A method for calculating the sliding surface of a finite-width soil body under active non-limit state

Abstract

The present application relates to geotechnical engineering technical field, especially to a kind of active non-limit state under the calculation method of sliding surface of finite width soil body, comprising the following steps: step 1, determine the gradual development law of internal friction angle, cohesion, wall-soil external friction angle and wall-soil interface cohesion in finite soil body between rigid gravity retaining wall and existing bedrock with retaining wall displacement development, establish the mathematical representation of the strength parameters of finite soil body non-limit state with retaining wall displacement development;Step 2, establish the relationship between horizontal stress, interlayer shear stress and average vertical stress;Step 3, establish the balance equation of rectangular area and triangular area in horizontal and vertical directions;Step 4, solve the sliding surface inclination angle that meets the plastic upper limit theorem by numerical method;Step 5, calculate the horizontal active earth pressure resultant force, the overturning moment generated by horizontal active earth pressure on wall heel.The present application provides theoretical calculation basis for the design of retaining structure under the working condition of finite width soil body.

Description

Technical Field

[0001] This invention relates to the field of geotechnical engineering design technology, specifically to a method for calculating the slip surface of a finite-width soil body under active non-limit conditions, and particularly to a method for calculating the slip surface angle when there is a finite-width soil body between an existing gravity retaining wall and existing bedrock. Background Technology

[0002] Retaining walls are the most common geotechnical retaining structures. Due to their excellent seepage prevention and structural support functions, they are widely used in practical projects such as roadbeds and slope protection. Determining the distribution law of earth pressure behind the retaining wall is a prerequisite for ensuring the toughness and service quality of underground engineering, and it is also one of the fundamental scientific issues in the field of geotechnical engineering.

[0003] Existing methods for determining earth pressure have made some progress. Some technologies have proposed numerical calculation methods for non-limit active earth pressure under rotational conditions around the wall top, deriving the relationship between soil strength parameters and wall displacement functions. Assuming the soil forms a circular arch and the slip surface is a logarithmic spiral surface, a numerical iterative format is constructed, and the calculation results are basically consistent with model tests, providing accurate solutions for rigid retaining wall design. However, the assumption of a logarithmic spiral surface slip surface can lead to complexity. Meanwhile, some technologies consider obtaining earth pressure for finite soil masses with varying widths. By determining multiple parameters and establishing energy balance relationships, the safety of foundation pit excavation can be evaluated and predicted in advance, providing a basis for internal support design. Furthermore, for passive earth pressure calculation methods in narrow fills, existing technologies have simulated failure under unsaturated seepage, considering the coupling effect of the seismic field and the unsaturated seepage field, accurately assessing seismic passive earth pressure, guiding engineering design, and improving stability and safety. These existing achievements have innovatively solved the earth pressure calculation problem in specific scenarios, possessing significant practical significance and application value. However, existing technologies still have certain gaps in determining the angle of the straight slip surface in the non-limit state of soil with finite width. At the same time, existing technologies for determining the slip surface angle of soil with finite width using the finite element method have certain limitations, because when using existing DEM and FEM methods to calculate the earth pressure and failure mode of retaining walls, a lot of time is often required to adjust material parameters and generate meshes.

[0004] Determining the slip surface angle of finite-width soil under non-limit conditions is of great value for accurately describing the earth pressure distribution. Therefore, there is an urgent need for a numerical calculation method that can effectively characterize the slip surface of finite-width soil under active non-limit conditions. Summary of the Invention

[0005] The purpose of this invention is to propose a numerical calculation method that can effectively characterize the slip surface of soil with finite width under active non-limit state, thereby solving the problems existing in the background technology.

[0006] The objective of this invention can be achieved through the following technical solutions:

[0007] A method for calculating the slip surface of finite-width soil under active non-limit state, comprising:

[0008] Step 1: Determine the gradual development law of the internal friction angle, cohesion, external friction angle between the wall and the soil, and the cohesion at the wall-soil interface of the rigid gravity retaining wall with the development of the finite soil between the wall and the existing bedrock, and establish a mathematical characterization of the non-limit state strength parameters of the finite soil with the development of the retaining wall displacement.

[0009] Step 2: Using the horizontal plane where the initial slip surface intersects with the existing bedrock on the right as the boundary line, divide the finite-width soil mass into an upper rectangular region (0 ≤ z ≤ 1). h = H - B tan β ) and the lower triangular area ( h = H - B tan β ≤ z ≤ H Two areas, of which H For a finite amount of soil, B For the width of a finite soil mass, h The height of the rectangular area. β Based on the principle of soil arching effect, the soil pressure coefficient and interlayer shear stress coefficient of the rectangular and triangular areas behind the retaining wall are determined according to the slip surface angle, and the relationship between horizontal stress, interlayer shear stress and average vertical stress is established.

