Method for calculating vertical additional stress and displacement of three-layer foundation under rectangular area load

Abstract

The invention discloses a method for calculating vertical additional stress and displacement of a three-layer foundation under a rectangular area load. The method comprises the following steps: step 100, setting a displacement intermediate calculation variable for the foundation; converting the Poisson ratio of each layer of soil body of the foundation and the thickness of the soil layer to form intermediate calculation parameters; step 200, solving an image function of a vertical displacement intermediate calculation variable and an image function of a vertical additional stress of each layer of soil body under any point load in the rectangular area load region by combining a state space theory with Hankel integral transformation; 300, the rectangular area load area is divided into four small rectangular areas on the plane according to the positions of the calculation points in the foundation, so that the four small rectangular areas have common angular points; carrying out Hankel integral inverse transformation and carrying out integration along the rectangular region to obtain vertical additional stress and vertical displacement at each depth under the corner points of the small rectangular region; and 400, according to the elastic theory superposition principle, vertical additional stress and vertical displacement at any calculation point in the three layers of foundations are obtained through calculation.

Description

Technical Field

[0001] The invention relates to the technical field of vertical additional stress and displacement of foundation, and in particular to a method for calculating vertical additional stress and displacement of a three-layer foundation under rectangular area load. Background Art

[0002] Buildings on the foundation transfer their own weight and the load they bear to the foundation, which generates vertical additional stress in the foundation. The foundation soil will deform and displace under the action of vertical additional stress, which is not only closely related to the bearing capacity of the foundation, but also affects the settlement and stability of the building. Therefore, the magnitude of the vertical additional stress of the foundation under the action of load is an especially important indicator in building design. In-depth research on the stress field and displacement field of the foundation has extremely high safety and economic significance for building design and maintenance.

[0003] The algorithm for calculating the vertical additional stress of the foundation is generally based on the elastic semi-infinite body theory, that is, the foundation is regarded as a continuous, homogeneous, elastic half-space body (which can be called a homogeneous semi-infinite foundation). This model can reflect the continuity of the foundation soil, and the stress solution theory of the elastic semi-infinite body under various loads is relatively mature, so this algorithm is widely used. The vertical additional stress coefficient table provided in various relevant specifications is based on this theory. However, compared with the elastic semi-infinite body model used in the specifications, the horizontal layered foundation model is more in line with reality. The actual foundation soil often presents the characteristics of multi-layered distribution. At this time, using the elastic semi-infinite body theoretical model to simulate the actual foundation will lead to an increase in the error between theory and practice.

[0004] However, considering the multi-layered nature of the foundation, solving the foundation stress by the traditional simultaneous equations and boundary condition method will become extremely difficult as the number of foundation layers increases. Although there have been methods of deriving stress solutions from transfer matrices, they are only stress solutions for double-layer foundations under circular area loads and belong to the category of implicit solutions. They cannot be applied to multi-layer foundations with three or more layers, nor do they belong to the problem of rectangular area loads. Therefore, the specific solution calculation expressions for the vertical additional stress and displacement in multi-layer foundations under rectangular area loads are unclear.

[0005] In summary, the current layered elastic theory is not yet mature, and the calculation and analysis mainly focus on double-layer foundations under axisymmetric circular area loads. However, there are no reports on research on rectangular area loads that are widely present in engineering practice and three-layer foundations that are more in line with reality. Summary of the invention

[0006] The technical problem to be solved by the present invention is to provide a method for calculating the vertical additional stress and displacement of a three-layer foundation under a rectangular area load, which has clear solution principle and calculation process, efficient numerical calculation, good rationality and practical operability. The technical solution is as follows:

[0007] Calculation method for vertical additional stress and displacement of a three-layer foundation under a rectangular area load, characterized in that it includes the following steps:

[0008] Step 100, set intermediate calculation variables for the displacement of the foundation; among them, the intermediate calculation variable u for the horizontal displacement * The calculation expression of is u * = u·2G, and the intermediate calculation variable w for the vertical displacement * The calculation expression of is w * = w·2G, where u is the horizontal displacement, w is the vertical displacement, and G represents the shear modulus of the foundation soil;

