CN113177288A - Analysis and calculation method for internal force and relative deformation of circular shield tunnel lining based on measured data and state space method - Google Patents
Analysis and calculation method for internal force and relative deformation of circular shield tunnel lining based on measured data and state space method Download PDFInfo
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- CN113177288A CN113177288A CN202110290577.0A CN202110290577A CN113177288A CN 113177288 A CN113177288 A CN 113177288A CN 202110290577 A CN202110290577 A CN 202110290577A CN 113177288 A CN113177288 A CN 113177288A
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Abstract
The invention discloses a circular shield tunnel internal force analysis method based on actual measurement data and a state space method, which comprises the following steps: establishing a curve coordinate system by using a middle lining curve of the circular shield tunnel, and taking the bottom of the middle lining curve as a coordinate origin; the load of the circular shield tunnel can include two conditions: a. designing common loads, namely, a stratum load and a soil body counterforce, wherein the stratum load is vertical soil body pressure at the top and the bottom of the tunnel and horizontal soil body pressure at two sides of the tunnel, and the soil body counterforce is force exerted on the lining after a stratum spring is stressed; b. and testing working conditions and loads are applied by the jack. And calculating to obtain state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement. The calculation method can improve the calculation efficiency of the shield tunnel structure, avoids the problem of the value of the joint rigidity parameter, and provides a quick and convenient analysis method for the evaluation of the internal force state of the shield tunnel lining during operation.
Description
Technical Field
The invention relates to the technical field of shield tunnel analysis, in particular to a method for analyzing and calculating internal force and relative deformation of a circular shield tunnel lining based on measured data and a state space method.
Background
The shield tunnel is transversely a flexible underground structure formed by connecting a plurality of segments through joints, and the mechanical property of the joints has obvious influence on the response of the lining. In practical engineering, after a force is applied to the joint, rotation, shearing and radial discontinuous displacement (namely, mutual bending, dislocation and compression of the joint of the section of the joint segment) can occur. On one hand, in a general mechanical model, a joint is usually simulated by a spring, but due to the complex contact relationship of the joint after being stressed, the evaluation of the rigidity of the joint is a difficulty in the application of the method. On the other hand, after the shield tunnel is built, the pipe sheet lining deforms after long-term operation, and the corresponding internal force of the cross section also changes to deviate from the initial state. The duct piece internal force is an important basis for evaluating the safety state of the section of the duct piece under the operation condition and the occasion of reinforcing design and reinforcement. The acquisition of the segment internal force is usually obtained by indirect conversion based on a flat section assumption through an embedded strain gauge. In actual engineering, strain gauges are prone to failure after long-term operation, and it is difficult to install strain gauges in all sections in consideration of economic efficiency. For the segment internal force, the apparent displacement of the segment lining, such as the corner, the shearing displacement and the axial displacement of the joint, can be easily obtained by observing through external equipment. The method for analyzing the force in the duct piece based on the actually measured displacement data and the mechanical model is developed, so that the problem of reasonable value of the joint rigidity can be avoided, and meanwhile, a quick and effective analysis method can be provided for safety assessment of the section of the duct piece during operation.
Disclosure of Invention
The invention aims to solve the technical problem of providing an analytical calculation method for the lining internal force and the relative deformation of a circular shield tunnel based on measured data and a state space method, wherein the analytical calculation method can provide important data and a mechanical model basis for the evaluation of the section safety state of a shield tunnel segment under the long-term operation condition.
Therefore, the invention provides a method for analyzing and calculating the lining internal force and relative deformation of a circular shield tunnel based on measured data and a state space method, which comprises the following steps:
(1) establishing a curve coordinate system by using a middle lining curve of the circular shield tunnel, wherein the curve coordinate system takes the bottom of the middle lining curve as an origin of coordinates;
(2) the load received by the circular shield tunnel can comprise two conditions: a. designing common loads, namely, a stratum load and a soil body counterforce, wherein the stratum load is vertical soil body pressure at the top and the bottom of the tunnel and horizontal soil body pressure at two sides of the tunnel, and the soil body counterforce is force exerted on the lining after a stratum spring is stressed; b. testing working conditions, wherein load is applied by a jack;
A. when the formation spring coefficient kzAnd ksWhen the initial force is zero, the formula (14) is adopted to calculate the three internal forces of the initial end of the first shield segment
Wherein the content of the first and second substances,
a matrix composed of elements of 1 st to 3 rd rows, 4 th to 6 th columns,
and
are vectors consisting of the elements of rows 4 to 6 of the vector, respectively, and
n is the total number of joints on the lining, and is consistent with the number of shield segments;
is a matrix;
the load integral vector of the lining is taken as the load integral vector of the lining;
actually measuring a relative displacement vector for the nth joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the nth shield segment is formed;
the nth shield segment load integral vector is obtained;
actually measuring a relative displacement vector for the n-1 th joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the (n-1) th shield segment is formed;
the (n-1) th shield segment load integral vector is obtained;
actually measuring a relative displacement vector for a 1 st joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the 1 st shield segment;
the 1 st shield segment load integral vector is obtained;
calculating three internal forces of the initial end of the first shield segment according to the formula (14)
Directly making three displacement quantities of the starting end of the first shield segment be zero according to the definition of rigid body displacement, and obtaining the internal force quantity of the starting end of the first shield segment according to the formula (14)
Form the initial end state vector of the first shield segment
j is the shield segment number, the first shield segment j is 1, and the state vector of any point of the first shield segment is calculated by adopting a formula (6)
The terminal state vector of the shield segment up to the first
when the j is equal to 1, the k value is equal to 1,
the 1 st shield segment coordinate is theta(1)One point and the shield segment coordinate is
The transfer matrix between the state vectors of the start terminals,
is the state vector of the starting end of the 1 st shield segment,
is composed of
To theta(1)Load integral vectors within the range;
then, the starting end state vector of the second shield segment is obtained by using the formula (10)
when the j is equal to 1, the k value is equal to 1,
wherein the content of the first and second substances,
is the starting end state vector of the 2 nd shield segment,
the end state vector of the 1 st shield segment,
a stiffness matrix for the 1 st joint on the lining;
and j sequentially taking 2, 3 and 4.. n to obtain state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement.
