CN115688225A - Failure mechanism for evaluating earthquake-resistant limit bearing capacity of strip foundation close to side slope - Google Patents
Failure mechanism for evaluating earthquake-resistant limit bearing capacity of strip foundation close to side slope Download PDFInfo
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- CN115688225A CN115688225A CN202211225000.2A CN202211225000A CN115688225A CN 115688225 A CN115688225 A CN 115688225A CN 202211225000 A CN202211225000 A CN 202211225000A CN 115688225 A CN115688225 A CN 115688225A
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Abstract
The invention provides a failure mechanism for evaluating the earthquake resistance limit bearing capacity of a strip foundation close to a side slope, which is characterized in that the theory and boundary conditions of an earthquake slip line field under the condition of basic load are deduced, and the slip line field under the earthquake action and a side slope curve in a limit state under the earthquake action are solved by adopting a finite difference method, namely the earthquake limit slope curve for short; whether the earthquake limit slope curve is tangent to the side slope surface or not is used as an earthquake limit state judgment standard, when only one intersection point exists between the earthquake limit slope curve and the side slope surface, whether the near side slope foundation is in a stable state is judged, when two intersection points exist between the earthquake limit slope curve and the side slope surface, whether the near side slope foundation is in a destabilization state is judged, and when the earthquake limit slope curve is tangent to the side slope surface, the foundation load is the near side slope foundation earthquake resistance limit bearing capacity.
Description
Technical Field
The invention belongs to the field of evaluation of foundation stability near a side slope, and particularly relates to a failure mechanism for evaluating the anti-seismic limit bearing capacity of a strip foundation near the side slope.
Background
Due to the need of engineering construction and the limitation of the surrounding environment, the foundations of bridge piers, retaining structures, high-rise buildings and the like are often arranged on the top of a side slope close to the slope surface to form a foundation close to the side slope. Unlike a horizontal foundation, the presence of a slope reduces the bearing capacity of the foundation adjacent to the slope. Meanwhile, in an earthquake-prone area, the earthquake action can also cause the foundation of the foundation close to the side slope to slide and damage. The evaluation of the earthquake-resistant limit bearing capacity of the foundation close to the side slope under the action of an earthquake is the geotechnical engineering subject of intersection of the side slope and the foundation.
The method for evaluating the earthquake resistance limit bearing capacity of the foundation adjacent to the side slope under the earthquake action comprises a limit balance method, a limit method, finite element limit analysis, discrete elements and the like. The determination of the failure mechanism is a key problem, wherein the limit balance and limit analysis adopt a hypothesis method to determine the failure mechanism, such as a multi-wedge model, a logarithmic spiral model and the like, and numerical analysis such as finite element limit analysis and the like adopts an optimization searching technology to determine the failure mechanism, such as self-adaptive mesh division, a discontinuous layout optimization technology and the like. The failure mechanism is an N-P type global optimization problem, and the application of the method has the problem of insufficient accuracy. The strength reduction technology does not need to determine a failure mechanism in advance, but the instability criterion of the method, such as calculating non-convergence and sudden change of the displacement value of an observation point, needs to be set or judged by human subjectivity. Therefore, a new failure mechanism is needed for the evaluation of the earthquake-resistant limit bearing capacity of the foundation adjacent to the side slope.
Disclosure of Invention
Aiming at the defects of the prior art, the invention aims to provide a failure mechanism for evaluating the earthquake-resistant limit bearing capacity of a strip foundation close to a side slope, which is scientific, reasonable, high in engineering practical value and good in effect.
