A method for calculating the vertical bearing capacity of a rigidly connected steel pipe pile at the top of a sunken road

Abstract

To solve the above problems, the application provides a kind of top and subsidence road rigid connection steel pipe pile vertical bearing capacity calculation method, comprising the following steps: measuring and obtaining the thickness of silty clay;Establish the moment balance equation and the differential equation of deflection curve of any point on steel pipe pile, and the moment balance equation is substituted into the differential equation of deflection curve to obtain the second-order constant coefficient non-homogeneous linear differential equation;The general solution of the second-order constant coefficient non-homogeneous linear differential equation is obtained;The derivative result is obtained by derivation to the general solution, and two groups of boundary conditions are substituted into the derivative result to obtain boundary equation;At the same time, the deflection value of the top is substituted into and associated with the boundary equation to obtain a five-element linear equation group;The value of the determinant of the coefficient matrix of the five-element linear equation group is 0, and the transcendental equation is obtained by expanding, and then the critical pressure of steel pipe pile is obtained.

Description

Technical Field

[0001] This invention belongs to the field of geotechnical design, specifically relating to a method for calculating the vertical bearing capacity of steel pipe piles rigidly connected to a subsided road at their top. Background Technology

[0002] To meet the needs of rapid socio-economic development, various regions are vigorously carrying out highway infrastructure construction. In southwestern my country, with its complex geological conditions and abundant rainfall, the stability of roadbeds faces significant challenges. Roadbeds, exposed to the natural environment for extended periods, are subject to uneven deformation due to soil weight, traffic loads, and rainwater infiltration, affecting driving quality and safety. In severe cases, road subsidence can occur, damaging the road surface and disrupting traffic. Road subsidence caused by soil liquefaction and softening presents even greater challenges for foundation treatment. Traditional reinforcement methods, such as large-scale replacement, grouting, dynamic compaction, granular material piles, and rigid piles, are often difficult to implement, require substantial investment in machinery, materials, and manpower, and are costly, making them unsuitable for emergency support in the event of sudden, localized road subsidence. Furthermore, existing basic theoretical design methods are often insufficient for emergency rescue projects involving sporadic and unpredictable disasters, particularly regarding the compressive stability of steel pipe piles with lateral fixing at the top. Therefore, research on the critical load theory for vertical support reinforcement design of steel pipe piles for subsided roads is of practical significance. Summary of the Invention

[0003] The purpose of this invention is to provide a method for calculating the vertical bearing capacity of steel pipe piles that are rigidly connected to the top of a subsided road. By calculating the critical pressure of the steel pipe pile, a suitable size of steel pipe pile can be determined for reinforcement.

[0004] To address the aforementioned problems, this invention provides a method for calculating the vertical bearing capacity of a steel pipe pile rigidly connected to a subsided road at its top, comprising the following steps: measuring and obtaining the thickness of the silty clay; establishing the moment balance equation and the deflection curve differential equation at any point on the steel pipe pile, and substituting the moment balance equation into the deflection curve differential equation to obtain a second-order non-homogeneous linear differential equation with constant coefficients; obtaining the general solution of the second-order non-homogeneous linear differential equation with constant coefficients; differentiating the general solution to obtain the derivative result, and substituting two sets of boundary conditions into the derivative result to obtain the boundary equation; simultaneously substituting the deflection value at the top and combining it with the boundary equation to obtain a system of five linear equations; setting the determinant of the coefficient matrix of the system of five linear equations to 0, and expanding it to obtain a transcendental equation, thereby obtaining the critical pressure of the steel pipe pile.

[0005] The moment balance equation described in the above-mentioned method for calculating the vertical bearing capacity of the steel pipe piles rigidly connected to the subsided road at the top is as follows:

[0006]

[0007] in, Let O be the distance from any point O on the steel pipe pile to the origin A. The bending moment at point O is... The critical pressure of the steel pipe pile is given. This represents the deflection value at the top of the steel pipe pile under combined load. Let O be the deflection at point O. The tensile force at the top of the steel pipe pile is... The length of the cantilever section. The bending moment at the top of the steel pipe pile; the differential equation of the deflection curve is as follows:

[0008]

[0009] in, The elastic modulus of the steel pipe pile material; The moment of inertia of the steel pipe pile section, , The outer diameter of the steel pipe pile. This refers to the inner diameter of the steel pipe pile.

