A method for predicting and calculating the active earth pressure of loess when the backfill surface is sloping.

Abstract

The present application belongs to the technical field of geotechnical engineering and geological engineering, and particularly relates to a method for predicting and calculating loess active earth pressure when the ground surface is inclined behind a wall. The present application obtains the parameter values of the shear strength, normal stress, cohesion strength, internal friction angle, tensile strength and loess unit weight of the loess to be predicted; establishes a joint strength envelope, establishes a shear strength-normal stress rectangular coordinate system and solves a stress Mohr circle equation; according to the stress Mohr circle equation, the predicted value of the loess active earth pressure is obtained, and the predicted value of the loess active earth pressure is applied to the design of loess slope and retaining wall engineering in five steps. The predicted and calculated value of the loess active earth pressure based on the joint strength theory is more in line with the engineering practice, is safer, and has a practical guiding value for the design of loess area engineering support.

Description

Technical Field

[0001] This invention belongs to the fields of geotechnical engineering and geological engineering technology, and specifically relates to a method for predicting and calculating the active earth pressure of loess when the surface of backfill is tilted. Background Technology

[0002] In recent years, with the vigorous development of my country's economy, there are a large number of geotechnical engineering structures and geological slopes such as mountain roads and high slopes in the Northwest region. This makes it necessary to place higher demands on the crack resistance of the structures in the project. In the process of pre-estimating and calculating the causes of cracks and engineering treatment, the testing of the tensile strength of the soil is essential.

[0003] Throughout western my country, steep loess slopes stand everywhere. Analysis of certain mechanical mechanisms reveals that shear strength alone is insufficient to support the stability of loess slopes; therefore, the unique tensile properties of loess play a crucial role. Furthermore, in actual working conditions, more complex and variable situations are often encountered. In many loess retaining wall projects, it has been found that the surface of the backfill (loess) behind the retaining wall often has a certain angle of inclination. This angle inevitably has a significant impact on the stress characteristics of the retaining wall, necessitating more accurate prediction and calculation of earth pressure when the backfill surface has an angle.

[0004] When predicting and calculating earth pressure on retaining walls in loess areas, since loess has relatively high tensile strength, the combined strength theory, which comprehensively considers the shear and tensile properties of loess, has gradually matured and begun to be applied in engineering technology. Therefore, based on the formula of the combined strength theory, it is of great engineering significance to explore the method of predicting and calculating active earth pressure on loess when there is an inclination angle on the backfill surface. Summary of the Invention

[0005] This invention provides a method for predicting and calculating the active earth pressure of loess when the backfill surface is tilted. The purpose is to provide a method for predicting the pressure of retaining walls and backfill soil that can be applied to the design of loess slope and retaining wall projects, and to provide support for the stability design of loess slope and retaining wall projects.

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

[0007] A method for predicting and calculating the active earth pressure of loess when the surface of the backfill land behind a wall is inclined includes the following steps.

[0008] Step 1: Obtain the parameter values ​​of shear strength, normal stress, cohesion strength, internal friction angle, tensile strength, and unit weight of the loess to be predicted;

[0009] Step 2: Establish a joint strength envelope;

[0010] Step 3: Establish a rectangular coordinate system of shear strength and normal stress, and derive the stress Mohr's circle equation;

[0011] Step 4: Based on the stress Mohr's circle equation obtained in Step 3, calculate the predicted value of active earth pressure on loess.

[0012] Step 5: Apply the predicted values ​​of active earth pressure of loess obtained in Step 4 to the design of loess slope and retaining wall projects.

[0013] The equation for the joint strength envelope in step two is as follows:

[0014]

[0015] In the formula, the test parameters are: τ is the shear strength; σ is the normal stress; and c is the cohesive strength. σ is the internal friction angle; t This refers to tensile strength.

