Modeling method for three-dimensional potential instability surface of foundation under action of complex foundation load

Abstract

The invention belongs to the field of foundation engineering, and relates to a modeling method for a three-dimensional potential instability surface of a foundation under the action of a complex foundation load, which comprises the following steps of: performing spatial expansion on a two-dimensional potential instability surface of the foundation along the y-axis direction to form the three-dimensional potential instability surface of the foundation, and reflecting the failure and damage characteristics of the foundation by using the two-dimensional potential instability surface of the foundation. The space expansion and the three-dimensional potential instability range of the foundation are limited within the sliding circumference of the three-dimensional potential instability surface of the foundation; and the three-dimensional potential instability surface of the foundation in any form is constructed by changing the sliding circumference parameters of the two-dimensional potential instability surface of the foundation and the three-dimensional potential instability surface of the foundation. According to the method, the three-dimensional potential instability surface of the foundation is generated in a discrete mode, so that the method is easily applicable to construction of the three-dimensional potential instability surface of the foundation under the action of a complex foundation load; therefore, the problems of universality, simplicity, high efficiency, practicability, effectiveness, reliability and the like of foundation three-dimensional potential instability surface modeling and foundation three-dimensional stability analysis under complex conditions are solved.

Description

Technical Field

[0001] The invention belongs to the field of foundation engineering, and in particular relates to a modeling method for a three-dimensional potential instability surface of a foundation under the action of complex foundation loads. Background Art

[0002] The foundation bears the entire weight and load of the upper structure, and its stability is directly related to the overall safety of the structure. When the foundation is unstable and damaged, the upper structure may tilt, crack, sink or even collapse, seriously threatening people's lives and property safety. Therefore, in order to prevent the occurrence of foundation instability accidents, an effective, reasonable and reliable method for constructing the potential instability surface of the foundation is urgently needed to achieve accurate assessment of foundation stability and provide scientific guidance for the reinforcement of the foundation of the structure.

[0003] Among the existing methods for constructing potential instability surface models of foundations, one is to consider the length of strip foundations much greater than their width, thus ignoring the length size effect of the foundation, simplifying the interaction between the foundation and the foundation into a plane strain problem in a horizontal semi-infinite space, and using a two-dimensional regular curve to simulate the potential instability surface of the foundation, i.e., a two-dimensional model; the other is to consider the spatial size effect of the foundation, and use a method similar to the construction of a three-dimensional slope sliding surface to generate a three-dimensional potential instability surface of the foundation, i.e., a three-dimensional model. However, the existing two-dimensional model cannot consider the arbitrariness of the foundation shape and the irregularity of the load acting on it. At the same time, simplifying the three-dimensional space problem of the interaction between the foundation and the foundation into a two-dimensional plane problem often leads to insufficient accuracy of the foundation stability analysis results. The existing three-dimensional model uses the three-dimensional slope sliding surface construction method to generate the potential instability surface of the foundation, which is mainly suitable for slope foundations. If it is a flat foundation, since the foundation is not close to the free boundary surface of the slope, the foundation failure characteristics are more of extrusion and uplift rather than sliding toward the free surface, which makes it unreasonable to use a three-dimensional slope sliding surface to simulate the potential instability surface of the foundation. In addition, the existing three-dimensional foundation instability surface model is only applicable to rectangular and circular foundations and foundations under symmetrical loads. This method is not currently applicable to foundations with asymmetric contours and applied loads.

[0004] With the acceleration of urbanization and industrialization, the demand for foundation construction projects is constantly expanding, and it is necessary to carry out foundation stability analysis accurately and reasonably. However, the existing construction method of the potential instability surface model of the foundation is still not perfect, and there are problems of insufficient accuracy and applicability, which makes it difficult to meet the needs of actual engineering for foundation stability analysis.

[0005] Therefore, there is a need in the art for a modeling method of a three-dimensional potential instability surface of a foundation under complex foundation loads. Summary of the invention

[0006] To this end, the present invention adopts a two-dimensional potential instability surface of the foundation to perform spatial expansion in a direction perpendicular to the two-dimensional potential instability surface to form a three-dimensional potential instability surface of the foundation, and uses the two-dimensional potential instability surface of the foundation to reflect the failure and damage characteristics of the foundation. At the same time, the spatial expansion and the three-dimensional potential instability range of the foundation are limited within the sliding contour of the three-dimensional potential instability surface of the foundation. Further, by changing the parameters of the two-dimensional potential instability surface of the foundation and the sliding contour of the three-dimensional potential instability surface of the foundation, it is used to construct a three-dimensional potential instability surface of the foundation in any form. In addition, the three-dimensional potential instability surface of the foundation is generated in a discrete manner, so that it is easy to apply to the construction of the three-dimensional potential instability surface of the foundation under the action of complex foundation loads, thereby solving the problems of versatility, simplicity, efficiency, practicality, effectiveness and reliability of the three-dimensional potential instability surface modeling of the foundation and the three-dimensional stability analysis of the foundation under complex conditions. The present invention has the advantages of simple implementation, wide application range, strong practicality, etc., and provides a reliable implementation path for the accurate evaluation of foundation engineering safety.

[0007] The present invention first provides a method for modeling a three-dimensional potential instability surface of a foundation under a complex foundation load, the method comprising constructing a three-dimensional potential instability surface model of the foundation by using a two-dimensional potential instability surface of the foundation and a sliding contour of the three-dimensional potential instability surface of the foundation, wherein the two-dimensional potential instability surface of the foundation is located in a rectangular coordinate system and is parallel to the three-dimensional potential instability surface of the foundation. y In a vertical plane perpendicular to the axis, the two-dimensional potential instability surface of the foundation is generated using a discrete method and is a general logarithmic spiral curve; the sliding contour of the three-dimensional potential instability surface of the foundation is a curve located on the plane of interaction between the foundation and the foundation; the method includes using the two-dimensional potential instability surface of the foundation along y The three-dimensional potential instability surface of the foundation is formed by spatial expansion in the axial direction, and the two-dimensional potential instability surface of the foundation is used to reflect the failure and damage characteristics of the foundation; and the two-dimensional potential instability surface of the foundation is along the y The spatial expansion in the axial direction and the three-dimensional potential instability range of the foundation are both limited within the sliding contour of the three-dimensional potential instability surface of the foundation; by changing the parameters of the two-dimensional potential instability surface of the foundation and the sliding contour of the three-dimensional potential instability surface of the foundation, an arbitrary shape of the three-dimensional potential instability surface of the foundation can be constructed.

[0008] In the present invention, regarding the interaction plane between the foundation and the subgrade, the one above it is the foundation, and the one below it is the subgrade.

[0009] In a specific embodiment, in the method, firstly given xyz Rotation angle in axis coordinate system r The range of the length magnification factor of the sliding contour of the three-dimensional potential instability surface of the given foundation l 1 ~ l 4 The shape influence factor of the sliding contour of the three-dimensional potential instability surface of the given foundation is a 2 , a3 , b 3 and oh The range of the initial polar diameter of the two-dimensional potential instability surface of the given foundation r 0 The range of the potential instability surface of the given foundation is given by the shape control parameters of the two-dimensional potential instability surface. g range; based on the mutual combination of these parameters, a series of three-dimensional potential instability surfaces of the foundation under the action of complex foundation loads are generated, and the safety factors corresponding to these three-dimensional potential instability surfaces of the foundation are calculated in combination with the limit equilibrium method, and the minimum value of the safety factor is taken as the optimization target, so as to determine the most dangerous three-dimensional potential instability surface of the foundation under the action of complex foundation loads.

[0010] In a specific embodiment, in the method, the rotation angle r The value range is 0°~180°, and the length magnification factor of the three-dimensional potential unstable surface sliding contour of the foundation l 1 ~ l 4 The value range of is 1~5, and the shape influencing factor of the sliding contour of the three-dimensional potential unstable surface of the foundation a 2 , a 3 and b 3 The value range is -5~5. oh The value range of is 3~10, and the initial polar diameter of the two-dimensional potential instability surface of the foundation r 0 The value range of is 1m~5m, and the shape control parameter of the two-dimensional potential instability surface of the foundation g The value range is -1~1.

[0011] In a specific implementation, the three-dimensional potential unstable surface sliding contour of the foundation is a closed curve formed by expanding the complex foundation contour outward at a certain ratio. Specifically, the length magnification factor l 1 ~ l 4 The contour lines of the complex base are x Negative axis direction, x Positive axis direction, y Negative axis direction, y The positive expansion in the axial direction forms the proportional coefficient of the three-dimensional potential instability surface sliding contour of the foundation; and the Fourier series is used to construct the function of the magnification coefficient of the three-dimensional potential instability surface sliding contour of the foundation, and the three-dimensional potential instability surface sliding contour of the foundation is discretized by equal angle increments, and the discrete points of the three-dimensional potential instability surface sliding contour of the foundation are established. tThe relationship with the complex foundation contour line is also determined by limiting the number of intersections between the direction line and the three-dimensional potential instability surface sliding contour of the foundation to ensure the rationality of the characteristics of the three-dimensional potential instability surface sliding contour of the foundation.

