Tunnel bottom load calculation method considering combined bearing of upper arch ring and inverted arch

Abstract

The invention discloses a tunnel bottom load calculation method considering combined bearing of an upper arch ring and an inverted arch, and the method comprises the following steps: S1, building a bearing model based on a two-parameter elastic foundation beam theory; s2, carrying out stress analysis on the upper structure of the inverted arch, and establishing a force normal equation; s3, obtaining boundary conditions of a lower inverted arch structure based on an internal force expression of redundant unknown force action of the upper structure in combination with a straight rod displacement calculation formula; s4, establishing a radial stress, tangential stress and moment balance equation of the inverted arch; and S5, on the basis of the balance equation, the boundary conditions and the elastic foundation beam deflection are combined, and inverted arch foundation counterforce is obtained. According to the method, the load transfer characteristics of the upper arch ring and the lower inverted arch are considered, and the interaction between the tunnel bottom structure and the stratum is associated, so that the calculation result better conforms to the actual engineering condition.

Description

Technical Field

[0001] The invention relates to the technical field of tunnel bottom load calculation, and in particular to a tunnel bottom load calculation method taking into account the joint bearing of an upper arch ring and an invert. Background Art

[0002] In the design process of tunnel structure, the calculation of surrounding rock pressure is the first issue to be considered. However, when solving the tunnel bottom load by traditional methods, it is usually considered as the sum of the surrounding rock pressure of the arch and the deadweight of the lining structure, and the interaction between the tunnel bottom structure and the stratum is rarely associated, so that the load acting on the tunnel structure by the surrounding rock at the tunnel base cannot be calculated more accurately, which affects the accuracy and comprehensiveness of the calculation of the surrounding rock load of the tunnel structure, resulting in great limitations in the design of the tunnel structure. Summary of the invention

[0003] The object of the present invention is to overcome the deficiencies in the above-mentioned prior art and provide a method for calculating the tunnel bottom load taking into account the joint bearing of the upper arch ring and the invert arch, taking into account the load transfer characteristics of the upper arch ring and the lower invert arch, introducing the Pasternak two-parameter elastic foundation beam theory that can respond to the continuous deformation of the foundation into the joint bearing model of the arch ring and the invert arch for analysis, and using the force method equation and the elastic foundation beam theory to comprehensively solve the invert arch foundation reaction force, i.e., the tunnel bottom load, so that the calculation result is more in line with the actual engineering situation.

[0004] To achieve the above object, the technical solution adopted by the present invention is: a method for calculating the tunnel bottom load considering the joint bearing of the upper arch ring and the invert, comprising the following steps:

[0005] Step S1: Based on the Pasternak dual-parameter elastic foundation beam theory, a joint bearing model of the upper arch ring and the invert is established, wherein the bearing model includes an upper structure and a lower structure;

[0006] Step S2: Based on the superstructure mechanical model in step S1, a force analysis is performed on the ABC segment of the invert superstructure, and a force method equation is established based on the vertical load and the horizontal load;

[0007] Step S3: Based on the internal force expression of each unit redundant unknown force of the upper structure in step S1 and the straight rod displacement calculation formula of structural mechanics, the rigid arm length y is obtained. s , based on the rigid arm length y s Simplify the force method equation in step S2 and obtain each unit redundant unknown force, and obtain the boundary conditions of the lower invert structure based on each unit redundant unknown force and the loads on both sides of the invert;

[0008] Step S4: establishing the radial stress, tangential stress and moment equilibrium equations of the invert based on the joint load-bearing model of the upper arch ring and the invert;

[0009] Step S5: Based on the equilibrium equation in step S4, combined with the boundary conditions of the lower inverted arch structure in step S3 and the deflection y of the elastic foundation beam x , and obtain the invert foundation reaction force p.

[0010] Preferably, the upper structure and the lower structure in step S1 include an upper structure EABCD and a lower structure E′D′, and the upper structure EABCD and the lower structure E′D′ are connected by a hingeless arch, and horizontal and vertical reaction forces and reaction couples are transmitted between the upper structure EABCD and the lower structure E′D′.

[0011] Preferably, the force law equation described in step S2 is expressed by the following formula:

[0012]

[0013] Where: 11 represents the displacement in the X1 direction caused by X1=1, δ 12 represents the displacement in the X1 direction caused by X2=1, δ 21 represents the displacement in the X2 direction caused by X1=1, δ 22 represents the displacement in the X2 direction caused by X2=1, δ 33 represents the displacement in the X3 direction caused by X3=1, Δ 1q represents the displacement of the external load q in the X1 direction, Δ 2q represents the displacement of the external load q in the X2 direction, Δ 3q Represents the displacement of the external load q in the X3 direction.

