The analytic representation of stress- and deformation fields in the ground surrounding a tunnel succeeds only in some extremely simplified special cases, which are rather academic. Nevertheless, analytical solutions offer the following benefits:
Being exact solutions, they provide insight into the basic mechanisms (i.e. displacements, deformation and stress fields) of the considered problem.
They provide insight of the role and the importance of the involved parameters.
They can serve as benchmarks to check numerical solutions.
In this section, some solutions are introduced which are based on
’s law, the simplest material law for solids. The underground is regarded here as linear-elastic, isotropic semi-infinite space, which is bound by a horizontal surface, the ground surface. The tunnel is idealised as a tubular cavity with circular cross section. Before its construction, the so-called primary stress state prevails. This stress state prevails also after the construction of the tunnel in a sufficiently large distance (so-called far field).