CN109101744B - Method for calculating tunnel deformation and stress under action of non-uniform explosive load - Google Patents

Method for calculating tunnel deformation and stress under action of non-uniform explosive load Download PDF

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CN109101744B
CN109101744B CN201810987095.9A CN201810987095A CN109101744B CN 109101744 B CN109101744 B CN 109101744B CN 201810987095 A CN201810987095 A CN 201810987095A CN 109101744 B CN109101744 B CN 109101744B
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lining
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displacement
soil
soil body
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CN109101744A (en
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陶彪
耿大新
胡文韬
刘豆
曾罗兰
王俊
刘鑫磊
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East China Jiaotong University
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Abstract

A method for calculating tunnel deformation and stress under the action of non-uniform explosive load comprises the following steps: assuming that the infinite medium hard rock tunnel has a non-uniform explosive load effect, simplifying the tunnel model into a rock mass-single layer lining plane strain model; establishing a rock mass control equation according to an elastic fluctuation theory, and deducing a rock mass motion equation and an constitutive relation again to enable transient load, annular strain and shear strain to follow
Figure DEST_PATH_IMAGE001
(ii) a change; introducing a potential function to decouple a rock mass balance equation; establishing a shell model equation under a polar coordinate to obtain a lining stress and displacement relation; and introducing boundary conditions, solving the set undetermined coefficient, and finally obtaining an internal force solution in the tunnel-rock mass.

Description

Method for calculating tunnel deformation and stress under action of non-uniform explosive load
Technical Field
The invention relates to a method for calculating tunnel deformation and stress under the action of non-uniform explosive load, belonging to the technical field of tunnel engineering.
Background
With the progress of urbanization in China, in order to relieve the pressure of traffic jam between cities and villages and drive the development of rural economic transformation, roads and railway tunnels are actively built in all places to promote the transformation and industrial upgrading of urban structures, thereby improving the economic benefit. However, the tunnel belongs to a special structure with closed periphery, and the condition that the building is damaged due to an accident explosion inside the tunnel gradually becomes a common concern in the engineering field.
When a highway or railway tunnel explodes, the explosion position is difficult to predict, most of the explosion positions are influenced by a random internal dynamic load, and the tunnel has larger influence on the position on the side closer to the explosion source. Therefore, the distribution form of the transient dynamic load under the action of explosion is not completely circumferentially uniform, but no very good method for calculating the internal force of the circular tunnel under the action of non-uniform explosion load exists at present, so that the method for calculating the deformation and stress of the tunnel under the action of the non-uniform explosion load is established, and the method has theoretical significance and higher practical value.
Disclosure of Invention
The invention aims to establish a method for calculating the deformation and stress of a tunnel under the action of non-uniform explosive load aiming at the problems of internal force calculation of the existing circular tunnel under the non-uniform explosive load.
The technical scheme for realizing the invention is as follows: a method for calculating tunnel deformation and stress under the action of non-uniform explosive load comprises the steps of firstly establishing a rock mass elastic wave equation for a medium-hard rock tunnel, and performing variable separation on the rock mass motion equation by assuming a soil mass displacement potential function to obtain a stress displacement expression of a soil mass in a lapalce domain; establishing a modal equation of the lining shell, carrying out dimensionless on the modal equation and carrying out Laplace transformation to obtain a shell stress-displacement relation; and introducing boundary conditions, solving the set undetermined coefficient, and finally obtaining the internal force solution of the tunnel rock mass.
The tunnel rock mass structure is regarded as elastic single-phase medium, neglects physical power existence, and its vector wave equation and constitutive relation can be expressed as:
Figure BDA0001779950660000021
Figure BDA0001779950660000022
σij＝λ1δijεkk+2μ1εij
in the formula, σijThe stress tensor of the soil framework is obtained; epsilonkkIs the volume strain of the soil framework; epsilonijIs the strain tensor of the soil framework; deltaijFor the Kronecker parameter, δ when i ≠ jijWhen j is equal to 0, delta is equal to iij＝1；λ1、μ1Is the soil framework Lame constant; u. ofrRadial displacement of soil skeleton medium; u. ofθCircumferential displacement of the soil framework medium; rho1Density of rock mass.