[0014] Preferably, the bending moment M of the lower structure in step S3 is B , shear force Q B and axial force N B The calculation process includes the following steps:

[0015] Step S301: establishing a reference coordinate system;

[0016] Step S302: in the reference coordinate system, based on different points, establish multiple internal force expressions of the redundant unknown forces of each unit;

[0017] Step S303: Based on the internal force expressions of the plurality of unit redundant unknown forces and the straight rod displacement calculation formula, an equation is established to obtain the rigid arm length y s Expression of

[0018] Step S304: Based on the rigid arm length y in step S303 s The assumptions in the calculation lead to the simplified force equation;

[0019] Step S305: Based on the simplified force method equation in step S304 and the displacement calculation formula of the straight rod, the redundant unknown force of each unit is obtained;

[0020] Step S306: Based on the unit redundant unknown forces in step S305 and the loads on both sides of the invert, obtain the bending moment M acting on any section at the arch foot position of the superstructure A , shear force Q A and axial force N A ;

[0021] Step S307: The bending moment M in step S306 A , shear force Q A and axial force N A Decompose and obtain the bending moment M in the lower invert direction B , shear force Q B and axial force N B .

[0022] Preferably, the unit redundant unknown forces in step S305 include redundant bending moment X1, redundant axial force X2 and redundant shear force X3, which are expressed by the following formula:

[0023]

[0024] Where: Δ 1q represents the displacement of the external load q in the X1 direction, Δ 2q represents the displacement of the external load q in the X2 direction, Δ 3q represents the displacement of the external load q in the X3 direction, δ 11 represents the displacement in the X1 direction caused by X1=1, δ 22 represents the displacement in the X2 direction caused by X2=1, δ 33 Indicates the displacement in the X3 direction caused by X3=1.

[0025] Preferably, the bending moment M in the lower invert direction in step S307 is B , shear force Q B and axial force N B It is expressed by the following formula:

[0026]

[0027] Where: θ is the angle between the cross section and the vertical direction.

[0028] Preferably, the radial stress, tangential stress and moment balance equations of the invert in step S4 are respectively expressed by the following equations:

[0029]

[0030] Where: Pdx is the foundation resistance, Q and Q+dQ are shear forces, N is the axial force, It is the angle between the two ends of the cross-section invert and the line connecting the center of the circle;

[0031]

[0032] Where: r is the radius of curvature;

[0033]

[0034] Where: M is the bending moment.

[0035] Preferably, the elastic foundation beam deflection y in step S5 is calculated x The following steps are involved:

[0036] Step S501: Based on the radial stress, tangential stress, moment equilibrium equation of the inverted arch and the expression of the inverted arch foundation reaction force p, the elastic foundation beam deflection y is obtained. x Expression of

[0037] Step S502: Introduce boundary conditions, take the axis of symmetry as the central axis of the tunnel, and solve the deflection y of the elastic foundation beam described in step S501 x The unknown constants C2 and C3;

[0038] Step S503: Based on the unknown constants C2 and C3 in step S502, calculate the elastic foundation beam deflection y x Simplify and solve for the bending moment M, shear force Q and axial force N of any section;

[0039] Step S504: based on the boundary conditions at both ends of the inverted arch, determine different conditions of the bending moment M, shear force Q and axial force N at both ends of the inverted arch, and obtain corresponding unknown constants C0, C1 and C4;

[0040] Step S505: Based on the unknown constants C0, C1, C2, C3 and C4 solved in steps S502 and S504, the elastic foundation beam deflection y is obtained. x ;

[0041] Step S506: The elastic foundation beam deflection y obtained in step S505 is converted to x Substitute it into the invert foundation reaction force p to obtain the invert foundation reaction force p.

[0042] Preferably, the expression of the invert foundation reaction force p in step S501 is:

[0043]

[0044] Where: p is the foundation reaction force; K is the foundation coefficient; y x is the deflection of the elastic foundation beam; b *is the equivalent width of the beam considering the continuity of the foundation, and b * =b[1+(G p / K) 12 / b],G p is the formation shear modulus, which is related to the shear deformation of the formation.

[0045] Preferably, the elastic foundation beam disturbance y in step S501 is x The expression is:

[0046]

[0047] In the formula: C0, C1, C2, C3, C4 are all unknown constants.

