A method for calculating internal forces of a tunnel secondary lining vault void lining

Abstract

The application discloses a tunnel secondary lining vault disengaging lining internal force calculation method, comprising the following steps: S1: establishing a tunnel vault underthickness disengaging lining internal force calculation model; S2: determining the vault disengaging lining calculation parameters; S3: listing the vault displacement coordination equation group, calculating the flexibility coefficient through piecewise integral, and calculating the displacement caused by the external load at the vault; S4: listing the arch foot displacement equation group, and calculating the arch foot elastic fixed coefficient and the displacement caused by the external load at the arch foot; S5: coupling the vault displacement coordination equation and the arch foot displacement equation, and calculating the vault bending moment and the vault axial force; S6: according to the relationship between the vault internal force and the section internal force, the internal force of any section on the lining axis is calculated; S7: according to the section internal force, the lining section safety coefficient is calculated, the unsafe section range is determined, and reinforcement is carried out. The internal force calculation method can realize the quantitative analysis and calculation of the disengaging lining internal force, and provides a guidance basis for the reinforcement parameters.

Description

Technical Field

[0001] This invention relates to the field of tunnel structure calculation technology, specifically to a method for calculating the internal forces of a tunnel secondary lining arch with a voided lining. Background Technology

[0002] Insufficient lining thickness and voids are common defects in tunnel construction, and are a major cause of lining cracks, spalling, and even arch collapse, seriously threatening tunnel structure and traffic safety. Under current technological conditions, approximately 10-30% (based on lining module statistics) of the lining has varying degrees of voids, and the proportion is even higher in tunnels constructed before anti-voidage measures were implemented.

[0003] For voided linings, the main treatment measures include grouting and filling, external bonding of steel plates or arch reinforcement. However, there is currently no calculation method for the internal forces of voided linings, which leads to a lack of basis for reinforcement parameters such as reinforcement range and strength requirements of reinforcement components. Reinforcement parameters often rely on empirical values ​​and lack sufficient theoretical support, which affects the reinforcement effect. Summary of the Invention

[0004] In view of this, the purpose of this invention is to provide a method for calculating the internal forces of the voided lining at the arch of a tunnel secondary lining, so as to realize the quantitative analysis and calculation of the internal forces of the voided lining and provide guidance for reinforcement parameters.

[0005] The present invention solves the above problems through the following technical means:

[0006] A method for calculating the internal forces of a voided lining in the arch of a tunnel secondary lining includes the following steps:

[0007] S1: Establish an internal force calculation model for the tunnel arch lining with insufficient thickness and void defects based on the site conditions;

[0008] S2: Determine the calculation parameters for the void lining at the arch crown based on the secondary lining design parameters and the geological conditions.

[0009] S3: Based on the superposition of displacements, list the arch crown displacement compatibility equations, calculate the flexibility coefficients through piecewise integration, and calculate the displacements caused by external loads on the arch crown.

[0010] S4: Based on the superposition of displacements, list the arch foot displacement equations and calculate the arch foot elastic fixation coefficient and the displacement generated at the arch foot by the external load.

[0011] S5: Couple the crown displacement compatibility equation and the arch foot displacement equation to calculate the crown bending moment and crown axial force.

[0012] S6: Based on the relationship between the internal forces at the crown and the internal forces in the cross section, calculate the internal forces at any cross section on the lining axis;

[0013] S7: Calculate the safety factor of the lining section based on the internal forces of the section, determine the range of unsafe sections, and carry out reinforcement.

[0014] Furthermore, step S1 includes the following steps:

[0015] S101: Establish a rectangular coordinate system with the axis of the arch crown where the lining has not been removed as the origin, the horizontal direction as the X-axis, and the vertical direction as the Y-axis;

[0016] S102: In the rectangular coordinate system described in step S101, the origin O of the coordinate system is at the axis of the lining arch, the center of the lining arch is O1, the vertical load of the lining is q, the horizontal load of the lining is e, the semi-central angle corresponding to the arc segment where the arch is located in the multi-centered circular tunnel is α0, the semi-central angle corresponding to the void range of the arch is α1, the outer diameter of the lining is r1, the radius of the lining axis is r2, and the inner diameter of the lining is r3.

