A method for evaluating the failure mechanism of the seismic ultimate bearing capacity of strip foundations adjacent to slopes

Abstract

The present invention provides a method for evaluating the failure mechanism of the seismic ultimate bearing capacity of a strip foundation adjacent to a slope. The method is characterized in that the seismic slip line field theory and boundary conditions under foundation load conditions are derived, and the finite difference method is used to solve the slip line field under earthquake action, as well as the limit state slope surface curve under earthquake action, referred to as the seismic ultimate slope surface curve; whether the seismic ultimate slope surface curve is tangent to the slope surface is used as the seismic ultimate state judgment standard; when the seismic ultimate slope surface curve and the slope surface have only one intersection point, the adjacent slope foundation is judged to be in a stable state; when the seismic ultimate slope surface curve and the slope surface have two intersection points, the adjacent slope foundation is judged to be in an unstable state; when the seismic ultimate slope surface curve is tangent to the slope surface, the foundation load is the seismic ultimate bearing capacity of the adjacent slope foundation.

Description

Technical Field

[0001] The present invention belongs to the field of foundation stability evaluation near a side slope, and in particular relates to a failure mechanism for evaluating the seismic ultimate bearing capacity of a strip foundation near a side slope. Background Art

[0002] Due to construction needs and environmental constraints, the foundations of bridge piers, retaining structures, and high-rise buildings are often located on the crest of adjacent slopes, forming adjacent slope foundations. Unlike horizontal foundations, the presence of a slope reduces the bearing capacity of adjacent slope foundations. Furthermore, in earthquake-prone areas, seismic action can cause sliding failures in adjacent slope foundations. Evaluating the ultimate seismic bearing capacity of adjacent slope foundations under earthquake action is a geotechnical engineering issue at the intersection of slopes and foundations.

[0003] Methods for evaluating the ultimate seismic bearing capacity of foundations adjacent to slopes under earthquakes include the limit equilibrium method, the bounded limit method, finite element limit analysis, and discrete element method. Determining the failure mechanism is a key issue. Limit equilibrium and limit analysis use hypothetical methods, such as the multi-wedge model and logarithmic spiral model, to determine the failure mechanism. Numerical analyses, such as finite element limit analysis, use optimization search techniques, such as adaptive meshing and discontinuous layout optimization, to determine the failure mechanism. The failure mechanism is an NP-type global optimization problem, and the application of the above methods suffers from insufficient accuracy. Strength reduction technology does not require the predetermination of the failure mechanism, but the instability criteria of this method, such as non-convergence of the calculation and sudden changes in the displacement value of the observation point, require subjective setting or judgment. Therefore, the evaluation of the ultimate seismic bearing capacity of foundations adjacent to slopes requires a new failure mechanism. Summary of the Invention

[0004] In view of the shortcomings of the existing technology, the purpose of the present invention is to provide a scientific and reasonable, high engineering practical value and good effect failure mechanism for evaluating the seismic ultimate bearing capacity of strip foundations adjacent to slopes.

[0005] To achieve the above object, the technical solution adopted by the present invention is:

[0006] 1. A failure mechanism for evaluating the ultimate seismic bearing capacity of a strip foundation adjacent to a slope, characterized by the following: 1) Seismic slip line field theory

[0007] The differential equation of soil stress under earthquake action is as follows:

[0008]

[0009] where σ x ,σ y represents the normal stress in the x and y directions, τ xy and τ yx represents the tangential stress in the x and y directions, fx =γ·k H , f y =γ·(1-k V ), γ represents the bulk density, k H and k v represents the seismic coefficient in the x and y directions;

[0010] The calculation formula of characteristic stress σ is derived using the Mohr-Coulomb criterion

[0011]

[0012] Where: c is the cohesive force, is the internal friction angle, σ1 is the maximum principal stress, and σ3 is the minimum principal stress;

[0013] The stress expression is as follows:

[0014]

[0015] Where θ is the angle between σ1 and the x-axis;