The soil displacement potential function is as follows:
the general solution of the soil displacement potential function can be expressed by an n-order Bessel function linear combination form:
Figure BDA0001779950660000023
in the formula, An、Bn、Cn、DnIs the undetermined coefficient; i isn(x) is a first class n-order virtual vector BeA ssel function; knAnd (, is a Bessel function of the second class of n-order imaginary vectors.
The expression of the stress and displacement of the soil body in the Laplace domain is as follows:
Figure BDA0001779950660000024
Figure BDA0001779950660000031
Figure BDA0001779950660000032
Figure BDA0001779950660000033
Figure BDA0001779950660000034
wherein r is the diameter of the pole; u. ofrThe soil body is radially displaced; u. ofθThe soil body is in circumferential displacement; a. then、Bn、Cn、DnIs the undetermined coefficient; sigmarThe radial stress of the soil body; sigmaθThe soil body hoop stress; sigmarθThe tangential stress of the soil body is adopted; mu 1 is a soil framework Lame constant; knIs a Bessel function of a second class of n-order imaginary vectors.
The lining shell modal equation expression is as follows:
Figure BDA0001779950660000035
Figure BDA0001779950660000036
in the formula of Uθn、UrnFourier expansion coefficient terms of lining displacement in theta and r directions are respectively; q. q.srn、qθnFourier expansion coefficient terms of lining stress in r and theta directions are respectively; esModulus of elasticity, v, for liningsIs the poisson's ratio of the lining; c. C1The lining longitudinal wave velocity; rhosdIs lining density; a is1A is a + h/2, and a is the radius of the tunnel; a is1The distance from the center of the tunnel to the middle curved surface of the lining; h is the lining thickness.
Carrying out dimensionless transformation on a modal equation and carrying out Laplace transformation to obtain a shell stress-displacement relation:
Figure BDA0001779950660000041
Figure BDA0001779950660000042
in the formula (I), the compound is shown in the specification,
Figure BDA0001779950660000043
is a lining dimensionless radial displacement coefficient item;
Figure BDA0001779950660000044
is a lining dimensionless circumferential displacement coefficient item; beta is a2＝h2/12a1 2；
Figure BDA0001779950660000045
Wherein c is1The wave velocity of the longitudinal wave of the lining is shown,
Figure BDA0001779950660000046
s is a Laplace transformation parameter;
Figure BDA0001779950660000047
a lining dimensionless radial stress coefficient term;
Figure BDA0001779950660000048
dimensionless liningA hoop stress coefficient term; e*＝Es/μs、h*＝h/a1(ii) a n is a positive integer.
The solving of the set waiting coefficient is expressed as follows:
the lining is assumed to be a thin-wall cylindrical shell, the radius of the tunnel is far larger than the thickness of the lining, and therefore a curved surface in the lining shell can be regarded as a contact surface between a soil body and the lining, namely r ═ a1At least one of (1) and (b); the conditions of stress coordination and continuity of the soil body and lining interface are considered to obtain:
when r is a1When the temperature of the water is higher than the set temperature,
Figure BDA0001779950660000049
the boundary conditions of the inner surface of the lining are as follows:
when the r is equal to a, the r,
Figure BDA00017799506600000410
in order to solve the dynamic response expression under the radial nonuniform transient load, the pulse load is subjected to Laplace transformation, which is expressed as follows:
Figure BDA00017799506600000411
wherein f (theta) is an arbitrary-shaped non-uniform load; load form oriented two-sided concentrated load
Figure BDA0001779950660000051
T is the pulse load period;
substituting the soil body and the lining stress displacement into the boundary condition to obtain a linear equation set of undetermined coefficients, and solving the undetermined coefficients in each potential function by using the boundary condition, wherein the matrix is as follows:
Figure BDA0001779950660000052
the undetermined coefficient solved by the matrix is substituted into the expression of soil body and lining stress displacement, and the solution of soil body and lining stress and displacement under the Laplace domain can be solved;
in the formula:
Figure BDA0001779950660000053
Figure BDA0001779950660000054
Figure BDA0001779950660000055
Figure BDA0001779950660000056
Figure BDA0001779950660000057
Figure BDA0001779950660000058
Figure BDA0001779950660000059
Figure BDA00017799506600000510
Figure BDA00017799506600000511
Figure BDA00017799506600000512
Figure BDA00017799506600000513
Figure BDA0001779950660000061
Figure BDA0001779950660000062
Figure BDA0001779950660000063
Figure BDA0001779950660000064
Figure BDA0001779950660000065
Figure BDA0001779950660000066
Figure BDA0001779950660000067
Figure BDA0001779950660000068
Figure BDA0001779950660000069
Figure BDA00017799506600000610
in the above formula:
Figure BDA00017799506600000611
the subscript j is 1,2,3,4, and indicates when r is a1The expansion coefficient term of the radial stress and displacement of the soil body is obtained;
Figure BDA00017799506600000612
representing a tangential stress coefficient term of a contact surface of the lining and the soil body;
Figure BDA00017799506600000613
a soil body annular displacement expansion coefficient term; p1i，N2iAnd when the subscript i is 1,2,3 and 4, the coefficient term of the lining net stress Laplace transform expansion is expressed when r is a.
