Analytical method for reasonable arch axis of underground arch structure expressed by stress function

Abstract

The invention relates to a stress function expressed analysis method for a reasonable arch axis of an underground arch structure, which specifically comprises the following steps: S1, solving a stress function around the underground arch structure in a specific load mode to obtain a reasonable arch axis general solution in the load mode; and S2, applying boundary conditions to solve undetermined parameters in the reasonable arch axis general solution to obtain a reasonable arch axis equation of the underground arch structure. The method for analyzing the reasonable arch axis of the underground arch structure expressed by the stress function, provided by the invention, has clear physical significance, is simple to operate and convenient to master, avoids complex structural mechanics solution, can be used for determining the reasonable arch axis under various load modes, and has good engineering application prospects and technical popularization prospects.

Description

Technical Field

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

[0002] The reasonable arch axis of the structure makes the structure in a uniformly compressed state across the entire cross section. For brittle materials such as concrete and stone that have high compressive strength and low tensile strength, the reasonable arch axis is often considered to be the best, and is widely used in the design of buildings such as arch bridges, dams, and domes. Similar arch structures also exist in underground engineering, such as culverts, tunnels, and shafts. This type of arch structure is generally in direct contact with the stratum and bears the water and soil pressure in the stratum environment. When the underground arch structure is a reasonable arch axis, the structural force performance is optimal.

[0003] Usually, determining the reasonable arch axis is a matter of structural mechanics principles. Under a known load mode, the bending moment of each section of the structure is obtained and set to zero to obtain the equation of the reasonable arch axis. The reasonable arch axis is different for different load modes. Common reasonable arch axes are parabolas, catenaries, circular arcs, etc. For complex loads, it is usually difficult to obtain directly. For example, when solving the reasonable arch axis under fill load, the horizontal arch reaction force is pre-assumed. The obtained catenary equation contains this unknown variable and cannot be directly applied. In order to facilitate engineering applications, engineers use high-order parabolas for fitting based on experience or adjust and optimize the pre-determined curves for multiple times. The optimization results meet the basic 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 a reasonable arch axis of an underground arch structure expressed by a stress function.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

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

[0007] 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;

[0008] 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.

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

[0010] 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;

[0011] In plane problems, ignoring the soil body force, the stress state at any position is , and Using stress function The expression is as follows:

[0012] ;

[0013] In the formula,

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

[0015] , and The stress functions are Coordinates and The second partial differential of ;

[0016] 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;

[0017] 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:

[0018] ;

[0019] ;

[0020] In the formula,

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

[0022] , and is the soil stress at that point;

[0023] is the tangent line to the arch axis The angle of the axis;

[0024] and The coordinates are and The microelement of

[0025] Introducing the stress function, we get the following total differential equation:

[0026] ;

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

[0028] ;

[0029] In the formula, and is an arbitrary constant;

[0030] Combined with the geometric equation, we get the new total differential equation:

[0031] ;

[0032] The integration results in the reasonable arch axis equation of the underground arch structure:

[0033] ;

[0034] In the formula, is an arbitrary constant;

[0035] Adding a first-order term to the stress function still gives the stress function of the stress. The above expression uses 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;

[0036] 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 reasonable arch axis of the underground arch structure:

[0037] .

[0038] Using partial differential equations The axial force at any position of the underground arch structure expressed by the stress function is obtained:

[0039] .

[0040] The stress of the underground arch structure on the stratum is expressed as:

[0041] ;

[0042] In the formula, and are the horizontal distributed force and vertical distributed force on the underground arch structure respectively;

[0043] Introducing the stress function, we get the partial differential equation about the stress function:

[0044] ;

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

[0046] when When , 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 an unknown number, determined by 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] Specific load modes include vertical uniformly distributed load mode, circumferential uniformly distributed load mode, and fill load mode.

[0052] The beneficial effects of the present invention are as follows: the analytical method for the reasonable arch axis of the underground arch structure expressed by the stress function provided by the present invention has clear physical meaning, 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 modes, and has good engineering application prospects and technology promotion prospects. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0054] Figure 2 It is a simplified diagram for calculating any position of the underground arch structure of the present invention;

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

[0056] Figure 4 It is a calculation diagram of case 1 of the present invention;

[0057] Figure 5 It is a calculation diagram of case 2 of the present invention;

[0058] Figure 6 It is a calculation diagram of case 3 of the present invention;

[0059] In the figure: 1-stratum where the underground arch structure is stored; 2-underground arch structure; 21-vertical distributed load; 22-horizontal distributed load; 3-calculation unit at any position of the underground arch structure.

[0060] The following is a detailed description of the embodiments of the present invention with reference to the accompanying drawings. DETAILED DESCRIPTION

[0061] The principles and features of the present invention are described below in conjunction with the accompanying drawings. The embodiments are only used to explain the present invention and are not used to limit the scope of the present invention. The present invention is described in more detail by way of example with reference to the accompanying drawings in the following paragraphs. The advantages and features of the present invention will become clearer according to the following description. It should be noted that the accompanying drawings are all in a very simplified form and are not in precise proportions, and are only used to facilitate and clearly assist in explaining the purpose of the embodiments of the present invention.

[0062] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by those skilled in the art to which the present invention belongs. The terms used herein in the specification of the present invention 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 in conjunction with the accompanying drawings and embodiments:

[0064] The present invention is based on elasticity theory and provides an analytical method for determining a reasonable arch axis of an underground arch structure expressed by a stress function. The present invention is implemented by the following technical solutions:

[0065] In the first aspect, the present invention starts from the elasticity theory and derives the relationship between the reasonable arch axis and the stress function of the underground arch structure. Assuming that the spatial force of the underground arch structure belongs to a plane strain problem, the arch plane can be simplified to a two-dimensional arch plane.