[0009] Transform the Poisson's ratio and soil layer thickness of each layer of soil in the foundation to form intermediate calculation parameters; among them, the intermediate calculation parameter s for the Poisson's ratio j The calculation expression of is s i = 3 - 4ν i and the intermediate calculation parameter h for the soil layer thickness m The calculation expression of is where i is the number of each layer of soil from top to bottom, i = 1, 2, 3, ν i is the Poisson's ratio, h1 is the thickness of the first layer of soil, and h2 is the thickness of the second layer of soil;

[0010] Step 200, solve the image function of the intermediate calculation variable of the vertical displacement of each layer of soil and the image function of the vertical additional stress under an arbitrary point load in the rectangular area load region by combining the state space theory with the Hankel integral transform;

[0011] Step 300, divide the rectangular area load region into four small rectangular regions on the plane according to the position of the calculation point in the foundation, so that the four small rectangular regions have a common corner point; use the inverse Hankel integral and integrate along the rectangular region to obtain the vertical additional stress and vertical displacement at each depth under the corner point of the small rectangular region;

[0012] Step 400, calculate the vertical additional stress and vertical displacement at any calculation point in the three-layer foundation by the superposition principle of elastic theory.

[0013] First, the method of the present invention is based on a layered inhomogeneous semi-infinite model. Compared with the traditional homogeneous semi-infinite foundation, it fully considers the multi-layered nature of the foundation soil and is closer to the actual foundation conditions. Secondly, taking the three-layer foundation under rectangular area load as a typical example, based on the elastic theory, intermediate calculation variables and intermediate calculation parameters are introduced. By means of Hankel integral transformation and using the transfer matrix method to solve, explicit expressions of the vertical additional stress and vertical displacement at each point in the three-layer foundation under rectangular area load can be obtained, making up for the deficiency of the existing layered foundation analysis theory which only focuses on the solution of the vertical additional stress of the double-layer foundation and enriching the elastic mechanics theory of the layered foundation. Finally, the method of the present invention is a high-efficiency numerical iterative solution calculation method that avoids numerical overflow. The solution principle and calculation process are clear, the numerical calculation is efficient, and it has good rationality and practical operability. It can conveniently realize the design analysis and calculation of the bearing capacity and deformation of the foundation under rectangular load, and can simply and efficiently and reasonably carry out the basic engineering design related to the rectangular area load, providing an effective method and scientific basis for the design calculation of such projects and having important technical significance and engineering application value.

[0014] The following further describes the present invention in conjunction with the drawings and specific embodiments. The additional aspects and advantages of the present invention will be partially given in the following description, partially become obvious from the following description, or be understood through the practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] The drawings constituting a part of the present invention are used to assist in understanding the present invention. The content provided in the drawings and the related explanations in the present invention can be used to explain the present invention, but do not constitute an improper limitation to the present invention. In the drawings:

[0016] Figure 1 It is a schematic diagram of the calculation analysis model of the rectangular area load on the ground surface acting on the three-layer foundation.

[0017] Figure 2 It is a schematic diagram of dividing the rectangular area load on the ground surface into four small rectangular areas according to the calculation points.

[0018] Figure 3 It is a schematic diagram of the axisymmetric load calculation model.

[0019] Figure 4 It is a schematic diagram of the integral calculation of the point load along the rectangular area load area.

[0020] Figure 5 It is a schematic diagram of calculating the vertical additional stress and vertical displacement at each depth on the central axis of the rectangular area load area by the superposition principle of elastic theory.

[0021] Figure 6The distribution curve of the vertical additional stress along the soil depth at the corner point of the small rectangle in the rectangular area load region of the embodiment.

[0022] Figure 7 The distribution curve of the vertical additional stress along the soil depth at the center point of the rectangular area load region of the embodiment.

[0023] Figure 8 The comparison diagram of the distribution curves of the vertical additional stress along the soil depth at the center point of the rectangular area load region obtained by the method of the present invention and the numerical simulation method.

[0024] Figure 9 The comparison diagram of the ground settlement amounts at the center point of the rectangular area load region obtained by the method of the present invention and the numerical simulation method. Specific embodiments

[0025] The present invention will be clearly and completely described below with reference to the accompanying drawings. Those of ordinary skill in the art will be able to implement the present invention based on these descriptions. Before describing the present invention with reference to the accompanying drawings, it should be particularly noted that:

[0026] The technical solutions and technical features provided in each part including the following description in the present invention can be combined with each other without conflict.

[0027] In addition, the embodiments of the present invention involved in the following description are usually only a part of the embodiments of the present invention, rather than all the embodiments. Therefore, all other embodiments obtained by those of ordinary skill in the art without creative efforts based on the embodiments in the present invention should fall within the protection scope of the present invention.