[0017] S103: Since the structure's geometry and stress are symmetrical, we will analyze a half-structure.

[0018] S104: There are unknown bending moment X1 and unknown axial force X2 at the break at the top of the arch. Since the structure and load are symmetrically distributed, the shear force X3 at the top of the arch is 0.

[0019] Furthermore, the calculation parameters for the voided lining in step S2 are as follows: vertical load q of the lining, horizontal load e of the lining, elastic modulus E of the lining concrete, and elastic foundation coefficient k at the voided boundary.

[0020] Furthermore, step S3 includes the following steps:

[0021] S301: Based on the superposition of displacements, the arch crown displacement compatibility equation is listed:

[0022] ;

[0023] Where: δ ik δ is the flexibility coefficient, which represents the displacement of the arch crown by a unit unknown force along the direction of the unknown force in the basic structure when the arch foot is rigidly fixed. According to the reciprocal displacement theorem, δ ik = δ ki Δ ip External loads caused by X i The displacements in the direction are β0 and μ0, respectively, representing the total elastic rotation angle and total horizontal displacement of the arch foot;

[0024] S302: Calculate the compliance coefficient through piecewise integration:

[0025] ;

[0026] S303: Calculate the displacement Δ caused by the external load at the crown. ip :

[0027]

[0028] .

[0029] Furthermore, in step S302, specifically, a cylindrical coordinate system is established with the center O1 of the lining arch as the origin, and piecewise integration is performed on the voided section AB and the non-voided section BC to calculate the compliance coefficient δ. ik :

[0030] ;

[0031] Where: h z To calculate the height difference between the cross section and the arch crown position of the offset axis of the hollow lining; r b R1 is the radius of the offset axis of the lining in the void section; E is the elastic modulus of the lining concrete; r2 is the radius of the lining axis; r3 is the inner diameter of the lining; H is the lining thickness; h1 is the void height at the arch crown; H1 is the lining thickness at the arch crown; dα is the area of ​​the infinitesimal element; y is the perpendicular distance between the infinitesimal element dα and the x-axis; b is the width of the cross section; h is the height of the cross section.

[0032] Furthermore, step S4 includes the following steps:

[0033] S401: Based on the superposition of displacements, derive the following set of equations for the arch foot displacements.

[0034] ;

[0035] Among them: β1, μ1, β2, μ2, β P μ P β1 and μ1 are the elastic constants at the arch foot, respectively: β1 and μ1 are the elastic rotation angle and horizontal displacement generated at the arch foot by a unit bending moment at the arch crown; β2 and μ2 are the elastic rotation angle and horizontal displacement generated at the arch foot by a unit axial force at the arch crown; β P μ P These represent the elastic rotation and horizontal displacement generated at the arch foot by a unit external load, respectively; f is the arch's rise.

[0036] S402: Calculate the elastic fixation coefficients β1, μ1, β2, μ2 of the arch foot:

[0037] The elastic constant of the arch foot under bending moment is:

[0038]

[0039]

[0040] ;

[0041] The elastic constant coefficient of the arch foot under axial force is:

[0042]

[0043] ;

[0044] S403: Calculate the deformation β caused by external loads at the arch foot. P μ P First, determine the external load of the secondary lining. The secondary lining load consists of two parts. One part is the uniformly distributed surrounding rock load borne by the non-voided section. In addition to the uniformly distributed surrounding rock load borne by the non-voided section, since the voided section is not in contact with the initial support, the uniformly distributed surrounding rock load that should have been borne by the voided section is transferred to the void boundary during the lining deformation process, forming a concentrated load F. T The horizontal and vertical components of the concentrated force are equal to the values ​​of the horizontal and vertical loads of the detached section, respectively; after obtaining the external load of the lining, the bending moment M generated by the external load at the arch foot is calculated. 0 P and axial force N 0 P Then, based on the elastic fixation coefficient of the arch foot, the displacement generated at the arch foot by the external load is calculated:

[0045]

[0046]

[0047] .

