A method for calculating earth pressure in circular foundation pits based on circumferential stress correction

Abstract

This invention relates to a method for calculating earth pressure in circular foundation pits considering circumferential stress. The method includes the following steps: determining the design parameters of the circular foundation pit and the physical and mechanical parameters of the soil surrounding the pit; determining the sliding failure surface of the soil outside the pit; assuming a linear relationship between the circumferential stress at any point within the sliding soil and its distance from the pit, and determining the functional expression of the corresponding circumferential stress and the distance *r* from the pit; establishing circumferential and radial equations based on force equilibrium conditions using the horizontal layer analysis method, and finally obtaining the distribution of active earth pressure with depth, ultimately calculating the earth pressure in the circular foundation pit. Using the calculation method of this invention, the distribution of earth pressure around a circular foundation pit can be obtained more accurately and reasonably, and it is applicable to multi-layered soil conditions and cohesive and non-cohesive soil layers, which is beneficial for the rational design of circular foundation pit projects.

Description

Technical Field

[0001] This invention relates to the field of foundation pit engineering technology, specifically to a method for calculating earth pressure in circular foundation pits based on circumferential stress correction. Background Technology

[0002] With the development of super high-rise buildings and the increasing utilization of urban underground space in my country, foundation pit engineering is also developing towards larger and deeper areas. Among them, circular foundation pits, due to their unique geometric spatial effects, can form a circular whole with the support structure, giving full play to the compressive strength of the concrete structure, making them widely used in deep foundation pit engineering.

[0003] However, due to the lack of a solid theoretical foundation for calculating earth pressure in circular foundation pits, coupled with an insufficient understanding of the spatial effects of circular foundation pits, it is difficult to determine the earth pressure reasonably. Currently, most circular foundation pit projects still use traditional plane earth pressure calculation theory to obtain the horizontal loads acting on the supporting structure during the design phase. However, plane earth pressure theory has significant shortcomings in solving the earth pressure problem of circular foundation pits. For example, it does not consider the triaxial stress state of soil elements under the circular spatial effect; nor does it consider the influence of soil arching effect on the magnitude of earth pressure. This leads to conservative results in the design, resulting in increased costs and wasted resources.

[0004] In conclusion, it is necessary to explore a theory and method for calculating earth pressure in circular foundation pits that takes into account spatial effects, which can improve the design and construction level of circular foundation pits. Summary of the Invention

[0005] This invention addresses the shortcomings of existing theories for calculating earth pressure in circular foundation pits by providing a method for calculating earth pressure in circular foundation pits based on circumferential stress correction. This method yields a more accurate and reasonable distribution of earth pressure around the circular foundation pit, thus improving the design and construction level of circular foundation pits.

[0006] To achieve the above technical objectives, this invention provides a method for calculating earth pressure in a circular foundation pit based on circumferential stress correction, characterized in that the method specifically includes the following steps:

[0007] Step 1: Determine the design parameters of the circular foundation pit and the physical and mechanical parameters of the surrounding soil, including: excavation depth D, excavation radius R, support structure length H; number of soil layers n, and soil weight γ. i Cohesion c i internal friction angle The external friction angle δ between the support structure and the soil i The cohesion between the support structure and the soil is c wiThe uniformly distributed load q acting on the soil surface (where i = 1 to n); the following calculations all use the above parameters.

[0008] Step 2: Determine the sliding failure surface of the soil outside the foundation pit, assuming the angle between the sliding surface and the horizontal plane is... The sliding surface intersects with the bottom of the support structure, and according to the different soil layer distributions, the sliding surfaces in each soil layer are connected to form a continuous sliding surface;

[0009] Step 3: Assuming that the circumferential stress at any point in the soil within the sliding surface under the limit state is linearly related to the distance r from that point to the support structure, the formula for calculating the circumferential stress at any point in the soil within the sliding surface is obtained:

[0010]

[0011] In the formula, x is the distance from the sidewall of the foundation pit to any point within the sliding soil body;

[0012] σ θ σ is the circumferential stress of the soil; R is the radius of the circular foundation pit excavation; v This refers to the vertical stress of the soil.

[0013] Step 4: Analyze the annular soil wedge formed after the support structure is completed under the ultimate condition. Using the horizontal layer analysis method, establish the radial equilibrium equation and vertical static equilibrium equation of the differential element in the polar coordinate system, and solve them simultaneously to obtain:

[0014]

[0015] The initial value condition for this differential equation is: when z = 0, σ v =q;

[0016] In the formula, z is the burial depth of the differential unit; A is the area of ​​the upper surface of the differential unit;

[0017] K aw Lateral earth pressure coefficient considering the soil arching effect:

[0018]

[0019] Step 5: Using numerical methods, Euler iteratively solves the first ordinary differential equation in Equation ② and obtains the corresponding active earth pressure distribution:

[0020]

[0021] In the formula, F(z,σ) v The unknown variable is the vertical earth pressure σ. v The function expression obtained by taking the first derivative with respect to the burial depth z; σ v(z＝0)=q is the initial condition of the ordinary differential equation, that is, the external load on the surface of the soil layer when the burial depth z = 0.