[0028] Regarding the terms and units in the present invention. The terms "comprising", "having" and any variations thereof in the specification, claims and relevant parts of the present invention are intended to cover non-exclusive inclusion.

[0029] The specific implementation manner of the calculation method of the vertical additional stress and displacement of the three-layer foundation under the rectangular area load of the present invention includes steps 100-500, which are as follows:

[0030] Step 100, set displacement-related intermediate calculation variables for the foundation; transform the Poisson's ratio and soil layer thickness of each soil layer of the foundation to form intermediate calculation parameters.

[0031] The reasons for introducing the displacement-related intermediate calculation variables and the resulting technical effects are as follows:

[0032] Figure 1 It is a schematic diagram of the calculation and analysis model for the rectangular area load acting on the three-layer foundation on the ground surface. Figure 2Schematic diagram for dividing the surface rectangular area load into four small rectangular areas at the calculation point. Figure 3 Schematic diagram of the axisymmetric load calculation model.

[0033] As Figure 1-2 shown, according to the superposition principle of elastic theory, the rectangular area load region is divided into four small rectangular areas on the plane according to the position of the calculation point M in the foundation, so that the four small rectangular areas have a common corner point A, and this corner point A is the projection point of the calculation point M on the ground. For the convenience of formula derivation, first solve the vertical additional stress σ z (The so-called vertical additional stress refers to the stress caused only by the surface load, excluding the self-weight stress of the stratum soil, the same below) and the vertical displacement w at each depth in the foundation under the corner point A of each small rectangular area. The main calculation parameters include: the rectangular area load p and its distributed width B and length L, the elastic modulus E i (i = 1, 2, 3), Poisson's ratio ν i and thickness h i , where i is the number of each layer of soil from top to bottom. From E i , ν i , the shear modulus of each layer of soil can be directly obtained

[0034] According to the axisymmetric load calculation model of a circular area load with a radius of r0 as shown in Figure 3 (a polar coordinate system is established with the center of the load action as the origin and extending outward along the ground surface, and the polar radius of any point on the ground surface is r) acting on the surface of a three-layer foundation, from the control equations of elasticity, a partial differential equation system about [u, w, σ z , τ zr T can be established in a single soil layer (u and τ zr are the horizontal displacement and additional shear stress respectively). Through the differential property of the Hankel integral transform, the partial differential equation system of [u, w, σ z , τ zr T can be converted into a corresponding ordinary differential equation system of the image function , where are the quantities after Hankel integral transform of u, w, σ z , τ zr respectively. Solving this ordinary differential equation system can obtain the relationship of in a single soil layer.

[0035] In the traditional method, using state space theory, ​​Regarding the state vector transmitted between each layer of soil, the sub-transfer matrix within each layer of soil can be determined. Then, by aggregating the sub-transfer matrices, the total transfer matrix of the entire foundation can be obtained, and the entire foundation can be solved based on the boundary conditions. However, since u and w are significantly affected by the elastic modulus and the elastic constants of each layer of soil are inconsistent, the total transfer matrix will become extremely complex and is not conducive to efficient calculation.

[0036] To eliminate this influence, the present invention first uses displacement-related intermediate calculation variables to replace the displacement in the calculation. Specifically, the displacement-related intermediate calculation variables include the horizontal displacement intermediate calculation variable u * and the vertical displacement intermediate calculation variable w * , where the calculation expression of the horizontal displacement intermediate calculation variable u * is u * = u·2G, and the calculation expression of the vertical displacement intermediate calculation variable w * is w * = w·2G, where G represents the shear modulus of the foundation soil.

[0037] After using the displacement-related intermediate calculation variables, perform the 0th-order Hankel transform on σ z and w * respectively to obtain the vertically additional stress after integral transformation and the vertical displacement intermediate calculation variable Perform the 1st-order Hankel transform on u * and τ zr respectively to obtain the horizontally displacement intermediate calculation variable after integral transformation and the additional shear stress There is:

[0038]

[0039] In the formula, α is the independent variable that needs to be introduced correspondingly when directly performing the Hankel integral transform on the polar radius r in the polar coordinate system, and can be regarded as the variable after the Hankel integral transform corresponding to the polar radius r; z is the depth of the calculation point M in the foundation from the ground surface; J0 and J1 respectively represent the zeroth-order and first-order Bessel functions of the first kind.

