Initiative soil pressure slip line solution calculation method for foundation instability limited width filling foundation pit support

Abstract

The invention relates to the technical field of geotechnical engineering foundation pit support design, in particular to a foundation instability limited width filling foundation pit support soil pressure slip line solution calculation method, and aims to solve the problems that a foundation is simplified into a rigid body and the combined action of the foundation and a structure is ignored in a traditional method. A unified calculation model fusing an active area behind a wall, a passive area in front of the wall and a foundation plastic area is established, differential numerical solution is performed by adopting a characteristic line method through a simultaneous static equilibrium equation and a yield criterion, and construction and stress state calculation of a slip line field are realized by utilizing three types of boundary value problems. According to the method, the active soil pressure and the passive soil pressure borne by the retaining wall and the ultimate bearing capacity of the foundation under the condition of foundation instability can be accurately solved, the accuracy and the engineering applicability of soil pressure calculation under the working conditions of limited width filling and soft soil foundations are remarkably improved, and a reliable theoretical basis is provided for optimal design of a supporting structure.

Description

Technical Field

[0001] This invention relates to the field of geotechnical engineering foundation pit support technology, specifically to a method for calculating the earth pressure slip line solution of finite-width fill foundation pit support for foundation instability. Background Technology

[0002] Retaining structures are widely used in geotechnical engineering, and their stability is significantly affected by foundation conditions, fill properties, external loads, and environmental factors (such as earthquakes). my country has a vast territory and complex geological conditions, and engineering construction often encounters weak foundations, such as soft clay and silty soil. These foundations typically have low bearing capacity, high water content, and high compressibility, posing serious challenges to the design and construction of retaining structures. The core issue in retaining wall design lies in accurately calculating the earth pressure distribution and foundation bearing capacity. However, when there is significant deformation or instability risk in the foundation, the applicability of traditional theoretical methods is limited.

[0003] Existing research indicates that the displacement mode of retaining walls is closely related to foundation deformation, which in turn affects the stress state and failure mechanism of the soil behind the wall. Furthermore, numerous studies, through numerical simulations and experiments, have confirmed the significant impact of foundation deformation on the lateral earth pressure of retaining walls. Although these studies emphasize the importance of foundation-structure interaction, most existing methods are still based on the assumption of a rigid foundation or only consider foundation bearing capacity or earth pressure distribution, failing to fully couple the interaction between foundation instability and the ultimate limit state of the fill. Especially under finite-width fill conditions (such as retaining walls near natural slopes), the width of the fill behind the wall is limited, and its failure mode often occurs simultaneously with overall foundation instability, forming a complex failure mechanism including active, passive, and plastic zones of the foundation. In this case, the applicability of traditional earth pressure theory is significantly reduced.

[0004] In the field of geotechnical limit analysis, the slip line method is widely used to solve earth pressure and foundation bearing capacity problems because it does not require prior assumptions about the slip surface morphology and can strictly satisfy the static equilibrium and yield conditions within the plastic zone. However, existing slip line theory research mostly focuses on retaining wall problems under the traditional assumption of infinite-width fill or rigid foundation, and research on the failure mechanism under the coupled action of finite-width fill and foundation instability is still relatively lacking. Most experimental and numerical models assume that the foundation is a rigid body or only focus on the plastic development of the fill zone, failing to fully reflect the combined failure characteristics of the foundation's plastic zone, active zone, and passive zone. In actual engineering, finite-width fill retaining walls near natural slopes often fail as a whole due to foundation instability, and the earth pressure distribution and failure mode in this case are significantly different from the traditional theoretical assumptions.

[0005] Therefore, it is necessary to develop an earth pressure calculation method that can simultaneously consider the coupling effect of foundation instability and finite-width fill, accurately reflect the stress characteristics and failure mechanism of retaining walls on soft foundations, and provide a more accurate theoretical basis for the design of retaining structures under similar geological and geometric boundary conditions. Summary of the Invention

[0006] Based on the aforementioned problems, this invention provides a method for calculating the slip line solution of active earth pressure in finite-width fill foundation pit support for foundation instability. By accurately reflecting the stress state of the soil and the force distribution of the retaining wall, it provides a more accurate theoretical basis for the design of retaining structures under similar geological and geometric boundary conditions.