[0010] Substituting the moment balance equation described in the above-mentioned method for calculating the vertical bearing capacity of the steel pipe piles rigidly connected to the subsided road into the deflection curve differential equation and rearranging it, we obtain the following second-order non-homogeneous linear differential equation with constant coefficients:

[0011]

[0012] make, Substituting into the above equation and rearranging, we obtain the following second-order non-homogeneous linear differential equation with constant coefficients:

[0013] .

[0014] The general solution of the second-order constant-coefficient non-homogeneous linear differential equation described in the above-mentioned method for calculating the vertical bearing capacity of the steel pipe piles rigidly connected to the subsided road is as follows:

[0015] .

[0016] In the above method for calculating the vertical bearing capacity of the steel pipe piles rigidly connected to the subsided road at the top, the general solution of the second-order non-homogeneous linear differential equation with constant coefficients can be differentiated to obtain:

[0017]

[0018] Substituting the two endpoints of the steel pipe pile as two sets of boundary conditions, the boundary equations are obtained as follows:

[0019]

[0020]

[0021]

[0022]

[0023] The deflection value at the top is At the same time, let Substituting the values, we get the following formula:

[0024] .

[0025] Combining the above equation with the boundary equation, we obtain the following system of five linear equations in five variables:

[0026] .

[0027] In the above method for calculating the vertical bearing capacity of the steel pipe piles rigidly connected to the subsided road at the top, setting the determinant of the coefficient matrix of the five-element linear equation system to 0, we obtain the following formula:

[0028]

[0029] After unfolding, we get:

[0030]

[0031] set up, The critical pressure of the steel pipe pile is as follows:

[0032]

[0033] in, The formula for the moment of inertia of a steel pipe pile section is as follows: .

[0034] The above-mentioned technical solution of the present invention has the following beneficial technical effects: Addressing road subsidence caused by subgrade soil liquefaction, the problem of the vertical bearing capacity of the cantilever section of a steel pipe pile located in liquefied soil is transformed into a column stability problem. By establishing an arbitrary moment equilibrium equation on the steel pipe pile after bending, an expression for the moment at any point is obtained. This expression is then substituted into the approximate differential equation of the deflection curve to obtain the deflection curve equation. Further differentiation of the deflection curve equation yields the deflection angle equation. By incorporating the deformation boundary conditions of the steel pipe pile, a system of five linear equations is obtained, leading to the conclusion that the determinant of the coefficient matrix is ​​zero. The expansion of the determinant is a transcendental equation. Solving the transcendental equation using auxiliary methods yields the expression for the maximum vertical bearing capacity. Finally, a complete calculation method for the vertical bearing capacity of a steel pipe pile rigidly connected to the subsided road is completed. Attached Figure Description

[0035] Figure 1 This is a schematic diagram and related geometric dimensions of the steel pipe pile support reinforcement for subsided roads in an embodiment of the present invention;

[0036] Figure 2 The diagram shows the constraint form and stress analysis of the steel pipe pile in this embodiment of the invention.

[0037] Figure 3 This is a deformation analysis diagram of a single steel pipe pile under combined load in an embodiment of the present invention. Detailed Implementation

[0038] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to specific embodiments and accompanying drawings. It should be understood that these descriptions are merely exemplary and not intended to limit the scope of the present invention. Furthermore, descriptions of well-known structures and technologies are omitted in the following description to avoid unnecessarily obscuring the concept of the present invention. In the description of the present invention, it should be noted that the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0039] Liquefaction of the subgrade soil can lead to road subsidence. In this case, steel pipe piles need to be installed at the subsidence end for reinforcement. The top of the steel pipe pile is connected to the road by drilling and grouting, and the bottom is embedded in stable rock. The vertical load of the road is transferred to the deep stable rock through the steel pipe pile. In this way, the steel pipe pile located in the liquefied subgrade soil can be transformed into a compression stability problem, thus forming a design method for steel pipe pile support of subsided roads based on the compression stability theory.

