U-shaped valley pile foundation displacement calculation method considering terrain effect

Abstract

The invention discloses a U-shaped valley pile foundation displacement calculation method considering a terrain effect, and particularly relates to the field of U-shaped valley aseismic design, which comprises the following steps of: 1, determining physical and mechanical parameters of valley terrain, a pile foundation and a soil body; 2, determining an inner-domain displacement field and an outer-domain displacement field, caused by seismic waves, of the valley terrain; 3, based on a Winkler foundation model, a U-shaped valley free field caused by seismic waves is converted into loads to be applied to the pile foundation, and a stress balance equation of the pile foundation is established; 4, solving the pile foundation displacement caused by uniformly distributed loads without considering the counterforce action of the pile foundation bottom end; and 5, solving the counterforce at the bottom of the pile foundation, and further deducing the bending moment and the shearing force borne by the pile foundation. According to the method, the Winkler foundation model is combined with the wave function expansion method to solve the U-shaped valley bridge pile displacement distribution function considering the terrain effect, the calculation method is more accurate, and good application and popularization value is achieved.

Description

Technical Field

[0001] The invention relates to the technical field of U-shaped valley seismic design, and in particular to a method for calculating the displacement of a U-shaped valley pile foundation taking terrain effects into consideration. Background Art

[0004] The existing valley pile foundation displacement is mostly calculated and predicted using the flat ground transverse wave model and the flat ground longitudinal wave model. However, the above calculation methods are not accurate enough to meet people's usage needs. Summary of the invention

[0005] The object of the present invention is to provide a method for calculating the displacement of a U-shaped valley pile foundation taking into account the terrain effect, so as to solve the problems raised in the above-mentioned background technology.

[0006] In order to solve the above technical problems, the present invention provides the following technical solutions: a method for calculating the displacement of a U-shaped valley pile foundation considering terrain effects, comprising the following steps:

[0007] Step 1: Determine the physical and mechanical parameters of valley topography, pile foundation and soil;

[0008] Step 2: Determine the inner and outer displacement fields of the valley terrain caused by the earthquake wave;

[0009] Step 3: Based on the Winkler foundation model, the U-shaped valley free field caused by the seismic wave is converted into a load and applied to the pile foundation to establish the force equilibrium equation of the pile foundation;

[0010] Step 4: When the reaction force at the bottom of the pile base is not considered, solve the pile foundation displacement caused by the uniformly distributed load;

[0011] Step 5: Solve the reaction force at the bottom of the pile foundation, and then derive the bending moment and shear force on the pile foundation.

[0012] In a preferred embodiment, the valley terrain parameters in step 1 include valley radius, seismic wave frequency, incident angle, the distance from the center of the canyon to both sides, and the vertical distance from the top of the canyon to the bottom of the slope; the pile foundation parameters include the equivalent diameter of the pile foundation, the depth of burial, the elastic modulus of the pile foundation, the moment of inertia of the pile foundation cross section, and the location of the pile foundation; the soil parameters include the elastic modulus of the soil and the Poisson's ratio of the soil.

[0013] In a preferred embodiment, the calculation formula of the external displacement field formed by the incident seismic wave in step 2 is as follows:

[0014] u outer (r,θ)=u f (r,θ)+u s1 (r,θ)+u s2(r,θ)

[0015] Where uf(r,θ) is the external free field caused by the U-shaped valley, u s1 (r,θ) is the external scattering field caused by the U-shaped valley, u s2 (r,θ) is the external scattered field II caused by the U-shaped valley;

[0016] where uf(r,θ), u s1 (r,θ) and u s2 (r,θ) are obtained by the following formulas:

[0017]

[0018]

[0019] In the formula, α is the incident angle of the seismic wave; r is the polar radius of the circle centered at point o, θ is the polar angle; k is the incident wave number, where η is the incident wave frequency, ε n is the Newman coefficient (ε0=1,n=0;ε n =2, n>0); H (1) n (.) is the nth-order third-kind Bessel function, J n (.) is the nth-order Bessel function of the first kind; A n , B n is the coefficient to be determined.

