Based on the nonlinear failure criterion and the upper bound theorem, the modified tangential technique method was proposed to derive the expression of supporting pressure acting on shallow tunnel. Instead of the same stress state, different normal stresses on element boundaries were used. In order to investigate the influence of different factors on supporting pressures, the failure mechanism was established. The solution of supporting pressure, with different parameters, was obtained by optimization theory. The corresponding failure mechanism and numerical results were presented. In comparison with the results using the single tangential technique method, it is found that the proposed method is effective, and the good agreement shows that the present solution of supporting pressure is reliable.
Linear failure criterion is widely used in calculation of earth pressure acting on shallow tunnels. However, experimental evidence shows that nonlinear failure criterion is able to represent fairly well the failure of almost all types of rocks, A nonlinear Hock-Brown failure criterion is employed to estimate the supporting pressures of shallow tunnels in limit analysis framework. Two failure mechanisms are proposed for calculating the work rate of extemal force and the internal energy dissipation. A tangential line to the nonlinear failure criterion is used to formulate the supporting pressure problem as a nonlinear programming problem. The objective function formulated in this way is minimized with respect to the failure mechanism and the location of tangency point. In order to assess the validity, the supporting pressures for the proposed failure mechanisms are calculated and compared with each other, and the present results are compared with previously published solutions when the nonlinear criterion is reduced to linear criterion. The agreement supports the validity of the proposed failure mechanisms. An experiment is conducted to investigate the influences of the nonlinear criterion on collapse shape and supporting pressures of shallow tunnels.
To analyze the stability of a shallow square tunnel, a new curved failure mechanism, representing the mechanical characteristics and collapsing form of this type of tunnel, is constructed. Based on the upper bound theorem of limit analysis and the Hoek-Brown nonlinear failure criterion, the supporting pressure derived from the virtual work rate equation is regarded as an objective function to achieve optimal calculation. By employing variational calculation to optimize the objective function, an upper bound solution for the supporting pressure and the collapsing block shape of a shallow square tunnel are obtained. To evaluate the validity of the failure mechanism proposed in this paper, the solutions computed by the curved failure mechanism are compared with the results calculated by the linear multiple blocks failure mechanism when the Hoek-Brown nonlinear failure criterion is converted into the Mohr-Coulomb linear criterion. The influences of rock mass parameters on the supporting pressure and collapsing block shape are discussed.
On the basis of upper bound theorem, non-associated flow rule and non-linear failure criterion were considered together.The modified shear strength parameters of materials were obtained with the help of the tangent method. Employing the virtual power principle and strength reduction technique, the effects of dilatancy of materials, non-linear failure criterion, pore water pressure,surface loads and buried depth, on the stability of shallow tunnel were studied. In order to validate the effectiveness of the proposed approach, the solutions in the present work agree well with the existing results when the non-associated flow rule is reduced to the associated flow rule and the non-linear failure criterion is degenerated to the linear failure criterion. Compared with dilatancy of materials, the non-linear failure criterion exerts greater impact on the stability of shallow tunnels. The safety factor of shallow tunnels decreases and the failure surface expands outward when the dilatancy coefficient decreases. While the increase of nonlinear coefficient, the pore water pressure coefficient, the surface load and the buried depth results in the small safety factor. Therefore, the dilatancy as well as non-linear failure criterion should be taken into account in the design of shallow tunnel supporting structure. The supporting structure must be reinforced promptly to prevent potential mud from gushing or collapse accident in the areas with abundant pore water, large surface load or buried depth.
The stability of natural slope was analyzed on the basis of limit analysis. The sliding model of a kind of natural slope was presented. A new kinematically admissible velocity field for the new sliding model was constructed. The stability factor formulation by the upper bound theorem leads to a classical nonlinear programming problem, when the external work rate and internal energy dissipation were solved, and the constraint condition of the programming problem was given. The upper bound optimization problem can be solved efficiently by applying a nonlinear SQP algorithm, and stability factor was obtained, which agrees well with previous achievements.
