CN112765825B - Roadbed surface displacement response determination method considering modulus nonlinear distribution - Google Patents

Roadbed surface displacement response determination method considering modulus nonlinear distribution Download PDF

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CN112765825B
CN112765825B CN202110108452.1A CN202110108452A CN112765825B CN 112765825 B CN112765825 B CN 112765825B CN 202110108452 A CN202110108452 A CN 202110108452A CN 112765825 B CN112765825 B CN 112765825B
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张军辉
范海山
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Abstract

The invention discloses a roadbed surface displacement response determining method considering modulus nonlinear distribution, which comprises the following steps: obtaining roadbed material parameters, applying dynamic load on the roadbed surface, and constructing a roadbed modulus function which is distributed in a vertical nonlinear manner; calculating a surface displacement response solution of the roadbed in the Laplace-Hankel domain, wherein the surface displacement response solution simultaneously considers the transverse isotropy and the nonlinear distribution of the roadbed modulus along the vertical direction; and calculating the surface displacement response of the current roadbed model in a time domain range based on a displacement response solution of the current roadbed mechanics model in a Laplace-Hankel domain. The invention fully reflects the non-uniform distribution of the modulus of the actual roadbed along the depth direction, simultaneously considers the transverse isotropy of the roadbed, improves the accuracy of the roadbed surface displacement response, has high precision and high speed, and can obtain the distribution condition of the modulus of the roadbed along the depth direction by combining with the roadbed nondestructive detection technology.

Description

Roadbed surface displacement response determination method considering modulus nonlinear distribution
Technical Field
The invention belongs to the technical field of road engineering, and relates to a roadbed surface displacement response determining method considering modulus nonlinear distribution, in particular to a roadbed surface displacement response determining method considering transverse isotropy and modulus nonlinear distribution, and application, equipment and a storage medium thereof.
Background
With the development of nondestructive testing technology, some soil-based rapid nondestructive testing instruments are in succession, and the application of portable hammer type deflectometer (PFWD) is the most widespread. The PFWD carries out roadbed parameter inversion calculation by applying impact load on the top surface of the roadbed and combining a certain mechanical model and a parameter adjustment algorithm. However, the current mainstream roadbed parameter inversion calculation is still established on the basis of an isotropic linear elastic system, and the actual characteristics of the roadbed cannot be reflected. At present, a great deal of research results show that the roadbed has stronger stress dependence characteristics, and the modulus of the roadbed generally shows an ascending trend along with the depth along with the increase of the depth. In addition, the transverse isotropy of the subgrade due to natural settlement cannot be ignored in practical calculations.
In part of the prior art, for a pavement structure, the modulus of a material in each layer is considered to be uniform and does not change with space, but the modulus of a roadbed is changed in the vertical direction, the modulus has a nonlinear distribution characteristic in the vertical direction, and the displacement response of the roadbed surface is difficult to accurately determine by adopting the existing method. In part of the prior art, iterative computation is performed through structural layer strain and modulus nonlinearity on the basis of considering roadbed modulus homogeneity, the method is essentially a mathematical iterative method, the nonlinear distribution characteristic of the modulus is not considered on a mechanical model, the iterative method can only approach to a real result, the computation precision is low, repeated iteration is needed, and the computation is slow.
Disclosure of Invention
In order to solve the problems, the invention provides a roadbed surface displacement response determining method considering the nonlinear distribution of modulus, which fully reflects the nonuniform distribution of the modulus of an actual roadbed along the depth direction, simultaneously considers the transverse isotropy of the roadbed, and improves the accuracy of determining the roadbed surface displacement response.
The invention adopts the technical scheme that a roadbed surface displacement response determining method considering modulus nonlinear distribution is specifically carried out according to the following steps:
step S1, obtaining roadbed material parameters, wherein the roadbed material parameters comprise: vertical modulus E of roadbed top surfacev0Modulus E of road bed infinite depthv∞Roadbed modulus non-uniform coefficient alpha and roadbed modulus ratio neHorizontal poisson's ratio muhVertical poisson's ratio muvRoad, roadBase density rho and roadbed thickness H;
s2, applying dynamic load p (r, t) to the roadbed surface, establishing a cylindrical coordinate system by taking the dynamic load center as the coordinate center, wherein r represents a radial coordinate, z represents a vertical coordinate, phi represents an annular coordinate, and constructing a roadbed modulus function E which is distributed along the vertical non-linearityv(z)＝Ev0+(Ev∞-Ev0)(1-e-αz) (ii) a Calculating a surface displacement response solution of the roadbed in the Laplace-Hankel domain, which simultaneously considers the transverse isotropy and the nonlinear distribution of the roadbed modulus along the vertical direction
Figure GDA0003633561620000021
Xi and s respectively represent Hankel transformation and Laplace integral transformation coefficients;
step S3, based on the displacement response solution of the current roadbed mechanics model in the Laplace-Hankel domain
Figure GDA0003633561620000022
And calculating the surface displacement response of the current roadbed model in the time domain range.
The method is used for obtaining the vertical modulus distribution state of the on-site roadbed.
An electronic device, comprising:
a memory for storing instructions executable by the processor; and
a processor for executing the instructions to implement the above-described method.
A computer-readable storage medium for storing a computer program which, when executed, is capable of carrying out the above-mentioned method.
The invention has the beneficial effects that:
embodiment of the invention through the construction of EvThe (z) function describes the modulus distribution condition of the roadbed along the depth direction, and meanwhile, a physical equation with transverse isotropy is introduced to derive a variable coefficient partial differential equation set considering the characteristics of the roadbed. And by Hankel-Laplace integral transformation and Frobenius method, a method considering transverse isotropy and modulus of the roadbed along the vertical direction is providedA nonlinear distributed roadbed dynamic response analytic solution and a corresponding determination method are provided. The transverse isotropy of the roadbed and the nonlinear distribution characteristics of the modulus along the depth direction are fully considered during roadbed dynamic response calculation, and compared with a traditional elastic system, the method can truly reflect the actual state of the roadbed. Compared with traditional methods such as a finite element method and the like, the method has the characteristics of high precision, high speed and the like.
