A method for calculating active earth pressure of wrapped end of reinforced soil embankment

Abstract

The present application relates to the technical field of reinforced soil foundation engineering, and particularly relates to a calculation method of active earth pressure of a reverse-packing end of a reinforced soil subgrade, comprising the following steps: establishing a calculation analysis model of the reinforced soil subgrade with the reverse-packing end under an active limit state, and obtaining a sliding soil wedge; dividing the sliding soil wedge into n strips along a horizontal plane in a horizontal strip manner, wherein the middle part of each strip contains a layer of reinforcing bars; establishing a balance equation set of each strip, and calculating to obtain active earth pressure P of the reverse-packing end of each strip i and a relationship function P i (β) of the inclination angle β of the sliding fracture surface, wherein i={1, 2, 3…, n}, wherein n is equal to the number of strips; adding all P i (β) to obtain P s (β), and obtaining the inclination angle value β m of the sliding fracture surface under the active limit state; and bringing β m into the balance equation set of each strip to obtain corresponding P i .

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of reinforced soil foundation engineering, and particularly to a calculation method of active earth pressure of a wrapped end of a wrapped end reinforced soil subgrade. BACKGROUND

[0002] In order to improve the properties of soil filling engineering and the overall stability of the retaining structure, many soil reinforcement technologies have emerged, one of which is geosynthetic reinforcement technology. Geosynthetic reinforced retaining walls can withstand lateral pressure and enhance structural stability. Reinforced soil subgrade is an engineering structure that adds reinforcement to the soil to improve its tensile and shear strength. In particular, wrapped reinforced soil subgrade is wrapped in reverse at the end of the reinforcement on the subgrade slope surface, forming a local wrapped end, resisting lateral soil slip, and better improving the bearing capacity of the reinforced soil subgrade. Compared with traditional pure filling subgrade, reinforced soil subgrade is particularly suitable for road engineering under conditions such as larger road load, uneven settlement of foundation, higher seismic grade requirement, etc. For the wrapped end reinforced soil subgrade structure with load on the top surface of the subgrade, the active earth pressure acting on the wrapped end is one of the key factors in engineering design. Among many methods for calculating soil pressure, the limit equilibrium principle is the most commonly used method in engineering practice due to its simple concept. Among them, the classic Rankine or Coulomb soil pressure theory is an important basic method for determining soil pressure. On this basis, other soil pressure calculation methods based on the limit equilibrium framework, such as limit analysis method, slip line method, etc., have also been developed.

[0003] In addition to these soil pressure calculation methods, the strip method is also an important method based on the limit equilibrium theory. According to the different ways of block division, the strip method mainly includes vertical strip method, inclined strip method and horizontal strip method. Among the above three methods, the horizontal strip method is preferred due to its simplicity and ease of calculation. In previous studies, some scholars have studied the seismic stability of reinforced slopes using the horizontal strip method within the pseudo-static framework, and determined the shape and position of the critical slip surface. Some other scholars have analyzed the internal stability of reinforced soil walls under seismic conditions based on the horizontal strip and pseudo-dynamic method, and obtained the required reinforcement tension and length to maintain the stability of the reinforced soil wall under seismic conditions, as well as the influence of parameters such as soil friction angle, horizontal and vertical seismic acceleration on the stability of the reinforced soil wall. Some other scholars have proposed a new slice analysis method for reinforced soil retaining walls to analyze their internal stability under horizontal and vertical seismic loads. The relationship between interlayer force and wall safety factor is considered, and the simplified critical slip surface is more easily applied to engineering than the logarithmic spiral failure surface.

[0004] Previous methods for analyzing the seismic stability of reinforced soil retaining walls have provided references for practical engineering projects. However, there are currently no references for calculating the earth pressure at the end of the reinforced soil subgrade with end wrapping, and in particular, previous studies lack sufficient evidence regarding the treatment of the tension action mode of the reinforcing bars. Summary of the Invention

[0005] The purpose of this invention is to address the existing problem that there is currently no reference for calculating the earth pressure at the wrapping end of end-wrapped reinforced soil subgrades, especially regarding the lack of sufficient evidence for handling the tension action mode of the reinforcing bars in previous studies. This invention provides a method for calculating the active earth pressure at the wrapping end of end-wrapped reinforced soil subgrades.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A method for calculating the active earth pressure at the reverse end of an end-reinforced soil subgrade includes the following steps:

[0008] A computational analysis model of end-reinforced soil subgrade under active limit state was established to obtain the sliding soil wedge.