[0040] refer to Figure 1 A highway is constructed of concrete. The road width is measured to be L, the thickness to be H, the weight to be G, and the uniformly distributed load to be q. According to the survey data, the roadbed bearing layer is a soil-rock mixture, consisting of silty clay and moderately weathered sandstone. The measured distribution range of the weathered sandstone is L1, and the distribution range of the silty clay is L2, with a thickness of l. Surface water infiltration and road vibration loads cause liquefaction of the silty clay, resulting in a loss of soil mechanical strength and inducing overturning and subsidence at the front end of the road. Within a width of m at the front end of the road, n steel pipe piles are installed for reinforcement and support. The elastic modulus of the steel pipe piles is E, the outer diameter is d1, and the inner diameter is d2. The top of each pile is embedded in the concrete road through a borehole with a grouting injection length of H and a borehole diameter of D. The bottom of each pile is embedded in the moderately weathered sandstone with a grouting injection length of l / 2, and the middle portion is located in the liquefied silty clay with a length of l.

[0041] refer to Figure 2 As shown, due to the liquefaction of the silty clay subgrade, the mechanical strength is lost. Therefore, a single steel pipe pile in the liquefied silty clay will not be constrained by the surrounding soil and can be simplified to the tensile force F at its top. R With bending moment M RThe stability mechanics problem of a compression column with a cantilever length (equal to the thickness of the silty clay) of l, and the existence of a critical pressure F. cr This ensures that the steel pipe piles are in a critical stable state under compression, which is the maximum vertical pressure that a single steel pipe pile can withstand, as designed. This can be expressed by the relevant physical and mechanical parameters of the steel pipe piles. The bottom of the steel pipe pile support system is fixedly constrained by moderately weathered sandstone, and the top is fixedly connected to the subsided concrete road. When a section of the road subsides and overturns, the top of the steel pipe piles will generate a superposition effect of horizontal lateral force and bending moment. The addition of lateral force and bending moment to the column stability can better fit the actual engineering situation.

[0042] After establishing the mechanical model, such as Figure 3 Let there be a point o on the steel pipe pile at a distance x from the origin A. Then the bending moment at point o can be expressed as: The deflection can be expressed as the value at point o. Establish the moment equilibrium equation at point O, as follows:

[0043] (1)

[0044] in, This represents the deflection at the top of the steel pipe pile under combined loads, i.e. .

[0045] Since the deflection of the steel pipe pile under compression is a small deformation problem, the second derivative of the deflection ω(x) at point o, which is x distance from the origin A, is... With bending moment There exists a linear proportional relationship, which is the approximate differential equation of the deflection curve, as follows:

[0046] (2)

[0047] in, The elastic modulus of the steel pipe pile material; The moment of inertia of the steel pipe pile section, in, The outer diameter of the steel pipe pile. The inner diameter of the steel pipe pile;

[0048] Substituting formula (1) into formula (2), we get:

[0049] (3)

[0050] After sorting, we get:

[0051] (4)

[0052] make Substituting into formula (4) and rearranging, we obtain a second-order non-homogeneous linear differential equation with constant coefficients:

[0053] (5)

[0054] The general solution of the second-order nonhomogeneous linear differential equation with constant coefficients in the above equation can be expressed as:

[0055] (6)

[0056] Differentiating formula (6), we get:

[0057] (7)

[0058] Analyzing point A, it satisfies the following boundary conditions: hour, ,Will Substituting into formulas (6) and (7) respectively, we obtain formulas (8) and (9).

[0059] (8)

[0060] (9)

[0061] Analyzing point B, it satisfies the following boundary conditions: hour, , , This is the deflection angle at the top of the steel pipe pile under combined loads, which can be obtained by referring to tables based on relevant knowledge of mechanics of materials. ,make Substituting into ,Will Substituting into formulas (6) and (7) respectively, we obtain formulas (10) and (11).

[0062] (10)

[0063] (11)

[0064] in, This refers to the deflection value at the top of the steel pipe pile under combined loads, which can be obtained by referring to tables based on relevant knowledge of mechanics of materials. ,make After substituting, we get By rearranging and simplifying, we can obtain formula (12).