According to the determining method provided by the embodiment of the invention, the modulus distribution condition of the roadbed can be described, so that the rigidity of the roadbed can be evaluated more accurately and quickly compared with the traditional method; the method is combined with roadbed nondestructive detection, a roadbed vertical modulus curve can be obtained, and the roadbed humidification can be evaluated through the vertical modulus change curve monitored for a long time; and a more reliable theoretical basis is provided for roadbed design, detection, evaluation, research and the like.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a schematic diagram of a roadbed mechanics system with transverse isotropy and vertically nonlinear distribution of modulus under the action of a rigid bearing plate according to an embodiment of the invention.
FIG. 2 is a flow chart of a dynamic response determination method of an embodiment of the present invention.
FIG. 3 is a diagram of an ABAQUS axisymmetric model in an embodiment of the present invention.
Figures 4a-4b are graphs comparing a dynamic response determination method of an embodiment of the present invention to a method using a finite element model, ABAQUS.
Fig. 5 is a comparison graph of a numerical simulation calculation result obtained by a humidity-stress dependent finite element method and an actual real value of the modulus distribution in the depth direction of the roadbed structure obtained by fitting in the embodiment of the invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The embodiment of the invention provides a subgrade surface displacement response determining method considering modulus nonlinear distribution, which is specifically carried out according to the following steps as shown in fig. 2:
step S1, obtaining roadbed material parameters, wherein the roadbed material parameters comprise: vertical modulus E of roadbed top surfacev0Modulus E of road bed infinite depthv∞Roadbed modulus non-uniform coefficient alpha and roadbed modulus ratio neHorizontal poisson's ratio muhVertical poisson's ratio muvSubgrade density rho and subgrade thickness H; the roadbed modulus non-uniformity coefficient alpha represents the change speed of the roadbed along with the modulus in the depth direction; the subgrade modulus ratio neThe uniform distribution is realized in the whole roadbed space; the thickness H may be finite or infinite.
Step S2, as shown in FIG. 1, applying dynamic load p (r, t) to the roadbed surface, establishing a cylindrical coordinate system with the dynamic load center as the coordinate center, wherein r represents a radial coordinate, z represents a vertical coordinate, and phi represents an annular coordinate, and establishing a roadbed modulus function E which is distributed along the vertical non-linear directionv(z)＝Ev0+(Ev∞-Ev0)(1-e-αz) In FIG. 1, R represents the radius of the rigid carrier plate; calculating a surface displacement response solution of the roadbed in the Laplace-Hankel domain, considering transverse isotropy and modulus along vertical nonlinear distribution
Figure GDA0003633561620000031
Roadbed surface displacement response solution of roadbed mechanics model considering transverse isotropy and modulus in vertical nonlinear distribution in Laplace-Hankel domain
Figure GDA0003633561620000041
Is calculated as:
Figure GDA0003633561620000042
in the formula (1), A1,A2,A3,A4For the boundary condition parameters, by solving the following matrix equation [ equation (2) ]]And (4) obtaining. Wherein, if the roadbed thickness is H → + ∞, there is A3＝A4If 0, then equation (2) may be degenerated into a two-dimensional matrix equation;
Figure GDA0003633561620000043
in the formula (2)
Figure GDA0003633561620000044
The result of Hankel-Laplace transformation for the dynamic load p (r, t),
Figure GDA0003633561620000045
xi and s are Hankel integral transformation and Laplace integral transformation coefficients respectively; ev0Representing the modulus of the top surface of the subgrade;
Figure GDA0003633561620000046
Figure GDA0003633561620000047
Figure GDA0003633561620000048
Figure GDA0003633561620000049
in the formula,alpha is the inhomogeneous coefficient of roadbed modulus; e is constant, xi is modulus E through the roadbed top surfacev0Modulus E at infinite depth of roadbedv∞Obtaining by calculation according to the formula (7); xi represents a dimensionless coefficient of the relative magnitude of modulus of the top surface and the bottom of the roadbed, and the larger the numerical value is, the larger the difference of the upper modulus and the lower modulus of the roadbed is;
Figure GDA00036335616200000410
C11，C12，C13，C33，C44a parameter characterizing the transverse isotropy of the road base material in the problem of axial symmetry, C11＝kne(1-neμv)，
Figure GDA00036335616200000411
C13＝kneμv(1+μh)，
Figure GDA00036335616200000412
C44＝0.5(1+μv)-1；
Figure GDA00036335616200000413
k represents an intermediate variable used for calculating each parameter of transverse isotropy and simplifying an expression; n iseIs the subgrade modulus ratio; mu.sv，μhN is the serial number of the series, namely the Poisson ratio of the roadbed in the vertical direction and the horizontal direction.
On the basis of the roadbed mechanics model shown in fig. 1, partial differential equations shown in formulas (32) to (33) can be obtained by combining dynamic balance equations [ formulas (26) to (27) ] of the axial symmetry problem, physical equations representing axial symmetry transverse isotropy and geometric equations [ formulas (28) to (31) ].
Figure GDA0003633561620000051
Figure GDA0003633561620000052
Figure GDA0003633561620000053
Figure GDA0003633561620000054
Figure GDA0003633561620000055
Figure GDA0003633561620000056
Wherein u isr＝ur(r,z,t)，uz＝uz(r, z, t) are displacement components in the r-direction and the z-direction, respectively; s2, obtaining a displacement analysis solution in a Hankel-Laplace domain, and S3 obtaining a displacement response result in a time domain through integral inverse transformation; ρ is the subgrade density; t is time; sigmar＝σr(r,z,t),σz＝σz(r,z,t)，σφ＝σφ(r, z, t) are stress components in the r direction, the z direction and the phi direction of the cylindrical coordinate system, respectively; tau iszt＝τzt(r, z, t) is the shear stress in the z-r plane.
