Active soil pressure calculation method for limited-width soil body considering soil arching effect

Abstract

The invention discloses a limited-width soil body active soil pressure calculation method considering a soil arching effect, it is assumed that a limited-width soil body exists between an L foundation pit and an R foundation pit which are distributed left and right, and calculation of the active soil pressure borne by a wall body, close to the limited-width soil body, in the L foundation pit comprises the following steps: determining the position of a sliding surface of the limited-width soil body between the foundation pits; the sliding soil body is divided into an upper rectangular area 1 and a lower triangular area 2 according to the sliding surface; taking a rectangular infinitesimal body from the soil body in the first area along the depth, and establishing an equilibrium equation by combining the stress between the wall body and the soil body and the vertical stress on the infinitesimal body according to an assumed condition when the sliding soil body is in an active limit equilibrium state; applying the vertical stress at the bottom of the area 1 as a load q1 to the top of the area 2, analyzing the stress of the trapezoidal infinitesimal body of the area 2 along the depth, and establishing an equilibrium equation; relevant parameters on the whole wall body are solved, and a more reasonable basis is provided for design of a foundation pit supporting structure under the condition of a limited width soil body.

Description

Technical Field

[0001] The present invention belongs to the technical field of foundation pit engineering in the construction industry, and relates to a method for calculating active earth pressure of a soil body of limited width taking into account soil arch effect, and in particular to a method for calculating active earth pressure when a soil body of limited width exists in a foundation pit group engineering. Background Art

[0002] In foundation pit engineering, the size of active earth pressure behind the wall is crucial to the design of 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 overly conservative design of the foundation pit support structure and increase engineering costs.

[0003] In order to reasonably calculate the active earth pressure, some researchers found that the calculation results after considering the soil arch 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 use of 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 limited width soil conditions. Summary of the invention

[0004] The purpose of the present invention is to provide a method for calculating active earth pressure of soil with limited width taking into account soil arch effect, so as to solve the problems existing in the background technology.

[0005] The technical solution adopted by the present invention is a method for calculating the active earth pressure of a finite width soil body considering the soil arch effect. Assuming that there is a finite width soil body between the L foundation pit and the R foundation pit distributed on the left and right, the active earth pressure on the wall of the L foundation pit (i.e., the right wall of the L foundation pit) adjacent to the finite width soil body is calculated, including the following steps:

[0006] (1) Determine the position 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;

[0007] (2) A rectangular microelement is taken along the depth of the soil in zone 1. Based on the assumption that the sliding soil is in an active limit equilibrium state, the stress σ between the wall and the soil is taken into consideration. w1 and vertical stress etc. to establish a balanced equation;

[0008] (3) The vertical stress at the bottom of zone 1 is applied as load q1 to the top of zone 2, and the stress of the trapezoidal micro-element along the depth of zone 2 is analyzed and the equilibrium equation is established;

[0009] (4) Combine the calculation results of steps (2) and (3) to solve the active earth pressure strength P on the entire wall ah and size E a , bending moment M and other parameters.

[0010] Furthermore, the active earth pressure is calculated, and step (1) is specifically as follows: for the finite width soil between the L foundation pit and the R foundation pit, determine whether equation (1) is established. If established, the finite width soil condition is satisfied, and the sliding surface of the finite width soil 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. The upper rectangular area of ​​the sliding soil is divided into zone 1, and the lower triangular area is divided 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 the L foundation pit, H = H1 + H2, H1 is the height of the wall in zone 1, H2 is the height of the wall in zone 2, is the internal friction angle, δ is the wall-soil friction angle, and γ is the soil weight;

[0013] Further, step (2) is as follows: Take a rectangular microelement of the soil in zone 1, and combine the stress σ between the wall and the soil on the microelement w1 and vertical stress etc. to establish a balanced equation;

[0014] Considering that the soil of limited width is subjected to the friction of the wall, the stress state of each point of the soil behind the wall will be deflected; for the soil in zone 1 in the active limit state, a microelement is taken, and the left and right end units of the microelement are recorded as A and B respectively:

[0015]

[0016] Among them, θ A1 ,θ B1 They are the angles between the direction of the major principal stress at point A and point B on the wall of zone 1 and the horizontal plane;

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

[0018]

