Method for evaluating surrounding rock stability of straight-wall arched tunnel with through cracks

Abstract

The invention relates to a method for evaluating the surrounding rock stability of a straight-wall arched tunnel containing through cracks, and belongs to the technical field of geotechnical engineering. The invention discloses a method for evaluating the surrounding rock stability of a straight-wall arched tunnel containing through cracks. The method comprises the steps that crack parameters and surrounding rock parameters of the straight-wall arched tunnel are obtained; determining first principal stress according to the surrounding rock parameters; determining a first type stress intensity factor and a second type stress intensity factor; determining a first dimensionless stress intensity factor and a second dimensionless stress intensity factor according to the crack parameter, the first type stress intensity factor, the second type stress intensity factor and the first principal stress; and evaluating the stability of the surrounding rock of the straight-wall arched tunnel according to the first dimensionless stress intensity factor and the second dimensionless stress intensity factor. According to the method, the blank of theoretical solutions of the stress intensity factors of the tunnel containing the through cracks in the prior art is filled, the method is suitable for the working condition of the 0-90-degree crack dip angle, calculation is accurate, the number of set parameters is small, and reliable theoretical support is provided for tunnel stability evaluation.

Description

Technical Field

[0001] This invention relates to a method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks, belonging to the field of geotechnical engineering technology. Background Technology

[0002] As a core carrier of major infrastructure projects such as transportation, water conservancy, and energy, tunnel engineering has gradually extended to complex geological conditions such as high ground stress, karst development, and large deformation of soft rock in recent years, with the advancement of deep development and cross-regional engineering construction in China. Its stability directly determines the safety and operational life of the project. In current tunnel construction, "through cracks" are a typical hidden danger that induces structural instability. These cracks are mostly caused by the extension of geological structural fractures, stress concentration during excavation, or long-term water pressure infiltration and operational vibration loads. They are characterized by penetrating the tunnel's radial or circumferential cross-section or the key bearing layer of the surrounding rock, directly cutting off the stress transmission path of the rock mass and weakening the structure's shear and tensile bearing capacity. If their stability is not accurately assessed, it can easily lead to rapid crack propagation, surrounding rock collapse, and even project shutdown and safety accidents. Therefore, the stability assessment of tunnels containing through cracks has become a core requirement in the field.

[0003] Current key technologies for tunnel stability analysis are based on fracture mechanics and engineering mechanics. Mainstream methods include theoretical analysis, numerical simulation, and in-situ monitoring and inversion. Core evaluation indicators include crack tip stress intensity factor, crack initiation angle, and ultimate bearing stress. However, existing technologies have significant limitations: First, the research focuses primarily on non-penetrating cracks, failing to adequately adapt to the specific characteristics of penetrating cracks—existing theoretical models often assume "semi-infinite isolated cracks" or "local non-penetrating cracks," neglecting the impact of penetrating cracks on the overall tunnel bearing system, leading to large deviations in stress calculations. Second, under complex load coupling, there is a lack of specific models for the mechanical response of penetrating cracks. Existing numerical simulations either simplify the homogeneous assumption, ignoring the heterogeneity after crack penetration, or use fine meshes resulting in extremely low efficiency, making it difficult to meet design requirements. Third, existing crack initiation criteria do not incorporate correction parameters for the penetration of penetrating cracks, leading to significant deviations in the predicted crack initiation angle and ultimate bearing capacity, failing to provide accurate support for support design. Therefore, dedicated calculation methods are urgently needed to fill this gap. Summary of the Invention

[0004] To address the aforementioned problems, this invention aims to overcome the shortcomings of existing technologies by proposing a method for evaluating the stability of surrounding rock in straight-walled arched tunnels with penetrating cracks.

[0005] The technical solution provided by this invention to solve the above-mentioned technical problems is: a method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks, comprising the following steps: Obtain crack parameters and surrounding rock parameters for a straight-walled arched tunnel; The first principal stress is determined based on the surrounding rock parameters. ; Determine the first type of stress intensity factor Type II stress intensity factor ; Based on crack parameters and Type I stress intensity factor Type II stress intensity factor First principal stress Determine the first dimensionless stress intensity factor Second dimensionless stress intensity factor ; According to the first dimensionless stress intensity factor Second dimensionless stress intensity factor The stability of the surrounding rock of a straight-walled arched tunnel is evaluated.

[0006] A further technical solution is that the crack parameters include crack length and crack inclination angle.

[0007] A further technical solution is that the surrounding rock parameters include vertical stress. Confining pressure .

[0008] A further technical solution is that the first principal stress Vertical stress and confining pressure The great ones in the village.