[0048] Compared with the prior art, the present invention has the following advantages:

[0049] 1. The calculation method of the present invention takes into account the load transfer characteristics of the upper arch ring and the lower invert arch, associates the interaction between the tunnel bottom structure and the stratum, and improves the accuracy of calculating the tunnel bottom load.

[0050] 2. The present invention introduces Pasternak dual-parameter elastic foundation beam theory into the joint bearing model of the arch ring and the invert for analysis, taking into account the continuous deformation characteristics of the stratum and overcoming the inaccurate selection of the end load of the invert.

[0051] 3. The calculation method of the present invention has the advantages that parameters are easy to obtain and the calculation is relatively simple.

[0052] 4. The calculation method of the present invention uses the force method equation and the elastic foundation beam theory to comprehensively solve the invert foundation reaction force, that is, the surrounding rock pressure of the tunnel, so that the calculation result is more in line with the actual engineering situation.

[0053] The present invention is further described in detail below through the accompanying drawings and examples. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 It is a schematic diagram of the combined bearing model of the upper arch ring and the inverted arch of the present invention;

[0055] Figure 2 It is a schematic diagram of the mechanical model of the upper structure of the present invention;

[0056] Figure 3 It is a schematic diagram of the force balance of the upper structure of the present invention;

[0057] Figure 4 It is a schematic diagram of the force balance of the arch structure under the action of the unit redundant unknown force of the present invention;

[0058] Figure 5It is a schematic diagram of the force balance of the arch and the side wall structure under the action of the unit redundant unknown force of the present invention;

[0059] Figure 6 It is a schematic diagram of the force conversion of the upper structure of the present invention;

[0060] Figure 7 It is a schematic diagram of the mechanical model of the inverted arch structure of the present invention;

[0061] Figure 8 It is a schematic diagram of the bending moment borne by the end of the inverted arch beam of the present invention;

[0062] Fig. 9 This is a schematic diagram of the inverted arch beam end bearing axial force of the present invention;

[0063] Fig.10 This is a schematic diagram of the inverted arch beam end bearing shear force of the present invention;

[0064] Fig.11 It is a schematic diagram comparing the theoretical calculated value and the measured value of the present invention. DETAILED DESCRIPTION

[0065] The present invention discloses a tunnel bottom load calculation method considering the joint bearing of the upper arch ring and the invert arch, comprising the following steps:

[0066] Step S1: Based on the Pasternak dual-parameter elastic foundation beam theory, a joint bearing model of the upper arch ring and the invert is established, wherein the bearing model includes an upper structure and a lower structure;

[0067] like Figure 1 As shown, in step S1, the upper structure and the lower structure include an upper structure EABCD and a lower structure E′D′, and the upper structure EABCD and the lower structure E′D′ are connected by a hingeless arch, and the upper structure EABCD and the lower structure E′D′ transmit horizontal and vertical reaction forces and reaction couples. There is no rotation and movement between the upper structure EABCD and the lower structure E′D′.

[0068] Step S2: Based on the superstructure mechanical model in step S1, a force analysis is performed on the ABC segment of the invert superstructure, and a force method equation is established based on the vertical load and the horizontal load;

[0069] like Figure 2 As shown in the figure, the ABC section of the superstructure is a three-fold hyperstatic hingeless arch structure, which is disconnected from the symmetry axis. The vertical load distribution form is based on the non-uniform load, and the horizontal load distribution form is based on the uniform load calculated in the highway tunnel design specification. The force method equation established based on the vertical load and horizontal load is:

[0070]

[0071] Where:12 represents the displacement in the X1 direction caused by X2=1, Δ 1q It represents the displacement of the external load q in the X1 direction. The external load q includes q1 and q2, and q1 includes q 11 and q 12 , since X1, X2, q are positively symmetric and X3 is antisymmetric, therefore, δ 13 =δ 31 =0,δ 23 =δ 32 =0, then the force method equation in step S2 is expressed as follows:

[0072]

[0073] Where: 11 represents the displacement in the X1 direction caused by X1=1, δ 12 represents the displacement in the X1 direction caused by X2=1, δ 21 represents the displacement in the X2 direction caused by X1=1, δ 22 represents the displacement in the X2 direction caused by X2=1, δ 33 represents the displacement in the X3 direction caused by X3=1, Δ 1q represents the displacement of the external load q in the X1 direction, Δ 2q represents the displacement of the external load q in the X2 direction, Δ 3q Represents the displacement of the external load q in the X3 direction.