[0010] Step 3: Take differential soil strip elements along the depth direction for the finite width cohesive fill between the rigid gravity retaining wall and the existing bedrock, and establish the equilibrium equations in the horizontal and vertical directions for the rectangular and triangular regions respectively, based on the average vertical stress and horizontal stress on the differential soil strips.

[0011] Step 4: Combining the results of Steps 2 and 3, use numerical methods to solve for the slip surface inclination angle that satisfies the plastic upper bound theorem, so that it meets the requirement of the continuity of the earth pressure curve.

[0012] Step 5: Calculate the resultant horizontal active earth pressure based on the slip surface inclination angle obtained in Step 4. The overturning moment generated by the horizontal active earth pressure on the wall heel .

[0013] Furthermore, in step 1, the calculation of the non-limit state earth pressure of finite soil is as follows: For a finite width of cohesive fill between a rigid gravity retaining wall and the existing bedrock, it is first necessary to determine the gradual development law of the internal friction angle, cohesion, external friction angle between the wall and the soil, and the cohesion parameters at the wall-soil interface as the retaining wall moves from a static state to an active limit state.

[0014] Specifically, in the parallel movement mode of the retaining wall, the internal friction angle of the soil in the non-limit state is determined by equation (1).

[0015] (1)

[0016] in, K 0 represents the coefficient of lateral earth pressure at rest; R f The failure ratio can be determined based on triaxial unloading tests; It is the ratio of the wall base displacement under non-limit active state to the wall base displacement under limit active state; The internal friction angle is the angle at the intermediate state. The internal friction angle is the limiting angle.

[0017] Furthermore, in the parallel movement mode of the retaining wall, the external friction angle between the gravity retaining wall on the left and the finite soil is... δ m The performance value is determined by formula (2).

[0018] (2)

[0019] in, δ 0 = / 2, δ To measure the external friction angle between the left retaining wall and the limited soil mass.

[0020] Furthermore, in the parallel movement mode of the retaining wall, the interfacial friction angle between the existing bedrock and the finite soil on the right side is utilized. α m Determined by equation (3).

[0021] (3)

[0022] in, α 0 = / 2, α This is the measured interfacial friction angle between the existing bedrock and the finite soil mass on the right side.

[0023] Furthermore, under the parallel movement mode of the retaining wall, the finite soil cohesion utilization value c mThe cohesion between the rigid gravity retaining wall on the left and the finite soil mass is determined by equation (4). c wm The cohesion value between the existing bedrock and the finite soil mass on the right side is determined by equation (5). c dm Determined by equation (6).

[0024] (4)

[0025] (5)

[0026] (6)

[0027] in, c w To measure the cohesion value of the retaining wall soil, c d This represents the measured cohesion value of the bedrock wall soil.

[0028] Formulas (1), (2), (3), (4), (5), and (6) establish calculation formulas for the development of soil non-limit state strength parameters with retaining wall displacement under the parallel movement mode of rigid retaining wall.

[0029] Furthermore, in step 2, the calculation of the non-ultimate earth pressure of the finite-width soil mass is as follows:

[0030] Analyze the differential soil strip element at any depth within the rectangular region, and determine the minor principal stress rotation angles on both sides of the differential soil strip element. and They are determined by equations (7) and (8) respectively.

[0031] (7)

[0032] (8)

[0033] Average vertical stress on a rectangular differential unit Determined by equation (9):

[0034] (9)

[0035] (10)

[0036] In the formula, For the major principal stresses of the soil element in the new coordinates, Let be the Rankine active earth pressure coefficient, the value of which is determined by equation (11).

[0037] (11)

[0038] Earth pressure coefficient of soil behind retaining wall in rectangular area Determined by equation (12).

[0039] (12)

[0040] (13)

[0041] in, and The small principal stress deflection angle at the slip surface in the rectangular region is determined by equations (7) and (8); The average vertical stress in the rectangular region under the new coordinate system is determined by equations (9) and (10); The Rankine active earth pressure coefficient is determined by equation (11); internal variables Determined by equation (13).

[0042] Interlayer shear stress coefficient of soil behind the retaining wall in the rectangular area To determine by equation (14):

[0043] (14)

[0044] Analyze the differential soil strip element at any depth in the triangular region, and determine the minor principal stress rotation angles on both sides of the differential soil strip element. and They are determined by equations (15) and (16) respectively.

[0045] (15)

[0046] (16)

[0047] In the formula, The angle between the slip surface and the horizontal direction is determined precisely by step (4).

[0048] Average vertical stress on the triangular region differential unit Determined by equation (17):

[0049] (17)

[0050] Earth pressure coefficient of soil behind retaining wall in triangular area It can be determined by equation (18).

[0051] (18)

[0052] (19)

[0053] in, and The small principal stress deflection angle at the slip surface in the triangular region is determined by equations (15) and (16); The average vertical stress in the triangular region under the new coordinate system is determined by equation (17); The Rankine active earth pressure coefficient is determined by equation (11); internal variables Determined by equation (19).

