Calculation method of vertical additional stress and displacement of three-layered ground under rectangular area load

Abstract

The application discloses a calculation method of vertical additional stress and displacement of a three-layer foundation under a rectangular area load, and comprises the following steps: step 100, setting a displacement intermediate calculation variable for the foundation; transforming Poisson's ratios and soil layer thicknesses of each soil body of the foundation to form intermediate calculation parameters; step 200, solving an image function of the vertical displacement intermediate calculation variable and an image function of the vertical additional stress of each soil body under any point load in a rectangular area load region by combining a state space theory with Hankel integral transformation; step 300, dividing the rectangular area load region into four small rectangular regions on a plane according to the calculation point positions in the foundation, so that the four small rectangular regions have a common corner point; obtaining the vertical additional stress and the vertical displacement at each depth under the corner point of the small rectangular region by using Hankel integral inverse transformation and integrating along the rectangular region; and step 400, calculating the vertical additional stress and the vertical displacement at any calculation point in the three-layer foundation by using the superposition principle of the elastic theory.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of vertical additional stress and displacement of foundation, in particular to a calculation method of vertical additional stress and displacement of three-layer foundation under rectangular area load. BACKGROUND

[0002] The building on the foundation will transmit the dead weight and the load it bears to the foundation, and the vertical additional stress in the foundation will be generated accordingly. The deformation and displacement of the foundation soil will occur under the action of the 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 size of the vertical additional stress of the foundation under the load is an important index in the design of the building. In-depth study of the stress field and displacement field of the foundation has a high safety and economic significance for the design and maintenance of the building.

[0003] For the algorithm of the vertical additional stress of the foundation, it is generally based on the elastic half-space theory, that is, the foundation is regarded as a continuous, homogeneous and elastic half-space (which can be called homogeneous half-space foundation). The model can reflect the continuity of the foundation soil, and the stress solution theory of the elastic half-space under various loads is relatively mature, so the algorithm is widely used, and the vertical additional stress coefficient table provided in the relevant specifications is based on this theory. However, compared with the elastic half-space model used in the specifications, the horizontally layered foundation model is more in line with the actual situation. The actual foundation soil often shows a multi-layered distribution, so using the elastic half-space theory model to simulate the actual foundation will increase the error between theory and practice.

[0004] However, considering the multi-layered nature of the foundation, it is difficult to solve the stress of the foundation by the traditional simultaneous equations and boundary conditions method as the number of foundation layers increases. Although there are methods for deriving stress solutions from transfer matrices, they are only applicable to double-layer foundation stress solutions under circular area load and belong to implicit solutions, which cannot be applied to multi-layer foundation with three or more layers, and are not applicable to rectangular area load problems. Therefore, the specific calculation expression of the vertical additional stress and displacement of multi-layer foundation under rectangular area load is not clear.

[0005] In summary, the layered elastic theory is not mature yet, and the calculation and analysis are mainly based on the double-layer foundation under the axisymmetric circular area load, while the research on rectangular area load and three-layer foundation which is more in line with the actual situation is not reported. SUMMARY

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

[0007] A method for calculating the vertical additional stress and displacement of a three-story foundation under a rectangular area load, characterized by the following steps:

[0008] Step 100: Set intermediate displacement calculation variables for the foundation; among which, the intermediate horizontal displacement calculation variable u * The calculation expression is u * =u·2G, the intermediate calculation variable w for vertical displacement * The calculation expression 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] The Poisson's ratio and soil layer thickness of each soil layer in the foundation are transformed to form intermediate calculation parameters; among them, the intermediate calculation parameter s of Poisson's ratio is... i The calculation expression is s i =3-4ν i The intermediate calculation parameter h for soil layer thickness m The calculation expression is as follows i represents the number of each soil layer from top to bottom, i = 1, 2, 3, ν i h1 is Poisson's ratio, h1 is the thickness of the first soil layer, and h2 is the thickness of the second soil layer.

[0010] Step 200: Using state-space theory combined with Hankel integral transform, solve for the image function of the intermediate calculation variable of vertical displacement of each layer of soil under arbitrary point load in the rectangular area load region, as well as the image function of vertical additional stress.

