CN116361902A - Method for designing optimal cross section of quasi-rectangular tunnel - Google Patents
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Abstract
The invention provides a design method of an optimal cross section of a rectangular-like tunnel, which can rapidly give out the optimal cross section of the rectangular-like tunnel with a large section by utilizing the efficient calculation advantage of a matrix displacement method in structural mechanics and provides reference for the design of the rectangular-like tunnel. Taking a double-line subway tunnel as an example, firstly determining a limit range required by the tunnel; on the criterion that the tunnel cross-section must completely contain the bounding area, a mathematical expression is derived that can take into account the shape of the tunnel cross-section. By changing the sizes of the cross sections, finding out the cross section corresponding to the minimum internal force in the tunnel lining, namely the optimal cross section form of the rectangular tunnel. The invention can provide efficient theoretical support for the design of the rectangular-like tunnel and has important practical significance for popularization of the application of the rectangular-like tunnel in practical engineering.
Description
Technical Field
The invention belongs to the field of tunnel design and research, and particularly relates to a method for designing an optimal cross section of a rectangular-like tunnel.
Background
The tunnel cross section is subjected to the development process from rectangular to circular to non-circular, and the development trend is to continuously improve the utilization rate of the tunnel cross section and meet various actual engineering requirements. Thus, more and more tunnels of large diameter and different cross-section are applied in practical tunnel engineering, where tunnels, especially in the form of rectangular-like cross-sections, develop most rapidly. However, the current research on the cross section design of the type of tunnel is less, and the related research stays in the model test exploration stage, so that a proper cross section design basis is required to be found for the type of tunnel in order to popularize and apply the rectangular-like tunnel.
Disclosure of Invention
The invention aims to provide a design method of an optimal cross section of a rectangular-like tunnel, which realizes the design of the optimal cross section of the rectangular-like tunnel and simultaneously provides the internal force and displacement value of a lining corresponding to an optimal cross section form. In order to achieve the above purpose, the following technical scheme is adopted:
the design method of the optimal cross section of the rectangular-like tunnel comprises the following steps:
step S1, determining a tunnel building limit M0, including the short axis length D of the limit 0 Length W of long axis 0 The shape is as follows; the shape is an asymmetric structure;
s2, simplifying the tunnel building limit M0 determined in the S1 into a symmetrical shape, wherein the simplified tunnel building limit M1 completely comprises the tunnel building limit M0 before simplification;
s3, constructing a rectangular-like tunnel contour mathematical model A per ：
Step S31, taking 1/4 of the tunnel building limit M1 simplified in the step S2 as a study object, setting the cross section of the quasi-rectangular tunnel:
setting the cross section profile of the 1/4 rectangular tunnel to be composed of a first arc, a second arc and a third arc which are sequentially connected;
setting the circle center O of the arc I 1 Located on the y-axis of the coordinates and having coordinates (0, y) 1 ) Circle center O of arc II 2 The coordinates are (x) 2 ，y 2 ) Circle center O of arc three 3 Is located on the x-axis of the coordinates and has a coordinate (x 3 0), the intersection point A of the arc I and the arc II 1 (d 2 ,h 1 ) Tangent to the arc II and the arc III at an intersection point A 2 (d 1 ,h 2 ) Tangent;
step S32, setting constraint conditions of the cross section of the quasi-rectangular tunnel:
d 1 ＝1/2W 0 ，
h 1 ＝1/2D 0 ；
d 1 -d 2 ＝h 1 -h 2 ＝s 1 ；
step S33, solving the circle center O 2 Center of circle O 1 Concerning x 3 Is represented by the expression:
step S331, solving the center of circle O 2 Concerning x 3 Is represented by the expression:
first, based on the setting in step S31, the center O of the second arc is solved 2 (x 2 ，y 2 ) Line segment A 1 A 2 The vertical equation for the midpoint is:
then solve O 3 A 2 The equation for the straight line is:
finally, due to O 2 The intersection point of the two straight lines is based on the two formulas:
step S332, solving the circle center O 1 With respect tox 3 Is represented by the expression:
first, point A 1 、O 2 Vertical equation solving O in step S331 1 A 1 Expression of the straight line:
thereafter, due to point O 1 (0，y 1 ) On the y-axis of coordinates, the following are available:
step S34, solving θ 1 =angle OO 1 O 2 、θ 2 =angle a 1 O 2 A 2 、θ 3 =angle OO 3 O 2 The radius of the arc I is R 1 The radius of the arc II is R 2 Radius of arc III is R 3 Respectively about x 3 Is represented by the expression:
step S35, solving the simplified cross-sectional area A of the tunnel building boundary M1 rect Rectangular-like tunnel cross section A sr ：
Wherein,,
A s1 ＝θ 1 *R 1 *R 1 /2-d 2 *(h 1 -y 1 )/2；
A s2 ＝θ 2 *R 2 *R 2 /2-R 2 *R 2 *sin(θ 2 /2)*cos(θ 2 /2)；
A s3 ＝θ 3 *R 3 *R 3 /2-h 2 *(d 1 -x 3 )/2；
step S36, solving A per ：
A per ＝(A sr -A rect )/A rect ；
Step S4, regarding x according to step S32 3 Is used to calculate the sum x of the values within the range of values by using the exhaustion method 3 Corresponding y 1 、x 2 、y 2 、R 1 、R 2 、R 3 、θ 1 、θ 2 、θ 3 Is a value of (2);
then, the rectangular tunnel cross section is discretized into a plurality of equally divided beam units, and the beam units are mutually connected through nodes based on the obtained x 3 、y 1 、x 2 、y 2 、R 1 、R 2 、R 3 、θ 1 、θ 2 、θ 3 ObtainingSolving the coordinates of each node;
s5, obtaining the external load F of the tunnel surrounding rock applied to the tunnel corresponding to the tunnel building limit M0 through calculation;
then, based on a matrix displacement method, calculating internal force values at all nodes on the cross section of the quasi-rectangular tunnel, and storing;
step S6, calculating x based on the obtained 3 、y 1 、x 2 、y 2 、R 1 、R 2 、R 3 、θ 1 、θ 2 、θ 3 Each x 3 Corresponding rectangular-like tunnel contour mathematical model A per Is a value of (2); when A is per If the value of (2) is greater than the set value, returning to the execution step S4, otherwise, executing the step S7;
step S7, screening out the minimum internal force value stored in the step S5, wherein the minimum internal force value corresponds to x 3 The determined cross section is the optimal cross section of the quasi-rectangular tunnel.
Preferably, the perpendicular in step S331 equation solving process includes:
triangle O 2 A 1 A 2 Is isosceles triangle, O 2 A 1 Length of (2) is equal to O 2 A 2 So the center O of the second arc 2 (x 2 ，y 2 ) Located at the line segment A 1 A 2 The vertical line of the midpoint;
line segment A 1 A 2 The slope of (2) is:
then cross line segment A 1 A 2 The slope of the midpoint plumb line is +.>
Assume that an line segment A is crossed 1 A 2 The equation of the perpendicular to the midpoint is
Carry-in line segment A 1 A 2 Coordinates of the midpoint
Available->
Preferably, O in step S331 3 A 2 The equation solving process of the straight line comprises the following steps:
let x be 3 Known, then O 3 A 2 The slope of the straight line is
Assuming the O 3 A 2 The straight line equation is
Carry over O 3 Coordinates (x) 3 0), can be obtained
Preferably, in step S332O 1 A 1 The solving process of the expression of the straight line is specifically as follows:
O 2 A 1 with O 1 A 1 The straight lines are identical and are based on point A 1 、O 2 Can calculate the slope
Let it be assumed that line segment O is crossed 1 A 1 Is given by the equation of
Carry over A 1 (d 2 ,h 1 ) The method can obtain:
preferably, x is as for in step S32 3 Is (are) constrainedThe specific solving process of the condition comprises the following steps:
O 2 is line O 3 A 2 Sum line O 1 A 1 Is a cross point of (2);
O 2 is positioned in the left lower region of the arc II and positioned in
On a straight line;
J 1 to be used in
Translate to point A 2 Then, the intersection point of the translation straight line and the x axis;
the cross section outline of the quasi-rectangular tunnel is convex;
bond O 2 The position and the outer convex requirement of the cross section outline of the quasi-rectangular tunnel acquire constraint conditions:
O 3 x of the abscissa of (2) 3 < Point J 1 Is the abscissa of (2)
Preferably J 1 Is the abscissa of (2)
The specific solving process is as follows:
set to A 2 The translation linear equation of (2) is
Carry over A 2 (d 1 ,h 2 ) Can be obtained
Carry over J 1 Obtaining
Preferably, y is the reference in step S32 1 The constraint conditions of (2) are:
O 1 is y of the ordinate of (2) 1 < Point J 2 Is the ordinate of (2)
J 2 To be used in
Translate to point A 1 Thereafter, the translation line intersects the y-axis.