[0009] A further technical solution is that the determination of the first type of stress intensity factor... Type II stress intensity factor include: Determine the undetermined constants using the collocation method. ; According to the undetermined constant Determine the first type of stress intensity factor Type II stress intensity factor .

[0010] A further technical solution is to determine the undetermined constants using the collocation method. include: Uniformly select on the boundary of the straight-walled arched tunnel N One point; The stress at the collocation point is collected using a boundary pressure sensor and then projected onto the calculation surface to obtain the corresponding surface force components; Substituting the surface force components into the linear equation yields the undetermined constants. .

[0011] A further technical solution is that the linear equation is:

[0012] In the formula: E jk These are the elements of the coefficient matrix; M The number of terms in the series; and External forces on the boundary x and y The directional component.

[0013] A further technical solution is that the first type of stress intensity factor Type II stress intensity factor The calculation formula is: When the crack tip is on the right side of a straight-walled arched tunnel:

[0014] When the crack tip is on the left side of a straight-walled arched tunnel:

[0015] In the formula: The length of the crack is half its length; It is a type I stress intensity factor; It is a type II stress intensity factor.

[0016] A further technical solution is that the first dimensionless stress intensity factor Second dimensionless stress intensity factor The calculation formula is: When the crack tip is on the right side of a straight-walled arched tunnel:

[0017] When the crack tip is on the left side of a straight-walled arched tunnel:

[0018] In the formula: The first dimensionless stress intensity factor; This is the second dimensionless stress intensity factor; This is the reference stress intensity factor.

[0019] A further technical solution is that the evaluation of the surrounding rock stability of a straight-walled arch tunnel based on the dimensionless stress intensity factor specifically involves: a first dimensionless stress intensity factor. Y I The smaller the absolute value, the stronger the tunnel's resistance to tensile failure; the second dimensionless stress intensity factor Y II The closer the absolute value of the value is to 0, the better the shear stability of the tunnel.

[0020] The beneficial effects of this invention are as follows: This invention fills the gap in the theoretical solution of stress intensity factor for tunnels with through-cracks in the prior art, and is applicable to crack inclination angles of 0°-90°. It is accurate in calculation and requires fewer parameters, providing reliable theoretical support for tunnel stability assessment. Attached Figure Description

[0021] Figure 1 A model diagram of a tunnel with a through-crack; Figure 2 The diagram shows the stress on the crack surface under vertical compressive stress and confining pressure. Figure 3 For Plemelj function Schematic diagram; Figure 4 This is a closed curve diagram containing the crack. Detailed Implementation

[0022] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] The derivation process of the formula for calculating the dimensionless stress intensity factor in this invention is as follows: For a straight-walled arched tunnel with a through crack, its stress diagram under vertical stress and confining pressure is as follows: Figure 1 As shown. The height and width of the straight-walled arched tunnel are... The crack surface passes through the center of the arch and is centered at the center, with an angle of inclination of . , It is half the crack length. There is interaction between the crack surfaces. It is the normal compressive stress on the crack surface. This represents the frictional stress on the crack surface. On the other hand, because the lining may exist on the tunnel surface, uniformly distributed pressure is used. P This represents the force exerted by the lining on the tunnel surface. Represents vertical compressive stress. Represents confining pressure.

[0024] As is well known, when using the complex function method to solve elasticity problems, the biharmonic equation expressed by the complex function of the stress function is: (1) According to the relevant definition of complex numbers, there exists formula (2): (2) in z = x + iy For complex coordinates, x and y These are Cartesian coordinates.

[0025] Formula (3) is derived from equation (2): (3) According to the relevant definition of differentiation in advanced mathematics, there exists formula (4): (4) From equation (4), its second-order partial derivative formula (5) can be derived: (5) According to the Laplace operator, there exists formula (6): (6) Combining equations (5) and (6) leads to formula (7): (7) Formula (8) can be derived from equations (6) and (7): (8) Integral formula (8), derived formula (9): (9) From equation (9), it can be seen that the biharmonic stress function U Since it is a real function, there must be pairwise conjugates on the right side of the equation, thus formula (10) can be derived: (10) Here, we assume formula (11): (11) Combining equations (9), (10), and (11), we derive equation (12): (12) Equation (12) is the stress function expressed by complex variable functions, which is essentially the Goursat formula.

[0026] If physical strength is neglected, the relationship between the stress function and stress components can be expressed by formula (13): (13) From the above equations, formulas (14), (15), and (16) can be derived: (14) (15) (16) Equations (14), (15), and (16) are the stress component forms expressed by using complex variable functions. At the same time, it can be seen that by combining equations (14) and (15), equation (16) can be derived. Therefore, each pair of equations (14), (15), and (16) is independent.