[0009] The sliding soil wedge is divided into n strips along the horizontal plane, with each strip containing a layer of reinforcing bars in the middle.

[0010] Establish the equilibrium equations for each block and calculate the active earth pressure P at the reverse end of each block. i The relationship function P with the dip angle β of the sliding fracture surface i (β), where i = {1,2,3…,n}, and n equals the number of strips;

[0011] All P i (β) Summing the results yields the total active earth pressure P at the reverse end of the sliding soil wedge. s The relationship function P with the dip angle β of the sliding fracture surface s (β), based on the total active earth pressure P s By finding the condition for maximization, the dip angle β of the sliding fracture surface under the active limit state is obtained. m and β m Substituting the equilibrium equations of each block into the equations, we can obtain the corresponding P. i Complete the calculation.

[0012] The active limit state is a special stress state of the soil body behind the retaining wall when the retaining wall and the soil body interact. When the retaining wall moves forward or rotates due to some reasons (such as construction error, foundation settlement, earthquake, etc.), the soil filling behind the wall starts to slide downward or flow plastically, but has not yet completely lost stability. In this state, the soil body behind the wall reaches the limit of bearing shear stress, i.e. significant shear failure has not yet occurred, and is still in a dynamic balance state that is not static but can still be maintained. Because the calculation model includes a sliding soil wedge, the sliding fracture surface passes through the roadbed slope foot and penetrates to the roadbed top surface. The angle between the sliding fracture surface and the roadbed bottom surface is an acute angle. The sliding soil wedge is an entity surrounded by the sliding fracture surface, the inside end surface of the reverse-packing end of the roadbed, and the roadbed top surface. In the calculation, the soil in the roadbed is regarded as an average soil for convenience. Because it is a theoretical calculation, the external force points on the boundaries of each block are located at the midpoints of the corresponding blocks. The earthquake force is an equivalent inertia force acting on the gravity center of each block in a quasi-static manner. When establishing the balance equation set of each block, the vertical balance equation, the horizontal balance equation, and the moment balance equation are included. For the calculation condition set in the active limit state, according to the Rankine or Coulomb soil pressure theory, it is assumed that the pressure of the soil on the retaining wall reaches the maximum when the soil reaches the limit of shear strength, so β m is calculated according to this principle. After Pi is calculated, the building above the reverse-packing type reinforced soil roadbed can be designed.

[0013] As a preferred scheme of the present application, the blocks are sequentially numbered from top to bottom, and when i = 1, the following balance equation set one is established:

[0014] S1cosβ-N1sinβ+P1cosδsinα+P1sinδcosα+H1-k h W1-H2+T1=0

[0015] S1sinβ+N1cosβ+P1sinδsinα-P1cosδcosα-L q q+V2-(1+k v )W1=0

[0016]

[0017] Wherein, S1 is the shear force of the sliding fracture surface of the first block; N1 is the normal force of the sliding fracture surface of the first block; P1 is the active soil pressure of the inside of the reverse-packing end of the first block; δ is the external friction angle of the soil; α is the angle between the reverse-packing end of the reinforcing bar and the horizontal surface inside the roadbed; H1 is the horizontal force of the top surface of the first block; H2 is the horizontal force of the bottom surface of the first block; k his the horizontal seismic influence coefficient; W1 is the dead weight of the first block; T1 is the tension of the first block corresponding to the tension bar; L g is the width of the effective load acting on the sliding soil wedge; q is the effective load acting on the top surface of the subgrade; V2 is the normal force of the bottom surface of the first block; k v is the vertical seismic influence coefficient; X q is the horizontal distance from the action point of the strip load acting on the top of the sliding soil wedge to the outer lateral surface of the subgrade towards the inverted end; X V2 is the horizontal distance from the action point of V2 to the outer lateral surface of the subgrade towards the inverted end; X G1 is the horizontal distance from the action point of W1 to the outer lateral surface of the subgrade towards the inverted end; h1 is the body height of the first block.