Preferably J 2 Is the ordinate of (2)
The specific solving process is as follows:
set to A 1 The translation linear equation of (2) is
Carry over A 1 (d 2 h 1 ) Can be obtained
Carry over J 2 Obtaining
Preferably, the setting basis in step S31 is specifically:
(1) The simplified tunnel construction boundary M1 is symmetrical about the centroid, and assuming that the coordinates x and y axes pass through the centroid of the graph, the simplified tunnel construction boundary M1 is also symmetrical about the x and y axes;
taking 1/4 of the tunnel construction limit M1 simplified in S2 as the subject:
the circular arc I and the part symmetrical about the y axis are determined by a sector with the same radius and circle center, and the sector is symmetrical about the y axis, then the circle center O of the circular arc I 1 Must lie on the y-axis of the coordinates and have a coordinate of (0, y) 1 )；
The third arc and the part symmetrical to the x axis are formed by radius and circle centerThe same sector is defined and symmetrical about the x-axis, then the centre of the arc O 3 Is located on the x-axis of the coordinates and has a coordinate (x 3 ，0)；
(2) The smoothness of the tunnel contour line is ensured, namely the same tangent line needs to exist at the intersection point A1 of the arc I and the arc II, and the same tangent line needs to exist at the intersection point A2 of the arc II and the arc III;
triangle O 2 A 1 A 2 Is isosceles triangle, so the circle center O of the arc II 2 (x 2 ，y 2 ) Located at the line segment A 1 A 2 The vertical line at the midpoint.
Compared with the prior art, the invention has the advantages that:
taking a double-line subway tunnel as an example, firstly determining a limit range required by the tunnel; on the criterion that the tunnel cross-section must completely contain the bounding area, a mathematical expression is derived that can take into account the shape of the tunnel cross-section. By changing the sizes of the cross sections, finding out the cross section corresponding to the minimum internal force in the tunnel lining, namely the optimal cross section form of the rectangular tunnel. The invention can provide efficient theoretical support for the design of the rectangular-like tunnel and has important practical significance for popularization of the application of the rectangular-like tunnel in practical engineering.
Drawings
FIG. 1 is a flow chart of an optimal cross section design of a rectangular-like tunnel;
FIG. 2 is a schematic diagram of a rectangular-like tunnel lining discrete unit;
FIG. 3 is a schematic diagram of a rectangular-like tunnel building boundary M0;
FIG. 4 is a simplified schematic diagram of a rectangular-like tunnel building boundary M1;
FIG. 5 is a schematic diagram of a rectangular-like tunnel profile mathematical model;
FIG. 6 is a schematic diagram of node coordinate solution;
FIG. 7 is a schematic diagram of the optimal cross-sectional profile of a rectangular-like tunnel designed by the method.
Detailed Description
The method of designing an optimal cross-section for a rectangular-like tunnel according to the present invention will be described in more detail with reference to the accompanying drawings, in which preferred embodiments of the present invention are shown, it being understood that the invention described herein can be modified by those skilled in the art while still achieving the advantageous effects of the invention. Accordingly, the following description is to be construed as broadly known to those skilled in the art and not as limiting the invention.
As shown in fig. 1 to 7, a rectangular-like tunnel optimum cross section design method is completed by MATLAB, and includes the following steps:
s1, ensuring driving safety in a tunnel, and determining the limit that any object cannot invade in a certain height and width range of a tunnel cross section according to the existing tunnel design specification and tunnel construction application.
Determining a tunnel building limit M0, including the short axis length D of the limit 0 Length W of long axis 0 The shape is as follows; the shape is an asymmetric structure. The tunnel construction boundary is 8880mm long, 5500mm high, and irregularly shaped as shown in fig. 3.
Step S2, simplifying the tunnel building limit M0 determined in the step S1 into a symmetrical shape, wherein the simplified tunnel building limit M1 completely comprises the tunnel building limit M0 before simplification. Wherein the length W 0 =8880 mm, width D 0 ＝5500mm，s 1 ＝760mm。
S3, constructing a rectangular-like tunnel contour mathematical model A per 。
As shown in fig. 5, h is known 1 ＝2750mm，h 2 ＝1990mm，d 1 ＝4400mm，d 2 ＝3640mm；
Step S31, taking 1/4 of the tunnel building limit M1 simplified in the step S2 as a study object, setting the cross section of the quasi-rectangular tunnel:
according to the principle that the tunnel contour line is as close to the building limit as possible (the minimum excavation area is sought) and smoothness is indicated, the cross section contour of the 1/4 rectangular tunnel is set to be composed of an arc I, an arc II and an arc III which are sequentially connected;
setting the circle center O of the arc I 1 Located on the y-axis of the coordinates and having coordinates (0, y) 1 ) Circle center O of arc II 2 The coordinates are (x) 2 ，y 2 ) Circle center O of arc three 3 Is located on the x-axis of the coordinates and has a coordinate (x 3 0), the intersection point A of the arc I and the arc II 1 (d 2 ,h 1 ) Tangent to the arc II and the arc III at an intersection point A 2 (d 1 ,h 2 ) And (5) tangential.