Preferably, the analysis of the internal force and the relative deformation of the lining of the circular shield tunnel depends on the expansion of a test device of a stress deformation indoor model of a shield tunnel structure, the test device comprises a model groove, a tunnel model, a soil layer and a pressure device, the tunnel model is positioned in the model groove and embedded in the soil layer, and the following analysis and calculation are carried out after the data are obtained through the test model; or the analysis of the internal force and the relative deformation of the lining of the circular shield tunnel depends on the field actual measurement data.
Preferably, the middle curve of the lining is a circumferential curve of the middle position of the inner circumference and the outer circumference of the circular shield tunnel, and the circular shield tunnel is composed of a plurality of shield segments with radian.
Preferably, a curvilinear coordinate system is established by using a middle lining curve of the circular shield tunnel, wherein the middle lining curve is a circumferential curve at the middle position of the inner circumference and the outer circumference of the circular shield tunnel, and the circular shield tunnel is composed of a plurality of shield segments with radian; the curve coordinate system takes the bottom of a curve in the lining as a coordinate origin; the following are descriptions of the elements that make up X:
wherein w is the radial displacement of each position point on the curve in the lining, u is the circumferential displacement of any point on the curve in the lining,
the bending moment of the shield segment is M, Q is the bending moment of the shield segment, N is the axial force of the shield segment, and Q is the shearing force of the shield segment. In the following description, a matrix or vector is represented by a bold font, and a scalar is represented by a normal font. The other physical quantities have the following meanings: r is the radius of the curve in the lining, h is the section height of the shield segment, E is the Young modulus of the shield segment material, A is the sectional area of the shield segment, I is the polar inertia moment of the circumferential section of the shield segment, theta is the angle coordinate of any point on the curve in the lining, qzThe distributed load in the z direction is borne by the shield segment, the z direction is the radial direction of a curve coordinate system, and q issThe distributed load in the s direction of the shield segment is the circumferential direction of a curve coordinate system, and the s direction is PzFor z-direction concentrated load, P, on shield segmentssIs a concentrated load in the s direction, Delta theta, to which the segment is subjected(j)、Δw(j)、Δu(j)The relative rotation displacement, the radial relative displacement and the axial relative displacement of the jth shield segment joint are respectively.
The invention also provides a method for analyzing and calculating the lining internal force and the relative deformation of the circular shield tunnel based on the measured data and the state space method, which comprises the following steps:
(1) establishing a curve coordinate system by using a middle lining curve of the circular shield tunnel, wherein the curve coordinate system takes the bottom of the middle lining curve as an origin of coordinates;
(2) the load received by the circular shield tunnel can comprise two conditions: a. designing common loads, namely, a stratum load and a soil body counterforce, wherein the stratum load is vertical soil body pressure at the top and the bottom of the tunnel and horizontal soil body pressure at two sides of the tunnel, and the soil body counterforce is force exerted on the lining after a stratum spring is stressed; b. and testing working conditions and loads are applied by the jack.
B. When the formation spring coefficient kzAnd ksWhen the initial force is zero, the formula (12) is adopted to calculate the internal force of the initial end of the first shield segment
Where the superscript-1 represents the inverse of the matrix, and
n is the total number of joints on the lining, and is consistent with the number of shield segments;
is a matrix;
the load integral vector of the lining is taken as the load integral vector of the lining;
actually measuring a relative displacement vector for the nth joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the nth shield segment is formed;
the nth shield segment load integral vector is obtained;
actually measuring a relative displacement vector for the n-1 th joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the (n-1) th shield segment is formed;
the (n-1) th shield segment load integral vector is obtained;
actually measuring a relative displacement vector for a 1 st joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the 1 st shield segment;
the 1 st shield segment load integral vector is obtained;
calculating the state vector of the starting end of the first shield segment according to the formula (12)
j is the shield segment number, the first shield segment j is 1, and the state vector of any point of the first shield segment is calculated by adopting a formula (6)
The terminal state vector of the shield segment up to the first
when the j is equal to 1, the k value is equal to 1,
the 1 st shield segment coordinate is theta(1)One point and the shield segment coordinate is
The transfer matrix between the state vectors of the start terminals,
is the state vector of the starting end of the 1 st shield segment,
is composed of
To theta(1)Load integral vectors within the range;
then, the starting end state vector of the second shield segment is obtained by using the formula (10)
when the j is equal to 1, the k value is equal to 1,
wherein the content of the first and second substances,
is the starting end state vector of the 2 nd shield segment,
the end state vector of the 1 st shield segment,
a stiffness matrix for the 1 st joint on the lining;
and j sequentially taking 2, 3 and 4.. n to obtain state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement.
And obtaining state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement, and relative deformation is obtained through displacement calculation.
