Method for calculating deformation of reinforced stone blocking wall under rockfall impact based on reaming theory

Abstract

The invention discloses a method for calculating deformation of a reinforced stone blocking wall under rockfall impact based on a reaming theory. The method comprises the following steps: 1) calculating a mass point displacement field and a velocity field of a plastic zone and an elastic zone; 2) determining boundary stress conditions of an elastic-plastic stage and a plastic stage based on a mass point displacement field and a velocity field; (3) constructing an expression of resistance borne by the rockfall in the x direction; 4) calculating the resistance of the rockfall in the elastic-plastic stage and the plastic stage; 5) based on the resistance, constructing a motion differential equation of the rockfall; and 6) solving the motion differential equation of the rockfall to obtain the penetration depth of the rockfall in the elastic-plastic stage and the plastic stage. According to the method, the elastic-plastic property of the soil body is considered, the range of disturbing the soil body is strictly assumed, the influence of boundary conditions on the rockfall penetration process is emphasized, and relatively accurate penetration depth and impact force prediction can be provided.

Description

Technical Field

[0001] The present invention relates to the field of calculation of retaining wall deformation, specifically a method for calculating the deformation of a reinforced retaining wall under the impact of falling rocks based on the reaming theory. Background Technique

[0002] Flexible reinforced retaining walls are applied in the prevention and control projects of falling rock disasters due to their excellent coordinated deformation ability and impact resistance. However, due to the variety of reinforced retaining walls and the complex internal action mechanism of the structure, the structural design of retaining walls still relies on engineering experience and lacks clear theoretical guidance. When evaluating the impact deformation ability of a reinforced retaining wall, the penetration depth of a falling rock impacting the wall is a key parameter.

[0003] Although the prior art provides various methods for calculating the impact force and penetration depth of falling rocks, most of them are based on the research results in other fields and are fitted to the test results under specific conditions. These methods are applied to the engineering practice of retaining walls after being simplified and corrected, and there are few research results directly based on the impact process of retaining walls. Considering the high-speed and high-energy characteristics of falling rock impact, the geometric irregularity and randomness of the contact surface, as well as the multi-degree-of-freedom boundary conditions of the retaining wall, the characteristics of the geogrid-reinforced soil composite material, the deformation process at high strain rates, and the mixed energy dissipation mechanism, etc., when these methods are applied to retaining walls, there will be errors of varying degrees. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for calculating the deformation of a reinforced retaining wall under the impact of falling rocks based on the reaming theory, including the following steps:

[0005] 1) Select the horizontal plane passing through the centroid of the concave pit of the reinforced retaining wall as the calculation section; the concave pit is the concave pit left on the impacted wall surface after the falling rock impacts the retaining wall;

[0006] 2) Construct a falling rock simulation model;

[0007] 3) Use a finite sphere to simulate the front tip of the falling rock, and use the expansion process of the finite sphere in the disturbed area to simulate the process of the falling rock impacting the retaining wall and forming a concave pit in the soil body; the expansion process of the finite sphere includes an elastic-plastic stage and a plastic stage;

[0008] 4) Divide the disturbed area soil body in the area where the concave pit is located into a plastic zone and an elastic zone, and calculate the particle displacement field and velocity field of the plastic zone and the elastic zone;

[0009] 5) Based on the particle displacement field and velocity field, determine the boundary stress conditions of the elastic-plastic stage and the plastic stage;

[0010] 6) Construct an expression for the resistance of the falling rock in the x direction;

[0011] 7) Calculate the resistance on the rockfall in the elastoplastic stage and the plastic stage;

[0012] 8) Based on the resistance, construct the motion differential equation of the rockfall;

[0013] 9) Solve the motion differential equation of the rockfall to obtain the penetration depth of the rockfall in the elastoplastic stage and the plastic stage.

[0014] Furthermore, in step 2), the steps of constructing the rockfall simulation model include:

[0015] 2.1) Divide the rockfall into a penetration tip and a cylinder; where the length of the penetration tip is L1, and the penetration tip has an elliptical cross-section; the length of the cylinder is L2 and the diameter is d p , and the cross-section is equal in size to the largest cross-section of the penetration tip;

[0016] 2.2) Use multiple spheres with a radius of r c that are inscribed in the surface of the rockfall to simulate the penetration tip; where the centers O of these spheres are the intersection points of the normal lines of the tangent points on the sphere surface and the axis of the rockfall;

[0017] 2.3) Construct the rockfall simulation model, including the spheres simulating the penetration tip and the cylinder.

[0018] Furthermore, in step 3), when the expansion process is in the elastoplastic stage, the finite sphere is divided into a pit cavity, a plastic zone, and an elastic zone; the radius of the pit cavity satisfies 0 ≤ r ≤ r c ; the radius of the plastic zone satisfies r c < r ≤ r p ; the radius of the elastic zone satisfies r p < r ≤ R;

[0019] When the expansion process is in the plastic stage, the finite sphere is divided into a pit cavity and a plastic zone;

[0020] Among them, the cavity radius r c and the radius R of the finite sphere are respectively as follows:

[0021] r c = y / cosα (1)

[0022] R = (T e - L p + x + y·tanα) / sinα (2)

[0023] In the formula, α is the angle between the normal line of the tangent point and the y-axis, L p is the penetration depth of the rockfall, T e is the thickness of the calculation section of the rockfall retaining wall, v p is the penetration speed of the rockfall; x, y are the coordinates of the contact point between the sphere surface and the pit;

[0024] The expansion velocity v of the pit cavity c is as follows:

[0025]

[0026] Furthermore, in step 4), the steps of calculating the particle displacement field and velocity field in the plastic zone and elastic zone include:

[0027] 4.1) Construct the stress-strain relationship, that is:

[0028]