[0007] To achieve the above objectives, the present invention provides the following technical solution: a method for calculating the slip line solution of soil pressure in finite-width fill foundation pit support for foundation instability, the method comprising: S1. Calculation and analysis of element stress state and slip line difference; S2, Analysis of stress discontinuity characteristics; S3. Boundary problem handling and analysis; S4. Establish the calculation model and calculation program; S5. Comparison and verification: Compare and verify the slip line solution with the finite element solution to prove the feasibility of the slip line solution; Wherein, S1 includes: The stress element of the soil satisfies different conditions when the soil reaches different states, as follows: In a two-dimensional plane problem, the soil is defined as a rigid-plastic body, and the three stress components are: s x , s z τ xz Satisfying the Mohr-Coulomb yield criterion: ; in, f The internal friction angle of the backfill soil behind the wall. c The soil cohesion (kPa) is such that plastic flow occurs in the plastic region of the soil under load; the soil element satisfies the equilibrium equation: ; in, a To disrupt the angle between the sliding surface and the horizontal line; When the soil reaches plastic limit equilibrium, a pair of shear planes appear in the soil element. Let... The angle between the axis and the vertical line is... a Major principal stress s 1 The angle with the X-axis is l , s 1 The angle with the horizontal line is ɵ , ɵ=λ + a Then the shear plane and the major principal stress s 1 The included angle is Then the three stress components of the stress unit s x , s z τ xz They are expressed as follows: ; In the formula, f The internal friction angle of the soil. The average normal reference stress; Connecting the principal stress directions at each point continuously forms the principal stress trajectory. When the soil is in a yielding state, there is a pair of shear surfaces at each point. Connecting the shear surfaces at each point on the plane continuously yields two families of slip lines. Using the major principal stress trajectory as the baseline, the family of slip lines that rotate clockwise at an acute angle to the baseline is called... a A family of slip lines that move counterclockwise at an acute angle to the baseline is called a slip line. β Wire, a lines and β The differential equation of the line is expressed as: ; Combining the above equations, we obtain the limit equilibrium equation, which is based on... x , z , s , ɵ The equation of a spatial surface with variables can be solved using the method of characteristics, and the equation of the characteristic lines is as follows: ; The above system of equations uses the finite difference method to obtain an approximate solution. When the difference step size is sufficiently close, the result is infinitely close to the true solution. However, the difference calculation always starts from the known boundary and proceeds to the unknown boundary point by point. Given points a and b x a , z a , s a , ɵ a and x b ,z b , s b , ɵ b Point a along α The line connects to point m, and point b follows... β If the line connects to point m, then the value of point m is... x m , z m , s m and ɵ m It can be calculated from the following difference equation: .

[0008] Further analysis of stress discontinuity characteristics: Within the overlapping area of ​​the slip network, the same points will exhibit different stress states. External forces acting on the plastic zone cannot satisfy the static equilibrium condition. This phenomenon constitutes a stress discontinuity. A stress discontinuity line is found in the soil, where the stress changes on both sides. Define line I as a stress discontinuity line. Take any unit cell spanning both sides of line I. The two sides of stress discontinuity line I represent the strongly plastic region and the weakly plastic region, respectively. The strongly plastic region is region I, and the weakly plastic region is region II. The major principal stresses... s 1 Ⅰ , s 1 Ⅱ The angles between the x and y axes are respectively ɵ Ⅰ and ɵ Ⅱ The stress components are respectively s n1 , s t1 , t t1 and s n2 , s τ2 , t t2 The unit satisfies the equilibrium equation and yield condition, and the tangential stresses in the strongly plastic region and the weakly plastic region are equal: , , ; The formula Substituting into the above equation, the relationship between the reference stresses in region I and region II is derived as follows: ; When the slip line passes through the stress discontinuity line to another plastic region, it will deflect and satisfy the following relationship: ; S3 specifically includes: boundary stress conversion and three types of boundary value problems; The boundary stress conversion is as follows: Normal stress on the known boundary s n and tangential stress t t Assuming c , f Given, then Figure 6 Stress referenced at point M p and its angle with the normal d It can be represented as: ; Based on the Mohr circle of stress in the soil element and Points M and M on the boundary correspond to the passive and active limit states of the stress element, respectively. The stress state at point M on the boundary can be represented by a Mohr circle. Figure 6 According to geometric relations: ; Where N is taken when point M is in a passive state. When point M is in an active state, N takes... ; From the above formula, we can obtain: ; in, , It is a positive integer; When the difference calculation starts from a known boundary or proceeds to an unknown boundary, it is directly given by the geometric relations on the Mohr circle. s n and t t : ; The three types of boundary value problems are as follows: When performing slip line difference calculations, there are three types of boundary value problems: Cauchy problem, Riemann problem, and mixed boundary value problem. (1) Cauchy's problem, such as Figure 7 As shown, at the known boundaryO 1 A Points on 1 x , z , s , ɵ The value is obtained by solving point by point using the difference formula. O 1 A 1 A The solution within area 2; (2) The Riemann problem, such as Figure 7 As shown, given two characteristic lines O 1 O 2 and O 1 A Points on 1 x , z , s , ɵ Values ​​are calculated point-by-point in the region using the difference formula. O 1 O 2 A 1 A 2 All the answers within, when O 1 O 2 When they coincide at a single point, the region O 1 O 2 A 1 A 2 It is fan-shaped; (3) Mixed problems, such as Figure 7 As shown, in the known characteristic lines O 1 A Points on 1 x , z , s , ɵ Values ​​and boundaries O 1 A 2 The two conditions above can be used to substitute the known boundary values ​​into the difference formula for solution. O 1 A All solutions on side 2.