[0022] A further technical solution of the present invention: In step four, the horizontal layer analysis method is used to establish the radial equilibrium equation of the differential unit volume in polar coordinates based on the force balance in the polar coordinate system:

[0023]

[0024] In the formula, N represents the corresponding differential unit. w =P w cosδ i P w The resultant force of contact between the support structure and the soil.

[0025] Solving the equation yields:

[0026]

[0027] In the formula, P θ The resultant force on the side of the differential element is expressed as follows:

[0028]

[0029] x is the distance from the sidewall of the foundation pit to any point within the sliding soil body;

[0030] Further establish the vertical static equilibrium equations for the differential unit:

[0031]

[0032] In the formula, P n The resultant force of soil contact at the sliding surface;

[0033] Solving by combining formulas ⑧ and ⑦ yields formula ② in step four.

[0034] A preferred technical solution of the present invention: When the soil body is a combination of multiple soil layers, the solution for each soil layer still uses the numerical method modified Euler method. The difference is that the initial conditions for each soil layer need to be modified according to the following formula:

[0035]

[0036] In the formula, σ v,i Z represents the vertical stress value in soil layer i; i Let i be the burial depth of the i-th soil layer;

[0037] σ v,i-1 This represents the vertical stress value at the bottom of the (i-1)th soil layer.

[0038] The preferred technical solution of the present invention is as follows: the soil can be cohesive soil or non-cohesive soil, and the soil surface is horizontal.

[0039] The preferred technical solution of the present invention is that the modified numerical method Euler is solved by programming on the Visual Studio platform.

[0040] The beneficial effects of this invention are as follows: the method for calculating earth pressure in a circular foundation pit can take into account the influence of circumferential stress on earth pressure calculation, and more reasonably considers the influence of three-dimensional spatial effects of the circular foundation pit on active earth pressure calculation. The resulting earth pressure distribution is more accurate and reasonable, and it can be applied to the calculation of active earth pressure in a circular foundation pit. This is beneficial to the design and cost control of foundation pit engineering and has certain promotion and application value. Attached Figure Description

[0041] Figure 1 This is a schematic diagram of the calculation process of the present invention;

[0042] Figure 2 This is a schematic diagram of the radial equilibrium of a differential unit;

[0043] Figure 3 This is a schematic diagram of the vertical equilibrium of a differential unit cell;

[0044] Figure 4 This is a schematic diagram of the geometric information of the example foundation pit;

[0045] Figure 5 This is a comparison chart of the active earth pressure distribution curves in the examples; Detailed Implementation

[0046] The invention will now be described in further detail with reference to the accompanying drawings and embodiments. The drawings are specific embodiments of the invention, and simplified drawings are used to concisely and clearly illustrate the specific objectives of the embodiments. However, it should be understood that the scope of the specific implementation of the invention is not limited to the corresponding content in the embodiments, and can be implemented in many different ways. The purpose of providing this embodiment is solely to more comprehensively and clearly disclose the content of the invention.

[0047] This invention addresses the shortcomings of existing theories for calculating earth pressure in circular foundation pits by providing a method for calculating earth pressure in circular foundation pits based on circumferential stress correction. The main calculation process of this method can be found in [reference needed]. Figure 1 The following are the specific implementation steps. In the following calculations, the same parameter appearing in different formulas has the same meaning:

[0048] Step 1: Determine the design parameters of the circular foundation pit and the physical and mechanical parameters of the surrounding soil, including: excavation depth D, excavation radius R, support structure length H; number of soil layers n, and soil weight γ. i Cohesion c iinternal friction angle The external friction angle δ between the support structure and the soil i The cohesion between the support structure and the soil is c wi The uniformly distributed load q acting on the soil surface (where i = 1 to n);

[0049] Step 2: Determine the sliding failure surface of the soil outside the foundation pit, assuming the angle between the sliding surface and the horizontal plane is... The sliding surface intersects with the bottom of the support structure, and according to the different soil layer distributions, the sliding surfaces within each soil layer are connected to form a continuous sliding surface. For specific sliding surface shapes, please refer to [reference needed]. Figure 2 ;

[0050] Step 3: Assuming that the circumferential stress of the soil on the side of the support structure is linearly related to the distance r from the support structure under the limit state, the formula for calculating the circumferential stress at any point in the soil within the sliding surface can be obtained:

[0051]

[0052] In the formula: x is the distance from the sidewall of the foundation pit to any point within the sliding soil body;

[0053] σ θ σ is the circumferential stress of the soil; R is the radius of the circular foundation pit excavation; v This refers to the vertical stress of the soil.