[0048] Furthermore, in step S5, the arch crown displacement compatibility equation and the arch foot displacement equation are calculated in a coupled manner:

[0049] Depend on:

[0050]

[0051]

[0052] get:

[0053] .

[0054] Furthermore, in step S6, based on the relationship between the internal forces at the arch crown and the internal forces at the cross section, the internal forces at any cross section j on the lining axis are calculated:

[0055]

[0056] Among them, M j , N j Q j h represents the internal force at section j; zTo calculate the height difference between the cross section and the arch crown position of the offset axis of the hollow lining; φ j M is the central angle corresponding to section j; 0 j P, N 0 j P,Q 0 j P is the internal force caused by the external load on section j.

[0057] Furthermore, step S7 includes the following steps:

[0058] S701: For plain concrete lining or reinforced concrete lining, the safety factor shall be calculated based on compressive strength and tensile strength, respectively.

[0059] S702: Compare the safety factor obtained in step S701 with the standard value to determine whether it is a dangerous section.

[0060] Furthermore, in step S701, for plain concrete lining, calculations are performed according to the Highway Tunnel Design Code JTG3370.1-2018; for reinforced concrete lining, calculations are performed according to the Highway Tunnel Design Code JTG 3370.1-2018.

[0061] The beneficial effects of this invention are:

[0062] The method for calculating the internal forces of the voided lining in the arch of a tunnel secondary lining according to the present invention can achieve quantitative analysis and calculation of the internal forces of the voided lining, providing guidance for reinforcement parameters. More specifically, the present invention has at least the following beneficial effects:

[0063] 1. This invention takes into account the influence of changes in the neutral axis of the structure on internal forces, resulting in more accurate results.

[0064] 2. This invention takes into account the influence of cross-sectional thickness variation on internal forces, resulting in more accurate results.

[0065] 3. This invention, by combining the void size parameters obtained from on-site testing, can analyze the internal forces of voided and insufficiently thick linings, determine the extent of lining damage, and quantitatively evaluate the safety of voided linings, providing a basis for void treatment. Attached Figure Description

[0066] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0067] Figure 1 This is a diagram of the computational model of the present invention;

[0068] Figure 2 This is a basic structural diagram of the present invention;

[0069] Figure 3 This is a schematic diagram showing the calculation of the heights of the separation section;

[0070] Figure 4 This is a schematic diagram of the arch foot under bending moment;

[0071] Figure 5 This is a schematic diagram of the arch foot under the action of horizontal force;

[0072] Figure 6 This is a flowchart of the method of the present invention. Detailed Implementation

[0073] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments. Through these descriptions, the features and advantages of the present invention will become clearer and more apparent. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them.

[0074] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," "counterclockwise," "axial," "radial," and "circumferential" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing this invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0075] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0076] like Figures 1-6 As shown, this invention discloses a method for calculating the internal forces of a tunnel secondary lining arch with a voided lining.

[0077] Step S1: See Figure 1 An internal force calculation model for the tunnel arch lining with insufficient thickness and void defects was established based on the site conditions. q represents the vertical load on the lining; e represents the lateral load on the lining; F... T α0 is the concentrated load at the void boundary; α1 is the semi-central angle corresponding to the arc segment where the arch crown is located in the multi-centered circular tunnel; r1 is the semi-central angle corresponding to the void range of the arch crown; r2 is the outer diameter of the lining; r3 is the radius of the lining axis; O is the origin of the coordinate system (at the axis of the lining arch crown), and O1 is the center of the lining segment.

