An analytical method for the reasonable arch axis of an underground arch structure expressed by a stress function

Abstract

The present invention is an analytical method for the rational arch axis of an underground arch structure expressed by stress functions, specifically including: S1. Solve the stress function around the underground arch structure under a specific load mode to obtain the general solution of the rational arch axis under this load mode; S2. Apply the boundary conditions to solve the undetermined parameters in the general solution of the rational arch axis to obtain the equation of the rational arch axis of the underground arch structure. The analytical method for the rational arch axis of the underground arch structure expressed by stress functions provided by the present invention has clear physical meanings, is simple to operate and easy to master, avoids complex structural mechanics solutions, can be used to determine the rational arch axis under various load modes, and has good engineering application prospects and technology promotion prospects.

Description

Technical Field

[0001] The present invention relates to the technical field of determining the reasonable arch axis of an underground arch structure, and particularly to an analytical method for the reasonable arch axis of an underground arch structure expressed by a stress function. Background Art

[0002] The reasonable arch axis of a structure makes the structure in a state of uniform compression of the entire cross-section. For brittle materials such as concrete and stone with high compressive strength and low tensile strength, the reasonable arch axis is often considered to be optimal and is widely used in the design of buildings such as arch bridges, dams, and domes. There are also similar arched structures in underground engineering, such as culverts, tunnels, and shafts. Such arch structures generally directly contact the formation and bear the water and soil pressure in the surrounding formation environment. When the underground arch structure has a reasonable arch axis, the mechanical properties of the structure are optimal.

[0003] Generally, determining the reasonable arch axis belongs to the category of structural mechanics principles. Under a known load pattern, the bending moment of each cross-section of the structure is obtained and set to zero to obtain the equation of the reasonable arch axis. Different load patterns result in different forms of the reasonable arch axis. Common forms of the reasonable arch axis include parabola, catenary, circular arc, etc. For complex loads, it is usually difficult to directly obtain. For example, when solving the reasonable arch axis under the action of filling load, the horizontal arch reaction is assumed in advance, and the obtained catenary equation contains this unknown variable and cannot be directly applied. For the convenience of engineering application, engineers use high-order parabolas for fitting according to experience or perform multiple adjustments and optimizations on a pre-determined curve. The optimization results basically meet the engineering requirements, but the physical meaning is not clear enough, and it is difficult to obtain the internal force distribution of the reasonable arch axis. Summary of the Invention

[0004] The present invention aims to solve the deficiencies of the prior art and provides an analytical method for the reasonable arch axis of an underground arch structure expressed by a stress function.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] An analytical method for the reasonable arch axis of an underground arch structure expressed by a stress function specifically includes:

[0007] S1. Solve the stress function around the underground arch structure under a specific load pattern to obtain the general solution of the reasonable arch axis under this load pattern;

[0008] S2. Apply the boundary conditions to solve the undetermined parameters in the general solution of the reasonable arch axis to obtain the equation of the reasonable arch axis of the underground arch structure.

[0009] In step S1, starting from the elastic theory, the relationship between the reasonable arch axis of the underground arch structure and the stress function is deduced, specifically as follows:

[0010] Assume that the spatial stress of the underground arch structure belongs to a plane strain problem, and the arch plane of the underground arch structure is simplified to a two-dimensional arch plane;

[0011] In the plane problem, without considering the body force of the soil mass, the stress state at any position 、 and is expressed by the stress function as follows:

[0012] ;

[0013] In the formula,

[0014] 、 are respectively the abscissa and ordinate of the two-dimensional arch plane;

[0015] 、 and are respectively the second-order partial differentials of the stress function with respect to the coordinates and ;

[0016] It is pre-assumed that the underground arch structure conforms to a reasonable arch axis, and a calculation unit is taken at any point in the arch structure position;

[0017] The stress is positive when the soil element is compressed, and the self-weight of the underground arch structure is ignored, and the differential equilibrium equation and geometric equation of the calculation unit are obtained as follows:

[0018] ;

[0019] ;

[0020] In the formula,

[0021] is the axial pressure of the underground arch structure at this point;

[0022] 、 and are the soil stresses at this point;

[0023] is the angle between the tangent of the arch axis and the axis;

[0024] and are respectively the infinitesimals of the coordinates and ;

[0025] By introducing the stress function, the following total differential equation is obtained:

[0026] ;

[0027] Integrating gives a set of partial differential equations:

[0028] ;

[0029] wherein, and are arbitrary constants;

[0030] Combining with the geometric equations gives a new total differential equation:

[0031] ;

[0032] Integrating gives the equation of the reasonable arch axis of the underground arch structure:

[0033] ;

[0034] wherein, is an arbitrary constant;

[0035] Adding a linear term to the stress function is still the stress function of the stress. The above expression is represented by the new stress function . The unknowns in the stress function are determined by the boundary conditions. Determining the reasonable arch axis of the underground arch structure is transformed into solving the stress function near the arch structure. The analytical solution of the reasonable arch axis of the underground arch structure expressed by the above stress function satisfies the biharmonic equation;

[0036] If the stress around the underground arch structure is decomposed into multiple groups of stresses , each group of stresses corresponds to its own stress function respectively. Adding these stress functions is still the stress function of the original stress. When the underground arch structure is subjected to multiple stresses, the reasonable arch axis of the underground arch structure is obtained by adding the stress functions obtained by solving each stress function separately:

[0037] .

[0038] Using the partial differential equation gives the axial force at any position of the underground arch structure expressed by the stress function:

[0039] .

[0040] The stress of the underground arch structure exerted by the formation is expressed as:

[0041] ;

[0042] wherein, and They are the horizontal distributed force and the vertical distributed force acting on the underground arch structure respectively;

[0043] Introduce the stress function to obtain the partial differential equation about the stress function:

[0044] ;

[0045] Therefore, the stress function obtained by solving this partial differential equation is the reasonable arch axis equation of the underground arch structure;

[0046] When it is, the general solution of the reasonable arch axis of the underground arch structure expressed by the stress function is obtained:

[0047] ;

[0048] In the formula, is the undetermined number, which is determined by the boundary conditions;

[0049] Under the action of multiple groups of loads, the reasonable arch axis of the underground arch structure is expressed as:

[0050] .

[0051] The specific load patterns include the vertical uniform load pattern, the circumferential uniform load pattern, and the fill load pattern.

[0052] The beneficial effects of the present invention are: The method for analyzing the reasonable arch axis of the underground arch structure expressed by the stress function provided by the present invention has clear physical meanings, is simple to operate, easy to master, and avoids complex structural mechanics solutions. The method provided by the present invention can be used to determine the reasonable arch axis under various load patterns, and has good engineering application prospects and technology promotion prospects. Brief Description of the Drawings

[0053] Figure 1 is the schematic diagram of the underground structure and the surrounding strata of the present invention;

[0054] Figure 2 is the calculation sketch of any position of the underground arch structure of the present invention;

[0055] Figure 3 is the load simplified model of the underground arch structure of the present invention;

[0056] Figure 4 is the calculation sketch of Case 1 of the present invention;

[0057] Figure 5 is the calculation sketch of Case 2 of the present invention;

[0058] Figure 6 is the calculation sketch of Case 3 of the present invention;

[0059] In the figure: 1 - The strata where the underground arch structure is located; 2 - The underground arch structure; 21 - The vertically distributed load; 22 - The horizontally distributed load; 3 - The calculation unit at any position of the underground arch structure.

[0060] The following will be described in detail with reference to the embodiments of the present invention and the accompanying drawings. Specific embodiments

[0061] The principles and features of the present invention will be described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention. In the following paragraphs, the present invention will be described more specifically by way of example with reference to the accompanying drawings. According to the following description, the advantages and features of the present invention will be clearer. It should be noted that the drawings are all in a very simplified form and use non-precise scales, only for the purpose of facilitating and clearly assisting in explaining the objectives of the embodiments of the present invention.