[0018] The balance equation set one comprises a lateral balance equation, a vertical balance equation and a moment balance equation. The effective load width refers to a concept introduced for the purpose of simplifying calculation in the design analysis of a specific structure or component, considering that the load (such as vehicle, crowd, equipment weight, etc.) is not uniformly distributed on the structure surface, but has a certain concentration or local effect. The effective load width is an equivalent width used to simulate the influence of actual load distribution on the structure response, so that the actual non-uniformly distributed load can be regarded as a load uniformly distributed on the entire effective width in the calculation, thereby facilitating the use of traditional structure analysis methods for processing. In the calculation method, the action points of external forces on the boundaries of each block are located at the midpoints of the corresponding blocks.

[0019] As a preferred scheme of the present application, L g and X q are obtained by the following formula:

[0020]

[0021]

[0022] wherein H is the height of the subgrade; d is the distance between the effective load q acting on the top surface of the subgrade and the top surface of the first block towards the inverted end; b' is the action width of the effective load q acting on the top surface of the subgrade.

[0023] As a preferred scheme of the present application, the blocks are sequentially numbered from top to bottom, and when i>1, the following balance equation set two is established:

[0024] S i cosβ-N i sinβ+P i cosδsinα+P i sinδcosα+H i -k h W i -Hi+1 +T i =0

[0025] S i sinβ+N i cosβ+P i sinδsinα-P i cosδcosα-W i +W i+1 -(1+k v )W i =0

[0026]

[0027] wherein, S i is the shear force of the sliding fracture surface of the ith block; N i is the normal force of the sliding fracture surface of the ith block; P i is the active earth pressure inside the inverted end of the ith block; δ is the external friction angle of the soil; α is the included angle between the inverted end of the tensile reinforcement and the horizontal plane inside the embankment; H i is the horizontal force of the top surface of the ith block; H i+1 is the horizontal force of the bottom surface of the ith block; W i is the dead weight of the ith block; k h is the horizontal seismic influence coefficient; T i is the tensile force of the tensile reinforcement corresponding to the ith block; W i is the normal force of the top surface of the ith block, V i+1 is the normal force of the bottom surface of the ith block; k v is the vertical seismic influence coefficient; X Vi is the horizontal distance from the action point of V i to the inverted end facing the outer surface of the embankment; X Gi is the horizontal distance from the action point of W i to the inverted end facing the outer surface of the embankment; h i is the body height of the ith block; j is the serial number of the block; h j is the body height of the jth block.

[0028] Because the top surface of the first block is provided with an effective load q acting on the top surface of the embankment, and the top surface of the first block is not subjected to the normal force V i from the top surface of other blocks, the vertical equilibrium equation and the moment equilibrium equation will be changed accordingly.

[0029] As a preferred scheme of the present application, the parameters including the shear force S i of the sliding fracture surface of the ith block, the body height h i of the ith block, and the normal force V i of the top surface of the ith block need to be used when establishing the balance equation set of each block.Horizontal distance X from the action point of the i-th block to the lateral surface of the subgrade facing the inverted end Vi Self-weight W of the i-th block i Horizontal distance X from the action point of the i-th block to the lateral surface of the subgrade facing the inverted end Gi Tension T of the i-th block corresponding to the tensile reinforcement i Horizontal force H of the top surface of the i-th block i is obtained by the following formula:

[0030]

[0031]

[0032]

[0033]

[0034]

[0035] wherein Ni is the normal force of the sliding fracture surface of the i-th block; is the internal friction angle of the soil body, c is the cohesive force of the sliding fracture surface of the soil body, l i is the sliding surface length of the i-th block; H is the height of the subgrade; L i is the top surface length of the i-th block; s xi is the lateral horizontal displacement parameter of the corresponding inverted end of the i-th block; s x0 is the maximum value of the lateral horizontal displacement parameters of the corresponding inverted ends of the n blocks; λ i is the ratio of the tangential force between the i-th block and the (i-1)-th block and the shear resistance between the i-th block and the (i-1)-th block.

[0036] As a preferred scheme of the present application, λ i is determined by the following formula:

[0037]

[0038] wherein z i is the distance from the top surface of the i-th block to the top surface of the sliding soil wedge.

[0039] The value of λ i is not greater than 1.

[0040] As a preferred scheme of the present application, L i is obtained by the following formula:

[0041]

[0042] The value of L i+1 is obtained by the following formula.

