Design method of optimal cross section of quasi-rectangular tunnel

Abstract

The application provides a kind of rectangular tunnel optimal cross section design method, using the efficient calculation advantage of matrix displacement method in structural mechanics can quickly give the optimal cross section of large section rectangular tunnel, provide reference for rectangular tunnel design.Taking double-track subway tunnel as an example, first determine the necessary limit range of the tunnel;With the tunnel cross section must completely contain limit area as the criterion, deduce the mathematical expression that can consider the shape of tunnel cross section.By changing the size of different cross section, find the corresponding cross section when the internal force in tunnel lining is minimum, which is the optimal cross section form of the rectangular tunnel.The application can provide efficient theoretical support for the design of rectangular tunnel, and has important practical significance for promoting the application of rectangular tunnel in practical engineering.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of tunnel design and research, and particularly relates to a method for designing an optimal cross section of a quasi-rectangular tunnel. BACKGROUND

[0002] The cross section form of a tunnel has undergone a development process from a rectangle to a circle and then to a non-circle, and the development trend is to continuously improve the utilization rate of the tunnel cross section and meet various actual engineering requirements. Therefore, more and more tunnels with large diameters and different cross section forms are applied to actual tunnel engineering, among which the tunnels with quasi-rectangular cross section forms develop the most rapidly. However, the research on the design of the cross section of the tunnels of this type is less, and the related research is mostly limited to the model test exploration stage. In order to popularize and apply the quasi-rectangular tunnels, it is necessary to find a suitable design basis for the cross section of the tunnels of this type. SUMMARY

[0003] The application aims to provide a method for designing an optimal cross section of a quasi-rectangular tunnel, to realize the design of an optimal cross section of a quasi-rectangular tunnel, and to give the internal force and displacement values of the lining corresponding to the optimal cross section form. In order to achieve the above-mentioned purpose, the following technical scheme is adopted:

[0004] A method for designing an optimal cross section of a quasi-rectangular tunnel, comprising the following steps:

[0005] Step S1, determining a tunnel building limit M0, including the short axis length D0, the long axis length W0 and the shape of the limit; the shape is a non-symmetrical structure;

[0006] Step S2, simplifying the tunnel building limit M0 determined in S1 into a symmetrical shape, and the simplified tunnel building limit M1 completely contains the tunnel building limit M0 before simplification;

[0007] Step S3, constructing a quasi-rectangular tunnel contour mathematical model A per :

[0008] Step S31, taking 1 / 4 of the simplified tunnel building limit M1 in S2 as a research object, and setting a quasi-rectangular tunnel cross section:

[0009] Setting the 1 / 4 quasi-rectangular tunnel cross section contour to be composed of a circular arc one, a circular arc two and a circular arc three connected in sequence;

[0010] Setting the center O1 of the circular arc one to be located on the coordinate y axis and the coordinate to be (0, y1), the center O2 of the circular arc two to be (x2, y2), and the center O3 of the circular arc three to be located on the coordinate x axis and the coordinate to be (x3, 0), the circular arc one to be tangent to the circular arc two at the intersection point A1(d2, h1), and the circular arc two to be tangent to the circular arc three at the intersection point A2(d1, h2);

[0011] Step S32, setting the constraint condition of the quasi-rectangular tunnel section:

[0012]

[0013] d1 = 1 / 2W0,

[0014] h1 = 1 / 2D0;

[0015] d1 - d2 = h1 - h2 = s1;

[0016] Step S33, solving the expression of the center O2 of the circle and the center O1 about x3:

[0017] Step S331, solving the expression of the center O2 about x3:

[0018] First, based on the setting in step S31, the equation of the perpendicular line of the midpoint of the line segment A1A2 where the center O2(x2, y2) of the circle arc two is located is solved as:

[0019]

[0020] Then, the equation of the straight line where O3A2 is located is solved as:

[0021]

[0022] Finally, since O2 is the intersection point of the above two straight lines, based on the above two equations, it is known that:

[0023]

[0024]

[0025] Step S332, solving the expression of the center O1 about x3:

[0026] First, the expression of the straight line where O1A1 is located is solved based on the perpendicular line equation in step S331 and the point A1 and O2:

[0027]

