Search

In practical engineering projects, bedrock is often encountered under a compressible soil layer. Compared with the soil layer, the hard bedrock has negligible deformation, so the displacement at the bottom of the soil layer is considered to be zero. In this case, it is difficult to use the Boussinesq solution to accurately calculate the stress and displacement of the soil layer under the action of shallow foundations. In the past, the solution has mostly been approximated by the layer-wise summation method or the finite-element method. So far, there is no analytic method to accurately calculate the stress and displacement of the elastic compressible soil layer. In this paper, an analytic method is proposed to solve the plane strain problem under the action of strip foundations. By using a mapping function to convert a soil layer with a finite thickness into the unit circle in the image plane, the stress boundary conditions at the ground surface and the displacement boundary conditions at the bottom of the soil layer are represented by a combined expression for the boundary values of the two analytic functions. A system of linear equations to find the coefficients of the analytic functions is established using a new power series method, and it is solved using the least-squares method. The stress and displacement at any point in the soil layer can be calculated quickly and efficiently using the derived analytic function. The effects of soil layer thickness on soil layer stress and ground surface settlement are discussed, and the results are compared with those obtained by the finite-element method. [ABSTRACT FROM AUTHOR]

• 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 analytical solution for a non‐circular tunnel excavated in an elastic homogeneous half‐plane is obtained, which considers gravity and different lateral pressure coefficients. The solution is deduced using the complex variable method, mapping the region containing a non‐circular tunnel onto a circular ring. The basic equations for the analytic functions are established using the stress boundary conditions at the ground surface and the edges of the tunnel and the analytical functions are solved by the power series method. In the process of solving, the Fourier series was used. To elaborate the solution process clearly and verify the correctness of the solution, a shallow quasi‐ellipse tunnel was analyzed as a computational example, and the results obtained using the presented analytical method are compared with that obtained by the numerical software ANSYS. [ABSTRACT FROM AUTHOR]

This paper proposes a method for solving the stress field of a deep circular tunnel affected by an unsteady temperature field. The tunnel is idealized as an infinite domain problem, and the solution of the unsteady temperature field can be regarded as solving the temperature field in the infinite domain under the first type of boundary conditions. The temperature field distribution inside the tunnel surrounding rock is obtained by the Laplace transform method. On this basis, through the stress condition at the boundary of the tunnel and the zero displacement boundary condition at infinity, the solution equations expressed by the analytic functions are listed, and then the power series method of the complex variable function method is used to obtain the desired analytic functions. From this, the temperature stress field caused by the temperature change can be calculated (the displacement field caused by the temperature change is also derived). By superimposing this stress field upon the surrounding rock stress caused by in situ stress, the total stress field inside the surrounding rock can be obtained. The example shows that when the time t is small and the range of the numerical model is suitably large, the results of the analytical method and the numerical method agree well. This paper also discusses two cases in which the tunnel boundary temperature is greater than and less than the initial temperature of the surrounding rock. [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]

An analytical solution is presented for an elastic infinite plate containing an ellipse hole and a circle hole, which undergoes the internal traction along the holes and uniform tensions and shears at infinity. The solution uses the conformal transformation in complex functions, with the multiple connected region mapping onto a circular ring. The stress distribution solved for the infinite plane containing two unequal circle holes in prior literatures is the special case of this article. And contrast between the results of two unequal circle holes obtained by using the present analytical method and the Hoang's results is made in the article. As there is no analytical solution for two non-circular holes, the results obtained using the present analytical method are compared with the numerical results calculated by ANSYS. Computational examples given in this article has been analyzed to study the effect of distance between two holes, the ratio of stress in two directions at infinity, and the ratio of the major and minor axis, on the stress distribution of the hole boundary. [ABSTRACT FROM AUTHOR]