[0016] Substituting equations (3a) and (4) into equation (1a), and substituting equations (3b) and (4) into equation (1b), we can obtain the α-family and β-family characteristic line differential equations according to the characteristic line method:

[0017]

[0018] in is the intersection angle of the two sets of slip lines;

[0019] The finite difference method approximates equations (5) and (6) as follows:

[0020]

[0021] Where M α (x α ,y ɑ ,θ α ,σ α ) is a point on the α family, M β (x β ,y β ,θ β ,σ β ) is a point on the β family, (x, y) is the coordinate value, and

[0022] The unknown point M(x, y, θ, σ) on the slip line is calculated by combining formulas (7) and (8):

[0023]

[0024]

[0025] The differential equation of the limit state slope surface curve under earthquake action, referred to as the earthquake limit slope surface curve, is: Solve the seismic limit slope curve coordinate point M by combining with the β family slip line equation ij (x ij ,y ij ,θ ij ,σ ij ):

[0026]

[0027] Where M b (x b ,y b ,θ b ,σ b ) is a known point on the seismic limit slope curve, M′ β (x′ β , y′ β ,θ′ β ,σ′ β ) is a known point on the β-family slip line;

[0028] 2) Boundary conditions of earthquake slip line field

[0029] (1) Cauchy boundary conditions in the active region O1AB

[0030] The known calculation points M of the αth and βth groups in the active region α and M β (x, y) is the coordinate value of the slope top O1A, where the horizontal coordinate x = Δx·i, Δx is the calculation step, i is a natural number, i = 0~N1, N1 is the number of steps, the vertical coordinate y is the slope height, and the intersection angle θ between the maximum principal stress at the active zone boundary and the x-axis is I for:

[0031]

[0032] In the formula σ0=P0·(1-k V ) is the earthquake normal stress, τ0=P0·k H is the earthquake shear stress, P0 is the foundation load at the top of the slope;

[0033] Characteristic stress at the active zone boundary σ I for:

[0034]

[0035] In the formula

[0036] (2) Degenerate Riemannian boundary conditions in the transition region O1BC

[0037] The (x, y) coordinates of the known boundary point O1 in the transition zone are the slope shoulder coordinates, and the characteristic stress in the transition zone is:

[0038]

[0039] In the formula k is a natural number, k = 0 ~ N2, N2 is the number of points in the transition zone, Δθ = θ III -θ I ,θ III is the angle between the maximum principal stress in the passive zone and the x-axis;

[0040] (3) O1CD mixed boundary conditions in the passive zone

[0041] The characteristic stress value of the passive zone is Substituting into equation (19) gives the angle θ between the maximum principal stress in the passive zone and the x-axis: III for:

[0042]

[0043] 3) Failure mechanism

[0044] Slope top foundation load variable P i for:

[0045] P i =P0+j·ΔP (21)

[0046] ΔP is the increment of P0, j is a natural number;

[0047] To calculate the seismic ultimate bearing capacity P of the adjacent slope su , P i Substitute it into the earthquake slip line field theory and boundary condition formula to calculate the earthquake limit slope curve: when the earthquake limit slope curve and the slope surface have only one intersection point, that is, the slope top, the foundation near the slope is in a stable state, then P i <P su , at this time, ΔP in formula (21) = 0.1kPa; when the seismic limit slope curve and the slope surface have a second intersection, the foundation near the slope is in an unstable state, then P i >P su , at this time ΔP in formula (21) = -0.1kPa; when the earthquake limit slope curve is tangent to the slope surface, the foundation adjacent to the slope is in the limit state, at this time P i =P su .

[0048] Compared with the prior art, the failure mechanism of the present invention for evaluating the seismic ultimate bearing capacity of a strip foundation adjacent to a slope has the following beneficial effects:

[0049] (1) The seismic slip line field theory and boundary value conditions under foundation load conditions were theoretically derived, and the ultimate limit state slope surface curve (hereinafter referred to as the seismic ultimate slope surface curve) was calculated. Whether the seismic ultimate slope surface curve is tangent to the slope surface was used as the seismic ultimate state evaluation standard, thus achieving an objective standard for quantifying the seismic ultimate bearing capacity of strip foundations adjacent to slopes.