[0054] Step 4: Analyze the annular soil wedge formed after the support structure is completed under the ultimate condition. Using the horizontal layer analysis method, establish the radial equilibrium equations of the differential element based on the force equilibrium in the polar coordinate system:

[0055]

[0056] In the formula, θ is the angle of the center of the differential unit circle;

[0057] N w =P w cosδ i P w For the resultant force of contact between the support structure and the soil, please refer to [reference needed]. Figure 2 ,beg

[0058] Solving the equation yields:

[0059]

[0060] In the formula, P θ The resultant force on the side of the differential unit body.

[0061] Its expression is

[0062] x is the distance from the sidewall of the foundation pit to any point within the sliding soil body;

[0063] Further: Establish the vertical static equilibrium equations for the differential unit:

[0064]

[0065] In the formula, P n The resultant force of the soil contact at the sliding surface can be found in [reference]. Figure 3 .

[0066] Solving equations ④ and ⑤ simultaneously yields:

[0067]

[0068] The initial value condition for this differential equation is: when z = 0, σ v =q;

[0069] K aw Lateral earth pressure coefficient considering the soil arching effect:

[0070]

[0071] Step 5: Using the numerical method to modify the Euler method, solve the first ordinary differential equation in equation ⑥ to calculate the active earth pressure at a certain burial depth z:

[0072]

[0073] In the formula, F(z,σ) v The unknown variable is the active earth pressure σ. v The function expression obtained by taking the first derivative of the burial depth z, σ v(z＝0) =q is the initial condition of the ordinary differential equation, that is, the external load on the surface of the soil layer when the burial depth z = 0.

[0074] When using the circular foundation pit earth pressure calculation method based on circumferential stress correction provided in this paper to solve for multi-layer soil cases, the numerical method modified Euler method is still used for solving different soil layers. The difference is that the initial conditions of each soil layer need to be corrected using the following formula:

[0075]

[0076] In the formula, σ v,i Z represents the vertical stress value in soil layer i; i Let i be the burial depth of the i-th soil layer;

[0077] σ v,i-1 This represents the vertical stress value at the bottom of the (i-1)th soil layer.

[0078] The following specific embodiments are used as a comparison to test whether the proposed calculation method can reasonably calculate the earth pressure on the support structure in the case of multi-layered cohesive soil. The embodiments use the same calculation case as in the references, namely a circular foundation pit with a radius of 2m, an excavation depth of 20m, and a cohesion between the support structure and the soil of c. w =0MPa, the friction angle between the support structure and the soil is δ=0°. The first soil layer is non-cohesive soil with a unit weight of γ1=18kN / m. 3 The internal friction angle is The second soil layer has a thickness of h1 = 10m and is cohesive with a unit weight of γ2 = 20kN / m³. 3 The internal friction angle is The cohesion is c2 = 5 kPa, and the soil layer thickness is h2 = 10 m.

[0079] The active earth pressure of a circular foundation pit under the above parameters is calculated using the method of this invention. First, the corresponding calculation parameters, namely the geometric dimensions of the foundation pit and the soil layer information and parameters, are input into the program. Second, the functional expressions of M(z) and N(z) with respect to depth z are obtained through the equilibrium equation of the differential unit, and substituted into the first term of equation ⑥ to obtain the differential equation expression. Then, the differential equation of equation ⑥ is solved by the numerical method—the modified Euler method. A comparison chart of the active earth pressure distribution when the foundation pit is excavated to the bottom, the calculated values ​​from the present invention, the references, and the Rankine earth pressure theory, is obtained, as shown in the figure. Figure 5 As shown.

[0080] To analyze the influence of the excavation radius of a circular foundation pit on the magnitude of active earth pressure, this specific embodiment considers excavation radii of 20m, 100m, and 200m simultaneously, and uses the calculation method of this invention for calculation. A comparison chart of the calculated active earth pressure distribution when the foundation pit is excavated to the bottom under different radii is obtained, as shown in the figure. Figure 5 As shown; by Figure 5 As can be seen, compared with Rankine's earth pressure theory in the standard, the calculated values ​​of this invention are significantly smaller and closer to the corresponding calculated values ​​in the references. Overall, the earth pressure variation shows that the magnitude of the earth pressure gradually increases with the depth of the foundation pit. However, due to the three-dimensional spatial effect of the circular foundation pit, the earth pressure distribution is significantly different from that in the two-dimensional plane. Furthermore, it is evident that the active earth pressure of the circular foundation pit gradually increases with the increase of the excavation radius, and the three-dimensional spatial effect of the circular foundation pit gradually weakens with the increase of the pit radius, ultimately remaining basically consistent with the two-dimensional plane calculation results. This also demonstrates that the calculation method of this invention effectively considers the three-dimensional spatial effect of the circular foundation pit, reasonably reflects the active earth pressure distribution of the circular foundation pit, and that calculating according to Rankine's earth pressure as used in the standard would result in significant waste and be detrimental to cost control.

