CN116484691A - Power response calculation method for pile foundation in V-shaped valley topography - Google Patents
Power response calculation method for pile foundation in V-shaped valley topography Download PDFInfo
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Abstract
The invention relates to a power response calculation method of pile foundations in V-shaped valley terrains, which comprises the following steps: (1) Obtaining physical and mechanical parameters of the valley, pile foundation and soil body; (2) Calculating a soil body displacement field caused by seismic wave vibration based on Graf addition theorem; (3) Based on a dynamic Winkler foundation model, converting a soil body displacement field caused by seismic wave vibration into a load to be applied to a pile foundation, and establishing a stress balance equation of the pile foundation; (4) And under the condition that the pile top and the pile bottom are both in free constraint, solving pile foundation displacement caused by uniformly distributed load and bending moment and shearing force born by the pile foundation displacement. According to the method, the closed solution of the pile foundation vibration response on the V-shaped valley is provided for the first time by combining the Graf addition theorem with the Winkler foundation Liang Moxing, and the influence of the V-shaped valley earthquake amplification effect on the pile foundation vibration response is accurately revealed.
Description
Technical Field
The invention relates to the technical field of earthquake-proof design, in particular to a power response calculation method of pile foundations in V-shaped valley terrains.
Background
The area of the domestic land is vast, but the area of the mountain land, hills and plateaus is more than 2/3 of the area of the domestic land. Therefore, the risks such as debris flow and landslide exist in many areas in China, and particularly the occurrence probability of the risks can be increased under the action of earthquakes, so that great difficulty is brought to road and bridge construction engineering, and the stability of the V-shaped valley pile foundation is affected. In this way, the dynamic response condition of the mountain V-shaped valley pile foundation under the action of the earthquake is accurately analyzed, so that the actual condition of the mountain V-shaped valley pile foundation is well known, a correct construction plan is formulated, the quality of the pile foundation is well improved, the life and property safety of people is guaranteed, and the due benefit is realized.
Compared with the traditional design method of pile foundations, the design of bearing the V-shaped valley pile foundations in mountain areas is more complex, and the construction difficulty is high. The research on the aspect of China is relatively few. Therefore, the displacement calculation method of the pile foundation in the V-shaped valley topography is provided under the action of the earthquake waves, so that the displacement of the pile foundation is calculated more accurately, and engineering accidents are prevented. In the prior art, a blank exists for researching the vibration response of the foundation of the V-shaped valley pile, the vibration response of the V-shaped valley pile is researched internationally at present to simulate the residual deformation of the V-shaped valley after vibration by a limit method, and the residual deformation is converted into a displacement load to calculate the static force stress and deformation of the V-shaped valley pile. This method has the following problems:
1. the method can only simulate the influence of the residual displacement of the V-shaped valley post-earthquake on the stress deformation of the pile foundation, and the pile foundation is often in an earthquake process rather than after the earthquake when the earthquake-proof safety is the lowest, so that the V-shaped valley pile foundation earthquake-proof design theory based on the method cannot calculate the most unfavorable situation of the V-shaped valley pile foundation earthquake-proof.
2.V valley topography effect can obviously amplify pile foundation vibration response in the earthquake process, but the existing research theory can not reveal the amplification rule, and the safety redundancy is seriously reduced when the V-shaped valley pile foundation earthquake-proof design is carried out based on the existing theory.
3. The vibration rule of the pile foundation in the earthquake process cannot be simulated, and the dynamic response time course curve of the pile foundation in the earthquake process cannot be obtained.
Disclosure of Invention
The method aims to solve the technical problems that in the prior art, blank and limit method simulation of residual deformation of the V-shaped valley after earthquake exists in V-shaped valley pile foundation vibration response research. The invention discloses a power response calculation method of pile foundations in V-shaped valley terrains, which comprises the following steps:
(1) Obtaining physical and mechanical parameters of V-shaped valleys, pile foundations and soil bodies according to the geological survey data;
(2) Calculating a soil body displacement field caused by seismic wave vibration based on Graf addition theorem;
(3) Based on a dynamic Winkler foundation model, converting a soil body displacement field caused by seismic waves into a load, applying the load to a pile foundation, and establishing a stress balance equation of the pile foundation;
(4) And under the condition that the pile top and the pile bottom are both in free constraint, solving pile foundation displacement caused by uniformly distributed load and bending moment and shearing force born by the pile foundation displacement.