In order to achieve the purpose, the invention adopts the technical scheme that:
1. a failure mechanism for evaluating the earthquake-resistant limit bearing capacity of a strip foundation close to a side slope is characterized by comprising the following contents:
1) Seismic glide line field theory
The differential equation of the stress of the soil body under the action of the earthquake is as follows:
wherein sigma x ，σ y Denotes the normal stress in the x and y directions, τ xy And τ yx Representing tangential stresses in the x and y directions, f x ＝γ·k H ，f y ＝γ·(1-k V ) γ represents volume weight, k H And k V Representing x and y direction seismic coefficients;
derivation of characteristic stress sigma calculation formula by Mohr-Coulomb criterion
In the formula: c is the cohesive force of the rubber composition,
angle of internal friction, σ 1 Is the maximum principal stress, σ 3 Is the minimum principal stress;
the stress expression is as follows:
where θ is σ 1 And the x-axis;
when formula (3 a) and formula (4) are substituted for formula (1 a) and formula (3 b) and formula (4) are substituted for formula (1 b), the system of a-group and β -group characteristic line differential equations can be obtained according to the characteristic line method:
wherein
Is the intersection angle of the two sets of slip lines;
the finite difference method approximately solves equations (5) and (6) as follows:
in the formula M α (x α ，y α ，θ α ，σ α ) Is a point on the alpha group, M β (x β ，y β ，θ β ，σ β ) Is a point in the beta family, (x, y) are coordinate values,
and
calculating a point M (x, y, θ, σ) to be solved on the slip line simultaneously from equations (7) and (8), which are:
limit state edge under earthquake actionThe differential equation of the slope surface curve, called earthquake limit slope surface curve for short, is:
simultaneous solving of earthquake extreme slope curve coordinate point M with beta family slip line equation ij (x ij ，y ij ，θ ij ，σ ij )：
In the formula M b (x b ，y b ，θ b ，σ b ) Is a known point on a seismic limit slope curve, M' β (x′ β ，y′ β ，θ′ β ，σ′ β ) Is a known point on the group beta slip line;
2) Seismic glide line field boundary conditions
(1) Active region O 1 AB Cauchy boundary conditions
Known calculation point M of alpha and beta families of active region α And M β (x, y) is the top of a slope O 1 A coordinate value, where x = Δ x · i, Δ x is a calculation step, i is a natural number, i =0 to N 1 ，N 1 The step number, the vertical coordinate y is the slope height, and the intersection angle theta of the maximum principal stress of the boundary of the active region and the x axis 1 Comprises the following steps:
in the formula
σ 0 ＝P 0 ·(1-k V ) Is the seismic normal stress, τ 0 ＝P 0 ·k H Is seismic shear stress, P 0 Is the slope top foundation load;
characteristic stress sigma of active zone boundary I Comprises the following steps:
in the formula
(2) Transition zone O 1 BC degenerated Riemann boundary condition
Known boundary point O of transition zone 1 And (x, y) is a slope shoulder coordinate value, and the characteristic stress of the transition region is as follows:
in the formula
k is a natural number, k =0 to N 2 ，N 2 For transition region point section, Δ θ = θ III -θ I ，θ m The intersection angle of the maximum main stress of the passive region and the x axis;
(3) Passive region O 1 CD hybrid boundary conditions
Characteristic stress value of passive region
Substituting equation (19) to give the angle theta between the maximum principal stress of the passive region and the x-axis III Comprises the following steps:
3) Mechanism of failure
Variable P of slope top foundation load i Comprises the following steps:
P i ＝P 0 +j·ΔP (21)
Δ P is P 0 J is a natural number;
for calculating the earthquake-resistant limit bearing capacity P of the adjacent slope su A 1 is to P i Substituting the obtained value into a seismic slip line field theory and boundary condition formula to calculate a seismic limit slope curve: when the curve of the earthquake limit slope surface and the surface of the side slope have only one intersection point, namely the top of the slope, the foundation close to the side slope is in a stable state, and P is i ＜P su When Δ P =0.1kPa in formula (21); when a second intersection point occurs between the earthquake limit slope surface curve and the side slope surface, the adjacent side slope foundation is in a destabilization state, and P is i ＞P su When Δ P = -0.1kPa in formula (21); when the earthquake limit slope curve is tangent to the surface of the side slope, the foundation close to the side slope is in a limit state, and P is at the moment i ＝P su 。
Compared with the prior art, the failure mechanism for evaluating the earthquake-resistant limit bearing capacity of the strip foundation close to the side slope has the beneficial effects that:
(1) The earthquake slip line field theory and the edge value condition under the condition of basic load are deduced theoretically, a slope curve of a side slope in a limit state (called earthquake limit slope curve for short) is obtained through calculation, whether the earthquake limit slope curve is tangent to the surface of the side slope serves as an earthquake limit state evaluation standard, and objective standard quantification of the earthquake resistance limit bearing capacity of the strip foundation close to the side slope is evaluated;
(2) In the conventional method, when the failure mechanism of the anti-seismic ultimate bearing capacity of the strip foundation close to the side slope is evaluated, the critical sliding surface needs to be assumed or determined by adopting an optimization method, but the method does not need to assume and search the critical sliding surface of the side slope, so that the calculation efficiency and the precision are improved;
(3) The factors influencing the unconvergence of the intensity reduction technology calculation are more, and the selection of the slope characteristic points and the judgment of the mutation points by the displacement reduction curve are easily influenced by artificial subjective factors. Compared with the existing strength reduction technology, the failure mechanism of the invention has no influence of non-convergence of calculation, and avoids artificial subjective judgment and value;
(4) The method is scientific and reasonable, the engineering practical value is high, and the effect is good.