Compared with the prior art, the invention has the following advantages:
the method for analyzing the internal force and the relative deformation of the circular shield tunnel lining based on the measured data and the state space method solves by adopting the state space method, can improve the calculation efficiency of the shield tunnel structure, avoids the problem of taking the value of the joint rigidity parameter, and provides a quick and convenient analysis method for the state evaluation of the internal force of the shield tunnel lining during operation.
Drawings
FIG. 1 is a schematic diagram of a duct piece coordinate system and related physical quantities in accordance with the present invention;
FIG. 2 is a schematic view of the segment load distribution of example 2 of the present invention;
FIG. 3 is a schematic view of the segment load distribution and its structural parameters in example 1 of the present invention;
FIG. 4 is a graph showing the shear results of example 1 of the present invention;
FIG. 5 is a schematic diagram showing the results of axial force calculation in example 1 of the present invention;
FIG. 6 is a schematic view showing the calculation results of bending moment in example 1 of the present invention;
FIG. 7 is a schematic diagram showing the results of shear force calculation according to example 2 of the present invention;
FIG. 8 is a diagram showing the results of axial force calculation in example 2 of the present invention;
FIG. 9 is a schematic diagram of the results of calculating the bending moment in example 2 of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples. In which like parts are designated by like reference numerals. It should be noted that the terms "front," "back," "left," "right," "upper" and "lower" used in the following description refer to directions in the drawings, and the terms "bottom" and "top," "inner" and "outer" refer to directions toward and away from, respectively, the geometric center of a particular component.
Referring to fig. 1-9, the following description is developed corresponding to the main contents of the invention, and the analysis method for the inner force of the segment lining based on the measured data and the state space method is developed by relying on a test device of a stress deformation indoor model of a shield tunnel structure, wherein the test device comprises a model groove, a tunnel model, a soil layer and a pressure device, the tunnel model is positioned in the model groove and embedded in the soil layer, and the following analysis calculation is developed after the data is obtained through the test model; or the analysis of the internal force and the relative deformation of the lining of the circular shield tunnel depends on the field actual measurement data, and the method comprises the following steps:
(1) establishing a curve coordinate system by using a middle lining curve of the circular shield tunnel, wherein the middle lining curve is a circumferential curve of the middle position of the inner circumference and the outer circumference of the circular shield tunnel, and the circular shield tunnel is composed of a plurality of shield segments with radians;
the curve coordinate system takes the bottom of a curve in the lining as a coordinate origin, and the following elements are used for describing the X: w is the radial displacement of each position point on the curve in the lining, u is the circumferential displacement of any point on the curve in the lining,
the bending moment of the shield segment is M, Q is the bending moment of the shield segment, N is the axial force of the shield segment, and Q is the shearing force of the shield segment. In the following description, a matrix or vector is represented by a bold font, and a scalar is represented by a normal font. The other physical quantities have the following meanings: r is the radius of the curve in the lining, h is the section height of the shield segment, E is the Young modulus of the shield segment material, A is the sectional area of the shield segment (namely the circumferential sectional area of the shield segment), I is the polar inertia moment of the circumferential section of the shield segment, theta is the angle coordinate of any point on the curve in the lining, q iszThe distributed load in the z direction is borne by the shield segment, the z direction is the radial direction of a curve coordinate system, and q issThe distributed load in the s direction of the shield segment is the circumferential direction of a curve coordinate system, and the s direction is PzFor the z-direction concentrated load to which the shield segments are subjected,Psis a concentrated load in the s direction, Delta theta, to which the segment is subjected(j)、Δw(j)、Δu(j)The relative rotation displacement, the radial relative displacement and the axial relative displacement of the jth shield segment joint are respectively.
(2) Assuming that any load borne by the circular shield tunnel comprises an active load and a passive load, wherein the active load is vertical soil pressure at the top of the tunnel and horizontal soil pressure at two sides of the tunnel, the passive load is soil counterforce, the soil counterforce of the circular shield tunnel comprises soil counterforce irrelevant to lining displacement (soil counterforce at the bottom of the tunnel) and soil counterforce relevant to lining displacement (soil counterforce at two sides of the tunnel),
A. when the formation spring coefficient kzAnd ksWhen the initial force is zero, the formula (14) is adopted to calculate the three internal forces of the initial end of the first shield segment
Wherein the content of the first and second substances,
a matrix of its row 1 to 3, column 4 to 6 elements, and similarly,
and
respectively, the vectors are composed of the elements of the 4 th to 6 th rows of the vectors. While
n is the total number of joints on the lining, and is consistent with the number of shield segments;
is a matrix;
the load integral vector of the lining is taken as the load integral vector of the lining;
actually measuring a relative displacement vector for the nth joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the nth shield segment is formed;
the nth shield segment load integral vector is obtained;
actually measuring a relative displacement vector for the n-1 th joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the (n-1) th shield segment is formed;
the (n-1) th shield segment load integral vector is obtained;
actually measuring a relative displacement vector for a 1 st joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the 1 st shield segment;
and the load integral vector of the 1 st shield segment is obtained.
