A method for calculating active earth pressure on finite-width soils considering soil arching effect

Abstract

This invention discloses a method for calculating active earth pressure on finite-width soil masses considering the soil arching effect. Assuming a finite-width soil mass exists between two foundation pits, L and R, distributed to the left and right, the calculation of the active earth pressure on the wall adjacent to the finite-width soil mass in foundation pit L includes the following steps: determining the location of the sliding surface of the finite-width soil mass between the foundation pits; dividing the sliding soil mass into an upper rectangular region 1 and a lower triangular region 2 based on the sliding surface; taking a rectangular micro-element along the depth of the soil mass in region 1; establishing an equilibrium equation based on the assumption that the sliding soil mass is in an active limit equilibrium state, combined with the stress and vertical stress on the micro-element subjected to the wall and soil; applying the vertical stress at the bottom of region 1 as a load q1 to the top of region 2; analyzing the force on the trapezoidal micro-element along the depth of region 2 and establishing an equilibrium equation; solving for relevant parameters on the entire wall, providing a more reasonable basis for the design of foundation pit support structures under finite-width soil conditions.

Description

Technical Field

[0001] This invention belongs to the field of foundation pit engineering technology in the construction industry, and relates to a method for calculating active earth pressure on finite-width soil considering the soil arching effect, especially a method for calculating active earth pressure in foundation pit group engineering where there is finite-width soil. Background Technology

[0002] In foundation pit engineering, the magnitude of the active earth pressure behind the wall is crucial for designing a reasonable foundation pit support structure. However, test results show that the calculation results of the classical earth pressure theory are too large, which will lead to an overly conservative design of the foundation pit support structure and increase the project cost.

[0003] To reasonably calculate active earth pressure, some researchers have found that the calculation results after considering the soil arching effect are closer to the experimental results. At the same time, with the emergence of large-scale projects such as urban commercial complexes and subway transfer station projects, the spacing between foundation pits is limited, and the classical earth pressure theory is no longer applicable. Therefore, it is extremely important to propose an effective active earth pressure calculation method to solve the design of foundation pit support structures under the condition of limited width soil. Summary of the Invention

[0004] The purpose of this invention is to provide a method for calculating active earth pressure on finite-width soils that takes into account the soil arching effect, thereby solving the problems existing in the background art.

[0005] The technical solution adopted in this invention is a method for calculating active earth pressure on finite-width soil considering the soil arching effect. Assuming that there is a finite-width soil mass between L-shaped and R-shaped foundation pits distributed to the left and right, the calculation of the active earth pressure on the wall adjacent to the finite-width soil mass in the L-shaped foundation pit (i.e., the right wall of the L-shaped foundation pit) includes the following steps:

[0006] (1) Determine the location of the sliding surface of the soil with limited width between the foundation pits, and divide the sliding soil into an upper rectangular area (zone 1) and a lower triangular area (zone 2) according to the sliding surface;

[0007] (2) Take a rectangular micro-element along the depth of the soil in Zone 1. Based on the assumption that the sliding soil is in an active limit equilibrium state, and considering the stress σ between the wall and the soil on the micro-element, w1 and vertical stress Establish the equilibrium equations;

[0008] (3) Apply the vertical stress at the bottom of zone 1 as a load q1 to the top of zone 2, analyze the force on the trapezoidal micro-element along the depth of zone 2 and establish the equilibrium equation.

[0009] (4) Combine the calculation results of steps (2) and (3) to solve for the active earth pressure intensity P on the entire wall. ah and size E a Parameters such as bending moment M.

[0010] Further, calculate the active earth pressure. Step (1) is as follows: For the soil with a limited width between the L foundation pit and the R foundation pit, determine whether equation (1) is valid. If it is valid, the condition of the soil with a limited width is met. The sliding surface of the soil with a limited width develops from the bottom of the right wall of the L foundation pit to the wall of the R foundation pit and extends vertically to the ground. Divide the upper rectangular area of ​​the sliding soil into zone 1 and the lower triangular area into zone 2.