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

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

[0013] First, the method of this invention is based on a layered heterogeneous semi-infinite model. Compared with traditional homogeneous semi-infinite foundations, it fully considers the multi-layered nature of the foundation soil and is closer to actual foundation conditions. Second, taking a three-layer foundation under rectangular area load as a typical example, this invention, based on elasticity theory, introduces intermediate calculation variables and parameters. Through Hankel integral transformation and the transfer matrix method, it obtains explicit expressions for the vertical additional stress and vertical displacement at each point in the three-layer foundation under rectangular area load. This overcomes the deficiency in existing layered foundation analysis theories, which only provide solutions for the vertical additional stress of two-layer foundations, and enriches the elasticity theory of layered foundations. Finally, the method of this invention is a highly efficient numerical iterative solution 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 problem of the foundation under rectangular load. It can perform the foundation engineering design related to rectangular area load in a simple, efficient and reasonable way, providing an effective method and scientific basis for such engineering design calculations. It has important technical significance and engineering application value.

[0014] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0015] The accompanying drawings, which form part of this invention, are used to aid in understanding the invention. The content provided in the drawings and their related descriptions can be used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings:

[0016] Figure 1 This is a schematic diagram of a calculation and analysis model for a rectangular area load on a three-layer foundation.

[0017] Figure 2 This is a schematic diagram showing how the rectangular area load on the ground surface is divided into four smaller rectangular areas based on the calculation point.

[0018] Figure 3 This is a schematic diagram of an axisymmetric load calculation model.

[0019] Figure 4 This is a schematic diagram for calculating the point load by integral along a rectangular area load region.

[0020] Figure 5 This is a schematic diagram illustrating the vertical additional stress and vertical displacement at various depths along the axis of the center point of a rectangular area load region, calculated using the superposition principle of elasticity theory.

[0021] Figure 6The figure shows the distribution curve of the vertical additional stress along the soil depth at the corner of the small rectangular area under the rectangular area load region in the example.

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

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

[0024] Figure 9 This is a comparison diagram of the surface settlement at the center point of the rectangular area load region obtained by the method of this invention and the numerical simulation method. Detailed Implementation

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

[0026] The technical solutions and features provided in the various parts of this invention, including the following description, can be combined with each other without conflict.

[0027] Furthermore, the embodiments of the present invention described below are generally only some, not all, of the embodiments of the present invention. Therefore, all other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort should fall within the scope of protection of the present invention.

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

[0029] The specific implementation method of the calculation method for vertical additional stress and displacement of a three-layer foundation under a rectangular area load of the present invention includes steps 100-500, 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 to form intermediate calculation parameters.

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

[0032] Figure 1 This is a schematic diagram of a calculation and analysis model for a rectangular area load on the ground surface acting on a three-layer foundation. Figure 2This is a schematic diagram showing how the rectangular area load on the ground surface is divided into four smaller rectangular areas based on the calculation point. Figure 3 This is a schematic diagram of an axisymmetric load calculation model.

[0033] like Figures 1-2 As shown, based on the superposition principle of elasticity theory, the rectangular area load region is divided into four smaller rectangular regions on the plane according to the location of the calculation point M in the foundation. These four smaller rectangular regions share a common corner point A, which is the projection of the calculation point M onto the ground. To facilitate formula derivation, the vertical additional stress σ at each depth in the foundation below corner point A of each smaller rectangular region is first calculated. z (The so-called vertical additional stress refers to the stress caused only by the surface load, excluding the self-weight stress of the soil strata, the same below) and vertical displacement w. The main calculation parameters include: rectangular area load p and its distribution width B and length L, elastic modulus E of each soil layer. i (i = 1, 2, 3), Poisson's ratio ν i and thickness h i , where i is the number of each soil layer from top to bottom. (From E) i ν i The shear modulus of each soil layer can be obtained directly.

[0034] According to such Figure 3 The circular area load of radius r0 (established in a polar coordinate system with the load center as the origin and the polar radius of any point on the ground surface as r) acts on the surface of a three-layer foundation. Based on the governing equations of elasticity, a calculation model can be established for the relationship between [u, w, σ] within a single soil layer. z τ zr ] T The partial differential equation system (u, τ) zr (representing horizontal displacement and additional shear stress, respectively), using the differential properties of the Hankel integral transform, [u, w, σ] can be expressed as... z τ zr ] T The partial differential equations are transformed into their corresponding image functions. A system of ordinary differential equations, where, They are u, w, and σ, respectively. z τ zr The quantity obtained after Hankel integral transformation can be used to solve this system of ordinary differential equations to obtain the quantity within a single soil layer. The relational expression.