The setting basis is specifically as follows:
(1) The simplified tunnel construction boundary M1 is symmetrical about the centroid, and assuming that the coordinates x and y axes pass through the centroid of the graph, the simplified tunnel construction boundary M1 is also symmetrical about the x and y axes;
taking 1/4 of the tunnel construction limit M1 simplified in S2 as the subject:
the circular arc I and the part symmetrical about the y axis are determined by a sector with the same radius and circle center, and the sector is symmetrical about the y axis, then the circle center O of the circular arc I 1 Must lie on the y-axis of the coordinates and have a coordinate of (0, y) 1 )；
The third arc and the part symmetrical about the x-axis are determined by a sector with the same radius and circle center, and the sector is symmetrical about the x-axis, then the circle center O of the third arc 3 Is located on the x-axis of the coordinates and has a coordinate (x 3 ，0)；
(2) The smoothness of the tunnel contour line is ensured, namely the same tangent line needs to exist at the intersection point A1 of the arc I and the arc II, and the same tangent line needs to exist at the intersection point A2 of the arc II and the arc III;
triangle O 2 A 1 A 2 Is isosceles triangle, so the circle center O of the arc II 2 (x 2 ，y 2 ) Located at the line segment A 1 A 2 The vertical line at the midpoint.
Step S32, setting constraint conditions of the cross section of the quasi-rectangular tunnel:
d 1 ＝1/2W 0 ，
h 1 ＝1/2D 0 ；
d 1 -d 2 ＝h 1 -h 2 ＝s 1 ；
regarding x in step S32 3 The specific solving process of the constraint condition of (2) comprises the following steps:
O 2 is line O 3 A 2 Sum line O 1 A 1 Is a cross point of (2);
O 2 is positioned in the left lower region of the arc II and positioned in
On a straight line;
J 1 to be used in
Translate to point A 2 Then, the intersection point of the translation straight line and the x axis;
the cross-sectional profile of the quasi-rectangular tunnel is convex (i.e. x needs to be satisfied 3 <d 1 )；
Bond O 2 The position and the outer convex requirement of the cross section outline of the quasi-rectangular tunnel acquire constraint conditions:
O 3 x of the abscissa of (2) 3 < Point J 1 Is the abscissa of (2)
I.e. x 3 <2410mm。
Wherein J is 1 Is the abscissa of (2)
The specific solving process is as follows:
set to A 2 The translation linear equation of (2) is
Carry over A 2 (d 1 ,h 2 ) Can be obtained
Carry over J 1 Obtaining
If it is
As can be seen from FIG. 5, line O 3 A 2 And straight line->
Will be located at the upper right of the second arc, which will result in the second arc being concave, clearly contrary to the intended target tunnel cross-section.
Similarly, regarding y 1 The constraint conditions of (2) are:
O 1 is y of the ordinate of (2) 1 < Point J 2 Is the ordinate of (2)
J 2 To be used in
Translate to point A 1 Thereafter, the translation line intersects the y-axis.
Wherein J is 2 Is the ordinate of (2)
The specific solving process is as follows:
set to A 1 The translation linear equation of (2) is
Carry over A 1 (d 2 h 1 ) Can be obtained
Carry over J 2 Obtaining
Step S33, solving the circle center O 2 Center of circle O 1 Concerning x 3 Is given (i.e., assuming x3 is known).
Step S331, solving the center of circle O 2 Concerning x 3 Is an expression of (2).