[0012] In a specific embodiment, the method comprises the following steps: Step S1: Given xyz Axis calculation coordinate system rotation angle r , the magnification factor of the length of the sliding contour of the three-dimensional potential unstable surface of the foundation l 1 ~ l 4 , the influencing factors of the sliding contour shape of the three-dimensional potential unstable surface of the foundation a 2 , a 3 , b 3 and oh , initial polar diameter of the two-dimensional potential instability surface of the foundation r 0 And the shape control parameters of the two-dimensional potential instability surface of the foundation g ; Step S2: Establish a x ' y ′ axis coordinate system, and use the gravity center calculation formula to obtain the center point of the complex foundation load O of x 'and y ′ axis coordinate and ; Step S3: Point O Center point and parallel x The positive direction of the 'axis is the starting direction, and the angle increment is counterclockwise. d Draw multiple rays, and then use the intersection of the rays and the complex basic contour line to discretize the complex basic contour line and obtain the discrete points k of x 'and y ′ axis coordinate and ,in, k = 0,1,2,…,Num_lk,Num_lk=360° / d – 1; Step S4: Load action center point O as the origin, and x 'and y 'Axis counterclockwise rotation angle r ,Establish xyzAxis calculation coordinate system, then, use equations (1) and (2) to calculate the discrete points of the complex basic contour line k of x and y Axis coordinates x k and y k ; (1) (2) Step S5: Using equations (3) and (4), determine the point A and Point B of x Axis coordinates x A and x B , where point A and Point B They are the three-dimensional potential unstable surface sliding contour of the foundation and x The left and right intersection of the axis; (3) (4) In the formula, x k and x k+1 are discrete points on the contour line of the complex basic shape k and k +1 x Axis coordinates, and in formula (3) , and in formula (4) , int is the rounding function; Step S6: Using the foundation two-dimensional potential instability surface parameter calculation step to solve the foundation two-dimensional potential instability surface parameter or 0 , or 1 , r 1 , x T , y T and z T ; in, or 0 is the starting polar angle of the two-dimensional potential instability surface of the foundation, that is, point A The extreme angle of the sliding surface; or 1 is the termination polar angle of the two-dimensional potential instability surface of the foundation, that is, point B The extreme angle of the sliding surface; r1 is the terminal pole diameter of the two-dimensional potential instability surface of the foundation, that is, point B The extreme diameter of the sliding surface at ; x T , y T and z T The polar coordinate center points T of x Axis coordinates, y Axis coordinates and z Axis coordinates; Step S7: Use equations (10) to (13) to solve the influencing factors of the sliding contour shape of the three-dimensional potential unstable surface of the foundation a 0 , a 1 , b 1 and b 2 ; (10) (11) (12) (13) In the formula, a 0 , a 1 , a 2 , a 3 , b 1 , b 2 , b 3 and oh All of them are the influencing factors of the shape of the sliding contour of the three-dimensional potential unstable surface of the foundation, among which, a 0 , a 1 , b 1 and b 2 Based on points A Length magnification factor l 1 ,point B Length magnification factor l 2 ,point C Length magnification factor l 3 and Point DLength magnification factor l 4 To solve, a 2 , a 3 and b 3 is any real number between -5 and 5, oh Any integer from 3 to 10; Step S8: Point O Center point and parallel x The positive direction of the axis is the starting direction, and the angle increment is counterclockwise. ψ Draw multiple rays, and then use the intersection of the rays and the three-dimensional potential instability surface sliding contour of the foundation to discretize the three-dimensional potential instability surface sliding contour of the foundation, and use equations (14) and (15) to obtain the discrete points t of x and y Axis coordinates x t and y t ,in, t = 0,1,2,…,Num_hz,Num_hz=360° / ψ – 1; (14) (15) in, t are discrete points; P is a continuous point; where x P and y P Points P of x and y Axis coordinates; l P Points on the contour line of the complex foundation P 'along OP ' extends outward to point P The length magnification factor of l P ≥1; i P for OP and x The counterclockwise angle between the positive axes; d is the angle increment; x k and y k are discrete points on the contour line of the complex basic shape k of x and yaxis coordinates, and , int is the rounding function; x k+1 and y k+1 They are discrete points on the contour line of the complex basic shape. k +1 x and y Axis coordinates; In equations (14) and (15), the subscript P use t Instead, at the same time, numerically i P use t × ψ Instead, the discrete points can be solved t of x and y Axis coordinates x t and y t ; Step S9: Discrete points based on the contour of the three-dimensional potential unstable surface sliding of the foundation t of x and y Axis coordinates, obtain the three-dimensional potential unstable surface sliding contour of the foundation xyz On the plane y Minimum axis coordinate y min and maximum value y max and its corresponding point E and Point F , and the three-dimensional potential instability surface sliding contour of the foundation xyz On the plane x Minimum axis coordinate x min and maximum value x max and its corresponding point G and Point H , at point E and Point F Divide equally between m +1 parallel to x The axis of the line, at point G and Point H Divide equally between n +1 parallel to y The columnar line of the axis corresponds to any point on the three-dimensional potential instability surface of the foundation s ij , which is xyz The projection on the plane is located at i Line and j At the intersection of the lines, where 0 ≤i ≤ n , 0 ≤ j ≤ m ; Step S10: Using j The steps for solving the intersection of the line and the three-dimensional potential instability surface sliding contour of the foundation are to determine the point A j and Point B j of x Axis coordinates and ; If the unstable sliding contour is unreasonable, the three-dimensional potential unstable surface of the foundation is unreasonable, and the calculation is terminated; Step S11: Using the two-dimensional curve on the three-dimensional potential instability surface of the foundation j Parameter calculation steps to solve two-dimensional curves j Related parameters r 0_j , r 0_j、 r 1_j , or 0_ j , or 1_j and g j ; Step S12: Use equations (29) and (30) to calculate the value of any point on the three-dimensional potential instability surface of the foundation: s ij of x Axis coordinates x ij and y Axis coordinates y ij ; (29) (30) Step S13: Using any point on the three-dimensional potential instability surface of the foundation s ij The calculation steps of the sliding surface polar angle are solved or ij , or ij is any point on the three-dimensional potential instability surface of the foundation s ij In vertical section j The corresponding sliding surface polar angle in ; Step S14: Use equation (31) to solve the problem of any point on the three-dimensional potential instability surface of the foundation s ij of z Axis coordinates zij ; (31) In the formula, For a two-dimensional curve j The polar coordinate center point of T j of z Axis coordinates; Step S15: output the calculation result of the three-dimensional potential instability surface of the foundation.

[0013] In a specific implementation, step S6 in the method specifically includes the following steps: Step S6-1: or 0 and or 1 Calculate variables for the iteration loop and take or 0 and or 1 The initial values ​​are or 0 (0) and or 1 (0) , and when first calculated or 0 (0) = 0 and or 1 (0) = 0; Step S6-2: Calculate using formula (19) r 1 ; (19) Step S6-3: Use equation (17) and equation (18) to calculate the new or 0 and or 1 ; (17) (18) Step S6-4: If | or 0 – or 0 (0) | ≤ e 1 and | or 1 – or 1 (0) | ≤ e1 , here take e 1 = 0.001°, then the calculated or 0 , or 1 and r 1 is the final result, otherwise, or 0 (0) = or 0 and or 1 (0) = or 1 , and repeat steps S6-2 and S6-3; Step S6-5: Use equations (20) to (22) to solve the center point T of x axis, y Axis and z Axis coordinates x T , y T and z T ; (20) (twenty one) (twenty two) Step S6-6: Output the two-dimensional potential instability surface parameters of the foundation or 0 , or 1 , r 1 , x T , y T and z T .

[0014] In a specific implementation, step S10 in the method specifically includes the following steps: Step S10-1: Discrete points on the three-dimensional potential instability surface sliding contour of the foundation t Placed on the three-dimensional potential instability surface sliding contour of the foundation x The intersection point of the positive axis, t = 0, and let the point A j and Point B j The identification factor isP AB ,and P AB = 0; Step S10-2: Obtaining discrete points on the three-dimensional potential unstable surface sliding contour of the foundation t of x and y Axis coordinates x t and y t And discrete points t +1 x and y Axis coordinates x t+1 and y t+1 ; Step S10-3: Determine whether y t ≤ y min + j × ( y max – y min ) / m < y t+1 or y t ≥ y min + j × ( y max – y min ) / m > y t+1 , if satisfied, then let j The intersection of the line of travel and the sliding contour of the three-dimensional potential instability surface of the foundation x The axis coordinates are x j ,and , if not satisfied, jump to step S10-5; Step S10-4: If P AB = 0, then let as well as P AB = 1; if P AB = 1, then let as well as P AB =2; if P ABIf = 2, then the sliding contour line of the three-dimensional potential instability surface of the foundation is unreasonable, and the calculation ends; Step S10-5: If t <Num_hz, then take t = t + 1, and repeat Steps S10-2 to S10-4; otherwise, end the calculation; Step S10-6: If , then let and , otherwise, let and ; Step S10-7: Output the A j and the B j of the x axis coordinates and .

[0015] In a specific embodiment, Step S11 in the method specifically includes the following steps: Step S11-1: Use Equation (24) to calculate the starting polar radius r 0_ j ; (24) Step S11-2: Use Equation (25) to calculate the ending polar radius r 1_ j ; (25) In the formula, and are respectively the A j and the B j of the x axis coordinates; Step S11-3: Use Equation (26) to calculate the starting polar angle or 0_ j ; (26) Step S11-4: Use Equation (27) to calculate the ending polar angle or 1_j ; (27) Step S11-5: Use Equation (28) to calculate the shape control parameter g j ; (28) Step S11-6: Output two-dimensional curve j Parameters, i.e. r 0_ j , r 1_ j , or 0_ j , or 1_j and g j .