[0054] interlayer shear stress coefficient of soil behind retaining wall in triangular zone Determined by equation (20):

[0055] (20)

[0056] Furthermore, in step 3, the calculation of the seismic earth pressure in the non-limit state of finite soil is as follows:

[0057] For a rectangular region of finite width cohesive fill, a differential soil strip element is taken along the depth direction. The average vertical stress on the differential soil strip element is: Normal force at the left wall Tangential force at the wall The normal force at the right interface is Tangential force at the right interface Differential soil strip horizontal interlayer shear stress The weight of the thin soil layer is The stress balance equations for the horizontal and vertical directions of the differential soil strip are established using equations (21) and (22):

[0058] (twenty one)

[0059] (twenty two)

[0060] Combining equations (21) and (22), we can obtain the equilibrium differential equation:

[0061] (twenty three)

[0062] Boundary conditions are at the surface ( z =0) Vertical stress is ground overload q 0, that is .in and The values ​​represent the average vertical stress on the differential soil strip elements in the rectangular and triangular regions, respectively.

[0063] The earth pressure distribution in the rectangular area was thus obtained. for:

[0064] (twenty four)

[0065] (25)

[0066] (26)

[0067] For a finite-width cohesive fill in the triangular region, a differential soil strip element is taken along the depth direction. The average vertical stress on the differential soil strip element is... Normal force at the left wall Tangential force at the wall The normal force at the right interface is Tangential force at the right interface Differential soil strip horizontal interlayer shear stress The weight of the thin soil layer is The stress balance equations for the horizontal and vertical directions of the differential soil strip are established using equations (27) and (28):

[0068] (27)

[0069] (28)

[0070] Combining equations (27) and (28), we obtain the equilibrium differential equation:

[0071] (29)

[0072] Among them, N, I, P and T are all intermediate variables, which are determined by equations (30), (31), (32) and (33) respectively:

[0073] (30)

[0074] (31)

[0075] (32)

[0076] (33)

[0077] The general solution of differential equation (29) is:

[0078] (34)

[0079] Boundary conditions are ,in and The earth pressure distributions in the rectangular and triangular regions are shown below. Substituting these into the boundary conditions yields the constant term. W :

[0080] (35)

[0081] Specifically, the distribution of non-limit active earth pressure under the parallel movement mode of the retaining wall towards the finite width of the fill is given by equation (36):

[0082] (36)

[0083] Furthermore, in step 4, the calculation of the seismic earth pressure in the non-limit state of finite soil is as follows:

[0084] The functional derivatives of the upper and lower interfaces of the rectangular and triangular regions can be calculated using equations (37) and (38):

[0085] (37)

[0086] (38)

[0087] Assuming the initial slip surface dip angle β Construct the error function σ = P' 1(z)- P' 2(z), and then use the error function to solve for the error value. If the minimum value condition is not met, readjust the dip angle of the slip surface until the minimum value requirement of the error function is met. The specific operation process is as follows: first, determine whether the current step number is less than the maximum step number. If so, increase the step number by 1. Then, calculate the upper boundary derivative according to formulas (37) and (38). P' 1(z) and lower boundary derivative P' 2(z) is calculated, and the difference between the two is taken as the current error. If the absolute value of the error exceeds the threshold of 10... -6 The numerical calculation workflow tool is invoked, and material parameters are calculated by iteratively calling functions. Then, the upper derivative is calculated using both the rectangular region model and the triangular region model. P' 1(z) and lower derivative P' 2(z), recalculate the error. If the error still exceeds the threshold, find a new slip surface dip angle. If the initial guess The error corresponding to the value is positive, indicating that the current... If the value is large, it needs to be reduced. Conversely, if the error is negative, then it needs to be increased. Through continuous iterative adjustments When the error approaches zero, find the optimal... The numerical tools are then driven to bring the error close to zero, and finally the slip surface inclination angle is updated. This process is repeated until the error meets the accuracy requirements, ultimately outputting a reasonable slip surface inclination angle. .

[0088] Furthermore, step 5 is detailed below:

[0089] Resultant earth pressure in the non-ultimate state of finite soil Determined by equation (39).

[0090] (39)

[0091] Overturning moment generated by horizontal active earth pressure on the wall heel in the non-ultimate state of finite soil Determined by equation (40).

[0092] (40).