Drawings
FIG. 1 is a schematic diagram of: a schematic diagram of earthquake limit slope curve calculation under the condition of foundation load;
FIG. 2 is a diagram: the failure mechanism of the invention judges the state schematic diagram of the foundation close to the side slope;
FIG. 3 is a diagram of: the technical flow chart of the invention;
FIG. 4 is a diagram of: the invention provides a failure mechanism evaluation calculation chart of the earthquake-resistant limit bearing capacity of a strip foundation close to a side slope.
Detailed Description
The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.
The schematic diagram of the curve of the earthquake limit slope calculated by the earthquake glide line field theory and the boundary condition is shown in figure 1.
1. A failure mechanism for evaluating the earthquake-resistant limit bearing capacity of a strip foundation close to a side slope is characterized by comprising the following contents:
1) Seismic glide line field theory
The differential equation of the stress of the soil body under the action of the earthquake is as follows:
wherein sigma x ，σ y Normal stress, τ, in the x and y directions xy And τ yx Denotes the tangential stress in the x and y directions, f x ＝γ·k H ，f y ＝γ·(1-k V ) γ represents volume weight, k H And k V Representing x and y direction seismic coefficients;
derivation of characteristic stress sigma calculation formula by Mohr-Coulomb criterion
In the formula: c is the cohesive force of the rubber composition,
angle of internal friction, σ 1 Is the maximum principal stress, σ 3 Is the minimum principal stress;
the stress expression is as follows:
where θ is σ 1 And the x-axis;
the differential equations of the alpha group and beta group eigenlines can be obtained by substituting the formula (3 a) and the formula (4) for the formula (1 a) and substituting the formula (3 b) and the formula (4) for the formula (1 b) according to the eigenline method:
wherein
The intersection angle of the two sets of slip lines;
finite difference method approximately solves equations (5) and (6) as follows:
in the formula M α (x α ，y α ，θ α ，σ α ) Is a point on the alpha group, M β (x β ，y β ，θ β ，σ β ) Is a point in the beta family, (x, y) are coordinate values,
and
calculating a point M (x, y, θ, σ) to be solved on the slip line simultaneously from equations (7) and (8), which are:
extreme state slope curve under earthquake action, called earthquake extreme slope curve for shortThe differential equation of (a) is:
simultaneous solving of earthquake extreme slope curve coordinate point M with beta family slip line equation ij (x ij ，y ij ，θ ij ，σ ij )：
In the formula M b (x b ，y b ，θ b ，σ b ) Is a known point on a seismic limit slope curve, M' β (x′ β ，y′ β ，θ′ β ，σ′ β ) Known points on the beta slip line;
3) Seismic glide line field boundary conditions
(1) Active region O 1 AB Cauchy boundary conditions
Known calculation point M of alpha and beta families of active region α And M β (x, y) is the crest O 1 A coordinate value, where x = Δ x · i, Δ x is a calculation step, i is a natural number, i =0 to N 1 ，N 1 The step number, the vertical coordinate y is the slope height, and the intersection angle theta of the maximum principal stress of the boundary of the active region and the x axis I Comprises the following steps:
in the formula
σ 0 ＝P 0 ·(1-k V ) Is the seismic normal stress, τ 0 ＝P 0 ·k H Is seismic shear stress, P 0 Is the slope top foundation load;
characteristic stress sigma of active zone boundary I Comprises the following steps:
in the formula
(4) Transition zone O 1 BC degenerated Riemann boundary condition
Known boundary point O of transition zone 1 And (x, y) is a slope shoulder coordinate value, and the characteristic stress of the transition region is as follows:
in the formula
k is a natural number, k = 0-N 2 ，N 2 For transition region point section, Δ θ = θ III -θ 1 ，θ III The intersection angle of the maximum main stress of the passive region and the x axis;