[0011]

[0012] Where α is the inclination angle of the Coulomb sliding surface. l1 is the width of the soil between the pits, H is the total height of the wall on the right side of pit L, H = H1 + H2, where H1 is the height of the wall in zone 1 and H2 is the height of the wall in zone 2. δ is the internal friction angle, γ is the wall-soil friction angle, and γ is the soil weight.

[0013] Furthermore, step (2) is as follows: Take a rectangular micro-element for the soil in zone 1, and combine the stress σ between the wall and the soil on the micro-element. w1 and vertical stress Establish the equilibrium equations;

[0014] Considering the frictional effect of the wall on the finite-width soil, the stress state at each point in the soil behind the wall will be deflected; for the soil in zone 1 under active limit state, consider a infinitesimal element, denoted as A and B at the left and right ends of the infinitesimal element, respectively:

[0015]

[0016] Where, θ A1 θ B1 The angles between the directions of the major principal stresses at points A and B on the wall edge of Zone 1 and the horizontal plane are respectively.

[0017] To simplify the derivation process, the Y-axis of the coordinate system σoτ, where the stress Mohr circle of any element of the soil behind the wall is located, is shifted to the left. The distance between the two coordinate systems gives us a new coordinate system σ′o′τ′, and the following relationship exists between the two coordinate systems:

[0018]

[0019] Where σ and σ′ are the values ​​of the normal stress σoτ and σ′o′τ′ in the coordinate system, respectively;

[0020] Any element in region 1 exists:

[0021] σ′ v1 =σ′ 1-1 sin 2 θ+σ′ 3-1 cos2 θ (5);

[0022] Where, σ′ v1 σ′ 1-1 σ′ 3-1 These represent the vertical stress, major principal stress, and minor principal stress of the element in region 1 in the σ′o′τ′ coordinate system.

[0023] The average vertical stress on the infinitesimal element in region 1 is:

[0024]

[0025] k a The active earth pressure coefficient, Let be the value of the average vertical stress of the infinitesimal element in region 1 in the σ′o′τ′ coordinate system;

[0026] Width, k a The active earth pressure coefficient, Let be the value of the average vertical stress of the infinitesimal element in region 1 in the σ′o′τ′ coordinate system;

[0027] Integrating equation (6), we get:

[0028]

[0029] Based on the coordinate relationship between σoτ and σ′o′τ′, we obtain:

[0030]

[0031] Where, σ ahw1 σ′ ahw1 The values ​​of the horizontal active earth pressure at point A along the wall in Zone 1 are respectively represented in the coordinate systems σoτ and σ′o′τ′. The value of σoτ, representing the average vertical stress of the infinitesimal element in region 1, is given in the coordinate system.

[0032] σ′ ahw1 =σ′ 1-1A cos 2 θ A1 +σ′ 3-1A sin 2 θ A1 (10);

[0033] Where, σ′ 1-1A σ′ 3-1A These are the values ​​of the principal stresses at point A on the wall side of zone 1 in the σ′o′τ′ coordinate system.

[0034]

[0035] k awn1The ratio of the active lateral pressure at the wall edge to the average vertical stress of the micro-unit in zone 1;

[0036] Substituting equations (9) and (10) into equation (11), we get:

[0037]

[0038] Right now

[0039] Where k1′ is a parameter,

[0040] Force analysis of the infinitesimal element AB in region 1, and establishment of the vertical equilibrium equation:

[0041]

[0042] Further, step (3) is as follows: establish equilibrium equations for force analysis of the infinitesimal element in region 2;

[0043] Considering the frictional effect of the wall on the finite-width soil, the stress state at each point in the soil behind the wall will be deflected; for the soil in zone 2 under active limit state, a trapezoidal infinitesimal element is used for analysis, and the left and right end elements of the infinitesimal element are denoted as A and B, respectively:

[0044]

[0045] Where θ A2 θ B2 The angles between the directions of the major principal stresses at points A and B on the wall edge of Zone 2 and the horizontal plane are respectively.