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

[0020] Any unit on the microelement in zone 1 exists:

[0021] σ v ′1=σ1′ -1 sin 2 θ+σ3′ -1 cos2 θ(5);

[0022] Among them, σ v ′1、σ1′ -1 ,σ3′ -1 They are the vertical stress, major principal stress and minor principal stress of the unit in the microelement of zone 1 in the σ′o′τ′ coordinate system;

[0023] Then the vertical average stress on the microelement in zone 1 is:

[0024]

[0025] k a is the active earth pressure coefficient, is the value of the average vertical stress of the microelement in zone 1 in the σ′o′τ′ coordinate system;

[0026] Width, k a is the active earth pressure coefficient, is the value of the average vertical stress of the microelement in zone 1 in the σ′o′τ′ coordinate system;

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

[0028]

[0029] According to the relationship between σoτ and σ′o′τ′ coordinate system, we get:

[0030]

[0031] Among them, σ ahw1 , σ a ' hw1 are the values ​​of the horizontal active earth pressure at point A on the wall of zone 1 in the σoτ and σ′o′τ′ coordinate systems, respectively. is the value of the average vertical stress σoτ of the microelement in zone 1 in the coordinate system;

[0032] σ a ' hw1 =σ1′ -1A cos 2 θ A1 +σ3′ -1A sin 2 θ A1 (10);

[0033] Among them, σ1′ -1A ,σ3′ -1A They are respectively the values ​​of the major and minor principal stresses at point A on the wall of zone 1 in the σ′o′τ′ coordinate system.

[0034]

[0035] k awn1 is the ratio of the active lateral pressure on the wall of the micro-unit in zone 1 to the average vertical stress;

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

[0037]

[0038] Right now

[0039] Among them, k1′ is a parameter,

[0040] The force analysis of microelement AB in zone 1 is carried out and the vertical equilibrium equation is established:

[0041]

[0042] Furthermore, step (3) is specifically as follows: establish the equilibrium equation for the force analysis of the micro-element in zone 2;

[0043] Considering the friction effect of the wall on the soil with limited width, the stress state of each point of the soil behind the wall will be deflected; the trapezoidal microelement analysis is taken for the soil in zone 2 in the active limit state, and the left and right end units of the microelement are recorded as A and B respectively:

[0044]

[0045] where θ A2 ,θ B2 They are the angles between the direction of the major principal stress at point A and point B on the wall of zone 2 and the horizontal plane;

[0046] Any unit on the microelement in zone 2 exists:

[0047] σ v ′2=σ1′ -2 sin 2 θ+σ3′ -2 cos 2 θ(17);

[0048] Among them, σ v ′2、σ1′ -2 ,σ3′ -2 They are respectively the vertical stress, major principal stress and minor principal stress of the unit in the microelement of zone 2 in the σ′o′τ′ coordinate system;

[0049]

[0050] Among them, dA2 is the width of the microelement unit in zone 2, l2 is the width of the microelement in zone 2, is the value of the average vertical stress of the microelement in zone 2 in the σ′o′τ′ coordinate system;

[0051] Substituting formula (17) into formula (18), the integral is:

[0052]

[0053] According to the relationship between σoτ and σ′o′τ′ coordinate system, we can get:

[0054]

[0055] Among them, σ ahw2 , σ a ' hw2 are the values ​​of the horizontal active earth pressure at point A on the wall of zone 2 in the σoτ and σ′o′τ′ coordinate systems, respectively. is the value of the average vertical stress of the microelement in zone 2 in the σoτ coordinate system;

[0056] σ a ' hw2 =σ1′ -2A cos 2 θ A2 +σ3′ -2A sin 2 θ A2 (twenty two);

[0057] Among them, σ1′ -2A ,σ3′ -2A are the values ​​of the major and minor principal stresses at A on the wall of zone 2 in the σ′o′τ′ coordinate system;

[0058]

[0059] Among them, k awn2 is the ratio of the active lateral pressure on the wall of the micro-unit in zone 2 to the average vertical stress;

[0060] Among them, σ1′ -2A ,σ3′ -2A are the values ​​of the major and minor principal stresses at point A on the wall of zone 2 in the σ′o′τ′ coordinate system;

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

[0062]

