Similarity solution is investigated for the synchronous grouting of shield tunnel under the vertical non-axisymmetric displacement boundary condition in the paper. The synchronous grouting process of shield tunnel was simplified as the cylindrical expansion problem, which was based on the mechanism between the slurry and stratum of the synchronous grouting. The stress harmonic function on the horizontal and vertical ground surfaces is improved. Based on the virtual image technique, stress function solutions and Boussinesq's solution, elastic solution under the vertical non-axisymmetric displacement boundary condition on the vertical surface was proposed for synchronous grouting problems of shield tunnel. In addition, the maximum grouting pressure was also obtained to control the vertical displacement of horizontal ground surface. The validity of the proposed approach was proved by the numerical method. It can be known from the parameter analysis that larger vertical displacement of the horizontal ground surface was induced by smaller tunnel depth, smaller tunnel excavation radius, shorter limb distance, larger expansion pressure and smaller elastic modulus of soils.

Based on the virtual image technique, stress function method and Boussinesq's solutions, an approximate solution of the cylindrical cavity expansion is investigated under the non-axisymmetric displacement boundary condition on hypotenuse. The stress harmonic functions on the ground surface are improved. At first, the elastic solution without any correction is deduced, considering the combined effects of the actual and image sources. Then, stress on the ground surface is revised by the improved stress harmonic function. Stress on the slope surface boundary is corrected by integrating Boussinesq's solution and then applying the coordinate transformation technique. Moreover, the newly induced stresses on the ground surface boundary (resulting from the correction process of the slope surface boundary) are further revised by the improved stress harmonic function. Finally, the linear elastic superposition principle is adopted to obtain the approximate solution under the non-axisymmetric and free displacement boundary conditions on hypotenuse. In addition, the proposed solution is validated by the numerical results. The numerical analysis results show that it is helpful to control the safety reserve of stress and displacement considering the boundary effects and stress revisions. Larger vertical displacement of the ground surface could result from larger slope inclination, shorter limb distance, smaller cylindrical cavity depth, larger expansion pressure and smaller elastic modulus of soils.

In order to investigate the influence of the intermediate principal stress on the stress and displacement of surrounding rock, a novel approach based on 3D Hoek-Brown (H-B) failure criterion was proposed. Taking the strain-softening characteristic of rock mass into account, the potential plastic zone is subdivided into a finite number of concentric annulus and a numerical procedure for calculating the stress and displacement of each annulus was presented. Strains were obtained based on the nonassociated and associated flow rule and 3D plastic potential function. Stresses were achieved by the stress equilibrium equation and generalized Hoek-Brown failure criterion. Using the proposed approach, we can get the solutions of the stress and displacement of the surrounding rock considering the intermediate principal stress. Moreover, the proposed approach was validated with the published results. Compared with the results based on generalized Hoek-Brown failure criterion, it is shown that the plastic radius calculated by 3D Hoek-Brown failure criterion is smaller than those solved by generalized H-B failure criterion, and the influences of dilatancy effect on the results based on the generalized H-B failure criterion are greater than those based on 3D H-B failure criterion. The displacements considering the nonassociated flow rule are smaller than those considering associated flow rules.