[0035] In traditional methods, state-space theory is used to... If we consider these as state vectors transferred between soil layers, we can determine the sub-transfer matrices within each soil layer. Then, we can combine these sub-transfer matrices into the overall transfer matrix of the entire foundation, and solve for the entire foundation using the boundary conditions. However, since u and w are greatly affected by the elastic modulus, and the elastic constants of each soil layer are inconsistent, the overall transfer matrix becomes extremely complex and is not conducive to efficient calculation.

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

[0037] After using displacement-related intermediate calculation variables, for σ z w * The vertical additional stress after integral transformation is obtained by performing the 0th-order Hankel transform respectively. Intermediate variables for vertical displacement calculation For u * τ zr The intermediate calculation variables of the horizontal displacement after integral transformation are obtained by performing the first-order Hankel transform respectively. With additional shear stress have:

[0038]

[0039] In the formula, α is the independent variable that needs to be introduced when performing a Hankel integral transformation on the polar radius r in the polar coordinate system. It can be regarded as the variable after the Hankel integral transformation 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 represent the zeroth and first order Bessel functions of the first kind, respectively.

[0040] Thus, the state vector is then set to Then the sub-transfer matrices within each soil layer do not contain the elastic modulus, and the overall transfer matrix will be greatly simplified.

[0041] And because u * w * Since the interlayer contact surface is discontinuous, this invention adds a transformation matrix when the state vector is transferred to the interlayer contact surface to satisfy u. * w *The continuity condition of u and w between layers is given 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 as follows:

[0045]

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

[0047] Thus, the intermediate calculation variable u is used. * w * Instead of directly using variables u and w, this method ensures that 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, greatly reducing computational complexity and significantly improving computational efficiency. Furthermore, by adding a transformation matrix when transferring the state vector to the interlayer contact surface, the condition u is satisfied. * w * The continuity condition of u and w between layers is obtained under the form, thereby improving the accuracy of calculation.

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

[0049] Intermediate calculation parameters include Poisson's ratio and intermediate calculation parameter s. i Intermediate calculation parameter h for soil layer thickness m They are respectively:

[0050] s i =3-4ν i (7)

[0051]

[0052] In the formula, ν i h1 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] Intermediate calculation parameters are combinations of the original foundation calculation parameters that appear multiple times. Calculating intermediate calculation parameters first will greatly reduce the number of iterations in numerical calculations, thereby improving calculation efficiency and reducing calculation errors.

[0054] Step 200: Using state-space theory combined with Hankel integral transform, solve for the image function of the intermediate calculation variable of vertical displacement of each soil layer under arbitrary point load in the rectangular area load region, as well as the image function of vertical additional stress.

[0055] First, calculate the vertical distance z from corner point A of the small rectangular area to point M in the i-th layer of soil in the foundation. i Based on the location 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, based on the technical effects described in step 100, the point load will be applied... Treating the state vector as a whole, a transfer matrix is ​​established from the surface to infinity using the Hankel integral transform and the interlayer continuity condition. A transformation matrix is ​​added to the transfer matrix when the state vector is transferred to the interlayer contact surface of the soil. The surface displacement is then calculated using linear theory and surface load, and substituted back into the initial state vector. Finally, the image function f' of the displacement-related intermediate variables of each soil layer under point load after the Hankel integral transform is obtained from the transfer matrix. wi (α h ,z m And the image function f' of the vertical additional stress σi (α h ,z m The details are as follows:

[0057] 1) For the calculation point M in the first layer of soil, its actual depth z from the ground surface i Satisfying 0≤z i ≤h1, then:

[0058]

[0059] Its vertical additional stress image function f' σ1 (α h ,z m The expression for ) consists of the sum of 7 terms, which can be represented as follows for ease of expression:

[0060]

[0061] In the formula, k is the number of terms constituting the expression; α h Let α be the independent variable after the Hankel integral transformation over the independent variable α. h =α·h1; By substituting the identity variable in the Hankel integral form, the integral with respect to α is transformed into the integral with respect to α. h The integral of the integral makes the calculation process mainly involve dimensionless quantities, which greatly reduces calculation errors and improves calculation efficiency.