[0027] Based on the fracture mechanics theory of Georgian mathematician Muskhelishvili, the following formulas (17) and (18) are assumed: (17) (18) The integral formula (18) can lead to formula (19): (19) In equation (19) C It is an integral constant. The complex functions in equations (17) and (19) φ ( z ), ω ( z ) is the stress potential function.

[0028] Combining the above equations, we can derive formula (20): (20) Substituting equations (17) and (20) into equations (14) and (16), we derive formulas (21) and (22): (twenty one) (twenty two) For equations (21) and (22), there is a relationship between the derivative of the stress potential function and the stress components. Based on the previous calculations, it can be shown that equations (21) and (22) conform to the equilibrium equation and the compatibility equation.

[0029] When the surrounding rock of the tunnel is under vertical compressive stress and confining pressure Under these conditions, the surface forces acting on the crack are as follows: Figure 2 As shown.

[0030] For the upper surface of the crack, its outer normal direction and The axis is perpendicular and points to Since the axis is in the negative direction, there exists a formula for the direction cosine value (23): (twenty three) Similar to equation (23), for the lower surface of the crack, its outer normal... Direction For the positive axis, there is also a formula for the direction cosine value (24): (twenty four) Substituting equations (23) and (24) into the force boundary condition formula for the two-dimensional case, we derive equation (25): (25) Therefore, the boundary stress formula (26) for the two surfaces of the crack can be derived: (26) Substituting equation (26) into equation (22), and considering the two crack surfaces, we have equation (27): (27) For an ideal crack with negligible crack width, it is assumed that the crack surface is... x There is a point on the axis ( t ,0), which satisfies the condition Formula (28) exists: (28) Executing (28)1+(28)2 and (28)1-(28)2, we have formulas (29) and (30): (29) (30) Vertical compressive stress Confining pressure Formulas (31) and (32) exist to represent the interaction forces between the crack surfaces: (31) (32) in f Let be the friction coefficient of the crack surface. According to the formula for shear stress on an oblique section in mechanics of materials, we have (33): (33) If the frictional stress on the crack surface Greater than shear stress Because of the inhibition of friction, the stress intensity factor at the crack tip is equal to zero. Therefore, only when... Under certain conditions, a stress intensity factor will only be present at the crack tip, which is used to characterize shear and tensile / compressive forces.

[0031] Vertical compressive stress Confining pressure From the right side of expression (29), expressed by the complex constant P, there exists formula (34): (34) Based on the single-valued stress case associated with multi-connected regions, the complex stress function Φ( t ) can be expressed as (35), Ω( t ) can be expressed as (36): (35) (36) In equations (35) and (36), v It is Poisson's ratio, and B, C, and D are constants. They can also be expressed using principal stresses, as given by formula (37): (37) In equation (37) yes Figure 1 First principal stress direction and x The angle between the axes, σ 1 and σ 2 represents the first and second principal stresses, respectively. Therefore, the lateral pressure coefficient... In this case, there exists , , Otherwise, exists , , .

[0032] exist In this case, based on equations (35) and (36), there exists equation (38): (38) Assume that equation (39) exists: (39) Then, from equations (29) and (30), we have equations (40) and (41): (40) (41) Therefore, it can be seen that equations (40) and (41) become simple Hilbert problems. To find the solutions to the above two equations, the Plemelj function related to this model should be introduced. The function is defined by formula (42): (42) As can be seen, if Then it exists Here we assume If a point on the crack surface, Below the crack At very close points, the Plemelj function has a unique limit. Similarly, above the crack, this function has a unique limit. ,like Figure 3 As shown.

[0033] It can be seen that the Plemelj function X ( t The property shown in formula (43) exists: (43) Using the property of the Plemelj function in equation (43), divide both sides of equation (40) by... Existential equation (44): (44) Assume that equation (45) exists: (45) Then formula (44) can be transformed into formula (46): (46) According to the Cauchy integral over the arc, the general solution of equation (46) can be expressed as equation (47): (47) In equation (47), It can be expressed as equation (48): (48) In equation (48), n It is the number of cracks. , It is an undetermined constant.

[0034] Using the stress function to express equation (47), equation (49) exists: (49) Treat the right side of equation (49) as Similarly, by using Cauchy integrals, the general solution form of its stress function can be derived, as shown in equation (50): (50) Combining equations (49) and (50), two formulas for the stress function can be derived: (51) (52) To obtain the stress function and The solution, first step, should be obtained by integrating the right-hand sides of equations (51) and (52). The first integral on the right-hand side of equations (51) and (52) is the curve where the crack is located. L The integral over, i.e., equation (53): (53) like Figure 4 As shown, using the closed curve around the crack C Integral expression over [a certain period of time].