[0062] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present invention belongs. The terms used in the description of the present invention herein are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" used herein includes any and all combinations of one or more of the related listed items.

[0063] The present invention will be further described below with reference to the accompanying drawings and embodiments:

[0064] Based on the elastic theory, the present invention provides an analytical method for determining the reasonable arch axis of an underground arch structure expressed by a stress function. The present invention is realized through the following technical solutions:

[0065] In the first aspect, starting from the elastic theory, the present invention derives the relationship between the reasonable arch axis of the underground arch structure and the stress function. It is assumed that the spatial stress of the underground arch structure belongs to a plane strain problem, and its arch plane can be simplified to a two-dimensional arch plane.

[0066] In the plane problem, without considering the body force of the soil mass, the stress state at any position , and can be expressed by the stress function as follows:

[0067] ;

[0068] In the formula,

[0069] , are respectively the abscissa and ordinate of the two-dimensional arch plane;

[0070] , and are the second-order partial derivatives of the stress function with respect to the coordinates and respectively.

[0071] The present invention presupposes that the underground arch structure conforms to a reasonable arch axis. A calculation unit is taken at any point in the position of the arch structure, as Figure 1 shown, including the strata 1 in which the underground arch structure is stored, the underground arch structure 2, and the calculation unit 3 at any position of the underground arch structure. The calculation unit 3 at any position of the underground arch structure is as Figure 2 shown. The stress is considered positive when the soil element is compressed, and the self-weight of the underground arch structure is ignored. The differential equilibrium equation and geometric equation of the calculation unit are obtained as follows:

[0072] ;

[0073] ;

[0074] In the formula, is the axial pressure of the underground arch structure at this point, , and are the soil stresses at this point, is the angle between the tangent of the arch axis and the axis, and are the infinitesimals of the coordinates and respectively.

[0075] By introducing the stress function, the following total differential equation can be obtained:

[0076] ;

[0077] Integrating gives a set of partial differential equations

[0078] ;

[0079] In the formula, and are arbitrary constants.

[0080] Combining with the geometric equation, a new total differential equation can be obtained:

[0081] ;

[0082] Integrating gives the reasonable arch axis equation of the underground arch structure.

[0083] ;

[0084] In the formula, is an arbitrary constant.

[0085] Adding a linear term to the stress function still results in a stress function for the same stress. The above expression can be represented by a new stress function The unknowns in the stress function are determined by the boundary conditions. Therefore, determining the reasonable arch axis of the underground arch structure is transformed into solving the stress function near the arch structure. The analytical solution of the reasonable arch axis of the underground arch structure expressed by the above stress function should satisfy the biharmonic equation.

[0086] If the stresses around the underground arch structure are decomposed into multiple groups of stresses each group of stresses corresponds to its own stress function respectively. Summing up these stress functions is still a stress function of the original stress. Therefore, when the underground arch structure is subjected to multiple stress actions, the reasonable arch axis of the underground arch structure can be obtained by separately solving the respective stress functions and then summing them up.

[0087] ;

[0088] This situation is used when the underground arch structure usually bears multiple loads such as water and soil pressure or when the load form borne by the underground arch structure is relatively complex. The load can be decomposed first and then the stress functions can be solved one by one.

[0089] In addition, using the above partial differential equation, the axial force at any position of the underground arch structure expressed by the stress function can also be obtained:

[0090] .

[0091] On the second aspect, the load - structure method is usually adopted for the structural analysis of the underground arch structure, and the action of the stratum on the structure is directly expressed in the form of load. When determining the reasonable arch axis of the underground arch structure, the present invention can be used to solve the reasonable arch axis under a given load pattern.