Figure GDA0003633561620000057
Figure GDA0003633561620000061
In order to solve the partial differential equation set shown in the equations (32) to (33), firstly, Laplace transformation about t is carried out on the equations (32) to (33); first order H about r is performed on the Laplace transformed expression (32)Performing ankel transformation; performing zeroth-order Hankel transformation on r for the Laplace transformed expression (33); obtaining a variable coefficient ordinary differential equation set of the formulas (34) to (35), wherein the independent variable z is omitted. Wherein the Laplace transform for x is defined:
Figure GDA0003633561620000062
define the zeroth order Hankel transform for x:
Figure GDA0003633561620000063
define a first order Hankel transform for x:
Figure GDA0003633561620000064
ξ and s represent the Hankel transform and Laplace integral transform coefficients, respectively.
Figure GDA0003633561620000065
Figure GDA0003633561620000066
Figure GDA0003633561620000067
Represents a pair urThe result after the first-order transformation of Hankel-Laplace is carried out,
Figure GDA0003633561620000068
represents a pair uzAnd (5) performing a result after Hankel-Laplace zeroth-order transformation.
To solve the system of constant differential equations with variable coefficients as shown in equations (34) to (35), the following variables are introduced:
ψ＝Ξe-αz (36)
xi calculated according to equation (7), and in subsequent derivations, E is obtained by replacing the independent variable z in equation by the variable ψvThe expression of (ψ) is shown in formula (38). Variable substitution of z by Ψ, Ev(psi) and Ev(z) has the same meaning except that an argument is replaced, and its value is not issuedAnd changed. The independent variable is z in (34) to (35) and ψ in (40) to (43); the purpose is to eliminate Ev(z) an exponential term present, which is converted to algebraic form as shown in (39); only the variable substitutions are made and the expressions themselves are not changed.
Meanwhile, the relation of the formula (39) can be obtained by applying a derivative theory; where f (z) denotes any function with z as argument:
Figure GDA0003633561620000071
Figure GDA0003633561620000072
by bringing expressions (38) to (39) into expressions (34) to (35), a system of constant differential equations with a variable coefficient having a ψ argument can be obtained as shown in expressions (40) to (43).
Wherein formulae (40) to (41) correspond to Ev0<Ev∞In the case of (1), expressions (42) to (43) correspond to Ev0>Ev∞The case (1). The argument ψ is omitted in the formula; where δ is an intermediate calculation parameter, δ ═ Ev0/Ev∞。
Figure GDA0003633561620000073
Figure GDA0003633561620000074
Figure GDA0003633561620000075
Figure GDA0003633561620000076
Based on the infinite series theory, approximation is carried out by infinite series, and the equations (40) - (43) are assumed) The shown ordinary differential equation system has displacement solutions shown in formulas (44) to (45); wherein
Figure GDA0003633561620000077
To represent
Figure GDA0003633561620000078
A first derivative of (a) is,
Figure GDA0003633561620000079
to represent
Figure GDA00036335616200000710
A second derivative of;
Figure GDA00036335616200000711
to represent
Figure GDA00036335616200000712
A first derivative of (a) is obtained,
Figure GDA00036335616200000713
represent
Figure GDA00036335616200000714
A second derivative of;
Figure GDA00036335616200000715
Figure GDA00036335616200000716
in the formula, m is a parameter, and n is a serial number of a series sequence;
Figure GDA0003633561620000081
corresponding m in infinite series representing z-direction displacement componentiThe coefficient of the nth order of terms of (a),
Figure GDA0003633561620000082
indicates the r directionCorresponding m in infinite series of displacement componentsiThe coefficient of the nth order of terms of (a); a isnIs composed of
Figure GDA0003633561620000083
Abbreviation of (a), (b)nIs composed of
Figure GDA0003633561620000084
The abbreviation of (1); the series form shown in the formulas (44) to (45) is subjected to displacement solution
Figure GDA0003633561620000085
Figure GDA0003633561620000086
By substituting the equations (40) to (43), the equations shown in the equations (46) to (49) can be obtained. The formulae (46) to (47) correspond to Ev0<Ev∞In the case of (1), expressions (48) to (49) correspond to Ev0>Ev∞The case (1). Where θ is an intermediate calculation parameter, and θ is ρ s2/Ev∞；
Figure GDA0003633561620000087
Figure GDA0003633561620000088
Figure GDA0003633561620000089
Figure GDA00036335616200000810
Let n be 0, and compare equations (46) to (49) on both sides, the following relationship can be obtained:
[α2C44m2-C11ξ2-θ]a0+αξ(C13+C44)mb0＝0 (50)
αξ(C13+C44)ma0-[α2C33m2-C44ξ2-θ]b0＝0 (51)
if equations (50) to (51) are satisfied, m must satisfy equation (52).
Figure GDA00036335616200000811
In the complex domain, there must be four solutions to the equation for m [ equation (52) ] followed by:
Figure GDA00036335616200000812
Figure GDA00036335616200000813
Figure GDA0003633561620000091
Figure GDA0003633561620000092
in formulae (8) to (11), miFor intermediate calculation of the parameters, theta1，Θ2，Θ3Respectively intermediate calculation parameters; theta1＝α2C33C44；
Figure GDA0003633561620000093
Θ3＝(C44ξ2+θ)(C11ξ2+ θ); where θ is an intermediate calculation parameter, θ ═ ρ s2/Ev∞(ii) a Rho is roadbed density, s in the theta expression is Hankel integral transformation and Laplace integral transformation coefficients, and after integral transformation, the coefficients become s equivalent to time t and can be regarded as a variable.
m has four solutions, and it can be seen from equation (44) that the final solution should be in the form of a combination of four infinite series, in order to distinguish the coefficients a in the four infinite seriesnTherefore, in the equation (12), the superscripts (1), (2), (3), and (4) correspond to 4 solutions of m, respectively. By bringing formula (12) into formula (50), the compound represented by formula (13) can be obtained
Figure GDA0003633561620000094
And (5) expression.