[0078] See Figure 2 Since the structure's geometry and stress are symmetrical, we analyze a half-structure, and the simplified basic structure is as follows: Figure 2 As shown. There are unknown bending moments X1 and unknown axial forces X2 at the break at the top of the arch. Since the structure and load are symmetrically distributed, the shear force X3 at the top of the arch is 0.

[0079] Step S2: Determine the calculation parameters for the voided lining at the arch crown based on the secondary lining design parameters and the geological conditions. Based on the site conditions, determine the following boundary condition parameters: vertical load q, lateral load e, elastic modulus E of the lining concrete, and elastic subgrade coefficient k at the voided boundary (the boundary is concrete, so the value here is also the elastic modulus of concrete).

[0080] Step S3: See Figure 3 Based on the superposition of displacements, a set of equations for the compatibility of the arch crown displacements is established, and the compliance coefficient δ is calculated through piecewise integration. ik And calculate the displacement Δ caused by the external load at the crown of the arch. ip :

[0081] Based on the superposition of displacements, the arch crown displacement compatibility equation is established, and the compliance coefficient is calculated through piecewise integration:

[0082]

[0083] Where: δ ik δ is the flexibility coefficient, which represents the displacement of the arch crown by a unit unknown force along the direction of the unknown force in the basic structure when the arch foot is rigidly fixed. According to the reciprocal displacement theorem, δ ik = δ ki ;

[0084] Δ ip External loads caused by X i Displacement in the direction;

[0085] β0 and μ0 are the total elastic rotation angle and total horizontal displacement of the arch foot, respectively.

[0086] Calculate the compliance coefficient δik:

[0087]

[0088] Since the span-to-span ratio f / l > 1 / 4, the influence of axial force can be ignored. For example... Figure 3 As shown, for voided lining, voiding causes changes in the cross-sectional thickness, cross-sectional eccentricity, and external load within the voided area. A cylindrical coordinate system is established with the center of the circle as the origin. Piecewise integration is performed on the voided section AB and the non-voided section BC to calculate the compliance coefficient δ. ik :

[0089]

[0090] In the formula:

[0091] h z To calculate the height difference between the cross section and the arch crown position offset from the axis of the hollow lining, the following formula is used:

[0092]

[0093] r b The radius of the offset axis of the lining in the void section is calculated using the following formula:

[0094]

[0095] For a rectangular cross-section, the moment of inertia is:

[0096]

[0097] Where: E is the elastic modulus of concrete; r2 is the radius of the lining axis; r3 is the inner diameter of the lining; H is the lining thickness; h1 is the void height at the arch crown; H1 is the lining thickness at the arch crown; h z The height difference between the cross section and the arch position of the offset axis of the hollow lining is calculated; dα is the area of ​​the infinitesimal element; y is the perpendicular distance between the infinitesimal element dα and the x-axis; b is the width of the cross section; h is the height of the cross section.

[0098] Step S4: Based on the superposition of displacements, establish a set of displacement equations for the arch foot, and calculate the elastic fixation coefficients β1, μ1, β2, μ2 of the arch foot and the displacement β produced at the arch foot by the external load. P μ P

[0099]

[0100] In the formula: β1, μ1, β2, μ2, β P μ P β1 and μ1 are the elastic constants at the arch foot, respectively: β1 and μ1 are the elastic rotation angle and horizontal displacement generated at the arch foot by a unit bending moment at the arch crown; β2 and μ2 are the elastic rotation angle and horizontal displacement generated at the arch foot by a unit axial force at the arch crown; β P μ P These represent the elastic rotation and horizontal displacement generated at the arch foot by a unit external load, respectively; f is the rise of the arch crown.