[0074] Step S3: Based on the internal force expression of each unit redundant unknown force of the upper structure in step S1 and the straight rod displacement calculation formula of structural mechanics, the rigid arm length y is obtained. s , based on the rigid arm length y s Simplify the force method equation in step S2 and obtain each unit redundant unknown force, and obtain the boundary conditions of the lower invert structure based on each unit redundant unknown force and the loads on both sides of the invert;

[0075] like Figures 3 to 5 As shown, due to the δ in formula (2) 12 =δ 21 ≠0, the force equations need to be solved simultaneously. If a rigid arm can be established, let δ 12 =δ 21 =0, then the auxiliary coefficients in formula (2) are all zero, which will simplify the calculation of the force method equation.

[0076] Step S301: establishing a reference coordinate system;

[0077] In order to calculate the length of the rigid arm, first establish the reference coordinate system X′O1Y1, with the original cutout of the symmetry axis as the coordinate origin O1, the horizontal right as the positive direction of the X′ axis, and the clockwise rotation of 90° as the positive direction of the Y1 axis. At the same time, establish the coordinate system XOY with the rigid arm endpoint O as the coordinate origin, and stipulate that the X axis is positive to the right and the Y axis is positive downward. Then the length of the rigid arm y s It is expressed as:

[0078] y s =y1-y (3)

[0079] Where y1 is the length of O1Y1 in the coordinate system X′O1Y1, and y is the length of OY in the coordinate system XOY.

[0080] The positive and negative signs of the structural forces are stipulated as follows: the selected bending moment is positive when the lower edge of the structure is tensile, the axial force is positive when the structure is compressed, and the shear force is positive when the supporting structure rotates clockwise. is the inclination angle of the tangent line at each point on the arch axis.

[0081] Step S302: in the reference coordinate system, based on different points, establish multiple internal force expressions of the redundant unknown forces of each unit;

[0082] L is the tunnel excavation radius. Take a point on the semi-arch structure BC. The horizontal coordinate of the point is x and the vertical coordinate is y. When -L<x<L, 0<y

[0083]

[0084]

[0085] Where: is the bending moment at the point BC when the unit bending moment X1=1 acts on the semi-arch structure. is the shear force at the point BC when the unit bending moment X1=1 acts on the semi-arch structure. is the axial force at point BC when the unit bending moment X1=1 acts on the semi-arch structure; is the bending moment at the point BC when the unit axial force X2=1 acts on the semi-arch structure. is the shear force at the point BC when the unit axial force X2=1 acts on the semi-arch structure. is the axial force taken at point BC when the unit axial force X2=1 acts on the semi-arch structure; is the bending moment at the point BC when the unit shear force X3=1 acts on the semi-arch structure. is the shear force at the point BC when the unit shear force X3=1 acts on the semi-arch structure. ​is the axial force at the point BC when the unit shear force X3=1 acts on the semi-arch structure; is the bending moment of the point BC of the semi-arch structure when the vertical load and horizontal load act together. is the shear force of the point BC of the semi-arch structure caused by the combined action of vertical load and horizontal load. It is the axial force at point BC of the semi-arch structure where the vertical load and horizontal load act together.

[0086] When x=L or x=-L, h1≤y≤h2, when X1=1, X2=1, X3=1 act on the semi-arch structure BC respectively, the internal force expression of each unit excess unknown force is:

[0087]

[0088] Step S303: Based on the internal force expressions of multiple units of redundant unknown forces and the straight rod displacement calculation formula, an equation is established to obtain the rigid arm length y s Expression of

[0089] According to the straight rod displacement calculation formula in structural mechanics:

[0090]

[0091] Where: E is the elastic modulus, I is the section moment of inertia, A is the cross-sectional area, G is the shear modulus, and k is the shear correction coefficient of the cross section.

[0092] Substituting formula (4) to formula (7) into formula (8), we get:

[0093]

[0094] If δ 12 =δ 21 =0, then the rigid arm length y in step S3 s It is expressed by the following formula:

[0095]

[0096] In formula (10), ∫y1ds is solved according to the Simpsen method, as follows: Figure 5 As shown, y1 is the axis length of Y1 in the X′O1Y1 coordinate system, and y is the length of the Y axis in the XOY coordinate system.