[0062] A(α h Let f' be an expression for the initial state vector, which is the sum of all soil layers. σi (α h ,z m ), f' wi (α h ,z m The common function term of ) is used in the calculation of vertical additional stress and vertical displacement of all soil layers, as detailed in equation (10a).

[0063] M k (α h ,z m ) is f' σ1 (α h ,z m The seven terms accumulated in ) are respectively shown in equations (10b) to (10h), where e is a natural constant.

[0064]

[0065]

[0066] The image function f' of the intermediate calculated vertical displacement variable w1 (α h ,z m )for:

[0067]

[0068] In the formula, D k (α h ,z m ) is f' w1 (α h ,z m The seven terms accumulated in the formula are respectively shown in equations (11a) to (11g):

[0069]

[0070]

[0071] 2) For the calculation point within the second soil layer, its actual distance from the ground surface is z. i satisfying h1≤z i If h1 ≤ h2, then:

[0072]

[0073] Its vertical additional stress image function f' σ2 (α h ,z m )for:

[0074]

[0075] The image function f' of the intermediate calculated vertical displacement variable w2 (α h ,z m )for:

[0076] 3) For the calculation point within the third soil layer, its actual distance from the ground surface is z. i Satisfy z i ≥h1+h2, then:

[0077]

[0078] Its vertical additional stress image function f' σ3 (α h ,z m )for:

[0079]

[0080] The image function f' of the intermediate calculated vertical displacement variable w3 (α h ,z m )for:

[0081]

[0082] Step 300, according to Figure 2 The small rectangular region, divided by the rectangular area load region, is used to obtain the vertical additional stress and vertical displacement at each depth below the corner point of the small rectangular region by applying the inverse Hankel integral transform and integrating along the rectangular region.

[0083] Figure 4 This diagram illustrates the integral calculation of a point load along a rectangular area load region. Through the inverse Hankel integral transform, the intermediate calculation variables of the vertical additional stress and vertical displacement at any location in the three-layer foundation under the action of any point load (a small area concentrated load) within the rectangular area load region can be obtained. Then, the integral is applied along... Figure 4 The rectangular area load region shown is integrated, and the relationship between the vertical displacement intermediate variable and the vertical displacement w is calculated. * =w·2G, yielding the vertical additional stress and vertical displacement at the corner of the small rectangular area. Based on the soil layer location of the calculation point M, the image function f' of the vertical additional stress is... σi (α h ,z m The image function f' of the intermediate calculation variable of vertical displacement. wi (α h ,z mSubstituting these values ​​into the corresponding formulas for calculating vertical additional stress and vertical displacement, we can obtain the explicit calculation expressions for the vertical additional stress and vertical displacement at each depth below the corner point of the small rectangular area, as follows:

[0084]

[0085] In the formula, θ is the integral variable in polar coordinates; 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.

[0086] Equations (18) and (19) are both double semi-infinite integrals containing Bessel functions. Therefore, MATLAB, which has powerful numerical computation capabilities, is used for numerical integration. When using the built-in integral2 function of MATLAB for numerical integration, the α... h The upper limit of integration is stipulated as follows: for non-surface calculation points (z... i >0), α h The upper limit of integration is set to infinity; for the surface calculation point (z... i =0), in order to balance computational accuracy and speed, α is set to 0. h The maximum score is set to or Where X is a large number, preferably X≥100. This method of selecting the upper limit of integration in this invention can satisfy the dual requirements of improving computational accuracy and efficiency.

[0087] Step 400: Using the superposition principle of elastic theory, calculate the vertical additional stress and vertical displacement at any calculation point in the three-layer foundation.

[0088] According to the superposition principle of elasticity theory, the sum of the vertical additional stress and vertical displacement at corner point A under the load of the four small rectangular areas is equal to the vertical additional stress and vertical displacement at calculation point M under the load of the rectangular area.

[0089] Figure 5 This is a schematic diagram illustrating the calculation of vertical additional stress and vertical displacement at various depths along the axis of the center point of a rectangular area load region using the superposition principle of elasticity theory. (See diagram for example.) Figure 5 As shown, specifically, when the calculation point M is located on the axis of the center point O of the rectangular area load region, only the vertical additional stress at the corner of a small rectangular area is calculated. and vertical displacement w j Where j is the number of the four small rectangular areas; then by w o =4w j The superimposed vertical additional stress at calculation point M and vertical displacement w o .

