Analytical method for local damage and leakage problems of retaining wall of foundation pit
By establishing a partial leakage model of foundation pit retaining wall and deriving the head analytical solution using the separation variable method and the orthogonality of the trigonometric function, the problem of difficult to analyze the leakage flow field after local failure of foundation pit retaining wall in the prior art is solved. It realizes efficient and accurate seepage field analysis.
Patent Information
- Application Number
- CN202411526284.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-10-30
- Publication Date
- 2025-02-14
AI Technical Summary
The prior art is difficult to efficiently and accurately analyze the seepage field of leakage after local damage to the foundation pit retaining wall, especially in the case of multiple leakages, making it difficult to achieve accurate analysis.
By establishing a local leakage model of foundation pit retaining wall, the head solution of each region is derived using the separation variable method, and the head analytical solution of each adjacent region is obtained through the orthogonality and integration of the trigonometric function, and the pores are obtained by combining the Bernoulli equation. Analytical solution to water pressure.
The precise analysis of the seepage field after local failure of the foundation pit retaining wall is achieved, the analysis efficiency and accuracy are improved, and the water inflow volume at the bottom of the foundation pit is reflected, providing a reliable basis for the stability evaluation of the foundation pit project.
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Figure CN119442416A_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the technical field of foundation pit engineering, and in particular relates to an analytical method for the problem of local damage and leakage of a foundation pit retaining wall. Background Art
[0002] In foundation pit projects in water-rich areas, the foundation pit retaining wall is a key structure to maintain structural stability and prevent leakage. However, due to the water level difference formed after the foundation pit excavation and the long-term seepage corrosion, some areas inside the retaining wall are prone to local leakage, which in turn affects the overall stability and safety of the foundation pit. In the prior art, the seepage field analysis of local leakage of the foundation pit retaining wall mostly relies on approximate solutions, numerical simulations or experimental methods, which makes it difficult to obtain efficient and accurate analytical solutions, and even more difficult to solve the situation where multiple leakages occur.
[0003] Therefore, there is an urgent need for a seepage field analysis method based on the continuous conditions of combined water head and flow velocity to achieve accurate analysis of the seepage field of local damage to the foundation pit retaining wall. Summary of the invention
[0004] The purpose of the present invention is to overcome the deficiencies in the prior art and provide an analytical method for the problem of local damage and leakage of a foundation pit retaining wall.
[0005] The analytical method for the local damage and leakage problem of the foundation pit retaining wall comprises the following steps:
[0006] Step 1: Establish a local leakage model of the foundation pit retaining wall based on the symmetry of the model;
[0007] Step 2: Based on the separation of variables method, the general solution H(x,z) of the water head in each area of the foundation pit is derived. The trigonometric function is multiplied and integrated in each interval to obtain the full rank expression of the coefficient matrix equation in the analytical solution of the water head in each adjacent area. The unknown coefficients in the water head expression are solved to obtain the analytical solution of the water head in the seepage field in each area of the foundation pit.
[0008] Step 3: Combine the analytical solution of the seepage field head in each area of the foundation pit with the Bernoulli equation to obtain the analytical solution of the foundation pit pore water pressure p.
[0009] Preferably, in step three, the analytical solution of the water head in the seepage field of each area of the foundation pit obtained in step two is combined with the hydraulic gradient to obtain the seepage volume in the leaking area of the vertical retaining wall of the foundation pit, the water inflow from the surface and the water outflow from the bottom of the foundation pit.
[0010] Preferably, in step 2, the orthogonality of trigonometric functions is utilized, each interval is multiplied by the corresponding trigonometric function sin() and integrated, the infinite series equation is integrated over the characteristic interval, and the infinite series is truncated to N terms in advance to obtain a strict analytical solution of the coefficients, thereby obtaining the full rank expression of the coefficient matrix equation in the analytical solution of the water head in each adjacent area.
[0011] Preferably, in step 2, a unified form of seepage calculation processing is performed on the seepage field in the damaged area of the foundation pit retaining wall and the seepage field of the soil, and the seepage field of the damaged retaining wall is simplified.
[0012] Preferably, in step one, the surface of the water-rich area and the bottom of the foundation pit are both constant heads. In step two, according to all model boundary conditions, each adjacent area in the solution area satisfies the head continuity and velocity continuity conditions, the foundation pit soil and the leaking parts of the retaining wall are in a stable seepage state, and the water flow obeys Darcy's law.