[0050] (2) Existing methods require assumptions or optimization methods to determine the critical sliding surface when evaluating the failure mechanism of the seismic ultimate bearing capacity of strip foundations near slopes. However, the present invention does not require assumptions or searches for the critical sliding surface of the slope, thereby improving calculation efficiency and accuracy.

[0051] (3) There are many factors that affect the non-convergence of strength reduction technology calculations. The selection of slope feature points and the judgment of mutation points in the displacement reduction curve are easily affected by human subjective factors. Compared with existing strength reduction technology, the failure mechanism of the present invention is not affected by non-convergence of calculations, thus avoiding human subjective judgment and value selection;

[0052] (4) It is scientific and reasonable, has high engineering practical value and good effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 Yes: Schematic diagram of calculation of earthquake limit slope curve under foundation load conditions;

[0054] Figure 2 It is: the schematic diagram of the failure mechanism of the present invention for judging the state of the adjacent slope foundation;

[0055] Figure 3 It is: the technical flow chart of the present invention;

[0056] Figure 4 Yes: Calculation diagram of the ultimate seismic bearing capacity of strip foundations adjacent to slopes based on the failure mechanism of the present invention. DETAILED DESCRIPTION

[0057] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0058] The present invention's earthquake slip line field theory and boundary condition calculation earthquake limit slope curve diagram is shown in Figure 1 .

[0059] 1. A failure mechanism for evaluating the ultimate seismic bearing capacity of a strip foundation adjacent to a slope, characterized by the following: 1) Seismic slip line field theory

[0060] The differential equation of soil stress under earthquake action is as follows:

[0061]

[0062] where σx ,σ y represents the normal stress in the x and y directions, τ xy and τ yx represents the tangential stress in the x and y directions, f x =γ·k H , f y =γ·(1-k V ), γ represents the bulk density, k H and k V represents the seismic coefficient in the x and y directions;

[0063] The calculation formula of characteristic stress σ is derived using the Mohr-Coulomb criterion

[0064]

[0065] Where: c is the cohesive force, is the internal friction angle, σ1 is the maximum principal stress, and σ3 is the minimum principal stress;

[0066] The stress expression is as follows:

[0067]

[0068] Where θ is the angle between σ1 and the x-axis;

[0069] Substituting equations (3a) and (4) into equation (1a), and substituting equations (3b) and (4) into equation (1b), we can obtain the α-family and β-family characteristic line differential equations according to the characteristic line method:

[0070]

[0071] in is the intersection angle of the two sets of slip lines;

[0072] The finite difference method approximates equations (5) and (6) as follows:

[0073]

[0074] Where M ɑ (x α ,y α ,θ α ,σ α ) is a point on the α family, M β (x β ,y β ,θ β ,σ β ) is a point on the β family, (x, y) is the coordinate value, and

[0075] The unknown point M(x, y, θ, σ) on the slip line is calculated by combining formulas (7) and (8):

[0076]

[0077] The differential equation of the limit state slope surface curve under earthquake action, referred to as the earthquake limit slope surface curve, is: Solve the seismic limit slope curve coordinate point M by combining with the β family slip line equation ij (x ij ,y ij ,θ ij ,σ ij ):

[0078]

[0079] Where M b (x b ,y b ,θ b ,σ b ) is a known point on the seismic limit slope curve, M′ β (x′ β , y′ β ,θ′ β ,σ′ β ) is a known point on the β-family slip line;

[0080] 3) Boundary conditions of earthquake slip line field

[0081] (1) Cauchy boundary conditions in the active region O1AB

[0082] The known calculation points M of the αth and βth groups in the active region ɑ and M β (x, y) is the coordinate value of the slope top O1A, where the horizontal coordinate x = Δx·i, Δx is the calculation step, i is a natural number, i = 0~N1, N1 is the number of steps, the vertical coordinate y is the slope height, and the intersection angle θ between the maximum principal stress at the active zone boundary and the x-axis is I for:

[0083]

[0084] In the formula σ0=P0·(1-k V ) is the earthquake normal stress, τ0=P0.k H is the earthquake shear stress, P0 is the foundation load at the top of the slope;

[0085] Characteristic stress at the active zone boundary σ I for:

[0086]

[0087] In the formula

[0088] (4) Degenerate Riemannian boundary conditions in the transition region O1BC

[0089] The (x, y) coordinates of the known boundary point O1 in the transition zone are the slope shoulder coordinates, and the characteristic stress in the transition zone is:

[0090]

[0091] In the formula k is a natural number, k = 0 ~ N2, N2 is the number of points in the transition zone, Δθ = θ III -θ I ,θ III is the angle between the maximum principal stress in the passive zone and the x-axis;

[0092] (5) Passive zone O1CD mixed boundary conditions

[0093] The characteristic stress value of the passive zone is Substituting into equation (19) gives the angle θ between the maximum principal stress in the passive zone and the x-axis: III for:

[0094]

[0095] 3) Failure mechanism

[0096] Slope top foundation load variable P i for:

[0097] P i =P0+j·ΔP (21)

[0098] ΔP is the increment of P0, j is a natural number;

[0099] To calculate the seismic ultimate bearing capacity P of the adjacent slope su P i Substitute it into the earthquake slip line field theory and boundary condition formula to calculate the earthquake limit slope curve: when the earthquake limit slope curve and the slope surface have only one intersection point, that is, the slope top, the foundation near the slope is in a stable state, then P i <P su , at this time, ΔP in formula (21) = 0.1kPa; when the seismic limit slope curve and the slope surface have a second intersection, the foundation near the slope is in an unstable state, then P i >P su , at this time ΔP in formula (21) = -0.1kPa; when the earthquake limit slope curve is tangent to the slope surface, the foundation adjacent to the slope is in the limit state, at this time P i =P su .

[0100] Table 1 gives the geometric and mechanical parameter values ​​of a strip foundation adjacent to a slope.

[0101] Table 1 Calculation parameters of the embodiment of the present invention

[0102]

[0103] When the horizontal and vertical seismic coefficients are k H = 0.1 and k V =0.05, according to the process Figure 3 Calculate and evaluate the conclusions. Figure 4 ; (1) When the base load P1 = 115kpa, the seismic limit slope curve has only one intersection with the slope surface; (2) When the base load P2 = 165kpa, the seismic limit slope curve is tangent to the slope surface; (3) When the base load P1 = 265kpa, the seismic limit slope curve has a second intersection with the slope surface; According to the failure mechanism of the present invention (see Figure 2 ), we can get the ultimate seismic bearing capacity Psu = P2 = 165kpa.

[0104] Through the examples, it can be seen that the failure mechanism of the present invention for evaluating the seismic ultimate bearing capacity of a strip foundation adjacent to a slope can provide a reliable seismic ultimate bearing capacity. The calculation process shows that the failure mechanism of the present invention provides an objective standard for evaluating the seismic ultimate bearing capacity of a strip foundation adjacent to a slope, that is, when the seismic ultimate slope curve is tangent to the slope surface, the foundation load is the seismic ultimate bearing capacity; compared with the strength reduction technology, there is no need to judge whether the slope is damaged by the displacement curve mutation point based on calculation non-convergence and human subjective judgment of the characteristic points; compared with the existing methods, the failure mechanism of the present invention does not require the assumption and search of the critical slip surface.

[0105] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, ordinary technicians in the field should understand that the specific implementation methods of the present invention can still be modified or replaced by equivalents. Any modification or equivalent replacement that does not depart from the spirit and scope of the present invention should be included in the scope of the claims of the present invention.