[0016] In a specific implementation, step S13 in the method specifically includes the following steps: Step S13-1: or ij Calculate variables for the iteration loop and take or ij The initial value is or ij (0) , and when first calculated, or ij (0) = 90°; Step S13-2: Use equation (32) and substitute or ij (0) , and thus obtain new or ij ; (32) Step S13-3: If | or ij – or ij (0) | ≤ e 2 , here e 2 Take 0.001°, then it is considered or ij is the final calculation result; otherwise, let or ij (0) = or ij , repeat step S13-2; Step S13-4: Output any point on the three-dimensional potential instability surface of the foundation s ij In vertical section j The corresponding sliding surface polar angle or ij .

[0017] The advantages of the invention are that the three-dimensional potential instability surface of the foundation is only controlled by the two-dimensional potential instability surface of the foundation and the sliding contour of the three-dimensional potential instability surface of the foundation. At the same time, the instability and damage characteristics of the foundation can be reflected by the shape of the two-dimensional potential instability surface of the foundation, and the three-dimensional potential instability range of the foundation can be rationalized based on the sliding contour of the three-dimensional potential instability surface of the foundation. Therefore, the three-dimensional potential instability surface model of the foundation is easy to be constructed and the effectiveness of its generation can be ensured. In addition, when different parameters of the two-dimensional potential instability surface of the foundation and the sliding contour of the three-dimensional potential instability surface of the foundation are selected, the arbitrary construction of the shape of the three-dimensional potential instability surface of the foundation can be realized, thereby providing a prerequisite for determining a reliable and most dangerous three-dimensional potential instability surface of the foundation. Further, the three-dimensional potential instability surface of the foundation is generated in a discrete manner, which can be combined with the limit equilibrium method, and is easy to carry out foundation stability assessment under the action of complex foundation loads. Therefore, the invention has the advantages of simple implementation, wide application range, strong practicality, etc., can effectively improve the accuracy of foundation stability assessment, and provide strong scientific guidance for theoretical analysis and construction disposal of foundation engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 It is a schematic diagram of the three-dimensional potential instability surface of the foundation under the action of complex foundation loads of the present invention.

[0019] Figure 2 A schematic diagram of discretizing the complex basic outline contour and establishing a calculation coordinate system for the present invention.

[0020] Figure 3 It is a schematic diagram for solving the position of the center point of the complex foundation load of the present invention.

[0021] Figure 4 It is a schematic diagram of the intersection of the three-dimensional potential unstable surface sliding contour of the foundation and the coordinate axis of the present invention.

[0022] Figure 5 It is a schematic diagram of any point on the sliding contour of the three-dimensional potential unstable surface of the foundation of the present invention.

[0023] Figure 6 It is a schematic diagram of the discrete contour of the three-dimensional potential unstable surface sliding of the foundation of the present invention.

[0024] Figure 7 This is a schematic diagram of the two-dimensional potential instability surface of the foundation and its spatial expansion in the present invention. Figure 7 a is the three-dimensional potential unstable surface sliding contour of the foundation x Coordinates and y Schematic diagram of the corresponding points of the maximum and minimum coordinates, Figure 7 b is a schematic diagram of the two-dimensional potential instability surface of the foundation in three-dimensional space. Figure 7 c is the potential instability surface of the foundation in two dimensions in three dimensions y Schematic diagram of axis forward expansion, Figure 7 d is the potential instability surface of the foundation in two dimensions in three dimensions.y Schematic diagram of negative axis expansion.

[0025] Figure 8 It is a schematic diagram of constructing a two-dimensional potential instability surface of the foundation according to the present invention.

[0026] Figure 9 The present invention is a flow chart of the steps for calculating the parameters of the two-dimensional potential instability surface of the foundation.

[0027] Figure 10 The vertical section of the present invention j Two-dimensional curve of three-dimensional potential instability surface of upper foundation j Schematic diagram.

[0028] Figure 11 The present invention j Flow chart for the implementation of coordinate calculation of the intersection point between the line of travel and the three-dimensional potential unstable surface sliding contour of the foundation.

[0029] Figure 12 The two-dimensional curve on the three-dimensional potential instability surface of the foundation of the present invention j Flowchart of parameter calculation steps.

[0030] Figure 13 Any point on the three-dimensional potential instability surface of the foundation of the present invention s ij Determine the schematic diagram.

[0031] Figure 14 Any point on the three-dimensional potential instability surface of the foundation of the present invention s ij Flowchart of the steps for calculating the sliding surface polar angle.

[0032] Figure 15 The present invention is a flow chart of the steps for modeling the three-dimensional potential instability surface of the foundation under the action of complex foundation loads.

[0033] In the figure: 1. Complex foundation contour line, 2. Complex foundation load, 3. Three-dimensional potential instability surface of foundation, 4. Foundation load area, 5. Adjacent unloaded area, 6. Foundation and foundation interaction plane, 7. Complex foundation load action center point, 8. Complex foundation contour line discrete points k , 9. Rotation angle r , 10. Unit load, 11. Three-dimensional potential instability surface sliding contour of foundation, 12. Discrete points of three-dimensional potential instability surface sliding contour of foundation t , 13. Two-dimensional potential instability surface of foundation, 14. Two-dimensional potential instability surface of foundation y The positive spatial expansion in the axial direction, 15. The two-dimensional potential instability surface of the foundation y Negative space expansion in the axial direction, 16, xz Plane, 17, j 18 vertical sectionsj , 19. Two-dimensional curve j , 20, no. i Column direction line, 21, any point on the three-dimensional potential instability surface of the foundation s ij , 22. Invalid point, 23. Valid point. DETAILED DESCRIPTION

[0034] In order to solve the above-mentioned technical method problems, the present invention adopts the spatial expansion of the two-dimensional potential instability surface of the foundation along the direction perpendicular to it to form a three-dimensional potential instability surface of the foundation, wherein the two-dimensional potential instability surface of the foundation reflects the failure and damage characteristics of the foundation, and is a general logarithmic spiral curve, and can be degenerated into a circular arc. The characteristic of the spatial expansion method is that when the spatial expansion is carried out in the vertical direction of the two-dimensional potential instability surface of the foundation, the two-dimensional curve of the three-dimensional potential instability surface of the foundation on any vertical section is the same type of curve as the two-dimensional potential instability surface of the foundation. At the same time, the spatial expansion and the three-dimensional potential instability range of the foundation are limited to the sliding perimeter of the three-dimensional potential instability surface of the foundation, wherein the sliding perimeter of the three-dimensional potential instability surface of the foundation is the intersection line of the three-dimensional potential instability surface of the foundation on the interaction plane between the foundation and the foundation, and the complex foundation shape can be used to wheel The contour is formed by expanding outward at a certain proportion and is a closed curve. The Fourier series is used to construct the function of the magnification coefficient of the sliding contour of the three-dimensional potential instability surface of the foundation. In addition, the sliding contour of the three-dimensional potential instability surface of the foundation is discretized by equal-angle increments, and the correlation between the discrete points of the sliding contour of the three-dimensional potential instability surface of the foundation and the contour line of the complex foundation is established, so as to realize the generalization, practicality and efficiency of the generation of the sliding contour of the three-dimensional potential instability surface of the foundation, and ensure the simplicity, practicality, effectiveness and reliability of the generation of the three-dimensional potential instability surface of the foundation. Furthermore, combined with the limit equilibrium method, within the reasonable range of the selected parameters of the three-dimensional potential instability surface of the foundation, the most dangerous three-dimensional potential instability surface of the foundation under the action of the complex foundation load corresponding to the minimum safety factor can be determined through optimization search.

[0035] The specific model building method and implementation process of the present invention are as follows: like Figure 1 As shown in the figure, a complex foundation is a foundation with a complex outer contour. The outer contour of a complex foundation is generally a convex shape with a limited geometric size range and is regular or irregular. The characteristic of a convex shape is that the line between any two points on a closed figure is completely within the figure, such as a circle and a rectangle. The complex foundation load is the vertical load acting on the foundation by the complex foundation. Taking into account the differences in superstructure layout and usage functions, the diversity of upper loads and the complexity of foundation types, the vertical load borne by the complex foundation and transmitted to the foundation (i.e., the complex foundation load) may be distributed in a non-uniform and asymmetric manner, thereby causing the three-dimensional potential instability surface of the foundation to exhibit complex geometric characteristics.

[0036] like Figure 1 As shown in the figure, according to a certain strength criterion, when the shear stress of the soil exceeds its shear strength, the soil will produce shear failure. For the foundation, under the action of complex foundation loads, the soil in the foundation load area tends to shear failure state under the action of load, and moves outward under the action of load compression, thereby forcing the soil in the adjacent unloaded area to bulge due to shear and form a connected three-dimensional unstable surface, which leads to the destruction of the foundation.

[0037] (3) If Figure 2 As shown, in the interaction plane between the foundation and the soil, an arbitrary point is taken as the origin to establish x ' y ′ axis coordinate system, let the center point of the complex foundation load be point O ,point O of x 'and y The '-axis coordinates are and At the same time, O Center point and parallel x The positive direction of the 'axis is the starting direction, and the angle increment is counterclockwise. d Draw multiple rays, the number of rays is Num_lk = 360° / d – 1. Usually d Take 1°, and then use this through the center point O The intersection of the ray and the complex basic contour line, the complex basic contour line is discretized into Num_lk + 1 points in sequence, and on this basis, the corresponding complex basic contour line discrete points are extracted k ( k = 0, 1, 2, …, Num_lk) x 'and y ' axis coordinates, that is and , further, x ' y The origin of the ′-axis coordinate system is moved to the center point of the complex foundation load O , and rotate counterclockwise x ' y The 'axis coordinate system is established based on this xyz Axis calculation coordinate system, where x Axis and x The angle between the axes is the rotation angle r ,and r The value range of is [0°, 180°], so we can get xyz Discrete points of complex basic contour lines in the axis coordinate system k The coordinates are: (1) (2) In the formula, x k and y k They are the discrete points of the complex basic contour line k of x and y Axis coordinates.