The physical mechanical parameters include:
(1) Valley parameters: the valley radius d, the dimensionless frequency of the seismic wave is eta, the angle of incidence alpha and the distances b1 and b2 between the center of the canyon and the two sides;
(2) Pile foundation parameters: equivalent diameter D, soil penetration depth L, pile foundation elastic modulus E p Moment of inertia I of pile p ；
(3) Soil parameters: modulus of elasticity E s Soil poisson ratio mu s 。
The soil body out-of-domain displacementField u outer The (r, θ) calculation method is:
u outer (r，θ)＝u f (r，θ)+u s1 (r，θ)+u s2 (r，θ)； (1)
wherein the outer field free field u f (r, θ), out-field fringe field 1u s1 (r, θ), out-field fringe field 2u s2 The calculation method of (r, θ) is as follows:
inner field fringe field u inner1 (r, θ) and inner field fringe field u inner2 The calculation formulas of (r, θ) are respectively:
wherein alpha is the incidence angle of the seismic wave; r is a polar radius taking an o point as a center of a circle, and θ is a polar angle; k is the wave number of the incident wave, whereinη is the dimensionless frequency of the incident wave; epsilon n Is a Newman coefficient (epsilon) 0 ＝1，n＝0；ε n ＝2，n>0)；H (1) (. Cndot.) is an n-th order Bessel function of the third class, J n (. Cndot.) is a Bessel function of the first class of n-th order; a is that n ，B n ，C n ，D n Is the coefficient to be determined, m and n represent the number of stages, and V represents the V-type valley parameter.
The stress balance equation of the pile foundation is as follows:
wherein W (y) is the displacement function of the pile foundation; e (E) p Is the pile foundation elastic modulus; i p The moment of inertia is the cross section of the pile foundation; d is the equivalent width; k (k) 1 Spring rate for pile side soil; e (E) s Is the elastic modulus of soil body; mu (mu) s Poisson ratio of soil body; p (y) is the vertical external load applied to the pile foundation.
P(y)＝k 1 ×w f (y)； (13)
Wherein w is f (y) is soil mass free field displacement; x is the position where the pile foundation is placed.
The pile foundation shear force Q and bending moment M caused by concentrated load are solved as follows:
the above equation can be written in the form of finite differences:
αw i-2 +βw i-1 +γw i +βw i+1 +αw i+2 ＝p i ； (15)
in the method, in the process of the invention,
the equation for single pile displacement can be deduced in combination with boundary conditions:
k can be seen here as a matrix from n+1 x n+1;
while stiffness matrixThe following are provided:
the above equation is solved by finite difference method. Dividing the pile foundation into n sections, wherein the height of each section is H=L/n, and L is the pile length. Pile foundation node numbers are 0,1, …, n-1 and n from pile top to pile end in sequence. Two virtual nodes-2, -1, n+1, n+2 are added at the pile top and the pile end respectively during calculation.
The pile body bending moment is as follows:
the pile body shear force is as follows:
the technical scheme of the invention has the beneficial effects that: the vibration response of the V-shaped valley pile foundation is solved by combining the dynamic Winkler foundation model with the wave function for the first time, the power time course curve obtained in the seismic process of the V-shaped valley pile foundation is obtained, and the amplification rule of the V-shaped valley topography effect on the pile vibration response can be revealed.
Drawings
In the following, by way of example, the drawings of exemplary embodiments of the invention are shown, the same or similar reference numbers being used in the various drawings to designate the same or similar elements. In the accompanying drawings:
fig. 1 is a flow chart of the method of the present invention.
Fig. 2 is a computational model diagram of the method of the present invention.
Fig. 3 is a graph of V-shaped valley mono-pile free field displacement based on terrain effects in an embodiment of the invention.
FIG. 4 is a pile foundation moment diagram in an embodiment of the invention.
FIG. 5 is a pile foundation shear diagram in an embodiment of the present invention.
Detailed Description
The invention will be better explained by the following detailed description of the embodiments with reference to the drawings.
As shown in fig. 1, the specific steps of the model of the power response calculation method of the pile foundation in the V-shaped valley terrain according to the embodiment are as follows:
(1) Obtaining physical and mechanical parameters of the valley, pile foundation and soil body according to the geological survey data:
as shown in fig. 2, the physical and mechanical parameters of a V-shaped valley near the pile foundation excavation project are as follows:
A. valley parameters: the valley radius d=1m, the seismic wave dimensionless frequency is η=1, the incident angle alpha=0°, the distance b1=0.75 m between the center of the canyon and the two sides, b2=0.6 m;
B. pile foundation parameters: equivalent diameter d=0.4m, soil penetration depth l=25m, pile foundation elastic modulus E p ＝2×10 10 Pa, pile foundation cross section moment of inertia I p ＝1.2566×10 -3 m 4 X= -2/3 is the position where the pile foundation is placed;
C. soil parameters: modulus of elasticity E s ＝2.5×10 7 Pa, soil body Poisson's ratio mu s ＝1/3。
(2) Substituting the parameters in A into the formula (1) to calculate the displacement field of the external field of the V-shaped valley soil:
u outer (r,θ)＝u f (r,θ)+u s1 (r,θ)+u s2 (r,θ)； (20)
wherein the V-shaped valley causes an ectodomain free field u f (r, θ) out-of-domain fringe field (1)u) s1 (r, θ) out-of-domain fringe field (2)u) s2 (r, θ) are respectively:
inner field displacement field (1)u) inner1 (r, θ) and an inner field displacement field (2)u) inner2 (r, θ) are respectively:
(3) Based on a Winkler foundation model, establishing a stress balance equation of the pile foundation:
P(y)＝k 1 ×w f (y)； (27)
wherein: determining pile foundation cross-section moment of inertia based on known parametersSpring rate of pile side soil>
Wherein: w (w) f And (y) is soil body free field displacement, wherein x is the position where the pile foundation is placed.