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

[0064] Right now

[0065] Among them, k2′ is a parameter,

[0066] Analyze the forces on the microelement in zone 2 and establish the equilibrium equation:

[0067]

[0068] Among them, M and N are parameters,

[0069]

[0070] Furthermore, step (4) is as follows: Solve the parameters such as active earth pressure strength, earth pressure size, bending moment, etc. on the wall:

[0071] Solve for zone 1:

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

[0073]

[0074] Among them, A is a parameter,

[0075]

[0076] Among them, p ah1 is the active earth pressure strength on the retaining wall in zone 1;

[0077] Let p ah1 is 0, substitute equation (13) and equation (27) into equation (28) and solve to obtain:

[0078]

[0079] Among them, z 01 is the crack depth of the top soil in zone 1;

[0080]

[0081] Substituting equation (13) and equation (27) into equation (30), the integral yields:

[0082]

[0083] Among them, E ah1 is the horizontal earth pressure value in zone 1, E a1 is the earth pressure value in zone 1, M1 is the bending moment in zone 1; solve for zone 2

[0084] Integrating equation (26), we get:

[0085]

[0086] Among them, Q is a parameter,

[0087]

[0088] Where P ah2 is the active earth pressure strength on the retaining wall in zone 2;

[0089] Due to the action of q1, the crack depth z of the top soil in zone 2 02 ≤0, so take z 02 =0;

[0090]

[0091] Substituting equation (25) and equation (35) into equation (36), the integral yields:

[0092]

[0093] Among them, E ah2 is the horizontal earth pressure value in zone 2, E a2 is the earth pressure value in zone 2, M2 is the bending moment in zone 2;

[0094] M=E ah1 ·(H2+h 01 )+E ah2 ·h 02 (40);

[0095] h0=M / (E ah1 +E ah2 )(41);

[0096] Where M is the total bending moment, h0 is the position where the resultant earth pressure acts, and h 01 is the point of action of the resultant earth pressure in zone 1, i.e., the distance from the bottom of zone 1, h 02 It is the point of action of the resultant earth pressure in zone 2, that is, the distance from the bottom of zone 2.

[0097] 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 limited width soil conditions, and considers the soil arch effect generated by the soil behind the wall caused by the friction between the wall and the soil. Compared with the existing calculation method, its calculation result is closer to the test result; 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, which provides a more reasonable basis for the setting of foundation pit support structure under limited width soil conditions, and has significant engineering application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0098] Figure 1 It is a schematic diagram of the present invention.

[0099] Figure 2This is a limited width soil partition model diagram of the present invention.

[0100] Figure 3 It is the Mohr stress circle diagram of the soil behind the wall of the present invention.

[0101] Figure 4 This is the microelement analysis model of zone 1 under the active limit equilibrium state of the present invention.

[0102] Figure 5 This is the two-zone microelement analysis model under the active limit equilibrium state of the present invention.

[0103] Figure 6 The following is a simplified calculation diagram of an embodiment of the present invention.

[0104] Figure 7 The experimental results of the embodiments of the present invention are compared with the theoretical solution. DETAILED DESCRIPTION

[0105] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

[0106] like Figure 1 As shown, the present invention provides a method for calculating the active earth pressure of a soil body with limited width considering the soil arch effect. Assuming that there is a soil body with limited width between the L foundation pit and the R foundation pit distributed on the left and right, the active earth pressure on the wall of the L foundation pit adjacent to the soil body with limited width (i.e., the right wall of the L foundation pit) is calculated, comprising the following steps:

[0107] Step 1: Determine whether equation (1) is valid. If so, the finite width soil condition is satisfied. The sliding surface of the finite width soil 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. The upper rectangular area of ​​the sliding soil is divided into zone 1, and the lower triangular area is zone 2. Figure 2 shown.