[0029] In the formula, c, are the cohesion and internal friction angle of the soil; E is the elastic modulus of the soil; ν is the Poisson's ratio of the soil; σ r , σ θ are the radial and circumferential stresses; ε r , ε θ are the radial and circumferential strains;

[0030] 4.2) Construct the displacement field equation of the incompressible soil element, that is:

[0031]

[0032] (r - u) 3 = r 3 + f(r), r c ≤ r ≤ R (7)

[0033] In the formula, f(r) is a function to be determined; ρ, ρ' are the densities of the soil before and after displacement; u is the radial displacement of the soil element; r is the radius;

[0034] 4.3) Let r = r c , u = r c , substitute into formula (7) to obtain the particle displacement field in the plastic zone and elastic zone, that is:

[0035]

[0036] 4.4) In the Euler coordinates, construct the particle velocity field in the plastic zone, that is:

[0037]

[0038] 4.5) Combine formulas (8) and (9) to obtain the particle velocity field in the elastic zone and plastic zone, that is:

[0039]

[0040] Specifically, in the elastic zone, the displacement equation of the soil mass still satisfies Equation 7. Since the displacement and velocity of the soil mass are continuous at the elastic-plastic boundary:

[0041]

[0042] Therefore, the displacement field and velocity field of the soil mass particles in the elastic zone are the same as those in the plastic zone.

[0043] Furthermore, in step 5), the steps to determine the boundary stress conditions in the elastic-plastic stage and the plastic stage include:

[0044] 5.1) Construct the motion equation of the soil element within a finite sphere, that is:

[0045]

[0046] where σ y is the yield stress of the soil mass, and σ θ is the lateral earth pressure at the current calculation section;

[0047] 5.2) In the plastic zone, substitute Equations (10) and (12) into Equation (11) to obtain:

[0048]

[0049] 5.3) Integrate Equation (13) and consider the boundary stress conditions of the cavity wall r = r c , σ r = σ rc to obtain:

[0050]

[0051] 5.4) At the elastic-plastic boundary, construct the boundary stress conditions, that is:

[0052]

[0053] where σ rp is the elastic-plastic boundary stress;

[0054] 5.5) Substitute Equation (15) into Equation (14) to obtain the radial stress σ rc on the cavity wall, that is:

[0055]

[0056] Among them, the radial stress at the elastic-plastic boundary is as follows:

[0057]

[0058] 5.6) In the elastic zone, construct the relationship between the radial and circumferential strains and the displacement of the particles σ rp, that is:

[0059]

[0060] 5.7) Substitute Equation (8) into Equation (18) to obtain:

[0061]

[0062] 5.8) Expand Equation (19) according to the Taylor series and neglect the high-order terms, then substitute it into Equation (4) to obtain:

[0063]

[0064] 5.9) Substitute the above equation and Equation (10) into Equation (12) to get:

[0065]

[0066] 5.10) Integrate Equation (21) to obtain the radial stress field in the elastic region, that is:

[0067]

[0068] 5.11) Based on the stress continuity condition, let r = r p , substitute Equation (22) into Equation (16) to obtain the radial stress on the cavity wall in the elastic-plastic stage, that is:

[0069]

[0070] 5.12) Combine Equation (12) and (20) to obtain:

[0071]

[0072] 5.13) Substitute Equation (25) into Equation (24) to obtain:

[0073]

[0074] 5.14) Calculate the cavity radius in the plastic stage, that is:

[0075]

[0076] 5.15) Use Equation (16) to represent the soil stress field in the plastic region, substitute the stress boundary condition r = R, σ r = 0 into Equation (16) to obtain the radial stress on the cavity wall in the plastic stage, that is:

[0077]

[0078] Furthermore, in step 6), the steps of constructing the expression for the resistance force suffered by the falling rock in the x direction include:

[0079] 6.1) Construct the penetration depth of the falling rock as L p , that is:

[0080]

[0081] 6.2) Let y'(x) = tanα, and transform Equation (29) to obtain:

[0082]

[0083] In the formula, y'(x) is the slope of the tangent line of the falling rock contour curve at this point;

[0084] 6.3) Calculate the penetration depth L of the falling rock p = L pu ; L pu is the critical penetration depth;

[0085] 6.4) Construct the expression of the resistance suffered by the falling rock in the x direction, that is:

[0086]

[0087] In the formula, σ r (x, L p ) is the radial stress suffered by the particle at the coordinate x on the surface of the falling rock when the penetration depth is L p .

[0088] Furthermore, in step 6.3), when the falling rock is an oval falling rock, the critical penetration depth is as follows:

[0089]

[0090] In the formula, L1 is the length of the penetration tip with an elliptical cross-section;

[0091] When the falling rock is a hemispherical falling rock, the critical penetration depth is as follows:

[0092]

[0093] Furthermore, the resistance suffered by the falling rock in the elastic-plastic stage and the plastic stage is as follows respectively:

[0094]

[0095] Among them, the parameters A1, B1, C1, D1, A, A2, B2, C2, D2 are as follows respectively:

[0096]

[0097] Furthermore, the motion differential equation of the falling rock is as follows:

[0098]

[0099] where m p is the mass of the falling rock.

[0100] The technical effect of the present invention is beyond doubt. The present invention proposes a calculation method based on the spherical cavity expansion theory, which is particularly suitable for predicting the impact deformation of falling rocks with a rounded contour line (such as spherical and oval). Considering the elastoplastic properties of the soil, there are strict assumptions about the range of the disturbed soil, and the influence of boundary conditions on the penetration process of the falling rock is emphasized, which can provide relatively accurate predictions of the penetration depth and impact force.

[0101] In this calculation method, the front tip of the falling rock is simulated by a series of spheres that are tangent to the surface of the falling rock. The process of the falling rock forming a pit in the soil is regarded as a series of spheres gradually expanding. The radius of the central pit sphere increases from zero, and the plastic zone radius of the spherical deformation zone outside the pit wall and the boundary radius of the finite sphere also gradually increase. The radius of the finite sphere is the distance from the intersection of the normal line of the tangent point and the back boundary line of the rockfall retaining wall to the center of the sphere. Considering the influence of the free boundary of the finite-thickness wall, the error is reduced.