[0097] Step S304: Based on the rigid arm length y in step S303 s The assumptions in the calculation lead to the simplified force equation:

[0098]

[0099] Step S305: Based on the simplified force method equation in step S304 and the displacement calculation formula of the straight rod, the redundant unknown force of each unit is obtained;

[0100] Solve the coefficient δ in formula (11) 11 , δ 22 , δ 33 and the free term Δ 1q , Δ 2q , Δ 3q :

[0101] Since the arch is a curved rod, the effect of curvature on deformation should be considered when calculating the coefficient and free term. However, the calculation results show that the effect of curvature on deformation is very small. Therefore, the coefficient and free term are solved based on the displacement calculation formula of the straight rod:

[0102]

[0103] In tunnel structure calculation, when When, h c is the tunnel lining section height, l is the tunnel arch span, f is the arch height, and the coefficient δ ii and the free term Δ iq In the calculation, the axial force N and the shear force Q are ignored and only the bending moment M is considered. Then the coefficients and free terms can be rewritten as:

[0104]

[0105] In step S305, each unit redundant unknown force includes redundant bending moment X1, redundant axial force X2 and redundant shear force X3, which are expressed by the following formula:

[0106]

[0107] Where: Δ 1q represents the displacement of the external load q in the X1 direction, Δ 2q represents the displacement of the external load q in the X2 direction, Δ 3q represents the displacement of the external load q in the X3 direction, δ 11 represents the displacement in the X1 direction caused by X1=1, δ 22 represents the displacement in the X2 direction caused by X2=1, δ 33 Indicates the displacement in the X3 direction caused by X3=1.

[0108] Step S306: Based on the unit redundant unknown forces in step S305 and the loads on both sides of the invert, obtain the bending moment M acting on any section at the arch foot position of the superstructure A , shear force Q A and axial force N A ;

[0109] like Figure 6 As shown in the figure, the loads on both sides of the inverted arch include bending moment M, shear force Q and axial force N. The load and the redundant unknown force act together on the ABC section of the arch ring, and the bending moment M acting on any section at the arch foot position of the superstructure EABCD is A , shear force Q A and axial force N A It is expressed by the following formula:

[0110]

[0111] Step S307: The bending moment M in step S306 A , shear force Q A and axial force N A Decompose and obtain the bending moment M in the lower invert direction B , shear force Q B and axial force N B .

[0112] like Figure 6 As shown, the bending moment M in the lower invert direction in step S307 B , shear force Q B and axial force N B They are:

[0113]

[0114] Where: θ is the angle between the cross section and the vertical direction.

[0115] Step S4: establishing the radial stress, tangential stress and moment equilibrium equations of the invert based on the joint load-bearing model of the upper arch ring and the invert;

[0116] According to the mechanical model of the inverted arch structure, a micro-element segment of the inverted arch structure is taken for force analysis. According to the force balance characteristics of the micro-element segment, the radial, tangential and moment balance equations are established.

[0117] In this application, the inverted arch is an elastic structure and follows Hooke's law; the thickness of the inverted arch structure is smaller than the radius of the arc, and the inverted arch can be regarded as a thin beam. The inverted arch is regarded as a curved beam with a circular arc cross section, and its radial foundation reaction force and continuous deformation characteristics of the foundation are considered at the same time.

[0118] like Figures 7 to 10 As shown in the figure, according to the mechanical model of the inverted arch structure, in the arc coordinate system, any micro segment with a curvature radius of r is cut from the inverted arch. and The cross section is respectively attached with shear force Q and Q+dQ, bending moment M and M+dM, axial force N and N+dN and foundation resistance Pdx, according to the force balance in radial and tangential directions and the moment balance of the micro-segment.

[0119] Then the radial stress, tangential stress and moment equilibrium equations of the invert in step S4 are respectively expressed by the following equations:

[0120] Radial stress equilibrium equation:

[0121]

[0122] Where: Pdx is the foundation resistance, Q and Q+dQ are shear forces, N is the axial force, It is the angle between the two ends of the cross-section invert and the line connecting the center of the circle;

[0123] Tangential stress equilibrium equation:

[0124]

[0125] Where: r is the radius of curvature;

[0126] Torque balance equation:

[0127]

[0128] Where: M is the bending moment.

[0129] Step S5: Based on the equilibrium equation in step S4, combined with the boundary conditions of the lower invert structure in step S3 and the deflection y of the elastic foundation beam x , and obtain the invert foundation reaction force p.

[0130] Step S501: Based on the radial stress, tangential stress, moment equilibrium equation of the inverted arch and the expression of the inverted arch foundation reaction force p, the elastic foundation beam deflection y is obtained. x Expression of

[0131] The Pasternak model in the dual-parameter elastic foundation beam model is used to characterize the interaction characteristics between the tunnel and the foundation soil. The model introduces a shear layer so that the shear force can be transmitted between the soil spring units. The expression of the invert foundation reaction force p in step S501 is:

[0132]

[0133] Where: p is the foundation reaction force; K is the foundation coefficient; y x is the deflection of the elastic foundation beam; b * is the equivalent width of the beam considering the continuity of the foundation, and b * =b[1+(G p / K) 12 / b],G p is the formation shear modulus, which is related to the shear deformation of the formation.