(5) Passive region O 1 CD hybrid boundary conditions
Characteristic stress value of passive region
Substituting equation (19) to give the angle theta between the maximum principal stress of the passive region and the x-axis III Comprises the following steps:
3) Mechanism of failure
Variable P of slope top foundation load i Comprises the following steps:
P i ＝P 0 +j·ΔP (21)
Δ P is P 0 J is a natural number;
for calculating the earthquake-resistant limit bearing capacity P of the adjacent slope su A 1 is to P i Substituting the obtained value into a seismic slip line field theory and boundary condition formula to calculate a seismic limit slope curve: when the curve of the earthquake limit slope surface and the surface of the side slope have only one intersection point, namely the top of the slope, the foundation close to the side slope is in a stable state, and P is i ＜P su When Δ P =0.1kPa in formula (21); when a second intersection point occurs between the earthquake limit slope surface curve and the side slope surface, the adjacent side slope foundation is in a destabilization state, and P is i ＞P su When Δ P = -0.1kPa in formula (21); when the earthquake limit slope curve is tangent to the surface of the side slope, the foundation close to the side slope is in a limit state, and P is at the moment i ＝P su 。
Table 1 shows the geometrical and mechanical parameters of a strip-shaped foundation of an adjacent slope.
TABLE 1 calculation parameters for an embodiment of the invention
When the horizontal and vertical seismic coefficients are k respectively H =0.1 and k V When =0.05, the calculation is performed according to the flowchart of fig. 3, and the evaluation conclusion is shown in fig. 4: (1) When the foundation load P 1 When the slope surface curve of the earthquake limit slope is not less than 115kpa, the curve of the earthquake limit slope and the surface of the side slope have only one intersection point; (2) When the foundation load P 2 When =165kpa, the earthquake limit slope curve is tangent to the side slope surface; (3) When the foundation load P 1 When the velocity is not less than 265kpa, the earthquake limit slope curve and the side slope surface have a second intersection point; according to the failure mechanism of the invention (see fig. 2), the earthquake resistant limit bearing capacity Psu = P can be obtained 2 ＝165kpa。
According to the embodiment, the failure mechanism for evaluating the earthquake-proof limit bearing capacity of the strip-shaped foundation close to the side slope can provide reliable earthquake-proof limit bearing capacity, and the calculation process shows that the failure mechanism of the invention provides an objective standard for evaluating the earthquake-proof limit bearing capacity of the strip-shaped foundation close to the side slope, namely when an earthquake limit slope curve is tangent to the surface of the side slope, the foundation load is the earthquake-proof limit bearing capacity; compared with the strength reduction technology, whether the slope is damaged or not is judged without calculating unconvergence and artificially and subjectively judging and selecting the displacement curve catastrophe point of the characteristic point; compared with the existing method, the failure mechanism of the invention does not need to assume and search the critical slip fracture surface.
Finally, it should be noted that the above-mentioned embodiments are only used for illustrating the technical solutions of the present invention and not for limiting the same, and although the present invention has been described in detail with reference to the above-mentioned embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions can be made on the specific embodiments of the present invention without departing from the spirit and scope of the present invention, and all the modifications or equivalent substitutions should be covered in the claims of the present invention.