Figure GDA0003633561620000095
Respectively representing the corresponding m in infinite seriesiThe coefficient of the nth order of the terms of (1) is a specific coefficient sequence;
Figure GDA0003633561620000096
Figure GDA0003633561620000097
the coefficients of the corresponding stages in (46) to (49) are equal, that is, the stages of the same power should be equal on both sides of the equation, thereby obtaining
Figure GDA0003633561620000098
See the formulas (14) and (15).
Figure GDA0003633561620000099
Figure GDA00036335616200000910
Calculation according to the recursion relationships shown in equations (12) to (15)
Figure GDA00036335616200000911
The expanded expressions of the parameters in the formulas (14) and (15) are shown in (16) to (21):
Figure GDA00036335616200000912
Figure GDA00036335616200000913
Figure GDA00036335616200000914
Figure GDA00036335616200000915
Figure GDA0003633561620000101
Figure GDA0003633561620000102
in the formulae (12) to (15),
Figure GDA0003633561620000103
for the intermediate calculation parameters, the calculation is performed in accordance with equations (16) to (21). Wherein,
Figure GDA0003633561620000104
is calculated with Ev0And Ev∞Are different, and are considered in two cases, when Ev0<Ev∞When E is calculated according to the formula (20)v0>Ev∞Then, the calculation is performed by the formula (21), where δ is an intermediate calculation parameter, and δ is equal to E in the formula (21)v0/Ev∞。
In formula (20)
Figure GDA0003633561620000105
Introducing intermediate parameters
Figure GDA0003633561620000106
The recursion formula can be obtained by the corresponding number coefficient in (46) - (49) being equal, because the result that n is 0 is known, and by this, the result that n is 1 can be obtained, and the recursion calculation can be carried out sequentially, and n is 2, n is 3, and … can be calculated.
According to the ordinary differential solution theory, the equation is solved into a series form, m has corresponding four results, so that the equation has four infinite series solutions in total, and the final general solution formula can be represented by linear combination of the four infinite series solutions, namely the final displacement general solution formula (53) and the final displacement general solution formula (1):
Figure GDA0003633561620000107
Figure GDA0003633561620000108
four undetermined coefficients A exist in the general solution1，A2，A3，A4. To solve for its value, certain boundary conditions need to be brought in. According to fig. 1, boundary conditions are introduced as shown in equations (54) to (57); wherein p (r, t) is a roadbed top surface load expression.
Figure GDA0003633561620000109
Figure GDA00036335616200001010
ur|z＝H＝0 (56)
uz|z＝H＝0 (57)
Laplace transform on t and Hankel transform on 0 th order of r are executed on the equations (54) and (57); for equation (55), equation (56) performs Laplace transform on t and Hankel transform of order 1 on r. Then, by introducing the conversion relationships shown in the expressions (7) and (38), the boundary conditions shown in the expressions (58) to (61) can be obtained.
Figure GDA0003633561620000111
Figure GDA0003633561620000112
Figure GDA0003633561620000113
Figure GDA0003633561620000114
The general solution expressions of the formula (53) and the formula (1) are brought into the boundary conditions shown in the formulas (58) to (61), so that a fourth-order matrix equation shown in the formula (2) can be obtained, and the matrix equation shown in the formula (2) is solved, so that the coefficient A to be determined can be obtained1，A2，A3，A4(ii) a In particular, in the case of roadbed thickness H → + ∞, if equations (60) to (61) need to be satisfied, a must be present3＝A4The matrix equation shown in equation (2) can be further degenerated into a second-order matrix equation. Determined A1，A2，A3，A4The analytical solution results in the Hankel-Laplace domain can be obtained by substituting the formula (53) and the formula (1). Further, a Hankel inverse transform and a Laplace inverse transform are performed on the result, and a time domain solution of the time domain can be obtained. The calculation process is shown in fig. 2.
In practical calculation, the vertical modulus E of the top surface of the roadbed is firstly calculatedv0Modulus E of road bed infinite depthv∞Roadbed modulus non-uniform coefficient alpha and roadbed modulus ratio neHorizontal poisson's ratio muhVertical Poisson ratio muvSubgrade density rho and subgradeThe thickness H and other parameters are calculated according to the formulas (8) to (11)1，m2，m3，m4(ii) a Further, coefficient series were calculated from equations (12) to (21)
Figure GDA0003633561620000115
Further establishing a fourth-order matrix equation according to the formulas (2) to (6), and solving the coefficient A on the basis of the fourth-order matrix equation1,A2,A3,A4. Finally, calculating the roadbed surface dynamic response solution in a Hankel-Laplace domain according to the formula (1)
Figure GDA0003633561620000116
Step S3, based on the displacement response solution of the current roadbed mechanics model in the Laplace-Hankel domain
Figure GDA0003633561620000117
And calculating the surface displacement response of the current roadbed model in the time domain range.