[0028] Then, since the point O1(0, y1) is located on the coordinate y-axis, it is known that:

[0029]

[0030] Step S34, solving the expressions of θ1 = angle OO1O2, θ2 = angle A1O2A2, θ3 = angle OO3O2, the radius of the circle arc one is R1, the radius of the circle arc two is R2, and the radius of the circle arc three is R3 about x3, respectively:

[0031]

[0032]

[0033]

[0034]

[0035]

[0036]

[0037] Step S35, solving the cross-sectional area A of the simplified tunnel building limit M1 rect , rectangular tunnel cross section A sr :

[0038]

[0039]

[0040] wherein,

[0041] A s1 = θ1*R1*R1 / 2-d2*(h1-y1) / 2;

[0042] A s2 = θ2*R2*R2 / 2-R2*R2*sin(θ2 / 2)*cos(θ2 / 2);

[0043] A s3 = θ3*R3*R3 / 2-h2*(d1-x3) / 2;

[0044] Step S36, solving A per :

[0045] A per = (A sr -A rect ) / A rect ;

[0046] Step S4, according to the constraint condition of x3 in step S32, the value of x3 is taken in its value range by using the exhaustion method, and the values of y1, x2, y2, R1, R2, R3, θ1, θ2, θ3 corresponding to x3 are calculated;

[0047] Then, the rectangular tunnel cross section is discretized into several equal beam units, the beam units are connected to each other through nodes, and based on the obtained x3, y1, x2, y2, R1, R2, R3, θ1, θ2, θ3, the coordinates of each node are solved;

[0048] Step S5, the external load F applied by the tunnel surrounding rock to the tunnel corresponding to the tunnel building limit M0 is obtained by calculation;

[0049] Then, the internal force values at each node on the cross section of the quasi-rectangular tunnel are calculated based on the matrix displacement method, and are saved;

[0050] Step S6, the value of A per is calculated based on the acquired x3, y1, x2, y2, R1, R2, R3, θ1, θ2, θ3, each x3 corresponding to a quasi-rectangular tunnel profile mathematical model A per ; when the value of A per is greater than a set value, return to execute step S4, otherwise, execute step S7;

[0051] Step S7, the smallest internal force value saved in step S5 is screened out, and the cross section determined by the corresponding x3 is the optimal cross section of the quasi-rectangular tunnel.

[0052] Preferably, the process of solving the equation of the perpendicular line in step S331 includes:

[0053] The triangle O2A1A2 is an isosceles triangle, and the length of O2A1 is equal to the length of O2A2, so the center O2(x2, y2) of the circular arc two is located on the perpendicular line passing through the midpoint of the line segment A1A2;

[0054] The slope of the line segment A1A2 is: The slope of the perpendicular line passing through the midpoint of the line segment A1A2 is

[0055] Suppose the equation of the perpendicular line passing through the midpoint of the line segment A1A2 is: Substitute the coordinates of the midpoint of the line segment A1A2 , and the following can be obtained:

[0056] Preferably, the process of solving the equation of the straight line where O3A2 is located in step S331 includes:

[0057] Suppose x3 is known, then the slope of the straight line where O3A2 is located is

[0058] Suppose the equation of the straight line where O3A2 is located is:

[0059] Substitute the coordinates of O3(x3, 0), and the following can be obtained:

[0060] Preferably, the process of solving the expression of the straight line where O1A1 is located in step S332 includes:

[0061] O2A1 and O1A1 are located on the same straight line, and based on points A1 and O2, the slope

[0062] Suppose the equation of the straight line passing through the line segment O1A1 is: Bring into A1(d2, h1) can get:

[0063]

[0064] Preferably, the constraint condition about x3 in step S32, the specific solving process includes:

[0065] O2 is the intersection of line O3A2 and line O1A1;

[0066] O2 is located in the lower left area of the circular arc two, and is located On a straight line;

[0067] J1 is the intersection of the translation line and the x-axis after the translation of to point A2;

[0068] The cross-sectional profile of the rectangular-like tunnel is convex outward;

[0069] Combined with the position of O2 and the convex requirement of the cross-sectional profile of the rectangular-like tunnel, the constraint condition is obtained:

[0070] The abscissa x3 of O3 is less than the abscissa of point J1

[0071] Preferably, the abscissa of J1 is The specific solving process is:

[0072] The translation line equation passing through A2 is set as Bring into A2(d1, h2), can get

[0073] Bring into J1, get

[0074] Preferably, the constraint condition about y1 in step S32 is:

[0075] The ordinate y1 of O1 is less than the ordinate of point J2

[0076] J2 is the intersection of the translation line and the y-axis after the translation of to point A1.