First, based on the setting in step S31, the center O of the second arc is solved 2 (x 2 ，y 2 ) Line segment A 1 A 2 The vertical equation for the midpoint is:
then solve O 3 A 2 The equation for the straight line is:
finally, due to O 2 The intersection point of the two straight lines is based on the two formulas:
the vertical equation solving process comprises the following steps:
triangle O 2 A 1 A 2 Is isosceles triangle, O 2 A 1 Length of (2) is equal to O 2 A 2 So that the length of arc twoCentre of circle O 2 (x 2 ，y 2 ) Located at the line segment A 1 A 2 The vertical line of the midpoint;
line segment A 1 A 2 The slope of (2) is:
then cross line segment A 1 A 2 The slope of the midpoint plumb line is +.>
Assume that an line segment A is crossed 1 A 2 The equation of the perpendicular to the midpoint is
Carry-in line segment A 1 A 2 Coordinates of the midpoint
Available->
O 3 A 2 The equation solving process of the straight line comprises the following steps:
let x be 3 Known, then O 3 A 2 The slope of the straight line is
Assuming the O 3 A 2 The straight line equation is
Carry over O 3 Coordinates (x) 3 0), can be obtained
Step S332, solving the circle center O 1 Concerning x 3 Is given (i.e., assuming x3 is known).
First, point A 1 、O 2 Vertical equation solving O in step S331 1 A 1 Expression of the straight line:
thereafter, due to point O 1 (0，y 1 ) On the y-axis of coordinates, the following are available:
o in step S332 1 A 1 The solving process of the expression of the straight line is specifically as follows:
O 2 A 1 with O 1 A 1 The straight lines are identical and are based on point A 1 、O 2 Can calculate the slope
Let it be assumed that line segment O is crossed 1 A 1 Is given by the equation of
Carry over A 1 (d 2 ,h 1 ) The method can obtain:
step S34, solving θ 1 =angle OO 1 O 2 、θ 2 =angle a 1 O 2 A 2 、θ 3 =angle OO 3 O 2 The radius of the arc I is R 1 The radius of the arc II is R 2 The radius of the arc III is R 3 Respectively about x 3 Is given (i.e., assuming x3 is known).
Step S35, solving the simplified cross-sectional area A of the tunnel building boundary M1 rect Rectangular-like tunnel cross section A sr 。
Wherein,,
A s1 ＝θ 1 *R 1 *R 1 /2-d 2 *(h 1 -y 1 )/2；
A s2 ＝θ 2 *R 2 *R 2 /2-R 2 *R 2 *sin(θ 2 /2)*cos(θ 2 /2)；
A s3 ＝θ 3 *R 3 *R 3 /2-h 2 *(d 1 -x 3 )/2；
step S36, solving A per ：
A per ＝(A sr -A rect )/A rect ；
Step S4, regarding x according to step S32 3 Is used to calculate the sum x of the constraint conditions (definition domain) of the (B) by using the exhaustion method 3 Corresponding y 1 、x 2 、y 2 、R 1 、R 2 、R 3 、θ 1 、θ 2 、θ 3 Is a value of (2). At x 3 In the definition domain, the iteration step length is 0.01, and x is continuously changed 3 Values.
Thereafter, the rectangular tunnel cross section is discretized into several equally divided beam units (x is taken 3 The cross section of the quasi-rectangular tunnel determined for constant value is discretized into a cross section formed by a certain number of beam units), the beam units are connected with each other through nodes, and the cross section is based on the obtained x 3 、y 1 、x 2 、y 2 、R 1 、R 2 、R 3 、θ 1 、θ 2 、θ 3 And solving the coordinates of each node. Knowing the coordinates of the node, the external load F applied to the node by the surrounding rock can be determined according to the burial depth position. (Prior Art)
As shown in fig. 6, the node coordinate calculation exemplifies: at x 3 Known to calculate a unique corresponding parameter y 1 ，x 2 ,y 2 ,R 1 ,R 2 ,R 3 ,θ 1 ,θ 2 ,θ 3 Values. Suppose that in FIG. 5 arc 1 is divided into 5 beam units, namely arc A 0 A 1 4 nodes are respectively numbered as P1, P2, P3 and P4, and the four nodes are respectively connected with the point A 0 And point A 1 At the same time on the arc 1, theta 1 Is equally divided into 5 points, point A 0 And point A 1 Coordinates, theta of 1 、y 1 、R 1 Are all known, e.g. point A 0 The coordinates are (0, R) 1 +y 1 ) The abscissa of the P1 point is R 1 *sin(θ 1 5), the ordinate is R 1 *cos(θ 1 /5)+y 1 And so onTo obtain the position coordinates of P2, P3 and P4 according to the geometric relationship.
And S5, calculating to obtain the external load F (which can be obtained in the prior art) of the tunnel surrounding rock applied to the tunnel corresponding to the tunnel building limit M0.