Displacement response solution of current roadbed mechanics model in Laplace-Hankel domain through formula (23)
Figure GDA0003633561620000118
And carrying out Hankel inverse transformation, and then carrying out Laplace inverse transformation on the result of the Hankel inverse transformation through a formula (22) to obtain the surface displacement response of the current roadbed mechanics system in the time domain range:
Figure GDA0003633561620000121
Figure GDA0003633561620000122
wherein,
Figure GDA0003633561620000123
is composed of
Figure GDA0003633561620000124
F (r,0, t)l) Is uz(r,0,tl)；uz(r,0,tl) For the surface displacement response of the current roadbed mechanical model in the time domain range, the corresponding time step is tl，tl＝l·T/NaT is the total length of solution, NaIs the total time increment step number of the solution, l is an increment step sequence, and l is less than or equal to N; r is the horizontal distance from the calculated point position to the load center; Δ t is the time increment step; the formula (22) shows that inverse Lpalace transformation is carried out by Durbin method, L and gamma are calculation parameters of Durbin method, and L.Na50-5000, and 5-10 of gamma and T; j is a unit imaginary number, j2-1; for the Durbin method, the final time solution of 1/4 can generate a larger system error, and the solution is usually solved by prolonging the total time length T of solution, through testing, the invention can meet the precision requirement by taking 2 times of time length (when inverse transformation is carried out on the solution in the Hankel-Laplace domain, the time length of calculation needs to be given, for example, 60ms dynamic response needs to be calculated, because the Durbin method has certain defects, the time length parameter of calculation is generally 2-3 times in actual calculation, the embodiment of the invention takes 2 times of time length, namely the time length parameter T of calculation takes 120ms, but the final result only takes 60 ms). χ is the upper limit of the numerical integration, AkThe weight of the kth integral node of the 20-node Gaussian interpolation; x is a radical of a fluorine atomikIs the kth integral node value, x, of the i integral sections corresponding to the Gaussian integralik＝(i-1)ΔL+xkΔ L is the Gaussian integration segment division length, xkA kth integrator node value that is a 20 node gaussian integral; n is a radical of0Is the total number of segments of the Gaussian integral, N0Ceil (χ/Δ L), obtained by pair calculation (χ/Δ L) and rounding up;
Figure GDA0003633561620000125
representing the result of the arbitrary function f (-) after the Hankel-Laplace transform;
Figure GDA0003633561620000126
representing the result of the arbitrary function f (-) after Laplace transformation; re (-) is the operation of taking a complex real part; j. the design is a square0(xikr) represents an independent variable of (x)ikZero order of r)And function results corresponding to the Bessel functions. The Laplace inverse transformation is carried out by Durbin method (first-order autocorrelation inspection method) in the formula (1) and the formula (22), the Hankel inverse transformation is carried out by Gaussian interpolation integration in the formula (23), the calculation processes are all realized by adopting a programming mode, and only a numerical value determination method is introduced here. The numerical value determination methods are all realized by programming, programming languages are not limited, and data calculated in the embodiment of the invention are all calculated by MATLAB 2016 a.
After verification, when the upper limit n of the infinite series sequence in the formula (1) is 100, the upper limit χ of the gaussian integral in the formula (23) and the length Δ L of the integral interval are calculated according to the formulas (24) to (25), the result can be converged to an accurate solution.
Figure GDA0003633561620000127
Figure GDA0003633561620000131
Effect verification:
the comparative calculation was performed using finite element software ABAQUS. The subgrade parameters are shown in table 1, and the defined subprogram UMAT is adopted in ABAQUS to realize the nonlinear distribution of the subgrade along the depth direction. p (r, t) is the load corresponding to the rigid bearing plate with the diameter of 30cm, the load is in a half sine wave form, the peak value is 100kPa, and the action time is 20 ms; and setting the calculation time length to be 40ms and the time step increment to be 1ms during calculation. When the invention is applied to calculation, according to the parameter recommended values of the equations (24) to (25), the calculation parameters are set as: n is 100, χ is 300, Δ L is 10.
Table 1 roadbed structure calculation parameter table
Name (R) Cell type modulus/MPa ne μv(＝μh) ρ/kg·m-3 H/m
Bearing plate CAX4 2.1×105 1.0 0.25 7500 -
Road bed CAX4/CINAX4 100+200(1-e-2z) 1.0 0.35 1816 1.0/2.0
As shown in fig. 3, the entire ABAQUS finite element model is modeled in an axisymmetric manner, and in order to simulate the horizontal direction infinity of the layered system, infinite CINAX4 cells are used for the boundary portion, and CAX4 cells are used for the finite size portion. The calculation result is shown in fig. 4, MATLAB in fig. 4 is a calculation result obtained by using the method of the embodiment of the present invention, ABAQUS is a calculation result using a finite element model, and the ABAQUS finite element can be used as a method for widely checking the calculation correctness. In the calculation, ABAQUS obtained by adopting an Intel i 78750H/8G notebook computer to test needs 5 minutes for completing the calculation, but the method only needs 2 minutes, thereby greatly reducing the calculation cost.
When the non-uniform distribution of the roadbed is calculated according to the prior method, local meteorological data need to be acquired, and the meteorological data mainly comprise: and calculating the non-uniform humidity field distribution of the roadbed by adopting calculation software according to the data of wind speed, temperature, precipitation and the like. On the basis, the non-uniform modulus field of the roadbed is calculated by considering the roadbed stress state and the non-uniform humidity field data and combining the roadbed soil dynamic resilience modulus and humidity coupling equation. The calculation consumption is large, and the method is not suitable for popularization.
The invention employs Ev(z)＝Ev0+(Ev∞-Ev0)(1-e-αz) Approximately describing the vertical modulus distribution of the roadbed, wherein E in the formulav0，Ev∞And α is a fitting parameter. For verifying the rationality of the formula, the roadbed modulus along the depth direction of the center of the driveway in the initial state, the operation 1 year, the operation 2 years and the balance humidity state is calculated by adopting meteorological data of nearly 16 years in Nanchang city and combining a soil-based modulus-humidity coupling equation, and E is adoptedv(z) fitting was performed in combination with least squares, as can be seen from FIG. 5, the fitting was good, indicating EvAnd (z) the modulus distribution rule of the roadbed along the depth direction in the operation process can be represented more ideally.
E can be applied no matter how long the roadbed is operatedvThe formula (z) fits the test sample, the fitting effect is good, and the result shows thatvAnd (z) the modulus distribution rule of the roadbed along the depth direction in the operation process can be represented more ideally.