[0020] In a preferred embodiment, the calculation formulas of the inner domain displacement field 1 and the inner domain displacement field 2 of the U-shaped valley terrain in step 2 are as follows:

[0021]

[0022] In the formula, C n , D n is the coefficient to be determined; d0 is the vertical distance from the top of the canyon to the bottom of the slope.

[0023] In a preferred embodiment, the formula for establishing the force balance equation of the pile foundation based on the Winkler foundation model in step 3 is:

[0024]

[0025] Where W(y) is the displacement function of the pile foundation; Ep is the elastic modulus of the pile foundation; Ip is the moment of inertia of the pile cross section; D is the equivalent width; k is the spring stiffness of the soil on the pile side, and P(y) is the vertical external load on the pile foundation.

[0026] In a preferred embodiment, the calculation formula for the spring stiffness k of the pile side soil is:

[0027]

[0028] Where Es is the elastic modulus of soil; μs is the Poisson's ratio of soil.

[0029] In a preferred embodiment, the calculation formula of the vertical external load P(y) on the pile foundation is:

[0030] p(y)=k*wf(y)

[0031] Where wf(y) is the free field of the pile foundation, which can be obtained by the following formula:

[0032]

[0033] In the formula, x represents the location where the pile foundation is placed.

[0034] In a preferred embodiment, the finite difference method is used to solve the pile foundation displacement w caused by the uniformly distributed load in step 4, and the formula is as follows:

[0035]

[0036] Where, {K} -1 (n+1)*(n+1) is the stiffness matrix.

[0037] In a preferred embodiment, when the reaction force at the bottom of the pile base is not considered in step 5, the calculation formula for solving the pile foundation shear force Q caused by the uniformly distributed load is as follows:

[0038]

[0039] In a preferred embodiment, in the step 5, when the reaction force at the bottom end of the pile base is not considered, the calculation formula for solving the pile foundation bending moment M caused by the uniformly distributed load is as follows:

[0040]

[0041] Compared with the prior art, the beneficial effects achieved by the present invention are:

[0042] The present invention adopts the Winkler foundation model combined with the wave function expansion method for the first time to solve the U-shaped valley bridge pile displacement distribution function considering the terrain effect. Compared with the flat ground shear wave model and the flat ground longitudinal wave model, it is more accurate and has great promotion and application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation of the present invention. In the accompanying drawings:

[0044] Figure 1 It is a flow chart of a method for calculating the displacement of a U-shaped valley pile foundation taking into account terrain effect according to the present invention;

[0045] Figure 2 It is a calculation model diagram of a method for calculating the displacement of a U-shaped valley pile foundation taking into account the terrain effect of the present invention;

[0046] Figure 3 It is a single pile soil displacement curve diagram of a U-shaped valley pile foundation displacement calculation method considering terrain effect of the present invention;

[0047] Figure 4 The present invention is a pile foundation bending moment diagram of a method for calculating the displacement of a U-shaped valley pile foundation considering terrain effect;

[0048] Figure 5 The present invention discloses a pile foundation shear force diagram of a method for calculating the displacement of a U-shaped valley pile foundation taking terrain effect into consideration. DETAILED DESCRIPTION

[0049] The following specific embodiments illustrate the implementation of the present invention, and the technical solutions in the embodiments of the present invention are clearly and completely described. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0050] Please refer to Figure 1 and Figure 2 As shown, the present invention provides a method for calculating the displacement of a U-shaped valley pile foundation taking into account terrain effects, comprising the following steps:

[0051] Step 1: Determine the physical and mechanical parameters of valley topography, pile foundation and soil;

[0052] The specific valley terrain parameters are valley radius d = 1m, seismic wave frequency η = 1, incident angle α = 0°, distances from the center of the canyon to the two sides b1 = 0.9m, b2 = 0.9m, vertical distance d0 = 0.9m from the top of the canyon to the bottom of the slope; the pile foundation parameters are equivalent diameter D = 0.4m, burial depth L = 25m, pile foundation elastic modulus Ep = 2*10 10 Pa, pile foundation cross section moment of inertia Ip=1.2566*10 -3 m 4, x = -2 / 3 is the location of the pile foundation; the soil parameter is elastic modulus Es = 2.5*10 7 , Poisson's ratio of soil μs = 1 / 3.