[0016] The method for establishing a shear strength-normal stress rectangular coordinate system in step three and obtaining the Mohr's circle equation is as follows:

[0017] In the shear strength-normal stress rectangular coordinate system, i.e., the σ-τ rectangular coordinate system, if the backfill soil behind the wall is in an active limit equilibrium state, then the stress Mohr's circle corresponding to the stress state at any point in the soil is tangent to the combined strength envelope. Two rays, OA and OC, are drawn on the orthogonal axis through the origin O, both with an angle of i to the σ axis. Ray OA intersects the stress Mohr's circle at two points, with the intersection point furthest from the origin O defined as point A. Ray OC intersects the stress Mohr's circle at points B and C, with the intersection point furthest from the origin O defined as point C. At this point, line segment OA represents the larger stress p in the active limit state. A The OB segment represents the stress parallel to the ground surface, that is, the earth pressure p acting on the vertical surface behind the retaining wall. B p B This refers to the predicted active earth pressure p when the slope angle of the backfill soil behind the wall is i, based on the combined strength theory. a Predicted values;

[0018] Assume M is the point of tangency between the stress Mohr circle and the joint strength envelope, N is the center of the stress Mohr circle on the σ axis, and line segment MF is orthogonal and perpendicular to the σ axis, with the foot of the perpendicular being F;

[0019] Since point M lies on the joint strength envelope, let's assume the x-coordinate of point M is σ. M The equation for the joint strength envelope in step two. The ordinate of point M can be obtained, that is, the coordinates of M are... The slope k of the tangent at point M can be obtained by differentiating the joint strength envelope. tfor:

[0020]

[0021] Therefore, we obtain the equation for the normal MN:

[0022]

[0023] Since the normal MN intersects the abscissa (τ=0) at point N, combining equation (3) with τ=0 yields the abscissa of the center N of the stress Mohr's circle:

[0024]

[0025] The coordinates of the center N of the stress Mohr circle are:

[0026] In triangle ΔMFN, the ordinate of the point of tangency M can be represented by the magnitude of the line segment MF, i.e., |MF| = τ. M |FN|=k t τ M ,but Let be the radius of the stress Mohr circle, then the equation of the stress Mohr circle can be expressed as:

[0027]

[0028] The origin of the shear strength-normal stress rectangular coordinate system, i.e., the σ-τ rectangular coordinate system, is selected at the zero stress point, and the stress on both the vertical and horizontal axes is zero.

[0029] The specific method for calculating the predicted value of active earth pressure on loess in step four, based on the Mohr's circle equation, is as follows:

[0030] Since the equation of line BC is τ = σtani, and the stress Mohr's circle intersects this line at points B and C, we can solve the problem by simultaneously solving the equations of the stress Mohr's circle and line BC:

[0031]

[0032] Since the x-coordinate of point B is σ B The x-coordinate of point C is equal to the x-coordinate of point A, and both can be represented as σ. A Based on this, by solving equation (6), σ can be obtained respectively. A With σ B The value of, that is:

[0033]

[0034]

[0035] Therefore, according to p Ap B respectively with σ A σ B From the geometric relationships, we can obtain:

[0036]

[0037]

[0038] And because p A p B The expressions all contain σ M Therefore, by p A Expression (9) can be inversely solved for σ. M ,have to:

[0039]

[0040] By p B Expression (10) can be solved by inversely solving σ. M ,have to:

[0041]

[0042] To simplify the expression of the formula, let:

[0043]

[0044] Then the above two equations (11) and (12) can be simplified as follows:

[0045]

[0046]

[0047] Since equations (14) and (15) are equal, we have:

[0048]

[0049] Therefore, based on the above formula, we can obtain the following:

[0050]

[0051] If the inclination angle of the backfill soil behind the wall is expressed as i, then when the backfill soil behind the wall reaches the active limit equilibrium state, the larger stress is the stress applied parallel to the inclined ground surface, i.e., the surface of the backfill soil behind the wall, which is p. A =γhcosi, the smaller stress is the stress applied to the vertical surface behind the retaining wall, and its direction is parallel to the ground surface, therefore p B =p a Then, from equation (16), the formula for predicting and calculating the active earth pressure when the inclination angle of the backfill soil behind the wall is i is:

[0052]

[0053] Where: p a γ represents the predicted value of active earth pressure when the inclination angle of the backfill slope is i, h represents the unit weight of the backfill soil, and i represents the depth of the backfill soil.