[0013] As a preferred method, in step 2, based on the separation of variables method, the general solution of the hydraulic head H(x,z) is obtained as
[0014]
[0015] Where h is the depth, J, K, L, A n , B n , C n , D n , k n 、n=1,2…、E e 、F e , G e , H e and e=1,2…are unknown coefficients in the general solution.
[0016] Preferably, in step 3, the surface z=h0+L+h1 outside the foundation pit is used as the starting position for water head calculation, and the foundation pit pore water pressure p is expressed by water head H(x,z) as follows:
[0017] p=γ w (H(x,z)-z+h0+L+h1)
[0018] In the formula, γ w is the gravity of water, L is the height of local leakage of foundation pit retaining wall, h0 is the height of foundation pit retaining wall below the local leakage of foundation pit retaining wall, and h1 is the height of foundation pit retaining wall above the local leakage of foundation pit retaining wall.
[0019] The beneficial effects of the present invention are:
[0020] 1) The seepage field analysis method provided by the present invention avoids the complexity of numerical simulation and the high cost of experimental methods in the form of analytical solutions, thereby improving the analysis efficiency and accuracy; it can accurately reflect the impact of local damage to the foundation pit retaining wall on the hydraulic gradient of the dangerous area, and can reflect the water gushing volume at the bottom of the foundation pit, providing a reliable basis for the stability assessment of the foundation pit project.
[0021] 2) The method proposed in the present invention is not only applicable to a single leakage damage area, but can also be extended to the working conditions of multiple leakage damage areas on the foundation pit retaining wall. The area where the foundation pit retaining wall is locally damaged and difficult to solve is treated in a manner similar to the seepage field in the soil. Finally, a medium continuous head and flow velocity continuous treatment method is adopted in the place where the soil and the retaining wall leakage area are continuous. The change of the overall seepage field of the foundation pit after the local damage of the retaining wall can be quickly obtained by analysis, which is helpful for the anti-seepage design of the foundation pit project and has stronger applicability. It can be applied to the seepage field analysis of the foundation pit of vertically layered soil foundation, which broadens the scope of application and has higher engineering application value. BRIEF DESCRIPTION OF THE DRAWINGS
[0022] Figure 1 The overall framework flow chart provided for the present invention;
[0023] Figure 2 Schematic diagram of the left half of the foundation pit retaining wall leakage model. DETAILED DESCRIPTION
[0024] The present invention is further described below in conjunction with embodiments. The description of the following embodiments is only used to help understand the present invention. It should be noted that for ordinary persons in the art, without departing from the principle of the present invention, the present invention can also be modified in some ways, and these improvements and modifications also fall within the scope of protection of the claims of the present invention.
[0025] Embodiment 1
[0026] As an example, Figure 1 As shown, the analytical method for the local damage and leakage problem of the foundation pit retaining wall includes the following steps:
[0027] Step 1: According to the symmetry of the model, the left half model of the local leakage of the foundation pit retaining wall is established, including the geometric parameters of the foundation pit, the water level on the surface and at the bottom of the foundation pit, the permeability coefficient of the soil and the permeability coefficient of the retaining wall, and other parameters, and the basic assumptions, boundary conditions, and model head and flow rate continuity conditions of the model are determined. The leakage area of the foundation pit retaining wall has been simplified to a rectangular area, and its upper and lower boundaries are impermeable, and its permeability coefficient is generally much smaller than the permeability coefficient of the soil in which it is located;
[0028] Step 2: Based on the separation of variables method, the general solution of the hydraulic head in each area of the foundation pit is derived, and the expressions of the hydraulic head and flow velocity continuity conditions of each adjacent area are solved by combining the model boundary conditions, hydraulic head and flow velocity continuity conditions. It should be noted that under the assumption of the foundation pit damage area in step 1, the seepage field in the foundation pit retaining wall damage area can be treated with the seepage field of the soil in a unified form of seepage calculation, which effectively and practically simplifies the solution of the complex retaining wall damage seepage field. Using the orthogonality of trigonometric functions, multiplying the corresponding trigonometric function sin() on the corresponding interval and integrating it, the full rank expression of the coefficient matrix equation in the analytical solution of the hydraulic head in each adjacent area can be further obtained. The infinite series equation is integrated on the characteristic interval, and the infinite series is truncated to N terms in advance to obtain a strict coefficient analytical solution.