[0093] Beneficial effects

[0094] Compared with existing technologies, this invention combines the soil arching effect, the differential element method, and numerical calculation methods to establish a progressive relationship between the non-limit state strength parameters of finite-width soil and the displacement development of the wall, characterizing the earth pressure distribution of finite-width soil along the wall depth direction. From the perspective of evaluating the service status of retaining structures, it proposes a method for effectively calculating the inclination angle of the slip surface of finite-width soil under active non-limit state, which has the following significant advantages:

[0095] (1) The method for calculating the inclination angle of the slip surface of finite soil provided by the present invention addresses the shortcomings of the non-limit state earth pressure calculation method under finite soil conditions. It not only considers the gradual development of the strength parameters of finite soil caused by the displacement of the wall, but also obtains the inclination angle of the slip surface that satisfies the plastic upper limit theorem. The resulting earth pressure function curve is continuous and differentiable. Compared with the existing calculation methods, the calculation results are closer to the experimental results.

[0096] (2) The method for calculating the inclination angle of the finite-width soil slip surface provided by the present invention can determine the magnitude of the change in earth pressure in the non-limit state caused by the asymptotic exertion of soil strength parameters under different conditions based on different soil parameters. For actual projects where the slip surface is unknown but the soil layer parameters are known (i.e., geological exploration conditions), the corresponding finite soil pressure distribution pattern can also be obtained, so as to meet the seismic design requirements of retaining walls under different conditions, and has significant engineering application value. Attached Figure Description

[0097] Figure 1 This is a schematic diagram of the algorithm flow of the method of the present invention;

[0098] Figure 2 This is a schematic diagram of a finite soil model between retaining walls and its displacement mode in an embodiment of the present invention;

[0099] Figure 3 This is a schematic diagram of the soil arching effect and minor principal stress trace in an active state finite soil according to an embodiment of the present invention;

[0100] Figure 4This is a schematic diagram of the Mohr circle of soil stress behind the retaining wall in an embodiment of the present invention;

[0101] Figure 5 This is a pseudocode diagram illustrating the numerical method in an embodiment of the present invention;

[0102] Figure 6 This is a comparison and verification diagram of experimental results and numerical calculation results in an embodiment of the present invention. Detailed Implementation

[0103] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are implemented based on the technical solution of the present invention, providing detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0104] Example

[0105] This embodiment provides a method for calculating the slip surface of finite-width soil under active non-limit conditions, which is used for the design calculation of retaining structures under finite-width soil conditions. The overall flowchart of the invention method is shown below. Figure 1 As shown. Assume there is a finite width of cohesive fill between the rigid gravity retaining wall on the left and the existing bedrock on the right, such as... Figure 2 The calculation of the slip surface in finite soil under non-ultimate earth pressure includes the following steps:

[0106] Step 1: Calculate the gradual development of parameters such as internal friction angle, cohesion, external friction angle between wall and soil, and cohesion at the wall-soil interface under the translational mode of the retaining wall as the retaining wall displacement develops;

[0107] Among them, the internal friction angle of soil in its non-limit state The sine value can be determined by equation (1); the external friction angle between the gravity retaining wall on the left and the finite soil. δ m The effective value can be determined by equation (2); the effective value of the interface friction angle between the existing bedrock and the finite soil on the right side. α m The finite soil cohesion utilization value can be determined by equation (3). c m The cohesion between the rigid gravity retaining wall on the left and the finite soil mass can be determined by equation (4). c wm The value of the cohesion between the existing bedrock and the finite soil mass on the right side can be determined by equation (5). c dm The non-limit state Mohr circle of the soil behind the retaining wall can be determined by equation (6). Figure 4 This is an illustration. The specific formula is as follows:

[0108] (1)

[0109] (2)

[0110] (3)

[0111] (4)

[0112] (5)

[0113] (6)

[0114] in, K 0 represents the coefficient of lateral earth pressure at rest; R f The failure ratio can be determined based on triaxial unloading tests; η It is the ratio of the wall base displacement under non-limit active state to the wall base displacement under limit active state; H This refers to the height of the gravity retaining wall; The internal friction angle is the angle at the intermediate state. The internal friction angle is the limiting angle. δ To measure the external friction angle between the rigid gravity retaining wall on the left and the finite soil mass; α This is the measured interfacial friction angle between the existing bedrock and the finite soil mass on the right side.

[0115] Step Two: According to Figure 3 The diagram shows the soil arching effect and minor principal stress trajectory of finite soil in active state. Based on the soil arching effect principle, the active earth pressure coefficient of the soil behind the retaining wall in the rectangular and triangular regions is calculated respectively. and and interlayer shear stress friction coefficient and .

[0116] For the rectangular region, the minor principal stress rotation angles on both sides of the differential soil element are first determined using equations (7) and (8). and .

[0117] (7)

[0118] (8)

[0119] Furthermore, the average vertical stress on the rectangular region differential unit is determined by equations (9), (10), and (11). .in For the major principal stresses of the soil element in the new coordinates, is Rankine's active earth pressure coefficient.

[0120] (9)

[0121] (10)

[0122] (11)

[0123] Furthermore, the coefficient of lateral earth pressure of the soil behind the retaining wall in the rectangular area It can be determined by equation (12).

