A method for calculating elastic stress of a slope body under support

Abstract

The present application is suitable for geotechnical engineering technical field, the present application provides a kind of slope body elastic stress calculation method under supporting action, comprising the following steps: the slope angle is β for the slope under the action of initial ground stress, slope top overload and supporting force is established mathematical model;The mathematical model is decomposed into: semi-infinite plane model α and gravity-free slope model β;The gravity-free slope model β is decomposed into: infinite plane inside semi-infinite hole model β1 under the action of virtual surface force V b Of hole edge, and, surface virtual surface force V v Under the action of semi-infinite plane model β2 without hole;The stress of semi-infinite plane model α, the stress of semi-infinite plane model β2, the stress of semi-infinite hole model β1 are calculated, and the stress analytical solution of model β is iteratively calculated.The present application can consider the influence of initial ground stress, overload, supporting pressure and anchoring force and other load actions simultaneously, to be suitable for the stress calculation of various angle slopes.

Description

Technical Field

[0001] This invention belongs to the field of geotechnical engineering technology, and in particular relates to a method for calculating the elastic stress of a slope under support. Background Technology

[0002] With the rapid development of my country's economy, the scale of infrastructure construction such as mines, highways and water conservancy has continued to expand, resulting in a large number of artificial high slopes. Their stability has become a hot research topic in the field of geotechnical engineering.

[0003] Slope instability can trigger geological disasters such as landslides and collapses, threatening not only the lives of people in the surrounding area but also causing significant damage to infrastructure and resulting in huge economic losses. Therefore, determining the distribution of elastic stress within the slope is essential in geotechnical engineering designs such as slope stability assessment and reinforcement design.

[0004] In recent years, the complex function method has been widely used to solve the stress field of slopes. However, the complex function method is only applicable to specific slope angles and cannot solve the stress field of slopes with arbitrary slope angles.

[0005] Meanwhile, the complex function method fails to consider the situation where there are distributed or concentrated forces acting on the slope surface. In actual engineering, slopes are usually reinforced by retaining walls, anchors, or soil nails, which will generate reinforcement support forces on the slope surface and bottom surface. Therefore, the complex function method cannot be used in actual engineering.

[0006] To address the aforementioned shortcomings, this invention aims to propose a method for calculating the elastic stress of a slope under support, which can simultaneously consider the effects of loads such as initial ground stress, surcharge, support pressure, and anchoring force, making it applicable to stress calculations for slopes at various angles. Summary of the Invention

[0007] The purpose of this invention is to provide a method for calculating the elastic stress of a slope under support, which can simultaneously consider the effects of loads such as initial ground stress, surcharge, support pressure and anchoring force, so as to be applicable to the stress calculation of slopes at various angles.

[0008] This invention provides a method for calculating the elastic stress of a slope under support, comprising the following steps:

[0009] S1: Establish a mathematical model for a slope with an angle of β subjected to initial ground stress, surcharge at the top of the slope and support force; decompose the mathematical model into: a semi-infinite plane model α and a zero-gravity slope model β; where β is an arbitrary value;

[0010] S2: Decompose the zero-gravity slope model β into: virtual surface force V at the edge of the hole. b The model β1 of a semi-infinite hole in an infinite plane, and the virtual surface force V.v The semi-infinite plane model β2 without holes under the action;

[0011] S3: Calculate the stress in the semi-infinite plane model α;

[0012] S4: Calculate: Virtual surface force V on the surface v Stress in a semi-infinite plane model β2 without holes under action;

[0013] S5: Calculation: Virtual surface force V acting on the edge of the hole b Stress in a semi-infinite hole model β1 within an infinite plane;

[0014] Step S6: Iteratively calculate the analytical solution of stress for model β.

[0015] Preferably, in S1: for a slope with an angle of β subjected to initial ground stress, surcharge at the top of the slope, and support force, it is decomposed into the following according to the superposition principle:

[0016] In a semi-infinite plane model α under initial ground stress and slope crest surload,

[0017] And, a zero-gravity slope model β under the combined action of excavation unloading, support force and excavation construction load.