[0081] Finally, it should be noted that the above embodiments are only used to illustrate specific implementations of the present invention and are not intended to limit it. Although the present invention has been described in detail through the above examples, those skilled in the art should understand that different modifications can be made in form and detail without departing from the inherent theory of the present invention, without departing from the scope defined by the claims of the present invention.

Claims

1. A circular foundation pit soil pressure calculation method based on annular stress correction, characterized by The method specifically comprises the following steps: Step one: determine the circular foundation pit design parameters and the physical and mechanical parameters of the soil around the foundation pit, including: foundation pit excavation depth D, foundation pit excavation radius R, support structure length H; soil layer number n, soil bulk density γ i , cohesion c i , internal friction angle Support structure and soil external friction angle δ i , support structure and soil contact cohesion c wi , uniform load acting on the surface of the soil q, where i = 1 ~ n; the following calculation process uses the above parameters for calculation; Step two: determine the sliding rupture surface of the soil outside the foundation pit, assuming that the angle between the sliding surface and the horizontal plane is The sliding surface intersects the bottom end of the supporting structure and connects the sliding surface in each layer of soil to form a continuous sliding surface according to the distribution of different soil layers. Step three: assuming that the hoop stress of an arbitrary point in the sliding surface under the limit state is linearly related to the distance r of the point from the supporting structure, a calculation formula of the hoop stress of an arbitrary point in the sliding surface is obtained: In the formula, x is the distance from the side wall of the foundation pit to an arbitrary point in the sliding soil body; σ θ σtis the circumferential stress of the soil; R is the radius of the circular foundation pit excavation; σ v σvtis the vertical stress of the soil; Step four: taking the annular soil wedge formed behind the supporting structure under the limit state for analysis, a horizontal layer analysis method is adopted, a radial balance equation and a vertical static balance equation of a differential unit body under a polar coordinate system are respectively established, and the following formula 2 is obtained by solving the equations simultaneously: The initial condition for this differential equation is: z = 0 when σ v = q; In the formula, z is the buried depth of the differential unit body; and A is the upper surface area of the differential unit body; K aw Lateral earth pressure coefficient considering soil arching effect: Step five: a modified numerical method Euler method is adopted to iteratively solve the first term ordinary differential equation in the formula 2 and the corresponding active earth pressure distribution is obtained: where F(z,σ v ) is the unknown vertical earth pressure σ v with respect to the depth z; and σ v(z＝0) = q is the initial condition of the ordinary differential equation, i.e., the external load on the surface of the soil layer when the depth z = 0.

2. The method according to claim 1, wherein the method is characterized in that: In the step four, the horizontal layer analysis method is adopted, the radial balance equation of the differential unit body under the polar coordinate system is established according to the force balance under the polar coordinate system, and the following formula 6 is obtained: Where θ is the differential unit body central angle; N w = P w cosδ i , P w is the contact force between the supporting structure and the soil body, The formula is solved to obtain the following formula 7: where P θ is the differential cell side force, which is expressed as: In the formula, x is the distance from the side wall of the foundation pit to an arbitrary point in the sliding soil body; A vertical static balance equation of the differential unit body is further established as follows: In the formula, P n is the contact force of the soil at the sliding surface; The formula 7 and the formula 6 are solved to obtain the formula 2 in the step four.

3. The method according to claim 1 or 2, characterized in that: When the soil body is a combination of multiple different soil layers, the numerical method modified Euler method is still adopted for solving different soil layers, and the difference is that the initial value condition of each soil layer needs to be modified according to the following formula: In the formula, σ v,i is the vertical stress value in the i-th layer of soil; z i is the burial depth of the i-th layer of soil; σ v,i-1 σi-1is the vertical stress value at the bottom of the i-1th layer of soil.

4. The method according to claim 1 or 2, characterized in that: When the soil body is cohesive soil or non-cohesive soil, the soil body surface is horizontal.

5. The method according to claim 1 or 2, wherein the method is characterized in that: The modified numerical method Euler method is programmed and solved based on a Visual studio platform.

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