[0134] Based on equation (21), we first solve the deflection y of the elastic foundation beam: x :

[0135] From formula (18), we can get:

[0136]

[0137] Substituting formula (22) into formula (21) yields:

[0138]

[0139] Take the first-order derivative of x in formula (23) and further simplify it to:

[0140]

[0141] It is known that the elastic differential equation for the bending of a curved beam is:

[0142]

[0143] The first and third order derivatives of formula (25) are obtained respectively:

[0144]

[0145] Substitute formula (26) and formula (27) into formula (24) and simplify to obtain:

[0146]

[0147] make Further variable substitution is performed, and the following formula is obtained from formula (28):

[0148]

[0149] make If the equation has the form , then substitute it into the equation and simplify it to get:

[0150] λ 5 +ψλ 3 +ζ 2 λ=0 (30)

[0151] Based on formula (30), five different roots are obtained: λ1 = 0; λ 2、3、4、5 = ±(α ± βi); where

[0152]

[0153] Then the elastic foundation beam disturbance y in step S501 is x The expression is:

[0154]

[0155] In the formula: C0, C1, C2, C3, C4 are all unknown constants.

[0156] make Then we have:

[0157]

[0158] Step S502: Introduce boundary conditions, take the axis of symmetry as the tunnel centerline, and solve the elastic foundation beam deflection y in step S501 x The unknown constants C2 and C3;

[0159] Introduce the corresponding boundary conditions, take the symmetry axis as the tunnel centerline, and the symmetric load as the working condition. Then the slope of the invert deformation on the symmetric section at the tunnel centerline is zero, that is: Substituting it into formula (31) and simplifying it, we get:

[0160]

[0161] Similarly, the shear force on the symmetrical section at the tunnel centerline is zero, that is: Substituting it into formula (31) and simplifying it, we get:

[0162]

[0163] Combining equations (33) and (34), we get: C2 = C3 = 0

[0164] Step S503: Based on the unknown constants C2 and C3 in step S502, calculate the elastic foundation beam deflection y x Simplify and solve for the bending moment M, shear force Q and axial force N of any section;

[0165] Since C2=C3=0, equation (32) can be simplified as:

[0166]

[0167] Since the excavation radius corresponding to the tunnel invert is relatively large, equation (26) can be simplified as:

[0168]

[0169] Combining formula (19), formula (35) and formula (36), the bending moment M, shear force Q and axial force N of any section on the beam can be obtained:

[0170]

[0171] Step S504: Based on the boundary conditions at both ends of the inverted arch, the bending moment M, shear force Q and axial force N at both ends of the inverted arch are determined to obtain corresponding unknown constants C0, C1 and C4;

[0172] When the same bending moment acts on both ends of the invert, according to the boundary conditions at both ends: Combining formulas (30) to (32) and formulas (37) to (39), we can get the unknown coefficient C0 M 、C1 M 、C4 M :

[0173]

[0174] in:

[0175]

[0176] When the same axial force acts on both ends of the invert, according to the boundary conditions at both ends: Get the unknown constant C0 N 、C1 N 、C4 N :

[0177]

[0178] C1 N =C4 N =0 (44)

[0179] When the same shear force acts on both ends of the invert, according to the boundary conditions at both ends: Get the unknown constant C0 Q 、C1 Q 、C4 Q :

[0180]

[0181] According to the above analysis, when the same bending moment, shear force and axial force act on both sides of the inverted arch structure at the same time, the unknown constants C0, C1 and C4 are obtained by using the superposition principle and simultaneous equations (40) to (47):

[0182]

[0183] Step S505: Based on the unknown constants C0, C1, C2, C3 and C4 solved in steps S502 and S504, the elastic foundation beam deflection y is obtained. x ;

[0184] Substituting formula (48), formula (49) and formula (50) into formula (35), the radial displacement of any section of the inverted arch structure can be obtained, that is, the deflection of the elastic foundation beam y x :

[0185]

[0186] Step S506: The elastic foundation beam deflection y obtained in step S505 is converted to x Substituting it into the invert foundation reaction force, we get the invert foundation reaction force p.

[0187] Substituting the calculation result of formula (51) into formula (21), the invert foundation reaction force p generated by the interaction between the invert and the foundation can be obtained, that is, the surrounding rock pressure at the bottom of the tunnel.