[0018] Preferably, step S3 includes the following steps:

[0019] S31: Calculate the surrounding rock stress at any point A within the semi-infinite plane model α under the surcharge at the slope crest. ,include

[0020] Determine the number of Gaussian points:

[0021] The overloaded area at the top of the slope w i (i=1,2,...,n) o ) divided into p o,i part;

[0022] Each segment has 5 Gaussian points, that is, the total number of Gaussian points is N. o,i =5·p o,i ;

[0023] Determine the number of equivalent concentrated force groups:

[0024] The number of equivalent concentrated force sets at the Gauss point is:

[0025] ;

[0026] Calculate the stress of surrounding rock under overload. :

[0027] ;

[0028] In the formula:

[0029] ;

[0030] , , Let x represent the normal stress, y represent the normal stress, and shear stress at any point A in the surrounding rock under the surcharge at the top of the slope;

[0031] ;

[0032] matrix Point A represents the stress in the surrounding rock to be determined under ground overload. With the i-th Gaussian point The positional relationship between them;

[0033] ;

[0034] ;

[0035] S32: Solving for the initial geostress at point A yields the solution. ;

[0036] S33: The stress at any point A in the semi-infinite plane model α under initial ground stress and slope crest surcharge is:

[0037] .

[0038] Preferably, step S4 includes the following steps:

[0039] S41: Determine the Gaussian point;

[0040] The load application area is divided into p according to a geometric sequence. v Each segment has 5 Gaussian points, for a total of N Gaussian points. v =5·p v ;

[0041] S42: Calculate the stress at any point A within model β2. :

[0042] ;

[0043] In the formula:

[0044] ;

[0045] ;

[0046] ;

[0047] in: The equivalent concentrated force at the i-th Gaussian point is calculated as follows:

[0048] ;

[0049] In the formula:

[0050] ;

[0051] Represents the distance to the Gaussian point. These are the Gaussian point coefficients;

[0052] ;

[0053] ;

[0054] ;

[0055] The direction of the outward normal to the ground; , Let be the virtual normal stress and shear stress at the i-th Gaussian point.

[0056] Preferably, step S5 includes the following steps:

[0057] S51: Divide the half-plane and number it;

[0058] The side, bottom, and top surfaces of the hole are respectively numbered as semi-infinite planes n1, n2, and n3;

[0059] The semi-infinite plane n1 is composed of the hole edge S1, the imaginary left extension edge S4, and the imaginary right extension edge S5;

[0060] The semi-infinite plane n2 is composed of the hole edge S2 and the imaginary extended edge S6.

[0061] The semi-infinite plane n3 is formed by the edge of the hole. It is composed of the imaginary extended edge S3;

[0062] S52: Determine the Gaussian point;

[0063] Divide surfaces S1, S2, S3, S4, S5, and S6 into several segments;

[0064] in,

[0065] Surface S1 is divided into P according to the principle of equal division. b Each segment has a length of d. b Each segment is assigned 5 Gaussian points, meaning that surface S1 has a total of N Gaussian points. b =5·p b ;

[0066] Faces S2, S3, S4, S5, and S6 are divided into p groups according to a geometric sequence. e Each segment has 5 Gaussian points, for a total of N Gaussian points. e =5·p e ;

[0067] S53: Determine the equivalent concentrated force;

[0068] Among them, the imaginary extension edge S3 of the semi-infinite plane n3 is located inside the semi-infinite plane n1;

[0069] Based on Gaussian theory and the superposition principle, the equivalent concentrated force F3 acting on the imaginary extended edge S3 is obtained as follows:

[0070] ;

[0071] in:

[0072] ;

[0073] i = 2, 3, 4, 5, 6;

[0074] ;

[0075] ;

[0076] ;

[0077] ;

[0078] , , This is the positional relationship matrix between Gaussian points;

[0079] Similarly, we can conclude that:

[0080] ;

[0081] ;

[0082] ;

[0083] Among them, equivalent concentration force , The results are obtained based on the virtual surface forces and the S42 process;

[0084] S54: Iterative solution for equivalent concentrated force;

[0085] assumed Substituting into S53, we get , , and And repeat this step until... , , , convergence;

[0086] S55: The stress of model β1 is obtained based on the equivalent concentrated force. .