[0038] like Figure 3 As shown, in x ' y In the rectangular coordinate system of the ′ axis, the complex foundation load is expressed as a unit load, and the distribution function of the unit load is q ( x ′, y ′), then the center point of the complex foundation load O of x 'and y ′ axis coordinate and The center of gravity calculation formula can be used to solve the problem, that is, using the unit load distribution function q ( x ′, y ′) replaces the density distribution function in the gravity center calculation formula, and then, within the complex foundation contour, the unit load distribution function is used q ( x ′, y ′) and its point of action x ' axis (or y The integral of the product of the coordinates of the load distribution function over the unit area divided by the unit load distribution function q ( x ′, y ′) is integrated over the unit area, and then the (or ).

[0039] like Figure 4 As shown, in xyz In the axis calculation coordinate system, the complex foundation contour line and x The left and right intersection points of the axis are points A ′ and dot B ′ and y The lower and upper intersection points of the axis are points C ′ and dot D ′, the three-dimensional potential instability surface of the foundation and xyz The intersection of the planes is the three-dimensional potential instability surface sliding contour of the foundation. x The left and right intersection points of the axis are points A and Point B andy The lower and upper intersection points of the axis are points C and Point D Compared with the complex foundation contour line, the three-dimensional potential instability surface sliding contour line of the foundation is larger than its range, and the three-dimensional potential instability surface sliding contour line of the foundation can be regarded as a closed curve formed by the complex foundation contour line expanding outward at a certain proportion. A , which can be a point on the contour line of the complex basic shape A 'along OA ' direction according to the length magnification factor l 1 The points obtained by extending outward are the points on the sliding contour of the three-dimensional potential instability surface of the foundation. B , which can be a point on the contour line of the complex basic shape B 'along OB ' direction according to the length magnification factor l 2 The points obtained by extending outward are the points on the sliding contour of the three-dimensional potential instability surface of the foundation. C , which can be a point on the contour line of the complex basic shape C 'along OC ' direction according to the length magnification factor l 3 The points obtained by extending outward are the points on the sliding contour of the three-dimensional potential instability surface of the foundation. D , which can be a point on the contour line of the complex basic shape D 'along OD ' direction according to the length magnification factor l 4 The points obtained by extending outward are, l 1 = OA / OA '、 l 2 = OB / OB '、 l 3 = OC / OC 'and l 4 = OD / OD ',and l 1 ≥ 1, l 2 ≥ 1, l 3 ≥ 1 and l 4 ≥ 1, further, based on the coordinates of discrete points of the complex basic contour line, and based on the point A For pointA 'along OA ' direction according to the length magnification factor l 1 The points obtained by extending outwards can be obtained by introducing the linear interpolation technique. A of x The axis coordinates are: (3) In the formula, x A For point A of x Axis coordinates; x k and x k+1 They are discrete points on the contour line of the complex basic shape. k and k +1 x axis coordinates, and , int is the rounding function.

[0040] Similarly, you can get some B of x The axis coordinates are: (4) In the formula, x B For point B of x Axis coordinates; x k and x k+1 They are discrete points on the contour line of the complex basic shape. k and k +1 x axis coordinates, and , int is the rounding function.

[0041] like Figure 5 As shown, the point P is any point on the three-dimensional potential instability surface sliding contour of the foundation, which can be obtained from the points on the contour line of the complex foundation P 'along OP ' direction according to the length magnification factor l P ( l P ≥ 1) is extended outward, that is OP = OP ′× l P At the same time, considering the length magnification factor l P and OP and xThe counterclockwise angle between the positive axes i P There is a functional relationship, and the sliding contour of the unstable surface is a continuous smooth closed curve. Therefore, the period of the function is [0, 2 π ], for any periodic function, it can be represented by a Fourier series. Here, the sum of the binomial Fourier series and the high-frequency remainder is used to construct l P about i P The function is: (5) In the formula, a 0 , a 1 , a 2 , a 3 , b 1 , b 2 , b 3 and oh All of them are the influencing factors of the shape of the sliding contour of the three-dimensional potential unstable surface of the foundation, among which, a 0 , a 1 , b 1 and b 2 Based on points A Length magnification factor l 1 ,point B Length magnification factor l 2 ,point C Length magnification factor l 3 and Point D Length magnification factor l 4 To solve, a 2 , a 3 and b 3 is any real number between -5 and 5, oh It is any integer from 3 to 10.

[0042] In formula (5), when i P = 0° (i.e. point B When l P = l 2 , and then we can get: (6) In formula (5), when i P = 90° (i.e. point D When l P = l 4 , and then we can get: (7) In formula (5), when i P = 180° (i.e. point A When l P = l 1 , and then we can get: (8) In formula (5), when i P = 270° (i.e. point C When l P = l 3 , and then we can get: (9) Combining equations (6) to (9) we can get a 0 , a 1 , b 1 and b 2 The calculation formulas are: (10) (11) (12) (13) Furthermore, based on the coordinates of the discrete points of the complex basic contour line, and based on the point P For point P 'along OP ' direction according to the length magnification factor l P The points obtained by extending outwards can be obtained by introducing the linear interpolation technique. P of x and yThe axis coordinates are: (14) (15) In the formula, x P and y P Points P of x and y Axis coordinates; x k and y k are discrete points on the contour line of the complex basic shape k of x and y axis coordinates, and , int is the rounding function; x k+1 and y k+1 are discrete points on the contour line of the complex basic shape k +1 x and y Axis coordinates.

[0043] like Figure 6 As shown, it is similar to the discretization of complex basic contour lines. O Center point and parallel x The positive direction of the axis is the starting direction, and the angle increment is counterclockwise. ψ Make multiple rays, and then use this to pass through the center point O The intersection of the ray and the three-dimensional potential instability surface sliding contour of the foundation is discretized into Num_hz points in sequence, and Num_hz = 360° / ψ – 1. Usually ψ It can also be taken as 1°. Then, for the discrete points of the sliding contour of the three-dimensional potential instability surface of the foundation t ( t = 0, 1, 2, ..., Num_hz), and compare it with the point O is connected, then the line is connected to x The counterclockwise angle between the positive axes is i t ,and i t = ( t / Num_hz) × or Then, according to any point on the sliding contour of the three-dimensional potential instability surface of the foundation P of x and y The axis coordinate calculation formula is it Replace with i P and x P and y P Replace with x t and y t , then the discrete points of the sliding contour of the three-dimensional potential instability surface of the foundation can be calculated using equations (14) and (15): t of x and y Axis coordinates, where x t and y t They are the discrete points of the three-dimensional potential unstable surface sliding contour of the foundation t of x and y Axis coordinates.

[0044] like Figure 7 As shown in the figure, based on the discrete points of the three-dimensional potential unstable surface sliding contour of the foundation t ( t = 0, 1, 2, ..., Num_hz) x and y Axis coordinates, obtain the three-dimensional potential unstable surface sliding contour of the foundation xyz On the plane y Minimum axis coordinate y min and maximum value y max and its corresponding point E and Point F , and the three-dimensional potential instability surface sliding contour of the foundation xyz On the plane x Minimum axis coordinate x min and maximum value x max and its corresponding point G and Point H At the same time, the three-dimensional potential instability surface of the foundation can be regarded as the two-dimensional potential instability surface of the foundation along the direction perpendicular to it (i.e. y The three-dimensional surface formed by the spatial expansion of the foundation (axis direction), among which the two-dimensional potential instability surface of the foundation is located in the xyz Axis calculation coordinate system x In the vertical plane of the axis, the three-dimensional space expansion method is characterized by y When the space is expanded in the axial direction, the two-dimensional curve of the three-dimensional potential instability surface of the foundation on any vertical section is the same type of curve as the two-dimensional potential instability surface of the foundation.

[0045] like Figure 8 As shown, the vertical direction is z Axis, xyz The axis two-dimensional calculation coordinate system is extended to x.y.z Axis three-dimensional calculation coordinate system, for the two-dimensional potential instability surface of the foundation, it is located at xz In the plane, and can be around the polar coordinate center point T The general logarithmic spiral curve is used to represent the point T To any point on the two-dimensional potential instability surface of the foundation S The length of the connecting line is the sliding surface polar diameter r And point T and Point S Connect at x The horizontal inclination angle in the axial direction is the sliding surface polar angle or , where the sliding surface polar angle or For counterclockwise angles, it takes a positive value. Based on this, xz The polar coordinate equation of the two-dimensional potential instability surface of the foundation on the plane is: (16) In the formula, r 0 is the starting pole diameter of the two-dimensional potential instability surface of the foundation, that is, point A The extreme diameter of the sliding surface; g is the shape control parameter of the two-dimensional potential instability surface of the foundation, when x = 0, the two-dimensional potential instability surface of the foundation is in the shape of an arc; or 0 is the starting polar angle of the two-dimensional potential instability surface of the foundation, that is, point A The extreme angle of the sliding surface.

[0046] Let the termination angle of the two-dimensional potential instability surface of the foundation be or 1 and the terminal diameter is r 1 , that is, the corresponding points B Then, based on the triangle TAB The geometric relationship between the two-dimensional potential instability surface of the foundation can be established to establish the starting polar angle or 0 and the end polar angle or 1 The calculation formulas are: (17) (18) In the formula, x A and x B Points A and PointB of x Axis coordinates.

