Method for calculating opening response of shield tunnel with supports

A method for calculating an opening response of a shield tunnel with supports, wherein the shield tunnel with supports comprises the shield tunnel and the supports arranged in the shield tunnel, the shield tunnel comprises segments, circumferential joints between adjacent rings, and longitudinal joints spliced and assembled to each other, and the shield tunnel is provided with a rectangular opening, wherein the method comprises following steps:

• step (1) treating each of the segments as a curved beam with a rectangular section, wherein a curve radius in the curved beam of the segments is recorded as R, and a coordinate axis z-s is established along a radial direction and a tangential direction, with a z direction being the radial direction, a s direction being the tangential direction, a radial displacement is recorded as w, a tangential displacement is recorded as u, a rotation angle of a segment cross-section is recorded as p, a shear force, an axial force and a bending moment on the segment cross-section is recorded as Q, N and M, respectively;
• and
• taking each of the segments as an analysis object, and establishing a segment mechanical model of the shield tunnel;
• step (2) considering a shear deformation of the segment cross-section under the segment mechanical model of the shield tunnel, simulating, by a Timoshenko beam theory, a mechanical behavior of a lining, and normalizing a mechanical behavior of an inter-ring seam between rings by a radial shear spring coefficient kfz and a tangential shear spring coefficient kfs to obtain a normalized radial shear spring coefficient kfz and a normalized tangential shear spring coefficient kfs;
• setting a radial soil spring coefficient kz and a tangential soil spring coefficient ks, simulating, by a Winkler soil spring, the interaction between the shield tunnel and a stratum, and performing normalization to obtain a normalized radial soil spring coefficient kz and a normalized tangential soil spring coefficient ks;
• obtaining a radial load qz and a tangential load qs borne by the segments, and performing normalization to obtain a normalized radial load qzi and a normalized tangential load qsi;
• establishing a segment state equation;
• and
• solving a standard solution according to the segment state equation to obtain a transfer equation and a transfer matrix inside the segments;
• step (3) setting a joint radial spring coefficient kw, a joint tangential spring coefficient ku and a joint rotational spring coefficient kφ, and simulating a mechanical behavior of the longitudinal joints of the shield tunnel connected to each other along a circumferential direction of the segments using a radial, tangential and rotational three-directional joint spring;
• considering staggered joint assembly of the segments of the shield tunnel, following a principle of a continuous internal force and a discontinuous displacement where at least one of the longitudinal joints connected to each other along the circumferential direction of the segments exists;
• setting virtual joints no joint exists, following a principle of both continuous internal force and displacement, establishing a longitudinal joint equation of the shield tunnel connected along the circumferential direction of the segments, transforming the longitudinal joint equation to obtain a transfer equation between a tail end of one segment and a starting end of a next segment, and arranging the transfer equation into a form of a matrix equation to obtain a joint transfer matrix;
• step (4) dividing the longitudinal joints connected to each other along the circumferential direction of the segments and connection positions between the supports and the segments of the shield tunnel into multiple segment sections, alternately using the transfer equation inside the segments obtained in the step (2) and the transfer equation between the tail end of one segment and the starting end of the next segment obtained in the step (3) along the circumferential direction of the shield tunnel according to a sequence of the segment sections, until the circumferential direction turns back to a circle, that is, establishing an initial integral ring matrix equation, wherein a left side of the initial integral ring matrix equation is an initial coefficient matrix right multiplied by a state vector comprising radial displacements, tangential displacements, section rotation angles, shear forces, axial forces and bending moments of the sections at the tail end and the starting end of all segments after normalization, a right side of the initial integral ring matrix equation is a vector comprising a load integral vector and a zero vector, the initial integral ring matrix equation comprises a transfer inside the segments and a joint transfer caused by the longitudinal joints, excluding a transfer at the connection positions between the supports and the segments;
• step (5) a length of the rectangular opening along an axial direction of the shield tunnel being an integer multiple of a width of the segments, and using a free boundary condition for simulation at the rectangular opening, with the internal force being 0, that is, a supplementary condition, constructing a corresponding supplementary coefficient matrix, screening out the internal force corresponding to an opening position in the state vector through the supplementary coefficient matrix, the internal force at the opening position being a corresponding zero vector to obtain a supplementary opening equation;
• step (6) taking each of the supports as a straight beam, considering a shear deformation of a cross section of each of the supports under a support mechanical model, and stimulating a mechanical behavior of the supports by the Timoshenko beam theory;
• and
• establishing a coordinate axis z-s along the straight beam, with a s direction being the axial direction and a z direction being perpendicular to the axial direction;
• denoting a displacement perpendicular to the axial direction as w and an axial displacement as u, considering that the section rotation angle of the segments is denoted as φ, and the shear force, the axial force and the bending moment of the sections are denoted as Q, N and M, respectively, taking each of the supports as an analysis object, establishing the support mechanical model to obtain a state equation of each of the supports, solving the state equation to obtain a transfer equation inside the supports, and integrating the transfer equation in multiple supports to obtain a total transfer equation inside the supports;
• step (7) segmenting from an intersection node, calculating and obtaining a balance equation of a support node at the intersection node during calculation, as rigid connection, according to a principle of same displacement and internal force balance;
• step (8) calculating the connection positions between the supports and the segments as rigid connection, and obtaining a balance equation at the intersection of the supports and the segments according to the principle of same displacement and internal force balance;
• and
• step (9) integrating the initial integral ring matrix equation obtained in the step (4) with the supplementary opening equation obtained in the step (5), the transfer equation inside the supports obtained in the step (6), the balance equation of the support node obtained in the step (7) and the balance equation at the intersection of the supports and the segments obtained in the step (8) to form a final integral ring matrix equation, and solving a final matrix equation to obtain the state vector comprising radial displacements, tangential displacements, section rotation angles, shear forces, axial forces and bending moments of the sections at the tail end and the starting end of all segments and the supports after normalization;
• and
• obtaining a state vector comprising radial displacements, tangential displacements, section rotation angles, shear forces, axial forces and bending moments of any section of all segments and the supports after normalization according to the transfer equation inside the segments obtained in the step (2) and the transfer equation inside the supports obtained in the step (6), performing reverse normalization on the obtained state vector comprising the radial displacements, the tangential displacements, the section rotation angles, the shear forces, the axial forces and the bending moments of each section after normalization to obtain the opening response of the shield tunnel, obtaining optimized structures, positions and quantities of all the segments and the supports, determining support and reinforcement structures of the shield tunnel and manufacturing all the segments and the supports based on the opening response of the shield tunnel, and constructing the shield tunnel with the segments and the supports based on the optimized structures, positions, quantities and the support and reinforcement structures.