CN109518573A - Method for calculating roadbed dynamic additional stress under action of vehicle load - Google Patents
Method for calculating roadbed dynamic additional stress under action of vehicle load Download PDFInfo
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- 238000000034 method Methods 0.000 title abstract description 8
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- 238000006073 displacement reaction Methods 0.000 claims abstract description 18
- 239000011159 matrix material Substances 0.000 claims abstract description 14
- 239000013598 vector Substances 0.000 claims abstract description 12
- 238000012546 transfer Methods 0.000 claims abstract description 7
- 238000004364 calculation method Methods 0.000 claims description 8
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- 238000011068 loading method Methods 0.000 description 2
- 238000005259 measurement Methods 0.000 description 2
- 239000004576 sand Substances 0.000 description 2
- 239000002689 soil Substances 0.000 description 2
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Classifications
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- E—FIXED CONSTRUCTIONS
- E01—CONSTRUCTION OF ROADS, RAILWAYS, OR BRIDGES
- E01C—CONSTRUCTION OF, OR SURFACES FOR, ROADS, SPORTS GROUNDS, OR THE LIKE; MACHINES OR AUXILIARY TOOLS FOR CONSTRUCTION OR REPAIR
- E01C23/00—Auxiliary devices or arrangements for constructing, repairing, reconditioning, or taking-up road or like surfaces
- E01C23/01—Devices or auxiliary means for setting-out or checking the configuration of new surfacing, e.g. templates, screed or reference line supports; Applications of apparatus for measuring, indicating, or recording the surface configuration of existing surfacing, e.g. profilographs
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- Engineering & Computer Science (AREA)
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- Civil Engineering (AREA)
- Structural Engineering (AREA)
- Control Of Driving Devices And Active Controlling Of Vehicle (AREA)
Abstract
The invention discloses a method for calculating roadbed dynamic additional stress under the action of vehicle load, which comprises the following steps: (1) establishing a two-dimensional elastic lamellar body model according to the actual condition of the roadbed and the pavement of the highway, and converting stress and displacement in an elastic dynamics control equation into state vectors under modal coordinates; (2) separating and eliminating the time variable in the dynamic additional stress expression; (3) performing Laplace transformation on the state vector, and obtaining a stiffness matrix of the layered foundation by adopting a transfer matrix method; (4) establishing a dynamic additional stress quantification model of the layered roadbed pavement system according to the initial condition and the boundary condition; (5) and (4) carrying out Laplace inverse transformation on the model by using a Durbin method to obtain the roadbed dynamic additional stress under the action of vehicle load. The method for calculating the dynamic additional stress of the roadbed is suitable for the selection of highway and railway roadbed filling materials and the structural combination design of the roadbed.
Description
Technical field
The present invention relates to the calculation methods that roadbed under Vehicle Load moves additional stress.
Background technique
For subgrade and pavement system under the action of vehicular load, internal structure will receive the dynamic effects such as vibration, impact, by
It is known as dynamic additional stress in the structural stress that dynamic stress effect generates.If roadbed system is during operation, carload institute
The dynamic additional stress generated is greater than the critical dynamic stress of the soil body inside roadbed, and roadbed will generate cumulative settling deformation, to lead
Road surface is caused to will appear the diseases such as cracking, track.Therefore, the regularity of distribution pair that roadbed under Vehicle Load moves additional stress is studied
It is of great significance in roadbed filling selection and road structure Combination Design.
Research at present both at home and abroad about the dynamic additional stress of subgrade and pavement internal system under Vehicle Load is also seldom, and
And most of the lamellar character of subgrade and pavement and the periodicity of Vehicle Load are not all accounted for.
Summary of the invention
The technical problem to be solved by the present invention is in view of the shortcomings of the prior art, provide road under a kind of Vehicle Load
Base moves the calculation method of additional stress, considers the lamellar character of subgrade and pavement and the periodicity of Vehicle Load, accurately asks
It solves subgrade and pavement internal system under Vehicle Load and moves additional stress.
