Explicit analytical solutions for stress and displacement of surrounding rock in shallow tunnels

The ground movements caused by urban shallow tunnelling can be passed to the ground surface and then form surface settlements. It is necessary to accurately predict the ground movements before tunnelling for effectively controlling the ground settlements. Based on the complex variable function method, the series forms of complex potential functions in z-plane are first obtained using the inverse mapping function. By taking the derivative of analytical functions proposed by Verruijt with Cauchy-Riemann equations, explicit analytical solutions for the stress and displacement induced by shallow tunnelling are obtained. The explicit solutions are intuitional and easily used by engineers, which have smaller amount of calculation than the solutions proposed by Verruijt. Moreover, by improving the previous researches on deformation modes of shallow tunnels, three deformation modes and two deformation ratios are put forward. Finaly, the explicit solutions are compared with the field test data of two tunnel projects, and the accuracy and practicability of the proposed explicit solutions are proved.