[0046] Any element in the 2nd region infinitesimal exists:

[0047] σ′ v2 =σ′ 1-2 sin 2 θ+σ′ 3-2 cos 2 θ (17);

[0048] Where, σ′ v2 σ′ 1-2 σ′ 3-2 These represent the vertical stress, major principal stress, and minor principal stress of the element in the σ′o′τ′ coordinate system within the 2nd region micro-element.

[0049]

[0050] Where dA2 is the width of the 2-zone infinitesimal element, and l2 is the width of the 2-zone infinitesimal element. Let be the value of the average vertical stress of the infinitesimal element in region 2 in the σ′o′τ′ coordinate system;

[0051] Substituting equation (17) into equation (18) and integrating, we get:

[0052]

[0053] Based on the coordinate relationship between σoτ and σ′o′τ′:

[0054]

[0055] Where, σ ahw2 σ′ ahw2 The values ​​of the horizontal active earth pressure at point A along the wall in zone 2 are respectively represented in the coordinate systems σoτ and σ′o′τ′. Let be the value of the average vertical stress of the infinitesimal element in region 2 in the σoτ coordinate system;

[0056] σ′ ahw2 =σ′ 1-2A cos 2 θ A2 +σ′ 3-2A sin 2 θ A2 (twenty two);

[0057] Where, σ′ 1-2A σ′ 3-2A These are the values ​​of the principal stresses at point A on the wall edge of zone 2 in the σ′o′τ′ coordinate system;

[0058]

[0059] Where, k awn2 The ratio of the active lateral pressure at the wall edge to the average vertical stress of the micro-unit in zone 2;

[0060] Where, σ′ 1-2A σ′ 3-2A These are the values ​​of the principal stresses at point A along the wall in zone 2, in the σ′o′τ′ coordinate system.

[0061] Substituting equations (19) and (22) into equation (23), we get:

[0062]

[0063] Where q1 is the vertical stress at the bottom of region 1.

[0064] Right now

[0065] Where k2′ is a parameter,

[0066] Force analysis of the infinitesimal element in region 2, and establishment of equilibrium equations:

[0067]

[0068] Where M and N are parameters,

[0069]

[0070] Furthermore, step (4) is as follows: Solve for parameters such as the active earth pressure intensity, earth pressure magnitude, and bending moment on the wall:

[0071] Solve for zone 1:

[0072] Integrating equation (14), we get:

[0073]

[0074] Where A is a parameter,

[0075]

[0076] Where, p ah1 The active earth pressure intensity on the retaining wall of Zone 1;

[0077] Let p ah1 If the value is 0, substituting equations (13) and (27) into equation (28), we obtain the following solution:

[0078]

[0079] Among them, z 01 The depth of the cracks in the top soil of Zone 1;

[0080]

[0081] Substituting equations (13) and (27) into equation (30) and integrating, we get:

[0082]

[0083] Among them, E ah1 E represents the horizontal earth pressure value in zone 1. a1 M1 represents the earth pressure value in zone 1, and M2 represents the bending moment in zone 1.

[0084] Solve for zone 2

[0085] Integrating equation (26), we get:

[0086]

[0087] Where Q is a parameter,

[0088]

[0089] Where P ah2 The active earth pressure intensity on the retaining wall of Zone 2;

[0090] Due to the effect of q1, the depth of the crack in the top soil of zone 2 is z. 02 ≤0, therefore take z 02 =0;

[0091]

[0092] Substituting equations (25) and (35) into equation (36) and integrating, we get:

[0093]

[0094] Among them, E ah2 E represents the horizontal earth pressure value in zone 2. a2 M1 represents the earth pressure value in zone 2, and M2 represents the bending moment in zone 2.

[0095] M all =E ah1 ·(H2+h 01 )+E ah2 ·h 02 (40);

[0096] h0 = M all / (E ah1 +E ah2 (41);

[0097] Among them, M all h is the total bending moment, h0 is the location where the resultant earth pressure acts, and h 01 h is the point of application of the resultant earth pressure in zone 1, i.e., the distance from the bottom of zone 1. 02 This is the point of application of the resultant earth pressure in zone 2, i.e., the distance from the bottom of zone 2.