[0077] Preferably, the ordinate of J2 is The specific solving process is:

[0078] The translation line equation passing through A1 is set as Bring into A1(d2h1), can get

[0079] Bring into J2, get

[0080] Preferably, the specific set in step S31 is:

[0081] (1) Simplified tunnel building limit M1 is symmetrical about the centroid, assuming that coordinate x and y axes pass through the centroid of the graph, then the simplified tunnel building limit M1 is also symmetrical about the x axis and the y axis;

[0082] When taking 1 / 4 of the simplified tunnel building limit M1 in S2 as the research object:

[0083] The circular arc one and its symmetrical part about the y axis are determined by a sector with the same radius and center, and the sector is symmetrical about the y axis, so the center O1 of the circular arc one must be located on the coordinate y axis and the coordinate is (0, y1);

[0084] The circular arc three and its symmetrical part about the x axis are determined by a sector with the same radius and center, and the sector is symmetrical about the x axis, so the center O3 of the circular arc three is located on the coordinate x axis and the coordinate is (x3, 0);

[0085] (2) To ensure the smoothness of the tunnel contour line, that is, the same tangent line should exist at the intersection point A1 of the circular arc one and the circular arc two, and the same tangent line should exist at the intersection point A2 of the circular arc two and the circular arc three;

[0086] The triangle O2A1A2 is an isosceles triangle, so the center O2 (x2, y2) of the circular arc two is located on the vertical line passing through the midpoint of the line segment A1A2.

[0087] Compared with the prior art, the advantages of the present application are:

[0088] Taking a double-track subway tunnel as an example, first, the necessary limit range of the tunnel is determined; taking the criterion that the tunnel cross section must completely contain the limit area, a mathematical expression considering the shape of the tunnel cross section is derived. By changing the size of the different cross sections, the cross section corresponding to the minimum internal force in the tunnel lining is found, which is the optimal cross section form of the rectangular tunnel. The present application can provide efficient theoretical support for the design of rectangular tunnels, and has important practical significance for promoting the application of rectangular tunnels in practical engineering. BRIEF DESCRIPTION OF DRAWINGS

[0089] Figure 1 Flow chart for optimal cross section design of rectangular tunnel;

[0090] Figure 2 Schematic diagram of discrete elements of tunnel lining of rectangular tunnel;

[0091] Figure 3 Schematic diagram of tunnel building limit M0 of rectangular tunnel;

[0092] Figure 4 Schematic diagram of simplified tunnel building limit M1 of rectangular tunnel;

[0093] Figure 5 Figure is a schematic diagram of a mathematical model of a quasi-rectangular tunnel profile;

[0094] Figure 6 Figure is a schematic diagram of a solution of node coordinates;

[0095] Figure 7 Figure is a schematic diagram of an optimal cross-sectional profile of a quasi-rectangular tunnel designed using the method. DETAILED DESCRIPTION

[0096] The quasi-rectangular tunnel optimal cross-sectional design method of the present application will be described in more detail below with reference to the accompanying schematic diagrams, which show preferred embodiments of the present application, it being understood that those skilled in the art can modify the present application described herein while still achieving the advantageous effects of the present application. Therefore, the following description should be understood as a broad general knowledge of those skilled in the art and not as a limitation on the present application.

[0097] As shown in Figures 1-7 , a quasi-rectangular tunnel optimal cross-sectional design method is completed by MATLAB and includes the following steps:

[0098] Step S1, to ensure driving safety in the tunnel, the boundary within which any object cannot intrude in the tunnel cross-sectional height and width range determined according to the existing tunnel design specifications and the purpose of tunnel construction.