The method for calculating the deformation and the stress of the tunnel has the advantages that when the tunnel is subjected to complex blasting, for example, the tunnel is internally provided with the blast impact action which is distributed unevenly to the periphery, and the deformation and the stress of the tunnel caused by the blast loads distributed in different types can be accurately calculated according to the working conditions. Can provide theoretical reference for the research of the action mechanism of the explosion impact tunnel and the disaster prevention and reduction of the explosion accident in the tunnel.
Drawings
FIG. 1 is a flow chart of a method for calculating the internal force of a tunnel under a transient blasting load according to the invention;
FIG. 2 is a schematic diagram of a tunnel being subjected to a non-uniform transient load;
fig. 3 is a schematic diagram of an excitation function time course curve.
Detailed Description
A specific embodiment of the present invention is shown in fig. 1.
Referring to fig. 1, the method for calculating the deformation and stress of the tunnel under the action of the non-uniform explosive load according to the embodiment is implemented by the following steps:
firstly, establishing a soil body motion equation in medium hard rock:
the tunnel rock mass structure is regarded as an elastic single-phase medium, the existence of physical strength is ignored, and a vector wave equation and an constitutive relation can be expressed as follows:
Figure BDA0001779950660000071
Figure BDA0001779950660000072
σij＝λ1δijεkk+2μ1εij (2)
in the formula, σijThe stress tensor of the soil framework is obtained; epsilonkkIs the volume strain of the soil framework; epsilonijIs the strain tensor of the soil framework; deltaijFor the Kronecker parameter, δ when i ≠ jijWhen j is equal to 0, delta is equal to iij＝1；λ1、μ1Is the soil framework Lame constant; u. ofrRadial displacement of soil skeleton medium; u. ofθThe circumferential displacement of the soil framework medium; rho1Is the density of the rock mass.
Introducing a two-bit potential shift function to simplify the equation, and obtaining potential function general solution expression by adopting separation variables:
according to Helmholtz vector decomposition theorem, soil displacement can be expressed by two potential functions:
Figure BDA0001779950660000081
in the formula (I), the compound is shown in the specification,
Figure BDA0001779950660000082
psi is scalar potential function and vector potential function of the soil skeleton, wherein dimensionless parameters are defined as
Figure BDA0001779950660000083
The compound represented by formula (3) may be substituted for formula (1):
Figure BDA0001779950660000084
Figure BDA0001779950660000085
taking Laplace transform to both sides of the formula (4),
Figure BDA0001779950660000086
are potential functions, respectively
Figure BDA0001779950660000087
Laplace transform form of ψ:
Figure BDA0001779950660000088
from equation (5) we can derive:
Figure BDA0001779950660000089
Figure BDA00017799506600000810
in the formula (I), the compound is shown in the specification,
k1、k2complex wave numbers of compressional waves and shear waves respectively;
Figure BDA0001779950660000091
Figure BDA0001779950660000092
Figure BDA0001779950660000093
and s is a Laplace transformation parameter.