In pair Ev(z) on the basis of the verification of the rationality,the embodiment of the invention further provides a roadbed surface displacement response determining method based on the roadbed nonlinear distribution, and the determining method can be used for back-calculating the modulus distribution of the roadbed, and specifically comprises the following steps:
actually measuring displacement response on site, and recording the displacement response and load data on site; randomly assuming a series of E within a certain rangev(z) parameter Ev0、Ev∞α and subgrade modulus ratio neWherein E isv0In the range of 20MPa to 100MPa, Ev∞The range of (a) is 100MPa to 300MPa, the range of the roadbed modulus non-uniform coefficient alpha is 0.05 to 5.0, neThe range of (1) to (4); while the remaining parameters generally take the following values: the poisson ratio is 0.35, and the density is 1.8-2.0 g/cm3At a sequence of assumptions EvOn the basis of the (z) parameter, the obtained surface load data is taken as p (r, t) in S2, and a series of theoretical displacement response calculation results can be obtained by combining and adopting the calculation method provided by the invention and are compared and matched with an actual displacement curve. According to the matching result, the parameter E is adjusted through a parameter adjusting algorithm (such as a genetic algorithm, a group intelligent algorithm and the like)v0、Ev∞、α、neAnd adjusting to enable the theoretical displacement response to be close to the field actual measurement displacement response to the maximum extent. And taking the parameter value corresponding to the most matched theoretical displacement response calculation result as an inversion result of the actual roadbed, thereby obtaining the modulus distribution rule of the roadbed along the depth direction and realizing roadbed rigidity evaluation.
In addition, a large number of theoretical displacement response building databases can be obtained in advance through the calculation method provided by the invention, and the field actual measurement displacement curve is quickly matched by combining the current common BP neural network and a database searching method, so that the vertical modulus distribution state of the field roadbed can be quickly obtained. Obtaining E corresponding to the theoretical displacement response with the closest actual displacement responsev(z) parameter and neAnd thus obtaining a modulus distribution function in the depth direction of the actual roadbed structure. Compared with the traditional method which can only obtain a single structural modulus, the method can more accurately reflect the vertical modulus distribution state of the roadbed by calculation.
Obtaining the result of the distribution of the vertical modulus of the roadbed by iterative calculation of climate data and a series of finite elements, and then using the method E of the inventionvAnd (z) fitting to obtain theoretically true roadbed vertical modulus distribution, and assuming a series of loads p (r, t) to obtain a series of theoretical displacement response calculation results for theoretical analysis and calculation. Such as the impact of overload, calculating subgrade operating areas, etc.
The method is combined with roadbed nondestructive detection, a roadbed vertical modulus curve can be obtained, and roadbed humidification can be evaluated by monitoring the vertical modulus change curve for a long time. The method for evaluating the wetting of the roadbed comprises the following steps: wetting the roadbed to reduce the vertical modulus curve of the roadbed as a whole; the humidity state of the roadbed can be qualitatively evaluated by regularly detecting the roadbed and comparing with roadbed vertical modulus distribution curves in different periods; if the modulus of the top of the roadbed is reduced, the humidity of the roadbed is integrally increased, and if the modulus of the bottom is basically unchanged, the bottom is proved to reach the equilibrium humidity, and the upper part does not reach the equilibrium humidity; the method is that the roadbed nondestructive test using a homogeneous elastomer system as a mechanical model can not be realized at present, and the traditional mechanical model can only obtain one modulus which approximately represents the modulus of the whole roadbed, so that the depth distribution rule can not be obtained.
The advantages of the invention are as follows:
1. at present, the nonuniform modulus distribution of the roadbed is acknowledged by scholars in the field, but in practical application, when the traditional method is used for solving and calculating the dynamic response considering the nonuniform modulus distribution of the roadbed along the vertical direction, a finite element method is mostly needed, although the method can obtain a more accurate result, the calculation speed is slower, the modeling is required to be carried out again every time, and particularly, for a certain specific situation, the method needs to be independently modeled, so that the time consumption is longer, the cost is high, and the calculation is inconvenient to carry out on site. The invention introduces the nonlinear distribution state of the modulus along the depth direction, combines the prior ordinary differential and partial differential solution theories to carry out the relevant derivation of the analytical solution, can be compiled into any program code to be embedded into small equipment in the derivation and calculation process, can be realized by programming, is simple and convenient to calculate, avoids the complicated process of finite elements, can realize the rapid calculation on site, and meets the relevant requirements of actual calculation.
2. Prior art 1 (multilayer displacement response determination method considering interlayer contact conditions and transverse isotropy) is to divide a pavement structure into N layers, but the modulus in each layer is uniform. The invention only considers the roadbed, but the modulus distribution of the roadbed is uneven, and the fundamental differences of the modulus in the mechanical model are larger. The invention provides a method for determining pavement displacement of a roadbed with isotropy in transverse view and nonlinear distribution of modulus under the impact load of nondestructive testing equipment such as PFWD (frequency-modulated wavelength distribution), and the like.
The subgrade surface displacement response determination method considering the modulus nonlinear distribution, which is disclosed by the embodiment of the invention, can be stored in a computer readable storage medium if the subgrade surface displacement response determination method is realized in the form of a software functional module and is sold or used as an independent product. Based on such understanding, the technical solution of the present invention may be embodied in the form of a software product, which is stored in a storage medium and includes instructions for causing a computer device (which may be a personal computer, a server, or a network device) to execute all or part of the steps of the subgrade surface displacement response determination method considering the nonlinear distribution of the modulus according to the embodiment of the present invention. And the aforementioned storage medium includes: various media capable of storing program codes, such as a U disk, a removable hard disk, a ROM, a RAM, a magnetic disk, or an optical disk.
The above description is only for the preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention shall fall within the protection scope of the present invention.