[0087] Preferably, step S6 includes the following steps:

[0088] S61: Assumption , Let S1 and S2 be the equivalent concentrated forces acting on the edges of the holes, respectively. , They are respectively equal to , ;

[0089] S62: Will , Substituting into S53, we get ;

[0090] S63: The equivalent concentrated force in the same region of model β2 is... To satisfy the stress boundary conditions applied to the top surface of model β2;

[0091] The stresses at the hole edges S1 and S2 are obtained according to step S42, so as to obtain the equivalent concentrated forces acting on the hole edges S1 and S2 of model β2. , ;

[0092] S64: The corrected result is obtained based on the above calculations. , ,Right now,

[0093] , ;

[0094] S65: Repeat steps S62-S64 until... , , convergence.

[0095] Compared with the prior art, the present invention has the following beneficial effects:

[0096] This invention can simultaneously consider the effects of loads such as initial ground stress, surcharge, support pressure and anchoring force, making it applicable to stress calculations for slopes at various angles.

[0097] The calculation results of this invention are accurate, and the stress concentration near the toe of the slope is high, which can determine the slope toe 10.-11 Up to 10 -10 Stress within the range of slope height.

[0098] This invention can fully reflect the effects of reinforcement force and overload, and is more in line with the actual stress situation of slopes, providing theoretical guidance for slope reinforcement design. Attached Figure Description

[0099] Figure 1 This is a schematic diagram of the slope model in this invention.

[0100] Figure 2 This is a schematic diagram of the semi-infinite plane model α in this invention.

[0101] Figure 3 This is a schematic diagram of the zero-gravity slope model β in this invention.

[0102] Figure 4 This is a schematic diagram of the semi-infinite hole model β1 in an infinite plane in this invention.

[0103] Figure 5 This is a schematic diagram of the semi-infinite plane model β2 without holes in this invention.

[0104] Figure 6 This is a schematic diagram of the extension edge division of the semi-infinite hole model β1 in the infinite plane in this invention.

[0105] Figure 7 This is a schematic diagram of one embodiment of the present invention.

[0106] Figure 8 As described in the embodiments of the present invention Stress cloud diagram.

[0107] Figure 9 These are stress cloud diagrams near the toe of the slope at different scales in embodiments of the present invention.

[0108] Figure 10 This is a schematic diagram of the stress distribution of principal stress under uniformly distributed support pressure in an embodiment of the present invention. Detailed Implementation

[0109] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0110] The specific implementation of the present invention will be described in detail below with reference to specific embodiments.

[0111] like Figure 1-6The diagram shown illustrates a structural method for calculating the elastic stress of a slope under support, according to an embodiment of the present invention. The method includes the following steps:

[0112] S1: Establish a mathematical model for a slope with an angle of β subjected to initial ground stress, surcharge at the top of the slope and support force; decompose the mathematical model into: a semi-infinite plane model α and a zero-gravity slope model β; where β is an arbitrary value;

[0113] Specifically, for a slope with an angle of β subjected to initial ground stress, surcharge at the top of the slope, and support force, it is decomposed into the following based on the superposition principle:

[0114] In a semi-infinite plane model α under initial ground stress and slope crest surload,

[0115] And, a zero-gravity slope model β under the combined action of excavation unloading, support force and excavation construction load.