[0099] Determine the tunnel construction boundary M0, including the short axis length D0, the long axis length W0 and the shape of the boundary; the shape is an asymmetric structure. As shown in Figure 3 the tunnel construction boundary is 8880 mm long and 5500 mm high, and the shape is irregular.

[0100] Step S2, simplify the tunnel construction boundary M0 determined in S1 to a symmetrical shape, and the simplified tunnel construction boundary M1 completely contains the simplified tunnel construction boundary M0. Wherein the length W0=8880 mm, the width D0=5500 mm, and s1=760 mm.

[0101] Step S3, construct a quasi-rectangular tunnel profile mathematical model A per .

[0102] As shown in Figure 5 , it is known that h1=2750 mm, h2=1990 mm, d1=4400 mm, and d2=3640 mm;

[0103] Step S31, take 1 / 4 of the simplified tunnel construction boundary M1 in S2 as the research object, and set the quasi-rectangular tunnel cross section:

[0104] According to the principle that the tunnel contour line should be as close as possible to the building limit (minimize the excavation area) and indicate smoothness, the 1 / 4 rectangular tunnel cross section profile is set to be composed of sequentially connected circular arc one, circular arc two and circular arc three;

[0105] The center O1 of the circular arc one is located on the coordinate y-axis and has coordinates (0, y1), the center O2 of the circular arc two has coordinates (x2, y2), and the center O3 of the circular arc three is located on the coordinate x-axis and has coordinates (x3, 0). The circular arc one and the circular arc two are tangent at the intersection point A1 (d2, h1), and the circular arc two and the circular arc three are tangent at the intersection point A2 (d1, h2).

[0106] Among them, the specific settings are:

[0107] (1) The simplified tunnel building limit M1 is symmetrical about the centroid, assuming that the coordinate x and y axes pass through the centroid of the figure, then the simplified tunnel building limit M1 is also symmetrical about the x and y axes;

[0108] When taking 1 / 4 of the simplified tunnel building limit M1 in S2 as the research object:

[0109] The circular arc one and its symmetrical part about the y-axis are determined by a sector with the same radius and center, and the sector is symmetrical about the y-axis, so the center O1 of the circular arc one must be located on the coordinate y-axis and has coordinates (0, y1);

[0110] The circular arc three and its symmetrical part about the x-axis are determined by a sector with the same radius and center, and the sector is symmetrical about the x-axis, so the center O3 of the circular arc three is located on the coordinate x-axis and has coordinates (x3, 0);

[0111] (2) To ensure the smoothness of the tunnel contour line, that is, the circular arc one and the circular arc two should have the same tangent at the intersection point A1, and the circular arc two and the circular arc three should have the same tangent at the intersection point A2;

[0112] The triangle O2A1A2 is an isosceles triangle, so the center O2 (x2, y2) of the circular arc two is located on the perpendicular line passing through the midpoint of the line segment A1A2.

[0113] Step S32, set the constraint conditions of the rectangular tunnel section:

[0114]

[0115]

[0116] d1 = 1 / 2W0,

[0117] h1 = 1 / 2D0;

[0118] d1-d2 = h1-h2 = s1;

[0119] The specific solution process for the constraint condition of x3 in step S32 includes:

[0120] O2 is the intersection of line O3A2 and line O1A1;

[0121] O2 is located in the lower left region of arc two, and is located in On a straight line;

[0122] J1 is the general After translating to point A2, the intersection of the translation line and the x-axis;

[0123] The cross-sectional profile of the rectangular tunnel is convex (i.e., it needs to satisfy x3). <d1);

[0124] Based on the location described in O2 and the outward convexity requirement of the rectangular tunnel cross-section profile, the constraint conditions are obtained as follows:

[0125] The x-coordinate of point O3 is x3 < the x-coordinate of point J1.

[0126] That is, x3 < 2410mm.

[0127] Where, the x-coordinate of J1 The specific solution process is as follows:

[0128] Let the equation of the translation line passing through A2 be: Substituting A2(d1,h2), we get

[0129] Substituting J1, we get

[0130] like according to Figure 5 It can be seen that line O3A2 and line The intersection point will be located to the upper right of arc two, causing arc two to be concave, which is obviously the opposite of the desired tunnel cross section.

[0131] Similarly, the constraints on y1 are:

[0132] The ordinate y1 of point O1 is less than the ordinate of point J2.