[0043] As a preferred scheme of the present application, through large-scale model test on the end-reverse-packed reinforced soil subgrade, s i and the expression of the relationship between the position height of the strip and the block:

[0044]

[0045] wherein y i is the position height parameter of the i-th block, Y i is the height of the i-th block to the subgrade bottom.

[0046] s xi is the 1000 times of the actual displacement of the i-th block divided by the dimensionless quantity of the subgrade height H. The model with the scale ratio not less than 1:10 is regarded as the large-scale model.

[0047] As a preferred scheme of the present application, through β m is obtained.

[0048] Because the analysis model of the present calculation method is the end-reverse-packed reinforced soil subgrade in the active limit state, the condition of the maximum active earth pressure is taken.

[0049] As a preferred scheme of the present application, P i is obtained. i After that, P ai is divided by the thickness of the i-th block to obtain the stress value P i of the active earth pressure of the i-th block.

[0050] In summary, due to the adoption of the above technical scheme, the present application has the following beneficial effects:

[0051] A calculation method of the active earth pressure of the reverse-packed end of the end-reverse-packed reinforced soil subgrade, first establishes the calculation analysis model of the end-reverse-packed reinforced soil subgrade in the active limit state to obtain the sliding soil wedge; then divides the sliding soil wedge into n strips in the horizontal direction, and sets a layer of reinforcing bar in the center of each strip; then establishes the balance equation set of each strip according to the force balance to obtain the active earth pressure P i of the reverse-packed end and the relationship function P i (β) of the dip angle β of the sliding fracture surface, then accumulates the n P s (β) to obtain P m (β), according to the maximum principle of the active earth pressure, obtains the dip angle value β m of the sliding fracture surface in the active limit state, and then brings β i into the balance equation set to obtain the corresponding P iThe present application solves the problem that there is no reference for the calculation method of the end soil pressure of the end-wrapping type reinforced soil subgrade in the prior art, and the treatment of the tensile reinforcement tensile force action mode is also lack of sufficient basis in the previous research. BRIEF DESCRIPTION OF DRAWINGS

[0052] Figure 1 is a step diagram of the calculation method of the active soil pressure of the wrapped end of an end-wrapping type reinforced soil subgrade in Example 1;

[0053] Figure 2 is a schematic diagram of the sliding soil wedge of the end-wrapping type reinforced soil subgrade in Example 1 under the active limit state;

[0054] Figure 3 is a schematic diagram of the sliding soil wedge and the overall mechanical analysis model of the end-wrapping type reinforced soil subgrade in Example 1;

[0055] Figure 4 is a mechanical analysis model diagram of the first block in Example 1;

[0056] Figure 5 is a mechanical analysis model diagram of the i-th block (i>1) in Example 1;

[0057] Figure 6 is a distribution curve diagram of the lateral horizontal displacement parameter s xi and the corresponding position height parameter y i of the wrapped end corresponding to the i-th block in Example 1;

[0058] Figure 7 is a schematic diagram of the cross section of the end-wrapping type reinforced soil subgrade of a single-track railway in Example 1;

[0059] Figure 8 is a distribution curve diagram of the active soil pressure inside the wrapped end along the height under the static working condition and the typical seismic working condition in Example 1;

[0060] Figure 9 is a variation curve diagram of the total active soil pressure inside the wrapped end under different seismic working conditions in Example 1.

[0061] Legend: 100-wrapped end, 200-tensile reinforcement, 300-sliding fracture surface. DETAILED DESCRIPTION

[0062] The present application will be described in detail below with reference to the drawings.

[0063] In order to make the purpose, technical scheme and advantages of the present application more clear and explicit, the present application will be further described in detail below with reference to the drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application and do not limit the present application.

[0064] In the following description of specific embodiments, the terms of orientation or positional relationship such as "upper", "lower", "left", "right", "center", "inner", "outer", and the like, are expressed based on the orientation or positional relationship shown in the drawings, or the orientation or positional relationship when the device / apparatus is normally used. These terms of orientation or positional relationship are only for the convenience of description or simplification of the description in the specific embodiments, for the quick understanding of the scheme by the skilled person, and do not indicate or imply that a specific device / component / element must have a specific orientation or be constructed and operated in a specific positional relationship, and therefore cannot be understood as a limitation on the present application.