[0092] The simplified model of the load of the stratum acting on the underground arch structure, as Figure 3 shown, includes the underground arch structure 2, the calculation unit 3 at any position of the underground arch structure, the vertical distributed load 21, and the horizontal distributed load 22. The stress of the underground arch structure subjected to the stratum can be expressed as:

[0093] ;

[0094] In the formula, and are the horizontal distributed force and the vertical distributed force respectively that the underground arch structure is subjected to;

[0095] By introducing the stress function, a partial differential equation about the stress function is obtained:

[0096] ;

[0097] Therefore, the stress function obtained by solving this partial differential equation is the reasonable arch axis equation of the underground arch structure.

[0098] After the load is determined, the reasonable arch axis of the arch structure is also determined accordingly. The coordinate variables of the arch axis are not independent of each other. Therefore, the vertical distributed force of the arch structure can be expressed in the form of or , and the horizontal distributed force can also be expressed as or . In particular, when , the general solution of the reasonable arch axis of the underground arch structure expressed by the stress function can be obtained:

[0099] ;

[0100] In the formula, are undetermined numbers, which are determined by the boundary conditions.

[0101] Under the action of multiple groups of loads, the reasonable arch axis of the underground arch structure can be expressed as:

[0102] .

[0103] The analytical method for the reasonable arch axis of the underground arch structure proposed by the present invention starts from the stress distribution around the underground arch structure and can be directly used to solve the reasonable arch axis equation under a specific load pattern.

[0104] Thirdly, the steps for applying the analytical method for the reasonable arch axis of the underground arch structure proposed by the present invention to solve the reasonable arch axis of the underground arch structure are as follows:

[0105] The first step: Solve the stress function around the underground arch structure under a specific load pattern to obtain the general solution of the reasonable arch axis under this load pattern;

[0106] The second step: Apply the boundary conditions to solve the undetermined parameters in the general solution of the reasonable arch axis to obtain the reasonable arch axis equation of the underground arch structure.

[0107] The specific load pattern includes but is not limited to the vertical uniform load pattern, the circumferential uniform load pattern, and the filling load pattern.

[0108] The boundary conditions vary according to the actual case.

[0109] Applying the technical solution proposed by the present invention, several cases with symmetric loads and the same boundary conditions are analyzed. The boundary conditions of the underground arch structure in the cases are:

[0110] Boundary condition 1: When ;

[0111] Boundary condition 2: When ;

[0112] Boundary condition 3: When ;

[0113] Boundary condition 4: When .

[0114] Case 1: The reasonable arch axis of the underground arch structure under vertical uniformly distributed load, as Figure 4 shown.

[0115] ;

[0116] The stress function under this stress state is calculated as:

[0117] ;

[0118] Applying the boundary conditions, the equation of the reasonable arch axis of the underground arch structure under vertical uniformly distributed load is obtained:

[0119] .

[0120] Case 2: The reasonable arch axis of the underground arch structure under circumferential uniformly distributed load, as Figure 5 shown.

[0121] ;

[0122] The stress function under this stress state is calculated as:

[0123] ;

[0124] Applying the boundary conditions, the equation of the reasonable arch axis of the underground arch structure under circumferential uniformly distributed load is obtained:

[0125] .

[0126] Case 3: The reasonable arch axis of the underground arch structure under fill load, as Figure 6 shown.

[0127] ;

[0128] The partial differential equation for the stress function is obtained:

[0129] .

[0130] The stress function under this stress state is calculated as follows:

[0131] .

[0132] Applying the boundary conditions, the equation of the rational arch axis of the underground arch structure under the fill load is obtained:

[0133] .

[0134] In the formula, , representing the horizontal thrust at the arch seat of the arch structure.

[0135] The analytical method for the rational arch axis of the underground arch structure expressed by the stress function provided by the present invention has clear physical meanings, is simple to operate and easy to master, and avoids complex structural mechanics solutions. The method provided by the present invention can be used to determine the rational arch axis under various load modes, and has good prospects for engineering application and technical popularization.

[0136] The present invention can solve the rational arch axis of the underground arch structure under specific load modes, including the vertical uniform load mode, the circumferential uniform load mode, the fill load mode, etc.