[0009] Furthermore, step S4 includes: Based on the three types of boundary value problems in step S2, a calculation model for the limit state of instability of the retaining wall foundation near the natural slope is established, such as... Figure 8 As shown, the X-axis extends horizontally to the right from point O, and the Z-axis extends vertically downwards from point O. The calculation steps are as follows: Within the active zone behind the wall: (1) From the known aboutSolve the Cauchy problem while solving the unknown region. oab ; (2) respectively by oh and sharpening o , ba and sharpening b Points, solving the sharpened Riemann problem, solving for unknown regions. oac and bad ; (3) respectively by oh and oh , bd and bf Solve the mixed boundary value problem to obtain the unknown region. oh and bdf ; (4) By yes and af Solve the Riemann problem and solve for the unknown region. aegf ; (5) respectively by eg and oh , fg and fi Solving mixed boundary value problems and solving unknown regions. oh yeah and fgi ; (6) By gh and gj Solve the Riemann problem and solve for the unknown region. hey ; (7) By hj and hk Solving mixed boundary value problems and solving unknown regions. hjk ; (8) From the convergence point l The boundary of the plastic region is obtained. lp； Within the passive zone in front of the wall and the plastic zone of the foundation: (1) From the known st Solve the Cauchy problem while solving the unknown region. student ; (2) By so and sharpening s Points, solving the sharpened Riemann problem, can solve for unknown regions. SUV ; (3) By sv and sw Solving mixed boundary value problems can solve unknown regions. swvw ; (4) From the known tx Solving the Cauchy problem allows us to solve for unknown regions. txy; (5) By you and tw Solving the Riemann problem allows for the solution of unknown regions. twzy ; (6) By wz and sharpening w Points, solving the sharpened Riemann problem, can solve for unknown regions. wzr ; (7) By wr and wl Solving mixed boundary value problems can solve unknown regions. wrl ; During the calculation process, the initial nodes are continuously subdivided until the calculation accuracy converges within the specified error. This allows us to determine the stress state at any point within the plastic zone of the soil under the ultimate limit state, thus obtaining the stress state of the retaining wall surface behind the wall. ol , retaining wall front sw and retaining wall base wl The ultimate pressure it is subjected to.

[0010] Furthermore, in step S1, when solving for the three stress components, an approximate solution is obtained by applying slip line theory.

[0011] Furthermore, based on the fact that the partial differentials of stress and strain rate in the plastic zone are hyperbolic when the soil reaches "infinite" plastic flow under plane strain conditions, we can apply the characteristic line theory to solve the limit solution of the plane strain problem.

[0012] Furthermore, the boundary problem handling in step S3 requires that the known boundaries be considered before the calculation begins. x, z The pressure value at a location and its orientation angle are required to be converted into differential values. s, ɵ After calculating to the unknown boundary, then put the unknown boundary... x, z Location s, ɵ The value is converted into the pressure value and its direction angle used in the design.

[0013] Furthermore, in step S4, during the calculation process, the initial calculation node is continuously subdivided until the calculation accuracy converges to within the limit error.

[0014] Furthermore, in step S5, to verify the feasibility of the slip line solution method and to further study the influence of various parameters on the ultimate state of the soil, the slip line solution is used to provide the soil slip line field, fill stress vector diagram, fill stress cloud diagram, and wall-soil contact surface pressure distribution diagram under various working conditions, which are then compared and verified with the finite element solution.

[0015] As can be seen from the above technical solution, the present invention has the following beneficial effects: This paper presents a slip line method for calculating earth pressure on retaining walls in finite-width fill foundation pits, considering foundation instability. Existing methods fail to accurately determine the active earth pressure from the finite-width fill behind the wall, the passive earth pressure in front of the wall, and the bearing capacity of the foundation under ultimate conditions. The slip line method systematically and effectively solves for the active earth pressure from the fill behind the retaining wall, the passive earth pressure from the soil in front of the wall, and the ultimate bearing capacity of the foundation after foundation instability. The slip line network visually defines the extent of the plastic zone, and the location of the failure sliding surface is determined by the envelope of the slip line network. Attached Figure Description