[0065] The terms "horizontal", "vertical", and the like do not mean that the corresponding device / component / element must be absolutely horizontal or vertical or overhanging, but can be slightly inclined. For example, "horizontal" only means that its direction is more horizontal relative to "vertical", and does not mean that the structure must be completely horizontal, but can be slightly inclined. Alternatively, it can be simplified to mean that the corresponding device / component / element is set in a specific direction such as "horizontal", "vertical", etc., and can have an error / deviation of ±10% with respect to the corresponding direction, more preferably an error / deviation of ±8% or less, more preferably an error / deviation of ±6% or less, more preferably an error / deviation of ±5% or less, and more preferably an error / deviation of ±4% or less. As long as the corresponding device / component / element is within the error / deviation range, it can still achieve its role in the scheme of the present application.

[0066] The terms "first", "second", "third", and the like are only used to distinguish the same or similar components for description, and should not be understood as emphasizing or implying the relative importance of a specific component.

[0067] The terms "set", "mounted", "connected", and the like should be broadly understood, for example, it can be a fixed connection, or a detachable connection, or an integral connection, which can be welding, riveting, bolting, screwing, etc. connection means commonly used in the art, which can be directly connected or indirectly connected through an intermediate medium, or the communication between two elements.

[0068] Embodiment 1

[0069] As shown in Figure 1 A calculation method of active soil pressure of a wrapped end of a reinforced soil subgrade with wrapped end reinforcement, comprising the following steps:

[0070] S1, a calculation and analysis model of the wrapped end of the reinforced soil subgrade with wrapped end reinforcement under the active limit state is established, and a sliding soil wedge is obtained;

[0071] In the case that the subgrade soil body inside the wrapped end 100 reaches the active limit state, the calculation and analysis model includes:

[0072] (1) As shown in Figures 2-3 , the sliding rupture surface 300 in the reinforced soil subgrade is a plane, and passes through the subgrade toe and penetrates to the subgrade top surface, the plane has a certain included angle with the subgrade bottom surface characterized by an acute angle, the body height of the sliding soil wedge is equal to the subgrade height H;

[0073] (2) The soil body in the subgrade is a homogeneous soil body.

[0074] Thus, the sliding soil wedge formed in the reinforced soil subgrade is a solid body surrounded by the sliding rupture surface 300, the inner side surface of the subgrade at the wrapped end 100, and the subgrade top surface.

[0075] S2, the sliding soil wedge is divided into n strips along the horizontal plane in a horizontal strip manner, wherein the middle part of each strip contains a layer of reinforcement 200;

[0076] The following conditions are included when establishing the strip:

[0077] (1) The ratio of the tangential force on the interfacial surface of each strip to its shear resistance is a real number λ not greater than 1 i , which is the ratio of the depth z i of the interfacial surface (the top surface of the ith soil strip) from the subgrade top surface to the subgrade height H;

[0078] (2) The external force point on the boundary of each strip is located at the midpoint.

[0079] (3) The seismic action force is taken as the equivalent inertia force, which acts on the center of gravity of each strip in a pseudo-static force manner.

[0080] S3, the balance equation set of each strip is established, and the relationship function P i (β) of the active soil pressure P i of the wrapped end 100 of each strip and the inclination angle β of the sliding rupture surface 300 is calculated, wherein i={1, 2, 3... n};

[0081] S301, the strips are marked in ascending order from top to bottom, when i=1, the force model is as shown in Figure 4 , and the following balance equation set one is established:

[0082] S1cosβ-N1sinβ+P1cosδsinα+P1sinδcosα+H1-k h W1-H2+T1=0

[0083] S1sinβ+N1cosβ+P1sinδsinα-P1cosδcosα-L g q+V2-(1+k v )W1=0

[0084]

[0085] Wherein, S1 is the shear force of the sliding fracture surface 300 of the first block; N1 is the normal force of the sliding fracture surface 300 of the first block; P1 is the active earth pressure inside the inverted end 100 of the first block; δ is the external friction angle of the soil; α is the included angle between the inverted end 100 of the tensile reinforcement 200 and the horizontal plane inside the roadbed; H1 is the horizontal force of the top surface of the first block; H2 is the horizontal force of the bottom surface of the first block; k h is the horizontal seismic influence coefficient, k h = 0 when the sliding soil wedge is in a static state; W1 is the dead weight of the first block; T1 is the tension of the tensile reinforcement 200 corresponding to the first block; L q is the width of the effective load acting on the sliding soil wedge; q is the effective load acting on the top surface of the roadbed; V2 is the normal force of the bottom surface of the first block; k v is the vertical seismic influence coefficient, k v = 0 when the sliding soil wedge is in a static state; X q is the horizontal distance from the action point of the strip load acting on the top of the sliding soil wedge to the inverted end 100 towards the outer surface of the roadbed; X V2 is the horizontal distance from the action point of V2 to the inverted end 100 towards the outer surface of the roadbed; X G1 is the horizontal distance from the action point of W1 to the inverted end 100 towards the outer surface of the roadbed; h1 is the body height of the first block.

