CN113609447B - Rapid calculation method for mechanical response of elliptical load asphalt pavement structure surface - Google Patents
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Abstract
The invention relates to a method for rapidly calculating mechanical response of an elliptic load asphalt pavement structure surface, belonging to the field of asphalt pavement mechanical calculation, and aiming at solving the problem of elliptic loadThe mechanical response calculation efficiency of the round load asphalt pavement structure surface is low. The calculation method comprises the following steps: 1. determining elliptical load parameters according to the load data of the actually-measured tire and the pavement, and further obtaining an analytic solution of the mechanical response of the asphalt pavement under elliptical load; 2. calculation by numerical integrationAndis calculated by adopting a simple calculation formula containing the infinite integral of a Bessel functionUsing the formulaAnd the mechanical response of the surface of the elliptical load asphalt pavement structure is rapidly calculated.
Description
Technical Field
The invention belongs to the field of bituminous pavement mechanics calculation, and particularly relates to a calculation method of mechanical response at the surface of an elliptical load bituminous pavement structure.
Background
The mechanical response of the asphalt pavement structure is the key for pavement structure design, and the mechanical response at the pavement surface is an important analysis parameter. The actually measured grounding pressure of the tire and the road surface under the load of the heavy vehicle is elliptical, and an elliptical load model is established by using a mathematical function. Based on lamellar elastic system mechanics, solving structural mechanics response analysis solution of the elliptical load asphalt pavement. However, the mechanical response calculation efficiency of the asphalt pavement structure surface under elliptical load is very low, and the mechanical response of each calculated surface needs to be nearly 100 seconds, so that the application of the mechanical analysis program in heavy-load pavement engineering practice is seriously affected.
Disclosure of Invention
The invention aims to solve the problem of low calculation efficiency of mechanical response at the surface of an elliptical load asphalt pavement structure, and provides a rapid calculation method of mechanical response at the surface of the elliptical load asphalt pavement structure.
The method for rapidly calculating the mechanical response of the elliptical load asphalt pavement structure surface is realized by the following steps:
1. determining elliptical load parameters according to load data of actually measured tires and road surfaces
Fitting the ground pressure data of the heavy-duty tire and the asphalt pavement by adopting an elliptic paraboloid load function (1) to obtain an elliptic load ground pressure constant term p 0 The first parameter a of the elliptic load function and the second parameter b of the elliptic load function;
in p 0 -elliptic load ground pressure constant term (MPa), angle of θ -polar coordinate system, distance of r-polar coordinate system (cm), a-elliptic load function first parameter, b-elliptic load function second parameter;
the elliptic paraboloid load function (1) is expressed in the form of a series of:
elliptical load in formula:
according to a first Sonne finite integral formula, a Hankel integral transformation formula of the load function is obtained as follows:
in the middle of-Hankel integral transformation of p (r, θ), delta-load circle radius, ζ -Hankel integral transformation variable, J 1 (ζδ) -first-order Bessel function, J 2 (ζδ) -second order Bessel function, J 3 (ζδ) -third order Bessel function;
based on a non-axisymmetric load multilayer elastic system theoretical solution, the structural stress and displacement solution of the asphalt pavement under elliptical load are obtained by utilizing a Hankel integral transformation formula (3) of a load function as follows:
middle sigma r 、σ θ 、σ z Positive stresses (MPa) in three directions (horizontal, auxiliary and vertical) in polar coordinate system, τ rθ 、τ θz 、τ zr Shear stress (MPa) in three directions under polar coordinate system, displacement (mm) in three directions (horizontal, auxiliary and vertical) of u, v and omega-polar coordinates, E i -modulus of layer (MPa), μ of pavement structure of layer i i -poisson's ratio, h of the i-th pavement structure layer i -i-th pavement structure layer thickness (cm), a i 、B i 、C i And D i The depth of the point position and the x-integral variable are calculated by the stress coefficient and the z-;
2. rapid calculation of mechanical response of asphalt pavement surface under elliptical load
The integral kernel of the structural mechanical response analysis solution (formulas (4) - (12)) of the asphalt pavement under elliptical load consists of the product of a rational expression and a Bessel function, and the form of the analysis solution is expressed as follows:
wherein R is i (r, z) -calculating the mechanical response of the point (r, z);
-integrating the first half of the kernel;
b (r, x) -the product of two Bessel functions;
derived, equation (13) is discretized into two integrals:
the former term of the formula (14) is calculated by adopting a numerical integration method, and the latter term can be calculated by adopting the following method:
in formula (15)Calculation can be performed by numerical integration,/->Is the infinite integral containing the Bessel function, the [0, ] infinite integral value containing the Bessel function in the integral kernel is solved by using Weber-Xia Fuhai T Lin Gongshi, and the obtained formula is as follows:
calculating the value of the infinite integral containing the Bessel function by the formula (16) to the formula (30)Calculating +.>And obtaining the mechanical response value of the elliptical load asphalt pavement structure surface through the formula (14).