[0133] J2 is the general After translating to point A1, the intersection of the translation line and the y-axis.

[0134] Where, the ordinate of J2 The specific solution process is as follows:

[0135] Let the equation of the translation line passing through A1 be: Substituting A1(d2h1), we get

[0136] Substitute J2 into the equation, we get

[0137] Step S33, solve the expression of the center O2 and the center O1 about x3 (i.e. assuming x3 is known).

[0138] Step S331, solve the expression of the center O2 about x3.

[0139] First, based on the setting in step S31, the equation of the vertical line passing through the midpoint of the line segment A1A2 in which the center O2 (x2, y2) of the circular arc two is located is:

[0140]

[0141] Then, the equation of the straight line in which O3A2 is located is:

[0142]

[0143] Finally, since O2 is the intersection point of the above two straight lines, based on the above two equations, we know that:

[0144]

[0145]

[0146] The process of solving the vertical line equation includes:

[0147] The triangle O2A1A2 is an isosceles triangle, and the length of O2A1 is equal to the length of O2A2, so the center O2 (x2, y2) of the circular arc two is located on the vertical line passing through the midpoint of the line segment A1A2;

[0148] The slope of the line segment A1A2 is: Therefore, the slope of the vertical line passing through the midpoint of the line segment A1A2 is

[0149] Assuming the equation of the vertical line passing through the midpoint of the line segment A1A2 is Substitute the coordinates of the midpoint of the line segment A1A2 We get

[0150] The process of solving the equation of the straight line in which O3A2 is located includes:

[0151] Assuming x3 is known, the slope of the straight line in which O3A2 is located is

[0152] Assuming the equation of the straight line in which O3A2 is located is

[0153] Substitute the coordinates of O3 into the equation of the circle, we get

[0154] Step S332, solve the expression of the center O1 about x3 (i.e. assuming x3 is known).

[0155] First, the point A1, O2, the vertical line equation in step S331 to solve the expression of the line where O1A1 is located:

[0156]

[0157] Then, since the point O1 (0, y1) is located on the coordinate y-axis, we get:

[0158]

[0159] The expression solving process of the line where O1A1 is located in step S332 is as follows:

[0160] O2A1 and O1A1 are located on the same line, based on the points A1 and O2, the slope

[0161] Assume the equation of the line segment O1A1 is Substitute A1 (d2, h1) to get:

[0162]

[0163] Step S34, solve the expression of θ1 = angle OO1O2, θ2 = angle A1O2A2, θ3 = angle OO3O2, the radius of the first circular arc R1, the radius of the second circular arc R2, and the radius of the third circular arc R3 about x3 (i.e. assuming x3 is known).

[0164]

[0165]

[0166]

[0167]

[0168]

[0169]

[0170] Step S35, solve the cross-sectional area A of the simplified tunnel construction limit M1 rect , the cross-sectional area A of the rectangular tunnel sr .

[0171]

[0172]

[0173] wherein,

[0174] A s1 = θ1 * R1 * R1 / 2 - d2 * (h1 - y1) / 2;

[0175] A s2 = θ2 * R2 * R2 / 2 - R2 * R2 * sin(θ2 / 2) * cos(θ2 / 2);

[0176] A s3 = θ3 * R3 * R3 / 2 - h2 * (d1 - x3) / 2;

[0177] Step S36, solving A per :

[0178] A per = (A sr - A rect ) / A rect ;

[0179] Step S4, according to the constraint condition (definition domain) of x3 in step S32, the value of y1, x2, y2, R1, R2, R3, θ1, θ2, θ3 corresponding to x3 is calculated by using the exhaustion method in its value range. In the definition domain of x3, the iterative step length is 0.01, and the value of x3 is changed constantly.