Claims (1)
1. A failure mechanism for evaluating the earthquake-resistant limit bearing capacity of a strip foundation close to a side slope is characterized by comprising the following contents:
1) Seismic glide line field theory
The differential equation of the stress of the soil body under the action of the earthquake is as follows:
wherein sigma x ，σ y Normal stress, τ, in the x and y directions xy And τ yx Representing tangential stresses in the x and y directions, f x ＝γ·k H ，f y ＝γ·(1-k V ) γ represents volume weight, k H And k V Representing x and y direction seismic coefficients;
derivation of characteristic stress sigma calculation formula by Mohr-Coulomb criterion
In the formula: c is the cohesive force of the adhesive,
angle of internal friction, σ 1 Is the maximum principal stress, σ 3 Is the minimum principal stress;
the stress expression is as follows:
where θ is σ 1 And the x-axis;
when formula (3 a) and formula (4) are substituted for formula (1 a) and formula (3 b) and formula (4) are substituted for formula (1 b), the system of a-group and β -group characteristic line differential equations can be obtained according to the characteristic line method:
wherein
Is the intersection angle of the two sets of slip lines;
the finite difference method approximately solves equations (5) and (6) as follows:
in the formula M α (x α ，y α ，θ α ，σ α ) Is a point on the alpha group, M β (x β ，y β ，θ β ，σ β ) Is a point in the beta family, (x, y) are coordinate values,
and
calculating a point M (x, y, θ, σ) to be solved on the slip line simultaneously from equations (7) and (8), which are:
the differential equation of the extreme slope curve of the side slope under the action of the earthquake, called the earthquake extreme slope curve for short, is as follows:
simultaneous solving of earthquake extreme slope curve coordinate point M with beta family slip line equation ij (x ij ，y ij ，θ ij ，σ ij )：
In the formula M b (x b ，y b ，θ b ，σ b ) Is a known point on a curve of the earthquake limit slope surface, M' β (x′ β ，y′ β ，θ′ β ，σ′ β ) Is a known point on the group beta slip line;
2) Boundary condition of seismic glide line field
(1) Active region O 1 AB Cauchy boundary conditions
Known calculation point M of alpha and beta families of active region α And M β (x, y) is the crest O 1 A coordinate value, whereinCoordinates x = Δ x · i, Δ x being the calculation step length, i being a natural number, i =0 to N 1 ，N 1 The step number, the vertical coordinate y is the slope height, and the intersection angle theta of the maximum principal stress of the boundary of the active region and the x axis I Comprises the following steps:
in the formula
σ 0 ＝P 0 ·(1-k V ) Is the seismic normal stress, τ 0 ＝P 0 ·k H Is seismic shear stress, P 0 Is the slope top foundation load;
characteristic stress sigma of active zone boundary I Comprises the following steps:
in the formula
(2) Transition zone O 1 BC degenerate Riemann boundary condition
Known boundary point O of transition zone 1 And (x, y) is a slope shoulder coordinate value, and the characteristic stress of the transition region is as follows:
in the formula
k is a natural number, k = 0-N 2 ，N 2 For transition region point-section fraction, Δ θ = θ III -θ I ，θ III The intersection angle of the maximum main stress of the passive region and the x axis;
(3) Passive region O 1 CD hybrid boundary conditions
Characteristic stress value of passive region
Substituting equation (19) to give the angle theta between the maximum principal stress of the passive region and the x-axis III Comprises the following steps:
3) Failure mechanism
Variable P of slope top foundation load i Comprises the following steps:
P i ＝P 0 +j·ΔP (21)
Δ P is P 0 J is a natural number;
for calculating the earthquake-resistant limit bearing capacity P of the adjacent slope su A 1 is to P i Substituting the obtained value into a seismic slip line field theory and boundary condition formula to calculate a seismic limit slope curve: when the earthquake limit slope surface curve and the side slope surface have only one intersection point, namely the top of the slope, the foundation close to the side slope is in a stable state, and P is i ＜P su When Δ P =0.1kPa in formula (21); when the second intersection point appears between the curve of the earthquake limit slope and the surface of the side slope, the foundation close to the side slope is in a destabilization state, and P is i ＞P su When Δ P = -0.1kPa in formula (21); when the earthquake limit slope curve is tangent to the surface of the side slope, the foundation close to the side slope is in a limit state, and P is at the moment i ＝P su 。
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117113516A (en) * | 2023-10-23 | 2023-11-24 | 西南交通大学 | Limit bearing capacity prediction method and related device for strip foundation of adjacent side slope |
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117113516A (en) * | 2023-10-23 | 2023-11-24 | 西南交通大学 | Limit bearing capacity prediction method and related device for strip foundation of adjacent side slope |
| CN117113516B (en) * | 2023-10-23 | 2024-01-26 | 西南交通大学 | Limit bearing capacity prediction method and related device for strip foundation of adjacent side slope |
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