[0090] The beneficial effects of the present invention will be illustrated below through specific embodiments.

[0091] like Figure 1 As shown, in this embodiment, the surface load of the three-layer foundation is a rectangular area load with a distribution range of width B = 20m and length L = 200m, and a p = 100kPa. The elastic modulus and Poisson's ratio of each soil layer from the surface downwards are as follows: E1 = 15MPa, ν1 = 0.4, h1 = 20m, E2 = 5MPa, ν2 = 0.45, h2 = 20m, E3 = 25MPa, ν3 = 0.35.

[0092] like Figure 5 As shown, the rectangular area load region is divided into four smaller rectangular regions of the same size along the center point O, so we have: b = 10m, l = 100m, ξ = 10.

[0093] The shear moduli of each soil layer are obtained from the elastic constants of each layer as follows: G1 = 5.357 MPa, G2 = 1.724 MPa and G3 = 9.259 MPa.

[0094] According to equations (5) to (8), we get: G m1 =0.322, G m2 =5.371, s1=1.4, s2=1.2, s3=1.6, h m =1.

[0095] Therefore, the transformation matrix from the first soil layer to the second soil layer is:

[0096]

[0097] The transformation matrix from the second soil layer to the third soil layer is as follows:

[0098]

[0099] For the calculation points within the first layer of soil: (where z is in the formula) i (Unit: m, same below) z m With G m1 G m2 s1, s2, s3, h m Substituting these values ​​into equations (10), (10a), and (10h) respectively, we obtain f'. σ1 (α h ,z m );

[0100] For the calculation points within the second soil layer: z m With G m1 G m2s1, s2, s3, h m Substituting these values ​​into equations (10a) and (13) respectively, we obtain f'. σ2 (α h ,z m );

[0101] For the calculation points within the third soil layer: z m With G m1 G m2 s1, s2, s3, h m Substituting these values ​​into equations (10a) and (16) respectively, we obtain f'. σ3 (α h ,z m ).

[0102] f' σi (α h ,z m Substituting (i = 1, 2, 3) and b, ξ, p into equation (18) yields the solution of the vertical additional stress in the i-th soil layer. Then, input this calculation formula into MATLAB, and perform numerical integration using the function "integral2" for each specified calculation point, with the upper limit of integration set to infinity.

[0103] The resulting calculation yields the vertical additional stress solution at various depths below the corner points of a small rectangular area with a distribution range of p = 100 kPa and a radius of 10 m × 100 m. (See...) Figure 6 By superimposing the calculation results of the common corner points of four small rectangular areas of the same size, the vertical additional stress solutions at various depths below the center point of a rectangular load area with a surface area of ​​p = 100 kPa and a distribution range of 20 m × 200 m are obtained. (See...) Figure 7 .

[0104] As can be seen, in the case presented in this embodiment where the first soil layer is a relatively hard stratum and the second soil layer is a relatively soft stratum, near the interface between the two strata, the calculated value by the method of the present invention is less than the result of the traditional homogeneous foundation method (the method in the "Railway Code"). Furthermore, near the interfaces between different strata, the deviation between the calculated value by the method of the present invention and the traditional homogeneous foundation calculation value is even more significant.

[0105] To further verify and illustrate the rationality of the method of the present invention, the vertical additional stress variation curve along the depth at the center point of the rectangular area load region and the surface settlement at the center point are used as examples when a rectangular uniformly distributed load is applied to the surface of a double-layer foundation (i.e., E2 = E3, ν2 = ν3). Figure 8 , Figure 9 Comparisons of calculation results using the method of this invention and numerical simulation methods are presented respectively. For example... Figure 8 , 9As shown, the calculation results of the two methods are basically consistent. Therefore, the calculation results of the algorithm of the present invention are in good agreement with those of existing methods, which demonstrates that the method of the present invention has good rationality.

[0106] The foregoing has described the relevant content of the present invention. Those skilled in the art will be able to implement the present invention based on these descriptions. All other embodiments obtained by those skilled in the art based on the above description of the present invention without inventive effort should fall within the scope of protection of the present invention.