[0066] In plane problems, ignoring the soil body force, the stress state at any position is , and The stress function can be used The expression is as follows:

[0067] ;

[0068] In the formula,

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

[0070] , and The stress functions are Coordinates and The second-order partial differential of .

[0071] The present invention presupposes that the underground arch structure conforms to the reasonable arch axis, and takes a calculation unit at any point of the arch structure, such as Figure 1 As shown, it includes an underground arch structure occurrence stratum 1, an underground arch structure 2, and a 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 follows: Figure 2 As shown. The stress is taken to make the soil unit under pressure positive, and the weight of the underground arch structure itself is ignored, and the differential equilibrium equation and geometric equation of the calculation unit are obtained:

[0072] ;

[0073] ;

[0074] In the formula, is the axial pressure of the underground arch structure at this point, , and is the soil stress at that point, is the tangent line to the arch axis The angle of the axis, and The coordinates are and The microelement of .

[0075] By introducing the stress function, we can obtain the following total differential equation:

[0076] ;

[0077] Integration gives a set of partial differential equations

[0078] ;

[0079] In the formula, and is an arbitrary constant.

[0080] Combined with the geometric equation, we can get the new total differential equation:

[0081] ;

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

[0083] ;

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

[0085] Adding a first-order term to the stress function still gives the stress function of the stress. The above expression can be expressed with the new stress function It means that the unknowns in the stress function are determined by the boundary conditions. Therefore, 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 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 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. Therefore, when the underground arch structure is subjected to multiple stresses, the stress functions of each can be solved separately and then superimposed to obtain the reasonable arch axis of the underground arch structure.

[0087] ;

[0088] This situation is used when the underground arch structure is usually subjected to multiple loads such as water and soil pressure, or when the underground arch structure is subjected to more complex loads. The load can be decomposed first, and then the stress function can be solved one by one.

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

[0090] .

[0091] Secondly, underground arch structures are usually analyzed using the load-structure method, in which the effect 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 mode.

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

[0093] ;

[0094] In the formula, and are the horizontal distributed force and vertical distributed force on the underground arch structure respectively;

[0095] Introducing the stress function, we get the partial differential equation about the stress function:

[0096] ;

[0097] Therefore, the stress function obtained by solving the 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. The coordinate variables of the arch axis are not independent of each other. Therefore, the vertical distribution force of the arch structure Pressable or The form of expression, horizontal distribution force It can also be expressed as or . In particular, when 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, is an unknown number determined by 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 in the present invention takes the stress distribution around the underground arch structure as the starting point and can be directly used to solve the reasonable arch axis equation under a specific load mode.

[0104] In the third aspect, the method for analyzing the reasonable arch axis of an underground arch structure proposed in the present invention is used to solve the reasonable arch axis of the underground arch structure as follows:

[0105] Step 1: 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 this load mode;

[0106] Step 2: 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.

[0107] Specific load modes include but are not limited to vertical uniformly distributed load mode, circumferential uniformly distributed load mode, and fill load mode.

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

[0109] The technical solution proposed in this invention is applied to analyze several cases with symmetrical loads and the same boundary conditions. The boundary conditions of the underground arch structure in the case are:

[0110] Boundary Condition 1: hour, ;

[0111] Boundary Condition 2: hour, ;

[0112] Boundary Condition 3: hour, ;

[0113] Boundary Condition 4: hour, .

[0114] Case 1: Reasonable arch axis of underground arch structure under vertical uniform load, such as Figure 4 shown.

[0115] ;

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

[0117] ;

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

[0119] .

[0120] Case 2: Reasonable arch axis of underground arch structure under circumferentially uniformly distributed load, such as Figure 5 shown.

[0121] ;

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

[0123] ;

[0124] By applying the boundary conditions, the reasonable arch axis equation of the underground arch structure under the circumferential uniform load is obtained:

[0125] .

[0126] Case 3: Reasonable arch axis of underground arch structure under fill load, such 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:

[0131] .

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

[0133] .

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

[0135] The analytical method for the reasonable arch axis of the underground arch structure expressed by the stress function provided by the present invention has clear physical meaning, 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 modes, and has good engineering application prospects and technology promotion prospects.

[0136] The present invention can solve the reasonable arch axis of the underground arch structure under specific load modes including vertical uniformly distributed load mode, circumferential uniformly distributed load mode, fill load mode and the like.

[0137] The present invention is described above by way of example in conjunction with the accompanying drawings. It is obvious that the specific implementation of the present invention is not limited to the above-mentioned method. As long as various improvements are made using the method concept and technical solution of the present invention, or are 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; 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: In step S1, starting from the elastic theory, the relationship between the reasonable arch axis and the stress function of the underground arch structure is derived, specifically: 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 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 point; , and is the soil stress at that point; is the tangent line to the arch axis The angle of the axes; 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 integration results in the reasonable arch axis equation of the underground arch structure: ； In the formula, is an arbitrary constant; Adding a first-order term to the stress function still gives the stress function of the stress. The above expression uses 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 reasonable arch axis of the underground arch structure: 。 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: Using partial differential equations The axial force at any position of the underground arch structure expressed by the stress function is obtained: 。 4. The analytical method for determining the reasonable arch axis of an underground arch structure expressed by a stress function according to claim 3 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 reasonable arch axis of the underground arch structure is expressed as: 。 5. The analytical method for determining the 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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