[0086] When i > 1, the force model is as shown in Figure 5 The following two sets of balance equations are established:

[0087] S i cosβ-N i sinβ+P i cosδsinα+P i sinδcosα+H i -k h W i -H i+1 +T i =0

[0088] S i sinβ+N i cosβ+P i sinδsinα-P i cosδcosα-W i +W i+1 -(1+k v )W i =0

[0089]

[0090] wherein, S i Shear force of the sliding fracture surface 300 of the ith block; N i Normal force of the sliding fracture surface 300 of the ith block; H i Horizontal force of the top surface of the ith block; H i+1 Horizontal force of the bottom surface of the ith block; W i Dead weight of the ith block; T i Tension of the corresponding tension bar 200 of the ith block; W i Normal force of the top surface of the ith block, V i+1 Normal force of the bottom surface of the ith block; X Vi Horizontal distance from the action point of W i to the outer surface of the subgrade facing side of the inverted end 100; X Gi Horizontal distance from the action point of W i to the outer surface of the subgrade facing side of the inverted end 100; h i Body height of the ith block; j is the serial number of the block; h j Body height of the jth block.

[0091] S302, the related geometric parameters of the balance equation set one are obtained through the first formula set shown as follows:

[0092]

[0093]

[0094] wherein, H is the height of the subgrade; d is the distance between q and the top surface of the first block facing the inverted end 100; b' is the action width of q;

[0095] The related parameters of the balance equation set one and the balance equation set two are obtained through the second formula set shown as follows:

[0096] At the sliding fracture surface 300, according to the Mohr-Coulomb strength failure criterion, the following can be obtained:

[0097]

[0098] The geometric relationship in which the ith block exists includes:

[0099]

[0100]

[0101]

[0102]

[0103]

[0104] wherein, is the internal friction angle of the soil, c is the cohesion of the soil of the slip surface 300, l i is the slip surface length of the ith block; L i is the top surface length of the ith block;

[0105] S303, through the large-scale model test of the end-turnup type reinforced soil subgrade, the lateral horizontal displacement parameter s of the turnup end 100 under the action of different strip loads q on the top surface of the subgrade is obtained xi The distribution curve along the position height of the subgrade is a parabolic pattern with large middle and small upper and lower ends. As shown in Figure 6 , the fitted curve is within the specified difference range compared with the actually measured data. The general expression of the corresponding parabola is:

[0106]

[0107] wherein, s xi is the lateral horizontal displacement parameter of the corresponding turnup end 100 of the ith block, which is 1000 times of the actual displacement divided by the dimensionless quantity of the height H of the subgrade. y i is the position height parameter of the ith block, Y i is the height of the ith block to the bottom surface of the subgrade.

[0108] S304, under the active limit state of the reinforced soil subgrade, the parabola top point shown in S303 can reach the tensile limit state, that is: the corresponding tensile force of the layer of the tensile reinforcement 200 is equal to the design limit tensile force T of the tensile reinforcement 200 material, and the tensile force T of the remaining layers of the tensile reinforcement 200 can be represented as: i

[0109]

[0110] s x0 is the maximum value of the lateral horizontal displacement parameter of the corresponding turnup end 100 of each of the n blocks, that is, the maximum value of multiple s xi , s xi and s x0 can be obtained according to the calculation formula shown in S303, or can be obtained according to the actual displacement condition.

[0111] S305, according to the force analysis model characteristics of each block in step S2, the condition (1) of the step S2 has

[0112] ​

[0113] λ i It is the ratio of the tangential force between the i-th block and the (i-1)-th block to the shear resistance between the i-th block and the (i-1)-th block;

[0114] in

[0115]

[0116] z i Let be the distance from the top surface of the i-th block to the top surface of the sliding soil wedge.