[0098] Compared with the prior art, the present invention has the following significant advantages: 1. The active earth pressure calculation method provided by the present invention addresses the shortcomings of the active earth pressure calculation method under finite width soil conditions. It considers the soil arching effect caused by the friction between the wall and the soil, and its calculation results are closer to the experimental results than the existing calculation methods; 2. The active earth pressure calculation method provided by the present invention takes into account the active limit state characteristics of the soil in different areas behind the wall, and proposes an active earth pressure calculation method under corresponding conditions. It provides a more reasonable basis for the setting of foundation pit support structures under finite width soil conditions and has significant engineering application value. Attached Figure Description

[0099] Figure 1 This is a schematic diagram of the present invention.

[0100] Figure 2 This is a diagram of the finite-width soil zoning model of the present invention.

[0101] Figure 3 This is a circle diagram of the molar stress of the soil behind the wall according to the present invention.

[0102] Figure 4 This is the analytical model of a small element in zone 1 under the active limit equilibrium state of the present invention.

[0103] Figure 5 This is the analytical model of a two-zone infinitesimal element under the active limit equilibrium state of the present invention.

[0104] Figure 6 This is a simplified calculation diagram for an embodiment of the present invention.

[0105] Figure 7 The experimental results of the embodiments of the present invention are compared with the theoretical solutions. Detailed Implementation

[0106] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0107] like Figure 1 As shown, this invention provides a method for calculating active earth pressure on finite-width soil considering the soil arching effect. Assuming a finite-width soil mass exists between L-shaped and R-shaped foundation pits distributed to the left and right, the method calculates the active earth pressure on the wall adjacent to the finite-width soil mass in the L-shaped foundation pit (i.e., the right wall of the L-shaped foundation pit), including the following steps:

[0108] Step 1: Determine whether equation (1) holds true. If it does, then the finite width soil condition is satisfied. The finite width soil sliding surface extends from the bottom of the right wall of pit L to the wall of pit R and vertically to the ground. Divide the upper rectangular area of ​​the sliding soil into zone 1 and the lower triangular area into zone 2. Figure 2 As shown.

[0109]

[0110] Where α is the inclination angle of the Coulomb sliding surface. l1 is the width of the soil between the pits, H is the height of the wall on the right side of pit L, H = H1 + H2, where H1 is the height of the wall in zone 1 and H2 is the height of the wall in zone 2. δ is the internal friction angle, γ is the wall-soil friction angle, and γ is the soil weight.

[0111] Step Two: Considering the frictional effect of the wall on the finite-width soil, the stress state at various points in the soil behind the wall will be deflected; for the soil in zone 1 under active limit state, consider a rectangular micro-element, such as... Figure 3 As shown, the element analysis is performed at points A and B:

[0112]

[0113]

[0114] Where, θ A1 θ B1 The angles between the direction of the major principal stress at points A and B on the wall edge of Zone 1 and the horizontal plane are respectively.

[0115] To simplify the derivation process, the Y-axis of the coordinate system σoτ, where the stress Mohr circle of any element of the soil behind the wall is located, is shifted to the left. The distance gives us a new coordinate system σ′o′τ′, as shown below. Figure 4 As shown, the following relationship exists between the two coordinate systems:

[0116]

[0117] Where σ and σ′ are the values ​​of the normal stress σoτ and σ′o′τ′ in the coordinate system, respectively;

[0118] Any element in region 1 exists:

[0119] σ′ v1 =σ′ 1-1 sin 2 θ+σ′ 3-1 cos 2 θ (5);

[0120] Where, σ′ v1 σ′ 1-1 σ′ 3-1 These represent the vertical stress, major principal stress, and minor principal stress of the element in region 1 in the σ′o′τ′ coordinate system.