[0016] Figure 1 This is a flowchart of the present invention; Figure 2 This is a diagram illustrating the analysis of the instability and failure modes of retaining wall foundations near natural slopes, based on the present invention. (a) Schematic diagram of embankment retaining wall; (b) Schematic diagram of a gravity-type wharf; Figure 3 This is a stress state analysis diagram of the soil element in this invention; Figure 4 This is a schematic diagram of the slip line difference calculation of the present invention; Figure 5 This is a diagram illustrating the stress discontinuity characteristics of the present invention. Figure 6 This is a schematic diagram of the boundary stress conversion for the present invention: Schematic diagram of boundary surface forces; Mohr's circle of stress in soil elements; Figure 7 This is an analysis diagram of three boundary problems in this invention; Figure 8 This is a schematic diagram of the calculation model of the present invention; Figure 9 Comparison and verification diagram of the stress on retaining walls under different backfill widths; Figure 10 Comparative verification of the stress on retaining walls under different soil conditions: Different natural slope angles; Different interface strengths. Detailed Implementation

[0017] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and 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.

[0018] like Figure 1-Figure 9As shown, the present invention provides a technical solution: a method for calculating the slip line solution of soil pressure in finite-width backfill foundation pit support for foundation instability, the method comprising: like Figure 2 The image shows the failure mode of a retaining wall foundation near a natural slope. Currently, research on the ultimate failure mode of finite-width backfill behind retaining walls mainly focuses on the plastic failure of the backfill. Experimental and analytical models typically do not consider foundation conditions or assume the foundation is rigid, using assumed displacement modes of the retaining wall to induce an active limit state in the backfill. In actual engineering, the active limit failure of finite-width backfill behind retaining walls often accompanies foundation instability. For example... Figure 2 When a retaining wall near a natural slope experiences foundation instability, the backfill behind the wall reaches an active limit state (referred to as the active zone), while the soil in front of the wall reaches a passive limit state (referred to as the passive zone), resulting in foundation instability (referred to as the foundation plastic zone). To fully investigate the ultimate failure mode of retaining wall foundation instability near a natural slope, this invention explores the ultimate failure mode of finite-width backfill considering foundation instability using finite element analysis and slip line theory, solving for the stress state of the soil and the stress condition of the retaining wall.

[0019] like Figure 3 The diagram shown illustrates the stress state analysis of a single element. In this two-dimensional plane problem, the soil is assumed to be a rigid-plastic body, with three stress components: s x , s z τ xz Satisfying the Mohr-Coulomb yield criterion: ; Under load, plastic flow occurs in the plastic region of the soil. The soil element satisfies the equilibrium equations: ; When the soil reaches plastic limit equilibrium (plastic yielding), a pair of shear planes will appear in the soil element. The angle between the axis and the vertical line is... a Major principal stress s 1 The angle between it and the X-axis is λ. s 1 The angle with the horizontal line is ɵ , ɵ= l + a Then the shear plane and the major principal stress s 1 The included angle is Then the three stress components of the stress element sx , s z τ xz These can be expressed as: ; In the formula, f The internal friction angle of the soil. The average normal reference stress.

[0020] In plane strain problems, every point has two orthogonal principal stresses. Connecting the directions of these principal stresses continuously forms the principal stress trajectory. When the soil is in a yielding state, each point has a pair of shear surfaces. Connecting these shear surfaces continuously on the plane yields two families of slip lines. Using the major principal stress trajectory as the baseline, the family of slip lines rotating clockwise at an acute angle to the baseline is called... a A family of slip lines that move counterclockwise at an acute angle to the baseline is called a slip line. β Wire, a lines and β The differential equation of a line can be expressed as: ; Combining the above equations, we can obtain the limit equilibrium equation, which is a... x , z , s , ɵ The equation of the spatial surface with variable denoted as is a system of first-order quasi-linear partial differential equations, which can be solved using the method of characteristics. The equation of the characteristic lines is as follows: ; In general, the above system of equations is difficult to solve analytically directly, but an approximate solution can be obtained using the finite difference method. When the difference step size is sufficiently close, the result is infinitely close to the true solution, and the difference calculation always starts from the known boundary and proceeds point by point towards the unknown boundary.

[0021] like Figure 4 The diagram shown illustrates the calculation of the slip line difference. Points a and b are known. x a , z a , s a , ɵ a and x b , z b , sb , ɵ b Point a along α The line connects to point m, and point b follows... β If the line connects to point m, then the value of point m is... x m , z m , s m and ɵ m It can be calculated from the following difference equation: ; like Figure 5 The diagram illustrates the characteristics of stress discontinuities. Overlapping folds may occur in slip nets, resulting in different stress states at the same points within overlapping areas. External forces acting on the plastic zone may not satisfy static equilibrium conditions. This phenomenon is considered a stress discontinuity, and the search should focus on finding stress discontinuities in the soil where stress changes abruptly on either side. Line I represents a stress discontinuity, with a unit cell spanning both sides. Regions I and II on either side of the discontinuity represent the strong plastic region and the weak plastic region, respectively, with the major principal stresses... s 1 Ⅰ , s 1 Ⅱ The angles between the x and y axes are respectively ɵ Ⅰ and ɵ Ⅱ The stress components are respectively s n1 , s t1 , t t1 and s n2 , s τ2 , t t2 Since the element satisfies the equilibrium equation and yield condition, the stress along the tangential direction in the two regions should also be equal. Only the normal stress along the tangential direction of the discontinuity line in the two regions can be discontinuous.