[0102] In this calculation method, the disturbed soil around the spherical-like pit is divided into a plastic zone and an elastic zone. The expansion process of the finite sphere is divided into two stages: an elastoplastic stage and a plastic stage. The soil is regarded as an incompressible elastoplastic material, and its stress-strain relationship satisfies Hooke's law in the elastic zone and follows the Mohr-Coulomb strength yield criterion in the plastic zone, reducing the difficulty of solving the calculation.

[0103] The calculation process of the present invention separately describes the cavity expansion process of the two stages. For the two-stage expansion process, the calculation expressions of the penetration resistance and penetration depth of the falling rock are derived and given, and the radial stress received by the mass point is calculated in stages. The deformation occurring during the impact process of the falling rock is fully analyzed, improving the calculation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0104] Figure 1 is the horizontal calculation cross-section view of the falling rock impacting the rockfall retaining wall of the present invention;

[0105] Figure 2 is the geometric model view of the interaction between the falling rock and the soil of the present invention;

[0106] Figure 3 is the soil deformation zoning view of the present invention;

[0107] Figure 4 is the schematic diagram of the displacement of the soil element of the present invention;

[0108] Figure 5 is the calculation roadmap of the present invention;

[0109] Figure 6 It is a comparison chart of the calculation results of the reaming method. Specific implementation manners

[0110] The present invention will be further described below in conjunction with embodiments, but it should not be understood that the above-mentioned subject scope of the present invention is limited to the following embodiments. Without departing from the above technical idea of the present invention, various substitutions and changes made according to ordinary technical knowledge and customary means in the art shall be included within the protection scope of the present invention.

[0111] Embodiment 1:

[0112] Refer to Figures 1 to 6 , a calculation method for the deformation of a reinforced rockfall retaining wall under the impact of a rockfall based on the reaming theory, includes the following steps:

[0113] 1) Select the horizontal plane passing through the centroid of the concave pit of the reinforced rockfall retaining wall as the calculation section; the concave pit is the concave pit left on the impacted wall surface after the rockfall impacts the retaining wall;

[0114] 2) Construct a rockfall simulation model;

[0115] 3) Use a finite sphere to simulate the front tip of the rockfall, and use the expansion process of the finite sphere in the disturbed area to simulate the process of the rockfall impacting the retaining wall and forming a concave pit in the soil body; the expansion process of the finite sphere includes an elastoplastic stage and a plastic stage;

[0116] 4) Divide the disturbed area soil body where the concave pit is located into a plastic zone and an elastic zone, and calculate the particle displacement field and velocity field of the plastic zone and the elastic zone;

[0117] 5) Based on the particle displacement field and velocity field, determine the boundary stress conditions of the elastoplastic stage and the plastic stage;

[0118] 6) Construct an expression for the resistance of the rockfall in the x direction;

[0119] 7) Calculate the resistance of the rockfall in the elastoplastic stage and the plastic stage;

[0120] 8) Based on the resistance, construct the motion differential equation of the rockfall;

[0121] 9) Solve the motion differential equation of the rockfall to obtain the penetration depth of the rockfall in the elastoplastic stage and the plastic stage.

[0122] Embodiment 2:

[0123] A calculation method for the deformation of a reinforced rockfall retaining wall under the impact of a rockfall based on the reaming theory, the technical content is the same as that of Embodiment 1. Further, in step 2), the steps of constructing a rockfall simulation model include:

[0124] 2.1) Divide the falling rock into a penetration tip and a cylinder; where the length of the penetration tip is L1, the penetration tip has an elliptical cross-section; the length of the cylinder is L2 and the diameter is d p , and the cross-section is equal in size to the maximum cross-section of the penetration tip;

[0125] 2.2) Use multiple spheres with a radius of r c that are inscribed in the surface of the falling rock to simulate the penetration tip; where the centers of the spheres O are the intersection points of the normal lines of the tangent points on the sphere surface and the axis of the falling rock;

[0126] 2.3) Construct a falling rock simulation model, including the spheres simulating the penetration tip and the cylinder.

[0127] Example 3:

[0128] A method for calculating the deformation of a reinforced rockfall retaining wall under the impact of a falling rock based on the reaming theory, the technical content is the same as any one of Examples 1-2. Further, in step 3), when the expansion process is in the elastoplastic stage, the finite sphere is divided into a pit cavity, a plastic zone, and an elastic zone; the radius of the pit cavity satisfies 0 ≤ r ≤ r c ; the radius of the plastic zone satisfies r c < r ≤ r p ; the radius of the elastic zone satisfies r p < r ≤ R;

[0129] When the expansion process is in the plastic stage, the finite sphere is divided into a pit cavity and a plastic zone;

[0130] Among them, the cavity radius r c and the radius R of the finite sphere are respectively as follows:

[0131] r c = y / cosα (1)

[0132] R = (T e - L p + x + y · tanα) / sinα (2)

[0133] In the formula, α is the angle between the normal line of the tangent point and the y-axis, L p is the penetration depth of the falling rock, T e is the thickness of the calculation section of the rockfall retaining wall, v p is the penetration speed of the falling rock; see Figure 2 , and the coordinate axes are established with the front vertex of the falling rock as the coordinate origin. The function y = y(x) is the curve equation of this section of the falling rock cross-section, which corresponds to a local two-dimensional rectangular coordinate system with the coordinate origin at the vertex of the falling rock. The x direction is opposite to the direction of the falling rock speed, and the y direction is perpendicular to the direction of the falling rock speed. x and y are the coordinates of the contact point between the sphere surface and the pit, that is, Figure 2 the point where the tangent line is drawn on the sphere surface in