[0054] Beneficial effects:

[0055] This invention proposes a method for predicting and calculating the active earth pressure of loess when the backfill surface is inclined, based on the framework of combined strength theory. This method is applicable to the stability design of loess slopes and retaining wall projects. In retaining wall engineering, this method comprehensively considers the shear and tensile properties of loess soil based on combined strength theory, and also takes into account the influence of the inclination angle of the backfill slope on the magnitude of the active earth pressure. This makes the predicted active earth pressure values ​​of loess based on combined strength theory more consistent with engineering realities and safer, providing practical guidance for the design of retaining structures in loess areas.

[0056] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it in accordance with the contents of the specification, the preferred embodiments of the present invention will be described in detail below. Attached Figure Description

[0057] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0058] Figure 1 This is a schematic diagram of the stress state at a certain point in the backfill soil behind the retaining wall.

[0059] Figure 2 The diagram shows the relationship between the stress Mohr's circle and the combined strength envelope when the backfill soil behind the wall is in an active limit equilibrium state.

[0060] Figure 3 The curves show the distribution of predicted active earth pressure on loess with varying backfill depth under sloping backfill conditions.

[0061] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it in accordance with the contents of the specification, the following detailed description is provided through preferred embodiments of the present invention. Detailed Implementation

[0062] The technical solution of the present invention will be clearly and completely described below with reference to the embodiments. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0063] Example 1:

[0064] Reference Figure 1 and Figure 2 The method for predicting and calculating the active earth pressure of loess when the backfill surface is sloping, as shown, includes the following steps:

[0065] Step 1: Obtain the parameter values ​​of shear strength, normal stress, cohesion strength, internal friction angle, tensile strength, and unit weight of the loess to be predicted;

[0066] Step 2: Establish a joint strength envelope;

[0067] Step 3: Establish a rectangular coordinate system of shear strength and normal stress, and derive the stress Mohr's circle equation;

[0068] Step 4: Based on the stress Mohr's circle equation obtained in Step 3, calculate the predicted value of active earth pressure on loess.

[0069] Step 5: Apply the predicted values ​​of active earth pressure of loess obtained in Step 4 to the design of loess slope and retaining wall projects.

[0070] This invention proposes a method for predicting and calculating the active earth pressure of loess when the surface of backfill is tilted, based on the framework of joint strength theory. It can be applied to the stability design of loess slopes and retaining wall projects.

[0071] In retaining wall engineering, this invention comprehensively considers the shear and tensile properties of loess soil based on the combined strength theory, and also takes into account the influence of the inclination angle of the backfill slope on the magnitude of active earth pressure. If the inclination angle is ignored, the obtained active earth pressure value will be too small, making the engineering calculation unsafe. Moreover, the larger the inclination angle of the backfill slope, the greater the active earth pressure. Therefore, considering the inclination angle of the backfill slope makes the predicted and calculated value of loess active earth pressure based on the combined strength theory more consistent with engineering reality and safer, providing practical guidance for the design of retaining walls in loess areas.

[0072] Example 2:

[0073] A method for predicting and calculating the active earth pressure of loess when the backfill surface is sloping.

[0074] When calculating the prediction expression for active earth pressure on loess under the condition of surface inclination of the backfill behind the retaining wall in the loess region, the excellent properties of loess's shear and tensile strength are first considered within the framework of combined strength theory. Then, it is assumed that the retaining wall is rigid, vertical, and has a smooth surface, and that the soil behind the wall is a homogeneous and continuous material with a certain inclination angle, denoted by i, where i is the design index of the backfill surface in the retaining wall project. This constitutes the premise and engineering design conditions for the prediction and calculation analysis of this invention. Figure 1 As shown, i represents the inclination angle of the slope of the soil behind the wall, and σ h Assuming the stress is acting on a vertical plane, σ v Then assume that the magnitude of the force is parallel to the plane of the earth's surface.

