Solution Of Hamiltonian System For Special Mechanical Problems Of Mine And Underground Engineering

With the development of coal mining from shallow to deep and the expansion of mining area,the geological environment around the shaft wall also changes,and the damage degree and damage mode of the shaft wall are also different.Accurately depicting the stress field around the wellbore and predicting the deformation of the wellbore has become an urgent engineering problem to be solved to ensure the safety of the wellbore and roadway.The essence of these complex engineering problems is mechanical problems.To solve these problems,we need not only the use of modern mathematical and physical methods and research tools,but also the need to integrate theory with practice,and close cooperation between engineers and researchers.In this paper,the symplectic elastic mechanics method is applied to the analysis of the mechanical problems of the circular and non-circular roadways,multi-layer thick wall cylinder,vertical shaft and other engineering structures with complex boundary conditions,as well as the stress and displacement of the surrounding rock.Starting from the basic differential equations of elasticity and based on the generalized energy variational principle,the Hamiltonian dual equations are established by introducing the dual variable of displacement according to Legendre transformation.Furthermore,the mechanical problem composed of displacement variables in the original Euclidean space is transformed into a new mechanical problem composed of dual variables in the symplectic geometric space.According to the characteristics of symplectic geometric space and Hamiltonian dual equations,the separation variable method is used to solve the mixed state equation under the homogeneous boundary condition of mixed variable representation,and the symplectic eigenvector and symplectic eigenvalue analytical expression of the problem are obtained.The symplectic system solution method for the mechanical problems of mine shaft and tunnel engineering established in this paper provides a new way for equivalent analysis of similar mechanical problems of mine and underground engineering.(1)In order to solve the plane strain problem of circular tunnel,the Hamiltonian mechanical solution model of circular sector region is established in polar coordinate system,and the general solution and special solution expressions of mixed state differential equation under homogeneous and nonhomogeneous boundary conditions are derived.By comparing the results of FEM and symplectic method to calculation of tunnel surrounding rock stress,the accuracy and reliability of symplectic method are verified.The variation of stress at the hole edge of circular tunnel with the coefficient of lateral pressure is discussed.The smaller the lateral pressure coefficient,the more uneven the stress distribution;when the lateral pressure coefficient is less than 0.3,tensile stress begins to appear at the edge of the hole.Especially when the coefficient of lateral pressure is equal to 0,the tensile stress at the hole edge reaches the extreme value.(2)According to the boundary condition and continuous smooth condition,the coordination equation is established for the mechanical problem of multi-layer thick wall cylinder.The difference of stress field and displacement field on the inner and outer contact surface of thick wall cylinder under the condition of smooth contact and close connection is discussed.The influences of the lateral pressure coefficient and the elastic modulus ratio of the thick wall material on the stress field are discussed.It is concluded that the softer the shaft wall material,the smaller the shared stress value,and the harder the wall material,the larger the shared stress value.The extreme value of the circumferential stress generally appears in the area of the wall with a larger elastic modulus.(3)While using conformal mapping to realize region transformation,the stress component,displacement component and boundary condition are changed correspondingly.The mechanical problem of non-circular tunnel is transformed into the boundary value problem of circular area.Combined with symplectic algorithm,the stress field distribution of surrounding rock of elliptical tunnel is given.The influences of internal pressure,shape coefficient and lateral pressure coefficient on the stress field of surrounding rock are discussed through examples.The results show that the increase of internal pressure can effectively reduce the pressure stress of surrounding rock and help to improve the strength of surrounding rock;with the increase of lateral pressure coefficient,the fluctuation range of surrounding rock circumferential stress becomes smaller;the minimum value of surrounding rock circumferential stress is independent of shape coefficient,and the maximum value is closely related to shape coefficient.(4)According to the problem of vertical shaft mechanics with spatial rotational symmetry,the Hamiltonian mixed state equation is established in the spatial cylindrical coordinate system,and the general solution form of the mixed state equation is given by the method of separating variables.The unknown parameters in the general solution are determined according to the boundary conditions of the side and end of the wellbore.The local solution of the shaft end is analyzed by an engineering example,and the applicable conditions and scope of Saint-Venant principle are discussed.The influence of lateral pressure coefficient,wellbore thickness and wellbore radius on stress distribution in different shaft depths is discussed.These conclusions provide an important theoretical basis for the analysis of shaft stress,the improvement of shaft wall design and the control of shaft deformation and damage.