[0116] S2: Decompose the zero-gravity slope model β into: virtual surface force V at the edge of the hole. b The model β1 of a semi-infinite hole in an infinite plane, and the virtual surface force V. v The semi-infinite plane model β2 without holes under the action;

[0117] S3: Calculate the stress in the semi-infinite plane model α;

[0118] S31: Calculate the surrounding rock stress at any point A within the semi-infinite plane model α under the surcharge at the slope crest. ,include

[0119] Determine the number of Gaussian points:

[0120] The overloaded area at the top of the slope w i (i=1,2,...,n) o ) divided into p o,i part;

[0121] Each segment has 5 Gaussian points, that is, the total number of Gaussian points is N. o,i =5·p o,i ;

[0122] Determine the number of equivalent concentrated force groups:

[0123] The number of equivalent concentrated force sets at the Gauss point is:

[0124] ;

[0125] Calculate the stress of surrounding rock under overload. :

[0126] ;

[0127] In the formula:

[0128] ;

[0129] , , Let x represent the normal stress, y represent the normal stress, and shear stress at any point A in the surrounding rock under the surcharge at the top of the slope;

[0130] ;

[0131] matrix Point A represents the stress in the surrounding rock to be determined under ground overload. With the i-th Gaussian point The positional relationship between them;

[0132] ;

[0133] ;

[0134] S32: Solving for the initial geostress at point A yields the solution. ;

[0135] S33: The stress at any point A in the semi-infinite plane model α under initial ground stress and slope crest surcharge is:

[0136] .

[0137] S4: Calculate: Virtual surface force V on the surface v Stress in a semi-infinite plane model β2 without holes under action;

[0138] S41: Determine the Gaussian point;

[0139] The load application area is divided into p according to a geometric sequence. v Each segment has 5 Gaussian points, for a total of N Gaussian points. v =5·p v ;

[0140] S42: Calculate the stress at any point A within model β2. :

[0141] ;

[0142] In the formula:

[0143] ;

[0144] ;

[0145] ;

[0146] in: The equivalent concentrated force at the i-th Gaussian point is calculated as follows:

[0147] ;

[0148] In the formula:

[0149] ;

[0150] Represents the distance to the Gaussian point. These are the Gaussian point coefficients;

[0151] ;

[0152] ;

[0153] ;

[0154] The direction of the outward normal to the ground; , Let be the virtual normal stress and shear stress at the i-th Gaussian point.

[0155] S5: Calculation: Virtual surface force V acting on the edge of the hole b Stress in a semi-infinite hole model β1 within an infinite plane;

[0156] S51: Divide the half-plane and number it;

[0157] The side, bottom, and top surfaces of the hole are respectively numbered as semi-infinite planes n1, n2, and n3;

[0158] The semi-infinite plane n1 is composed of the hole edge S1, the imaginary left extension edge S4, and the imaginary right extension edge S5;

[0159] The semi-infinite plane n2 is composed of the hole edge S2 and the imaginary extended edge S6.

[0160] The semi-infinite plane n3 is formed by the edge of the hole. It is composed of the imaginary extended edge S3;

[0161] S52: Determine the Gaussian point;

[0162] Divide surfaces S1, S2, S3, S4, S5, and S6 into several segments;

[0163] in,

[0164] Surface S1 is divided into P according to the principle of equal division. b Each segment has a length of d. bEach segment is assigned 5 Gaussian points, meaning that surface S1 has a total of N Gaussian points. b =5·p b ;

[0165] Faces S2, S3, S4, S5, and S6 are divided into p groups according to a geometric sequence. e Each segment has 5 Gaussian points, for a total of N Gaussian points. e =5·p e ;

[0166] S53: Determine the equivalent concentrated force;

[0167] Among them, the imaginary extension edge S3 of the semi-infinite plane n3 is located inside the semi-infinite plane n1;

[0168] Based on Gaussian theory and the superposition principle, the equivalent concentrated force F3 acting on the imaginary extended edge S3 is obtained as follows:

[0169] ;

[0170] in:

[0171] ;

[0172] i = 2, 3, 4, 5, 6;

[0173] ;

[0174] ;

[0175] ;

[0176] ;

[0177] , , This is the positional relationship matrix between Gaussian points;

[0178] Similarly, we can conclude that:

[0179] ;

[0180] ;

[0181] ;

[0182] Among them, equivalent concentration force , The results are obtained based on the virtual surface forces and the S42 process;

[0183] S54: Iterative solution for equivalent concentrated force;

[0184] assumed Substituting into S53 yields , , and And repeat this step until... , , , convergence;

[0185] S55: The stress of model β1 is obtained based on the equivalent concentrated force. .