[0047] Using the polar coordinate equation of the two-dimensional potential instability surface of the foundation, that is, equation (16), the termination polar diameter of the sliding surface can be established: r 1 The calculation formula is: (19) At the same time, the sliding surface parameters (i.e. the starting polar angle) of the two-dimensional potential instability surface of the foundation are obtained. or 0 , End polar angle or 1 , starting diameter r 0 and the end pole diameter r 1 ) can be further used to TAB The geometric relationship of T of x , y and z Axis coordinates, the specific calculation formulas are: (20) (twenty one) (twenty two) In the formula, x T , y T and z T Center point T of x , y and z Axis coordinates.

[0048] like Figure 9 As shown, according to formula (17) to formula (19), given r 0 and g Under this condition, the two-dimensional potential instability surface parameter iteration cycle calculation strategy can be used to solve or 0 , or 1 and r 1 , further, use equations (20) to (22) to solve the center point T of x , y and z The specific calculation steps (called the calculation steps of the two-dimensional potential instability surface parameters of the foundation) are as follows: ① Let or 0 and or 1 Calculate variables for the iteration loop and take or 0 and or 1 The initial values ​​are or 0 (0) and or 1 (0) , and when first calculated or 0 (0) = 0 and or 1 (0) = 0; ② Calculate using formula (19) r 1 ; ③ Use equation (17) and equation (18) to calculate the new or 0 and or 1 ;④ If | or 0 – or 0 (0) | ≤ e 1 and | or 1 – or 1 (0) | ≤ e 1 (Here is taken e 1 = 0.001°), then the calculated or 0 , or 1 and r 1 is the final result, otherwise, or 0 (0) = or 0 and or 1 (0) = or 1 , and repeat steps ② and ③; ⑤ Use equations (20) to (22) to solve the center point T of x axis, y Axis and z Axis coordinates x T ,y T and z T ⑥ Output the two-dimensional potential instability surface parameters of the foundation ( or 0 , or 1 , r 1 , x T , y T and z T ).

[0049] like Figure 10 As shown, in xyz In a plane, based on a point E (The three-dimensional potential instability surface sliding contour of the foundation is xyz On the plane y Minimum axis coordinate y min The corresponding point) and point F (The three-dimensional potential unstable surface sliding contour of the foundation is xyz On the plane y Maximum axis coordinate y max corresponding points), divided into equal intervals m +1 parallel to x The line of the axis, for the j The line y The axis coordinates are y min + j × ( y max – y min ) / m At the same time, j Vertical section along the line j , then the vertical section j The two-dimensional curve of the three-dimensional potential instability surface of the upper foundation (referred to as the two-dimensional curve j ) is the same type of curve as the two-dimensional potential instability surface of the foundation, thus, the two-dimensional curve j In vertical section j The polar coordinate equation on is: (twenty three) In the formula, r j For a two-dimensional curve j The polar coordinate center point of T j To 2D Curve j Any point onS j The length of the connection; g j For a two-dimensional curve j Shape control parameters; or j For point T j and Point S j Connect at x Horizontal inclination in the axial direction; r 0_ j and or 0_j Two-dimensional curves j The starting polar diameter and initial polar angle.

[0050] In the past j Vertical section of the row line j Up, click A j and Point B j Respectively j The left and right intersections of the line of travel and the sliding contour of the three-dimensional potential instability surface of the foundation. At this time, point T j Solstice A j The length of the connection and its x The horizontal inclination angles in the axial direction represent the two-dimensional curves j The starting diameter r 0_ j and the starting polar angle or 0_j ,point T j Solstice B j The length of the connection and its x The horizontal inclination angles in the axial direction represent the two-dimensional curves j The terminal diameter r 1_ j and the end polar angle or 1_j .

[0051] Further, considering the two-dimensional curve j Based on the same type of curve as the two-dimensional potential instability surface of the foundation, let the point T j Located at the polar coordinate center point of the two-dimensional potential instability surface through the foundation T And with y On a straight line parallel to the axis, at this time, there is and ,in, and Points T j of x Axis and z Axis coordinates. Thus, based on the triangle T j A j B j The geometric relationship can be used to establish a two-dimensional curve j Middle starting diameter r 0_ j , Ending pole diameter r 1_ j , starting polar angle or 0_j , End polar angle or 1_j and shape control parameters g j The calculation formulas are: (twenty four) (25) (26) (27) (28) In the formula, and Points A j and Point B j of x Axis coordinates.

[0052] like Figure 11 As shown, for the j The intersection points of the left and right sides of the line of travel and the sliding contour of the three-dimensional potential instability surface of the foundation (i.e., point A j and Point B j ),That x The axis coordinate calculation can be implemented as follows (referred to as the first j The solution steps for the intersection of the three-dimensional potential instability surface sliding contour of the foundation are as follows: ① The discrete points on the three-dimensional potential instability surface sliding contour of the foundation are t Placed on the three-dimensional potential instability surface sliding contour of the foundation x The intersection point of the positive axis, t = 0, and let the point A j and Point B j The identification factor is PAB , and P AB = 0; ② Obtain the discrete points on the slip circumference of the three-dimensional potential instability surface of the foundation t of x and y axis coordinates x t and y t and the discrete point t + 1 of x and y axis coordinates x t+1 and y t+1 ; ③ Determine whether it satisfies y t ≤ y min + j × ( y max – y min ) / m < y t+1 or y t ≥ y min + j × ( y max – y min ) / m > y t+1 , if satisfied, let the j -th row direction line and the x axis coordinate of the intersection point of the slip circumference of the three-dimensional potential instability surface of the foundation be x j , and , if not satisfied, jump to step ⑤; ④ If P AB = 0, then let and P AB = 1, if P AB = 1, then let and P AB = 2, if P AB = 2, then the slip circumference of the three-dimensional potential instability surface of the foundation is unreasonable, and the calculation ends; ⑤ If t < Num_hz, then take t = t+ 1, and repeat steps ② to ④, otherwise, end the calculation; ⑥ If , then let and , otherwise, let and ; ⑦ Output point A j and Point B j of x Axis coordinates and .

[0053] like Figure 12 As shown, in determining the j The intersection of the left and right sides of the line of the line and the three-dimensional potential instability surface sliding contour of the foundation x Axis coordinates (i.e. and ), the two-dimensional curve can be solved based on equations (24) to (28) j The starting diameter r 0_ j , Ending pole diameter r 1_ j , starting polar angle or 0_ j , End polar angle or 1_j and shape control parameters g j , the specific calculation steps (called the two-dimensional curve on the three-dimensional potential instability surface of the foundation j The parameter calculation steps are as follows: ① Using formula (24), calculate the starting diameter r 0_ j ② Using formula (25), calculate the terminal pole diameter r 1_ j ; ③ Using formula (26), calculate the starting polar angle or 0_ j ; ④ Using formula (27), calculate the end polar angle or 1_j ⑤ Using formula (28), calculate the shape control parameters g j ;⑥ Output two-dimensional curve j Parameters, i.e. r 0_ j , r 1_ j , or 0_ j , or 1_j and g j .

[0054] like Figure 13As shown, based on the point G (The three-dimensional potential unstable surface sliding contour of the foundation is xyz On the plane x Minimum axis coordinate x min The corresponding point) and point H (The three-dimensional potential unstable surface sliding contour of the foundation is xyz On the plane x Maximum axis coordinate x max corresponding points), divided into equal intervals n +1 parallel to y The column direction line of the axis, for the i The strip line x The axis coordinates are x min + i × ( x max – x min ) / n , corresponding to any point on the three-dimensional potential instability surface of the foundation s ij , which is xyz The projection on the plane is located at i Line and j The intersection of the three-dimensional potential instability surface of the foundation s ij of x and y The axis coordinates are: (29) (30) In the formula, x ij and y ij are any points on the three-dimensional potential instability surface of the foundation s ij of x and y Axis coordinates.

[0055] Furthermore, combined with the two-dimensional curve j In vertical section j The shape characteristics on the foundation can be obtained by s ij of z The axis coordinates are: (31) In the formula, z ijis any point on the three-dimensional potential instability surface of the foundation s ij of z Axis coordinates; or ij is any point on the three-dimensional potential instability surface of the foundation s ij In vertical section j The corresponding sliding surface polar angle.

[0056] At the same time, any point on the three-dimensional potential instability surface of the foundation can be used s ij and center point T j of x Axis coordinates to solve or ij , the specific calculation formula is: (32) It should be noted that if any point on the three-dimensional potential instability surface of the foundation s ij The projection on the interaction plane between the foundation and the soil is outside the sliding contour of the three-dimensional potential instability surface of the foundation (that is, any point on the three-dimensional potential instability surface of the foundation s ij Located above the interaction plane between the foundation and the subgrade, z ij >0), then point s ij It is not a point on the three-dimensional potential instability surface of the foundation, that is, point s ij is an invalid point, otherwise it is a valid point.

[0057] like Figure 14 As shown, for any point on the three-dimensional potential instability surface of the foundation s ij In vertical section j The corresponding sliding surface polar angle or ij , the iterative cycle calculation strategy can be used to solve the problem using formula (32). The specific solution steps (referred to as any point on the three-dimensional potential instability surface of the foundation) are: s ij The calculation steps of the sliding surface polar angle are as follows: ① Let or ij Calculate variables for the iteration loop and take or ij The initial value is or ij (0) , and when first calculated, or ij (0)= 90°; ② Use formula (32) and substitute or ij (0) , and thus obtain new or ij ;③ If | or ij – or ij (0) | ≤ e 2 (Here e 2 Take 0.001°), then it is considered or ij is the final calculation result, otherwise, let or ij (0) = or ij , repeat step ②; ④ Output any point on the three-dimensional potential instability surface of the foundation s ij In vertical section j The corresponding sliding surface polar angle or ij .