[0101] Calculate the elastic fixation coefficients β1, μ1, β2, and μ2 of the arch foot:

[0102] ①See Figure 4 The elastic constant coefficient of the arch foot under bending moment:

[0103]

[0104]

[0105] ;

[0106] ②See also Figure 5 The elastic constant coefficient of the arch foot under horizontal force:

[0107]

[0108]

[0109] Calculate the deformation β caused by external loads at the arch foot. P μ P :

[0110] First, determine the external load on the secondary lining. The secondary lining load consists of two parts. One part is the uniformly distributed rock load borne by the non-voided section. Besides the uniformly distributed rock load borne by the non-voided section, since the voided section is not in contact with the initial support, the uniformly distributed rock load that should have been borne by the voided section is transferred to the void boundary during lining deformation, forming a concentrated load F. T F T The horizontal and vertical components are equal to the values ​​of the horizontal and vertical loads of the detached section, respectively. After obtaining the external load on the lining, calculate the bending moment M generated by the external load at the arch foot. 0 P and axial force N 0 P Then, based on the elastic fixation coefficients at the arch foot, the displacement caused by the external load at the arch foot is calculated:

[0111]

[0112]

[0113] .

[0114] Step S5: Couple the crown displacement coordination equation set (1) and the arch foot displacement equation set (2) to calculate the crown bending moment and crown axial force.

[0115]

[0116]

[0117] have to:

[0118]

[0119] Step S6: Based on the relationship between the internal forces at the crown and the internal forces in the cross sections, calculate the internal forces at any cross section j on the lining axis:

[0120]

[0121] Where: M j , N j Q j h represents the internal force at section j; z To calculate the height difference between the cross section and the arch crown position of the offset axis of the hollow lining; φ j M is the central angle corresponding to section j; 0 j P, N 0 j P,Q 0 j P is the internal force caused by the external load on section j.

[0122] Step S7: Calculate the safety factor of the lining section based on the internal forces of the section, determine the range of unsafe sections, and carry out reinforcement:

[0123] Safety factor calculation method:

[0124] ① For plain concrete lining, according to the specification (Highway Tunnel Design Specification JTG 3370.1-2018), the safety factor is calculated based on compressive strength as the standard:

[0125]

[0126] Where: K is the safety factor; N is the axial force (kN); R1 is the tensile ultimate strength of concrete; b is the cross-sectional width (m); φ is the longitudinal bending coefficient of the component, which can be taken as 1 for tunnel lining, open-cut arch ring and sidewall with tight backfill; α is the eccentricity influence coefficient of axial force, which is taken according to the table in the specification; Ra is the compressive ultimate strength of concrete or masonry; b is the cross-sectional width (m); h is the cross-sectional thickness (m).

[0127] The safety factor is calculated based on tensile strength as the standard:

[0128]

[0129] In the formula: K is the safety factor; N is the axial force (kN); R1 is the tensile ultimate strength of concrete; b is the cross-sectional width (m); h is the cross-sectional thickness (m).

[0130] Sections with a safety factor greater than the data in Table 1 are considered safe sections, while sections with a safety factor less than the data in Table 1 are considered dangerous sections.

[0131]

[0132] ② For reinforced concrete lining, according to the "Specifications for Design of Highway Tunnels" (JTG3370.1-2018), the calculation of the strength safety factor of each section of the secondary lining needs to be carried out in two cases: large eccentric compression and small eccentric compression.

[0133] The formula for calculating the section safety factor of a reinforced concrete rectangular section eccentrically compressed member is as follows:

[0134]

[0135]

[0136] The formula for calculating the section safety factor of a reinforced concrete rectangular section eccentrically compressed member is as follows:

[0137]

[0138] When axial force Acting on reinforcing bars Center of gravity and steel bars When the center of gravity is between the two points, the following requirements should be met:

[0139]

[0140] in: For safety factor;

[0141] The bending moment of the cross section;

[0142] The axial force is the force across the cross section.

[0143] This refers to the standard value of the ultimate flexural and compressive strength of concrete. concrete The value is 28.1 ;

[0144] The standard value for the tensile or compressive strength of reinforcing bars is used; longitudinal reinforcing bars are adopted. Its calculated tensile and compressive strength standard values ​​are ;

[0145] C35 concrete represents the ultimate compressive strength of concrete or masonry. The value is 22.5 ;

[0146] This represents the cross-sectional area of ​​the reinforcement in the tension zone;

[0147] This represents the cross-sectional area of ​​the reinforcing steel in the compression zone;

[0148] This is the total height of the cross-section;

[0149] The effective height of the cross section, ;

[0150] The height of the compression zone of the cross section. , Take 0.8;

[0151] The width of the cross section;

[0152] For steel reinforcement and The distance from the center of gravity to the point of application of the axial force.