[0108]

[0109] 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 the L foundation pit, H=H1+H2, H1 is the height of the wall in zone 1, H2 is the height of the wall in zone 2, is the internal friction angle, δ is the wall-soil friction angle, and γ is the soil weight;

[0110] Step 2: Considering the friction effect of the wall on the soil with limited width, the stress state of each point of the soil behind the wall will be deflected; take a rectangular microelement for the soil in zone 1 in the active limit state, such as Figure 3 As shown, the analysis of units at A and B is as follows:

[0111]

[0112]

[0113] Among them, θ A1 ,θ B1 They are the angles between the direction of the major principal stress at points A and B on the wall of zone 1 and the horizontal plane;

[0114] To simplify the derivation process, the Y-axis of the coordinate system σoτ where the stress Mohr circle of any unit of the soil behind the wall is located is translated to the left. The distance is used to obtain a new coordinate system σ′o′τ′, such as Figure 4 As shown, there is the following relationship between the two coordinate systems:

[0115]

[0116] Among them, σ and σ′ are the values ​​of normal stress σoτ and σ′o′τ′ in the coordinate system respectively;

[0117] Any unit on the microelement in zone 1 exists:

[0118] σ v ′1=σ1′ -1 sin 2 θ+σ3′ -1 cos 2 θ(5);

[0119] Among them, σ v ′1、σ1′ -1 ,σ3′ -1 They are the vertical stress, major principal stress and minor principal stress of the unit in the microelement of zone 1 in the σ′o′τ′ coordinate system;

[0120]

[0121] Among them, dA1 is the width of the microelement unit in zone 1, k a is the active earth pressure coefficient, is the value of the average vertical stress of the microelement in zone 1 in the σ′o′τ′ coordinate system;

[0122] Substituting equation (5) and equation (6) into equation (8), we get:

[0123]

[0124] Integrating formula (9), we get:

[0125]

[0126] according to Figure 4 The relationship between σoτ and σ′o′τ′ coordinate system is:

[0127]

[0128] Among them, σ ahw1 , σ a ' hw1 are the values ​​of the horizontal active earth pressure at point A on the wall of zone 1 in the σoτ and σ′o′τ′ coordinate systems, respectively. is the value of the average vertical stress σoτ of the microelement in zone 1 in the coordinate system;

[0129]

[0130] Substituting equation (11) and equation (12) into equation (13), we obtain:

[0131]

[0132] σ a ' hw1 =σ1′ -1A cos 2 θ A1 +σ3′ -1A sin 2 θ A1 (15);

[0133] Among them, σ1′ -1A ,σ3′ -1A They are the values ​​of the major and minor principal stresses at point A on the wall of zone 1 in the σ′o′τ′ coordinate system.

[0134] Substituting equation (11) and equation (16) into equation (15), we obtain:

[0135]

[0136] Right now

[0137] Among them, k1′ is a parameter,

[0138] According to Figure 3 Force analysis of microelement AB:

[0139]

[0140] Among them, σ w1 is the total stress on the cross section at point A, σ ahw1 is w1 Horizontal component, τ w1 is w1 vertical component;

[0141] Establish the vertical equilibrium equation:

[0142]

[0143] Substituting formula (17) into formula (20), we get:

[0144]

[0145] Step 3: Considering the friction effect of the wall on the soil with limited width, the stress state of each point of the soil behind the wall will be deflected; take a trapezoidal microelement for the soil in zone 2 in the active limit state, such as Figure 5 As shown, point A and point B are analyzed:

[0146]

[0147] where θ A2 ,θ B2 They are the angles between the direction of the major principal stress at point A and point B on the wall of zone 2 and the horizontal plane;

[0148] Any unit on the microelement in zone 2 exists:

[0149] σ v ′2=σ1′ -2 sin 2 θ+σ3′ -2 cos 2 θ(24);

[0150] Among them, σ v ′2、σ1′ -2 ,σ3′ -2 They are the horizontal stress, vertical stress, major principal stress and minor principal stress of the unit in the microelement of zone 2 in the σ′o′τ′ coordinate system;

[0151] dA2 = Rdθsinθ(25);

[0152]

[0153] Among them, dA2 is the width of the microelement unit in zone 2, R is the radius of the small principal stress arch in zone 2, l2 is the width of the microelement in zone 2, is the value of the average vertical stress of the microelement in zone 2 in the σ′o′τ′ coordinate system;

[0154] Substituting equation (24) and equation (25) into equation (27), the integral yields:

[0155]

[0156] according to Figure 4 The relationship between σoτ and σ′o′τ′ coordinate system is obtained:

[0157]

[0158]