[0124] (12)

[0125] (13)

[0126] in, and The small principal stress deflection angle at the slip surface in the rectangular region is determined by equations (7) and (8); The average vertical stress in the rectangular region under the new coordinate system is determined by equations (9) and (10); The Rankine active earth pressure coefficient is determined by equation (11); internal variables Determined by equation (13).

[0127] Interlayer shear stress coefficient of soil behind the retaining wall in the rectangular area It can be determined by equation (14):

[0128] (14)

[0129] For the triangular region, a differential soil strip element at any depth in the triangular region is analyzed. First, the minor principal stress rotation angles on both sides of the differential soil strip element are determined by equations (15) and (16). and ,

[0130] (15)

[0131] (16)

[0132] In the formula, The angle between the slip surface and the horizontal direction is determined precisely by step (4).

[0133] Average vertical stress on the triangular region differential unit It can be determined by equation (17):

[0134] (17)

[0135] The coefficient of lateral earth pressure on the soil behind the retaining wall in the triangular area is determined by equation (18). .

[0136] (18)

[0137] (19)

[0138] in, and The small principal stress deflection angle at the slip surface in the triangular region is determined by equations (15) and (16); The average vertical stress in the triangular region under the new coordinate system is determined by equation (17); The Rankine active earth pressure coefficient is determined by equation (11); internal variables Determined by equation (19).

[0139] Furthermore, the interlayer shear stress coefficient of the soil behind the retaining wall in the triangular region It can be determined by equation (20).

[0140] (20)

[0141] Step 3:

[0142] For a rectangular region of finite width cohesive fill, a differential soil strip element is taken along the depth direction. The average vertical stress on the differential soil strip element is: Normal force at the left wall Tangential force at the wall The normal force at the right interface is Tangential force at the right interface Differential soil strip horizontal interlayer shear stress The weight of the thin soil layer is The stress balance equations for the horizontal and vertical directions of the differential soil strip are established using equations (21) and (22):

[0143] (twenty one)

[0144] (twenty two)

[0145] Combining equations (21) and (22), we can obtain the equilibrium differential equation:

[0146] (twenty three)

[0147] Boundary conditions are at the surface ( z =0) Vertical stress is ground overload q 0, that is .

[0148] Therefore, the earth pressure distribution in the rectangular region can be obtained as follows:

[0149] (twenty four)

[0150] (25)

[0151] (26)

[0152] Furthermore, for the cohesive fill of finite width in the triangular region, differential soil strip elements are taken along the depth direction. The average vertical stress on the differential soil strip element is... ,in and The average vertical stress on the differential soil strip elements representing the rectangular and triangular regions, respectively, and the normal force at the left wall surface. Tangential force at the wall The normal force at the right interface is Tangential force at the right interface Differential soil strip horizontal interlayer shear stress The weight of the thin soil layer is The stress equilibrium equations for the horizontal and vertical directions of the differential soil strip can be established using equations (27) and (28):

[0153] (27)

[0154] (28)

[0155] Combining equations (27) and (28), we can obtain the equilibrium differential equation:

[0156] (29)

[0157] Among them, N, I, P and T are all intermediate variables, which are determined by equations (30), (31), (32) and (33) respectively:

[0158] (30)

[0159] (31)

[0160] (32)

[0161] (33)

[0162] Furthermore, the general solution of differential equation (29) is:

[0163] (34)

[0164] Boundary conditions are The constant term can be obtained by substituting the boundary conditions. W :

[0165] (35)

[0166] Specifically, the distribution of non-limit active earth pressure under the parallel movement mode of the retaining wall towards the finite width of the fill is given by equation (36):

[0167] (36)

[0168] Step 4: Calculate the dip angle of the slip surface that satisfies the plasticity theorem, and calculate the resultant horizontal earth pressure. and the overturning moment generated by horizontal earth pressure on the wall heel The pseudocode for the calculation method is as follows: Figure 5 As shown.

[0169] The functional derivatives of the upper and lower interfaces of the rectangular and triangular regions can be calculated using equations (37) and (38):

[0170] (37)

[0171] (38)

[0172] Assuming the initial slip surface dip angle β Construct the error function σ = P' 1(z)- P' 2(z), and then use the error function to solve for the error value. If the minimum value condition is not met, readjust the dip angle of the slip surface until the minimum value requirement of the error function is met. First, check if the current step number is lower than the maximum step number. If so, increase the step number by 1. Then, calculate the upper boundary derivative according to equations (37) and (38). P' 1(z) and lower boundary derivative P' 2(z), and calculate the difference between the two as the current error. If the absolute value of the error is greater than 10... -6 Then, an iterative numerical solution method is invoked. Material parameters are calculated by iteratively calling functions, and the upper derivative is recalculated using both rectangular and triangular region models. P' 1(z) and lower derivative P' 2(z), recalculate the error. If the error still exceeds the threshold, adjust the slip surface inclination angle. If the initial guess The error corresponding to the value is positive, indicating that the current... If the value is large, it needs to be reduced. Conversely, if the error is negative, then it needs to be increased. Through continuous iterative adjustments When the error approaches zero, find the optimal... The numerical tools are driven to bring the error close to 0, and the slip surface inclination angle is updated. This process is repeated until the error meets the accuracy requirements, ultimately outputting a reasonable slip surface inclination angle. .