[0180] After that, the rectangular tunnel cross section is discretized into several equal beam elements (the cross section composed of a certain number of beam elements discretized from the rectangular tunnel cross section determined when x3 is a constant value), each beam element is connected to each other through the node, and based on the obtained x3, y1, x2, y2, R1, R2, R3, θ1, θ2, θ3, the coordinates of each node are solved. After the node coordinates are known, the external load F applied to the node by the surrounding rock can be determined according to the buried depth position of the node. (Prior art)

[0181] As Figure 6 shown, the node coordinate calculation example: when x3 is known, the unique parameter y1, x2, y2, R1, R2, R3, θ1, θ2, θ3 value corresponding to it is calculated. It is assumed Figure 5The middle circular arc 1 is divided into 5 beam units, that is, there are 4 nodes between the circular arcs A0A1, which are numbered as P1, P2, P3 and P4, since the four points and points A0 and A1 are located on the circular arc 1 simultaneously, θ1 is divided into 5 parts, the coordinates of points A0 and A1, θ1, y1 and R1 are known, for example, the coordinates of point A0 are (0, R1+y1), the horizontal coordinate of P1 is R1*sin(θ1 / 5), and the vertical coordinate of P1 is R1*cos(θ1 / 5)+y1, and the position coordinates of P2, P3 and P4 can be obtained according to the geometric relationship.

[0182] Step S5, the external load F applied to the tunnel building boundary M0 of the tunnel by the surrounding rock of the tunnel is obtained by calculation (the prior art can be obtained).

[0183] Then, the internal force values at each node on the cross section of the rectangular tunnel are calculated based on the matrix displacement method (prior art), and are saved.

[0184] The matrix displacement method is efficient and reliable in the calculation process, and therefore, the method is particularly suitable for the design process in the early stage of tunnel construction. However, the matrix displacement method has been more used for the design of the cross section of a circular or horseshoe-shaped tunnel, and has not been applied to the design of the cross section of a rectangular tunnel. Therefore, by considering the special cross section form of the rectangular tunnel, the present application provides how to use the matrix displacement method to design the cross section of the rectangular tunnel.

[0185] Specifically,

[0186] [K]·[S]=[F] (prior art)

[0187] F=[F1,F2,…,F n ] T Each node on the tunnel lining structure includes three external force vectors, that is, F i including X-direction external force, Y-direction external force and bending moment, which are input parameters.

[0188] S=[S1,S2,…,S n ] T, Each node includes three displacement vectors, that is, S i including X-direction displacement, Y-direction displacement and angular displacement; F=[F1,F2,…,F n ] T is three external force vectors on each node;

[0189] n is the number of nodes; i=1~n.

[0190] K is the overall stiffness matrix, which is known, mainly depends on the elastic modulus E of the lining structure material, the area A of the lining section, the moment of inertia I of the section, the length Li of the divided unit. That is, as long as the lining material and thickness are known, the number of divided units is known, then the overall stiffness matrix K can be uniquely determined.

[0191] From the prior art: the matrix displacement method in structural mechanics needs to use the overall stiffness matrix K before calculation, and this K is obtained by converting the stiffness matrix of the unit in the local coordinate system to the overall coordinate system and then collecting it.

[0192] Similarly, the [S] calculated by [K]*[S]=[F] is in the overall coordinate system, and the output internal force needs to be in the local coordinate system, so here it is necessary to convert the [S] in the overall coordinate system to the local coordinate system, and then use the stiffness matrix in the local coordinate system to multiply the displacement in the local coordinate system to obtain the internal force of the unit in the local coordinate system.

[0193] Specifically, based on the matrix displacement method in structural mechanics, the beam element information obtained by discretizing the tunnel contour is used to calculate the stiffness matrix of each beam element in the local coordinate system, the stiffness matrix of the beam element in the overall coordinate system is obtained through coordinate transformation, and the stiffness matrix of all units in the overall coordinate system is collected to obtain the total stiffness matrix K of all units in the overall coordinate system.

[0194] The external load F applied by the tunnel surrounding rock to the rectangular tunnel is calculated, and the displacement value of each beam element node in the overall coordinate system is calculated according to the formula [K]·[S]=[F].

[0195] The displacement value of the beam element in the overall coordinate system is converted to the displacement value in the local coordinate system, combined with the stiffness matrix of the unit in the local coordinate system, and the internal force value of each unit node can be obtained by using the matrix displacement method.

[0196] Figure 1 In the formula, M is the bending moment of the lining (at each node), and N is the axial force (internal force) of the lining (at each node).