Claims (10)

1. A subgrade surface displacement response determination method considering modulus nonlinear distribution is characterized by comprising the following steps:
step S1, obtaining roadbed material parameters, wherein the roadbed material parameters comprise: vertical modulus E of roadbed top surfacev0Modulus E of road bed infinite depthv∞Roadbed modulus non-uniform coefficient alpha and roadbed modulus ratio neHorizontal poisson's ratio muhVertical poisson's ratio muvSubgrade density rho and subgrade thickness H;
s2, applying dynamic load p (r, t) to the roadbed surface, establishing a cylindrical coordinate system by taking the dynamic load center as the coordinate center, wherein r represents a radial coordinate, z represents a vertical coordinate, phi represents an annular coordinate, and constructing a roadbed modulus function E which is distributed along the vertical non-linearityv(z)＝Ev0+(Ev∞-Ev0)(1-e-αz) (ii) a Calculating a surface displacement response solution of the roadbed in the Laplace-Hankel domain, which simultaneously considers the transverse isotropy and the nonlinear distribution of the roadbed modulus along the vertical direction
Figure FDA0003639737170000011
Xi and s are Hankel integral transformation and Laplace integral transformation coefficients respectively;
step S3, based on the displacement response solution of the current roadbed mechanics model in the Laplace-Hankel domain
Figure FDA0003639737170000012
And calculating the surface displacement response of the current roadbed model in the time domain range.
2. The subgrade surface displacement response determination method considering the nonlinear distribution of modulus according to claim 1, characterized in that, in step S2,
Figure FDA0003639737170000013
Figure FDA0003639737170000014
in the formula (1-1), A1,A2,A3,A4In order to be a parameter of the boundary condition,
Figure FDA0003639737170000015
corresponding m in infinite series expressing r-direction displacement componentiI is 1,2,3, 4; n is the number of the series of stages, m1～m4Is an intermediate parameter; xi denotes a dimensionless coefficient of the relative magnitude of the modulus of the top and bottom subgrade surfaces;
Figure FDA0003639737170000016
Figure FDA0003639737170000017
Figure FDA0003639737170000018
Figure FDA0003639737170000021
in the formulae (1-3) to (1-6), Θ1，Θ2，Θ3Respectively intermediate calculation parameters;
Θ1＝α2C33C44；
Figure FDA0003639737170000022
Θ3＝(C44ξ2+θ)(C11ξ2+θ)；
where θ is an intermediate calculation parameter, θ ═ ρ s2/Ev∞(ii) a Xi and s are Hankel integral transformation and Laplace product respectivelyDividing a transformation coefficient, wherein the time t becomes s after integral transformation; c11，C12，C13，C33，C44Parameters for characterizing the transverse isotropy characteristics of the road base material in the axial symmetry problem; c11＝kne(1-neμv)，
Figure FDA0003639737170000023
C13＝kneμv(1+μh)，
Figure FDA0003639737170000024
C44＝0.5(1+μv)-1；
Figure FDA0003639737170000025
k represents an intermediate variable.
3. The subgrade surface displacement response determination method considering modulus nonlinear distribution according to claim 2, characterized in that in step S2, boundary condition parameter A1,A2,A3,A4The method is specifically determined according to the following method:
Figure FDA0003639737170000026
Figure FDA0003639737170000027
Figure FDA0003639737170000028
Figure FDA0003639737170000029
Figure FDA00036397371700000210
Figure FDA00036397371700000211
the result of the Hankel-Laplace transform for the load p (r, t), Ψ1i、Ψ2i、Ψ3i、Ψ4iIn order to calculate the parameters in the middle of the run,
Figure FDA00036397371700000212
corresponding m in infinite series representing z-direction displacement componentiThe coefficient of the nth order of terms of (1).
4. The subgrade surface displacement response determination method in consideration of the modulus nonlinear distribution according to claim 3, characterized in that, in step S2,
Figure FDA00036397371700000213
Figure FDA00036397371700000214
the method is specifically determined according to the following method:
Figure FDA00036397371700000215
Figure FDA00036397371700000216
Figure FDA0003639737170000031
Figure FDA0003639737170000032
in the formulae (1-12) to (1-15),
Figure FDA0003639737170000033
calculating parameters for intermediate calculation according to the formulas (1-16) to (1-21); when E isv0<Ev∞When E is calculated according to the formula (1-20)v0>Ev∞Then, the calculation is performed according to the formula (1-21), where δ is an intermediate calculation parameter, and δ is equal to Ev0/Ev∞；
Figure FDA0003639737170000034
Figure FDA0003639737170000035
Figure FDA0003639737170000036
Figure FDA0003639737170000037
Figure FDA0003639737170000038
Figure FDA0003639737170000039
5. The subgrade surface displacement response determination method considering modulus nonlinear distribution according to claim 3, characterized in that the intermediate parameter m1～m4The determination method comprises the following steps:
on the basis of a roadbed mechanics model, a dynamic balance equation of an axisymmetric problem, a physical equation representing axisymmetric transverse isotropy and a geometric equation are combined to obtain a partial differential equation set shown in formulas (1-22) to (1-23):
Figure FDA00036397371700000310
Figure FDA00036397371700000311
wherein u isr＝ur(r,z,t)，uz＝uz(r, z, t) are displacement components along the radius r direction of the cylindrical coordinate system and the roadbed depth z direction respectively; t is time; sigmar＝σr(r,z,t),σz＝σz(r,z,t),σφ＝σφ(r, z, t) are stress components in the r direction, the z direction and the phi direction, respectively; tau iszrIs the shear stress in the z-r plane;
firstly, Laplace conversion of t is carried out on the formulas (1-22) to (1-23); performing first-order Hankel transformation on r on the Laplace transformed expression (1-22); performing zeroth-order Hankel transformation on r on the Laplace transformed (1-23); obtaining a variable coefficient ordinary differential equation set of formulas (1-24) to (1-25), wherein an independent variable z is omitted;
Figure FDA0003639737170000041
Figure FDA0003639737170000042
wherein,
Figure FDA0003639737170000043
to representFor u is pairedrThe result after the first-order transformation of Hankel-Laplace is carried out,
Figure FDA0003639737170000044
represents a pair uzPerforming Hankel-Laplace zeroth order transformation;