[0022] , , ; The formula Substituting into the above equation, the relationship between the reference stresses in region I and region II can be derived as follows: ; When the slip line passes through the stress discontinuity line to another plastic region, it will deflect and satisfy the following relationship: ; like Figure 6 The diagram shown illustrates the boundary stress conversion. The normal stress on the boundary is known. s n and tangential stress t t Assuming c , f Given the stress at point M, p and its angle with the normal d It can be represented as: ; Based on the Mohr circle of stress in the soil element and Points M and M on the boundary correspond to the passive and active limit states of the stress element, respectively. The stress state at point M on the boundary can be represented by a Mohr circle. From the geometric relationship in the figure, we can see that: ; Where N is taken when point M is in a passive state. When point M is in an active state, N takes... .

[0023] Simplifying the above equation, we can obtain: ; in, , It is a positive integer.

[0024] When the difference calculation starts from a known boundary or proceeds to an unknown boundary, it can be directly given by the geometric relations on the Mohr circle. s n and t t , ; like Figure 7 The diagram shows the analysis of three types of boundary value problems. When performing slip line difference calculations, the three common boundary value problems include the Cauchy problem, the Riemann problem, and the mixed boundary value problem.

[0025] (1) Cauchy problem, with known boundaries O 1 APoints on 1 (non-characteristic line) x , z , s , ɵ The value can be solved point by point using the difference formula. O 1 A 1 A The solution within area 2.

[0026] (2) The Riemann problem, given two characteristic lines O 1 O 2 and O 1 A Points on 1 x , z , s , ɵ The value can be calculated point-by-point within the region using a difference formula. O 1 O 2 A 1 A 2 All the answers are within. It's worth noting that when... O 1 O 2 When they coincide at a single point, the region O 1 O 2 A 1 A 2 It is fan-shaped.

[0027] (3) Mixed problems, where the characteristic lines are known O 1 A Points on 1 x , z , s , ɵ Values ​​and boundaries O 1 A 2 The two conditions above are as follows ( x , ɵ The value of the boundary condition can be substituted into the difference formula to solve the problem. O 1 A All solutions on side 2.

[0028] Preferably, S4 specifically includes: A calculation model for the instability limit state of retaining wall foundations near natural slopes can be established based on the three types of boundary value problems mentioned above. The X-axis extends horizontally to the right from point O, and the Z-axis extends vertically downwards from point O.

[0029] like Figure 8The diagram shown is a schematic of the computational model. It establishes a computational model for the instability limit state of a retaining wall foundation near a natural slope, based on the three types of boundary value problems mentioned above. The X-axis extends horizontally to the right from point O, and the Z-axis extends vertically downwards from point O.

[0030] The specific calculation steps are as follows: Within the active zone behind the wall: (1) From the known about Solving the Cauchy problem allows us to solve for unknown regions. oab ; (2) respectively by oh and sharpening o , ba and sharpening b Points, solving the sharpened Riemann problem, can solve for unknown regions. oac and bad ; (3) respectively by oh and oh , bd and bf Solve the mixed boundary value problem to obtain the unknown region. oh and bdf ; (4) By yes and af Solving the Riemann problem allows for the solution of unknown regions. aegf ; (5) respectively by eg and oh , fg and fi Solving mixed boundary value problems can solve unknown regions. oh yeah and fgi ; (6) By gh and gj Solving the Riemann problem allows for the solution of unknown regions. hey ; (7) By hj and hk Solving mixed boundary value problems can solve unknown regions. hjk ; (8) From the convergence point l The boundary of the plastic region is obtained. lp .

[0031] Within the passive zone in front of the wall and the plastic zone of the foundation: (1) From the known st Solving the Cauchy problem allows us to solve for unknown regions. student ; (2) By soand sharpening s Points, solving the sharpened Riemann problem, can solve for unknown regions. SUV ; (3) By sv and sw Solving mixed boundary value problems can solve unknown regions. swvw ; (4) From the known tx Solving the Cauchy problem allows us to solve for unknown regions. txy ; (5) By you and tw Solving the Riemann problem allows for the solution of unknown regions. twzy ; (6) By wz and sharpening w Points, solving the sharpened Riemann problem, can solve for unknown regions. wzr ; (7) By wr and wl Solving mixed boundary value problems can solve unknown regions. wrl ; During the calculation process, to achieve the target accuracy, the initial nodes can be continuously subdivided until the calculation accuracy converges within the specified error. This allows us to solve for the stress state at any point within the plastic zone of the soil under the ultimate limit state, thus obtaining the stress state of the retaining wall surface behind the wall. ol , retaining wall front sw and retaining wall base wl The ultimate pressure it is subjected to.

