Analytical Method For Stress Analysis Of The Opening Anisotropic Structures

Anisotropy is an important property of the natural materials,such as rock mass,wood and bamboo,and the anisotropic plate made of composite materials.With the construction of the underground engineering and the requirements of the structural assembly or functions of the composite plates.different tunnels or holes are often made in the rock masses or in the plates.For these opening anisotropic structures,the stress concentration caused by the tunnels or holes becomes a particularly important problem.In order to ensure the safety and stability.the stress analysis for these opening anisotropic structures has been a research focus.In this paper,by means of the theoretical derivation,supplemented by the ANSYS numerical verification,several researches have been carried out for the infinite anisotropic plate and the deep-buried anisotropic surrounding rock mass:the establishment of the analytical method for stress analysis,the optimum design of fiber angle and hole orientation,and the determination of the flexibility matrix for skew anisotropic materials.The main research contents are as follows.(1)For the stress concentration problem of the infinite orthotropic plate,the analytical stress solutions are analyzed for an arbitrary-shaped hole under uniform in-plane loadings at infinity by using the complex variables function method.On this basic,the hexagonal hole and an irregular-shaped hole with an obvious sharp corner are taken as examples.The stress distributions at the edge and in the outer region of the hole are studied for different conditions,such as different fiber angles and external loading directions.And the results are also compared with that for isotropic materials.The research shows that for orthotropic plates,the tangential stresses are always exactly the same value-? at the intersections of the hole boundary and the axis,along which the uniaxial loadings ? are applied.Besides,when the uniaxial loadings are perpendicular to the pointing direction of the sharp corner,the maximum tangential stress in the orthotropic plate may occurs at the hole boundary(fibers arranged at[0°/-90°]s)or in the adjacent area of the hole boundary(fibers arranged at[45°/-45°]s).And when it occurs at the hole boundary,it is in the sharp corner point.However,the position of the maximum stress will get farther from the sharp corner point with the rotating of the fiber angles,and its value also decreases correspondingly.Therefore,the stress concentration of orthotropic plates can be reduced by adjusting the fiber angles.(2)With the goal of decreasing the stress concentration along the hole boundary,an optimum design of fiber angle and hole orientation is presented for orthotropic plates with complicated shapes of holes under given loading conditions.The optimization process is based on the analytical stress solution,and the Differential Evolution algorithm is used.The optimization criterion is to make the maximum absolute value of the tangential stress along the hole boundary reach its minimum value.And the optimized tangential stress distribution was also analyzed.The results show that the anisotropic degree of the plate(E2/E1)has great influence on the optimization results of fiber angles,but little effect on the orientation of the elliptical hole.Besides,compared with the holes with smooth edges,the stress concentration of the holes with sharp corners is more obvious,and its maximum values are located at the sharp corners or the adjacent points.In addition,for given hole shapes and loading conditions,the tangential stress concentration is much smaller when the fiber angle and hole orientation are optimized jointly.(3)For deep-buried hydraulic tunnels,considering both the anisotropic properties of the surrounding rock mass and the effects of the internal water pressure,the analytical solutions of stress around an arbitrary-shaped tunnel excavated in an orthotropic rock mass are derived by using the conformal transformation method and the power-series method.The analytical solutions are verified by numerical simulation of ANSYS,and the influence of the thin support on the stress results is also discussed.On this basic,taking the circular,horse hoof-shaped and inverted U-shaped tunnels as examples,the influence of the internal water pressure and the coefficient of the lateral pressure on the stress distributions are also analyzed.The results show that when the hole shapes and external loadings are all symmetrical,the stress field around the hole shows obvious asymmetry,especially when the elastic moduli in the two directions of the rock mass are different apparently.Besides,the increasing of the lateral pressure coefficient will significantly increase the tangential stresses near the sharp corners of the U-shaped tunnel,but the stresses in the two large curvature positions of the horse hoof-shaped tunnel are almost unchanged.In addition,the internal water pressure can reduce the stress concentration along the boundary,which is more obvious on the regions where the curvature is greater.However,excessive internal pressure will cause obvious tensile stress,which is not conducive to the tunnel safety.(4)The anisotropic materials are considered with only one elastic symmetry plane,in which the fiber angles or joints of the rock mass are in general conditions of skew anisotropic(non-orthogonal).By using the engineering constants and the conversion formulas between coordinate systems,the analytical method of solving the flexibility coefficient matrix of skew anisotropic materials is put forward for the first time.And the flexibility matrices of the skew anisotropic plates and skew anisotropic rock masses are derived.(5)Take the examples as an elliptical,hexagonal or square hole in plates and a circular,horse hoof-shaped or inverted U-shaped tunnel in rock masses.On basic that the flexibility matrix of the skew anisotropic material is known,the stress distributions of the opening skew anisotropic structures are analyzed analytically by using the above-mentioned complex variables function method.The analysis not only considered the influence of the hole shapes and external loadings on the stress solutions,but also compared with the isotropic materials.The theoretical solutions are also verified by ANSYS numerical results.The research shows that the stress distributions of the opening skew anisotropic structures also have obvious asymmetry,and the maximum tangential stresses along the hole boundary are much larger than that of the isotropic structures.Besides,when the external loadings are tensile(compressive)stresses,the stresses at the hole boundary are mainly tensile(compressive)stresses too,and the compressive(tensile)stresses are very small.What's more,the small compressive(tensile)stress region deviates from the intersection area of the hole boundary and the y-axis due to the skew anisotropic properties of the material.In addition,there are always four points at the hole boundary,where the tangential stresses are equal to zero for the uniform uniaxial loadings at infinity.