[0134] The expansion velocity v of the pit cavity c is as follows:

[0135]

[0136] Example 4:

[0137] A calculation method for the deformation of a reinforced rockfall retaining wall under rockfall impact based on the hole expansion theory, the technical content is the same as any one of Examples 1-3. Further, in step 4), the steps of calculating the particle displacement field and velocity field in the plastic zone and elastic zone include:

[0138] 4.1) Construct the stress-strain relationship, that is:

[0139]

[0140] In the formula, c, is the cohesion and internal friction angle of the soil; E is the elastic modulus of the soil; ν is the Poisson's ratio of the soil; σ r , σ θ are the radial and circumferential stresses; ε r , ε θ are the radial and circumferential strains;

[0141] 4.2) Construct the displacement field equation of the incompressible soil element, that is:

[0142]

[0143] (r - u) 3 = r 3 + f(r), r c ≤ r ≤ R (7)

[0144] In the formula, f(r) is a function to be determined; ρ, ρ' are the densities of the soil before and after displacement; u is the radial displacement of the soil element; r is the radius;

[0145] 4.3) Let r = r c , u = r c , substitute into formula (7) to obtain the particle displacement field in the plastic zone and elastic zone, that is:

[0146]

[0147] 4.4) Under the Euler coordinates, construct the particle velocity field in the plastic zone, that is:

[0148]

[0149] 4.5) Combine formulas (8) and (9) to obtain the particle velocity field in the elastic zone and plastic zone, that is:

[0150]

[0151] Specifically, in the elastic region, the displacement equation of the soil mass still satisfies Equation 7. Since the displacement and velocity of the soil mass are continuous at the elastic-plastic boundary:

[0152]

[0153] Therefore, the displacement field and velocity field of the soil mass particles in the elastic region are the same as those in the plastic region.

[0154] Example 5:

[0155] A calculation method for the deformation of a reinforced rockfall retaining wall under rockfall impact based on the cavity expansion theory, the technical content is the same as any one of Examples 1-4. Further, in step 5), the steps of determining the boundary stress conditions in the elastic-plastic stage and the plastic stage include:

[0156] 5.1) Construct the motion equation of the soil element within a finite sphere, that is:

[0157]

[0158] where σ y is the yield stress of the soil mass, and σ θ is the lateral earth pressure at the current calculation section;

[0159] 5.2) In the plastic region, substitute Equations (10) and (12) into Equation (11) to obtain:

[0160]

[0161] 5.3) Integrate Equation (13) and consider the boundary stress conditions of the cavity wall r = r c , σ r = σ rc to obtain:

[0162]

[0163] 5.4) At the elastic-plastic boundary, construct the boundary stress conditions, that is:

[0164]

[0165] where σ rp is the elastic-plastic boundary stress;

[0166] 5.5) Substitute Equation (15) into Equation (14) to obtain the radial stress σ rc on the cavity wall, that is:

[0167]

[0168] Among them, the radial stress σ at the elastic-plastic boundaryrp As follows:

[0169]

[0170] 5.6) In the elastic region, establish the relationship between radial and circumferential strains and the displacement of the mass point, that is:

[0171]

[0172] 5.7) Substitute Equation (8) into Equation (18) to obtain:

[0173]

[0174] 5.8) Expand Equation (19) according to the Taylor series and neglect the higher-order terms, then substitute it into Equation (4) to obtain:

[0175]

[0176] 5.9) Substitute the above equation and Equation (10) into Equation (12) to get:

[0177]

[0178] 5.10) Integrate Equation (21) to obtain the radial stress field in the elastic region, that is:

[0179]

[0180] 5.11) Based on the stress continuity condition, let r = r p , substitute Equation (22) into Equation (16) to obtain the radial stress on the cavity wall in the elastic-plastic stage, that is:

[0181]

[0182] 5.12) Combine Equation (12) and (20) to obtain:

[0183]

[0184] 5.13) Substitute Equation (25) into Equation (24) to obtain:

[0185]

[0186] 5.14) Calculate the cavity radius in the plastic stage, that is:

[0187]

[0188] 5.15) Use Equation (16) to represent the soil stress field in the plastic region, and apply the stress boundary condition r = R, σ rSubstituting \(= 0\) into Equation (16), the radial stress on the cavity wall in the plastic stage is obtained, i.e.:

[0189]

[0190] Example 6:

[0191] A calculation method for the deformation of a reinforced rockfall retaining wall under rockfall impact based on the cavity expansion theory, the technical content is the same as any one of Examples 1 - 5. Further, in step 6), the steps of constructing the expression of the resistance force received by the rockfall in the x - direction include:

[0192] 6.1) Construct the penetration depth of the rockfall as L p , i.e.:

[0193]

[0194] 6.2) Let \(y'(x)=\tan\alpha\), and transform Equation (29) to obtain:

[0195]

[0196] In the formula, \(y'(x)\) is the slope of the tangent line of the rockfall contour curve at this point;

[0197] 6.3) Calculate the penetration depth L of the rockfall p = L pu ; L pu is the critical penetration depth;

[0198] 6.4) Construct the expression of the resistance force received by the rockfall in the x - direction, i.e.:

[0199]

[0200] In the formula, \(\sigma\) r (x, L p ) is the radial stress received by the particle at the coordinate x on the surface of the rockfall when the penetration depth is L p .