Claims

1. A method for evaluating the failure mechanism of the seismic ultimate bearing capacity of a strip foundation adjacent to a slope, characterized in that it Includes the following: 1) Earthquake slip line field theory The differential equation of soil stress under earthquake action is as follows: where σ x , σ y represents the normal stress in the x and y directions, τ xy and τ yx represents the tangential stress in the x and y directions, f x =γ·k H , f y =γ·(1-k V ), γ represents the bulk density, k H and k V represents the seismic coefficient in the x and y directions; The calculation formula of characteristic stress σ is derived using the Mohr-Coulomb criterion Where: c is the cohesive force, is the internal friction angle, σ1 is the maximum principal stress, and σ3 is the minimum principal stress; The stress expression is as follows: Where θ is the angle between σ1 and the x-axis; Substituting equations (3a) and (4) into equation (1a), and substituting equations (3b) and (4) into equation (1b), we can obtain the α-family and β-family characteristic line differential equations according to the characteristic line method: in is the intersection angle of the two sets of slip lines; The finite difference method approximates equations (5) and (6) as follows: Where M ɑ (x α ,y α ,θ α ,σ α ) is a point on the α family, M β (x β ,y β ,θ β ,σ β ) is a point on the β family, (x, y) is the coordinate value, and The unknown point M(x, y, θ, σ) on the slip line is calculated by combining formulas (7) and (8): The differential equation of the limit state slope surface curve under earthquake action, referred to as the earthquake limit slope surface curve, is: Solve the seismic limit slope curve coordinate point M by combining with the β family slip line equation ij (x ij ,y ij ,θ ij , σ ij ): Where M b (x b ,y b ,θ b , σ b ) is a known point on the seismic limit slope curve, M′ β (x′ β , y′ β ,θ′ β ,σ′ β ) is a known point on the β-family slip line; 2) Boundary conditions of earthquake slip line field (1) Cauchy boundary conditions in the active region O1AB The known calculation points M of the αth and βth groups in the active region α and M β (x, y) is the coordinate value of the slope top O1A, where the horizontal coordinate x = Δx·i, Δx is the calculation step, i is a natural number, i = 0~N1, N1 is the number of steps, the vertical coordinate y is the slope height, and the intersection angle θ between the maximum principal stress at the active zone boundary and the x-axis is I for: In the formula σ0=P0·(1-k V ) is the earthquake normal stress, τ0=P0·k H is the earthquake shear stress, P0 is the foundation load at the top of the slope; Characteristic stress at the active zone boundary σ I for: In the formula (2) Degenerate Riemannian boundary conditions in the transition region O1BC The (x, y) coordinates of the known boundary point O1 in the transition zone are the slope shoulder coordinates, and the characteristic stress in the transition zone is: In the formula k is a natural number, k = 0 ~ N2, N2 is the number of points in the transition zone, Δθ = θ III -θ I ,θ III is the angle between the maximum principal stress in the passive zone and the x-axis; (3) Passive zone O1CD mixed boundary conditions The characteristic stress value of the passive zone is Substituting into equation (19) gives the angle θ between the maximum principal stress in the passive zone and the x-axis: III for: 3) Failure mechanism Slope top foundation load variable P i for: P i =P0+j·ΔP (21) ΔP is the increment of P0, j is a natural number; To calculate the seismic ultimate bearing capacity P of the adjacent slope su , P i Substitute it into the earthquake slip line field theory and boundary condition formula to calculate the earthquake limit slope curve: when the earthquake limit slope curve and the slope surface have only one intersection point, that is, the slope top, the foundation near the slope is in a stable state, then P i <P su , at this time, ΔP in formula (21) = 0.1kPa; when the seismic limit slope curve and the slope surface have a second intersection, the foundation near the slope is in an unstable state, then P i >P su , at this time ΔP in formula (21) = -0.1kPa; when the earthquake limit slope curve is tangent to the slope surface, the foundation adjacent to the slope is in the limit state, at this time P i =P su .

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