[0197] Step S6, calculate the value of the rectangular tunnel contour mathematical model A corresponding to each x3 based on the acquired x3, y1, x2, y2, R1, R2, R3, θ1, θ2, θ3. per When the value of A per is greater than 5%, return to step S4, otherwise, execute step S7.

[0198] Step S7, the minimum internal force value saved in step S5 is screened out, and the cross section determined by the corresponding x3 is the optimal cross section of the quasi-rectangular tunnel. Specifically, the Max function in MATLAB is used to find the minimum internal force value, and the cross section determined by the corresponding x3 is the optimal cross section of the quasi-rectangular tunnel, as shown in Figure 7 .

[0199] The above is only a preferred embodiment of the present application, and does not limit the present application in any way. Any person skilled in the art can make any form of equivalent replacement, modification or change of the technical solutions and technical contents disclosed by the present application without departing from the scope of the technical solutions of the present application, and still belongs to the protection scope of the present application.

Claims

1. A method for optimal cross section design of quasi-rectangular tunnels, characterized by, The method comprises the following steps: Step S1, determining a tunnel building limit M0, including a short axis length D0, a long axis length W0 and a shape of the limit; the shape is an asymmetric structure; Step S2, simplifying the tunnel building limit M0 determined in S1 into a symmetric shape, and the simplified tunnel building limit M1 completely contains the tunnel building limit M0 before simplification; Step S3, constructing a quasi-rectangular tunnel profile mathematical model A per : Step S31, taking 1 / 4 of the simplified tunnel building limit M1 in S2 as a research object, and setting a rectangular tunnel cross section: The 1 / 4 rectangular tunnel cross section profile is composed of a first circular arc, a second circular arc and a third circular arc which are connected in sequence; The center O1 of the first circular arc is located on the coordinate y-axis and has a coordinate (0, y1), the center O2 of the second circular arc has a coordinate (x2, y2), and the center O3 of the third circular arc is located on the coordinate x-axis and has a coordinate (x3, 0); the first circular arc is tangent to the second circular arc at intersection point A1 (d2, h1), and the second circular arc is tangent to the third circular arc at intersection point A2 (d1, h2); Step S32, setting constraint conditions of the rectangular tunnel cross section: d1 = 1 / 2W0, h1 = 1 / 2D0; d1-d2 = h1-h2 = s1; Step S33, solving expressions of the center O2 and the center O1 about x3: Step S331, solving the expression of the center O2 about x3: Firstly, based on the setting in step S31, the vertical line equation of the center O2 (x2, y2) of the second circular arc passing through the midpoint of the line segment A1A2 is solved as follows: Then, the equation of the straight line where O3A2 is located is solved as follows: Finally, since O2 is the intersection point of the above two straight lines, based on the above two equations, it can be known that: Step S332, solving the expression of the center O1 about x3: Firstly, the expression of the straight line where O1A1 is located is solved based on the vertical line equation in step S331 and point A1 and O2: Then, since point O1 (0, y1) is located on the coordinate y-axis, the following can be obtained: Step S34, solving expressions of θ1 = angle OO1O2, θ2 = angle A1O2A2, θ3 = angle OO3O2, a radius R1 of the first circular arc, a radius R2 of the second circular arc and a radius R3 of the third circular arc about x3 respectively: Step S35, solving the cross-sectional area A of the simplified tunnel construction limit M1 rect , the cross-sectional area A of the rectangular tunnel sr : Wherein, A s1 = θ1 * R1 * R1 / 2 - d2 * (h1 - y1) / 2; A s2 = θ2 * R2 * R2 / 2 - R2 * R2 * sin(θ2 / 2) * cos(θ2 / 2); A s3 = θ3 * R3 * R3 / 2 - h2 * (d1 - x3) / 2; Step S36, solving A per : A per = (A sr - A rect ) / A rect ; Step S4, according to the constraint conditions about x3 in step S32, the values of y1, x2, y2, R1, R2, R3, θ1, θ2, θ3 corresponding to x3 are calculated by using the exhaustion method in the value range; Then, the rectangular tunnel cross section is discretized into a plurality of equal beam elements, the beam elements are connected to each other through nodes, and the coordinates of the nodes are solved based on the obtained x3, y1, x2, y2, R1, R2, R3, θ1, θ2, θ3; Step S5, the external load F applied to the tunnel building limit M0 corresponding to the tunnel by the tunnel surrounding rock is obtained by calculation; Then, based on the matrix displacement method, the internal force values of the nodes on the rectangular tunnel cross section are calculated and saved; Step S6, calculate the value of the mathematical model A of the quasi-rectangular tunnel profile corresponding to each x3 based on the acquired x3, y1, x2, y2, R1, R2, R3, θ1, θ2, θ3; when the value of A per is greater than a set value, return to execute step S4, otherwise, execute step S7; per ​ Step S7, the smallest internal force value saved in step S5 is screened out, and the cross section determined by the corresponding x3 is the optimal cross section of the rectangular tunnel.