And then calculating and storing internal force values at each node on the cross section of the quasi-rectangular tunnel based on a matrix displacement method (prior art).
The matrix displacement method has the advantages of high calculation efficiency and reliable result. Therefore, the method is particularly suitable for the design process of the initial stage of tunnel construction. But the matrix displacement method was previously used more for the design of circular or horseshoe tunnel cross-sections and was not applied to the design of rectangular-like tunnel cross-sections. Therefore, by considering the special cross section form of the quasi-rectangle, the invention provides how to design the cross section of the quasi-rectangle tunnel by using the matrix displacement method.
Specific:
[K] [ S ] = [ F ] (Prior Art)
F＝[F 1 ,F 2 ,…,F n ] T To be applied to the tunnel lining structure at each junction comprises 3 external force vectors, namely F i Comprises X-direction external force, Y-direction external force and bending moment as input parameters.
S＝[S 1 ,S 2 ,…,S n ] T, Comprising 3 displacement vectors for each node, i.e. S i Including X-direction displacement, Y-direction displacement and corner displacement; f= [ F ] 1 ,F 2 ,…,F n ] T Is 3 external force vectors on each node;
n is the number of nodes; i=1 to n.
K is the overall stiffness matrix, and is known to be mainly dependent on four parameters, namely the elastic modulus E of the lining structural material, the area A of the lining section, the moment of inertia I of the section and the length Li of the dividing unit. I.e. the overall stiffness matrix K can be uniquely determined as long as the lining material and thickness are known and the number of dividing elements is known.
From the prior art, it is known that: before the matrix displacement method in structural mechanics is calculated, an integral rigidity matrix K is needed, and the rigidity matrix K is needed to be converted into an integral coordinate system by a unit under a local coordinate system and then is assembled.
Similarly, the [ S ] calculated by [ K ] = [ S ] = [ F ] is under the global coordinate system, and the output internal force is required to be under the local coordinate system, so that the [ S ] under the global coordinate system needs to be converted into the local coordinate system, and then the internal force of the unit under the local coordinate system is obtained by multiplying the displacement under the local coordinate system by the rigidity matrix under the local coordinate system.
Specifically, based on a matrix displacement method in structural mechanics, beam unit information obtained by tunnel profile discretization is utilized to calculate a rigidity matrix of each beam unit under a local coordinate system, the rigidity matrix of the beam unit under an integral coordinate system is obtained through coordinate conversion, rigidity matrixes of all units under the integral coordinate system are collected, and then a total rigidity matrix K of all units under the integral coordinate system is obtained.
And calculating to obtain the external load F applied to the quasi-rectangular tunnel by the surrounding rock of the tunnel, and calculating to obtain the displacement value of each beam unit node under the integral coordinate system according to the formula [ K ] [ S ] = [ F ].
And converting the displacement value of the lower beam unit of the integral coordinate system into the displacement value under the local coordinate system, and combining the rigidity matrix of the units under the local coordinate system, and obtaining the internal force value on each unit node by utilizing a matrix displacement method.
In fig. 1, M is a bending moment of the lining (at each node), and N is an axial force (internal force) of the lining (at each node).
Step S6, calculating x based on the obtained 3 、y 1 、x 2 、y 2 、R 1 、R 2 、R 3 、θ 1 、θ 2 、θ 3 Each x 3 Corresponding rectangular-like tunnel contour mathematical model A per Is a value of (2); when A is per If the value of (2) is greater than 5%, the process returns to the step S4, otherwise, the step S7 is executed.
Step S7, screening out the minimum internal force value stored in the step S5, wherein the minimum internal force value corresponds to x 3 The determined cross section is the optimal cross section of the quasi-rectangular tunnel. Specifically, the Max function of the MATLAB is utilized to find outMinimum internal force value, corresponding to x 3 The determined section is the optimal cross section of the quasi-rectangular tunnel, as shown in fig. 7.
The foregoing is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Any person skilled in the art will make any equivalent substitution or modification to the technical solution and technical content disclosed in the invention without departing from the scope of the technical solution of the invention, and the technical solution of the invention is not departing from the scope of the invention.