The first is obtained from equation (14)Three internal forces of shield segment starting end
Directly making three displacement quantities of the starting end of the first shield segment be zero according to the definition of rigid body displacement, and obtaining the internal force quantity of the starting end of the first shield segment according to the formula (14)
Form the initial end state vector of the first shield segment
j is the shield segment number, the first shield segment j is 1, and the state vector of any point of the first shield segment is calculated by adopting a formula (6)
The terminal state vector of the shield segment up to the first
when the j is equal to 1, the k value is equal to 1,
the 1 st shield segment coordinate is theta(1)One point and the shield segment coordinate is
The transfer matrix between the state vectors of the start terminals,
is the state vector of the starting end of the 1 st shield segment,
is composed of
To theta(1)Load integral vectors within the range;
then, the starting end state vector of the second shield segment is obtained by using the formula (10)
when the j is equal to 1, the k value is equal to 1,
wherein the content of the first and second substances,
is the starting end state vector of the 2 nd shield segment,
the end state vector of the 1 st shield segment,
a stiffness matrix for the 1 st joint on the lining;
and j sequentially taking 2, 3 and 4.. n to obtain state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement.
The last three items of the state vector are sequentially shear force, axial force and bending moment, and the first three items of the state vector are sequentially radial displacement, annular displacement and corner displacement.
B. When the formation spring coefficient kzAnd ksWhen the initial force is zero, the formula (12) is adopted to calculate the internal force of the initial end of the first shield segment
Where the superscript-1 represents the inverse of the matrix, and
n is the total number of joints on the lining, and is consistent with the number of shield segments;
is a matrix;
the load integral vector of the lining is taken as the load integral vector of the lining;
actually measuring a relative displacement vector for the nth joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the nth shield segment is formed;
the nth shield segment load integral vector is obtained;
actually measuring a relative displacement vector for the n-1 th joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the (n-1) th shield segment is formed;
the (n-1) th shield segment load integral vector is obtained;
actually measuring a relative displacement vector for a 1 st joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the 1 st shield segment;
the 1 st shield segment load integral vector is obtained;
calculating the state vector of the starting end of the first shield segment according to the formula (12)
j is the shield segment number, the first shield segment j is 1, and the state vector of any point of the first shield segment is calculated by adopting a formula (6)
The terminal state vector of the shield segment up to the first
when the j is equal to 1, the k value is equal to 1,
the 1 st shield segment coordinate is theta(1)One point and the shield segment coordinate is
The transfer matrix between the state vectors of the start terminals,
is the state vector of the starting end of the 1 st shield segment,
is composed of
To theta(1)Load integral vectors within the range;
then, the starting end state vector of the second shield segment is obtained by using the formula (10)
when the j is equal to 1, the k value is equal to 1,
wherein the content of the first and second substances,
is the starting end state vector of the 2 nd shield segment,
the end state vector of the 1 st shield segment,
a stiffness matrix for the 1 st joint on the lining;
and j sequentially taking 2, 3 and 4.. n to obtain state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement.
And obtaining state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement, and relative deformation is obtained through displacement calculation.
The last three items of the state vector are sequentially shear force, axial force and bending moment, and the first three items of the state vector are sequentially radial displacement, annular displacement and corner displacement.
Referring to fig. 1-9, the duct piece lining internal force analysis method based on the measured data and the state space method provided by the invention is described in detail again by depending on a complete derivation process, for convenience of description, the model is based on a curved surface coordinate system, and each physical quantity and the positive direction thereof are shown in fig. 1. Wherein 1 is a lining middle curve, w is the radial displacement of a segment (namely a shield segment), u is the circumferential displacement of the segment,
is the corner of the segment, Q is the shear force of the segment, Q0And Q1Shear force of a certain shield segment at the starting end and the tail end, N is axial force of the segment, and N is axial force of the segment0And N1Axial force of the starting end and the tail end of a certain shield segment, wherein M is bending moment of the segment, and M is bending moment of the segment0And M1Bending moments of the starting end and the tail end of a certain shield segment. In the following description, a matrix or vector is represented by a bold font, and a scalar is represented by a normal font. The other physical quantities have the following meanings: r is the radius of the pipe piece (namely the middle curve of the lining), h is the section height of the pipe piece, E is the Young modulus of the material of the pipe piece, A is the sectional area of the pipe piece, I is the polar moment of inertia of the pipe piece, theta is the angle coordinate, theta0And theta1As angular coordinate, qzFor the distributed load in the z-direction to which the duct pieces are subjected, qsFor the distributed load of the segment in the s direction, PzFor concentrated z-direction loads on the segments, PsIs the s-direction concentrated load, k, to which the segment is subjectedzAnd ksRespectively the radial stiffness and the tangential stiffness of the formation springs,
Δw(j)、Δu(j)the segment joint relative rotation displacement, the radial relative displacement and the axial relative displacement are respectively adopted.
The invention provides a segment lining internal force analysis method based on actual measurement data and a state space method, which comprises the following specific steps:
(1) without loss of generality, assume that there are n shield segments. For the jth shield segment, based on the curved beam theory and the minimum potential energy principle, a deformation equation and a motion control equation (formulas (1) and (2)) of the shield segment are deduced.
Wherein the superscript (j) represents the physical quantity of the jth segment.
(2) Based on a state space method, as shown in a formula (4) first formula, the circumferential displacement w of the jth shield segment is determined(j)Radial displacement u(j)And corner
And cross-sectional axial force N of its energy couple(j)Shear force Q(j)And bending moment M(j)As a state vector x(j)The motion equation and the deformation equation in the equations (1) and (2) are rewritten into a control equation in the form of a matrix, that is, a state equation (3)).
Wherein the content of the first and second substances,
and
for the consideration of numerical stability, dimensionless displacement, internal force and coordinates are introduced according to the following formula, and the addition of one horizontal line to the physical quantity represents the normalized physical quantity:
(3) solving a state equation (formula (2)) according to a matrix theory to obtain the coordinate theta of the jth shield piece(j)State variable of a point
And its coordinates are
State variable of the start terminal of
The transfer relation between them.