According to the separation variable method, the general solution of the soil potential function can be expressed by an n-order Bessel function linear combination form:
Figure BDA0001779950660000094
in the formula, An、Bn、Cn、DnIs the undetermined coefficient; i isn(x) is a Bessel function of the first class of n-order imaginary vectors; knAnd (, is a Bessel function of the second class of n-order imaginary vectors.
Thirdly, the relation between the soil stress displacement and the coefficient to be determined
Substituting the soil potential function (7) into the equation (3), considering the soil stress-strain relation, and defining that the displacement is dimensionless by a and the stress is μ1Dimensionless, wherein
Figure BDA0001779950660000095
The stress and displacement expression of the soil body in the Laplace domain under the polar coordinate can be obtained:
Figure BDA0001779950660000096
Figure BDA0001779950660000097
Figure BDA0001779950660000098
Figure BDA0001779950660000101
Figure BDA0001779950660000102
wherein r is the diameter of the pole; u. ofrThe soil body is radially displaced;uθthe soil body is in circumferential displacement; a. then、Bn、Cn、DnIs the undetermined coefficient; sigmarThe radial stress of the soil body; sigmaθThe soil body hoop stress; sigmarθThe tangential stress of the soil body is adopted; mu.s1Is the soil framework Lame constant; knIs a Bessel function of a second class of n-order imaginary vectors.
Fourthly, establishing a lining shell modal equation
Assuming that the lining is in complete contact with the soil body, considering the interaction between the lining and the soil body, the lining is equivalent to a thin-wall cylindrical shell, and according to the thin-shell moment-free theory, the lining motion equation can be expressed as follows:
Figure BDA0001779950660000103
Figure BDA0001779950660000104
in the formula of Uθn、UrnFourier expansion coefficient terms of lining displacement in theta and r directions are respectively; q. q.srn、qθnFourier expansion coefficient terms of lining stress in r and theta directions are respectively; es、νsRespectively representing the elastic modulus and the Poisson ratio of the lining; c. C1The lining longitudinal wave velocity; rhosdIs lining density; a is1A is a + h/2, and a is the radius of the tunnel; a is1The distance from the center of the tunnel to the middle curved surface of the lining; h is the lining thickness.
Fifthly, performing Laplace transformation on the lining motion equation to obtain a shell stress-displacement relation:
Figure BDA0001779950660000105
Figure BDA0001779950660000111
Figure BDA0001779950660000112
Figure BDA0001779950660000113
in the formula of Uθ(θ,t*)、Ur(θ,t*) Respectively circumferential displacement and radial displacement outside the shell; qr(θ,t*)、Qθ(θ,t*) The radial net stress and the hoop stress of the outer part of the shell are respectively.
The shell stress-displacement relation can be obtained by carrying out dimensionless transformation on the formula (9) and carrying out Laplace transformation:
Figure BDA0001779950660000114
Figure BDA0001779950660000115
wherein the dimensionless number is defined as E*＝Es/μs、h*＝h/a1、
Figure BDA0001779950660000116
Figure BDA0001779950660000117
μsIs the lining lame constant.
Figure BDA0001779950660000118
Figure BDA0001779950660000119
And (3) representing lining stress and displacement in the Laplace domain.
Solving undetermined coefficients by utilizing boundary conditions to obtain internal forces of different positions of the tunnel
Assuming the lining is a thin-walled cylindrical shell,the radius of the tunnel is far larger than the thickness of the lining, so that the curved surface in the lining shell can be regarded as a contact surface between a soil body and the lining, namely r ═ a1To (3). The conditions of stress coordination and continuity of the soil body and lining interface are considered to obtain:
when r is a1When the temperature of the water is higher than the set temperature,
Figure BDA00017799506600001110
the boundary condition of the inner surface of the lining is
When the r is equal to a, the r,
Figure BDA0001779950660000121
for solving the dynamic response expression under the radial non-uniform transient load, as shown in fig. 3, Laplace transform is performed on the pulse load, which can be expressed as:
Figure BDA0001779950660000122
wherein f (theta) is random-shaped non-uniform load, and the load form of the embodiment is oriented to concentrate loads on two sides
Figure BDA0001779950660000123
T is the pulse load period.