[0040] In this way, by setting the state vector as the sub-transfer matrix within each layer of soil does not contain the elastic modulus, and the total transfer matrix will be greatly simplified.

[0041] Also, since u * and w * are discontinuous at the interface between layers, the present invention adds a conversion matrix when the state vector is transmitted to the interface between layers to satisfy u * and w *The continuity conditions of u and w between layers in this form. The added transformation matrix is as follows:

[0042] The transformation matrix from the first soil layer to the second soil layer is:

[0043]

[0044] The transformation matrix from the second soil layer to the third soil layer is:

[0045]

[0046] In the formula, represents the ratio of the shear modulus G2 of the second soil layer to the shear modulus G1 of the first soil layer; represents the ratio of the shear modulus G3 of the third soil layer to the shear modulus G2 of the second soil layer.

[0047] In this way, by using the intermediate calculation variables u * , w * instead of directly using the variables u and w, the elastic modulus or shear modulus of each layer only appears in the integrated total transfer matrix and when finally converted to u and w, which can greatly reduce the complexity of the calculation and significantly improve the calculation efficiency. At the same time, by adding a transformation matrix when the state vector is transferred to the interface between layers, the continuity conditions of u * , w * in the form of u and w between layers are satisfied, thereby improving the calculation accuracy.

[0048] The reasons for introducing the intermediate calculation parameters and the resulting technical effects are as follows:

[0049] The intermediate calculation parameters include the Poisson's ratio intermediate calculation parameter s i and the soil layer thickness intermediate calculation parameter h m , which are respectively:

[0050] s i = 3 - 4ν i (7)

[0051]

[0052] In the formula, ν i is the Poisson's ratio of the i-th soil layer; h1 is the thickness of the first soil layer; h2 is the thickness of the second soil layer.

[0053] The intermediate calculation parameters are combinations of the original foundation calculation parameters that appear multiple times. Calculating the intermediate calculation parameters first will greatly reduce the number of iterations in numerical calculations, thereby improving the calculation efficiency and reducing the calculation error.

[0054] Step 200: Solve the image functions of the vertical displacement intermediate calculation variables and the image function of the vertical additional stress of each soil layer under any point load in the rectangular area load region by combining the state space theory with the Hankel integral transform.

[0055] First, the vertical distance from the corner point A of the small rectangular area to the calculation point M in the i-th soil layer of the foundation is the depth z. i , and according to the position of the calculation point in the foundation, locate the soil layer to which it belongs and determine the relative calculation depth z of the calculation point. m .

[0056] Then, according to the technical effects described in Step 100, regard the under the point load as the state vector. Through the Hankel integral transform and combined with the interlayer continuity condition, establish the transfer matrix of the state vector from the ground surface to each depth to infinity. Add the transformation matrix to the transfer matrix when the state vector is transferred to the soil layer interface. Then, inversely calculate the ground surface displacement from the linear theory and the ground surface load, substitute it back into the initial state vector, and then obtain the image functions f′ wi (α h , z m ) of the displacement-related intermediate calculation variables of each soil layer under the point load after the Hankel integral transform and the image function f′ σi (α h , z m ) of the vertical additional stress, specifically as follows:

[0057] 1) For the calculation point M in the first soil layer, its actual depth z from the ground surface i satisfies 0 ≤ z i ≤ h1, and there is:

[0058]

[0059] The expression of the image function f′ σ1 (α h , z m ) of its vertical additional stress is composed of the sum of 7 terms. For the convenience of expression, it can be expressed as:

[0060]

[0061] In the formula, k is the number of terms constituting the expression; α h is the independent variable after the Hankel integral transform of the independent variable α, and α h = α·h1; through the identity variable substitution in the form of the Hankel integral, transform the integral with respect to α into the integral with respect to α h , so that mainly dimensionless quantities participate in the calculation process, reducing the calculation error to a large extent and improving the calculation efficiency.

[0062] A(α h ) is an expression regarding the initial state vector and is a common function term of f′ for all soil layers σi (α h , z m ), f′ wi (α h , z m ). It will be cited in the calculation of the vertical additional stress and vertical displacement of all soil layers. See the following formula (10a) for details.

[0063] M k (α h , z m ) is the sum of 7 terms in f′ σ1 (α h , z m ). See formulas (10b) to (10h) respectively. Here, e is the natural constant.