(4) Deriving a single pile displacement equation under the condition that the pile top and the pile bottom are both in free constraint according to a formula (26) and combining boundary conditions:
k can be seen here as a matrix from n+1 x n+1.
While stiffness matrixThe following are provided:
the pile foundation displacement curve of this embodiment is shown in fig. 3.
(5) The above equation is solved by finite difference method. Dividing the pile foundation into n sections, wherein the height of each section is H=L/n, and L is the pile length. Pile foundation node numbers are 0,1, …, n-1 and n from pile top to pile end in sequence. Two virtual nodes-2, -1, n+1, n+2 are added at the pile top and the pile end respectively during calculation.
The pile body bending moment is as follows:
the pile body shear force is as follows:
the pile foundation bending moment and shearing force curves of the embodiment are shown in fig. 4 and 5.
It will be understood that the invention has been described in terms of several embodiments, and that various changes and equivalents may be made to these features and embodiments by those skilled in the art without departing from the spirit and scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed, but that the invention will include all embodiments falling within the scope of the appended claims.
Claims (6)
1. A power response calculation method of pile foundations in V-shaped valley terrain is characterized by comprising the following steps:
(1) Obtaining physical and mechanical parameters of V-shaped valleys, pile foundations and soil bodies; the physical mechanical parameters include:
valley parameters: the valley radius d, the dimensionless frequency of the seismic wave is eta, the angle of incidence alpha and the distances b1 and b2 between the center of the canyon and the two sides;
pile foundation parameters: equivalent diameter D, soil penetration depth L, pile foundation elastic modulus E p Moment of inertia I of pile p ；
Soil parameters: modulus of elasticity E s Soil poisson ratio mu s ；
(2) Calculating a soil body displacement field caused by seismic wave vibration based on Graf addition theorem;
(3) Based on a dynamic Winkler foundation model, converting a soil body displacement field caused by seismic wave vibration into a load, applying the load to a pile foundation, and establishing a stress balance equation of the pile foundation;
(4) And under the condition that the pile top and the pile bottom are both in free constraint, solving pile foundation displacement caused by uniformly distributed load and bending moment and shearing force born by the pile foundation displacement.
2. The method for calculating the dynamic response of pile foundations in V-shaped valley terrains according to claim 1, wherein in the step (2), the soil body displacement field caused by seismic wave vibration comprises an outer domain displacement field formed by seismic wave incidence, and the method for calculating the outer domain displacement field formed by seismic wave incidence comprises the following steps:
u outer (r,θ)＝u f (r,θ)+u s1 (r,θ)+u s2 (r,θ)； (1)
wherein the outer field free field u f (r, θ), out-field fringe field u s1 (r, θ), out-field fringe field u s2 The calculation formulas of (r, θ) are respectively:
wherein: alpha is the incidence angle of the seismic wave; r is a polar radius taking an o point as a center of a circle, and θ is a polar angle; k is the wave number of the incident wave, whereinη is the dimensionless frequency of the incident wave; epsilon n Is a Newman coefficient (epsilon) 0 ＝1，n＝0；ε n ＝2，n＞0)；H (1) (. Cndot.) is an n-th order Bessel function of the third class, J n (. Cndot.) is a Bessel function of the first class of n-th order; a is that n ，B n Is the coefficient to be set, m and n representSeries, V, represents a V-type valley parameter.
3. The power calculation method of pile foundation in V-shaped valley terrain as claimed in claim 2, wherein in step (2), said soil displacement field includes an in-field displacement field u inner1 (r, θ) and an in-field displacement field u inner2 (r, θ), the inner-field displacement field u inner1 (r, θ) and an in-field displacement field u inner2 The calculation method of (r, θ) is as follows:
wherein C is n ，D n Is a set coefficient.
4. A method of calculating the dynamic response of a pile foundation in V-valley terrain as claimed in claim 3, wherein in step (3), the stress balance equation of the pile foundation is:
wherein W (y) is the displacement function of the pile foundation; e (E) p Is the pile foundation elastic modulus; i p The moment of inertia is the cross section of the pile foundation; d is the equivalent width; k (k) 1 Spring rate for pile side soil; e (E) s Is the elastic modulus of soil body; mu (mu) s Poisson ratio of soil body; p (y) is the vertical external load received by the pile foundation;
P(y)＝k×w f (y)； (13)
wherein w is f (y) is a soil body free field; x is the position where the pile foundation is placed.
5. The method for calculating the dynamic response of the pile foundation in the V-shaped valley topography according to claim 4, wherein in the step (4), when the pile top and the pile bottom are both in free constraint, a calculation formula for solving the pile foundation displacement w caused by uniformly distributed loads is as follows:
wherein { K }, is (n+1)×(n+1) K is the moment of n+1×n+1, which is the stiffness matrix.
6. The method for calculating the dynamic response of the pile foundation in the V-shaped valley topography according to claim 5, wherein in the step (4), when the pile top and the pile bottom are both in free constraint, the calculation formulas for solving the bending moment M and the pile foundation shear force Q caused by uniform load are respectively as follows:
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