[0201] Example 7:

[0202] A calculation method for the deformation of a reinforced rockfall retaining wall under rockfall impact based on the cavity expansion theory, the technical content is the same as any one of Examples 1 - 6. Further, in step 6.3), when the rockfall is an oval rockfall, the critical penetration depth is as follows:

[0203]

[0204] In the formula, L1 is the length of the penetration tip with an elliptical cross - section;

[0205] When the rockfall is a hemispherical rockfall, the critical penetration depth is as follows:

[0206]

[0207] Example 8:

[0208] The deformation calculation method of the reinforced rockfall retaining wall under the impact of rockfall based on the cavity expansion theory, the technical content is the same as any one of Examples 1-7. Further, the resistances suffered by the rockfall in the elastic-plastic stage and the plastic stage are respectively as follows:

[0209]

[0210] Among them, the parameters A1, B1, C1, D1, A, A2, B2, C2, D2 are respectively as follows:

[0211]

[0212]

[0213] Example 9:

[0214] The deformation calculation method of the reinforced rockfall retaining wall under the impact of rockfall based on the cavity expansion theory, the technical content is the same as any one of Examples 1-8. Further, the motion differential equation of the rockfall is as follows:

[0215]

[0216] In the formula, m p is the mass of the rockfall.

[0217] Example 10:

[0218] The deformation calculation method of the reinforced rockfall retaining wall under the impact of rockfall based on the cavity expansion theory. This method is proposed based on the spherical cavity expansion theory and is applicable to the impact deformation calculation of spherical and ellipsoidal rockfalls such as spherical and ellipsoidal. The calculation model does not consider the penetration failure situation. This method assumes the process of the rockfall impacting the reinforced rockfall retaining wall as the penetration process of a rigid sphere into finite soil, and this penetration process is divided into two stages: In the first stage, the impact kinetic energy of the rockfall is small, and the plastic range of the expanded deformed soil is within the free boundary surface of the wall back. There are both plastic zones and elastic zones in the rockfall retaining wall. In the second stage, the impact kinetic energy of the rockfall is large enough, and the boundary of the plastic zone of the expanded soil reaches the wall back surface and continues to spread outward. Only the plastic zone remains in the rockfall retaining wall.

[0219] First, select the horizontal plane passing through the centroid of the pit as the calculation section. After the rockfall impacts the rockfall retaining wall, a pit with a regular shape (hemispherical or conical) is left on the impacted wall surface, and the whole pit is axisymmetric with any plane perpendicular to the wall surface. The soil in the disturbed deformation zone inside the wall radiates roughly in all directions, but the vertical direction is restricted by the horizontal reinforcing bars laid by each reinforcing body, and the deformation in this direction is not as obvious as that in the horizontal direction. The boundary of the plastic deformation zone will also reach the wall back surface first in the horizontal direction.

[0220] The rockfall is divided into two sections. One section is the penetration tip with an elliptical cross-section and a length of L1, which is the main research object in contact with the soil during the penetration process. Its cross-section is mostly elliptical, corresponding to a conical or oval rockfall bulge, and can also have an increased curvature to become circular, corresponding to a spherical rockfall. The function y = y(x) is the curve equation of the cross-section of this section of the rockfall, which corresponds to a local two-dimensional rectangular coordinate system with the origin at the vertex of the rockfall. The x-direction is opposite to the direction of the rockfall velocity, and the y-direction is perpendicular to the direction of the rockfall velocity. The other section is a cylinder with a length of L2 and a diameter of d p whose cross-section is equal to the maximum cross-section of the front tip. Generally, only when the impact kinetic energy of the rockfall is large enough will this section enter the soil.

[0221] In the calculation, the front tip of the rockfall is simulated by a series of spheres with a radius of r c that are inscribed in the rockfall surface (see the dashed circles inside the rockfall in the figure). The center of the sphere O is the intersection of the normal line of the tangent point on the sphere surface and the axis of the rockfall (i.e., the x-axis). The process of the rockfall forming a pit in the soil can be regarded as the process of this series of spheres gradually expanding. The radius of the central pit sphere increases from zero to r c , and the plastic zone radius r p of the spherical deformation zone outside the pit wall, and the boundary radius R of the finite sphere also gradually increases. The radius R of the finite sphere is the distance from the intersection of the normal line of the tangent point and the boundary line of the retaining wall back to the center of the sphere.

[0222] Figure 2 For other parameters in , the angle α is the angle between the normal line of the tangent point and the y-axis, L p is the penetration depth of the rockfall, T e is the thickness of the calculation section of the retaining wall, and v p is the penetration velocity of the rockfall.

[0223] For the soil in the disturbed area around the pit, according to the basic assumptions of the spherical cavity theory, it is divided into a plastic zone and an elastic zone. The expansion process of this finite sphere is divided into two stages: The first stage is the elastoplastic stage. When the plastic zone radius r p < R, the sphere is divided into a pit cavity (0 ≤ r ≤ r c ), a plastic zone (r c < r ≤ r p ), and an elastic zone (r p < r ≤ R). As the rockfall penetrates, the expanding sphere moves towards the retaining wall back. When the boundary of the plastic zone reaches the surface of the retaining wall back, i.e., r = R, the first stage ends. The second stage is the plastic stage. In this stage, the sphere only has a pit cavity and a plastic zone (r c ≤ r ≤ R).

[0224] When the penetration depth of the rockfall is L pAt this time, the cavity radius r corresponding to the position x on the surface of the falling rock c and the radius R of the finite sphere are respectively:

[0225] r c = y / cosα (1)

[0226] R = (T e - L p + x + y·tanα) / sinα (2)

[0227] The expansion velocity v of the cavity c is equal to the radial velocity of the falling rock surface, and its relationship with the penetration velocity v of the current falling rock p is:

[0228]

[0229] For the deformed soil mass within the finite sphere, using the spherical coordinate system, the Euler coordinate is r, the radial displacement u of the particle, and the radial expansion velocity v c Taking the normal direction of the tangent point as positive, the radial and circumferential stresses σ r 、σ θ and the radial and circumferential strains ε r 、ε θ are all positive in compression. To reduce the difficulty of solution, the soil mass is regarded as an incompressible elastoplastic material, and its stress-strain relationship satisfies Hooke's law in the elastic region and follows the Mohr-Coulomb strength yield criterion in the plastic region:

[0230]

[0231] In the formula, c, is the cohesion and internal friction angle of the soil mass; E is the elastic modulus of the soil mass; ν is the Poisson's ratio of the soil mass. Since it is assumed that the soil mass is incompressible, here ν = 0.5.

