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• An analytical solution for a lined circular tunnel at shallow depths is presented. • The interaction between surrounding rock and lining is considered. • A new approach is proposed based on complex variable method. • The mechanical contribution of lining is analyzed quantitatively. In this study, a new approach is presented to determine the stress and displacement field of a shallow buried circular tunnel subject to gravity and arbitrary lateral stress based on the complex variable method. The soil mass and lining are conformally mapped onto two ring regions. A more effective method than the power series method is used to solve the undetermined coefficients of complex potential functions. The interaction mechanism between the lining and the soil mass is fully considered. The boundary conditions along the inner boundary of the lining and the ground surface and the stress and displacement continuity conditions along the excavation line are used in the analytical derivation. In the verification, the stress and displacement distributions obtained in the semi-analytical solution agree with the numerical results and satisfy the boundary and continuity conditions. Using examples, the stress distributions along the inner boundary of the lining and interface are presented to examine the influence of key parameters including the Young's modulus and buried depth. The effect of the lining in reducing stress concentration and restricting tunnel convergence is quantitatively analyzed. [ABSTRACT FROM AUTHOR]

An analytic stress solution is presented for a circular tunnel problem in a half plane with a concentrated force acting on any position in the field under gravity. The solution uses the complex variable method and the power series method. The influence of the unbalanced force system on the tunnel boundary is considered. The relationship between two analytic functions is established by using surface stress boundary condition. The analytic functions can be determined from the tunnel stress boundary condition. Based on the principle of superposition, the stresses of the surrounding rock can be calculated by superimposing three partial solutions which are obtained separately. The examples give contour plots of the principal stresses in the surrounding rock, focus on the stress distribution on the ground surface and the tunnel boundary and analyze the effect on the stress distribution of some main parameters. [ABSTRACT FROM AUTHOR]

The problem refers to an elastic infinite plate bounded by two oval holes, which is subjected to uniform surface tractions on the boundary of the holes and uniform tension and shear stress at infinity. By looking for a new mapping function, the considered region is conformally mapped onto an annulus, and a complex variable method is used to obtain the stress solution. The problem that an elastic infinite plate contains two circular holes or a circular hole and an oval hole is the special case of this article. The present solution has been validated by both published solutions for special cases and ANSYS software. Some computational examples are given to investigate the effects of the separation distance of the two holes, relative ratio of applied stresses, surface tractions on the boundaries, the size of a hole, shear stress at infinity and the loading orientation angle on the stress distribution of the hole boundaries. Practical significance of the solution is placed in the fact that it could be used as a quick-solver with high accuracy. [ABSTRACT FROM AUTHOR]

• Analytical solution for shallow tunnel with point anchored rockbolts is presented. • The mechanical contribution of rockbolts is analyzed quantitatively. • The effect of parameters on axial load is discussed. • The interaction between surrounding rock and rockbolts is considered. In this paper, an analytical approach is presented, by focusing on the shallow buried tunnel reinforced by two-point anchored rockbolts, by considering the rockbolts as two identical concentrated forces acting in opposite directions, applied at both the ends of the rockbolts. It was assumed that the surrounding rock and the bolts were in an elastic state and the problem existed in a plane strain condition. The complex variable method was applied to reach a solution. While solving the problem, first, the problem of a circular shallow buried tunnel, with a concentrated force acting on the surrounding rock mass, was addressed. Afterwards, the solution for a tunnel reinforced by multiple rockbolts was obtained, based on the previous solution, considering that displacements between the rockbolts and rock mass were compatible. The case of pre-stressed rockbolts was also considered in this paper. The solution obtained in this paper provides the results for the axial load of the rockbolts and the stress and displacement of the surrounding rock. The analytical solution results were found to be similar to that of the finite element method obtained by ANSYS software. In addition, to quantify the mechanical contribution of the rockbolts, the stress and displacement distributions of the surrounding rock, for reinforced tunnel and non-reinforced tunnel, were also compared. [ABSTRACT FROM AUTHOR]

This paper presents an effective optimization approach based on the equi-stress criterion and conformal transformation to determine two equi-stress holes with different areas in an infinite plate subjected to biaxial loads at infinity and internal uniform pressure on hole boundaries. Conformal transformation in the complex variable method is used to map the outer domain of two holes in the physical plane onto an annulus in the image plane. Mathematically, the hole shapes are parameterized and expressed in one mapping function, therefore, the shape of two holes can be determined by obtaining the coefficients of the mapping function. In the process of optimization, the coefficients are used as design variables in the differential evolution (DE) algorithm to determine the hole shapes. As examples, the results has high accuracy, the optimal hole shapes under different conditions are given and the effect of each parameter are analyzed. What's more, the contour plots of principal stresses are presented to clearly show the stress distribution in the domain for the optimal hole shape. • Obtaining the shape of two equi-stress holes with different areas. • The equi-stress criterion and conformal transformation were used. • The process is simple and the scope of application is wide. • Presenting the hole shapes for different parameters. • Presenting the stress filed of the domain. [ABSTRACT FROM AUTHOR]