[0058] like Figure 15 As shown in Figure 2, the steps for modeling the three-dimensional potential instability surface of the foundation under complex foundation loads are as follows: ① Given xyz Axis calculation coordinate system rotation angle r , the magnification factor of the length of the sliding contour of the three-dimensional potential unstable surface of the foundation l 1 ~ l 4 , the influencing factors of the shape of the sliding contour of the three-dimensional potential unstable surface of the foundation ( a 2 , a 3 , b 3 and oh ), initial polar diameter of the two-dimensional potential instability surface of the foundation r 0 And the shape control parameters of the two-dimensional potential instability surface of the foundation g ; ② In the interaction plane between the foundation and the subgrade, establish x ' y ′ axis coordinate system, and use the gravity center calculation formula to obtain the center point of the complex foundation load O of x 'and y ′ axis coordinate and ; ③ Point O Center point and parallel xThe positive direction of the 'axis is the starting direction, and the angle increment is counterclockwise. d Draw multiple rays, and then use the intersection of the rays and the complex basic contour line to discretize the complex basic contour line and obtain the discrete points k ( k = 0, 1, 2, …, Num_lk) x 'and y ′ axis coordinate and , where Num_lk = 360° / d – 1; ④ At the center of load action O as the origin, and x 'and y 'Axis counterclockwise rotation angle r ,Establish xyz Axis calculation coordinate system, then, use equations (1) and (2) to calculate the discrete points of the complex basic contour line k of x and y Axis coordinates x k and y k ⑤ Using equations (3) and (4), determine the point A and Point B of x Axis coordinates x A and x B , where point A and Point B They are the three-dimensional potential unstable surface sliding contour of the foundation and x The intersection of the left and right sides of the axis; ⑥ Use the two-dimensional potential instability surface parameter calculation steps to solve the two-dimensional potential instability surface parameters of the foundation or 0 , or 1 , r 1 , x T , y T and z T ⑦ Use equations (10) to (13) to solve the influencing factors of the sliding contour shape of the three-dimensional potential instability surface of the foundation a 0 , a 1 , b 1 and b 2 ⑧ By point O Center point and parallelx The positive direction of the axis is the starting direction, and the angle increment is counterclockwise. ψ Draw multiple rays, and then use the intersection of the rays and the three-dimensional potential instability surface sliding contour of the foundation to discretize the three-dimensional potential instability surface sliding contour of the foundation, and use equations (14) and (15) to obtain the discrete points t ( t = 0, 1, 2, ..., Num_hz) x and y Axis coordinates x t and y t , where Num_hz = 360° / ψ – 1; ⑨ Discrete points based on the sliding contour of the three-dimensional potential unstable surface of the foundation t of x and y Axis coordinates, obtain the three-dimensional potential unstable surface sliding contour of the foundation xyz On the plane y Minimum axis coordinate y min and maximum value y max and its corresponding point E and Point F , and the three-dimensional potential instability surface sliding contour of the foundation xyz On the plane x Minimum axis coordinate x min and maximum value x max and its corresponding point G and Point H , at point E and Point F Divide equally between m +1 parallel to x The axis of the line, at point G and Point H Divide equally between n +1 parallel to y The columnar line of the axis corresponds to any point on the three-dimensional potential instability surface of the foundation s ij , which is xyz The projection on the plane is located at i Linear direction (0 ≤ i ≤ n ) and j Lines (0 ≤ j ≤ m ) at the intersection point; ⑩ Using the j The steps for solving the intersection of the line and the three-dimensional potential instability surface sliding contour of the foundation are to determine the point Aj and Point B j of x Axis coordinates and If the instability sliding contour is unreasonable, the three-dimensional potential instability surface of the foundation is unreasonable, and the calculation is terminated; Using the two-dimensional curve on the three-dimensional potential instability surface of the foundation j Parameter calculation steps to solve two-dimensional curves j Related parameters r 0_ j , r 1_ j , or 0_ j , or 1_j and g j ; Using equations (29) and (30), we can calculate the potential instability surface of the foundation at any point on the foundation. s ij of x Axis coordinates x ij and y Axis coordinates y ij ; Using any point on the three-dimensional potential instability surface of the foundation s ij The calculation steps of the sliding surface polar angle are solved or ij ; Formula (31) is used to solve the problem of any point on the three-dimensional potential instability surface of the foundation s ij of z Axis coordinates z ij ; Output the calculation results of the three-dimensional potential instability surface of the foundation.

[0059] In addition, when given xyz Axis calculation coordinate system rotation angle r , the magnification factor of the length of the sliding contour of the three-dimensional potential unstable surface of the foundation l 1 ~ l 4 , shape control parameters of potential two-dimensional instability surface of foundation g , the influencing factors of the shape of the sliding contour of the three-dimensional potential unstable surface of the foundation ( a 2 , a 3 , b 3 and oh), initial polar diameter of the two-dimensional potential instability surface of the foundation r 0 And the shape control parameters of the two-dimensional potential instability surface of the foundation g When the parameters are within a reasonable range, a series of three-dimensional potential instability surfaces of the foundation under complex foundation loads can be generated based on the combination of these parameters. At the same time, the safety factors corresponding to these three-dimensional potential instability surfaces of the foundation are calculated in combination with the limit equilibrium method. Furthermore, the minimum value of the safety factor is taken as the optimization target, and finally the most dangerous three-dimensional potential instability surface of the foundation under complex foundation loads can be determined.

[0060] The characteristics of the three-dimensional potential instability surface of the foundation of the present invention are: the three-dimensional potential instability surface of the foundation is a three-dimensional surface formed by spatially expanding the two-dimensional potential instability surface of the foundation in a direction perpendicular to it, and at the same time, the spatial expansion is controlled within the sliding contour of the three-dimensional potential instability surface of the foundation, thereby, the instability and destruction characteristics of the foundation can be reflected by applying the morphology of the two-dimensional potential instability surface of the foundation, and the three-dimensional potential instability range of the foundation can be rationalized based on the sliding contour of the three-dimensional potential instability surface of the foundation, thereby ensuring the simplicity, practicality, effectiveness and reliability of the generation of the three-dimensional potential instability surface of the foundation.

[0061] The characteristics of the three-dimensional potential instability surface sliding contour of the foundation of the present invention are: the three-dimensional potential instability surface sliding contour of the foundation is a closed curve formed by expanding the contour line of the complex foundation outward according to a certain proportion. Further, the function of the magnification coefficient of the three-dimensional potential instability surface sliding contour of the foundation is constructed by using Fourier series. At the same time, the three-dimensional potential instability surface sliding contour of the foundation is discretized by using equal angle increments, and the correlation relationship between the discrete points of the three-dimensional potential instability surface sliding contour of the foundation and the contour line of the complex foundation is established. In addition, by limiting the number of intersections between the direction line and the three-dimensional potential instability surface sliding contour of the foundation, the rationality of the characteristics of the three-dimensional potential instability surface sliding contour of the foundation is guaranteed, thereby realizing the generalization, practicality and efficiency of the generation of the three-dimensional potential instability surface sliding contour of the foundation. The characteristics of the two-dimensional potential instability surface of the foundation of the present invention are: the two-dimensional potential instability surface of the foundation is a general logarithmic spiral curve, which is simple in form, convenient in calculation, and can be degenerated into a circular arc, which conforms to the actual foundation failure and damage characteristics. Example