[0173] Step 5: Based on the optimized slip surface inclination angle, recalculate the earth pressure distribution and obtain the magnitude of the resultant earth pressure force and the magnitude of the moment.

[0174] Furthermore, the resultant earth pressure in the non-limit state of finite soil is determined by equation (39). .

[0175] (39)

[0176] Furthermore, the overturning moment generated by the horizontal active earth pressure on the wall heel in the non-limit state of finite soil is determined by equation (40). .

[0177] (40)

[0178] This invention proposes a framework for determining the slip surface of non-limit earth pressure in finite soil under rigid retaining wall translation mode, comprehensively considering the influence of soil strength parameters, wall displacement, and soil arching effect. Based on the differential element method, a mathematical representation of non-limit earth pressure is established, enabling non-limit earth pressure analysis of finite soil between retaining structures and providing insights for the rational design of retaining structures. This invention can calculate the development of earth pressure behind the retaining wall with wall displacement, and the resulting earth pressure curve is continuous and differentiable. Compared with classical earth pressure theory, it can provide more accurate calculation results, offering theoretical reference value for the rational design of retaining structures under finite width soil conditions.

[0179] This invention selects typical test results of Deng Bo et al. on the lateral earth pressure of unsaturated sand behind the wall under the active translation mode, Rankine's classical earth pressure solution, and Lai Fengwen et al.'s analytical solution for cross-comparison to verify the rationality. This verifies the accuracy of the calculation results of this embodiment for the actual earth pressure distribution of soil with finite width. The selection of calculation parameters, soil cohesion... c The soil friction angle is 0 kPa. φ The angle is 33.6°, and the displacement ratio is... η Equal to 0.66; Soil weight Aspect Ratio B / H The value is 0.5. The non-limit state earth pressure distribution calculated in this embodiment is compared with the measured and theoretical solutions by Deng Bo, Lai Fengwen, and Rankine et al., such as... Figure 6 As shown.

[0180] It can be seen that the calculation results of this embodiment are in good agreement with the industry-recognized typical benchmark test results of Deng Bo et al. (refer to [1], Deng Bo, Yang Minghui, Zhao Minghua. Experimental study on failure mode and lateral earth pressure distribution of unsaturated sand behind the wall under active translation mode [J]. Chinese Journal of Geotechnical Engineering, 2023, 45(01): 94-102.). Figure 6 The paper also presents a corrected solution for active earth pressure in finite soil obtained by Lai Fengwen et al. using theoretical methods. It can be seen that Lai Fengwen et al.'s analytical solution, which only considers the limit state, results in a smaller active earth pressure value. The classical Rankine earth pressure, on the other hand, is also undervalued because it does not consider the soil arching effect. The earth pressure distribution obtained by this method basically conforms to the trend of the curve obtained from experiments. This embodiment uses iterative calculation to determine whether the slip surface inclination angle meets the upper bound theorem conditions of plasticity. If not, the slip surface inclination angle is adjusted until the requirements are met. This approach can improve the inaccuracy of earth pressure calculation results when the slip surface inclination angle is assumed to be based on empirical formulas.

[0181] Compared with existing technologies, this embodiment considers the influence of soil arching effect and explores a method for determining the optimal slip surface inclination angle that conforms to the principle of mechanical equilibrium. This method can meet the design requirements of retaining walls under finite soil conditions and helps to provide a more refined calculation method for the rational design of retaining structures.

[0182] The above description is merely a description of preferred embodiments of this application and is not intended to limit the scope of this application in any way. Any changes or modifications made by those skilled in the art based on the above-disclosed technical content should be considered as equivalent and valid embodiments and fall within the scope of protection of the technical solution of this application.

[0183] Appendix: References:

[0184] [1] Deng Bo, Yang Minghui, Zhao Minghua. Experimental study on failure mode and lateral earth pressure distribution of unsaturated sand behind wall under active translation mode [J]. Chinese Journal of Geotechnical Engineering, 2023, 45(01):94-102.

[0185] [2] Lai Fengwen, Liu Songyu, Yang Dayu, et al. Corrected solution of active earth pressure for finite fill retaining wall [J]. Journal of Southeast University (Natural Science Edition), 2022, 52(03): 557-563.

[0186] [3]Rankine WJ M. II. On the stability of loose earth[J]. Philosophical transactions of the Royal Society of London, 1857 (147): 9-27.