[0121]

[0122] Where dA1 is the width of the micro-element unit in region 1, and k a The active earth pressure coefficient, Let be the value of the average vertical stress of the infinitesimal element in region 1 in the σ′o′τ′ coordinate system;

[0123] Substituting equations (5) and (6) into equation (8), we get:

[0124]

[0125] Integrating equation (9), we get:

[0126]

[0127] according to Figure 4 The relationship between the coordinate systems of σoτ and σ′o′τ′ is obtained as follows:

[0128]

[0129] Where, σ ahw1 σ′ ahw1 The values ​​of the horizontal active earth pressure at point A along the wall in Zone 1 are respectively represented in the coordinate systems σoτ and σ′o′τ′. The value of σoτ, representing the average vertical stress of the infinitesimal element in region 1, is given in the coordinate system.

[0130]

[0131] Substituting equations (11) and (12) into equation (13), we get:

[0132]

[0133] σ′ ahw1 =σ′ 1-1A cos 2 θ A1 +σ′ 3-1A sin 2 θ A1 (15);

[0134] Where, σ′ 1-1A σ′ 3-1A The values ​​of the principal stresses at point A along the wall in zone 1 in the σ′o′τ′ coordinate system are respectively.

[0135] Substituting equations (11) and (16) into equation (15), we get:

[0136]

[0137] Right now

[0138] Where k1′ is a parameter,

[0139] According to Figure 3 Force analysis of the intermediate element AB:

[0140]

[0141] Where, σ w1 Let σ be the total stress on section A. ahw1 For σ w1 Horizontal component, τ w1 For σ w1 Vertical component;

[0142] Establish the vertical equilibrium equations:

[0143]

[0144] Substituting equation (17) into equation (20), we get:

[0145]

[0146] Step 3: Considering the frictional effect of the wall on the finite-width soil, the stress state at various points in the soil behind the wall will be deflected; for the soil in zone 2 under active limit state, consider a trapezoidal micro-element, such as... Figure 5 As shown, let's analyze points A and B:

[0147]

[0148] Where θ A2 θ B2 The angles between the directions of the major principal stresses at points A and B on the wall edge of Zone 2 and the horizontal plane are respectively.

[0149] Any element in the 2nd region infinitesimal exists:

[0150] σ′ v2 =σ′ 1-2 sin 2 θ+σ′ 3-2 cos 2 θ (24);

[0151] Where, σ′ v2 σ′ 1-2 σ′ 3-2 These represent the horizontal stress, vertical stress, major principal stress, and minor principal stress of the element in the σ′o′τ′ coordinate system within the 2nd region micro-element.

[0152] dA2=Rdθsinθ(25);

[0153]

[0154] Where dA2 is the width of the 2-zone infinitesimal element, R is the radius of the 2-zone minor principal stress arch, and l2 is the width of the 2-zone infinitesimal element. Let be the value of the average vertical stress of the infinitesimal element in region 2 in the σ′o′τ′ coordinate system;

[0155] Substituting equations (24) and (25) into equation (27) and integrating, we get:

[0156]

[0157] according to Figure 4 The relationship between the coordinate systems of σoτ and σ′o′τ′ is obtained as follows:

[0158]

[0159] Where, σ ahw2 σ′ ahw2The values ​​of the horizontal active earth pressure at point A along the wall in zone 2 are respectively represented in the coordinate systems σoτ and σ′o′τ′. Let be the value of the average vertical stress of the infinitesimal element in region 2 in the σoτ coordinate system;

[0160]

[0161] Substituting equations (29) and (30) into equation (31), we get:

[0162]

[0163] σ′ ahw2 =σ′ 1-2A cos 2 θ A2 +σ′ 3-2A sin 2 θ A2 (33);

[0164] Where, σ′ 1-2A σ′ 3-2A These are the values ​​of the principal stresses at point A along the wall in zone 2, in the σ′o′τ′ coordinate system.

[0165] Substituting equations (28) and (33) into equation (32), we get:

[0166]

[0167] Where q1 is the vertical stress at the bottom of region 1.

