It sounds to me like this model is similar to a hole in an elastic solid. The analytical solution of which is the well-known Kirsch solution. Given your parameters the maximum hoop stress will occur at the tunnel wall and the minimum hoop stress will occur at the roof (compression negative, this is consistent with your first image). The Kirsch solution gives the hoop stress and radial displacement at the tunnel periphery: This is consistent with your second image.

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