Soil and lining stress displacement is substituted into the equations (12) and (13), a linear equation set of undetermined coefficients can be obtained, the undetermined coefficients in each potential function can be solved by utilizing boundary conditions, and the matrix is as follows:
Figure BDA0001779950660000124
and (3) solving undetermined coefficients obtained by the matrix, and substituting the undetermined coefficients into equations (8) and (11) to obtain solutions of soil body and lining stress and displacement under the Laplace domain.
In the formula, the parameters in the coefficient matrix are expressed as follows:
Figure BDA0001779950660000125
Figure BDA0001779950660000126
Figure BDA0001779950660000127
Figure BDA0001779950660000128
Figure BDA0001779950660000129
Figure BDA0001779950660000131
Figure BDA0001779950660000132
Figure BDA0001779950660000133
Figure BDA0001779950660000134
Figure BDA0001779950660000135
Figure BDA0001779950660000136
Figure BDA0001779950660000137
Figure BDA0001779950660000138
Figure BDA0001779950660000139
Figure BDA00017799506600001310
Figure BDA00017799506600001311
Figure BDA00017799506600001312
Figure BDA00017799506600001319
Figure BDA00017799506600001313
Figure BDA00017799506600001314
Figure BDA00017799506600001315
in the above formula:
Figure BDA00017799506600001316
the subscript j is 1,2,3,4, whenr＝a1In time, the expansion coefficient term of the radial stress and displacement of the soil body,
Figure BDA00017799506600001317
representing the tangential stress coefficient term of the contact surface of the lining and the soil body,
Figure BDA00017799506600001318
a soil body annular displacement expansion coefficient term; p1i，N2iAnd when the subscript i is 1,2,3 and 4, the coefficient term of the lining net stress Laplace transform expansion is expressed when r is a.

Claims (1)

1. A method for calculating tunnel deformation and stress under the action of non-uniform explosive load is characterized in that for a medium-hard rock tunnel, firstly, a rock mass elastic wave equation is established, and the rock mass elastic wave equation is subjected to variable separation by assuming a soil displacement potential function to obtain a stress and displacement expression of a soil in a lapalce domain; establishing a modal equation of the lining shell, carrying out dimensionless on the modal equation and carrying out Laplace transformation to obtain a shell stress-displacement relation; introducing boundary conditions, solving the set undetermined coefficient, and finally obtaining an internal force solution of the tunnel rock mass;
the rock mass elastic wave equation:
Figure FDA0003546769560000011
Figure FDA0003546769560000012
σij＝λ1δijεkk+2μ1εij
in the formula, σijThe stress tensor of the soil framework is obtained; epsilonkkIs the volume strain of the soil framework; epsilonijIs the strain tensor of the soil framework; deltaijFor the Kronecker parameter, δ when i ≠ jijWhen j is equal to 0, delta is equal to iij＝1；λ1、μ1Is the soil framework Lame constant; u. ofrRadial displacement of soil skeleton medium; u. ofθCircumferential displacement of the soil framework medium; rho1Density of rock mass;
the soil displacement potential function is as follows:
the general solution of the soil displacement potential function is expressed by an n-order Bessel function linear combination form:
Figure FDA0003546769560000013
in the formula, An、Bn、Cn、DnIs the undetermined coefficient; i isn(x) is a Bessel function of the first class of n-order imaginary vectors; kn(xi) is a Bessel function of the second class of n-order imaginary vectors;
the expression of the stress and displacement of the soil body in the Laplace domain is as follows:
Figure FDA0003546769560000021
Figure FDA0003546769560000022
Figure FDA0003546769560000023
Figure FDA0003546769560000024
Figure FDA0003546769560000025
wherein r is the diameter of the pole; u. ofrThe soil body is radially displaced; u. ofθIs a soil body ringDisplacing in the direction; a. then、Bn、Cn、DnIs the undetermined coefficient; sigmarThe radial stress of the soil body; sigmaθThe soil body hoop stress; sigmarθThe tangential stress of the soil body is adopted; mu.s1Is the soil framework Lame constant; knIs a Bessel function of a second class of n-order virtual vectors;
the lining shell modal equation expression is as follows:
Figure FDA0003546769560000026
Figure FDA0003546769560000027