Claims

1. A method for calculating the slip surface of finite-width soil under active non-limit state, characterized in that, Includes the following steps: Step 1: Determine the gradual development law of the internal friction angle, cohesion, external friction angle between the wall and the soil, and the cohesion at the wall-soil interface of the rigid gravity retaining wall with the development of the finite soil between the wall and the existing bedrock, and establish a mathematical characterization of the non-limit state strength parameters of the finite soil with the development of the retaining wall displacement. Step 2: Using the horizontal plane where the initial slip surface intersects with the existing bedrock on the right as the boundary line, divide the finite-width soil mass into an upper rectangular region (0 ≤ z ≤ 1). h = H - B tan β ) and the lower triangular area ( h = H - B tan β ≤ z ≤ H Two areas, of which H For a finite amount of soil, B For the width of a finite soil mass, h The height of the rectangular area. β Based on the principle of soil arching effect, the soil pressure coefficient and interlayer shear stress coefficient of the rectangular and triangular areas behind the retaining wall are determined according to the slip surface angle, and the relationship between horizontal stress, interlayer shear stress and average vertical stress is established. Step 3: Take differential soil strip elements along the depth direction for the finite width cohesive fill between the rigid gravity retaining wall and the existing bedrock, and establish the equilibrium equations in the horizontal and vertical directions for the rectangular and triangular regions respectively, based on the average vertical stress and horizontal stress on the differential soil strips. Step 4: Combining the results of Steps 2 and 3, use numerical methods to solve for the slip surface inclination angle that satisfies the plasticity upper bound theorem, so that it meets the requirement of the continuity of the earth pressure curve. Step 5: Calculate the resultant horizontal active earth pressure based on the slip surface inclination angle obtained in Step 4. The overturning moment generated by the horizontal active earth pressure on the wall heel ; The calculation of seismic earth pressure in the non-limit state of finite soil masses is as follows: For a rectangular region of finite width cohesive fill, a differential soil strip element is taken along the depth direction. The average vertical stress on the differential soil strip element is: Normal force at the left wall Tangential force at the wall The normal force at the right interface is Tangential force at the right interface Differential soil strip horizontal interlayer shear stress The weight of the thin soil layer is The stress balance equations for the horizontal and vertical directions of the differential soil strip are established using equations (21) and (22): （21） （22） Combining equations (21) and (22), we obtain the equilibrium differential equation: （23） Boundary conditions are at the surface ( z =0) Vertical stress is ground overload q 0, that is ;in and The average vertical stresses experienced by the differential soil strip elements in the rectangular and triangular regions are represented respectively. The earth pressure distribution in the rectangular area was thus obtained. for: （24） （25） （26） For a finite-width cohesive fill in the triangular region, a differential soil strip element is taken along the depth direction. The average vertical stress on the differential soil strip element is... Normal force at the left wall Tangential force at the wall ; The normal force at the right interface is Tangential force at the right interface Differential soil strip horizontal interlayer shear stress The weight of the thin soil layer is The stress balance equations for the horizontal and vertical directions of the differential soil strip are established using equations (27) and (28): （27） （28） Combining equations (27) and (28), we obtain the equilibrium differential equation: （29） Among them, N, I, P and T are all intermediate variables, which are determined by equations (30), (31), (32) and (33) respectively: （30） （31） （32） （33） The general solution of differential equation (29) is: （34） Boundary conditions are ,in and The earth pressure distributions in the rectangular and triangular regions are shown below. Substituting these into the boundary conditions yields the constant term. W : （35） The distribution of non-limit active earth pressure under the parallel movement mode of the retaining wall towards the finite width of the fill is given by equation (36): （36） in: δ m This represents the external friction angle between the gravity retaining wall on the left and the finite soil mass. α m This represents the effective value of the interface friction angle between the existing bedrock and the finite soil mass on the right side. c wm This represents the cohesion value between the rigid gravity retaining wall on the left and the finite soil mass. c dm This represents the value of cohesion between the existing bedrock and the finite soil mass on the right side. This represents the lateral earth pressure coefficient of the soil behind the retaining wall in the rectangular area. This represents the interlayer shear stress coefficient of the soil behind the retaining wall in the rectangular area. The calculation of seismic earth pressure in the non-limit state of finite soil masses is as follows: The functional derivatives of the upper and lower interfaces of the rectangular and triangular regions are calculated using equations (37) and (38): （37） （38） Assuming the initial slip surface dip angle β Construct the error function σ = P' 1(z)- P' 2(z), then use the error function to solve for the error value. If the minimum value condition is not met, readjust the dip angle of the slip surface until the minimum value requirement of the error function is met. The specific operation process is as follows: first, determine whether the current step number is less than the maximum step number. If so, increase the step number by 1. Then, calculate the upper boundary derivative according to formulas (37) and (38). P' 1(z) and lower boundary derivative P' 2(z), and calculate the difference between the two as the current error; if the absolute value of the error exceeds the threshold 10 -6 The numerical calculation workflow tool is invoked, and material parameters are calculated by iteratively calling functions. Then, the upper derivative is calculated using both the rectangular region model and the triangular region model. P' 1(z) and lower derivative P' 2(z), recalculate the error; if the error still exceeds the threshold, find a new slip surface dip angle. β m If the initial guess β m The error corresponding to the value is positive, indicating that the current... β m If the value is large, it needs to be reduced. β m Conversely, if the error is negative, then it needs to be increased. β m Through continuous iterative adjustments β m When the error approaches zero, find the optimal... β m The numerical tools are then driven to bring the error close to zero, and the slip surface inclination angle is updated. This process is repeated until the error meets the accuracy requirements, ultimately outputting a reasonable slip surface inclination angle. β m .