The method for quickly calculating the mechanical response of the asphalt pavement structure surface under elliptical load comprises the following beneficial effects:
the mechanical response of the surface points of the asphalt pavement is always an important index for structural design of the asphalt pavement. In the latest mechanical-empirical design specifications (MEPDG) in the United states, the vertical stress and the radial stress of a road surface are used as important indexes for researching rutting and Top-Down fatigue cracking of asphalt pavement. In JTG D50-2017 of highway asphalt pavement design Specification in China, the adoption of a layered sum method for checking the deformation of an asphalt pavement requires the adoption of vertical stress at the surface, and the adoption of vertical displacement at the surface is required during roadbed acceptance and acceptance after pavement construction. The low calculation efficiency of mechanical response at the surface is always one of the biggest barriers of the layered elastic system theory in the popularization and application of the asphalt pavement structural design.
The invention provides a method for quickly calculating mechanical response of an asphalt pavement structure surface under elliptical load.
The method for quickly calculating the mechanical response at the surface has clear concept of the deduction process, is simple and convenient to calculate, can quickly finish the calculation of the mechanical response at the surface of the asphalt pavement structure under elliptical load, improves the calculation efficiency, and ensures excellent calculation precision.
Drawings
FIG. 1 is a schematic diagram of a model of an elliptic paraboloid load pattern in step one of the present invention;
FIG. 2 is a chart showing the ground contact pressure test of heavy-duty vehicle tires and road surfaces (tire pressure 700 kPa/1430 kg);
FIG. 3 is a graph showing the calculated radial stress (MPa) response for a single circle of load with or without surface points;
FIG. 4 is a graph showing the response of the example with or without surface points under single circle load to calculate vertical displacement (0.01 mm);
FIG. 5 is a schematic diagram of a model structure of a single-axle double-wheel set under multi-circle load in an embodiment;
FIG. 6 is a schematic diagram of a model structure of a twin four wheel set under multi-circle load in an embodiment;
FIG. 7 is a schematic diagram of a model structure of a triple six-wheel set under multi-circle load in an embodiment;
FIG. 8 is a graph showing the calculated radial stress (MPa) response for a plurality of surface points under a circular load in the example;
FIG. 9 is a graph showing the comparison of the calculated vertical displacement (0.01 mm) response with or without surface points under multi-circle load in the example.
Detailed Description
The first embodiment is as follows: the method for quickly calculating the mechanical response of the elliptical load asphalt pavement structure surface according to the embodiment is implemented by the following steps:
1. determining elliptical load parameters according to load data of actually measured tires and road surfaces
Fitting the ground pressure data of the heavy-duty tire and the asphalt pavement by adopting an elliptic paraboloid load function (1) to obtain an elliptic load ground pressure constant term p 0 The first parameter a of the elliptic load function and the second parameter b of the elliptic load function;
in p 0 Elliptic load ground pressure constant term (MPa), angle of theta-polar coordinate system, r-polar seatDistance (cm) of the standard system, a-elliptic loading function first parameter, b-elliptic loading function second parameter;
the elliptic paraboloid load function (1) is expressed in the form of a series of:
elliptical load in formula:
according to a first Sonne finite integral formula, a Hankel integral transformation formula of the load function is obtained as follows:
in the middle of-Hankel integral transformation of p (r, θ), delta-load circle radius, ζ -Hankel integral transformation variable, J 1 (ζδ) -first-order Bessel function, J 2 (ζδ) -second order Bessel function, J 3 (ζδ) -third order Bessel function;
based on a non-axisymmetric load multilayer elastic system theoretical solution, the structural stress and displacement solution of the asphalt pavement under elliptical load are obtained by utilizing a Hankel integral transformation formula (3) of a load function as follows:
/>
middle sigma r 、σ θ 、σ z Positive stresses (MPa) in three directions (horizontal, auxiliary and vertical) in polar coordinate system, τ rθ 、τ θz 、τ zr Shear stress (MPa) in three directions under polar coordinate system, displacement (mm) in three directions (horizontal, auxiliary and vertical) of u, v and omega-polar coordinates, E i -modulus of layer (MPa), μ of pavement structure of layer i i -poisson's ratio, h of the i-th pavement structure layer i -i-th pavement structure layer thickness (cm), a i 、B i 、C i And D i The depth of the point position and the x-integral variable are calculated by the stress coefficient and the z-;
2. rapid calculation of mechanical response of asphalt pavement surface under elliptical load
The integral kernel of the structural mechanical response analysis solution (formulas (4) - (12)) of the asphalt pavement under elliptical load consists of the product of a rational expression and a Bessel function, and the form of the analysis solution is expressed as follows:
wherein R is i (r, z) -calculating the mechanical response of the point (r, z);
-integrating the first half of the kernel;
b (r, x) -the product of two Bessel functions;
derived, equation (13) is discretized into two integrals:
the former term of the formula (14) is calculated by adopting a numerical integration method, and the latter term can be calculated by adopting the following method:
in formula (15)Calculation can be performed by numerical integration,/->Is the infinite integral containing Bessel function, and Weber-Xia Fuhai T Lin Gongshi is used for solving the value of the infinite integral containing Bessel function in the integral kernel +.>Calculation by numerical integration/>And obtaining the mechanical response value of the elliptical load asphalt pavement structure surface through the formula (14).