[0159] Among them, σ ahw2 , σ a ' hw2 are the values ​​of the horizontal active earth pressure at point A on the wall of zone 2 in the σoτ and σ′o′τ′ coordinate systems, respectively. is the value of the average vertical stress of the microelement in zone 2 in the σoτ coordinate system;

[0160]

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

[0162]

[0163] σ a ' hw2 =σ1′ -2A cos 2 θ A2 +σ3′ -2A sin 2 θ A2 (33);

[0164] Among them, σ1′ -2A ,σ3′ -2A are the values ​​of the major and minor principal stresses at point A on the wall of zone 2 in the σ′o′τ′ coordinate system;

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

[0166]

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

[0168] Right now

[0169] Among them, k2′ is a parameter,

[0170] right Figure 5 Force analysis of microelement in middle zone 2, point A:

[0171]

[0172] Among them, σ w2 is the total stress on the section at A, σ ahw2 is w2 Horizontal component, τ w2 is w2 vertical component;

[0173] B:

[0174]

[0175] Among them, σ F2 is the total stress on the section at B, σ F2x is F2 Horizontal component, σ F2y is F2 vertical component;

[0176] Establish the horizontal and vertical equilibrium equations:

[0177]

[0178] Right now

[0179] Among them, M and N are parameters,

[0180]

[0181] Step 4: Solve the equilibrium equation in step 2 and integrate equation (21) to obtain:

[0182]

[0183] Among them, A is a parameter,

[0184]

[0185] Among them, p ah1 is the active earth pressure strength on the retaining wall in zone 1;

[0186] Let p ah1 is 0, and we substitute equation (17) and equation (43) into equation (44) to obtain:

[0187]

[0188] Solving (45) yields:

[0189]

[0190] Among them, z 01 is the crack depth of the top soil in zone 1;

[0191]

[0192] Substituting equation (17) and equation (43) into equation (47), the integral yields:

[0193]

[0194]

[0195] Among them, E ah1 is the horizontal earth pressure value in zone 1, E a1 is the earth pressure value of zone 1, M1 is the bending moment of zone 1; Substitute equation (17) and equation (43) into equation (50) and integrate to obtain:

[0196]

[0197] h 01 =M1 / E ah1 (52);

[0198] Among them, h 01 is the point of action of the resultant earth pressure in zone 1, i.e., the distance from the bottom of zone 1;

[0199] Solve the equilibrium equation in step 2 and integrate equation (42) to obtain:

[0200]

[0201] Among them, Q is a parameter,

[0202]

[0203] Where P ah2 is the active earth pressure strength on the retaining wall in zone 2;

[0204] Due to the action of q1, the crack depth z of the top soil in zone 2 02 ≤0, so take z 02 =0;

[0205]

[0206] Substituting equation (35) and equation (53) into equation (55), the integral yields:

[0207]

[0208] Among them, E ah2 is the horizontal earth pressure value in zone 2, E a2 is the earth pressure value of zone 2, M2 is the bending moment of zone 2; Substitute equation (35) and equation (53) into equation (58) and integrate to obtain:

[0209]

[0210] h 02 =M2 / E ah2 (60);

[0211] Among them, h 02is the point of action of the resultant earth pressure in zone 2, i.e., the distance from the bottom of zone 2;

[0212] M=E ah1 ·(H2+h 01 )+E ah2 ·h 02 (61);

[0213] h0=M / (E ah1 +E ah2 )(62);

[0214] Where M is the total bending moment and h0 is the position where the resultant earth pressure acts.

[0215] Example

[0216] (1) Test overview and calculation parameters

[0217] Below I will verify the above calculation method by combining the previous experimental results (TAKE) and theoretical solutions (Liu Xinxi).

[0218] In this model test, the wall height H = 5m, the soil width behind the wall l = 1.36m, the wall back is vertical, the soil is dense sand, and the gravity γ = 15.87kN / m 3 , soil internal friction angle The wall-soil friction angle δ=23°, and the cohesion c=0.

[0219] The calculation diagram is as attached Figure 6 shown.

[0220] The experimental results of the embodiment are compared with the theoretical solution. Figure 7 As shown in the figure, it can be seen that the active earth pressure distribution calculated by this method is slightly larger than the theoretical value calculated by Liu Xinxi et al., but it is more consistent with the test results as a whole. Therefore, under the condition of limited width soil, the active earth pressure calculated by this method can more accurately reflect the actual stress condition of the wall.