[0029] Then, the unknown coefficients in the water head expression are explicitly solved by the left division algorithm of MATLAB, so as to obtain the analytical solution of the water head of the seepage field in the whole area of the foundation pit. It should be noted that this coefficient solution method is much more accurate than directly using the non-full rank matrix equation to solve the corresponding matrix equation according to the boundary collocation method. The obtained analytical solution can also better reflect the seepage field of the actual foundation pit soil and the leakage damage area.
[0030] Step 3: Combine the analytical solution of the seepage field head in step 2 with the Bernoulli equation to obtain the analytical solution of the foundation pit pore water pressure; it should be noted that the vertical width of the local damaged area of the foundation pit is generally much smaller than other model sizes. Therefore, when displaying the analytical solution, it is necessary to increase the vertical grid spacing of the local damaged area of the foundation pit and its adjacent rectangular area so that the vertical distribution of the seepage field at this location can be displayed more clearly, otherwise the head contour line may oscillate up and down;
[0031] Combined with the analytical solution of the seepage field head in each area of the foundation pit in step 2 and the hydraulic gradient, the seepage volume in the leaking area of the vertical retaining wall of the foundation pit, the water inflow from the surface and the water outflow from the bottom of the foundation pit are obtained.
[0032] Embodiment 2
[0033] As another embodiment, this embodiment 2 proposes, based on the embodiment 1, a more specific method for analyzing the leakage problem of local damage of the foundation pit retaining wall, such as Figure 2 As shown:
[0034] In step 1, the basic assumptions of the model are:
[0035] 1) Constant water head on the surface in water-rich areas and constant water head at the bottom of the foundation pit;
[0036] 2) The foundation pit model is symmetrical about the vertical axis, and its left half is taken for analysis. The soil in the center of the foundation pit is impermeable in the horizontal direction;
[0037] 3) The soil is homogeneous and isotropic, and the permeability coefficient remains constant;
[0038] 4) The soil of the foundation pit and the seepage parts of the retaining wall are in a stable seepage state, and the water flow obeys Darcy's law.
[0039]
[0040] Where: (x,z) is Figure 2 The coordinates of any point in , H(x,z) is Figure 2 The total head value at any coordinate point (x, z) in the foundation pit plane.
[0041] Furthermore, the boundary conditions of the model in step 1 are:
[0042]
[0043] Where: a, b, c, d, h0, h1, h2, L are Figure 2 The size parameter of the model. h w1 and h w2 They are the constant water head of the ground surface and the constant water head at the bottom of the foundation pit. BC1 is the constant water head boundary condition of the ground surface, BC2 is the constant water head boundary condition at the bottom of the foundation pit, BC3 is the horizontal impermeable boundary condition of the intact part of the foundation pit retaining wall, the left wall of the foundation pit, and the vertical symmetric center axis of the foundation pit, BC4 is the impermeable boundary condition at the bottom of the model, and BC5 is the vertical impermeable boundary condition at the bottom of the retaining wall.
[0044] Furthermore, the model head and flow rate continuity conditions in step 1 are:
[0045]
[0046] The first and second equations in formula (3) are Figure 2 The continuity conditions of the head and flow velocity at the vertical connection of two adjacent areas in the solution area divided by , the third and fourth equations are Figure 2 The continuity conditions of the water head and flow velocity at the horizontal connections between two adjacent areas in the solution area divided.
[0047] It should be noted that the parts in this embodiment that are the same or similar to those in the first embodiment can be referenced to each other and will not be described in detail in this application.
[0048] Embodiment 3
[0049] As another embodiment, this embodiment 3 proposes a more specific method for analyzing the leakage problem of local damage of the foundation pit retaining wall on the basis of the embodiments 1 and 2:
[0050] In step 2, based on the separation of variables method, the general solution of H(x,z) in formula (1) is:
[0051]
[0052] Where: J, K, L, A n , B n , C n , D n , n=1,2...,E e , F e , G e , H e , e=1,2..., are all unknown parameters in the general solution.