[0032] The slip line method was used to study the influence of different geometric parameters of the backfill soil on the ultimate failure of the soil. To control for variables, soft plastic soil was used for the backfill soil, and stiff plastic soil was used for the foundation soil. The height of the retaining wall was [not specified]. H =10.0m, foundation depth D =3.0m, base width B =1 = 3.0 m, wall-soil contact strength reduction factor R = 0.67, base friction coefficient u =0.67, uniformly distributed load on the soil surface behind the wall q =10 kPa, the influence of changes in fill geometry is as follows: like Figure 9 The figure shows the stress conditions of the retaining wall under different backfill widths. Since the calculations for the passive zone and the plastic zone of the foundation are consistent under the three conditions, therefore... Figure 9 The passive earth pressure in front of the wall and the ultimate bearing capacity of the foundation slip linear solutions for the three working conditions in (a) and (b) coincide, and the calculation results are largely consistent with the finite element results. Since the soil being calculated is cohesive, in Figure 9In (c), the active earth pressure behind the wall has negative lateral pressure within the shaded area. In reality, the wall and soil will separate under very small tensile force; therefore, the earth pressure in the shaded area should be ignored. As can be seen from the figure, the slip line solution and the finite element results are in excellent agreement. As the width of the backfill behind the wall decreases, the active earth pressure on the retaining wall decreases accordingly. (Comparison) and In both working conditions, the reduction in active earth pressure at the wall heel reached 53.5%, and the nonlinear characteristics of the earth pressure distribution became increasingly apparent.

[0033] like Figure 10 As shown, Figure 10 (a) The distribution of active earth pressure behind the retaining wall under different natural slope inclination angles is presented and compared with finite element results. As the natural slope inclination angle decreases, the active earth pressure on the retaining wall decreases. and In both operating conditions, the reduction at the base reached The slip line solution and the finite element solution are in excellent agreement. Figure 10 (b) The earth pressure on the retaining wall under different wall-soil interface strength conditions is presented and compared with the finite element results. It can be seen from the figure that as the wall-soil interface strength increases, the active earth pressure behind the wall decreases. Comparing the two working conditions of R=0 and R=1, the active earth pressure decreases by 40.0%.

[0034] This invention is based on the study of the ultimate failure mode of retaining wall foundation instability near a natural slope using the finite element method. To accurately calculate the active earth pressure behind the retaining wall, the passive earth pressure in front of the wall, and the ultimate bearing capacity of the foundation, the slip line method is used for modeling and calculation, and corresponding calculation code is developed. This code can be used to analyze the influence of different parameters, such as the geometry of the backfill behind the wall, the wall-soil interface friction, soil strength, and foundation conditions, on the soil failure mode and the ultimate earth pressure on the retaining wall.

[0035] This invention discloses a method for calculating the slip line solution of earth pressure in finite-width fill foundation pits considering foundation instability. Based on slip line theory, this method constructs a unified slip line calculation model consisting of an active zone behind the wall, a passive zone in front of the wall, and a plastic zone of the foundation. By simultaneously applying static equilibrium equations and the Mohr-Coulomb yield criterion, a differential numerical solution is performed using the method of characteristics combined with three types of boundary value problems (Cauchy problem, Riemann problem, and mixed boundary value problem). The calculation process is automated using Matlab. The embodiments compare the calculation results under various working conditions, such as different fill widths, natural slope angles, and wall-soil interface friction coefficients, demonstrating that this method accurately reflects the nonlinear distribution characteristics of earth pressure under coupled foundation instability conditions, and highly agrees with the finite element results. This method effectively overcomes the limitations of the traditional rigid foundation assumption, providing a reliable theoretical tool and practical calculation approach for the accurate design and stability analysis of retaining structures under finite-width fill and weak foundation conditions.