[0062] A kind of Figure 1~Figure 15 The modeling method of the three-dimensional potential instability surface of the foundation under the action of complex foundation loads is shown in Figure 1. The case foundation is a circular foundation with a corresponding radius of 1 m. The circular foundation is subjected to a conical distributed load formed by the action of the superstructure. The load size is linearly related to the distance between its action position and the center of the circle. The distributed load at the center of the circular foundation is 300 kPa, and the distributed load at the edge of the circular foundation is 200 kPa. When an arbitrary point is used as the origin in the interaction plane between the foundation and the foundation, a three-dimensional potential instability surface modeling method of the foundation under the action of complex foundation loads is shown in Figure 1. x ' y ′ axis coordinate system, the load distribution function is , the unit is kPa, where and The center points of the circular base are x 'and y ' axis coordinates. In order to ensure the safety of the upper building structure and provide a scientific basis for carrying out necessary foundation reinforcement measures, it is necessary to accurately and effectively analyze the stability of the foundation and obtain the most dangerous three-dimensional potential instability surface of the foundation when it is unstable and damaged. To this end, the three-dimensional potential instability surface modeling method of the foundation under complex foundation loads of the present invention is applied, and the specific operations are as follows: According to the requirements of the Code for Geotechnical Engineering Investigation (GB50021-2001), the foundation was field-tested to obtain the strength parameters of the rock and soil corresponding to the foundation bearing layer. The foundation rock and soil can be regarded as a homogeneous body, and the shear failure of the rock and soil obeys the linear MC strength criterion. The soil strength parameter is c = 33 kPa and f = 15°; set up xyz Axis calculation coordinate system rotation angle r The value range is 0° ~ 180°, which sets the shape control parameters of the potential two-dimensional instability surface of the foundation. g The value range is -1 ~ 1, the starting diameter r 0 The value range of is 1 m ~ 5 m. In addition, the magnification factor of the length of the sliding contour of the three-dimensional potential unstable surface of the foundation is set. l 1 , l 2 , l 3 and l 4 The value range is 1 ~ 5, and the influencing factor of the shape of the sliding contour of the three-dimensional potential unstable surface of the foundation is set. a 2 , a 3 , b 3 The value range is -5 to 5. oh The value range is 3 ~ 10, and the angle increment is given d = 1° and ψ = 1°, and at the same time, the three-dimensional potential unstable surface sliding contour of the foundation is divided into equal intervals m +1 parallel to x Axis lines and equally spaced divisions n +1 parallel to y The columnar lines of the axis, where m = 50 and n= 50, and then, within the given parameter range, a series of three-dimensional potential instability surfaces of the foundation can be generated for foundation stability analysis; Combined with the limit equilibrium method and embedded in the mathematical optimization algorithm, the foundation stability analysis is carried out. In this process, when any set of potential two-dimensional instability surface shape control parameters of the foundation is given, g , starting diameter r 0 And the magnification factor of the length of the sliding contour of the three-dimensional potential unstable surface of the foundation ( l 1 , l 2 , l 3 , l 4 ) and shape influence factor ( a 2 , a 3 , b 3 and oh ), a reasonable three-dimensional potential instability surface of the foundation can be generated according to the modeling steps of the three-dimensional potential instability surface of the foundation under the action of complex foundation loads. Subsequently, the limit equilibrium method is used to solve the corresponding safety factor. Then, with the assistance of the mathematical optimization algorithm, the minimum safety factor is used as the optimization target, and the most dangerous three-dimensional potential instability surface of the foundation is searched out from the generated series of three-dimensional potential instability surfaces. The calculation results of this case show that the minimum safety factor of the foundation is 1.88, and the parameters of the most dangerous three-dimensional potential instability surface of the foundation are: shape control parameters of the two-dimensional potential instability surface of the foundation g = 0.09 and the starting pole diameter r 0 = 1.9m, magnification factor of the length of the three-dimensional potential unstable surface sliding contour of the foundation l 1 = 1.74, l 2 = 1.82, l 3 = 1.79, l 4 = 1.77 and shape influence factor a 2 = 0.08, a 3 = -0.01, b 3 = 0.02 and oh = 7. It should be noted that in this case, since the foundation contour and foundation load are symmetrical and the center point of the foundation load is located at the center of the circular foundation, xyz Axis calculation coordinate system rotation angle rIt will not affect the calculation results and can be any value within the calculation range; Based on the foundation stability analysis results and in accordance with the foundation stability requirements of the Code for Design of Building Foundations (GB50007-2011), the foundation is assessed to be in a stable state. At the same time, the calculation results identify the potential instability area of ​​the foundation, which can provide a scientific basis for the implementation of subsequent foundation reinforcement measures.

[0063] The present invention relates to a modeling method for a three-dimensional potential instability surface of a foundation under a complex foundation load, comprising: y The three-dimensional potential instability surface of the foundation is formed by spatial expansion in the axial direction, and the two-dimensional potential instability surface of the foundation is used to reflect the failure and damage characteristics of the foundation, and the spatial expansion and the three-dimensional potential instability range of the foundation are limited within the sliding contour of the three-dimensional potential instability surface of the foundation; by changing the parameters of the two-dimensional potential instability surface of the foundation and the sliding contour of the three-dimensional potential instability surface of the foundation, it is used to construct a three-dimensional potential instability surface of the foundation in any shape. The present invention generates the three-dimensional potential instability surface of the foundation in a discrete manner, making it easy to apply to the construction of the three-dimensional potential instability surface of the foundation under complex foundation loads, thereby solving the problems of universality, simplicity, efficiency, practicality, effectiveness and reliability of the three-dimensional potential instability surface modeling of the foundation and the three-dimensional stability analysis of the foundation under complex conditions.

[0064] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A modeling method for three-dimensional potential instability surface of foundation under complex foundation load, characterized in that: The method comprises constructing a three-dimensional potential instability surface model of a foundation by using a two-dimensional potential instability surface (13) of the foundation and a three-dimensional potential instability surface sliding contour (11), wherein the two-dimensional potential instability surface (13) of the foundation is located in a rectangular coordinate system and is parallel to the three-dimensional potential instability surface of the foundation. y In a vertical plane perpendicular to the axis, the two-dimensional potential instability surface (13) of the foundation is generated in a discrete manner and is a general logarithmic spiral curve; the three-dimensional potential instability surface sliding curve (11) of the foundation is a curve located on the interaction plane (6) between the foundation and the foundation; the method comprises using the two-dimensional potential instability surface (13) of the foundation to y The three-dimensional potential instability surface of the foundation is formed by spatial expansion in the axial direction, and the two-dimensional potential instability surface of the foundation (13) is used to reflect the failure and damage characteristics of the foundation; The two-dimensional potential instability surface (13) of the foundation is along y The spatial expansion in the axial direction and the three-dimensional potential instability range of the foundation are both limited within the sliding contour of the three-dimensional potential instability surface of the foundation (11); by changing the parameters of the two-dimensional potential instability surface of the foundation (13) and the sliding contour of the three-dimensional potential instability surface of the foundation (11), a three-dimensional potential instability surface of the foundation in any shape can be constructed.

2. The modeling method of three-dimensional potential instability surface of foundation under complex foundation load according to claim 1 is characterized in that: In the method, firstly given xy Rotation angle in axis coordinate system ρ The range of the length magnification factor of the sliding contour (11) of the three-dimensional potential instability surface of the given foundation λ 1~ λ 4, the shape influence factor of the sliding contour (11) of the three-dimensional potential instability surface of the given foundation a 2. a 3. b 3 and ω The range of the initial polar diameter of the two-dimensional potential instability surface (13) of the given foundation r 0, and the shape control parameters of the two-dimensional potential instability surface (13) of the given foundation ζ range; based on the mutual combination of these parameters, a series of three-dimensional potential instability surfaces of the foundation under the action of complex foundation loads are generated, and the safety factors corresponding to these three-dimensional potential instability surfaces of the foundation are calculated in combination with the limit equilibrium method, and the minimum value of the safety factor is taken as the optimization target, so as to determine the most dangerous three-dimensional potential instability surface of the foundation under the action of complex foundation loads.

3. The modeling method of three-dimensional potential instability surface of foundation under complex foundation load according to claim 2 is characterized in that: In the method, the rotation angle ρ The value range is 0°~180°, and the length magnification factor of the three-dimensional potential instability surface sliding contour (11) of the foundation is λ 1~ λ The value range of 4 is 1~5. The shape influence factor of the sliding contour of the three-dimensional potential unstable surface of the foundation (11) a 2. a 3 and b The value range of 3 is -5~5. ω The value range of is 3~10, and the initial polar diameter of the two-dimensional potential instability surface (13) of the foundation r The value range of 0 is 1m~5m, and the shape control parameter of the two-dimensional potential instability surface (13) of the foundation ζ The value range is -1~1.

4. The modeling method of three-dimensional potential instability surface of foundation under complex foundation load according to claim 2 is characterized in that: The three-dimensional potential unstable surface sliding contour (11) of the foundation is a closed curve formed by expanding the complex foundation contour line (1) outward at a certain ratio. Specifically, the length magnification factor λ 1~ λ 4 are respectively the contour lines of the complex foundation (1) and x Negative axis direction, x Positive axis direction, y Negative axis direction, y The positive expansion in the axial direction forms the proportional coefficient of the three-dimensional potential instability surface sliding contour (11) of the foundation; and the Fourier series is used to construct the function of the magnification coefficient of the three-dimensional potential instability surface sliding contour (11) of the foundation, and the three-dimensional potential instability surface sliding contour (11) of the foundation is discretized by using the equal angle increment method, and the discrete points of the three-dimensional potential instability surface sliding contour of the foundation are established. t The correlation relationship with the complex foundation outer contour line (1) is also achieved by limiting the number of intersections between the direction line and the three-dimensional potential instability surface sliding contour line (11) of the foundation to ensure the rationality of the characteristics of the three-dimensional potential instability surface sliding contour line (11) of the foundation.