2. The method of claim 1, wherein, The vertical line equation solving process in step S331 comprises: The triangle O2A1A2 is an isosceles triangle, the length of O2A1 is equal to the length of O2A2, so the center O2(x2, y2) of the circular arc two is located on the vertical line passing through the midpoint of the line segment A1A2; The slope of the line segment A1A2 is: The slope of the perpendicular line through the midpoint of the line segment A1A2 is Assume the equation of the perpendicular line passing through the midpoint of the segment A1A2 is Substitute the coordinates of the midpoint of the segment A1A2 We obtain 3. The method of claim 1, wherein, The solving process of the equation of the straight line on which O3A2 is located in step S331 includes: The solving process of the expression of the straight line on which O1A1 is located in step S332 is as follows: Assuming x3 is known, the slope of the line on which O3A2 lies is Assuming the O3A2 straight line equation is With the coordinates (x3, 0) of O3, we have 4. The method of claim 1, wherein, The solving process of the constraint condition about x3 in step S32 includes: O2A1 is the same as the line on which O1A1 lies, and based on the point A1, O2, the slope can be found Assume the equation of the line segment O1A1 is Substitute A1(d2, h1) to get:

5. The method of claim 1, wherein, O2 is the intersection point of the line O3A2 and the line O1A1; The cross-sectional profile of the rectangular-like tunnel is convex outward; O2 is located in the lower left region of the circular arc and is located on the straight line the straight line; J1 is a line segment connecting the intersection point of the translation line with the x-axis after translation to point A2; In combination with the position of O2 and the convex outward requirement of the cross-sectional profile of the rectangular-like tunnel, the constraint condition is obtained: The constraint condition about y1 in step S32 is: x3 < x-coordinate of point J1 6. The method of claim 5, wherein, The abscissa of J1 The specific solving process is: Set the translation straight line equation of A2 as Put in A2(d1, h2), we can get bring in J1, get the x coordinate of J1 7. The method of claim 1, wherein, The setting in step S31 is as follows: the ordinate y1 of O1 < the ordinate of point J2 J2 is the general After translating to point A1, the intersection of the translation line and the y-axis.

8. The method of claim 7, wherein, The ordinate of J2 The specific solving process is: Set the translation straight line equation of A1 as Put in A1(d2h1), can get Bring in J2, get the ordinate of J2 9. The method of claim 1, wherein, (1) The simplified tunnel construction limit M1 is symmetrical about the centroid, assuming that the coordinate x and y axes pass through the centroid of the graph, then the simplified tunnel construction limit M1 is also symmetrical about the x and y axes; When 1 / 4 of the simplified tunnel construction limit M1 in S2 is taken as the research object: The circular arc one and its symmetrical part about the y axis are determined by a sector with the same radius and center, and the sector is symmetrical about the y axis, so the center O1 of the circular arc one must be located on the coordinate y axis and the coordinate is (0, y1); The circular arc three and its symmetrical part about the x axis are determined by a sector with the same radius and center, and the sector is symmetrical about the x axis, so the center O3 of the circular arc three is located on the coordinate x axis and the coordinate is (x3, 0); (2) To ensure the smoothness of the tunnel profile line, it is required that the circular arc one and the circular arc two have the same tangent at the intersection point A1, and the circular arc two and the circular arc three have the same tangent at the intersection point A2; The triangle O2A1A2 is an isosceles triangle, so the center O2(x2, y2) of the circular arc two is located on the vertical line passing through the midpoint of the line segment A1A2. ​

Images