[0053] Step 2: Determine the inner and outer displacement fields of the valley terrain caused by the earthquake wave;

[0054] The calculation formula of the external displacement field formed by the incident seismic wave is as follows:

[0055] u outer (r,θ)=u f (r,θ)+u s1 (r,θ)+u s2 (r,θ) (1)

[0056] Where uf(r,θ) is the external free field caused by the U-shaped valley, u s1 (r,θ) is the external scattering field caused by the U-shaped valley, u s2 (r,θ) is the external scattered field 2 caused by the U-shaped valley.

[0057] Substituting the valley terrain parameters into equation (1) we can obtain the external displacement field.

[0058] Where uf(r,θ), us1(r,θ) and us2(r,θ) are obtained by the following formulas:

[0059]

[0060] In the formula, α is the incident angle of the seismic wave; r is the polar radius of the circle centered at point o, θ is the polar angle; k is the incident wave number, where η is the incident wave frequency, εn is the Newman coefficient (ε0=1, n=0; εn=2, n>0); H (1) n (.) is the nth order Bessel function of the third kind, Jn(.) is the nth order Bessel function of the first kind; An, Bn are the coefficients to be determined.

[0061] The calculation formulas for the inner displacement field 1 and inner displacement field 2 of the U-shaped valley terrain are as follows:

[0062]

[0063]

[0064] In the formula, C n , D n is the coefficient to be determined;

[0065] Step 3: Based on the Winkler foundation model, the U-shaped valley free field caused by the seismic wave is converted into a load and applied to the pile foundation to establish the force equilibrium equation of the pile foundation;

[0066] The formula for establishing the force equilibrium equation of pile foundation based on the Winkler foundation model is:

[0067]

[0068] In the formula, the cross-sectional area of ​​the pile foundation can be determined based on the known parameters: The spring stiffness of the soil around the pile is

[0069] The calculation formula of the vertical external load P(y) on the pile foundation is:

[0070] p(y)=k*wf(y) (8)

[0071] Where wf(y) is the free field of the pile foundation, as follows:

[0072]

[0073] Step 4: When the reaction force at the bottom of the pile base is not considered, solve the pile foundation displacement caused by the uniformly distributed load;

[0074] The above equation can be written in finite difference form:

[0075] αw i-2 +βw i-1 +γw i +βw i+1 +αw i+2 =p i (10)

[0076] In the formula, Combined with the boundary conditions, the equation for the displacement of a single pile can be derived as follows:

[0077]

[0078] Where, {K} -1 (n+1)*(n+1) is the stiffness matrix, which is as follows:

[0079]

[0080] The displacement curve of the pile foundation in this embodiment is obtained by using the single pile displacement equation as follows: Figure 3 shown.

[0081] Step 5: Solve the reaction force at the bottom of the pile foundation, and then derive the bending moment and shear force on the pile foundation.

[0082] The finite difference method is used to solve the above equation, and the pile foundation is divided into n sections. The height of each section is H = L / n, and L is the length of the pile. The pile foundation node numbers are 0, 1, ..., n-1, n from the top of the pile to the end of the pile. Two virtual nodes -2, -1, n+1, and n+2 are added at the top and end of the pile respectively during the calculation.

[0083] The bending moment of the pile foundation is as follows:

[0084]

[0085] The bending moment curve of the pile foundation in this embodiment is as follows: Figure 4 shown.

[0086] The pile foundation shear force is as follows:

[0087]

[0088] The shear force curve of the pile foundation in this embodiment is as follows: Figure 5 shown.

[0089] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art can still modify the technical solutions described in the aforementioned embodiments or replace some of the technical features therein by equivalents. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for calculating the displacement of U-shaped valley pile foundation considering terrain effect, characterized in that: The following steps are involved: Step 1: Determine the physical and mechanical parameters of valley topography, pile foundation and soil; Step 2: Determine the inner and outer displacement fields of the valley terrain caused by the earthquake wave; Step 3: Based on the Winkler foundation model, the U-shaped valley free field caused by the seismic wave is converted into a load and applied to the pile foundation to establish the force equilibrium equation of the pile foundation; Step 4: When the reaction force at the bottom of the pile base is not considered, solve the pile foundation displacement caused by the uniformly distributed load; Step 5: Solve the reaction force at the bottom of the pile foundation, and then derive the bending moment and shear force on the pile foundation.