[0075] This invention uses the combined strength theory as a premise for prediction and calculation analysis, and conducts testing and predictive calculation analysis on the active earth pressure of loess in retaining wall projects that can be simplified to plane strain conditions: when the retaining wall gradually moves away from the soil behind the wall and causes a certain displacement, the earth pressure between the wall and the soil will gradually decrease, that is... Figure 1 Stress σ parallel to the Earth's surface h It becomes smaller, while σ v It remains unchanged, when σ h When reduced to a certain value, the stress Mohr's circle corresponding to the stress state at a point in the backfill immediately adjacent to the wall becomes tangent to the combined strength envelope. This indicates that the soil is in an active limit equilibrium state under the combined strength condition. In this case, the stress value σ in the direction parallel to the ground surface is... h It refers to the active earth pressure p of loess. a Predicted values.

[0076] The equation for the joint strength envelope in step two is:

[0077]

[0078] The test parameters in the formula are: τ is the shear strength; σ is the normal stress; and c is the cohesive strength. σ is the internal friction angle; t This refers to tensile strength.

[0079] In the shear strength-normal stress rectangular coordinate system, i.e., the σ-τ rectangular coordinate system, if the backfill soil behind the wall is in an active limit equilibrium state, then the stress Mohr's circle corresponding to the stress state at any point in the soil is tangent to the combined strength envelope. Two rays, OA and OC, are drawn on the orthogonal axis through the origin O, both with an angle of i to the σ axis. Ray OA intersects the stress Mohr's circle at two points, with the intersection point furthest from the origin O defined as point A. Ray OC intersects the stress Mohr's circle at points B and C, with the intersection point furthest from the origin O defined as point C. At this point, line segment OA represents the larger stress p in the active limit state. A The OB segment represents the stress parallel to the ground surface, that is, the earth pressure p acting on the vertical surface behind the retaining wall. B p B This refers to the predicted active earth pressure p when the slope angle of the backfill soil behind the wall is i, based on the combined strength theory. a Predicted values;

[0080] Assume M is the point of tangency between the stress Mohr circle and the joint strength envelope, N is the center of the stress Mohr circle on the σ axis, and line segment MF is orthogonal and perpendicular to the σ axis, with the foot of the perpendicular being F;

[0081] Since point M lies on the joint strength envelope, let's assume the x-coordinate of point M is σ. M The equation for the joint strength envelope in step two. The ordinate of point M can be obtained, that is, the coordinates of M are... The slope k of the tangent at point M can be obtained by differentiating the joint strength envelope. t for:

[0082]

[0083] Therefore, we obtain the equation for the normal MN:

[0084]

[0085] Since the normal MN intersects the abscissa (τ=0) at point N, combining equation (3) with τ=0 yields the abscissa of the center N of the stress Mohr's circle:

[0086]

[0087] The coordinates of the center N of the stress Mohr circle are:

[0088] In triangle ΔMFN, the ordinate of the point of tangency M can be represented by the magnitude of the line segment MF, i.e., |MF| = τ. M |FN|=k t τ M ,but Let be the radius of the stress Mohr circle, then the equation of the stress Mohr circle can be expressed as:

[0089]

[0090] Furthermore, the origin of the shear strength-normal stress rectangular coordinate system, i.e., the σ-τ rectangular coordinate system, is selected at the zero stress point, and the stress on both the vertical and horizontal axes is zero.

[0091] Furthermore, the specific method for calculating the predicted value of active earth pressure on loess in step four, based on the stress Mohr's circle equation, is as follows:

[0092] Since the equation of line BC is τ = σtani, and the stress Mohr's circle intersects this line at points B and C, we can solve the problem by simultaneously solving the equations of the stress Mohr's circle and line BC:

[0093]

[0094] Since the x-coordinate of point B is σ B The x-coordinate of point C is equal to the x-coordinate of point A, and both can be represented as σ. A Based on this, by solving equation (6), σ can be obtained respectively. A With σ B The value of, that is:

[0095]

[0096]

[0097] Therefore, according to p A p B respectively with σ A σ B From the geometric relationships, we can obtain:

[0098]

[0099]

[0100] And because p A p B The expressions all contain σ M Therefore, by p A Expression (9) can be inversely solved for σ. M ,have to:

[0101]

[0102] By p B Expression (10) can be solved by inversely solving σ. M ,have to:

[0103]

[0104] To simplify the expression of the formula, let:

[0105]

[0106] Then the above two equations (11) and (12) can be simplified as follows:

[0107]

[0108]

[0109] Since equations (14) and (15) are equal, we have:

[0110]

[0111] Therefore, based on the above formula, we can obtain the following:

[0112]

[0113] If the inclination angle of the backfill soil behind the wall is expressed as i, then when the backfill soil behind the wall reaches the active limit equilibrium state, the larger stress is the stress applied parallel to the inclined ground surface, i.e., the surface of the backfill soil behind the wall, which is p. A =γhcosi, the smaller stress is the stress applied to the vertical surface behind the retaining wall, and its direction is parallel to the ground surface, therefore p B =p a Then, from equation (16), the formula for predicting and calculating the active earth pressure when the inclination angle of the backfill soil behind the wall is i is:

[0114]

[0115] Where: p a γ represents the predicted value of active earth pressure when the inclination angle of the backfill slope is i; γ represents the unit weight of the backfill soil; h represents the depth of the backfill soil; and i represents the inclination angle of the backfill slope, which is a design parameter for the backfill surface of the retaining wall project.

[0116] In practical applications, this method is used to predict and calculate the active earth pressure of loess when the backfill surface tilts, providing support for the stability design of loess slopes and retaining wall projects.

[0117] Example 3:

[0118] This embodiment selects a retaining wall project with a height of 15m. The backfill behind the wall is loess, and the surface slope angle (design index) of the backfill is i = 20°. The test index is: loess unit weight γ = 17kN / m 3 Cohesion c = 40 kPa, internal friction angle φ = 20°, and tensile strength σt =25kPa. Based on the design and testing parameters of the project, soil depths h of 0m, 0.5m, 1.0m, 1.5m, 2.0m, 2.5m, 3.0m, 4.0m, 5.0m, 7.0m, 8.0m, 10.0m, 12.5m, and 15.0m are substituted into formula (17) to calculate the soil pressure at different depths h as -23.44kPa, -23.43kPa, -22.46kPa, -20.81kPa, -18.65kPa, -16.08kPa, -13.18kPa, -6.59kPa, 0.83kPa, 17.45kPa, 26.44kPa, 45.46kPa, 70.73kPa, and 97.26kPa, respectively. Based on the different depths and corresponding predicted soil pressure values, the following plots are drawn. Figure 3 It can provide earth pressure loads for the design and verification of loess slopes and retaining wall projects.

[0119] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

[0120] Where there is no conflict, those skilled in the art can combine the relevant technical features in the above examples according to the actual situation to achieve the corresponding technical effects. Specific details of the various combinations will not be elaborated here.

[0121] It should be noted that all directional indications (such as up, down, left, right, front, back, etc.) in the embodiments of the present invention are only used to explain the relative positional relationship and movement of each component in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indication will also change accordingly.

[0122] The above description is merely a preferred embodiment of the present invention. The present invention is not limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein. Any simple modifications, equivalent variations, and alterations made to the above embodiments based on the technical essence of the present invention shall still fall within the scope of the present invention.