[0036] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for calculating the slip line solution of soil pressure in finite-width fill foundation pit support for foundation instability, characterized in that... The method includes: S1. Calculation and analysis of element stress state and slip line difference; S2, Analysis of stress discontinuity characteristics; S3. Boundary problem handling and analysis; S4. Establish the calculation model and calculation program; S5. Comparison and verification: Compare and verify the slip line solution with the finite element solution to prove the feasibility of the slip line solution; Wherein, S1 includes: The stress element of the soil satisfies different conditions when the soil reaches different states, as follows: In a two-dimensional plane problem, the soil is defined as a rigid-plastic body, and the three stress components are: σ x , σ z τ xz Satisfying the Mohr-Coulomb yield criterion: ； in, φ The internal friction angle of the backfill soil behind the wall. c The soil cohesion (kPa) is such that plastic flow occurs in the plastic region of the soil under load; the soil element satisfies the equilibrium equation: ； in, a To disrupt the angle between the sliding surface and the horizontal line; When the soil reaches plastic limit equilibrium, a pair of shear planes appear in the soil element. Let... The angle between the axis and the vertical line is... a Major principal stress σ 1 The angle with the X-axis is λ , σ 1 The angle with the horizontal line is ɵ , ɵ=λ + a Then the shear plane and the major principal stress σ 1 The included angle is Then the three stress components of the stress unit σ x , σ z τ xz They are expressed as follows: ； In the formula, φ The internal friction angle of the soil. The average normal reference stress; Connecting the principal stress directions at each point continuously forms the principal stress trajectory. When the soil is in a yielding state, there is a pair of shear surfaces at each point. Connecting the shear surfaces at each point on the plane continuously yields two families of slip lines. Using the major principal stress trajectory as the baseline, the family of slip lines that rotate clockwise at an acute angle to the baseline is called... a A family of slip lines that move counterclockwise at an acute angle to the baseline is called a slip line. β Wire, a lines and β The differential equation of the line is expressed as: ； Combining the above equations, we obtain the limit equilibrium equation, which is based on... x , z , σ , ɵ The equation of a spatial surface with variables can be solved using the method of characteristics, and the equation of the characteristic lines is as follows: ； The above system of equations uses the finite difference method to obtain an approximate solution. When the difference step size is sufficiently close, the result is infinitely close to the true solution. However, the difference calculation always starts from the known boundary and proceeds to the unknown boundary point by point. Given points a and b x a , z a , σ a , ɵ a and x b , z b , σ b , ɵ b Point a along α The line connects to point m, and point b follows... β If the line connects to point m, then the value of point m is... x m , z m , σ m and ɵ m It can be calculated from the following difference equation: .

2. The method for calculating the slip line solution of soil pressure in finite-width backfill foundation pit support for foundation instability according to claim 1, characterized in that: Step S2 includes: Analysis of stress discontinuity line characteristics: Within the overlapping area of ​​the slip network, the same points will exhibit different stress states. External forces acting on the plastic zone cannot satisfy the static equilibrium condition. This phenomenon constitutes a stress discontinuity. A stress discontinuity line is found in the soil, where the stress changes on both sides. Define line I as a stress discontinuity line. Take any unit cell spanning both sides of line I. The two sides of stress discontinuity line I represent the strongly plastic region and the weakly plastic region, respectively. The strongly plastic region is region I, and the weakly plastic region is region II. The major principal stresses... σ 1 Ⅰ , σ 1 Ⅱ The angles between the x and y axes are respectively ɵ Ⅰ and ɵ Ⅱ The stress components are respectively σ n1 , σ t1 , τ t1 and σ n2 , σ τ2 , τ t2 The unit satisfies the equilibrium equation and yield condition, and the tangential stresses in the strongly plastic region and the weakly plastic region are equal: ， ， ； The formula Substituting into the above equation, the relationship between the reference stresses in region I and region II is derived as follows: ； When the slip line passes through the stress discontinuity line to another plastic region, it will deflect and satisfy the following relationship: ； S3 specifically includes: boundary stress conversion and three types of boundary value problems; The boundary stress conversion is as follows: Normal stress on the known boundary σ n and tangential stress τ t Assuming c , φ Given that the stress at point M in Figure 6 is... p and its angle with the normal δ It can be represented as: ； Based on the Mohr circle of stress in the soil element and Points M and M on the boundary correspond to the passive and active limit states of the stress element, respectively. The stress state at point M on the boundary can be represented by a Mohr circle. As shown in the geometric relationship in Figure 6: ； Where N is taken when point M is in a passive state. When point M is in an active state, N takes... ; From the above formula, we can obtain: ； in, , It is a positive integer; When the difference calculation starts from a known boundary or proceeds to an unknown boundary, it is directly given by the geometric relations on the Mohr circle. σ n and τ t : ； The three types of boundary value problems are as follows: When performing slip line difference calculations, there are three types of boundary value problems: Cauchy problem, Riemann problem, and mixed boundary value problem. (1) Cauchy problem, as shown in Figure 7, under the known boundary O 1 A Points on 1 x , z , σ , ɵ The value is obtained by solving point by point using the difference formula. O 1 A 1 A The solution within area 2; 2) The Riemann problem, as shown in Figure 7, involves given two characteristic lines... O 1 O 2 and O 1 A Points on 1 x , z , σ , ɵ Values ​​are calculated point-by-point in the region using the difference formula. O 1 O 2 A 1 A 2 All the answers within, when O 1 O 2 When they coincide at a single point, the region O 1 O 2 A 1 A 2 It is fan-shaped; (3) Mixed problems, as shown in Figure 7, where the characteristic lines are known O 1 A Points on 1 x , z , σ , ɵ Values ​​and boundaries O 1 A 2 The two conditions above can be used to substitute the known boundary values ​​into the difference formula for solution. O 1 A All solutions on side 2.