The second embodiment is as follows: the first difference between the present embodiment and the specific embodiment is that in the first step, the ground pressure data of the heavy duty tire and the asphalt pavement are obtained by using a method of actual measurement by a test instrument or numerical simulation.
And a third specific embodiment: the second difference between this embodiment and the second embodiment is that the test device is a pressure sensor.
The specific embodiment IV is as follows: the difference between this embodiment and one of the first to third embodiments is that the derivation of the formula (15) in the second step is as follows:
in formula (13)Transition to a constant value with increasing argument x, expressed in the form of equation (14), while the latter half B (r, x) oscillates continuously with increasing x (generally x s The requirement can be met by setting the standard to 100);
wherein m (E) 1 ,μ 1 ) -a constant value associated with a first layer material parameter.
Fifth embodiment: the present embodiment differs from the first to fourth embodiments in that weber-Xia Fuhai t Lin Gongshi described in the second step is as follows:
in the middle of
F (,, x) -a super-geometric function;
c. d, mu, v and t, namely inputting parameters;
J μ () -a Bessel function of order μ;
J ν () -v order Bessel function;
Γ () -gamma function.
Examples: the method for rapidly calculating the mechanical response of the elliptical load asphalt pavement structure surface is implemented according to the following steps:
1. determining elliptical load parameters according to load data of actually measured tires and road surfaces
Based on the actual measurement data (shown in fig. 2) of the ground contact pressure of the heavy-duty automobile tire and the road surface, the ellipse load parameters are determined to be a=15, b=20 and p by using data fitting software in matlab 0 =0.897 MPa, a load circle radius of 10.65cm;
in p 0 -elliptic load ground pressure constant term (MPa), a-elliptic load function first parameter, b-elliptic load function second parameter;
the elliptic paraboloid load function (1) is expressed in the form of a series of:
elliptical load in formula:
according to a first Sonne finite integral formula, a Hankel integral transformation formula of the load function is obtained as follows:
in the middle of-Hankel integral transformation of p (r, θ), delta-load circle radius, ζ -Hankel integral transformation variable, J 1 (ζδ) -first-order Bessel function, J 2 (ζδ) -second order Bessel function, J 3 (ζδ) -third order Bessel function;
based on a non-axisymmetric load multilayer elastic system theoretical solution, the structural stress and displacement solution of the asphalt pavement under elliptical load are obtained by utilizing Hankel integral transformation of a load function as follows:
/>
/>
/>
middle sigma r 、σ θ 、σ z Positive stresses (MPa) in three directions (horizontal, auxiliary and vertical) in polar coordinate system, τ rθ 、τ θz 、τ zr Shear stress (MPa) in three directions under polar coordinate system, displacement (mm) in three directions (horizontal, auxiliary and vertical) of u, v and omega-polar coordinates, E i 、μ i 、h i -layer modulus (MPa), poisson's ratio and thickness (cm) of the i-th pavement structure, a i 、B i 、C i And D i -stress coefficients, z-depth of the calculated points, x-integral variables;
2. rapid calculation of mechanical response of asphalt pavement surface under elliptical load
The integral core of the analytic solution of the structural mechanical response of the asphalt pavement under the elliptic load is composed of the product of a rational expression and a Bessel function, and the form of the analytic solution is expressed as follows:
wherein R is i (r, z) -calculating the mechanical response of the point (r, z);
-integrating the first half of the kernel;
b (r, x) -the product of two Bessel functions;
derived, equation (13) is discretized into two integrals:
the former term of the formula (14) is calculated by adopting a numerical integration method, and the latter term can be calculated by adopting the following method:
in formula (15)Calculation can be performed by numerical integration,/->Is the infinite integral containing the Bessel function, the [0, ] infinite integral value containing the Bessel function in the integral kernel is solved by using Weber-Xia Fuhai T Lin Gongshi, and the obtained formula is as follows:
/>
/>
calculating the value of the infinite integral containing the Bessel function by the formula (16) to the formula (30)Calculating +.>And obtaining the mechanical response value of the structural surface of the elliptical load asphalt pavement.