In order to solve the above technical problems, the technical scheme adopted by the invention is that: roadbed is dynamic under a kind of Vehicle Load
The calculation method of additional stress, comprising the following steps:
1) according to the actual conditions of highway subgrade pavement, two-dimension elastic stratified model is established, it is assumed that each layer of subgrade and pavement is
Elastic Layered System, in Elasticity the equation of motion and physical equation derive, obtain the dynamics of stress and displacement
Partial differential equations；
2) modal coordinate transformation is carried out with displacement to the stress in above-mentioned dynamics partial differential equations, to eliminate the power
The time variable in partial differential equations is learned, is not considered the stress of time variable and the dynamics partial differential equation of displacement
Group；
3) the dynamics partial differential equations of the stress for not considering time variable and displacement are converted using Laplace transformation
For the system of linear equations containing integration variable；According to matrix theory, the transmitting square of stratiform subgrade and pavement system in the transform domain as illustrated is acquired
Battle array；
4) state vector that surface is found out using primary condition and boundary condition, then establishes Vehicle Load Layered
Dynamic additional stress model in subgrade and pavement system；
5) dynamic additional stress is acquired to the dynamic additional stress model solution in transform domain using Laplace inverse transformation.
In step 2), the expression of the time variable in partial differential equations is separated and eliminated are as follows:
Wherein, ρ is the density of medium；ζ is vehicular load frequency, and the π of ζ=2 ν/l, ν are the travel speed of vehicle, and l is two vehicles
Travel kept distance；For the shear stress under modal coordinate；For the direct stress under modal coordinate；E is elasticity modulus；μ
For Poisson's ratio；WithHorizontal displacement and length travel respectively under modal coordinate；Respectively σz(), w (), u (), τxzThe partial derivative of ()；X is vertical
The direction of roadbed middle line；Z is road surface downwardly direction.
In step 3), transfer matrix expression formula are as follows:
Wherein,
Compared with prior art, the advantageous effect of present invention is that: the stratiform that the present invention considers subgrade and pavement is special
Property and Vehicle Load periodicity, can accurately solve under Vehicle Load that subgrade and pavement internal system is dynamic additional to answer
Power.
Detailed description of the invention
Fig. 1 is subgrade and pavement layered system under Vehicle Load.
Specific embodiment
Subgrade and pavement system is multi-level layer structure, is divided into three surface layer, base and soil matrix levels, therefore
The dynamic additional stress of Vehicle Load Layered structure is analyzed using multilayer elastic system theory.Due to road length and
Depth bounds are far longer than its width, therefore tally with the actual situation in two-dimensional space modeling.1 equation of motion
Due to considering dynamic additional stress, physical strength is disregarded.Under two-dimensional coordinate system, the power of elastic half-space problem is flat
Weigh equation are as follows:
In formula: ρ is the density of medium, σz() is direct stress, τxz() is shear stress, and w () is length travel, and u () is
Horizontal displacement.
2 physical equations
In formula: λ and G is Lame constant；
U and w is respectively horizontal displacement and length travel.
Relationship between Lame constant is as follows:
Wherein, μ is Poisson's ratio；
E is elasticity modulus.
Displacement and stress are sought local derviation to coordinate z by 3
It can be obtained by formula (1-a):
It can be obtained by formula (1-b), (2-a), (2-b):
It can be obtained by formula (2-b):
It can be obtained by formula (2-c):
4 disengaging time variables
In cartesian coordinate system, stress and the writeable accepted way of doing sth (4-a) of displacement in formula (3-a)~(3-d) include circular frequency
The form stable of ω.According to the actual conditions of problem, stress and displacement are all the variables of real number space, therefore in separating variables mistake
Real part is taken to be calculated in journey.
E in formula (4-a)itω=iteζ, ζ is vehicular load frequency, and the π of ζ=2 ν/l, ν are the travel speed of vehicle, and l is two vehicles
Kept distance is travelled, value can refer to the regulation on highway.