Particularly when the duct piece
To theta(j)Within the range of m effects on
Acting to concentrate the load
When the function is performed (k is less than or equal to m),
middle element
Is explicitly expressed as follows:
wherein
And
is a matrix
Row 4 column and row 5 column elements of (i),
is a vector
The ith element of (1).
The terminal angle coordinate of the jth shield segment
Substituting the formula (6) to obtain the state vector of the starting end of the jth shield segment
And end state variables
The transfer relationship therebetween, equation (9) is obtained.
(4) The relative displacement of the segment joint in three directions is obtained by actual measurement and is known, so that the j-th joint has
Wherein the content of the first and second substances,
in the formula (I), the compound is shown in the specification,
the normalized radial relative displacement and the normalized axial relative displacement of the joint are respectively.
(5) Starting from the origin of coordinates, transmitting along the circular direction, respectively using a formula (7) for the tube sheet, using a formula (10) for the joint, and finally returning to the origin, so as to obtain,
where the superscript-1 represents the inverse of the matrix, and
when the formation spring coefficient kzAnd ksWhen the values are all zero, the formula (13) is a full-rank non-homogeneous equation set, and a state vector consisting of three internal forces and displacements of the starting end of the first segment can be obtained by the full-rank non-homogeneous equation set
And then, the state vector of each position of the whole lining can be obtained by repeatedly using the formulas (6) and (10).
In particular, the spring constant k of the stratumzAnd ksZero, in equation (12)
The inverse of the matrix does not exist, the problem has infinite solution, namely, rigid body displacement exists, but when self-balancing force is applied, the internal force of the structure is still certain. At this time, equation (12) may be degraded to
Wherein the content of the first and second substances,
a matrix of its row 1 to 3, column 4 to 6 elements, and similarly,
and
respectively, the vectors are composed of the elements of the 4 th to 6 th rows of the vectors. The three internal forces at the beginning of the first segment are obtained from equation (14). And three displacements of the section of the starting end of the first lost segment represent rigid body displacements, so that the value can be assigned at will. Thus, the state vector of the section of the starting end of the first pipe sheet is formed
The remaining steps and kzAnd ksThe same is true when not zero, and the description is omitted. The displacement at this time includes rigid body displacement, but since it is rigid body displacement, it does not affect the accuracy of the internal force calculation.
The invention is described below by way of example:
example 1
A circular shield tunnel lining internal force analysis method based on measured data and a state space method comprises the following steps:
(1) establishing a curvilinear coordinate system by using a middle lining curve of the circular shield tunnel, wherein the middle lining curve is a circumferential curve at the middle position of the inner circumference and the outer circumference of the circular shield tunnel, the circular shield tunnel consists of a plurality of shield segments with radians, and the curvilinear coordinate system takes the bottom of the middle lining curve as an origin of coordinates;
(2) the load received by the circular shield tunnel can comprise two conditions: referring to a shown in fig. 2, a common load is designed, namely, a stratum load and a soil body counterforce are provided, wherein the stratum load is vertical soil body pressure at the top and the bottom of a tunnel and horizontal soil body pressure at two sides of the tunnel, and the soil body counterforce is force exerted on a lining after a stratum spring is stressed; b. referring to fig. 3, the conditions were tested with a load applied by a jack.
When the formation spring coefficient kzAnd ksWhen the initial force is zero, the formula (14) is adopted to calculate the three internal forces of the initial end of the first shield segment
Wherein the content of the first and second substances,
a matrix of its row 1 to 3, column 4 to 6 elements, and similarly,
and
respectively, the vectors are composed of the elements of the 4 th to 6 th rows of the vectors. While
n is the total number of joints on the lining, and is consistent with the number of shield segments;
is a matrix;
the load integral vector of the lining is taken as the load integral vector of the lining;
actually measuring a relative displacement vector for the nth joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the nth shield segment is formed;
the nth shield segment load integral vector is obtained;
actually measuring a relative displacement vector for the n-1 th joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the (n-1) th shield segment is formed;
the (n-1) th shield segment load integral vector is obtained;
actually measuring a relative displacement vector for a 1 st joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the 1 st shield segment;
the 1 st shield segment load integral vector is obtained;
calculating three internal forces of the initial end of the first shield segment according to the formula (14)
Directly making three displacement quantities of the starting end of the first shield segment be zero according to the definition of rigid body displacement, and obtaining the internal force quantity of the starting end of the first shield segment according to the formula (14)
Form the initial end state vector of the first shield segment
j is the shield segment number, the first shield segment j is 1, and the state vector of any point of the first shield segment is calculated by adopting a formula (6)
The terminal state vector of the shield segment up to the first
when the j is equal to 1, the k value is equal to 1,
the 1 st shield segment coordinate is theta(1)One point and the shield segment coordinate is
The transfer matrix between the state vectors of the start terminals,
is the state vector of the starting end of the 1 st shield segment,
is composed of
To theta(1)Load integral vectors within the range;
then, the starting end state vector of the second shield segment is obtained by using the formula (10)
when the j is equal to 1, the k value is equal to 1,
wherein the content of the first and second substances,
is the starting end state vector of the 2 nd shield segment,
the end state vector of the 1 st shield segment,
a stiffness matrix for the 1 st joint on the lining;
and j sequentially taking 2, 3 and 4.. n to obtain state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement.