[0137] The present invention has been described exemplarily above in conjunction with the accompanying drawings. Obviously, the specific implementation of the present invention is not limited by the above methods. As long as various improvements are made by adopting the method concept and technical solution of the present invention, or directly applied to other occasions without improvement, they are all within the protection scope of the present invention.

Claims

1. An analytical method for determining a reasonable arch axis of an underground arch structure expressed by a stress function, characterized in that: Specifically include: S1. Solve the stress function around the underground arch structure under a specific load mode and obtain the general solution of the reasonable arch axis under the load mode; Based on the elasticity theory, the relationship between the reasonable arch axis and stress function of the underground arch structure is derived as follows: Assuming that the spatial force of the underground arch structure belongs to a plane strain problem, the arch plane of the underground arch structure is simplified to a two-dimensional arch plane; In plane problems, ignoring the soil body force, the stress state of the soil at any position is , and Using stress function The expression is as follows: ； In the formula, , are the abscissa and ordinate of the two-dimensional arch plane respectively; , and The stress functions are Coordinates and The second partial differential of ; It is pre-assumed that the underground arch structure conforms to the reasonable arch axis, and the calculation unit is taken at any point of the arch structure; The stress is such that the soil unit is compressed as positive, and the weight of the underground arch structure itself is ignored. The differential equilibrium equation and geometric equation of the calculation unit are obtained as follows: ； ； In the formula, is the axial pressure of the underground arch structure at that location; , and is the soil stress state at that location; is the tangent line to the arch axis The angle of the axis; and The coordinates are and The microelement of Introducing the stress function, we get the following total differential equation: ； Integrating gives a set of partial differential equations: ； In the formula, and is an arbitrary constant; Combined with the geometric equation, we get the new total differential equation: ； The general solution of the reasonable arch axis of the underground arch structure is obtained by integration: ； In the formula, is an arbitrary constant; Adding a first-order term to the stress function still gives the stress function of the stress. With the new stress function It means that the unknowns in the stress function are determined by the boundary conditions, and the determination of the reasonable arch axis of the underground arch structure is transformed into the solution of the stress function near the arch structure. The reasonable arch axis of the underground arch structure expressed by the above stress function analytically satisfies the biharmonic equation; If the stress around the underground arch structure is decomposed into multiple groups of stress When each group of stress Corresponding to their respective stress functions , the superposition of these stress functions is still the stress function of the original stress. When the underground arch structure is subjected to multiple stresses, the stress functions of each are solved separately and then superimposed to obtain the general solution of the reasonable arch axis of the underground arch structure: ； S2. Apply boundary conditions to solve the unknown parameters in the general solution of the reasonable arch axis and obtain the reasonable arch axis equation of the underground arch structure.

2. The analytical method for determining the reasonable arch axis of an underground arch structure expressed by a stress function according to claim 1 is characterized in that: Using partial differential equations The axial force at any position of the underground arch structure expressed by the stress function is obtained: 。 3. The analytical method for determining the reasonable arch axis of an underground arch structure expressed by a stress function according to claim 2 is characterized in that: The stress of the underground arch structure on the stratum is expressed as: ； In the formula, and are the horizontal distributed force and vertical distributed force on the underground arch structure respectively; Introducing the stress function, we get the partial differential equation about the stress function: ； Therefore, the stress function obtained by solving the partial differential equation is the reasonable arch axis equation of the underground arch structure; when When , the general solution of the reasonable arch axis of the underground arch structure expressed by the stress function is obtained: ； In the formula, is an unknown number, determined by boundary conditions; Under the action of multiple groups of loads, the general solution of the reasonable arch axis of the underground arch structure is: 。 4. The analytical method for determining a reasonable arch axis of an underground arch structure expressed by a stress function according to claim 1, characterized in that: Specific load modes include vertical uniformly distributed load mode, circumferential uniformly distributed load mode, and fill load mode.

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