It brings formula (4-a) into formula (3-a)~(3-d), can be obtained using the relationship between Lame constant:
5Laplace transformation
The formula of Laplace transformation are as follows:
Wherein:It is pairThe pull-type transformation of Ying Yu；
S is Laplace variable,
It is inversely transformed into accordingly:
With defined above, can derive:
Therefore:
It can similarly derive:
AndThe equation about variable z can be regarded as, peer-to-peer both sides are sought local derviation, had:
To:
Therefore:
Laplace transformation is all carried out to the right and left in formula (4-b), and brings relation above into, can be obtained:
The matrix form that above formula can be written as follow
6 state equations
If enabling state vectorFormula (5-f) can be write as matrix
The partial differential equation of form:
In formula,
According to modern control theory, the solution of above formula are as follows:
In formula, exponential matrix eA (ζ, s) zAs transfer matrix is indicated with T；Transfer matrix is established to be converted through Laplace
The boundary condition vector of original state at z=0 afterwardsWith the state vector at any depth zIt
Between relationship.
The characteristic equation of A in formula (6-b) are as follows: | A- λ ' E |=0 (6-c)
Wherein: E is 4 rank unit matrixs；
Expansion above formula can obtain:
Formula (6-d) is solved by MATLAB software programming, eigenvalue λ ' and corresponding feature vector P can be obtained.If
It enables
#5=2 ρ ζ2μ2+ρζ2μ-ρζ2+Es2μ-Es2
Then characteristic value are as follows:
Feature vector are as follows:
Enable A=P Λ P-1, according to Cayley-Hamilton theorem, transfer matrix is represented by
Formula (6-e) is brought into formula (6-b), each stress of single layer upper and lower surface and displacement in transform domain can be acquired.
7 ask stress and displacement
By derivation above, following state equation is established:
According to the analysis of front, vehicular load is simplified to linear load, load width is ab, and load is to roadnet
Loading method meets half-wave cosine loading rule, establishes plane coordinate system, then subgrade and pavement layered system under Vehicle Load
As shown in Figure 1.
For multilayer roadbed system shown in FIG. 1, completely attach between layers and continuous condition are as follows:
By separating variables and Laplace transformation after, between layers completely attach to and continuous condition are as follows:
Wherein: i indicates i-th layer.
The transitive relation that entire multilayer system can be obtained successively is transmitted just by contact conditions.
If enablingThen
The state vector of the random layer in transform domain can be acquired according to above formulaWith the state of surface layer upper epidermis
VectorBetween relationship.Then Laplace inverse transformation is carried out to required state vector, vehicular load can be acquired
Act on the dynamic additional stress of lower random layer.
8Laplace inverse transformation numerical solution
For simple transformation problem, Laplace can be directly carried out according to the definition that Laplace is converted and convert to obtain inversion
The mathematic(al) representation changed, and it is studied for the present invention the problem of, the expression formula of dynamic additional stress is extremely complex in the transform domain as illustrated,
It is difficult to be indicated with analytic expression, needs to carry out inverse transformation by numerical method.Since the exact numerical of inverse transformation realizes that difficulty is larger, it is
This present invention seeks Laplace inverse transformation numerical solution with the following method.
If function X (t, x, z) is in xjThe complex series that place can be expressed as:
Wherein: X be investigate point from coordinate origin always away from
From；N is total calculating step number.
It for L × N=50~5000, is learnt by calculating, as a*X=5~10, calculated result is preferable.
Embodiment
Section near the green mountain the Jiang Zhizhong sand small stream section K47+795 pile No. in the present embodiment ether Australia's highway Shuande illustrates
The present invention, multilayer subgrade and pavement system under Vehicle Load of verifying move the correctness of additional stress model.
One, the physical parameter of original position and indoor test
Pavement Dynamic Load is 500kN, and being converted into road surface load is 50kPa, travel speed 10m/s, it is assumed that two vehicles traveling
The safe distance kept is 100m.It is as shown in the table to test each layer physical parameter of layer.
Table 1 studies each layer physical and mechanical parameter of section
| Material | Thickness/m | Density/(kg.m3) | Determination of Dynamic Elastic Modulus/Mpa | Poisson's ratio |
| Base bankets | 4.0 | 1500 | 2300 | 0.30 |
| Sand bedding course | 0.5 | 1800 | 27 | 0.35 |
| Artificial earth fill | 1.0 | 1900 | 60 | 0.35 |
| Mud (after processing) | 15.0 | 1600 | 3 | 0.40 |
Two, additional stress calculated result and analysis are moved
The vertical dynamic additional stress value and field measurement value that model calculates are as shown in table 2.