[0064]

[0065]

[0066] The image function f′ of its intermediate calculation variable of vertical displacement w1 (α h , z m ) is as follows:

[0067]

[0068] In the formula, D k (α h , z m ) is the sum of 7 terms in f′ w1 (α h , z m ). See formulas (11a) to (11g) respectively:

[0069]

[0070]

[0071] 2) For the calculation point in the second soil layer, the actual depth z from the ground surface i satisfies h1 ≤ z i ≤ h1 + h2, and there is:

[0072]

[0073] The image function f′ of its vertical additional stress σ2 (α h , z m ) is as follows:

[0074]

[0075] The image function f′ of the intermediate calculation variable of its vertical displacement w2 (α h , z m ) is as follows:

[0076]

[0077] 3) For the calculation point in the third layer of soil, the actual depth z from the ground surface i satisfies z i ≥ h1 + h2, and there is:

[0078]

[0079] The image function f′ of its vertical additional stress σ3 (α h , z m ) is as follows:

[0080]

[0081] The image function f′ of the intermediate calculation variable of its vertical displacement w3 (α h , z m ) is as follows:

[0082]

[0083] Step 300, according to Figure 2 the small rectangular areas divided from the rectangular area load region in, use the inverse Hankel integral and integrate along the rectangular region to obtain the vertical additional stress and vertical displacement at each depth under the corner points of the small rectangular areas.

[0084] Figure 4 is a schematic diagram of the integral calculation of the point load along the rectangular area load region. Through the inverse Hankel integral, the vertical additional stress and the intermediate calculation variable of the vertical displacement at any position in the three-layer foundation under the action of any point load (concentrated load in a small area) in the rectangular area load region can be obtained. Then, integrate it along the Figure 4 rectangular area load region shown, and from the relationship between the intermediate calculation variable of the vertical displacement and the vertical displacement w * = w·2G, obtain the vertical additional stress and vertical displacement under the corner points of the small rectangular area. According to the soil layer position where the calculation point M is located, the image function f′ of the vertical additional stress σi (α h , z m ), the image function f′ of the intermediate calculation variable of the vertical displacement wi (α h , zm ) Substitute them into the corresponding calculation formulas of the vertical additional stress and vertical displacement respectively, and the explicit calculation expressions of the vertical additional stress and vertical displacement at each depth under the corner point of the small rectangular area can be obtained, which are respectively:

[0085]

[0086] In the formula, θ is the integration variable in polar coordinate form; ξ is the ratio of the length L to the width B of the rectangular area load distribution, ξ = L / B; b is the width of the small rectangular area.

[0087] Both formulas (18) and (19) are double semi-infinite integrals containing Bessel functions. Therefore, MATLAB with powerful numerical calculation capabilities is used for numerical integration. When using the built-in integral2 function of MATLAB for numerical integration, for α h The following regulations are made for the upper limit of integration: For the calculation point (z i > 0) that is not on the ground surface, the upper limit of integration of α h is set to infinity; For the calculation point on the ground surface (z i = 0), in order to balance the calculation accuracy and speed, the upper limit of integration of α h is set to or where X is a large number, and preferably X ≥ 100 can be taken. This way of selecting the upper limit of integration in the present invention can meet the dual requirements of improving the calculation accuracy and calculation efficiency.

[0088] Step 400, calculate the vertical additional stress and vertical displacement at any calculation point in the three-layer foundation according to the superposition principle of elastic theory.

[0089] According to the superposition principle of elastic theory, the sum of the vertical additional stress and vertical displacement at the corner point A under the action of the loads in the four small rectangular areas is equal to the vertical additional stress and vertical displacement at the calculation point M under the action of the rectangular area load.

[0090] Figure 5 is a schematic diagram for calculating the vertical additional stress and vertical displacement at each depth on the central axis of the center point of the rectangular area load according to the superposition principle of elastic theory. As Figure 5 shown, particularly, when the calculation point M is on the axis where the center point O of the rectangular area load is located, only calculate the vertical additional stress and the vertical displacement w j under the corner point of one small rectangular area, where j is the number of the 4 small rectangular areas; and then from w o = 4w j superimpose to obtain the vertical additional stress and the vertical displacement w o under the calculation point M.

[0091] The beneficial effects of the present invention will be described below through specific embodiments.