in the formula of Uθn、UrnFourier expansion coefficient terms of lining displacement in theta and r directions are respectively; q. q.srn、qθnFourier expansion coefficient terms of lining stress in r and theta directions are respectively; esModulus of elasticity, v, for liningsIs the poisson's ratio of the lining; c. C1The lining longitudinal wave velocity; rhosdIs lining density; a is1A is a + h/2, and a is the radius of the tunnel; a is1The distance from the center of the tunnel to the middle curved surface of the lining; h is the lining thickness;
carrying out dimensionless transformation on a modal equation, and carrying out Laplace transformation to obtain a relation between shell stress and a displacement coefficient term:
Figure FDA0003546769560000031
Figure FDA0003546769560000032
in the formula (I), the compound is shown in the specification,
Figure FDA0003546769560000033
for lining dimensionless radialA displacement coefficient term;
Figure FDA0003546769560000034
is a lining dimensionless circumferential displacement coefficient item; beta is a2＝h2/12a1 2；
Figure FDA0003546769560000035
Wherein c is1The wave velocity of the longitudinal wave of the lining is shown,
Figure FDA0003546769560000036
s is a Laplace transformation parameter;
Figure FDA0003546769560000037
a lining dimensionless radial stress coefficient term;
Figure FDA0003546769560000038
a lining dimensionless hoop stress coefficient term; e*＝Es/μs、h*＝h/a1(ii) a n is a positive integer;
the solving of the set waiting coefficient is expressed as follows:
assuming that the lining is a thin-wall cylindrical shell, the radius of the tunnel is greater than the thickness of the lining, so that the curved surface in the lining shell is regarded as a contact surface between a soil body and the lining, namely r ═ a1At least one of (1) and (b); considering the conditions of coordination and continuity of the soil body and lining interface stress:
when r is a1When the temperature of the water is higher than the set temperature,
Figure FDA0003546769560000039
the boundary conditions of the inner surface of the lining are as follows:
when the r is equal to a, the r,
Figure FDA00035467695600000310
in order to solve the dynamic response expression under the radial nonuniform transient load, the pulse load is subjected to Laplace transformation, which is expressed as follows:
Figure FDA0003546769560000041
wherein f (theta) is an arbitrary-shaped non-uniform load; load form oriented two-sided concentrated load
Figure FDA0003546769560000042
T is the pulse load period;
substituting the soil body and the lining stress displacement into the boundary conditions to obtain a linear equation set of undetermined coefficients, and solving the undetermined coefficients in each potential function by using the boundary conditions, wherein the matrix is as follows:
Figure FDA0003546769560000043
the undetermined coefficient solved by the matrix is substituted into a soil body and lining stress displacement expression, and the solution of soil body and lining stress and displacement under the Laplace domain is solved;
in the formula:
Figure FDA0003546769560000044
Figure FDA0003546769560000045
Figure FDA0003546769560000046
Figure FDA0003546769560000047
Figure FDA0003546769560000048
Figure FDA0003546769560000049
Figure FDA00035467695600000410
Figure FDA0003546769560000051
Figure FDA0003546769560000052
Figure FDA0003546769560000053
Figure FDA0003546769560000054
Figure FDA0003546769560000055
Figure FDA0003546769560000056
Figure FDA0003546769560000057
Figure FDA0003546769560000058
Figure FDA0003546769560000059
Figure FDA00035467695600000510
Figure FDA00035467695600000511
Figure FDA00035467695600000512
Figure FDA00035467695600000513
Figure FDA00035467695600000514
in the above formula:
Figure FDA00035467695600000515
the subscript j is 1,2,3,4, and indicates when r is a1The expansion coefficient term of the radial stress and displacement of the soil body is obtained;
Figure FDA00035467695600000516
representing a tangential stress coefficient term of a contact surface of the lining and the soil body;
Figure FDA00035467695600000517
soil body annular displacement expansion coefficientAn item; p1i，N2iAnd when the subscript i is 1,2,3 and 4, the coefficient term of the lining net stress Laplace transform expansion is expressed when r is a.
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