Claims

1. A method for predicting and calculating the active earth pressure of loess when the ground surface is inclined behind a wall, characterized in that, It comprises the following steps, Step one: obtaining the parameter values of the loess to be predicted, including the shear strength, normal stress, cohesion strength, internal friction angle, tensile strength and loess unit weight; Step two: establishing a joint strength envelope; Step three: establishing a shear strength-normal stress rectangular coordinate system, and solving the stress Mohr circle equation; Step four: according to the stress Mohr circle equation solved in step three, solving the predicted value of the loess active earth pressure; Step five: applying the predicted value of the loess active earth pressure obtained in step four to the design of loess slope and retaining wall engineering; The equation of the joint strength envelope in step two is （1） wherein the test parameters are: τ is the shear strength; σ is the normal stress; c is the cohesion strength; is the internal friction angle; σ t is the tensile strength; The method for establishing a shear strength-normal stress rectangular coordinate system and solving the Mohr circle equation in step three is as follows: In the anti-shear strength-normal stress rectangular coordinate system, i.e. σ - τ In the rectangular coordinate system, if the soil body behind the wall is in the active limit equilibrium state, then the stress state of any point in the soil body corresponds to the stress Mohr circle tangent to the joint strength envelope; draw two rays OA and OC with the coordinate origin O on the orthogonal axes, both of which have an angle i with the σ-axis; the ray OA intersects the stress Mohr circle at two points, and the intersection point far from the origin O is defined as point A; the ray OC intersects the stress Mohr circle at points B and C, and the intersection point far from the origin O is defined as point C, at this time, the line segment OA represents the greater stress in the active limit state , and the line segment OB represents the stress parallel to the ground surface, i.e. the earth pressure acting on the vertical surface of the wall back , , i.e. the predicted active earth pressure of the soil body behind the wall with the joint strength theory as the premise and the wall back fill soil body inclined angle i predicted value Suppose M is the tangent point of the stress Mohr circle tangent to the joint strength envelope, N is the center of the stress Mohr circle on the σ axis, the line segment MF is orthogonal to the σ axis, and the foot is F; Since point M is on the joint strength envelope, assume the horizontal coordinate of point M is , the vertical coordinate of point M can be obtained by the equation of the joint strength envelope in step two , i.e. the coordinates of point M are , . The slope of the tangent line at point M can be obtained by differentiating the joint strength envelope : （2） Thus, the equation expression of the normal line MN is obtained: （3） Because the normal MN is perpendicular to the x-coordinate ( If they intersect at point N, then equation (3) and... By combining the equations, we can obtain the x-coordinate of the center N of the stress Mohr circle: （4） The coordinates of the center N of the stress Mohr circle are ( ,0); In triangle ΔMFN, the ordinate of the point M can be expressed by the length of the segment MF, i.e. , , is the size of the stress Mohr circle radius, and the stress Mohr circle equation can be expressed as: （5）； The specific method for solving the predicted value of the loess active earth pressure according to the stress Mohr circle equation in step four is as follows: Because the equation of straight line BC is , the stress Mohr circle intersects with this straight line at points B and C. Thus, by combining the equation of the stress Mohr circle with the equation of straight line BC, the following can be obtained: （6） Since the abscissa of B is , the abscissa of C is equal to that of A, and can be expressed as , according to which the equation (6) is solved, the values of and are obtained, i.e. （7） （8） Thus, according to , the geometric relationship with , , respectively, it can also be obtained: （9） （10） And because , The expression contains Therefore, Expression (9) can be inversely solved. ,have to: （11） By The expression (10) can be solved , and we get: （12） In order to facilitate and simplify the expression of the formula, let （13） Then the above two formulas (11) and (12) can be simplified as （14） （15） From (14) and (15), we have , Thus, according to the above formula, the following can be solved: （16） If the size of the inclined angle of the soil body of the backfill behind the wall is expressed by i, in the case where the backfill behind the wall reaches the state of active limit equilibrium, the larger stress is the stress applied on the surface parallel to the inclined ground, i.e. the surface of the backfill behind the wall, and the smaller stress is the stress applied on the vertical surface of the back of the retaining wall, the direction of which is parallel to the ground, thus , and , then the formula for predicting the active earth pressure when the size of the inclined angle of the soil body of the backfill behind the wall is i is obtained from equation (16) as follows: （17） wherein: represents the predicted value of active earth pressure when the inclination of the wall backfill soil body is i, γ represents the specific weight of the wall backfill soil body, h represents the depth of the wall backfill soil body, and i represents the inclination of the wall backfill soil body.

2. The method for predicting and calculating the active earth pressure of loess when the ground surface is inclined behind a wall according to claim 1, characterized in that: The anti-shear strength-normal stress rectangular coordinate system is σ - τ The coordinate origin in the rectangular coordinate system is selected at the stress zero point, and the stresses of the longitudinal axis and the transverse axis are both zero.

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