[0153] Sections with a safety factor greater than the data in Table 2 are considered safe sections, while sections with a safety factor less than the data in Table 2 are considered dangerous sections.

[0154]

[0155] In summary, the internal force calculation method for the voided lining at the arch of the tunnel secondary lining in this embodiment can realize the quantitative analysis and calculation of the internal force of the voided lining, providing guidance for reinforcement parameters.

[0156] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for calculating the internal forces of a tunnel secondary lining arch with a voided lining, characterized in that: Includes the following steps: S1: Establish an internal force calculation model for the tunnel arch lining with insufficient thickness and void defects based on the site conditions; S2: Determine the calculation parameters for the void lining at the arch crown based on the secondary lining design parameters and the geological conditions. S3: Based on the superposition of displacements, list the arch crown displacement compatibility equations, calculate the flexibility coefficients through piecewise integration, and calculate the displacements caused by external loads on the arch crown. S4: Based on the superposition of displacements, list the arch foot displacement equations and calculate the arch foot elastic fixation coefficient and the displacement generated at the arch foot by the external load. S5: Couple the crown displacement compatibility equation and the arch foot displacement equation to calculate the crown bending moment and crown axial force. S6: Based on the relationship between the internal forces at the crown and the internal forces in the cross section, calculate the internal forces at any cross section on the lining axis; S7: Calculate the safety factor of the lining section based on the internal forces of the section, determine the range of unsafe sections, and carry out reinforcement; Step S4 includes the following steps: S401: Based on the superposition of displacements, derive the following set of equations for the arch foot displacements. ； Among them: β1, μ1, β2, μ2, β P μ P Let β1 and μ1 be the elastic constants at the arch foot, where β1 and μ1 are the elastic rotation and horizontal displacement at the arch foot caused by the unit bending moment on the arch lining, respectively; β2 and μ2 are the elastic rotation and horizontal displacement at the arch foot caused by the unit axial force on the arch lining, respectively; β P μ P α and β are the elastic rotation and horizontal displacement generated at the arch foot by the external load, respectively; f is the arch rise; β0 and μ0 are the total elastic rotation and total horizontal displacement at the arch foot, respectively. S402: Calculate the elastic fixation coefficients β1, μ1, β2, μ2 of the arch foot: The elastic constant of the arch foot under bending moment is: ； ； ； The elastic constant coefficient of the arch foot under axial force is: ； ； S403: Calculate the deformation β caused by external loads at the arch foot. P μ P First, determine the external load of the secondary lining. The secondary lining load consists of two parts. One part is the uniformly distributed surrounding rock load borne by the non-voided section. In addition to the uniformly distributed surrounding rock load borne by the non-voided section, since the voided section is not in contact with the initial support, the uniformly distributed surrounding rock load that should have been borne by the voided section is transferred to the void boundary during the lining deformation process, forming a concentrated load F. T F T The horizontal and vertical components are equal to the values ​​of the horizontal and vertical loads of the detached section, respectively; after obtaining the external load of the lining, calculate the bending moment M generated by the external load at the arch foot. 0 P and axial force N 0 P Then, based on the elastic fixation coefficient of the arch foot, the displacement generated at the arch foot by the external load is calculated: ； ； 。 2. The method for calculating the internal forces of the voided lining at the arch of the secondary tunnel lining according to claim 1, characterized in that: Step S1 includes the following steps: S101: Establish a rectangular coordinate system with the axis of the unremoved lining at the top of the arch as the origin, the horizontal direction as the X-axis, and the vertical direction as the Y-axis; S102: In the rectangular coordinate system described in step S101, the origin O of the coordinate system is at the axis of the lining arch, the center of the lining arch is O1, the vertical load of the lining is q, the horizontal load of the lining is e, the semi-central angle corresponding to the arc segment where the arch is located in the multi-centered circular tunnel is α0, the semi-central angle corresponding to the void range of the arch is α1, the outer diameter of the lining is r1, the radius of the lining axis is r2, and the inner diameter of the lining is r3. S103: Since the structure's geometry and stress are symmetrical, we will analyze a half-structure. S104: There are unknown bending moment X1 and unknown axial force X2 at the break at the top of the arch. Since the structure and load are symmetrically distributed, the shear force X3 at the top of the arch is 0.