[0232] For the incompressible soil element, in the radial displacement plane, due to mass conservation, there is:

[0233]

[0234] In the formula, ρ, ρ' are the densities of the soil mass before and after displacement. If the change in density is ignored, the above formula can be transformed into:

[0235]

[0236] Integrating both sides of the above formula with respect to r, the equation of the displacement field of the soil element is obtained:

[0237] (r - u) 3 = r 3 + f(r), r c≤ r ≤ R (7)

[0238] Where f(r) is a function to be determined and can be determined by the boundary conditions of the deformation zone.

[0239] In the plastic zone, there are boundary conditions on the cavity wall, r = r c , u = r c , Substituting into the above formula, the particle displacement field in the plastic zone is obtained:[[]]END]]

[0240]

[0241] In the Eulerian coordinates, the following relationship exists between the velocity and displacement of the particle:[[]]END]]

[0242]

[0243] Combining Eqs. (8) and (9), the particle velocity field in the plastic zone can be obtained:[[]]END]]

[0244]

[0245] In the elastic zone, the displacement equation of the soil still satisfies Eq. (7). Since the displacement and velocity of the soil are continuous at the elastic-plastic boundary:[[]]END]]

[0246]

[0247] Therefore, the displacement field and velocity field of the soil particles in the elastic zone are the same as Eqs. (8) and (9).[[]]END]]

[0248] In addition, within a finite sphere, the motion of the soil element should satisfy the conservation of momentum, so there is:[[]]END]]

[0249]

[0250] Here, in order to simplify the subsequent formulas, the yield criterion in Eq. (4) is transformed:[[]]END]]

[0251]

[0252] Where σ y is the yield stress of the soil, and σ θ can take the lateral earth pressure of the current calculation section.[[]]END]]

[0253] Next, the analysis and calculation of the cavity expansion process are carried out in two stages respectively.[[]]END]]

[0254] 1) Elastic-plastic stage[[]]END]]

[0255] In the plastic zone (r c ≤ r ≤ r p ), substituting Eqs. (10) and (13) into Eq. (12), we can get:[[]]END]]

[0256]

[0257] Integrate the above equation and consider the boundary stress condition of the cavity wall r = r c , σ r = σ rc , to obtain

[0258]

[0259] On the plastic zone side of the elastoplastic boundary, there is a boundary stress condition:

[0260]

[0261] Substitute the above equation into Equation (15) to obtain the radial stress on the cavity wall as:

[0262]

[0263] Among them, the radial stress on the elastoplastic boundary can be directly obtained from Equation (13):

[0264]

[0265] r c / r p Determined according to the boundary stress condition in the subsequent elastic zone analysis.

[0266] In the elastic zone (r p <r ≤ R), at time t, the displacements of the particles with spatial coordinates r and r + dr are u and Their initial spatial coordinates are r - u and The following geometric relationship exists between them:

[0267]

[0268] Adopt the small strain theory in the elastic zone, u << r, and the above equation can take the first-order approximation:

[0269]

[0270] Substitute Equation (8) into the above equation to obtain:

[0271]

[0272] In the elastic zone, r c << r, expand the above equation according to the Taylor series and ignore the high-order terms, and substitute it into Equation (4) to obtain:

[0273]

[0274] Substitute the above equation and Equation (10) into Equation (12) to obtain:

[0275]

[0276] Integrating the above equation and applying the boundary conditions \(r = R\) and \(\sigma = 0\) on the back surface of the wall, the radial stress field in the elastic region is obtained as follows: r equals 0, and the radial stress field in the elastic region is obtained as:

[0277]

[0278] On the boundary of the elastic - plastic region, from the stress - continuity condition:

[0279]

[0280] Therefore, when \(r = r\) p , substituting Equation (24) into Equation (17), the radial stress on the cavity wall in the elastic - plastic stage is obtained:

[0281]

[0282] Here, combining Equations (13) and (22) gives:

[0283]

[0284] Substituting the above equation back into Equation (26), finally we have:

[0285]

[0286] 2) Plastic stage

[0287] When the radius \(r\) of the elastic - plastic boundary p equals \(R\), the elastic - plastic stage ends and the plastic stage begins. The cavity radius in this stage is:

[0288]

[0289] The soil stress field in the plastic region is still expressed by Equation (17). Substituting the stress boundary conditions \(r = R\) and \(\sigma = 0\) into Equation (17), the radial stress on the cavity wall in the plastic stage is obtained: r equals 0 into Equation (17), the radial stress on the cavity wall in the plastic stage is obtained:

[0290]

[0291] The above equation is applicable to the case where \(r\) c < R < ∞, \(r\) c ≥ \(r\) c1 of the situation.

[0292] For the two - stage expansion process described above, in the first stage, the plastic region has not reached the back surface of the wall. When the boundary of the plastic region just reaches the back surface of the wall, the first stage ends, and the penetration depth at this time is the critical penetration depth \(L\) puIf the remaining velocity of the falling rock is zero, the corresponding initial impact velocity of the falling rock is the critical impact velocity v pu .

[0293] When the penetration depth of the falling rock is L p , the cavity radius of the finite sphere at the surface x of the falling rock is r c , the radius of the outer boundary is R, and the corresponding radius of the plastic zone is r p = r p (x), and 0 ≤ x ≤ min(L p , L1). When the plastic zone boundary of a finite sphere corresponding to a certain cavity radius r c (x) just reaches the back surface of the wall, r p (x) = R. From equations (1), (2) and (27), we get:

[0294]

[0295] To reduce the calculation parameters, tanα can be replaced by the slope of the tangent line of the falling rock contour curve at this point, that is, y'(x) = tanα. So the above formula can be converted to:

[0296]

[0297] Assume that the plastic zone boundary of the finite sphere corresponding to x = x u reaches the back surface of the wall first. Then when x = x u , the first penetration stage ends, and L p in the above formula has a minimum value and is equal to the critical penetration depth L pu , that is

[0298]

[0299] In the formula, x u is related to the shape of the falling rock.