[0117] By simultaneously solving the first set of equilibrium equations, the first set of formulas, and the second set of formulas, we can find the relationship between P1 and β for the first block. By simultaneously solving the second set of equilibrium equations and the second set of formulas, we can find the relationship between P1 and β for the corresponding block when i>1. i The relationship with β.

[0118] S4, put all P i (β) Summation yields P s (β) gives the total active earth pressure P acting on the entire inner side of the reverse end 100. s The function of the dip angle β of the sliding fracture surface 300 is expressed as:

[0119]

[0120] According to the principle of maximum active earth pressure, the condition for the total active earth pressure to reach its maximum value is as follows:

[0121]

[0122] The inclination angle β of the sliding fracture surface 30° under the active limit state is obtained. m and β m Substituting into equilibrium equations one and two, we obtain the corresponding P. i Find P i Then, P i Dividing by the thickness of the i-th block yields the stress value P of the active earth pressure on the i-th block. ai .

[0123] The following calculations are based on specific data:

[0124] like Figure 7 As shown, the embankment slope angle α is 90°, and the unit weight γ of the sandy fill is 20 kN / m. 3 The cohesion c is 0 kPa, and the internal friction angle is... The angle of friction is 30°, the external friction angle δ is 10°, the horizontal tie bar 200 uses HDPE (High Density Polyethylene) geogrid material, its design ultimate tensile force T is 15kN / m, the top surface of the subgrade has a surface load q=10kPa, d=3.3m, b′=4m.

[0125] According to the aforementioned method of the present invention, the static condition (k) is calculated. h =0, k v =0) and under typical seismic conditions (k h =0.15, k v =k h / 2) Sliding fracture surface with an inclination angle of 30° β m The stress distribution curves along the height of the active earth pressure inside the 100mm inverted end are as follows: (The values ​​are 56.73° and 51.21° respectively). Figure 8 As shown. It can be seen that under static conditions, the earth pressure at the reverse end (100mm) exhibits a non-linear distribution that gradually increases downwards along the height; under seismic conditions, the earth pressure at the reverse end (100mm) follows a parabolic distribution with a smaller middle section and larger ends along the height. Simultaneously, it can be obtained that under static conditions (k... h =0, k v Under condition P = 0), the total active earth pressure inside the reverse end 100 is P. s =152.9kN / m; at k h =0.15, k v =k h Under the condition of / 2, the total active earth pressure on the inner side of the reverse end 100 is P. s =212.8kN / m.

[0126] Furthermore, Figure 9 The curves showing the total active earth pressure within 100 mm of the reverse end as a function of the seismic coefficient are presented. For ease of comparison and verification, Figure 9 The results of the numerical simulation method are also given. It can be seen that, for a typical k... h In the three cases of 0.05, 0.15, and 0.25, k v =0, k h / 2 and k h When the results were obtained, the errors between the method of the present invention and the numerical simulation were 3.9%, 0.6%, 5.0%, 4.8%, 1.1%, 3.3%, and 4.1%, 0.4%, 2.7%, respectively. Therefore, the calculation results of the algorithm of the present invention are in good agreement with those of existing methods, which demonstrates the correctness of the method of the present invention.

[0127] The above merely preferred embodiments of the present application are not used to limit the present application, any modification, equivalent replacement and improvement etc. made within the spirit and principle of the present application should be included in the protection scope of the present application.