[0053] As a further solution of this scheme, the general solution of the seepage head in each area of the foundation pit in step 2 is specifically expressed as:
[0054] Substituting all model boundary conditions (2) into the general solution (4) respectively, we can obtain the partial coefficient values of each region in regions ① to ⑩ and the eigenvalues of the corresponding coordinate axis directions:
[0055] ① Regional head general solution:
[0056]
[0057] ② Regional head general solution:
[0058]
[0059] ③ Regional head general solution:
[0060]
[0061] ④General solution of regional water head:
[0062]
[0063] ⑤General solution of regional water head:
[0064]
[0065] ⑥ Regional head general solution:
[0066]
[0067] ⑦ Regional head general solution:
[0068]
[0069] ⑧ Regional head general solution:
[0070]
[0071] ⑨General solution for regional head:
[0072]
[0073] ⑩General solution of regional head:
[0074]
[0075] As a further solution of this scheme, the expressions of the continuity conditions of the water head and flow velocity of each adjacent area in step 2 can be obtained by substituting the continuity condition formula (3) into formulas (5) to (14), and the expressions of the continuity conditions of the water head and flow velocity of each adjacent area are respectively expressed as follows:
[0076] Areas ① and ②:
[0077] Continuous head:
[0078]
[0079] Continuous flow rate:
[0080]
[0081] Areas ② and ③:
[0082] Continuous head:
[0083]
[0084] Continuous flow rate:
[0085]
[0086] Areas ② and ⑩: Continuous head:
[0087]
[0088] Continuous flow rate:
[0089]
[0090] Areas ③ and ④: Continuous head:
[0091] Continuous flow rate:
[0092] Areas ④ and ⑤: Continuous head:
[0093]
[0094] Continuous flow rate:
[0095]
[0096] Areas ⑤ and ⑥: Continuous head:
[0097]
[0098] Continuous flow rate:
[0099]
[0100] Areas ⑥ and ⑦: Continuous head:
[0101]
[0102] Continuous flow rate:
[0103]
[0104] Areas ⑦ and ⑧: Continuous head:
[0105]
[0106] Continuous flow rate:
[0107]
[0108] Areas ⑧ and ⑨: Continuous head:
[0109]
[0110] Continuous flow rate:
[0111]
[0112] Areas ⑧ and ⑩: Continuous head:
[0113]
[0114] Continuous flow rate:
[0115]
[0116] It should be noted that the parts in this embodiment that are the same or similar to those in Embodiments 1 and 2 can be referenced to each other and will not be described in detail in this application.
[0117] Embodiment 4
[0118] As another embodiment, this embodiment proposes a more specific analytical method for the problem of local damage and leakage of foundation pit retaining wall on the basis of embodiments 1 to 3: In step 2, another form of expression of the head and flow velocity continuity conditions of each adjacent area is obtained by using the orthogonality of trigonometric functions, which is as follows:
[0119] Using the orthogonality of trigonometric functions, equations (15) to (34) are multiplied by the corresponding trigonometric function sin() in the corresponding interval and integrated to obtain another form of expression for the head and flow velocity continuity conditions of each adjacent area, which is specifically expressed as:
[0120] Areas ① and ②:
[0121] Continuous head:
[0122]
[0123] Continuous flow rate:
[0124]
[0125] Areas ② and ③: Continuous head:
[0126]
[0127] Continuous flow rate:
[0128]
[0129] Areas ② and ⑩: Continuous head:
[0130]
[0131] Continuous flow rate:
[0132]
[0133] Areas ③ and ④: Continuous head:
[0134]
[0135] Continuous flow rate:
[0136]
[0137] Areas ④ and ⑤: Continuous head:
[0138]
[0139] Continuous flow rate:
[0140]
[0141] Areas ⑤ and ⑥: Continuous head:
[0142]
[0143] Continuous flow rate:
[0144]
[0145] Areas ⑥ and ⑦: Continuous head:
[0146]
[0147] Continuous flow rate:
[0148]
[0149] Areas ⑦ and ⑧: Continuous head:
[0150]
[0151]
[0152] Continuous flow rate:
[0153]
[0154] Areas ⑧ and ⑨:
[0155] Continuous head:
[0156]
[0157] Continuous flow rate:
[0158]
[0159] Areas ⑧ and ⑩:
[0160] Continuous head:
[0161]
[0162] Continuous flow rate:
[0163]
[0164] As a further solution of this scheme, in step 2, the undetermined coefficients in the expression are solved by MATLAB to obtain the analytical solution of the water head of the seepage field in the whole area of the foundation pit, which is specifically:
[0165] Combining equations (35) to (54), MATLAB is used to completely analyze the unknown coefficients in equations (5) to (14), and the analytical solution of the seepage field in the entire foundation pit area is obtained, in which the number of series terms must be truncated at N0.