In this embodiment, four-layer, five-layer, six-layer and 8-layer structural combinations and material parameters are selected as the checking cases, and the material parameters of each structural layer are shown in tables 1-4. Considering that the vertical displacement and the radial stress of the surface point are often used as analysis indexes of the bearing capacity of the asphalt pavement structure and the top-down fatigue cracking, and taking the two mechanical responses as verification indexes of the surface point theory. The horizontal distance (from the center of a load circle) is 0-20 cm, the interval is 1cm, the total number of calculation points in each working condition is 21, the total calculation time T and the maximum error M-err of the 21 calculation points are used as analysis indexes of calculation efficiency and accuracy, a calculation program without surface point theory is named A, and a calculation program with surface point is named B.
N in the formula is the number of calculated points;
y Ai 、y Bi -calculated values for both methods at point i A, B;
table 1 four-layer asphalt pavement structural assembly and material parameters
Table 2 five-layer asphalt pavement structural combinations and material parameters
Table 3 six layer asphalt pavement structural combinations and material parameters
Table 4 8 layer material parameters for each layer of the layer structure
(1) Single circle loading
The calculation accuracy and efficiency of the surface point mechanical response calculation method are shown in table 5. As can be seen from Table 5, as the number of layers of the structure increases, the computation time of the program without surface points increases, and the computation time increases by about 15s by adding one structure layer. When the surface point mechanical response rapid calculation method is adopted, the calculation time length needs about 0.1s, and the calculation efficiency can be improved by about 1000 times; in addition, as can be seen from the calculation result of the M-err, the calculation error is controlled within 0.5% by the surface point calculation theory; the method for rapidly calculating the description surface points not only can ensure the calculation precision, but also can greatly improve the calculation efficiency.
Table 5 calculation accuracy and efficiency of the theory program containing surface points
(2) Multiple circle loading
And the pavement structure design is carried out by applying double-circle uniformly-distributed vertical load in JTG D50-2017 of the design specification of the highway asphalt pavement. Meanwhile, the specification refers to the United states asphalt pavement design specification, and 7 types of axle, 11 types of vehicles and 5 types of axle loads are considered. The axle type of the double-axle type single-wheel machine relates to a single-axle single-wheel set, a single-axle double-wheel set, a double-axle single-wheel set double-shaft double-wheel sets, three-shaft single-wheel sets and three-shaft double-wheel sets. Considering the single-side load combination as shown in fig. 5-7, the calculation precision and efficiency of the surface point mechanical response calculation method under the action of multi-circle load are analyzed.
TABLE 6 calculation accuracy and efficiency of surface Point program with or without surface Point programs under different wheel sets
As shown in table 6, fig. 8 and fig. 9, the time consumption of the mechanical response calculation program is continuously increased along with the increase of the number of structural layers and the increase of load circles, the time consumption of calculating the mechanical response of the surface points of the eight-layer pavement structure with three-shaft six-wheel set is close to 460s, the time used for calculating the mechanical response of the surface points by adopting the mechanical response calculation method of the surface points is controlled within 0.3s, and the maximum error M-err is controlled within 1%, which indicates that the rapid calculation method of the surface points not only can ensure the calculation precision of the mechanical response of the surface points of the lamellar elastic system under elliptical load, but also can greatly improve the calculation efficiency.