Claims

1. A method for calculating the active earth pressure at the reverse-wrap end of an end-reinforced soil subgrade, characterized in that, It includes the following steps: A computational analysis model of end-reinforced soil subgrade under active limit state was established to obtain the sliding soil wedge. The sliding soil wedge is divided into horizontal strips along the horizontal plane. Each strip contains a layer of reinforcing rib (200) in the middle. Establish the equilibrium equations for each block and calculate the active earth pressure at the reverse end (100) of each block. Angle of inclination with the sliding fracture surface (300) relational functions ,in ,in Equal to the number of strips; All of The total active earth pressure at the reverse end (100) of the sliding soil wedge is obtained by summing the values. Angle of inclination with the sliding fracture surface (300) relational functions According to the total active earth pressure Given the condition for maximizing the value, the dip angle of the sliding fracture surface (300) under the active limit state is obtained. and will Substituting the equilibrium equations of each block into the equations yields the corresponding... Complete the calculation; Number the bars sequentially from top to bottom, when... At that time, the following system of equilibrium equations is established: in, The shear force at the sliding fracture surface (300) of the first block; The normal force is at the sliding fracture surface (300) of the first block; The active earth pressure inside the reverse end (100) of the first block; The external friction angle of the soil; The angle between the reverse end (100) of the tie rod (200) and the horizontal plane inside the roadbed; The horizontal force on the top surface of the first block; The horizontal force on the bottom surface of the first block; This represents the horizontal earthquake influence coefficient. The weight of the first block; The tension of the tie rod (200) corresponding to the first block; The width of the effective load acting on the sliding soil wedge; This refers to the effective load acting on the top surface of the roadbed. The normal force on the bottom surface of the first block; This is the vertical seismic influence coefficient; The horizontal distance from the point of application of the strip load acting on the top of the sliding soil wedge to the outer surface of the roadbed at the reverse end (100); for The horizontal distance from the point of action to the reverse end (100) towards the outer surface of the roadbed; for The horizontal distance from the point of action to the reverse end (100) towards the outer surface of the roadbed; This is the height of the first block. and The following formula can be used to obtain: in, This refers to the roadbed height; Effective load acting on the top surface of the roadbed The distance between the top surface of the first block and the side facing the reverse end (100); Effective load acting on the top surface of the roadbed The width of the function.

2. The method for calculating the active earth pressure at the reverse end of an end-reinforced soil subgrade according to claim 1, characterized in that, Number the bars sequentially from top to bottom, when... At that time, the following system of equilibrium equations is established: in, No. Shear force at the sliding fracture surface (300) of the strip; For the first Normal force at the sliding fracture surface (300) of the strip; For the first Horizontal force on the top surface of the strip; For the first Horizontal force on the bottom surface of the strip; For the first The weight of the strip; For the first The tensile force of the tie rod (200) corresponding to the strip; For the first Normal force on the top surface of the strip, For the first Normal force on the bottom surface of the strip; for The horizontal distance from the point of action to the reverse end (100) towards the outer surface of the roadbed; for The horizontal distance from the point of action to the reverse end (100) towards the outer surface of the roadbed; For the first The height of the block itself; The number of the strip. For the first The height of the block itself.

3. The method for calculating the active earth pressure at the reverse-wrap end of an end-reinforced soil subgrade according to claim 1, characterized in that, Establishing the equilibrium equations for each block requires parameters including the first... Shear force at the sliding fracture surface (300) of the strip , No. The height of the strip , No. Normal force on the top surface of the strip The horizontal distance from the point of action to the reverse end (100) towards the outer surface of the roadbed , No. The weight of the strip The horizontal distance from the point of action to the reverse end (100) towards the outer surface of the roadbed , No. Tensile force of the tie rod (200) corresponding to the strip And the horizontal force on the top surface of the nth block The following formula can be used to obtain: in, For the first Normal force on the sliding fracture surface (300) of the strip; The internal friction angle of the soil. For the soil cohesion at the sliding fracture surface (300), For the first The length of the smooth surface of the strip; For the first The length of the top surface of the strip; For the first Lateral horizontal displacement parameters of the corresponding reverse end (100) of the strip; for The maximum value of the lateral horizontal displacement parameter of the corresponding reverse end (100) in each strip; For the first strips and blocks Tangential forces between strips and the first strips and blocks The ratio of shear resistance between strips.

4. The method for calculating the active earth pressure at the reverse end of an end-reinforced soil subgrade according to claim 3, characterized in that, Determined by the following formula: in, For the first The distance from the top surface of the strip to the top surface of the sliding soil wedge.

5. The method for calculating the active earth pressure at the reverse end of an end-reinforced soil subgrade according to claim 3, characterized in that, The following formula can be used to obtain: 。 6. The method for calculating the active earth pressure at the reverse end of an end-reinforced soil subgrade according to claim 3, characterized in that, Through large-scale model tests on end-reinforced soil subgrade, the following results were obtained. The expression relating the height of the bar to its position is as follows: in, For the first The position and height parameters of the strip. , For the first The height of the strip from the roadbed surface.

7. A method for calculating the active earth pressure at the reverse end of an end-reinforced soil subgrade according to any one of claims 1-6, characterized in that, pass Seek .

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