5. The modeling method of three-dimensional potential instability surface of foundation under complex foundation load according to claim 2 is characterized in that: The method comprises the following steps, Step S1: Given xy Axis calculation coordinate system rotation angle ρ , the magnification factor of the length of the sliding contour of the three-dimensional potential unstable surface of the foundation λ 1~ λ 4. Influencing factors of the shape of the sliding contour of the three-dimensional potential unstable surface of the foundation a 2. a 3. b 3 and ω , the initial polar diameter of the two-dimensional potential instability surface of the foundation r 0 and the shape control parameters of the two-dimensional potential instability surface of the foundation ζ ; Step S2: Establish a x ' y ′ axis coordinate system, and use the gravity center calculation formula to obtain the center point of the complex foundation load O of x 'and y ′ axis coordinate and ; Step S3: Point O Center point and parallel x The positive direction of the 'axis is the starting direction, and the angle increment is counterclockwise. δ Draw multiple rays, and then use the intersection of the rays and the complex basic contour line to discretize the complex basic contour line and obtain the discrete points k of x 'and y ′ axis coordinate and ,in, k = 0,1,2,…,Num_lk,Num_lk=360° / δ – 1; Step S4: Load action center point O as the origin, and x 'and y 'Axis counterclockwise rotation angle ρ ,Establish xy Axis calculation coordinate system, then, use equations (1) and (2) to calculate the discrete points of the complex basic contour line k of x and y Axis coordinates x k and y k ; (1) (2) Step S5: Using equations (3) and (4), determine the point A and Point B of x Axis coordinates x A and x B , where point A and Point B They are the three-dimensional potential unstable surface sliding contour of the foundation and x The left and right intersection of the axis; (3) (4) In the formula, x k and x k+1 are discrete points on the contour line of the complex basic shape k and k +1 x Axis coordinates, and in formula (3) , and in formula (4) , int is the rounding function; Step S6: Using the foundation two-dimensional potential instability surface parameter calculation step to solve the foundation two-dimensional potential instability surface parameter η 0. η 1. r 1. x T , y T and z T ; in, η 0 is the starting polar angle of the two-dimensional potential instability surface of the foundation, that is, point A The extreme angle of the sliding surface; η 1 is the termination polar angle of the two-dimensional potential instability surface of the foundation, that is, point B The extreme angle of the sliding surface; r 1 is the termination pole diameter of the two-dimensional potential instability surface of the foundation, that is, point B The extreme diameter of the sliding surface at ; x T , y T and z T The polar coordinate center points T of x Axis coordinates, y Axis coordinates and z Axis coordinates; Step S7: Use equations (10) to (13) to solve the influencing factors of the sliding contour shape of the three-dimensional potential unstable surface of the foundation a 0. a 1. b 1 and b 2; (10) (11) (12) (13) In the formula, a 0. a 1. a 2. a 3. b 1. b 2. b 3 and ω All of them are the influencing factors of the shape of the sliding contour of the three-dimensional potential unstable surface of the foundation, among which, a 0. a 1. b 1 and b 2 Based on the point A Length magnification factor λ 1. Point B Length magnification factor λ 2. Point C Length magnification factor λ 3 and Dot D Length magnification factor λ 4 Solved and obtained, a 2. a 3 and b 3 is any real number between -5 and 5, ω Any integer from 3 to 10; Step S8: Point O Center point and parallel x The positive direction of the axis is the starting direction, and the angle increment is counterclockwise. ψ Draw multiple rays, and then use the intersection of the rays and the three-dimensional potential instability surface sliding contour of the foundation to discretize the three-dimensional potential instability surface sliding contour of the foundation, and use equations (14) and (15) to obtain the discrete points t of x and y Axis coordinates x t and y t ,in, t = 0,1,2,…,Num_hz,Num_hz=360° / ψ – 1; (14) (15) in, t are discrete points; P is a continuous point; where x P and y P Points P of x and y Axis coordinates; λ P Points on the contour line of the complex foundation P 'along OP ' extends outward to point P The length magnification factor of λ P ≥1; θ P for OP and x The counterclockwise angle between the positive axes; δ is the angle increment; x k and y k are discrete points on the contour line of the complex basic shape k of x and y axis coordinates, and , int is the rounding function; x k+1 and y k+1 are discrete points on the contour line of the complex basic shape k +1 x and y Axis coordinates; In equations (14) and (15), the subscript P use t Instead, at the same time, numerically θ P use t × ψ Instead, the discrete points can be solved t of x and y Axis coordinates x t and y t ; Step S9: Discrete points based on the contour of the three-dimensional potential unstable surface sliding of the foundation t of x and y Axis coordinates, obtain the three-dimensional potential unstable surface sliding contour of the foundation xy On the plane y Minimum axis coordinate y min and maximum value y max and its corresponding point E and Point F , and the three-dimensional potential instability surface sliding contour of the foundation xy On the plane x Minimum axis coordinate x min and maximum value x max and its corresponding point G and Point H , at point E and Point F Divide equally between m +1 parallel to x The axis of the line, at point G and Point H Divide equally between n +1 parallel to y The columnar line of the axis corresponds to any point on the three-dimensional potential instability surface of the foundation s ij , which is xy The projection on the plane is located at i Line and j At the intersection of the lines, where 0 ≤ i ≤ n , 0 ≤ j ≤ m ; Step S10: Using j The steps for solving the intersection of the line and the three-dimensional potential instability surface sliding contour of the foundation are to determine the point A j and Point B j of x Axis coordinates and ; If the unstable sliding contour is unreasonable, the three-dimensional potential unstable surface of the foundation is unreasonable, and the calculation is terminated; Step S11: Using the two-dimensional curve on the three-dimensional potential instability surface of the foundation j Parameter calculation steps to solve two-dimensional curves j Related parameters r 0_j , r 0_j、 r 1_j , η 0_ j , η 1_j and ζ j ; Step S12: Use equations (29) and (30) to calculate the value of any point on the three-dimensional potential instability surface of the foundation: s ij of x Axis coordinates x ij and y Axis coordinates y ij ; (29) (30) Step S13: Using any point on the three-dimensional potential instability surface of the foundation s ij The calculation steps of the sliding surface polar angle are solved η ij , η ij is any point on the three-dimensional potential instability surface of the foundation s ij In vertical section j The corresponding sliding surface polar angle in ; Step S14: Use equation (31) to solve the problem of any point on the three-dimensional potential instability surface of the foundation s ij of z Axis coordinates z ij ; (31) In the formula, For a two-dimensional curve j The polar coordinate center point of T j of z Axis coordinates; Step S15: output the calculation result of the three-dimensional potential instability surface of the foundation.

6. The modeling method of three-dimensional potential instability surface of foundation under complex foundation load according to claim 5 is characterized in that: Step S6 in the method specifically includes the following steps: Step S6-1: η 0 and η 1 is the iterative loop calculation variable, and takes η 0 and η The initial values ​​of 1 are η 0 (0) and η 1 (0) , and when first calculated η 0 (0) = 0 and η 1 (0) = 0; Step S6-2: Calculate using formula (19) r 1; (19) Step S6-3: Use equation (17) and equation (18) to calculate the new η 0 and η 1; (17) (18) Step S6-4: If | η 0 – η 0 (0) | ≤ ε 1 and | η 1 – η 1 (0) | ≤ ε 1, here take ε 1 = 0.001°, then the calculated η 0. η 1 and r 1 is the final result, otherwise, η 0 (0) = η 0 and η 1 (0) = η 1, and repeat steps S6-2 and S6-3; Step S6-5: Use equations (20) to (22) to solve the center point T of x axis, y Axis and z Axis coordinates x T , y T and z T ; (20) (21) (22) Step S6-6: Output the two-dimensional potential instability surface parameters of the foundation η 0. η 1. r 1. x T , y T and z T .

7. The method for modeling a three-dimensional potential instability surface of a foundation under complex foundation loads according to claim 5, characterized in that: Step S10 in the method specifically includes the following steps: Step S10-1: Discrete points on the three-dimensional potential instability surface sliding contour of the foundation t Placed on the three-dimensional potential instability surface sliding contour of the foundation x The intersection point of the positive axis, t = 0, and let the point A j and Point B j The identification factor is P AB ,and P AB = 0; Step S10-2: Obtaining discrete points on the three-dimensional potential unstable surface sliding contour of the foundation t of x and y Axis coordinates x t and y t And discrete points t +1 x and y Axis coordinates x t+1 and y t+1 ; Step S10-3: Determine whether y t ≤ y min + j × ( y max – y min ) / m < y t+1 or y t ≥ y min + j × ( y max – y min ) / m > y t+1 , if satisfied, then let j The intersection of the line of travel and the sliding contour of the three-dimensional potential instability surface of the foundation x The axis coordinates are x j ,and , if not satisfied, jump to step S10-5; Step S10-4: If P AB = 0, then let as well as P AB = 1; if P AB = 1, then let as well as P AB = 2; if P AB = 2, the three-dimensional potential unstable surface sliding contour of the foundation is unreasonable, and the calculation ends; Step S10-5: If t < Num_hz, then take t = t + 1, and repeat steps S10-2 to S10-4; Otherwise, end the calculation; Step S10-6: If , then let and , otherwise, let and ; Step S10-7: Output point A j and Point B j of x Axis coordinates and .

8. The method for modeling a three-dimensional potential instability surface of a foundation under complex foundation loads according to claim 5, characterized in that: Step S11 in the method specifically includes the following steps: Step S11-1: Calculate the starting diameter using equation (24) r 0_ j ; (24) Step S11-2: Calculate the final diameter using equation (25) r 1_ j ; (25) In the formula, and Points A j and Point B j of x Axis coordinates; Step S11-3: Calculate the starting polar angle using equation (26): η 0_ j ; (26) Step S11-4: Calculate the end polar angle using equation (27): η 1_j ; (27) Step S11-5: Calculate the shape control parameters using equation (28) ζ j ; (28) Step S11-6: Output two-dimensional curve j Parameters, i.e. r 0_ j , r 1_ j , η 0_ j , η 1_j and ζ j .

9. The method for modeling a three-dimensional potential instability surface of a foundation under complex foundation loads according to claim 5, characterized in that: Step S13 in the method specifically includes the following steps: Step S13-1: η ij Calculate variables for the iteration loop and take η ij The initial value is η ij (0) , and when first calculated, η ij (0) = 90°; Step S13-2: Use equation (32) and substitute η ij (0) , and thus obtain new η ij ; (32) Step S13-3: If | η ij – η ij (0) | ≤ ε 2. Here ε 2 Take 0.001°, then it is considered η ij is the final calculation result; otherwise, let η ij (0) = η ij , repeat step S13-2; Step S13-4: Output any point on the three-dimensional potential instability surface of the foundation s ij In vertical section j The corresponding sliding surface polar angle η ij .

Images