Claims (9)
1. The design method of the optimal cross section of the rectangular-like tunnel is characterized by comprising the following steps of:
step S1, determining a tunnel building limit M0, including the short axis length D of the limit 0 Length W of long axis 0 The shape is as follows; the shape is an asymmetric structure;
s2, simplifying the tunnel building limit M0 determined in the S1 into a symmetrical shape, wherein the simplified tunnel building limit M1 completely comprises the tunnel building limit M0 before simplification;
s3, constructing a rectangular-like tunnel contour mathematical model A per ：
Step S31, taking 1/4 of the tunnel building limit M1 simplified in the step S2 as a study object, setting the cross section of the quasi-rectangular tunnel:
setting the cross section profile of the 1/4 rectangular tunnel to be composed of a first arc, a second arc and a third arc which are sequentially connected;
setting the circle center O of the arc I 1 Located on the y-axis of the coordinates and having coordinates (0, y) 1 ) Circle center O of arc II 2 The coordinates are (x) 2 ，y 2 ) Circle center O of arc three 3 Is located on the x-axis of the coordinates and has a coordinate (x 3 0), the intersection point A of the arc I and the arc II 1 (d 2 ,h 1 ) Tangent to the arc II and the arc III at an intersection point A 2 (d 1 ,h 2 ) Tangent;
step S32, setting constraint conditions of the cross section of the quasi-rectangular tunnel:
d 1 ＝1/2W 0 ，
h 1 ＝1/2D 0 ；
d 1 -d 2 ＝h 1 -h 2 ＝s 1 ；
step S33, solving the circle center O 2 Center of circle O 1 Concerning x 3 Is represented by the expression:
step S331, solving the center of circle O 2 Concerning x 3 Is represented by the expression:
first, based on the setting in step S31, the center O of the second arc is solved 2 (x 2 ，y 2 ) Line segment A 1 A 2 The vertical equation for the midpoint is:
then solve O 3 A 2 The equation for the straight line is:
finally, due to O 2 The intersection point of the two straight lines is based on the two formulas:
step S332, solving the circle center O 1 Concerning x 3 Is represented by the expression:
first, point A 1 、O 2 Vertical equation solving O in step S331 1 A 1 Expression of the straight line:
thereafter, due to point O 1 (0，y 1 ) On the y-axis of coordinates, the following are available:
step S34, solving θ 1 =angle OO 1 O 2 、θ 2 =angle a 1 O 2 A 2 、θ 3 =angle OO 3 O 2 The radius of the arc I is R 1 The radius of the arc II is R 2 Radius of arc III is R 3 Respectively about x 3 Is represented by the expression:
step S35, solving the simplified cross-sectional area A of the tunnel building boundary M1 rect Rectangular-like tunnel cross section A sr ：
Wherein,,
A s1 ＝θ 1 *R 1 *R 1 /2-d 2 *(h 1 -y 1 )/2；
A s2 ＝θ 2 *R 2 *R 2 /2-R 2 *R 2 *sin(θ 2 /2)*cos(θ 2 /2)；
A s3 ＝θ 3 *R 3 *R 3 /2-h 2 *(d 1 -x 3 )/2；
step S36, solving A per ：
A per ＝(A sr -A rect )/A rect ；
Step S4, regarding x according to step S32 3 Is used to calculate the sum x of the values within the range of values by using the exhaustion method 3 Corresponding y 1 、x 2 、y 2 、R 1 、R 2 、R 3 、θ 1 、θ 2 、θ 3 Is a value of (2);
then, the rectangular tunnel cross section is discretized into a plurality of equally divided beam units, and the beam units are mutually connected through nodes based on the obtained x 3 、y 1 、x 2 、y 2 、R 1 、R 2 、R 3 、θ 1 、θ 2 、θ 3 Solving the coordinates of each node;
s5, obtaining the external load F of the tunnel surrounding rock applied to the tunnel corresponding to the tunnel building limit M0 through calculation;
then, based on a matrix displacement method, calculating internal force values at all nodes on the cross section of the quasi-rectangular tunnel, and storing;
step S6, calculating x based on the obtained 3 、y 1 、x 2 、y 2 、R 1 、R 2 、R 3 、θ 1 、θ 2 、θ 3 Rectangular-like tunnel contour mathematical model A corresponding to each x3 per Is a value of (2); when A is per If the value of (2) is greater than the set value, returning to the execution step S4, otherwise, executing the step S7;
step S7, screening out the minimum internal force value stored in the step S5, wherein the minimum internal force value corresponds to x 3 The determined cross section is the optimal cross section of the quasi-rectangular tunnel.