Example 1 was derived from a model test and subjected to three different sets of thrust forces, P1-117 kN, P2-71 kN, and P3-91.1 kN, with segment structure and load distribution as shown in fig. 3. The radius of the test tube piece is 2.925m, the width of the tube piece is 1.2m, the section height is 0.35m, and the elastic modulus is 43.478 GPa. The measured joint displacement data obtained by the test are shown in table 1.
The specific calculation steps are as follows:
A. and carrying out parameter normalization according to the input parameters.
B. The normalized parameters and the coordinates of the initial end of each segment
And end coordinates
Substituting the formula (6) to obtain a matrix of transmission state vectors of the starting end and the tail end of each segment
And load integral vector of segment
C. Substituting the measured displacement of each joint into formula (11) to generate stiffness matrix of each joint
D. Will be provided with
Substituting into equation (13), find the matrix
And
then, the three internal forces of the first segment start end can be obtained by substituting the equation (14)
Order to
And
synthesizing the state vector of the starting end of the first segment
And (5) calculating the internal force of each segment section by combining the formulas (6) and (14).
F. Drawing according to actual needs.
The calculated shearing force, axial force and bending moment of example 1 are shown in FIGS. 4-6.
Example 2
A circular shield tunnel lining internal force analysis method based on measured data and a state space method comprises the following steps:
(1) establishing a curvilinear coordinate system by using a middle lining curve of the circular shield tunnel, wherein the middle lining curve is a circumferential curve at the middle position of the inner circumference and the outer circumference of the circular shield tunnel, the circular shield tunnel consists of a plurality of shield segments with radians, and the curvilinear coordinate system takes the bottom of the middle lining curve as an origin of coordinates;
(2) the load received by the circular shield tunnel can include three conditions: a. designing common loads, namely, a stratum load and a soil body counterforce, wherein the stratum load is vertical soil body pressure at the top and the bottom of the tunnel and horizontal soil body pressure at two sides of the tunnel, and the soil body counterforce is force exerted on the lining after a stratum spring is stressed; b. and testing working conditions and loads are applied by the jack.
When the formation spring coefficient kzAnd ksWhen the initial force is zero, the formula (12) is adopted to calculate the internal force of the initial end of the first shield segment
Where the superscript-1 represents the inverse of the matrix, and
n is the total number of joints on the lining, and is consistent with the number of shield segments;
is a matrix;
the load integral vector of the lining is taken as the load integral vector of the lining;
actually measuring a relative displacement vector for the nth joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the nth shield segment is formed;
the nth shield segment load integral vector is obtained;
actually measuring a relative displacement vector for the n-1 th joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the (n-1) th shield segment is formed;
the (n-1) th shield segment load integral vector is obtained;
actually measuring a relative displacement vector for a 1 st joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the 1 st shield segment;
the 1 st shield segment load integral vector is obtained;
calculating the state vector of the starting end of the first shield segment according to the formula (12)
j is the shield segment number, the first shield segment j is 1, and the state vector of any point of the first shield segment is calculated by adopting a formula (6)
The terminal state vector of the shield segment up to the first
when the j is equal to 1, the k value is equal to 1,
the 1 st shield segment coordinate is theta(1)One point and the shield segment coordinate is
The transfer matrix between the state vectors of the start terminals,
is the state vector of the starting end of the 1 st shield segment,
is composed of
To theta(1)Load integral vectors within the range;
then, the starting end state vector of the second shield segment is obtained by using the formula (10)
when the j is equal to 1, the k value is equal to 1,
wherein the content of the first and second substances,
is the starting end state vector of the 2 nd shield segment,
the end state vector of the 1 st shield segment,
a stiffness matrix for the 1 st joint on the lining;
and j sequentially taking 2, 3 and 4.. n to obtain state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement.
Example 2 structural parameters of a certain actual project, the load form is shown in figure 2, the external diameter of the lining: rout2.5m, segment cross-sectional height: h 0.3m, modulus of elasticity of the tube sheet: e ═ 3.5 × 107kPa, the lining cutting comprises 6 sections of jurisdiction, and the section of jurisdiction symmetry evenly distributed, each section of jurisdiction central angle is 60 degrees. The load parameters were as follows: p is a radical of1＝165.8kPa，p2＝187.9kPa，p3＝116.1kPa，p4＝60.2kPa，p57.5kPa, radial soil reaction coefficient kzIs 5000kN/m3Coefficient of reaction force k of tangential earth springsIs radial soil body counter-force coefficient k z1/3 of (1). The relative displacement values of the joints are shown in Table 1. The specific calculation steps are as follows:
A. and carrying out parameter normalization according to the input parameters.
B. The normalized parameters and the coordinates of the initial end of each segment
And end coordinates
Substituting the formula (6) to obtain a matrix of transmission state vectors of the starting end and the tail end of each segment
And load integral vector of segment
C. Substituting the measured displacement of each joint into formula (11) to generate stiffness matrix of each joint
D. Will be provided with
Substituting into equation (13), find the matrix
And
then, the state vector of the starting end of the first pipe piece can be obtained by substituting the equation (12)
And (5) calculating the internal force of each segment section by combining the formulas (6) and (15).
E. Drawing according to actual needs.
The calculated radial displacement, circumferential displacement, cross-sectional rotational displacement, shear force, axial force and bending moment of example two are shown in fig. 7-9.
The above description is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may occur to those skilled in the art without departing from the principle of the invention, and are considered to be within the scope of the invention.