3. The method for calculating the internal forces of the voided lining at the arch of the secondary tunnel lining according to claim 2, characterized in that: The calculation parameters for the arch caving lining in step S2 are as follows: vertical load q of the lining, horizontal load e of the lining, elastic modulus E of the lining concrete, and elastic foundation coefficient k at the caving boundary.

4. The method for calculating the internal forces of the voided lining at the arch of the secondary tunnel lining according to claim 3, characterized in that: Step S3 includes the following steps: S301: Based on the superposition of displacements, the arch crown displacement compatibility equation is listed: ； Where: δ ik δ is the flexibility coefficient, which represents the displacement of the arch crown along the direction of the unknown force when the arch foot is rigidly fixed in the basic structure. According to the reciprocal displacement theorem, δ ik = δ ki Δ ip External loads caused by X i Displacement in the direction; S302: Calculate the compliance coefficient through piecewise integration: ； S303: Calculate the displacement Δip caused by the external load at the crown: ； 。 5. The method for calculating the internal forces of the voided lining at the arch of the secondary tunnel lining according to claim 4, characterized in that: In step S302, specifically, a cylindrical coordinate system is established with the center O1 of the lining arch as the origin, and piecewise integration is performed on the voided section AB and the non-voided section BC to calculate the flexibility coefficient δ. ik : ； Where: h z To calculate the height difference between the cross section and the arch crown position of the offset axis of the hollow lining; r b H is the radius of the offset axis of the lining in the void section; H is the lining thickness; h1 is the void height of the arch crown; H1 is the lining thickness at the arch crown; hz is the height difference between the calculated section and the arch crown position of the offset axis of the void lining; dα is the area of ​​the infinitesimal element; y is the perpendicular distance between the infinitesimal element dα and the x-axis; b is the width of the section; h is the height of the section.

6. The method for calculating the internal forces of the voided lining at the arch of the secondary tunnel lining according to claim 5, characterized in that: In step S5, the crown displacement compatibility equation and the arch foot displacement equation are calculated in a coupled manner: Depend on: ； ； get: 。 7. The method for calculating the internal forces of the voided lining at the arch of the secondary tunnel lining according to claim 6, characterized in that: In step S6, the internal forces at any section j on the lining axis are calculated based on the relationship between the internal forces at the arch crown and the internal forces at the cross section. ； Among them, M j , N j Q j φ represents the internal force at section j; j M is the central angle corresponding to section j; 0 j P, N 0 j P,Q 0 j P is the internal force caused by the external load on section j.

8. The method for calculating the internal forces of the voided lining at the arch of the secondary tunnel lining according to claim 7, characterized in that: Step S7 includes the following steps: S701: For plain concrete lining or reinforced concrete lining, the safety factor shall be calculated based on compressive strength and tensile strength, respectively. S702: Compare the safety factor obtained in step S701 with the standard value to determine whether it is a dangerous section.

9. The method for calculating the internal forces of the voided lining at the arch of the secondary tunnel lining according to claim 8, characterized in that: In step S701, for plain concrete lining, calculations are performed according to the Highway Tunnel Design Code JTG 3370.1-2018; for reinforced concrete lining, calculations are performed according to the Highway Tunnel Design Code JTG 3370.1-2018.

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