[0092] As Figure 1 shown, in this embodiment, a rectangular area load with a distributed range of p = 100 kPa acts on the ground surface of the three-layer foundation, and the rectangular area has a width B = 20 m and a length L = 200 m. The elastic modulus and Poisson's ratio of each layer of soil from the ground surface downwards are: E1 = 15 MPa, ν1 = 0.4, h1 = 20 m, E2 = 5 MPa, ν2 = 0.45, h2 = 20 m, E3 = 25 MPa, ν3 = 0.35.

[0093] As Figure 5 shown, the rectangular area load region is divided into four small rectangular areas of the same size along the center point O. Therefore, b = 10 m, l = 100 m, and ξ = 10.

[0094] The shear moduli of each layer of soil are obtained from the elastic constants of each layer of soil as: G1 = 5.357 MPa, G2 = 1.724 MPa, and G3 = 9.259 MPa.

[0095] According to equations (5) to (8), G m1 = 0.322, G m2 = 5.371, s1 = 1.4, s2 = 1.2, s3 = 1.6, h m = 1.

[0096] Thus, the transformation matrix from the first layer of soil to the second layer of soil is:

[0097]

[0098] The transformation matrix from the second layer of soil to the third layer of soil is:

[0099]

[0100] For the calculation point in the first layer of soil: (where z i , the unit is m, the same below), substituting z m and G m1 , G m2 , s1, s2, s3, h m into equations (10), (10a) to (10h) respectively, f′ σ1 (α h , z m ) is obtained;

[0101] For the calculation point in the second layer of soil: Substituting z m and G m1 , G m2, s1, s2, s3, h m After substituting them into equations (10a) and (13) respectively, f′ is obtained σ2 (α h , z m );

[0102] For the calculation point in the third layer of soil mass: Substitute z m and G m1 , G m2 , s1, s2, s3, h m into equations (10a) and (16) respectively, and f′ σ3 (α h , z m ) is obtained.

[0103] Substitute f′ σi (α h , z m ) (i = 1, 2, 3), b, ξ, and p into equation (18) to obtain the solution of the vertical additional stress in the i-th layer of soil mass. Then input this calculation formula into MATLAB, and for each specified calculation point, use the function "integral2" for numerical integration respectively, where the upper limit of integration is set to infinity.

[0104] The calculation results obtained hereby are the solutions of the vertical additional stress at each depth under the corner point of the small rectangular area with a distribution range of 10m×100m for p = 100kPa, as shown in Figure 6 . Superimpose the calculation results of the common corner points of the four small rectangular areas with the same size, and the solutions of the vertical additional stress at each depth under the center point of the rectangular area load region with a distribution range of 20m×200m on the ground surface for p = 100kPa are obtained, as shown in Figure 7 .

[0105] It can be seen that for the case where the first layer of soil mass in this embodiment is a relatively hard stratum and the second layer of soil mass is a relatively soft stratum, near the interface between the two strata, the calculated value of the method of the present invention is less than the result of the traditional homogeneous foundation method (the "Railway Code" method). In addition, near the interface of each stratum, the deviation between the calculated value of the method of the present invention and the calculated value of the traditional homogeneous foundation is more obvious.

[0106] To further verify and illustrate the rationality of the method of the present invention, taking the vertical additional stress variation curve along the depth under the center point of the rectangular area load region and the ground surface settlement amount at the center point when a rectangular uniformly distributed load acts on the surface of a double-layer foundation (i.e., E2 = E3, ν2 = ν3) as an example, Figure 8 , Figure 9 the calculation result comparisons between the method of the present invention and the numerical simulation method are given respectively. As shown in Figure 8 , 9As shown, the calculation results of the two are basically the same. Therefore, the calculation results of the algorithm of the present invention are in good agreement with those of the existing method, indicating that the method of the present invention has good rationality.

[0107] The above describes the relevant content of the present invention. Those of ordinary skill in the art will be able to implement the present invention based on these descriptions. Based on the above content of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the scope of protection of the present invention.