[0035] With the help of Figure 4 Closed curve C Integral over, we can easily obtain equation (54): (54) In equation (54), due to the crack tip a , b Nearby, the radius of curvature of the curve Therefore, the second and fourth integral limits on the right side of the equation are almost equal to 0. Furthermore, according to the properties of the Plemelj function, i.e., equation (43), equation (55) exists: (55) Using equation (55), we can transform equations (51) and (52) into equations (56) and (57): (56) (57) Equations (56) and (57) are general expressions of the two stress functions. In equations (56) and (57) Existential formula (58): (58) From equation (48), we derive equation (59): (59) Since equations (49) and (50) are under vertical pressure and confining pressure The expressions under the influence of cracks and tunnels are (56) and (57), respectively. Therefore, they are equivalent. Thus, from (59), we have (60): (60) Equation (61) is derived from equation (60): (61) for Other undetermined constants in ( Except for the case where there are too many, the displacement single-value condition cannot be used to derive the result. Therefore, the boundary point method is used to derive the result using the boundary condition.

[0036] Because the crack studied in this paper has only two crack tips, and they are equidistant from the center of the arch, therefore... Inside On the other hand, the Plemelj function It becomes equation (62): (62) The functions in the stress function formulas (56) and (57) P ( t If ), then it is a constant. P ,therefore Q ( t ) naturally equals 0.

[0037] Based on the above derivation, and with the help of the residue theorem, the rightmost side of equation (56) follows the closed curve. C The integral can be expressed as in z The sum of the residues at the point and the point at infinity can therefore be rewritten as formula (63): (63) By using the residue at infinity in the solution of equation (63), we can simplify to equation (64): (64) In equation (64), as well as Existential form (65): (65) the remaining It is an undetermined constant. In fact, it is easy to know... It is a purely imaginary number.

[0038] Similarly, there exists another stress function, namely equation (66): (66) In equation (66) Existential form (67): (67) Equations (64) and (66) are two simplified stress functions expressed using power series.

[0039] Next, the formula for the stress intensity factor of the through-tunnel crack was derived. Typically, the relationship between the stress intensity factor and the complex stress function can be expressed as shown in equation (68): (68) In equation (68), It is the tip of the crack coordinate, Is the direction of the crack and The included angle of the shaft, and at the tip of the crack on the right side, such as Figure 1 As shown, ;otherwise, .

[0040] Combined with Φ ( z The expression, at the crack tip B on the right, exists as equation (69): (69) Similarly, at the crack tip A on the left side, equation (70) exists: (70) The dimensionless stress intensity factor at the corresponding two crack tips exists as Y. I Y II Expressions (71) and (72): (71) (72) In equations (71) and (72), It is the first principal stress, that is Figure 1 Vertical stress and confining pressure The great one in the village, at the same time As expressed in equation (73): (73) This invention provides a method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks, comprising the following steps: Step 1: Obtain the crack parameters and surrounding rock parameters of the straight-walled arch tunnel; The crack parameters include crack length and crack dip angle, and the surrounding rock parameters include vertical stress. Confining pressure ; Step 2: Determine the first principal stress based on the surrounding rock parameters. The first principal stress Vertical stress and confining pressure The greatest among them; Step 3: Calculate the undetermined constants using the collocation method. ; Uniformly select on the boundary of the straight-walled arched tunnel N Let there be a coordinate point, denoted as . ,in The coordinates of the boundary points are the complex coordinates.

[0041] The known external force components at each point are: and , respectively represent x and y In engineering, the surface force components in the direction can be obtained by using boundary pressure sensors to collect the stress at the distribution points and then projecting it onto the calculation surface. Substituting the surface force components into the linear equation yields the undetermined constants. ; For each point Substituting into formulas (64) and (66), we get: (74) in It is possible This is expressed as (see formulas (65) and (67)), therefore the equation contains only... .