2. The method for calculating the slip surface of finite-width soil under active non-limit state according to claim 1, characterized in that, In step 1, the earth pressure in the non-limit state of finite soil is calculated as follows: For a finite width of cohesive fill between a rigid gravity retaining wall and the existing bedrock, it is necessary to determine the gradual development of the internal friction angle, cohesion, external friction angle between the wall and the soil, and the cohesion parameters at the wall-soil interface as the retaining wall moves from a static state to the active limit state. Specifically, in the parallel movement mode of the retaining wall, the internal friction angle of the soil in the non-limit state is determined by equation (1); （1） in, K 0 represents the coefficient of lateral earth pressure at rest; R f The failure ratio can be determined based on triaxial unloading tests; It is the ratio of the wall base displacement under non-limit active state to the wall base displacement under limit active state; φ m The internal friction angle is the angle at the intermediate state. φ u The internal friction angle is the limiting angle. The external friction angle between the gravity retaining wall and the finite soil mass on the left side δ m The performance value is determined by formula (2); （2） in, δ 0 =φ u / 2, δ To measure the external friction angle between the left retaining wall and the finite soil mass; The effective value of the interfacial friction angle between the existing bedrock and the finite soil mass on the right side α m Determined by equation (3); （3） in, α 0 =φ u / 2, α To measure the interfacial friction angle between the existing bedrock and the finite soil mass on the right side; Limited soil cohesion performance value c m The cohesion between the rigid gravity retaining wall on the left and the finite soil mass is determined by equation (4). c wm The cohesion value between the existing bedrock and the finite soil mass on the right side is determined by equation (5). c dm Determined by equation (6); （4） （5） （6） in, c w To measure the cohesion value of the retaining wall soil, c d This represents the measured cohesion value of the bedrock wall soil.

3. The method for calculating the slip surface of finite-width soil under active non-limit state according to claim 2, characterized in that, In step 2, the calculation of the non-ultimate earth pressure of a finite-width soil mass is as follows: Analyze the differential soil strip element at any depth within the rectangular region, and determine the minor principal stress rotation angles on both sides of the differential soil strip element. and Determined by equations (7) and (8) respectively: （7） （8） Average vertical stress on a rectangular differential unit Determined by equation (9): （9） （10） In the formula, For the major principal stresses of the soil element in the new coordinates, The active earth pressure coefficient of Rankine is determined by equation (11): （11） Earth pressure coefficient of soil behind retaining wall in rectangular area Determined by equation (12): （12） （13） in, and The small principal stress deflection angle at the slip surface in the rectangular region is determined by equations (7) and (8); The average vertical stress of the rectangular region under the new coordinate system is determined by equations (9) and (10); The Rankine active earth pressure coefficient is determined by equation (11); internal variables Determined by equation (13); Interlayer shear stress coefficient of soil behind the retaining wall in the rectangular area To determine by equation (12): （14） Analyze the differential soil strip element at any depth in the triangular region, and determine the minor principal stress rotation angles on both sides of the differential soil strip element. and Determined by equations (15) and (16) respectively: （15） （16） In the formula, The angle between the slip surface and the horizontal direction is determined precisely through step (4); Average vertical stress on the triangular region differential unit Determined by equation (17): （17） Earth pressure coefficient of soil behind retaining wall in triangular area It can be determined by equation (18): （18） （19） in, and The small principal stress deflection angle at the slip surface in the triangular region is determined by equations (15) and (16); The average vertical stress in the triangular region under the new coordinate system is determined by equation (17); The Rankine active earth pressure coefficient is determined by equation (11); internal variables Determined by equation (19); interlayer shear stress coefficient of soil behind retaining wall in triangular zone Determined by equation (20): （20）。 4. The method for calculating the slip surface of finite-width soil under active non-limit state according to claim 1, characterized in that, Step 5 is as follows: Resultant earth pressure in the non-ultimate state of finite soil Determined by equation (39): （39） Overturning moment generated by horizontal active earth pressure on the wall heel in the non-ultimate state of finite soil Determined by equation (40): （40）。

Images