[0186] Step S6: Iteratively calculate the analytical solution of stress for model β.

[0187] S61: Assumption , Let S1 and S2 be the equivalent concentrated forces acting on the edges of the holes, respectively. , They are respectively equal to , ;

[0188] S62: Will , Substituting into S53, we get ;

[0189] S63: Let the equivalent concentrated force acting on the same region in model β2 be... To satisfy the stress boundary conditions at the top of the slope of model β2;

[0190] The stresses at the hole edges S1 and S2 are obtained according to step S42, so as to obtain the equivalent concentrated forces acting on the hole edges S1 and S2 of model β2. , ;

[0191] S64: The corrected result is obtained based on the above calculations. , ,Right now,

[0192] , ;

[0193] S65: Repeat steps S62-S64 until... , , convergence.

[0194] This invention can simultaneously consider the effects of loads such as initial ground stress, surcharge, support pressure, and anchoring force, making it applicable to stress calculations for slopes at various angles.

[0195] Example 1

[0196] like Figure 7 As shown, consider a vertical slope with a height of 10m, where an overload acts at the top of the slope. The effective width is 10m, and the initial ground stress consists of horizontal and vertical stresses.

[0197] With a Poisson's ratio of 0.33 and a specific weight of 20 kN / m³, 3 Slope internal stress Stress cloud diagram, such as Figure 8 As shown, the stress cloud diagrams near the toe of the slope at different scales are as follows: Figure 9 As shown.

[0198] Depend on Figure 8 , Figure 9 It can be seen that the calculation results of this invention are accurate, and the stress concentration near the toe of the slope is relatively high, even to the point that the toe of the slope is 10. -11 Up to 10 -10 Stress within the range of slope height.

[0199] Assume the slope is subjected to uniform pressure along its vertical plane. The reinforcement support was calculated using the elastic stress calculation method for the slope under support provided by this invention. The stress cloud diagrams of the major and minor principal stresses within the reinforced slope after calculation are shown below. Figure 10 As shown.

[0200] Depend on Figure 10 It can be seen that the present invention can fully reflect the effects of reinforcement force and overload, and is more in line with the actual situation of slope stress, and can provide theoretical guidance for slope reinforcement design.

[0201] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for calculating the elastic stress of a slope under support, characterized in that, Includes the following steps: S1: Establish a mathematical model for a slope with an angle of β subjected to initial ground stress, surcharge at the top of the slope and support force; decompose the mathematical model into: a semi-infinite plane model α and a zero-gravity slope model β; where β is an arbitrary value; S2: Decompose the zero-gravity slope model β into: virtual surface force V at the edge of the hole. b The model β1 of a semi-infinite hole in an infinite plane, and the virtual surface force V. v The semi-infinite plane model β2 without holes under the action; S3: Calculate the stress in the semi-infinite plane model α; S4: Calculate the virtual surface force V on the surface. v Stress in a semi-infinite plane model β2 without holes under action; S5: Calculate the virtual surface force V at the edge of the hole. b Stress in a semi-infinite hole model β1 within an infinite plane; Step S6: Iteratively calculate the analytical solution of stress for model β.

2. The method for calculating the elastic stress of a slope under support according to claim 1, characterized in that, In S1: For a slope with an angle of β subjected to initial ground stress, surcharge at the top of the slope, and support force, it can be decomposed into the following based on the superposition principle: In a semi-infinite plane model α under initial ground stress and slope crest surload, And, a zero-gravity slope model β under the combined action of excavation unloading, support force and excavation construction load.