2 the model calculation of table and field measurement value
From Table 2, it can be seen that calculated result of the invention is close to measured value, relative error than document [1] than it is small, say
Bright algorithm proposed by the present invention is higher than the arithmetic accuracy of document [1].The present invention consider subgrade and pavement system lamellar character and
The periodicity of Vehicle Load, tallies with the actual situation.
Calculation method of the invention is derived by based on theory of elastic mechanics, is suitable for being assumed to elasticity
The subgrade and pavement system of system.
Claims (3)
1. roadbed moves the calculation method of additional stress under a kind of Vehicle Load, which comprises the following steps:
1) according to the actual conditions of highway subgrade pavement, two-dimension elastic stratified model is established, it is assumed that each layer of subgrade and pavement is elasticity
Layered system, in Elasticity the equation of motion and physical equation derive, the dynamics for obtaining stress and displacement is partially micro-
Divide equation group；
2) modal coordinate transformation is carried out with displacement to the stress in above-mentioned dynamics partial differential equations, it is inclined to eliminate the dynamics
Time variable in differential equation group is not considered the stress of time variable and the dynamics partial differential equations of displacement；
3) the dynamics partial differential equations of the stress for not considering time variable and displacement are converted to using Laplace transformation and are contained
The system of linear equations of integration variable；According to matrix theory, the transfer matrix of stratiform subgrade and pavement system in the transform domain as illustrated is acquired；
4) then the state vector that surface is found out using primary condition and boundary condition establishes Vehicle Load Layered roadbed
Dynamic additional stress model in pavement system；
5) dynamic additional stress is acquired to the dynamic additional stress model solution in transform domain using Laplace inverse transformation.
2. roadbed moves the calculation method of additional stress under Vehicle Load according to claim 1, which is characterized in that step
It is rapid 2) in, separate and eliminate the expression of the time variable in partial differential equations are as follows:
Wherein, ρ is the density of medium；ζ is vehicular load frequency, and the π of ζ=2 ν/l, ν are the travel speed of vehicle, and l is two vehicles traveling
The distance kept；For the shear stress under modal coordinate；For the direct stress under modal coordinate；E is elasticity modulus；μ is pool
Loose ratio；WithHorizontal displacement and length travel respectively under modal coordinate； Respectively σz(), w (), u (), τxzThe partial derivative of ()；X is the direction of vertical roadbed middle line；Z is road
Direction downwards.
3. roadbed moves the calculation method of additional stress under Vehicle Load according to claim 2, which is characterized in that step
It is rapid 3) in, transfer matrix expression formula are as follows:
Wherein,
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN111222196A (en) * | 2020-04-24 | 2020-06-02 | 中国铁道科学研究院集团有限公司铁道建筑研究所 | Method for designing high-speed railway roadbed structure |
| CN111783275A (en) * | 2020-06-02 | 2020-10-16 | 中煤科工开采研究院有限公司 | Transmission matrix method based additional stress calculation method for laminated medium foundation in subsidence area |
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| CN108166545A (en) * | 2018-01-14 | 2018-06-15 | 华东交通大学 | A kind of settlement calculation method of the lower multistrata foundation of Arbitrary Load effect |
| CN108517735A (en) * | 2018-04-12 | 2018-09-11 | 长沙理工大学 | Durability asphalt pavement design method based on double-modulus theory and pavement structure thereof |
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2018
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Patent Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108166545A (en) * | 2018-01-14 | 2018-06-15 | 华东交通大学 | A kind of settlement calculation method of the lower multistrata foundation of Arbitrary Load effect |
| CN108517735A (en) * | 2018-04-12 | 2018-09-11 | 长沙理工大学 | Durability asphalt pavement design method based on double-modulus theory and pavement structure thereof |
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN111222196A (en) * | 2020-04-24 | 2020-06-02 | 中国铁道科学研究院集团有限公司铁道建筑研究所 | Method for designing high-speed railway roadbed structure |
| CN111783275A (en) * | 2020-06-02 | 2020-10-16 | 中煤科工开采研究院有限公司 | Transmission matrix method based additional stress calculation method for laminated medium foundation in subsidence area |
| CN111783275B (en) * | 2020-06-02 | 2021-02-02 | 中煤科工开采研究院有限公司 | Transmission matrix method based additional stress calculation method for laminated medium foundation in subsidence area |
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