[0166] As a further solution of this scheme, the pore water pressure of the foundation pit in step 3 is calculated starting from the position head z=h0+L+h1 on the surface outside the foundation pit, and the pore water pressure p is expressed by the head H as follows:
[0167] p=γ w (H(x,z)-z+h0+L+h1) (55)
[0168] As a further solution of this scheme, in step 4, the seepage rate per unit length of the vertical retaining wall leakage area of the foundation pit, the water inflow from the surface and the water outflow from the bottom of the foundation pit are obtained according to the hydraulic gradient, which can be specifically expressed as formulas (56), (57) and (58):
[0169]
[0170] The corresponding total leakage at the bottom of the foundation pit is Q2 or Q3.
[0171] It should be noted that the parts in this embodiment that are the same or similar to those in Embodiments 1 to 3 can be referenced to each other and will not be described in detail in this application.
[0172] The various embodiments in this specification are described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the various embodiments can be referenced to each other.
Claims
1. A method for analyzing the local damage and leakage problem of foundation pit retaining wall, characterized in that: The following steps are involved: Step 1: Establish a local leakage model of the foundation pit retaining wall based on the symmetry of the model; Step 2: Based on the separation of variables method, the general solution H(x,z) of the water head in each area of the foundation pit is derived. The trigonometric function is multiplied and integrated in each interval to obtain the full rank expression of the coefficient matrix equation in the analytical solution of the water head in each adjacent area. The unknown coefficients in the water head expression are solved to obtain the analytical solution of the water head in the seepage field in each area of the foundation pit. Step 3: Combine the analytical solution of the seepage field head in each area of the foundation pit with the Bernoulli equation to obtain the analytical solution of the foundation pit pore water pressure p.
2. The analytical method for local damage and leakage of foundation pit retaining wall according to claim 1 is characterized in that: In step three, the analytical solution of the seepage field head in each area of the foundation pit obtained in step two is combined with the hydraulic gradient to obtain the seepage volume in the leaking area of the vertical retaining wall of the foundation pit, the water inflow from the surface, and the water outflow from the bottom of the foundation pit.
3. The analytical method for local damage and leakage of foundation pit retaining wall according to claim 1 is characterized in that: In step 2, the orthogonality of trigonometric functions is utilized to multiply each interval by the corresponding trigonometric function sin() and integrate, integrate the infinite series equation on the characteristic interval, and truncate the infinite series to N terms in advance to obtain a strict coefficient analytical solution, and obtain the full rank expression of the coefficient matrix equation in the analytical solution of the water head in each adjacent area.
4. The analytical method for local damage and leakage of foundation pit retaining wall according to claim 1 is characterized in that: In step 2, the seepage field in the damaged area of the foundation pit retaining wall and the seepage field of the soil are processed by unified seepage calculation, and the seepage field of the damaged retaining wall is simplified.
5. The analytical method for local damage and leakage of foundation pit retaining wall according to claim 1 is characterized in that: In step one, the surface of the water-rich area and the bottom of the foundation pit are both constant heads. In step two, according to all the model boundary conditions, each adjacent area in the solution area meets the head continuity and velocity continuity conditions, the foundation pit soil and the leakage part of the retaining wall are in a stable seepage state, and the water flow obeys Darcy's law.
6. The analytical method for local damage and leakage of foundation pit retaining wall according to claim 1 is characterized in that: In step 2, based on the separation of variables method, the general solution of the hydraulic head H(x,z) is obtained as In the formula, J, K, L, A n , B n , C n , D n , k n 、n=1,2…、E e 、F e , G e , H e and e=1,2…are unknown coefficients in the general solution.
7. The analytical method for local damage and leakage of foundation pit retaining wall according to claim 1 is characterized in that: In step 3, the surface z=h0+L+h1 outside the foundation pit is used as the starting position for the water head calculation, and the foundation pit pore water pressure p is expressed by the water head H(x,z) as follows: p=γ w (H(x,z)-z+h0+L+h1) In the formula, γ w is the gravity of water, L is the height of local leakage of foundation pit retaining wall, h0 is the height of foundation pit retaining wall below the local leakage of foundation pit retaining wall, and h1 is the height of foundation pit retaining wall above the local leakage of foundation pit retaining wall.