Claims

1. A calculation method for vertical additional stress and displacement of a three-layer foundation under a rectangular area load, characterized in that: Including the following steps: Step 100, set intermediate calculation variables for the displacement of the foundation; among them, the intermediate calculation variable u for horizontal displacement * The calculation expression of is u * = u·2G, and the intermediate calculation variable w for vertical displacement * The calculation expression of is w * = w·2G, where u is the horizontal displacement, w is the vertical displacement, and G represents the shear modulus of the foundation soil mass; Transform the Poisson's ratio and soil layer thickness of each layer of the foundation soil to form intermediate calculation parameters; among them, the intermediate calculation parameter s of the Poisson's ratio j The calculation expression of is s i = 3 - 4ν i , the intermediate calculation parameter h of the soil layer thickness m The calculation expression of is i is the number of each layer of soil from top to bottom, i = 1, 2, 3, ν i is the Poisson's ratio, h1 is the thickness of the first layer of soil, and h2 is the thickness of the second layer of soil; Step 200: Solve the image functions of the intermediate calculation variables of the vertical displacements of each soil layer under any point load in the rectangular area load region and the image function of the vertical additional stress by combining the state space theory with the Hankel integral transform; Step 300: Divide the rectangular area load region into four small rectangular areas on the plane according to the position of the calculation point in the foundation, so that the four small rectangular areas have a common corner point; Use the inverse Hankel integral and integrate along the rectangular region to obtain the vertical additional stress and vertical displacement at each depth under the corner point of the small rectangular area; Step 400: Calculate the vertical additional stress and vertical displacement at any calculation point in the three-layer foundation by the superposition principle of elastic theory.

2. The calculation method according to claim 1, characterized in that: In step 200, using the state space theory, the under point load is regarded as the state vector. Through the Hankel integral transform and by combining the interlayer continuity conditions, the transfer matrix of the state vector from the ground surface to each depth at infinity is established. Then, based on the linear theory and the surface load, the surface displacement is inversely calculated and substituted back into the initial state vector. Subsequently, the image functions of the intermediate calculation variables related to the displacements of each layer of soil under the point load after the Hankel integral transform and the image function of the vertical additional stress are obtained respectively by the transfer matrix; Among them, a transformation matrix is added to the transfer matrix when the state vector is transferred to the soil layer interface; Among them, σ z is the vertical additional stress; τ zr is the additional shear stress; perform the 0th-order Hankel integral transform on σ z , w * respectively to obtain Perform the 1st-order Hankel integral transform on u * , τ zr respectively to obtain 3. The calculation method according to claim 2, wherein: In the said transformation matrix: The transformation matrix from the first soil layer to the second soil layer is: The transformation matrix from the second soil layer to the third soil layer is: In the formula, represents the ratio of the shear modulus G2 of the second layer of soil to the shear modulus G1 of the first layer of soil; represents the ratio of the shear modulus G3 of the third layer of soil to the shear modulus G2 of the second layer of soil.

4. The calculation method according to claim 2, wherein: In step 300, the vertical additional stress at each depth below the corner points of the small rectangular area and the vertical displacement w (i) (z i ) are respectively expressed as follows: where f σ ' i (α h , z m ) is the image function of the vertical additional stress; f w ' i (α h , z m ) is the image function of the intermediate calculation variable of the vertical displacement; J1 represents the first-kind first-order Bessel function; θ is the integral variable in polar coordinate form; ξ is the ratio of the length L to the width B of the rectangular area load distribution, ξ = L / B; α is the independent variable that needs to be introduced correspondingly when performing the Hankel integral transform on the polar radius r in the polar coordinate system established with the center of the load action as the origin and extending outward along the ground surface; α h is the independent variable after performing the Hankel integral transform on the independent variable α, α h = α·h1; G i represents the shear modulus of the i-th soil layer; z i is the vertical distance from the corner point of the small rectangular area to the calculation point in the i-th soil layer of the foundation; z m is the soil layer position where the calculation point is located and the determined relative calculation depth; b is the width of the small rectangular area.

5. The calculation method according to claim 4, wherein: In step 300, the integral2 function built in MATLAB is used for numerical integration; for calculation points other than the ground surface, the upper limit of the integral of α h is set to infinity; for ground surface calculation points, the upper limit of the integral of α h is set to or where X ≥ 100.

6. The calculation method according to claim 4, wherein: In Step 400, according to the superposition principle of elastic theory, the sum of the vertical additional stress and vertical displacement at the corner point under the action of the loads of the four small rectangular areas is equal to the vertical additional stress and vertical displacement at the calculation point under the action of the rectangular area load.

7. The calculation method according to claim 6, characterized in that: When the calculation point is located on the axis where the center point of the rectangular area load is located, only calculate the vertical additional stress under the corner point of a small rectangle and the vertical displacement w j , and then by w o = 4w j superimpose to obtain the vertical additional stress and the vertical displacement w o .

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