Claims (5)
1. The quick calculation method for the mechanical response of the elliptical load asphalt pavement structure surface is characterized by comprising the following steps:
1. determining elliptical load parameters according to load data of actually measured tires and road surfaces
Fitting the ground pressure data of the heavy-duty tire and the asphalt pavement by adopting an elliptic paraboloid load function (1) to obtain an elliptic load ground pressure constant term p 0 The first parameter a of the elliptic load function and the second parameter b of the elliptic load function;
in p 0 -elliptic load ground pressure constant term MPa, θ -polar system angle, r-distance of polar system cm, a-elliptic load function first parameter, b-elliptic load function second parameter;
the elliptic paraboloid load function (1) is expressed in the form of a series of:
elliptical load in formula:
according to a first Sonne finite integral formula, a Hankel integral transformation formula of the load function is obtained as follows:
in the middle ofHankel integral transformation, delta-load radius, delta-Hankel integral transformation variable, J 1 (ζδ) -first order Bessel function, J 2 (ζδ) -second order Bessel function, J 3 (ζδ) -third order Bessel function;
based on a non-axisymmetric load multilayer elastic system theoretical solution, the structural stress and displacement solution of the asphalt pavement under elliptical load are obtained by utilizing Hankel integral transformation of a load function as follows:
middle sigma r 、σ θ 、σ z Positive stresses in three directions under polar coordinate system, MPa, τ rθ 、τ θz 、τ zr Shear stress in three directions under polar coordinate system, namely three directions of u, v and omega-polar coordinate displacement mm and E i -modulus of layer of pavement structure of layer i MPa, μ i -Poisson's ratio of the ith pavement structure layer, h i -i-th pavement structure layer thickness cm, A i 、B i 、C i And D i All are stress coefficients, z-the depth of the point location is calculated, and x-the integral variable;
2. rapid calculation of mechanical response of asphalt pavement surface under elliptical load
The integral core of the analytic solution of the structural mechanical response of the asphalt pavement under the elliptic load is composed of the product of a rational expression and a Bessel function, and the form of the analytic solution is expressed as follows:
wherein R is i (r, z) -calculating the mechanical response of the point (r, z);
-integrating the first half of the kernel;
b (r, x) -the product of two Bessel functions;
derived, equation (13) is discretized into two integrals:
the former term of the formula (14) is calculated by adopting a numerical integration method, and the latter term can be calculated by adopting the following method:
in formula (15)Calculation can be performed by numerical integration,/->Is the infinite integral containing the Bessel function, the [0, ] infinite integral value containing the Bessel function in the integral kernel is solved by using Weber-Xia Fuhai T Lin Gongshi, and the obtained formula is as follows:
calculating the value of the infinite integral containing the Bessel function by the formula (16) to the formula (30)Calculating +.>And obtaining the mechanical response value of the elliptical load asphalt pavement structure surface through the formula (14).
2. The method for rapidly calculating the mechanical response of the elliptical load asphalt pavement structure according to claim 1, wherein the grounding pressure data of the heavy-duty tire and the asphalt pavement are obtained by using a method of actual measurement of a test instrument or numerical simulation in the first step.
3. The method for rapidly calculating the mechanical response of the elliptical load asphalt pavement structure surface according to claim 2, wherein the test instrument is a pressure sensor.
4. The method for rapidly calculating the mechanical response of the elliptical load asphalt pavement structure surface according to claim 1, wherein the derivation process of the formula (15) in the second step is as follows:
in formula (13)Transition to a constant value with increasing argument x, expressed in the form of equation (31), while the latter half B (r, x) continuously oscillates and decays with increasing x;
wherein m (E) 1 ,μ 1 ) -a constant value associated with a first layer material parameter.
5. The method for rapidly calculating the mechanical response of the surface of the elliptical load asphalt pavement structure according to claim 1, wherein the weber-Xia Fuhai t Lin Gongshi in the second step is as follows:
in the middle of
F (,, x) -a super-geometric function;
c. d, mu, v, t-input parameters;
J μ () -a Bessel function of order μ;
J v () -a Bessel function of order v;
Γ () -gamma function.
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN107742018A (en) * | 2017-09-30 | 2018-02-27 | 交通运输部公路科学研究所 | The Analysis of Asphalt Pavement Structure increment method of model is relied on based on ground surface material modulus stress and strain |
| CN107764644A (en) * | 2017-09-30 | 2018-03-06 | 交通运输部公路科学研究所 | The Analysis of Asphalt Pavement Structure equivalent method of model is relied on based on ground surface material modulus stress and strain |
| CN112214817A (en) * | 2020-09-29 | 2021-01-12 | 长沙理工大学 | Multilayer displacement response determination method considering interlayer condition and transverse isotropy |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN107742018A (en) * | 2017-09-30 | 2018-02-27 | 交通运输部公路科学研究所 | The Analysis of Asphalt Pavement Structure increment method of model is relied on based on ground surface material modulus stress and strain |
| CN107764644A (en) * | 2017-09-30 | 2018-03-06 | 交通运输部公路科学研究所 | The Analysis of Asphalt Pavement Structure equivalent method of model is relied on based on ground surface material modulus stress and strain |
| CN112214817A (en) * | 2020-09-29 | 2021-01-12 | 长沙理工大学 | Multilayer displacement response determination method considering interlayer condition and transverse isotropy |
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