Closed-form solution for plastic zone formation around a circular tunnel in half-space obeying Mohr–Coulomb criterion

A closed-form solution for the problem of the plastic zone and stress distribution around a circular tunnel in an elastic–plastic half space, derived using bipolar coordinates, is the main scope of this paper. By assuming a uniformly applied surface loading, the whole semi-infinite space is under uniform pressure: thus the gravitational effect is neglected, while the plastic zone formation around the circular tunnel is controlled by the applied internal support pressure. The plastic behaviour of the half space is described by the Mohr–Coulomb yield criterion, and the soil is assumed to be homogeneous and isotropic with earth pressure coefficient K₀ equal to unity. The critical internal pressure, where the initial yielding occurs at the tunnel wall, is derived, along with equations describing the plastic zone and plastic stresses. These equations are functions of the soil properties, which are cohesion and friction angle. This derived closed-form solution is validated through mathematical and computational analysis, and is also compared with numerical models under gravitational load, solved using the finite difference method. This innovative closed-form solution has a significant impact in practical problems by introducing a simple and effective method that allows the quick estimation of a shallow tunnel's behaviour, since it gives, in principle, the opportunity for quick and accurate calculation of the plastic zone and stress distribution around the circular tunnel. By applying different values of support pressure, the tunnel designer can easily evaluate the feasibility of different design alternatives, such as shotcrete shell and tunnel-boring machine support pressure. As a result, an efficient and innovative method of solving shallow tunnelling problems in cohesive-frictional soil is introduced.