Claims (7)
1. A method for analyzing and calculating the lining internal force and relative deformation of a circular shield tunnel based on measured data and a state space method is characterized by comprising the following steps of: the method specifically comprises the following steps:
(1) establishing a curve coordinate system by using a middle lining curve of the circular shield tunnel, wherein the curve coordinate system takes the bottom of the middle lining curve as an origin of coordinates;
(2) the load received by the circular shield tunnel can comprise two conditions: a. designing common loads, namely, a stratum load and a soil body counterforce, wherein the stratum load is vertical soil body pressure at the top and the bottom of the tunnel and horizontal soil body pressure at two sides of the tunnel, and the soil body counterforce is force exerted on the lining after a stratum spring is stressed; b. testing working conditions, wherein load is applied by a jack;
A. when the formation spring coefficient kzAnd ksWhen the initial force is zero, the formula (14) is adopted to calculate the three internal forces of the initial end of the first shield segment
Wherein the content of the first and second substances,
a matrix composed of elements of 1 st to 3 rd rows, 4 th to 6 th columns,
and
are vectors consisting of the elements of rows 4 to 6 of the vector, respectively, and
n is the total number of joints on the lining, and is consistent with the number of shield segments;
is a matrix;
the load integral vector of the lining is taken as the load integral vector of the lining;
actually measuring a relative displacement vector for the nth joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the nth shield segment is formed;
the nth shield segment load integral vector is obtained;
actually measuring a relative displacement vector for the n-1 th joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the (n-1) th shield segment is formed;
the (n-1) th shield segment load integral vector is obtained;
for the 1 st joint actual measurement relative displacement vector on the lining；
A transfer matrix between state vectors of the starting end and the tail end of the 1 st shield segment;
the 1 st shield segment load integral vector is obtained;
calculating three internal forces of the initial end of the first shield segment according to the formula (14)
Directly making three displacement quantities of the starting end of the first shield segment be zero according to the definition of rigid body displacement, and obtaining the internal force quantity of the starting end of the first shield segment according to the formula (14)
Form the initial end state vector of the first shield segment
j is the shield segment number, the first shield segment j is 1, and the state vector of any point of the first shield segment is calculated by adopting a formula (6)
The terminal state vector of the shield segment up to the first
when the j is equal to 1, the k value is equal to 1,
the 1 st shield segment coordinate is theta(1)One point and the shield segment coordinate is
The transfer matrix between the state vectors of the start terminals,
is the state vector of the starting end of the 1 st shield segment,
is composed of
To theta(1)Load integral vectors within the range;
then, the starting end state vector of the second shield segment is obtained by using the formula (10)
when the j is equal to 1, the k value is equal to 1,
wherein the content of the first and second substances,
is the starting end state vector of the 2 nd shield segment,
the end state vector of the 1 st shield segment,
a stiffness matrix for the 1 st joint on the lining;
and j sequentially taking 2, 3 and 4.. n to obtain state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement.
2. The method for analyzing and calculating the lining internal force and the relative deformation of the circular shield tunnel based on the measured data and the state space method according to claim 1, wherein the method comprises the following steps: the analysis of the internal force and the relative deformation of the lining of the circular shield tunnel depends on the expansion of a stress deformation indoor model test device of a shield tunnel structure, the test device comprises a model groove, a tunnel model, a soil layer and a pressure device, the tunnel model is positioned in the model groove and embedded in the soil layer, and the following analysis calculation is expanded after data are obtained through the test model; or the analysis of the internal force and the relative deformation of the lining of the circular shield tunnel depends on the field actual measurement data.
3. The method of claim 1, wherein the curve in the lining is a circumferential curve at a middle position between an inner circumference and an outer circumference of the circular shield tunnel, and the circular shield tunnel is composed of a plurality of shield segments with radian.
4. The method for analyzing and calculating the internal force and the relative deformation of the lining of the circular shield tunnel based on the measured data and the state space method according to claim 1, 2 or 3, is characterized in that: establishing a curve coordinate system by using a middle lining curve of the circular shield tunnel, wherein the middle lining curve is a circumferential curve of the middle position of the inner circumference and the outer circumference of the circular shield tunnel, and the circular shield tunnel is composed of a plurality of shield segments with radians; the curve coordinate system takes the bottom of a curve in the lining as a coordinate origin; the following are descriptions of the elements that make up X:
wherein w is the radial displacement of each position point on the curve in the lining, u is the circumferential displacement of any point on the curve in the lining,
is the corner of the shield segment and Q is the shear force of the shield segmentN is the axial force of the shield segment, and M is the bending moment of the shield segment. In the following description, a matrix or vector is represented by a bold font, and a scalar is represented by a normal font. The other physical quantities have the following meanings: r is the radius of the curve in the lining, h is the section height of the shield segment, E is the Young modulus of the shield segment material, A is the sectional area of the shield segment, I is the polar inertia moment of the circumferential section of the shield segment, theta is the angle coordinate of any point on the curve in the lining, qzThe distributed load in the z direction is borne by the shield segment, the z direction is the radial direction of a curve coordinate system, and q issThe distributed load in the s direction of the shield segment is the circumferential direction of a curve coordinate system, and the s direction is PzFor z-direction concentrated load, P, on shield segmentssIs a concentrated load in the s direction, Delta theta, to which the segment is subjected(j)、Δw(j)、Δu(j)The relative rotation displacement, the radial relative displacement and the axial relative displacement of the jth shield segment joint are respectively.