3. The method for calculating the elastic stress of a slope under support according to claim 2, characterized in that, S3 includes the following steps: S31: Calculate the surrounding rock stress at any point A within the semi-infinite plane model α under the surcharge at the slope crest. ,include Determine the number of Gaussian points: The overloaded area at the top of the slope w i (i=1,2,...,n) o ) divided into p o,i part; Each segment has 5 Gaussian points, that is, the total number of Gaussian points is N. o,i =5·p o,i ; Determine the number of equivalent concentrated force groups: The number of equivalent concentrated force sets at the Gauss point is: ； Calculate the stress of surrounding rock under overload. : ； In the formula: ； , , Let x represent the normal stress, y represent the normal stress, and shear stress at any point A in the surrounding rock under the surcharge at the top of the slope; ； matrix Point A represents the stress in the surrounding rock to be determined under ground overload. With the i-th Gaussian point The positional relationship between them; ； ； S32: Solving for the initial geostress at point A yields the solution. ; S33: The stress at any point A in the semi-infinite plane model α under initial ground stress and slope crest surcharge is: 。 4. The method for calculating the elastic stress of a slope under support according to claim 3, characterized in that, S4 includes the following steps: S41: Determine the Gaussian point; The load application area is divided into p according to a geometric sequence. v Each segment has 5 Gaussian points, for a total of N Gaussian points. v =5·p v ; S42: Calculate the stress at any point A within model β2. : ； In the formula: ； ； ； in: The equivalent concentrated force at the i-th Gaussian point is calculated as follows: ； In the formula: ； Represents the distance to the Gaussian point. These are the Gaussian point coefficients; ； ； ； The direction of the outward normal to the ground; , Let be the virtual normal stress and shear stress at the i-th Gaussian point.

5. The method for calculating the elastic stress of a slope under support according to claim 4, characterized in that, In S5, the following steps are included: S51: Divide the half-plane and number it; The side, bottom, and top surfaces of the hole are respectively numbered as semi-infinite planes n1, n2, and n3; The semi-infinite plane n1 is composed of the hole edge S1, the imaginary left extension edge S4, and the imaginary right extension edge S5; The semi-infinite plane n2 is composed of the hole edge S2 and the imaginary extended edge S6. The semi-infinite plane n3 is formed by the edge of the hole. It is composed of the imaginary extended edge S3; S52: Determine the Gaussian point; Divide surfaces S1, S2, S3, S4, S5, and S6 into several segments; in, Surface S1 is divided into P according to the principle of equal division. b Each segment has a length of d. b Each segment is assigned 5 Gaussian points, meaning that surface S1 has a total of N Gaussian points. b =5·p b ; Faces S2, S3, S4, S5, and S6 are divided into p groups according to a geometric sequence. e Each segment has 5 Gaussian points, for a total of N Gaussian points. e =5·p e ; S53: Determine the equivalent concentrated force; Among them, the imaginary extension edge S3 of the semi-infinite plane n3 is located inside the semi-infinite plane n1; Based on Gaussian theory and the superposition principle, the equivalent concentrated force F3 acting on the imaginary extended edge S3 is obtained as follows: ； in: ； ，i=2，3，4，5，6； ； ； ； ； , , This is the positional relationship matrix between Gaussian points; Similarly, we can conclude that: ； ； ； Among them, equivalent concentration force , The results are obtained based on the virtual surface forces and the S42 process; S54: Iterative solution for equivalent concentrated force; assumed Substituting into S53 yields , , and And repeat this step until... , , , convergence; S55: The stress of model β1 is obtained based on the equivalent concentrated force. .

6. The method for calculating the elastic stress of a slope under support according to claim 2, characterized in that, In S6, the following steps are included: S61: Assumption , Let S1 and S2 be the equivalent concentrated forces acting on the edges of the holes, respectively. , They are respectively equal to , ; S62: Will , Substituting into S53, we get ; S63: Let the equivalent concentrated force acting on the same region in model β2 be... To satisfy the stress boundary conditions at the top of the slope of model β2; The stresses at the hole edges S1 and S2 are obtained according to step S42, so as to obtain the equivalent concentrated forces acting on the hole edges S1 and S2 of model β2. , ; S64: The corrected result is obtained based on the above calculations. , ,Right now, ， ； S65: Repeat steps S62-S64 until... , , convergence.

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