[0300] For an oval falling rock with a diameter of d p , its surface curve equation is:

[0301]

[0302] When x u = 0, from equation (33), we can get:

[0303]

[0304] For a hemispherical falling rock with a diameter of d p , its surface curve equation is:

[0305]

[0306] When x u = 0, from Equation (33), we can obtain:

[0307]

[0308] When the penetration depth of the falling rock is L p , the resistance of the falling rock in the x-direction can be calculated according to the surface stress of the falling rock that has fallen into the pit:[[]]

[0309]

[0310] In the formula, σ r (x, L p ) is the radial stress on the particle at the surface coordinate x of the falling rock when the penetration depth is L p , and it should be calculated in stages.[[]]

[0311] 1) Elastic-plastic stage (L p ≤ L pu )

[0312] When the boundary of the plastic zone has not reached the back surface of the retaining wall, σ r (x, L p ) should be calculated according to Equation (28). After substituting Equation (3) into Equation (28) and then substituting it into Equation (38), we have:[[]]

[0313]

[0314] In the formula, the subscript 1 represents the calculation parameters in the first stage. A1, B1, C1, D1 are related to the shape of the falling rock, the penetration depth, and the thickness of the rock retaining wall, and their respective expressions are as follows:[[]]

[0315]

[0316] In the formula, r c / R is calculated from Equations (1) and (2):[[]]

[0317]

[0318] 2) Plastic stage (L p > L pu )

[0319] First, assume a finite sphere corresponding to x * , whose plastic zone boundary just reaches the back surface of the retaining wall, and the shape of the falling rock is a conventional shape, that is, x u = 0. From Equation (32), we get:[[]]

[0320]

[0321] When x < x *When the finite sphere enters the plastic stage, σ r (x, L p ) should be calculated according to Equation (30), and substitute each equation into Equation (38):

[0322]

[0323] Wherein:

[0324]

[0325] After knowing the penetration resistance of the falling rock, considering dv p / dt = v p ·dv p / dL p , according to the momentum theorem, the differential equation of motion of the falling rock can be obtained:

[0326]

[0327] Substitute Equations (39) and (43) into the above formula to obtain the calculation expressions for the penetration depth in two stages respectively, and solve them with the help of numerical software.

[0328] Example 11:

[0329] Verification of the deformation calculation method of the reinforced rockfall retaining wall under the impact of rockfall based on the reaming theory is as follows:

[0330] The theoretical calculation values of spherical and ellipsoidal spherical rockfalls calculated by the present invention are in good agreement with the test values, as shown in Figure 6 . Most of the calculations of the penetration depth at home and abroad do not consider the shape of the rockfall and the distribution of the contact surface. The calculation methods for rockfalls of any shape are based on the fitting of test results under specific conditions, and most of them are the development results of other research fields related to rockfalls. This calculation method directly based on the interaction mechanism between the rockfall and the soil in the disturbed area during the process of the rockfall impacting and penetrating the retaining wall, and obtains a relatively accurate formula for calculating the penetration depth.

[0331] Compare the theoretical calculation results of each method in domestic and foreign literatures in Table 1. Different methods have different limitations and consider too few factors, resulting in large deviations in specific applications. The method of the present invention has obvious advantages in terms of special application scope and limiting conditions, is more in line with the actual situation of spherical rockfalls, and can provide more reliable calculation support for engineering design.

[0332] Table 1 Comparison of the results of various calculation methods

[0333]

[0334]

Claims

1. A method for calculating the deformation of reinforced rock retaining wall under rockfall impact based on hole expansion theory, characterized in that: The following steps are involved: 1) The horizontal plane of the reinforced rock retaining wall passing through the centroid of the pit is selected as the calculation section; the pit is the pit left on the impacted wall surface after the falling rocks hit the rock retaining wall; 2) Construct a rockfall simulation model; 3) Using a finite sphere to simulate the front tip of the falling rock, and using the expansion process of the finite sphere in the disturbance zone to simulate the process of the falling rock impacting the rock retaining wall and forming a pit in the soil; the expansion process of the finite sphere includes an elastic-plastic stage and a plastic stage; 4) Divide the disturbed soil in the pit area into plastic zone and elastic zone, and calculate the particle displacement field and velocity field in the plastic zone and elastic zone; 5) Based on the particle displacement field and velocity field, determine the boundary stress conditions of the elastic-plastic stage and the plastic stage; 6) Construct an expression for the resistance force on the falling rock in the x direction; 7) Calculate the resistance of rockfall in the elastic-plastic stage and the plastic stage; 8) Construct the differential equation of motion of the falling rock based on the resistance it encounters; 9) Solve the differential equation of motion of falling rocks and obtain the penetration depth of falling rocks in the elastic-plastic stage and the plastic stage.

2. The deformation calculation method of reinforced rock retaining wall under rockfall impact based on hole expansion theory according to claim 1 is characterized in that: In step 2), the steps of constructing a rockfall simulation model include: 2.1) The rockfall is divided into a penetration tip and a cylinder; the penetration tip is L1 in length and has an elliptical cross section; the cylinder is L2 in length and d in diameter. p , the cross section is equal to the maximum cross section of the penetration tip; 2.2) Use multiple inscribed holes with a radius of r that are inscribed on the rockfall surface. c The spheres simulate the penetration tip; the center O of these spheres is the intersection of the normal line of the tangent point on the sphere surface and the axis of the rockfall; 2.3) Construct a rockfall simulation model, including a sphere and a cylinder to simulate the penetration tip.