2. The method for designing an optimal cross section of a rectangular-like tunnel according to claim 1, wherein the perpendicular-in-the-vertical equation solving process in step S331 comprises:
triangle O 2 A 1 A 2 Is isosceles triangle, O 2 A 1 Length of (2) is equal to O 2 A 2 So the center O of the second arc 2 (x 2 ，y 2 ) Located at the line segment A 1 A 2 The vertical line of the midpoint;
line segment A 1 A 2 The slope of (2) is:
then cross line segment A 1 A 2 The slope of the midpoint plumb line is +.>
Assume that an line segment A is crossed 1 A 2 Perpendicular to the midpointCheng Wei
Carry-in line segment A 1 A 2 Coordinates of the midpoint
Available->
3. The method for designing an optimal cross-section of a rectangular-like tunnel according to claim 1, wherein O in step S331 3 A 2 The equation solving process of the straight line comprises the following steps:
let x be 3 Known, then O 3 A 2 The slope of the straight line is
Assuming the O 3 A 2 The straight line equation is
Carry over O 3 Coordinates (x) 3 0), can be obtained
4. The method for designing an optimal cross-section of a rectangular-like tunnel according to claim 1, wherein O in step S332 1 A 1 The solving process of the expression of the straight line is specifically as follows:
O 2 A 1 with O 1 A 1 The straight lines are identical and are based on point A 1 、O 2 Can calculate the slope
Let it be assumed that line segment O is crossed 1 A 1 Is given by the equation of
Carry over A 1 (d 2 ,h 1 ) The method can obtain:
5. the method for designing an optimal cross-section of a rectangular-like tunnel according to claim 1, wherein x is as defined in step S32 3 The specific solving process of the constraint condition of (2) comprises the following steps:
O 2 is line O 3 A 2 Sum line O 1 A 1 Is a cross point of (2);
O 2 is positioned in the left lower region of the arc II and positioned in
On a straight line;
J 1 to be used in
Translate to point A 2 Then, the intersection point of the translation straight line and the x axis;
the cross section outline of the quasi-rectangular tunnel is convex;
bond O 2 The position and the outer convex requirement of the cross section outline of the quasi-rectangular tunnel acquire constraint conditions:
O 3 x of the abscissa of (2) 3 < Point J 1 Is the abscissa of (2)
6. The method for designing an optimal cross-section of a rectangular tunnel according to claim 5, wherein J 1 Is the abscissa of (2)
The specific solving process is as follows:
set to A 2 The translation linear equation of (2) is
Carry over A 2 (d 1 ,h 2 ) Can be obtained
Carry over J 1 Obtaining J 1 Is the abscissa of (2)
7. The method for designing an optimal cross-section of a rectangular-like tunnel according to claim 1, wherein y is defined in step S32 1 The constraint conditions of (2) are:
O 1 is y of the ordinate of (2) 1 < Point J 2 Is the ordinate of (2)
J 2 To be used in
Translate to point A 1 Thereafter, the translation line intersects the y-axis.
8. The method for designing an optimal cross-section of a rectangular tunnel according to claim 7, wherein J 2 Is the ordinate of (2)
The specific solving process is as follows:
set to A 1 The translation linear equation of (2) is
Carry over A 1 (d 2 h 1 ) Can be obtained
Carry over J 2 Obtaining J 2 Is the ordinate of (2)
9. The method for designing an optimal cross section of a rectangular tunnel according to claim 1, wherein the setting in step S31 is specifically:
(1) The simplified tunnel construction boundary M1 is symmetrical about the centroid, and assuming that the coordinates x and y axes pass through the centroid of the graph, the simplified tunnel construction boundary M1 is also symmetrical about the x and y axes;
taking 1/4 of the tunnel construction limit M1 simplified in S2 as the subject:
the circular arc I and the part symmetrical about the y axis are determined by a sector with the same radius and circle center, and the sector is symmetrical about the y axis, then the circle center O of the circular arc I 1 Must lie on the y-axis of the coordinates and have a coordinate of (0, y) 1 )；
The third arc and the part symmetrical about the x-axis are determined by a sector with the same radius and circle center, and the sector is symmetrical about the x-axis, then the circle center O of the third arc 3 Is located on the x-axis of the coordinates and has a coordinate (x 3 ，0)；
(2) The smoothness of the tunnel contour line is ensured, namely the same tangent line needs to exist at the intersection point A1 of the arc I and the arc II, and the same tangent line needs to exist at the intersection point A2 of the arc II and the arc III;
triangle O 2 A 1 A 2 Is equal toWaist triangle, so circle center O of arc two 2 (x 2 ，y 2 ) Located at the line segment A 1 A 2 The vertical line at the midpoint.
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