[0168] Right now

[0169] Where k2′ is a parameter,

[0170] right Figure 5 Force analysis of the infinitesimal element in region 2, at point A:

[0171]

[0172] Where, σ w2 Let σ be the total stress on the cross section at point A. ahw2 For σ w2 Horizontal component, τ w2 For σ w2 Vertical component;

[0173] Point B:

[0174]

[0175] Where, σ F2 Let σ be the total stress on the section at point B.F2x For σ F2 Horizontal component, σ F2y For σ F2 Vertical component;

[0176] Establish the equilibrium equations in the horizontal and vertical directions:

[0177]

[0178] Right now

[0179] Where M and N are parameters,

[0180]

[0181] Step 4: Solve the equilibrium equation in Step 2. Integrate equation (21) to obtain:

[0182]

[0183] Where A is a parameter,

[0184]

[0185] Where, p ah1 The active earth pressure intensity on the retaining wall of Zone 1;

[0186] Let p ah1 If the value is 0, substituting equations (17) and (43) into equation (44) yields:

[0187]

[0188] Solving (45) yields:

[0189]

[0190] Among them, z 01 The depth of the cracks in the top soil of Zone 1;

[0191]

[0192] Substituting equations (17) and (43) into equation (47) and integrating, we get:

[0193]

[0194] Among them, E ah1 E represents the horizontal earth pressure value in zone 1. a1 M1 is the earth pressure value in zone 1, and M1 is the bending moment in zone 1; substituting equations (17) and (43) into equation (50) and integrating, we get:

[0195]

[0196] h 01 =M1 / E ah1 (52);

[0197] Among them, h 01 This is the point of application of the resultant earth pressure in zone 1, i.e., the distance from the bottom of zone 1;

[0198] Solving the equilibrium equation in step two, and integrating equation (42), we get:

[0199]

[0200] Where Q is a parameter,

[0201]

[0202] Where P ah2 The active earth pressure intensity on the retaining wall of Zone 2;

[0203] Due to the effect of q1, the depth of the crack in the top soil of zone 2 is z. 02 ≤0, therefore take z 02 =0;

[0204]

[0205] Substituting equations (35) and (53) into equation (55) and integrating, we get:

[0206]

[0207] Among them, E ah2 E represents the horizontal earth pressure value in zone 2. a2 M1 is the earth pressure value in zone 2, and M2 is the bending moment in zone 2; substituting equations (35) and (53) into equation (58) and integrating, we get:

[0208]

[0209] h 02 =M2 / E ah2 (60);

[0210] Among them, h 02 This is the point of application of the resultant earth pressure in zone 2, i.e., the distance from the bottom of zone 2;

[0211] M = E ah1 ·(H2+h 01 )+E ah2 ·h 02 (61);

[0212] h0=M / (Eah1 +E ah2 (62);

[0213] Where M is the total bending moment and h0 is the location where the resultant earth pressure acts.

[0214] Example

[0215] (1) Experimental overview and calculation parameters

[0216] Below, I will combine previous experimental results (TAKE) and theoretical solutions to verify the above calculation method.

[0217] In this model test, the wall height H = 5m, the width of the soil behind the wall l = 1.36m, the wall back is vertical, and the soil is dense sand with a unit weight γ = 15.87kN / m³. 3 Internal friction angle of soil The wall-soil friction angle δ = 23°, and the cohesion c = 0.

[0218] A simplified calculation diagram is attached. Figure 6 As shown.

[0219] Comparison of experimental results and theoretical solutions for the examples is attached. Figure 7 As shown, the active earth pressure distribution calculated by this method is slightly larger than the theoretical value, but overall it matches the experimental results better. Therefore, under finite width soil conditions, the active earth pressure calculated by this method can accurately reflect the actual stress condition of the wall.

[0220] [1]TAKE WA,VALSANGKARA J.Earth pressures on unyielding retainingwalls of narrow backfill width[J].Canadian Geotechnical Journal,2001,38(6):1220-1230.