Claims

1. A method for calculating the vertical additional stress and displacement of a three-layer foundation under a rectangular area load, characterized in that: Includes the following steps: Step 100: Set intermediate displacement calculation variables for the foundation; among which, the intermediate horizontal displacement calculation variable u * The calculation expression is u * =u·2G, the intermediate calculation variable w for vertical displacement * The calculation expression 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; The Poisson's ratio and soil layer thickness of each soil layer in the foundation are transformed to form intermediate calculation parameters; among them, the intermediate calculation parameter s of Poisson's ratio is... i The calculation expression is s i =3-4ν i The intermediate calculation parameter h for soil layer thickness m The calculation expression is as follows i represents the number of each soil layer from top to bottom, i = 1, 2, 3, ν i h1 is Poisson's ratio, h1 is the thickness of the first soil layer, and h2 is the thickness of the second soil layer. Step 200: Using state-space theory combined with Hankel integral transform, solve for the image function of the intermediate calculation variable of vertical displacement of each layer of soil under arbitrary point load in the rectangular area load region, as well as the image function of vertical additional stress. Step 300: Divide the rectangular area load region into four smaller rectangular regions on the plane according to the calculation point position in the foundation, so that the four smaller rectangular regions have a common corner point; use the inverse Hankel integral transform and integrate along the rectangular region to obtain the vertical additional stress and vertical displacement at each depth below the corner point of the smaller rectangular region. Step 400: Using the superposition principle of elastic theory, calculate the vertical additional stress and vertical displacement at any calculation point in the three-layer foundation. In step 200, state-space theory is used to apply the point load... Treating the state vector as a whole, a transfer matrix is ​​established from the surface to infinity at each depth using Hankel integral transformation and combined with interlayer continuity conditions. Then, the surface displacement is calculated using linear theory and surface load, and substituted back into the initial state vector. The transfer matrix is ​​then used to obtain the image functions of the intermediate displacement-related variables and the image function of the vertical additional stress for each soil layer under point load after Hankel integral transformation. A transformation matrix is ​​added to the transfer matrix when the state vector is transferred to the interlayer contact surface of the soil. σ z For vertical additional stress; τ zr For additional shear stress; for σ z w * Performing the 0th order Hankel integral transform respectively yields For u * τ zr Performing the first-order Hankel integral transform respectively yields In the 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 as follows: In the formula, It is expressed as the ratio of the shear modulus G2 of the second soil layer to the shear modulus G1 of the first soil layer; It is expressed as the ratio of the shear modulus G3 of the third soil layer to the shear modulus G2 of the second soil layer; In step 300, the vertical additional stress at each depth below the corner of the small rectangular area. With vertical displacement w (i) (z i The calculation expressions for ) are as follows: In the formula, f′ σi (α h ,z m f' is the aspect function of the vertical additional stress; wi (α h ,z m ) represents the image function of the intermediate calculation variable for vertical displacement; J1 represents the first-order Bessel function of the first kind; 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 required to be introduced when performing a Hankel integral transformation on the polar radius r established directly outward from the ground surface with the load center as the origin; α h Let α be the independent variable after the Hankel integral transformation over the independent variable α. h =α·h1;G i z represents the shear modulus of the i-th soil layer; i This is the vertical distance from the corner of the small rectangular area to a point in the i-th layer of soil in the foundation; z m 'b' represents the location of the soil layer to which the calculation point belongs and the relative calculation depth; 'b' represents the width of the small rectangular area.

2. The calculation method as described in claim 1, characterized in that: In step 300, numerical integration is performed using the built-in integral2 function of MATLAB; for calculation points that are not on the ground surface, α h The upper limit of integration is set to infinity; for calculation points on the ground, α is... h The maximum score is set to or Where X ≥ 100.

3. The calculation method as described in claim 1, characterized in that: In step 400, according to the superposition principle of elasticity theory, the sum of the vertical additional stress and vertical displacement at the corner points under the load of the four small rectangular areas is equal to the vertical additional stress and vertical displacement at the calculation point under the load of the rectangular area.

4. The calculation method as described in claim 3, characterized in that: When the calculation point is located on the axis of the center point of the rectangular area load region, only the vertical additional stress at the corner point of a small rectangular area is calculated. and vertical displacement w j Then by w o =4w j Superimposed as vertical additional stress at the calculation point and vertical displacement w o .

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