Stress fields of a lined non-circular tunnel subjected to in situ stress are derived based on the complex variable method and on the assumption that the interface between the liner and surrounding rock is full-slip. The basic equations for solving the stress solutions are obtained according to the stress boundary condition along the inner boundary of the lining and the stress and normal displacement continuity conditions along the rock-lining interface. In the solving process, the support delay is also considered. The basic equations can be solved by the power series method, and the stresses in the surrounding rock mass and lining can be calculated. The distributions of the tangential stresses (also known as the circumferential stresses) along the excavation boundary and the inner boundary of the lining and the contact stresses along the rock-lining interface are analysed. An example demonstrates that the results are significantly affected by the number of terms in the power series. When the number of terms is greater than 100, the boundary conditions can be well satisfied, and the results of the stresses and displacements are highly accurate. The tangential stress results along the inner boundary of the lining for the full-slip condition are compared with those for the perfect bond condition, and the analysis indicates that the maximum value of the tangential stress for the full-slip condition is smaller than that for the perfect bond condition, which gives that the full-slip condition is superior to the perfect bond condition. Thus, the carrying capacity of the lining can be increased if sliding materials are installed between the lining and the surrounding rock mass. The analytic solutions are verified using computer simulation software. [ABSTRACT FROM AUTHOR]

Abstract: By using the semi-inverse method, an elastic plane stress solution of a lined vertical borehole in an isotropic ground due to non-axisymmetric in situ loads is presented. As two extreme cases, pure bond and pure slip boundary conditions, along the liner/ground interfaces have been implemented. The effect of elastic modulus of liner and ground and thickness of liner upon the stresses and displacements are analyzed. The result indicates that soft liner may alleviate the normal stress and shear stress acting on the liner outer surface. The maximum tangential compressive stresses and tensile stresses in liner for two extreme boundary cases occur on the inner and outer boundaries of liner. The shear stress reaches the maximum value at θ=45° and the harder the liner is, the greater the maximum value is. [Copyright &y& Elsevier]

Abstract: Analytical solutions are derived for the elastic–plastic stresses and strains around a circular tunnel subjected to an axial in situ stress, in a rock mass governed by the Mohr–Coulomb failure criterion. The calculation results show that the stresses, strains, and failure zones in the surrounding rock depend not only on the transverse in situ stress p but also on the axial in situ stress . The pattern of failure zones vary with different combinations of p and , which is different from the results produced by earlier plane strain analyses. If is less than some critical value, the radius of the plastic zones are the same as give by the plane strain analysis, and depends only on p. If exceeds the critical value, the radius of the plastic zone increases with increasing . The analytical solutions are verified by three-dimensional numerical analyses using FLAC3D software. [Copyright &y& Elsevier]

This paper presents a method to solve the stress field of a deep buried circular lining tunnel under the combined action of an unsteady temperature field and in situ stress. The solution of the temperature field can be reduced to a problem of solving the unsteady temperature field in the composite medium under the first type of boundary conditions in the infinite domain, and the distribution of the temperature field in the lining and surrounding rock can be obtained by the Laplace transform method. Through the stress boundary condition of the inner boundary of the lining, the continuous condition of the contact surface and the zero displacement boundary condition at infinity, equations expressed by the boundary values of the analytical functions are established. Then the temperature stresses caused by temperature changes can be obtained. By superimposing these stresses with the stresses caused by in situ stress, the total stresses inside the lining and surrounding rock can be obtained. This article compares the analytical solutions with the numerical solutions obtained by ANSYS. In addition, the influence of temperature field on the stress field of lining and surrounding rock is discussed in the case of temperature decrease and increase through calculation examples. [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]

The complex variable method is used for an analytical solution for an arbitrary cavity in a homogeneous elastic half-plane. Proposing a method for solving the coefficients of the function by conformally mapping the region that contains an arbitrary cavity onto a circular ring utilizing a mapping function, which can be used for arbitrary shape cavity. Considering the stress-free condition of the upper surface, the relationship between two analytic functions is derived; consequently, the problem of solving two analytic functions is attributed to solving one analytic function, and the solve process for the analytic function in general case is given in detail, finally, achieving the solution of an arbitrary shape cavity in an elastic half-plane. The effects of cavity shape and depth were analyzed to the required terms of analytic function under a given accuracy. Examples for a vertical-wall semicircle cavity and a horseshoe cavity are given through the present method, and verification example is given by ANSYS software. [ABSTRACT FROM AUTHOR]