in order to solve the variable coefficient ordinary differential equation set shown in equations (1-24) to (1-25), a variable ψ is introduced:
ψ＝Ξe-αz (1-26)
replacement of subgrade modulus function E with variable psiv(z)＝Ev0+(Ev∞-Ev0)(1-e-αz) Medium independent variable z, to obtain EvThe expression of (ψ) is as shown in the formula (1-27):
Figure FDA0003639737170000045
meanwhile, through derivative theory, the formula (1-28) can be obtained; where f (z) denotes any function with z as argument:
Figure FDA0003639737170000046
the equations (1-27) to (1-28) are taken into the equations (1-24) to (1-25) to obtain a system of constant differential equations with a variable ψ as shown in the equations (1-29) to (1-32); wherein formulae (1-29) to (1-30) correspond to Ev0<Ev∞In the case where the formulae (1-31) to (1-32) correspond to Ev0>Ev∞The case (1); the argument ψ is omitted in the formula; where δ is an intermediate calculation parameter, δ ═ Ev0/Ev∞；
Figure FDA0003639737170000051
Figure FDA0003639737170000052
Figure FDA0003639737170000053
Figure FDA0003639737170000054
In order to solve the ordinary differential equation set shown in the formulas (1-29) to (1-32), approximation is carried out by using infinite series based on infinite series theory, and the displacement solution shown in the formulas (1-33) to (1-34) is assumed to exist in the ordinary differential equation set shown in the formulas (1-29) to (1-32);
Figure FDA0003639737170000055
Figure FDA0003639737170000056
in the formula, m is a parameter, and n is a serial number of a series sequence; the series form shown in the formulas (1-33) to (1-34) is subjected to displacement solution
Figure FDA0003639737170000057
The equations are introduced into the equations (1-29) to (1-32) to obtain the equation relations shown in the equations (1-35) to (1-38); formulae (1-35) to (1-36) correspond to Ev0<Ev∞In the case where the formulae (1-37) to (1-38) correspond to Ev0>Ev∞The case (1);
Figure FDA0003639737170000058
Figure FDA0003639737170000059
Figure FDA0003639737170000061
Figure FDA0003639737170000062
when n is 0, the following relationships can be obtained by comparing equations (1-35) to (1-38):
[α2C44m2-C11ξ2-θ]a0+αξ(C13+C44)mb0＝0 (1-39)
αξ(C13+C44)ma0-[α2C33m2-C44ξ2-θ]b0＝0 (1-40)
if the expressions (1-39) to (1-40) are satisfied, m must satisfy the expression (1-41);
Figure FDA0003639737170000063
in the complex domain, there must be four solutions for equation (1-41) for m, see equations (1-3) through (1-6).
6. The subgrade surface displacement response determination method considering modulus nonlinear distribution according to claim 5, characterized in that the boundary conditions are shown in equations (1-42) to (1-45):
Figure FDA0003639737170000064
Figure FDA0003639737170000065
ur|z＝H＝0 (1-44)
uz|z＝H＝0 (1-45)
performing Laplace transformation on t and Hankel transformation of 0 order on r on the equations (1-42) and (1-45); performing Laplace transformation on t and 1-order Hankel transformation on r for equations (1-43) and equations (1-44); on the basis, the transformation relations shown in the formulas (1-2) and (1-27) are introduced, so that the boundary conditions shown in the formulas (1-46) to (1-49) can be obtained;
Figure FDA0003639737170000066
Figure FDA0003639737170000067
Figure FDA0003639737170000071
Figure FDA0003639737170000072
Figure FDA0003639737170000073
is composed of
Figure FDA0003639737170000074
The first derivative of (a) is,
Figure FDA0003639737170000075
is composed of
Figure FDA0003639737170000076
The first derivative of (1) can be obtained by bringing the general solution expression into the boundary conditions shown in the formulas (1-46) to (1-49) to obtain a fourth-order matrix equation shown in the formula (1-7), and the matrix equation shown in the formula (1-7) can be solved to obtain the coefficient to be determinedA1，A2，A3，A4。
7. The method for determining a subgrade surface displacement response considering the nonlinear distribution of the modulus according to claim 1, wherein the step S3 is specifically as follows: displacement response solution of current roadbed mechanics model pair in Laplace-Hankel domain through formula (1-51)
Figure FDA0003639737170000077
Namely, the formula (1-1) is used for carrying out Hankel inverse transformation, then Laplace inverse transformation is carried out on the result of the Hankel inverse transformation through the formula (1-50), and the surface displacement response u of the current mechanical model in the time domain range is obtainedz(r,0,tl)；
Figure FDA0003639737170000078
Figure FDA0003639737170000079
Wherein,
Figure FDA00036397371700000710
to represent
Figure FDA00036397371700000711
f(r,0,tl) Represents uz(r,0,tl) (ii) a Time step of tl，tl＝l·T/NaT is the total length of solution, NaIs the total time increment step number of the solution, l is an increment step sequence, and l is less than or equal to N; r is the horizontal distance from the calculated point position to the load center; Δ t is the time increment step; the expression (1-50) shows that inverse Lpalace transform is performed by Durbin method, L and gamma are Durbin method calculation parameters, j is unit imaginary number2-1; χ is the upper limit of the numerical integration, AkThe weight of the kth integral node of the 20-node Gaussian interpolation; x is the number ofikIs of GaussIntegrating the kth integral node value, x, of the corresponding i integral sectionsik＝(i-1)ΔL+xkΔ L is the Gaussian integration segment division length, xkA kth integrator node value that is a 20 node gaussian integral; n is a radical of0Is the total number of segments of the Gaussian integral, N0Ceil (χ/Δ L), obtained by pair calculation (χ/Δ L) and rounding up; re (-) is the operation of taking a complex real part; j. the design is a square0(xikr) represents an independent variable of (x)ikr) function result corresponding to the zero order Bessel function.
8. A method according to any one of claims 1 to 7 for obtaining a vertical modulus profile of a subgrade in a site.
9. An electronic device, comprising:
a memory for storing instructions executable by the processor; and
a processor for executing the instructions to implement the method of any one of claims 1-7.
10. A computer-readable storage medium for storing a computer program which, when executed, is capable of carrying out the method of any one of claims 1-7.
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