3. The deformation calculation method of reinforced rock retaining wall under rockfall impact based on hole expansion theory according to claim 1 is characterized in that: In step 3), when the expansion process is in the elastic-plastic stage, the finite sphere is divided into a concave cavity, a plastic zone, and an elastic zone; the radius of the concave cavity satisfies 0≤r≤r c ; The radius of the plastic zone satisfies r c <r≤r p ; The radius of the elastic zone satisfies r p <r≤R; When the expansion process is in the plastic stage, the finite sphere is divided into a concave cavity and a plastic zone; Where, the cavity radius r c and the radius R of the finite sphere are as follows: r c =y / cosα (1) R=(T e -L p +x+y·tanα) / sinα (2) Where α is the angle between the tangent point normal and the y-axis, L p is the penetration depth of the rockfall, T e is the thickness of the calculated section of the stone wall, v p is the penetration velocity of the falling rock; x, y are the coordinates of the contact point between the sphere surface and the pit; The expansion velocity of the pit cavity v c As shown below:

4. The method for calculating deformation of reinforced rock retaining wall under rockfall impact based on hole expansion theory according to claim 1 is characterized in that: In step 4), the steps of calculating the particle displacement field and velocity field in the plastic zone and the elastic zone include: 4.1) Construct the stress-strain relationship, namely: In the formula, c, is the cohesion and internal friction angle of the soil; E is the elastic modulus of the soil; ν is the Poisson's ratio of the soil; σ r , σ θ is the radial and hoop stress; ε r , ε θ is the radial and hoop strain; 4.2) Construct the displacement field equation of the incompressible soil unit, namely: (r-u) 3 =r 3 +f(r),r c ≤r≤R (7) Where f(r) is the function to be determined; ρ and ρ' are the densities of the soil before and after displacement; u is the radial displacement of the soil unit; r is the radius; 4.3) Let r = r c , u=r c , substituting into formula (7), we can obtain the particle displacement field in the plastic zone and the elastic zone, namely: 4.4) In Euler coordinates, construct the particle velocity field in the plastic zone, that is: 4.5) Combining equations (8) and (9), we can obtain the particle velocity field in the elastic region and the plastic region, namely:

5. The method for calculating deformation of reinforced rock retaining wall under rockfall impact based on hole expansion theory according to claim 1, characterized in that in step 5), the step of determining boundary stress conditions in the elastic-plastic stage and the plastic stage comprises: 5.1) Construct the motion equation of the soil unit in the finite sphere, namely: In the formula, σ y is the yield stress of the soil, σ θ is the lateral earth pressure of the current calculation section; 5.2) In the plastic zone, substituting equations (10) and (12) into equation (11), we obtain: 5.3) Integrate equation (13) and consider the boundary stress condition of the cavity wall r = r c , σ r =σ rc ,get: 5.4) At the elastic-plastic boundary, construct the boundary stress condition, namely: In the formula, σ rp is the elastic-plastic boundary stress; 5.5) Substituting equation (15) into equation (14), we obtain the radial stress σ on the cavity wall: rc ,Right now: Among them, the radial stress σ on the elastic-plastic boundary rp As shown below: 5.6) In the elastic region, the relationship between radial and annular strains and the displacement of the particle is constructed, that is: 5.7) Substituting equation (8) into equation (18), we obtain: 5.8) Expand equation (19) according to the Taylor series, ignore the higher-order terms, and substitute it into equation (4) to obtain: 5.9) Substituting the above equation and equation (10) into equation (12), we get: 5.10) Integrate equation (21) to obtain the radial stress field in the elastic region, namely: 5.11) Based on the stress continuity condition, let r = r p , substituting equation (22) into equation (16), we can obtain the radial stress on the cavity wall in the elastic-plastic stage, namely: 5.12) Combining equations (12) and (20), we obtain: 5.13) Substituting equation (25) into equation (24), we obtain: 5.14) Calculate the cavity radius in the plastic stage, that is: 5.15) Using equation (16) to express the soil stress field in the plastic zone, the stress boundary condition r = R, σ r = 0 Substituting into equation (16), we can obtain the radial stress σ on the cavity wall in the plastic stage. rc2 ,Right now:

6. The method for calculating deformation of reinforced rock retaining wall under rockfall impact based on hole expansion theory according to claim 1, characterized in that: In step 6), the step of constructing an expression for the resistance force on the falling rock in the x direction includes: 6.1) Construct the penetration depth of rockfall to be L p ,Right now: 6.2) Let y'(x) = tanα, transform equation (29) to obtain: Where y'(x) is the slope of the tangent line of the rockfall profile curve at that point; 6.3) Calculate the penetration depth L of the rockfall p =L pu ; L pu is the critical penetration depth; 6.4) Construct an expression for the resistance force on the falling rock in the x direction, namely: In the formula, σ r (x,L p ) is the penetration depth L p The radial stress on the particle at coordinate x on the rockfall surface is .

7. The method for calculating deformation of reinforced rock retaining wall under rockfall impact based on hole expansion theory according to claim 6, characterized in that: In step 6.3), when the rockfall is oval, the critical penetration depth L pu As shown below: Where L1 is the length of the penetration tip with an elliptical cross section; When the rockfall is hemispherical, the critical penetration depth is as follows:

8. The method for calculating deformation of reinforced rock retaining wall under rockfall impact based on hole expansion theory according to claim 1, characterized in that: The resistances encountered by falling rocks in the elastic-plastic stage and the plastic stage are as follows: Among them, the parameters A1, B1, C1, D1, A, A2, B2, C2, D2 are as follows:

9. The method for calculating deformation of reinforced rock retaining wall under rockfall impact based on hole expansion theory according to claim 1, characterized in that: The differential equation of motion for a falling rock is given by: In the formula, m p The mass of falling rocks.

Images