3. The method for calculating the slip line solution of soil pressure in finite-width backfill foundation pit support for foundation instability according to claim 2, characterized in that: Step S4 includes: Based on the three types of boundary value problems in step S2, a calculation model for the instability limit state of the retaining wall foundation near the natural slope is established, as shown in Figure 8. The X-axis extends horizontally to the right from point O, and the Z-axis extends vertically downwards from point O. The calculation steps are as follows: Within the active zone behind the wall: (1) From the known ob Solve the Cauchy problem while solving the unknown region. oab ; (2) respectively by oa and sharpening o , ba and sharpening b Points, solving the sharpened Riemann problem, solving for unknown regions. oac and bad ; (3) respectively by oc and oe , bd and bf Solve the mixed boundary value problem to obtain the unknown region. oce and bdf ; (4) By ae and af Solve the Riemann problem and solve for the unknown region. aegf ; (5) respectively by e.g. and eh , fg and fi Solving mixed boundary value problems and solving unknown regions. egh and fgi ; (6) By gh and gj Solve the Riemann problem and solve for the unknown region. ghji ; (7) By hj and HK Solving mixed boundary value problems and solving unknown regions. hjk ; (8) From the convergence point l The boundary of the plastic region is obtained. lp； Within the passive zone in front of the wall and the plastic zone of the foundation: (1) From the known st Solve the Cauchy problem while solving the unknown region. stu ; (2) By su and sharpening s Points, solving the sharpened Riemann problem, can solve for unknown regions. SUV ; (3) By sv and sw Solving mixed boundary value problems can solve unknown regions. svw ; (4) From the known tx Solving the Cauchy problem simultaneously allows for the solution of unknown regions. txy ; (5) By ty and tw Solving the Riemann problem allows for the solution of unknown regions. twzy ; (6) By wz and sharpening w Points, solving the sharpened Riemann problem, can solve for unknown regions. wzr ; (7) By wr and wl Solving mixed boundary value problems can solve unknown regions. wrl ; During the calculation process, the initial nodes are continuously subdivided until the calculation accuracy converges within the specified error. This allows us to determine the stress state at any point within the plastic zone of the soil under the ultimate limit state, thus obtaining the stress state of the retaining wall surface behind the wall. ol , retaining wall front sw and retaining wall base wl The ultimate pressure it is subjected to.

4. The method for calculating the slip line solution of soil pressure in finite-width backfill foundation pit support for foundation instability according to claim 1, characterized in that: In step S1, when solving for the three stress components, an approximate solution is obtained by applying slip line theory.

5. The method for calculating the slip line solution of soil pressure in finite-width backfill foundation pit support for foundation instability according to claim 1, characterized in that: Based on the fact that the partial differentials of stress and strain rate in the plastic zone of soil under plane strain state are hyperbolic when "infinite" plastic flow is achieved, the limit solution of the plane strain problem is solved by applying the characteristic line theory.

6. The method for calculating the slip line solution of soil pressure in finite-width backfill foundation pit support for foundation instability according to claim 1, characterized in that: Step S3, boundary problem handling, requires addressing the known boundaries before starting the calculation. x, z The pressure value at a location and its orientation angle are required to be converted into differential values. σ,ɵ ; After calculating to the unknown boundary, then put the unknown boundary... x, z Location σ,ɵ The value is converted into the pressure value and its direction angle used in the design.

7. The method for calculating the slip line solution of soil pressure in finite-width backfill foundation pit support for foundation instability according to claim 1, characterized in that: In step S4, during the calculation process, the initial nodes are continuously subdivided until the calculation accuracy converges to within the limit error.

8. The method for calculating the slip line solution of soil pressure in finite-width backfill foundation pit support for foundation instability according to claim 1, characterized in that: In step S5, to verify the feasibility of the slip line solution method and to further study the influence of various parameters on the ultimate state of the soil, the slip line solution is used to provide the soil slip line field, fill stress vector diagram, fill stress cloud diagram and wall-soil contact surface pressure distribution diagram under various working conditions, which are then compared and verified with the finite element solution.

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