5. A method for analyzing and calculating the internal force and relative deformation of a circular shield tunnel lining based on measured data and a state space method is characterized by comprising the following steps of:
(1) establishing a curve coordinate system by using a middle lining curve of the circular shield tunnel, wherein the curve coordinate system takes the bottom of the middle lining curve as an origin of coordinates;
(2) the load received by the circular shield tunnel can comprise two conditions: a. designing common loads, namely, a stratum load and a soil body counterforce, wherein the stratum load is vertical soil body pressure at the top and the bottom of the tunnel and horizontal soil body pressure at two sides of the tunnel, and the soil body counterforce is force exerted on the lining after a stratum spring is stressed; b. and testing working conditions and loads are applied by the jack.
B. When the formation spring coefficient kzAnd ksWhen the initial force is zero, the formula (12) is adopted to calculate the internal force of the initial end of the first shield segment
Where the superscript-1 represents the inverse of the matrix, and
n is the total number of joints on the lining, and is consistent with the number of shield segments;
is a matrix;
the load integral vector of the lining is taken as the load integral vector of the lining;
actually measuring a relative displacement vector for the nth joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the nth shield segment is formed;
the nth shield segment load integral vector is obtained;
actually measuring a relative displacement vector for the n-1 th joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the (n-1) th shield segment is formed;
is as followsn-1 shield segment load integral vectors;
actually measuring a relative displacement vector for a 1 st joint on the lining;
a transfer matrix between state vectors of the starting end and the tail end of the 1 st shield segment;
the 1 st shield segment load integral vector is obtained;
calculating the state vector of the starting end of the first shield segment according to the formula (12)
j is the shield segment number, the first shield segment j is 1, and the state vector of any point of the first shield segment is calculated by adopting a formula (6)
The terminal state vector of the shield segment up to the first
when the j is equal to 1, the k value is equal to 1,
the 1 st shield segment coordinate is theta(1)One point and the shield segment coordinate is
The transfer matrix between the state vectors of the start terminals,
is the state vector of the starting end of the 1 st shield segment,
is composed of
To theta(1)Load integral vectors within the range;
then, the starting end state vector of the second shield segment is obtained by using the formula (10)
when the j is equal to 1, the k value is equal to 1,
wherein the content of the first and second substances,
is the starting end state vector of the 2 nd shield segment,
the end state vector of the 1 st shield segment,
a stiffness matrix for the 1 st joint on the lining;
and j sequentially taking 2, 3 and 4.. n to obtain state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement.
And obtaining state vectors of all positions of the whole lining, wherein the state vectors comprise internal force and displacement, and relative deformation is obtained through displacement calculation.
6. The method of claim 5, wherein the curve in the lining is a circumferential curve at the middle position between the inner circumference and the outer circumference of the circular shield tunnel, and the circular shield tunnel is composed of a plurality of shield segments with radian.
7. The method for analyzing and calculating the internal force and the relative deformation of the lining of the circular shield tunnel based on the measured data and the state space method according to claim 5 or 6, wherein the method comprises the following steps: establishing a curve coordinate system by using a middle lining curve of the circular shield tunnel, wherein the middle lining curve is a circumferential curve of the middle position of the inner circumference and the outer circumference of the circular shield tunnel, and the circular shield tunnel is composed of a plurality of shield segments with radians; the curve coordinate system takes the bottom of a curve in the lining as a coordinate origin; the following are descriptions of the elements that make up X:
wherein w is the radial displacement of each position point on the curve in the lining, u is the circumferential displacement of any point on the curve in the lining,
the bending moment of the shield segment is M, Q is the bending moment of the shield segment, N is the axial force of the shield segment, and Q is the shearing force of the shield segment. In the following description, a matrix or vector is represented by a bold font, and a scalar is represented by a normal font. The other physical quantities have the following meanings: r is the radius of the curve in the lining, h is the section height of the shield segment, E is the Young modulus of the shield segment material, A is the sectional area of the shield segment, I is the polar inertia moment of the circumferential section of the shield segment, theta is the angle coordinate of any point on the curve in the lining, qzThe distributed load in the z direction is borne by the shield segment, the z direction is the radial direction of a curve coordinate system, and q issThe distributed load in the s direction of the shield segment is the circumferential direction of a curve coordinate system, and the s direction is PzFor z-direction concentrated load, P, on shield segmentssIs a concentrated load in the s direction, Delta theta, to which the segment is subjected(j)、Δw(j)、Δu(j)The relative rotation displacement, the radial relative displacement and the axial relative displacement of the jth shield segment joint are respectively.
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117540480A (en) * | 2024-01-08 | 2024-02-09 | 中铁南方投资集团有限公司 | Method for calculating stress deformation of tunnel lining structure under shield attitude adjustment |
| CN117951799A (en) * | 2024-03-26 | 2024-04-30 | 北京城建设计发展集团股份有限公司 | Long-term treatment method for transverse convergence deformation of subway shield tunnel |
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117540480A (en) * | 2024-01-08 | 2024-02-09 | 中铁南方投资集团有限公司 | Method for calculating stress deformation of tunnel lining structure under shield attitude adjustment |
| CN117540480B (en) * | 2024-01-08 | 2024-04-19 | 中铁南方投资集团有限公司 | Method for calculating stress deformation of tunnel lining structure under shield attitude adjustment |
| CN117951799A (en) * | 2024-03-26 | 2024-04-30 | 北京城建设计发展集团股份有限公司 | Long-term treatment method for transverse convergence deformation of subway shield tunnel |
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