[0065] (12)

[0066] Combining formulas (8), (9), (10), (11), and (12), we can obtain:

[0067] (13)

[0068] The system of equations (13) can be viewed as being about , , , , A system of five linear homogeneous equations, and , , , , Since the coefficients cannot all be zero, this system of equations must have a non-zero solution, and therefore the determinant of its coefficient matrix is ​​0, as shown in the following equation:

[0069] (14)

[0070] Expanding the determinant (14), we get:

[0071] (15)

[0072] Formula (15) is about The transcendental equations can be obtained using MATLAB or a Casio calculator, satisfying the minimum pressure condition. Value, set The critical pressure of the compression bar can be expressed as follows:

[0073] (16)

[0074] in, The formula for the moment of inertia of a steel pipe pile section is as follows: .

[0075] It should be understood that the specific embodiments described above are merely illustrative or explanatory of the principles of the invention and do not constitute a limitation thereof. Therefore, any modifications, equivalent substitutions, improvements, etc., made without departing from the spirit and scope of the invention should be included within the protection scope of the invention. Furthermore, the appended claims are intended to cover all variations and modifications falling within the scope and boundaries of the appended claims, or equivalent forms of such scope and boundaries.

Claims

1. A method for calculating the vertical bearing capacity of a steel pipe pile with its top rigidly connected to a subsided road, characterized in that, Includes the following steps: The thickness of the silty clay was measured. Establish the moment balance equation and the deflection curve differential equation at any point on the steel pipe pile, and substitute the moment balance equation into the deflection curve differential equation to obtain a second-order non-homogeneous linear differential equation with constant coefficients. The general solution of the second-order nonhomogeneous linear differential equation with constant coefficients is obtained; The derivative of the general solution is obtained, and the two endpoints of the steel pipe pile are used as two sets of boundary conditions and substituted into the derivative to obtain the boundary equations. Simultaneously, the deflection value of the top of the steel pipe pile under the combined load is substituted into the equation and combined with the boundary equation to obtain a system of five linear equations. The determinant of the coefficient matrix of the five-element linear equation system is set to 0, and the equation is expanded to obtain a transcendental equation, from which the critical pressure of the steel pipe pile is derived.

2. The method for calculating the vertical bearing capacity of the steel pipe pile with rigid connection between the top and the subsided road as described in claim 1, characterized in that: The torque balance equation is as follows: ， in, Let O be the distance from any point O on the steel pipe pile to the origin A. The bending moment at point O is... The critical pressure of the steel pipe pile is given. This represents the deflection value at the top of the steel pipe pile under combined load. Let O be the deflection at point O. The tensile force at the top of the steel pipe pile is... The length of the cantilever section. The bending moment at the top of the steel pipe pile; The differential equation for the torsion curve is as follows: ， in, The elastic modulus of the steel pipe pile material; The moment of inertia of the steel pipe pile section, , The outer diameter of the steel pipe pile. This refers to the inner diameter of the steel pipe pile.

3. The method for calculating the vertical bearing capacity of the steel pipe pile with rigid connection between the top and the subsided road as described in claim 2, characterized in that: Substituting the torque balance equation into the deflection curve differential equation and rearranging, we obtain the following second-order non-homogeneous linear differential equation with constant coefficients: ， make, Substituting into the above equation and rearranging, we obtain the following second-order nonhomogeneous linear differential equation with constant coefficients: 。 4. The method for calculating the vertical bearing capacity of the steel pipe pile with rigid connection between the top and the subsided road as described in claim 3, characterized in that: The general solution of the second-order nonhomogeneous linear differential equation with constant coefficients is as follows: 。 5. The method for calculating the vertical bearing capacity of the steel pipe pile with rigid connection between the top and the subsided road as described in claim 4, characterized in that: Differentiating the general solution of the second-order nonhomogeneous linear differential equation with constant coefficients yields: ， Substituting the two endpoints of the steel pipe pile as two sets of boundary conditions, the boundary equations are obtained as follows: ， ， ， ， The deflection value at the top is At the same time, let Substituting the values, we get the following formula: ， Combining the above equation with the boundary equation, we obtain the following system of five linear equations in five variables: 。 6. The method for calculating the vertical bearing capacity of the steel pipe pile with rigid connection between the top and the subsided road as described in claim 5, characterized in that: Setting the determinant of the coefficient matrix of the five-variable linear equation system to 0, we obtain the following equation: ， After unfolding, we get: ， set up, The critical pressure of the steel pipe pile is as follows: ， in, The formula for the moment of inertia of a steel pipe pile section is as follows: .

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