[0042] because yes The linear function (see formulas (65) and (67)) can be simplified to obtain: (75) in E jk For the elements of the coefficient matrix, by , , k and D k and C k The relationship is established; M The number of terms in the series to be selected (this needs to be set manually; usually 3-5 terms are sufficient for engineering accuracy, but can be increased to 7 terms under complex geological conditions). When (i.e., the number of collocations needs to be greater than twice the number of terms in the power series plus one; if M is 1, then the number of collocations needs to be greater than 3 to ensure that the system of linear equations has a unique solution), the system of linear equations can be solved to obtain C k ; Step 4: Based on the undetermined constants Determine the first type of stress intensity factor Type II stress intensity factor ; The series obtained from step three will be among them. Then substitute into formulas (69) to (70), note that in this formula, all those containing The terms were summed, thus obtaining the formula. The real part of this imaginary number is the first-order stress intensity factor. The absolute value of the imaginary part is the second-order stress intensity factor. , Step 5: Based on crack parameters and Type I stress intensity factor Type II stress intensity factor First principal stress Determine the first dimensionless stress intensity factor Second dimensionless stress intensity factor ; Among them, the first type of stress intensity factor Type II stress intensity factor Substituting into formulas (71) to (72) and performing dimensionless transformation yields... , I Therefore, the actual series obtained The final calculated values ​​are only two dimensionless stress intensity factors. , .

[0043] Step Six: Based on the first dimensionless stress intensity factor Second dimensionless stress intensity factor Evaluation of the surrounding rock stability of a straight-walled arched tunnel; The first dimensionless stress intensity factor Y I A negative value indicates crack closure; the smaller the absolute value, the stronger the tunnel's resistance to tensile failure. The second dimensionless stress intensity factor... Y II The absolute value directly reflects the degree of shear stress concentration. The closer the absolute value is to 0, the better the shear stability of the tunnel.

[0044] The above description is not intended to limit the present invention in any way. Although the present invention has been disclosed through the above embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some changes or modifications to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall fall within the scope of the present invention.

Claims

1. A method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks, characterized in that, Includes the following steps: Obtain crack parameters and surrounding rock parameters for a straight-walled arched tunnel; The first principal stress is determined based on the surrounding rock parameters. ; Determine the first type of stress intensity factor Type II stress intensity factor ; Based on crack parameters and Type I stress intensity factor Type II stress intensity factor First principal stress Determine the first dimensionless stress intensity factor Second dimensionless stress intensity factor ; According to the first dimensionless stress intensity factor Second dimensionless stress intensity factor The stability of the surrounding rock of a straight-walled arched tunnel is evaluated.

2. The method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks according to claim 1, characterized in that, The crack parameters include crack length and crack inclination angle.

3. The method for evaluating the surrounding rock stability of a straight-walled arched tunnel with penetrating cracks according to claim 1, characterized in that, The surrounding rock parameters include vertical stress. Confining pressure .

4. The method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks according to claim 3, characterized in that, First principal stress Vertical stress and confining pressure The great ones in the village.

5. The method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks according to claim 1, characterized in that, The determination of the first type of stress intensity factor Type II stress intensity factor include: Determine the undetermined constants using the collocation method. ; According to the undetermined constant Determine the first type of stress intensity factor Type II stress intensity factor .

6. A method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks according to claim 5, characterized in that, The method of determining undetermined constants based on collocation method include: Uniformly select on the boundary of the straight-walled arched tunnel N One point; The stress at the collocation point is collected using a boundary pressure sensor and then projected onto the calculation surface to obtain the corresponding surface force components; Substituting the surface force components into the linear equation yields the undetermined constants. .

7. A method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks according to claim 6, characterized in that, The linear equation is: In the formula: E jk These are the elements of the coefficient matrix; M The number of terms in the series; and External forces on the boundary x and y The directional component.

8. A method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks according to claim 6, characterized in that, Type I stress intensity factor Type II stress intensity factor The calculation formula is: When the crack tip is on the right side of a straight-walled arched tunnel: When the crack tip is on the left side of a straight-walled arched tunnel: In the formula: The length of the crack is half its length; It is a type I stress intensity factor; It is a type II stress intensity factor.

9. A method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks according to claim 8, characterized in that, First dimensionless stress intensity factor Second dimensionless stress intensity factor The calculation formula is: When the crack tip is on the right side of a straight-walled arched tunnel: When the crack tip is on the left side of a straight-walled arched tunnel: In the formula: The first dimensionless stress intensity factor; This is the second dimensionless stress intensity factor; This is the reference stress intensity factor.

10. A method for evaluating the stability of surrounding rock in a straight-walled arched tunnel with penetrating cracks according to claim 1, characterized in that, The evaluation of the surrounding rock stability of a straight-walled arched tunnel based on the dimensionless stress intensity factor specifically involves: the first dimensionless stress intensity factor. Y I The smaller the absolute value, the stronger the tunnel's resistance to tensile failure; Second dimensionless stress intensity factor Y II The closer the absolute value of the value is to 0, the better the shear stability of the tunnel.

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