Claims

1. A method for calculating active earth pressure on finite-width soil masses considering the soil arching effect, characterized in that, Assuming there is a finite-width soil mass between two foundation pits L and R, respectively, calculate the active earth pressure on the wall adjacent to the finite-width soil mass in foundation pit L, i.e., the right wall of foundation pit L; including the following steps: Step (1): Determine the location of the sliding surface of the soil with limited width between the foundation pits, and divide the sliding soil into an upper rectangular area (area 1) and a lower triangular area (area 2) according to the sliding surface. Step (2): Take a rectangular micro-element along the depth of the soil in Zone 1. Based on the assumption that the sliding soil is in an active limit equilibrium state, and combined with the stress σ between the wall and the soil on the micro-element... w1 and vertical stress Establish the equilibrium equation; Step (3): Apply the vertical stress at the bottom of zone 1 as a load q1 to the top of zone 2, perform force analysis on the trapezoidal micro-element along the depth of zone 2, and establish the equilibrium equation. Step (4): Combine the calculation results from steps (2) and (3) to solve for the active earth pressure intensity P on the entire wall. ah and size E a Total bending moment M parameter; Step (1) is as follows: For the finite-width soil between excavation pits L and R, determine whether equation (1) holds. If it does, the finite-width soil condition is satisfied. The finite-width soil sliding surface extends from the bottom of the right wall of excavation pit L to the wall of excavation pit R and vertically to the ground. Divide the upper rectangular area of ​​the sliding soil into zone 1 and the lower triangular area into zone 2. Where α is the inclination angle of the Coulomb sliding surface. l1 is the width of the soil between the pits, H is the height of the wall on the right side of pit L, H = H1 + H2, where H1 is the height of the wall in zone 1 and H2 is the height of the wall in zone 2. δ is the internal friction angle, γ is the wall-soil friction angle, and γ is the soil weight. Step (2) is as follows: Considering the frictional effect of the wall on the finite-width soil, the stress state of each point in the soil behind the right wall of the L-shaped foundation pit will be deflected; for the soil in zone 1 under active limit state, a rectangular infinitesimal element is taken, and the left and right end elements of the infinitesimal element are denoted as A and B, respectively: Where, θ A1 θ B1 These are the angles between the directions of the major principal stresses at points A and B of the micro-element in region 1 and the horizontal plane, respectively. To simplify the derivation process, the Y-axis of the coordinate system σoτ, where the two-dimensional stress Mohr circle of any element of the soil behind the wall is located, is shifted to the left. The distance between the two coordinate systems gives us a new coordinate system σ′o′τ′, and the following relationship exists between the two coordinate systems: Where σ and σ′ are the values ​​of the normal stress σoτ and σ′o′τ′ in the coordinate system, respectively; Any element in region 1 exists: in v1 =σ′ 1-1 sin 2 θ+σ′ 3-1 cos 2 θ (5); Where, σ′ v1 σ′ 1-1 σ′ 3-1 These represent the vertical stress, major principal stress, and minor principal stress of the element in region 1 in the σ′o′τ′ coordinate system. k a The active earth pressure coefficient, Let be the value of the average vertical stress of the infinitesimal element in region 1 in the σ′o′τ′ coordinate system; Integrating equation (6), we get: Based on the coordinate relationship between σoτ and σ′o′τ′, we obtain: Where, σ ahw1 σ′ ahw1 The values ​​of the horizontal active earth pressure at point A along the wall in zone 1 are respectively represented in the coordinate systems σoτ and σ′o′τ′. Let be the value of the average vertical stress of the infinitesimal element in region 1 in the σoτ coordinate system; in ahw1 =σ′ 1-1A cos 2 i A1 +s′ 3-1A sin 2 i A1 (10); Where, σ′ 1-1A σ′ 3-1A These are the values ​​of the principal stresses at point A on the wall edge of zone 1 in the σ′o′τ′ coordinate system; k awn1 The ratio of the active lateral pressure at the wall edge to the average vertical stress of the micro-unit in zone 1; Substituting equations (9) and (10) into equation (11), we get: Right now Where k1′ is a parameter, Force analysis of the infinitesimal element in region 1, establishing the vertical equilibrium equation:

2. The method for calculating active earth pressure on finite-width soil considering the soil arching effect according to claim 1, characterized in that, Step (3) is as follows: Considering the frictional effect of the wall on the finite-width soil, the stress state of each point in the soil behind the right wall of the L-shaped foundation pit will be deflected; for the soil in zone 2 under active limit state, a trapezoidal infinitesimal element is taken, and the left and right end elements of the infinitesimal element are denoted as A and B, respectively: Where θ A2 θ B2 The angles between the direction of the principal stress at points A and B along the wall in Zone 2 and the horizontal plane are respectively. Any element in the 2nd region infinitesimal exists: in v2 =σ′ 1-2 sin 2 θ+σ′ 3-2 cos 2 θ(17); Where, σ′ ah2 σ′ v2 σ′ 1-2 σ′ 3-2 These represent the horizontal stress, vertical stress, major principal stress, and minor principal stress of any element in the σ′o′τ′ coordinate system within the 2nd region micro-element; Where dA2 is the width of any element in the 2-zone infinitesimal, and l2 is the width of the 2-zone infinitesimal. Let be the value of the average vertical stress of the infinitesimal element in region 2 in the σ′o′τ′ coordinate system; Substituting equation (17) into equation (18) and integrating, we get: Based on the coordinate relationship between σoτ and σ′o′τ′: Where, σ ahw2 σ′ ahw2 The values ​​of the horizontal active earth pressure at point A along the wall in zone 2 are respectively represented in the coordinate systems σoτ and σ′o′τ′. Let be the value of the average vertical stress of the infinitesimal element in region 2 in the σoτ coordinate system; in ahw2 =σ′ 1-2A cos 2 i A2 +s′ 3-2A sin 2 i A2 (22); Where, σ′ 1-2A σ′ 3-2A These are the values ​​of the principal stresses at point A on the wall edge of zone 2 in the σ′o′τ′ coordinate system; Where, k awn2 The ratio of the active lateral pressure at the wall edge to the average vertical stress of the micro-unit in zone 2; Substituting equations (19) and (22) into equation (23), we get: Where q1 is the vertical stress at the bottom of region 1. Right now Where k2′ is a parameter, Force analysis of the infinitesimal element in region 2, and establishment of equilibrium equations: Right now Where M and N are parameters, 3. The method for calculating active earth pressure on finite-width soil considering the soil arching effect according to claim 2, characterized in that, Step (4) is as follows: Solve for zone 1: Integrating equation (14), we get: Where A is a parameter, Where, p ah1 The active earth pressure intensity on the retaining wall of Zone 1; Let p ah1 If the value is 0, substituting equations (13) and (27) into equation (28), we obtain the following solution: Among them, z 01 The depth of the cracks in the top soil of Zone 1; Substituting equations (13) and (27) into equation (30) and integrating, we get: Among them, E ah1 E represents the horizontal earth pressure value in zone 1. a1 This represents the earth pressure value for zone 1. Solve for zone 2 Integrating equation (26), we get: Where Q is a parameter, Where P ah2 The active earth pressure intensity on the retaining wall of Zone 2; Due to the effect of q1, the depth of the crack in the top soil of zone 2 is z. 02 ≤0, take z 02 =0; Substituting equations (25) and (34) into equation (35) and integrating, we get: Among them, E ah2 E represents the horizontal earth pressure value in zone 2. a2 The earth pressure value for zone 2; M all =E ah1 ·(H2+h 01 )+E ah2 h 02 (38); h0=M all / (AND ah1 +E ah2 )(39); Among them, M all h is the total bending moment, h0 is the location where the resultant earth pressure acts, and h 01 h is the point of application of the resultant earth pressure in zone 1, i.e., the distance from the bottom of zone 1. 02 This is the point of application of the resultant earth pressure in zone 2, i.e., the distance from the bottom of zone 2.

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