2. The method for calculating the displacement of a U-shaped valley pile foundation considering terrain effect according to claim 1, characterized in that: The valley terrain parameters in step 1 include valley radius, seismic wave frequency, incident angle, distance from the center of the canyon to both sides, and vertical distance from the top of the canyon to the bottom of the slope; the pile foundation parameters include equivalent diameter of the pile foundation, depth of burial, elastic modulus of the pile foundation, moment of inertia of the pile foundation cross section, and location of the pile foundation; the soil parameters include elastic modulus of the soil and Poisson's ratio of the soil.

3. The method for calculating the displacement of a U-shaped valley pile foundation considering terrain effect according to claim 1, characterized in that: The calculation formula of the external displacement field formed by the incident seismic wave in step 2 is as follows: you outer (r,θ)=u f (r,θ)+u s1 (r,θ)+u s2 (r,θ) Where uf(r,θ) is the external free field caused by the U-shaped valley, u s1 (r,θ) is the external scattering field caused by the U-shaped valley, u s2 (r,θ) is the external scattered field II caused by the U-shaped valley; where uf(r,θ), u s1 (r,θ) and u s2 (r,θ) are obtained by the following formulas: In the formula, α is the incident angle of the seismic wave; r is the polar radius of the circle centered at point o, θ is the polar angle; k is the incident wave number, where η is the incident wave frequency, ε n is the Newman coefficient (ε0=1,n=0;ε n =2, n>0); H (1) n (.) is the nth-order third-kind Bessel function, J n (.) is the nth-order Bessel function of the first kind; A n , B n is the coefficient to be determined.

4. The method for calculating the displacement of a U-shaped valley pile foundation considering terrain effect according to claim 3 is characterized by: The calculation formulas for the inner displacement field 1 and the inner displacement field 2 of the U-shaped valley terrain in step 2 are as follows: In the formula, C n , D n is the coefficient to be determined; d0 is the vertical distance from the top of the canyon to the bottom of the slope.

5. The method for calculating the displacement of U-shaped valley pile foundation considering terrain effect according to claim 4 is characterized by: The formula for establishing the force balance equation of the pile foundation based on the Winkler foundation model in step 3 is: Where W(y) is the displacement function of the pile foundation; Ep is the elastic modulus of the pile foundation; Ip is the moment of inertia of the pile cross section; D is the equivalent width; k is the spring stiffness of the soil on the pile side, and P(y) is the vertical external load on the pile foundation.

6. The method for calculating the displacement of U-shaped valley pile foundation considering terrain effect according to claim 5, characterized in that: The calculation formula of the spring stiffness k of the pile side soil is: Where Es is the elastic modulus of soil; μs is the Poisson's ratio of soil.

7. The method for calculating the displacement of U-shaped valley pile foundation considering terrain effect according to claim 5, characterized in that: The calculation formula of the vertical external load P(y) on the pile foundation is: p(y)=k*wf(y) Where wf(y) is the free field of the pile foundation, which can be obtained by the following formula: In the formula, x represents the location where the pile foundation is placed.

8. The method for calculating the displacement of U-shaped valley pile foundation considering terrain effect according to claim 7 is characterized by: In step 4, the finite difference method is used to solve the pile foundation displacement w caused by the uniformly distributed load, and the formula is as follows: Where, {K} -1 (n+1)*(n+1) is the stiffness matrix.

9. The method for calculating the displacement of U-shaped valley pile foundation considering terrain effect according to claim 8, characterized in that: In step 5, when the reaction force at the bottom of the pile foundation is not considered, the calculation formula for solving the pile foundation shear force Q caused by the uniformly distributed load is as follows:

10. The method for calculating the displacement of U-shaped valley pile foundation considering terrain effect according to claim 8, characterized in that: In the step 5, when the reaction force at the bottom of the pile base is not considered, the calculation formula for solving the pile foundation bending moment M caused by the uniformly distributed load is as follows:

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