[0188] The following is a specific example to apply a tunnel bottom load calculation method proposed by the present invention that takes into account the joint bearing of the upper arch ring and the invert:

[0189] Wangjiagou Tunnel is a single-hole double-track tunnel located in Yuzhong County, Lanzhou, in the Huangtuliangmao gully area. It is 2,621 meters long and the tunnel is approximately 214 meters deep. The strata that the tunnel passes through are mainly: Quaternary Holocene alluvial sandy loess, Upper Pleistocene aeolian, alluvial sandy loess, alluvial gravel sand, Quaternary Middle Pleistocene aeolian clay loess, diluvial medium sand, and Tertiary Upper Pliocene mudstone. The tunnel was mainly constructed using the three-step method. Due to the high water content of the loess in the shallow buried section of the tunnel entrance, the tunnel settlement was too large, causing serious deformation of the arch frame and cracking of the shotcrete. Some dangerous sections were constructed using the CRD method. The tunnel radius r = 6m, and the load above the tunnel arch q 11 =12.7kPa, both sides of the tunnel bear vertical load q 12 ≈70kPa, lateral pressure q2≈300kPa, invert radius r≈15m, and the bed coefficient K=3.5×10 6 MPa / m, formation shear modulus G p =2.5×10 6 N / m, according to the vertical load and horizontal load, establish the force method equation, that is, equation (1), and substitute it into the simplified force method equation, that is, equation (11) to solve the redundant bending moment X1, redundant axial force X2 and redundant shear force X3. Given X1, X2 and X3, solve the bending moment M in the invert direction through equations (16) and (17): B , shear force Q B and axial force N B That is, the boundary condition, and then solve C0 through equations (40) to (42) M 、C1 M 、C4 M , C0 is obtained by solving equations (43) to (44) N 、C1 N 、C4 N , C0 is obtained by solving equations (45) to (47) Q 、C1 Q 、C4 Q, the unknown coefficients C0, C1, and C4 can be obtained by using the superposition principle. Substituting C0, C1, and C4 into equation (31) can obtain the deflection y x , y x Substitute into the foundation reaction force calculation formula (21) proposed in the present invention to obtain the foundation reaction force p; the calculated and measured tunnel bottom surrounding rock pressure is shown in Fig.11 The tunnel bottom surrounding rock pressure calculation method proposed in the present invention has a high degree of consistency with the measured data, the theoretical values ​​of some measuring points are greater than the measured values, and there is a certain safety reserve, which shows that the tunnel bottom surrounding rock pressure calculation method proposed in the present invention is accurate and can be used to guide engineering practice.

[0190] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any way. Any simple modification, change and equivalent structural transformation made to the above embodiment based on the technical essence of the present invention still falls within the protection scope of the technical solution of the present invention.

Claims

1. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert, characterized in that: The following steps are involved: Step S1: Based on the Pasternak dual-parameter elastic foundation beam theory, a joint bearing model of the upper arch ring and the invert is established, wherein the bearing model includes an upper structure and a lower structure; Step S2: Based on the superstructure mechanical model in step S1, a force analysis is performed on the ABC segment of the invert superstructure, and a force method equation is established based on the vertical load and the horizontal load; Step S3: Based on the internal force expression of each unit redundant unknown force of the upper structure in step S1 and the straight rod displacement calculation formula of structural mechanics, the rigid arm length y is obtained. s , based on the rigid arm length y s Simplify the force method equation in step S2 and obtain each unit redundant unknown force, and obtain the boundary conditions of the lower invert structure based on each unit redundant unknown force and the loads on both sides of the invert; Step S4: establishing the radial stress, tangential stress and moment equilibrium equations of the invert based on the joint load-bearing model of the upper arch ring and the invert; Step S5: Based on the equilibrium equation in step S4, combined with the boundary conditions of the lower inverted arch structure in step S3 and the deflection y of the elastic foundation beam x , and obtain the invert foundation reaction force p.

2. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert according to claim 1, characterized in that: The upper structure and the lower structure in step S1 include an upper structure EABCD and a lower structure E′D′, wherein the upper structure EABCD and the lower structure E′D′ are connected by a hingeless arch, and horizontal and vertical reaction forces and reaction couples are transmitted between the upper structure EABCD and the lower structure E′D′.

3. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert according to claim 1, characterized in that: The force equation described in step S2 is expressed as follows: Where: 11 represents the displacement in the X1 direction caused by X1=1, δ 12 represents the displacement in the X1 direction caused by X2=1, δ 21 represents the displacement in the X2 direction caused by X1=1, δ 22 represents the displacement in the X2 direction caused by X2=1, δ 33 represents the displacement in the X3 direction caused by X3=1, Δ 1q represents the displacement of the external load q in the X1 direction, Δ 2q represents the displacement of the external load q in the X2 direction, Δ 3q Represents the displacement of the external load q in the X3 direction.

4. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert according to claim 1, characterized in that: The bending moment M of the lower structure in step S3 B , shear force Q B and axial force N B The calculation process includes the following steps: Step S301: establishing a reference coordinate system; Step S302: in the reference coordinate system, based on different points, establish multiple internal force expressions of the redundant unknown forces of each unit; Step S303: Based on the internal force expressions of the plurality of unit redundant unknown forces and the straight rod displacement calculation formula, an equation is established to obtain the rigid arm length y s Expression of Step S304: Based on the rigid arm length y in step S303 s The assumptions in the calculation lead to the simplified force equation; Step S305: Based on the simplified force method equation in step S304 and the displacement calculation formula of the straight rod, the redundant unknown force of each unit is obtained; Step S306: Based on the unit redundant unknown forces in step S305 and the loads on both sides of the invert, obtain the bending moment M acting on any section at the arch foot position of the superstructure A , shear force Q A and axial force N A ; Step S307: The bending moment M in step S306 A , shear force Q A and axial force N A Decompose and obtain the bending moment M in the lower invert direction B , shear force Q B and axial force N B .

5. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert according to claim 4, characterized in that: The unit redundant unknown forces in step S305 include redundant bending moment X1, redundant axial force X2 and redundant shear force X3, which are expressed by the following formula: Where: Δ 1q represents the displacement of the external load q in the X1 direction, Δ 2q represents the displacement of the external load q in the X2 direction, Δ 3q represents the displacement of the external load q in the X3 direction, δ 11 represents the displacement in the X1 direction caused by X1=1, δ 22 represents the displacement in the X2 direction caused by X2=1, δ 33 Indicates the displacement in the X3 direction caused by X3=1.

6. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert according to claim 4, characterized in that: The bending moment M in the lower invert direction in step S307 B , shear force Q B and axial force N B It is expressed by the following formula: Where: θ is the angle between the cross section and the vertical direction.

7. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert according to claim 1, characterized in that: The radial stress, tangential stress and moment equilibrium equations of the invert described in step S4 are respectively expressed by the following equations: Where: Pdx is the foundation resistance, Q and Q+dQ are shear forces, N is the axial force, It is the angle between the two ends of the cross-section invert and the line connecting the center of the circle; Where: r is the radius of curvature; Where: M is the bending moment.

8. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert according to claim 7, characterized in that: Calculate the elastic foundation beam deflection y in step S5 x The following steps are involved: Step S501: Based on the radial stress, tangential stress, moment equilibrium equation of the inverted arch and the expression of the inverted arch foundation reaction force p, the elastic foundation beam deflection y is obtained. x Expression of Step S502: Introduce boundary conditions, take the axis of symmetry as the central axis of the tunnel, and solve the deflection y of the elastic foundation beam described in step S501 x The unknown constants C2 and C3; Step S503: Based on the unknown constants C2 and C3 in step S502, calculate the elastic foundation beam deflection y x Simplify and solve for the bending moment M, shear force Q and axial force N of any section; Step S504: Based on the boundary conditions at both ends of the inverted arch, the bending moment M, shear force Q and axial force N at both ends of the inverted arch are determined to obtain corresponding unknown constants C0, C1 and C4; Step S505: Based on the unknown constants C0, C1, C2, C3 and C4 solved in steps S502 and S504, the elastic foundation beam deflection y is obtained. x ; Step S506: The elastic foundation beam deflection y obtained in step S505 is converted to x Substitute it into the invert foundation reaction force p to obtain the invert foundation reaction force p.

9. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert according to claim 8, characterized in that: The expression of the invert foundation reaction force p in step S501 is: Where: p is the foundation reaction force; K is the foundation coefficient; y x is the deflection of the elastic foundation beam; b * is the equivalent width of the beam considering the continuity of the foundation, and b * =b[1+(G p / K) 1 / 2 / b],G p is the formation shear modulus, which is related to the shear deformation of the formation.

10. A method for calculating tunnel bottom loads taking into account the combined bearing of the upper arch ring and the invert according to claim 8, characterized in that: The elastic foundation beam disturbance y in step S501 x The expression is: In the formula: C0, C1, C2, C3, C4 are all unknown constants.

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