[0221] [1]TAKE WA,VALSANGKARA J.Earth pressures on unyielding retainingwalls of narrow backfill width[J].Canadian Geotechnical Journal,2001,38(6):1220-1230.

[0222] [2] Liu Xinxi, Li Bin, Wang Weiwei, et al. Calculation of earth pressure in finite soil based on principal stress trace stratification[J]. Rock and Soil Mechanics, 2022, 43(05): 1175-1186.

Claims

1. A method for calculating active earth pressure of soil with limited width considering soil arching effect, characterized in that: Assuming that there is a soil body of finite width between the L foundation pit and the R foundation pit distributed on the left and right, the active earth pressure on the wall of the L foundation pit adjacent to the soil body of finite width, i.e., the right wall of the L foundation pit, is calculated; the following steps are included: Step (1), determining the position of the sliding surface of the soil with limited width between the foundation pits, and dividing the sliding soil into an upper rectangular area, i.e., area 1, and a lower triangular area, i.e., area 2, according to the sliding surface; Step (2): Take a rectangular microelement along the depth of the soil in zone 1, and according to the assumption that the sliding soil is in an active limit equilibrium state, combine the stress σ between the wall and the soil on the microelement w1 and vertical stress Establish balanced equations; Step (3), applying the vertical stress at the bottom of zone 1 as load q1 to the top of zone 2, analyzing the stress of the trapezoidal micro-element along the depth of zone 2 and establishing the equilibrium equation; Step (4): Combine the calculation results of step (2) and step (3) to solve the active earth pressure strength P on the entire wall. ah and size E a , bending moment M parameters.

2. The method for calculating active earth pressure of a soil with limited width considering soil arching effect according to claim 1 is characterized in that: Step (1) is as follows: For the finite width soil between the L foundation pit and the R foundation pit, determine whether formula (1) is valid. If it is valid, the finite width soil condition is satisfied. The sliding surface of the finite width soil 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. The upper rectangular area of ​​the sliding soil is divided into zone 1, and the lower triangular area is 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 the L foundation pit, H=H1+H2, H1 is the height of the wall in zone 1, H2 is the height of the wall in zone 2, is the internal friction angle, δ is the wall-soil friction angle, and γ is the soil weight.

3. The method for calculating active earth pressure of a soil with limited width considering soil arching effect according to claim 2 is characterized in that: Step (2) is as follows: Considering the friction effect of the wall on the soil with limited width, the stress state of each point of the soil behind the right wall of the L foundation pit will be deflected; a rectangular microelement is taken for the soil in zone 1 in the active limit state, and the left and right end units of the microelement are recorded as A and B respectively: Among them, θ A1 ,θ B1 They are the angles between the major principal stress direction and the horizontal plane at points A and B of the microelement in zone 1; To simplify the derivation process, the Y-axis of the coordinate system σoτ where the two-dimensional stress Mohr circle of any unit behind the wall is located is translated to the left. The distance is used to obtain a new coordinate system σ′o′τ′. The following relationship exists between the two coordinate systems: Among them, σ and σ′ are the values ​​of normal stress σoτ and σ′o′τ′ in the coordinate system respectively; Any unit on the microelement in zone 1 exists: s v ′1=σ1′ -1 sin 2 θ+σ3′ -1 cos 2 θ(5); Among them, σ v ′1、σ1′ -1 ,σ3′ -1 They are respectively the vertical stress, major principal stress and minor principal stress of the unit in the microelement of zone 1 in the σ′o′τ′ coordinate system; k a is the active earth pressure coefficient, is the value of the average vertical stress of the microelement in zone 1 in the σ′o′τ′ coordinate system; Integrating formula (6), we get: According to the relationship between σoτ and σ′o′τ′ coordinate system, we get: Among them, σ ahw1 , σ a ' hw1 are the values ​​of the horizontal active earth pressure at point A on the wall of zone 1 in the σoτ and σ′o′τ′ coordinate systems, respectively. is the value of the average vertical stress of the microelement in zone 1 in the σoτ coordinate system; s a ′ hw1 =σ1′ -1A cos 2 i A1 +σ3′ -1A sin 2 i A1 (10); Among them, σ1′ -1A ,σ3′ -1A are the values ​​of the major and minor principal stresses at A on the wall of zone 1 in the σ′o′τ′ coordinate system; k awn1 is the ratio of the active lateral pressure on the wall of the micro-unit in zone 1 to the average vertical stress; Substituting equation (9) and equation (10) into equation (11), we obtain: Right now Among them, k1′ is a parameter, The force analysis of the microelement in zone 1 is conducted and the vertical equilibrium equation is established:

4. The method for calculating active earth pressure of a soil with limited width considering soil arching effect according to claim 3 is characterized in that: Step (3) is as follows: Considering the friction effect of the wall on the soil with limited width, the stress state of each point of the soil behind the right wall of the L foundation pit will be deflected; a trapezoidal microelement is taken for the soil in zone 2 in the active limit state, and the left and right end units of the microelement are recorded as A and B respectively: where θ A2 ,θ B2 They are the angles between the direction of the largest principal stress at points A and B on the wall of zone 2 and the horizontal plane; Any unit on the microelement in zone 2 exists: s v ′2=σ1′ -2 sin 2 θ+σ3′ -2 cos 2 θ(17); Among them, σ a ' h2 , σ v ′2、σ1′ -2 ,σ3′ -2 They are the horizontal stress, vertical stress, major principal stress and minor principal stress of any unit in the microelement of zone 2 in the σ′o′τ′ coordinate system; Among them, dA2 is the width of any unit in the 2-zone microelement, l2 is the width of the 2-zone microelement, is the value of the average vertical stress of the microelement in zone 2 in the σ′o′τ′ coordinate system; Substituting formula (17) into formula (18), the integral is: According to the relationship between σoτ and σ′o′τ′ coordinate system, we can get: Among them, σ ahw2 , σ a ' hw2 are the values ​​of the horizontal active earth pressure at point A on the wall of zone 2 in the σoτ and σ′o′τ′ coordinate systems, respectively. is the value of the average vertical stress of the microelement in zone 2 in the σoτ coordinate system; s a ′ hw2 =σ1′ -2A cos 2 i A2 +σ3′ -2A sin 2 i A2 (22); Among them, σ1′ -2A ,σ3′ -2A are the values ​​of the major and minor principal stresses at A on the wall of zone 2 in the σ′o′τ′ coordinate system; Among them, k awn2 is the ratio of the active lateral pressure on the wall of the micro-unit in zone 2 to the average vertical stress; Substituting equation (19) and equation (22) into equation (23), we obtain: Where q1 is the vertical stress at the bottom of zone 1 Right now Among them, k2′ is a parameter, Analyze the forces on the microelement in zone 2 and build the equilibrium equation: Right now Among them, M and N are parameters, 5. The method for calculating active earth pressure of soil with limited width considering soil arching effect according to claim 4 is characterized in that: Step (4) is as follows: Solve for zone 1: Integrating equation (14), we get: Among them, A is a parameter, Among them, p ah1 is the active earth pressure strength on the retaining wall in zone 1; Let p ah1 is 0, substitute equation (13) and equation (27) into equation (28) and solve to obtain: Among them, z 01 is the crack depth of the top soil in zone 1; Substituting equation (13) and equation (27) into equation (30), the integral yields: Among them, E ah1 is the horizontal earth pressure value in zone 1, E a1 is the earth pressure value in zone 1; Solve Area 2 Integrating equation (26), we get: Among them, Q is a parameter, Where P ah2 is the active earth pressure strength on the retaining wall in zone 2; Due to the action of q1, the crack depth z of the top soil in zone 2 02 ≤0, take z 02 =0; Substituting equation (25) and equation (34) into equation (35), the integral is: Among them, E ah2 is the horizontal earth pressure value in zone 2, E a2 is the earth pressure value in zone 2; M=E ah1 ·(H2+h 01 )+E ah2 h 02 (38); h0=M / (E ah1 +E ah2 )(39); Where M is the total bending moment, h0 is the position where the resultant earth pressure acts, and h 01 is the point of action of the resultant earth pressure in zone 1, i.e., the distance from the bottom of zone 1, h 02 It is the point of action of the resultant earth pressure in zone 2, that is, the distance from the bottom of zone 2.

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