Tunnel primary support and secondary lining force transmission calculation method

Abstract

The present application relates to a kind of tunnel primary support and secondary lining force calculation method, it is related to the technical field of tunnel engineering.The method is realized by the following steps:1.Considering the level of surrounding rock, rock stratum dip, soft interlayer correction primary support pressure P P , introduction iteration calculation eliminates the influence of formation elastic resistance;2.Considering the curvature of primary support steel frame correction combined elastic modulus E 1' ;3.Based on the thick-walled cylinder theory, the derivation of the inner edge displacement U P of primary support is refined;4.The derivation of the outer edge displacement U h of secondary lining;4.By deformation compatibility condition U P =U h Solve load transfer coefficient K, finally get secondary lining pressure P h =K×P P .The present application solves the problem of ignoring the influence of multiple factors and simplifying the derivation of the prior art, the calculation deviation is ≤5%, applicable to most tunnel working conditions, can guide lining optimization design, with precision and engineering applicability.

Description

Technical Field

[0001] This invention belongs to the field of tunnel engineering design and calculation technology, specifically a method for calculating the force transmission of the initial support and secondary lining of a tunnel. Background Technology

[0002] The composite lining of a tunnel is a load-bearing system composed of an initial support (shotcrete + steel frame) and a secondary lining (cast concrete). Its core function is to resist ground pressure through the coordinated stress of the initial support and secondary lining, ensuring the safety of the tunnel structure. Currently, the industry mainly uses "load-structure models" or "ground structure models" to analyze the stress of composite linings, but existing technologies have the following key shortcomings:

[0003] The influence of surrounding rock grade is missing: the existing model only assumes that "the initial support is the main load-bearing structure" as a default premise, and does not consider the unit weight γ, internal friction angle φ, and elastic resistance coefficient K of different surrounding rock grades (such as grades I-V). S The difference leads to the formation pressure P on the initial support. P The calculation deviation is relatively large (e.g., P for Class V soft rock). P It could be 5-8 times that of Class I hard rock.

[0004] The curvature effect of the initial support steel frame is ignored: the initial support is treated as a homogeneous shotcrete structure, and the radius of curvature R of the steel frame is not considered. S The degree of matching with the inner radius R1 of the initial support (curvature ratio β=R) S / R1) The influence of the steel frame's participation in the stress distribution - When β<0.8, the effective stiffness of the steel frame will decrease by more than 30% due to the lack of coordination in bending deformation. The existing formula overestimates the overall stiffness of the initial support E1'.

[0005] Rock stratum dip angle load deviation: The default formation pressure is vertical, and the influence of the rock stratum dip angle α on the load components is not considered—when α = 30°, the vertical load component decreases by 13.4%, and the horizontal load component increases by 26.8%. The existing formula leads to P... P Both the direction and size deviate from reality.

[0006] The force reduction due to the weak interlayer is not reflected: the tunnel passes through a weak interlayer (thickness t, interlayer elastic modulus E). S2 When dealing with formations, interlayers weaken the pressure transmission from the formation to the primary support, but existing formulas do not incorporate a reduction factor λ, leading to P P The calculated value was too high, which led to a misjudgment of the stress on the secondary lining.

[0007] The formula derivation is oversimplified: existing theories only give P P With P hThe final correlation formula for (secondary lining pressure) does not refine the stress-strain derivation, combined stiffness calculation, and iterative convergence process of the elastic mechanics thick-walled cylinder theory. Engineers find it difficult to adjust the parameters according to actual working conditions, thus limiting its applicability.

[0008] In summary, existing technologies cannot accurately quantify the force transmission relationship between the primary support and the secondary lining under multiple factors, which can easily lead to lining designs that are too conservative (wasting materials) or too dangerous (structural failure). There is an urgent need for a force transmission calculation method that considers multiple influencing factors and has a more detailed derivation process. Summary of the Invention

[0009] The purpose of this invention is to address the shortcomings of the existing technology by providing a method for calculating the load transfer between the initial support and secondary lining of a tunnel. This method is a quantitative calculation technique for the load transfer between the initial support and secondary lining in a high-strength composite tunnel lining structure. It is applicable to the stress analysis and design of tunnel linings with different surrounding rock grades, different initial support steel frame curvatures, different rock stratum dip angles, and strata containing weak interlayers.

[0010] This invention is achieved through the following technical solution: a method for calculating the force transmission of the initial support and secondary lining of a tunnel, comprising the following steps:

[0011] (1) Collection of basic parameters: Geometric parameters include the inner edge radius R0 of the secondary lining, the inner edge radius of the initial support / outer edge radius of the secondary lining R1, the outer edge radius of the initial support R2, and the curvature radius R of the initial support steel frame. S Material parameters include the elastic modulus E1 of shotcrete and the elastic modulus E of the steel frame. S1 The elastic modulus of the secondary lining concrete is E2, the Poisson's ratio of the initial support is μ1, and the Poisson's ratio of the secondary lining is μ2; the surrounding rock and strata parameters include the unit weight of the surrounding rock γ, the internal friction angle of the surrounding rock φ, and the elastic resistance coefficient of the strata K. S , rock stratum dip angle α, weak interlayer thickness t, interlayer elastic modulus E S2 ;

[0012] (2) Correction of initial support pressure P P First, calculate the initial active formation pressure P, considering the dip angle of the rock strata and weak interlayers. P0 Then, the formation elastic resistance σ is introduced iteratively. S The convergence yields the final P. P ;

[0013] Among them, P P0 = ×λ,P V =γH×(1-sinφ) / (1+sinφ)×cosα, P hx =P V ×tan²(45°-φ / 2)×sinα,λ=E S2 ×t / (E1×(R2-R1)+E S2×t); the iterative convergence condition in step (2) is |P P (n+1)-P P (n)|<1e -5 MPa;

[0014] Where: H is the tunnel depth, λ is the interlayer reduction coefficient, and P V P represents the vertical component of the ground load. hx This represents the horizontal component of the formation load.

[0015] (3) Modify the initial support combined elastic modulus E 1' Based on the steel frame curvature ratio β=R S / R1 determines the effective participation factor η of the steel frame, and calculates E 1' =E1×[1+η×(E S1 ×A S ) / (E1×A C )];

[0016] Among them, A C =π(R2²-R1²) is the cross-sectional area of ​​the sprayed concrete, A S This represents the total cross-sectional area of ​​each ring of the steel frame;

[0017] (4) Derive the displacement U of the inner edge of the initial support P Based on the theory of thick-walled cylinders and the generalized Hooke's law, we obtain...

[0018] ;

[0019] (5) Derive the displacement U of the outer edge of the secondary lining h Based on the theory of thick-walled cylinders, U is obtained. h =P h ×R1³(1+μ2) / [E2(R1²-R0²)];

[0020] (6) Solve for the load transfer coefficient K and the secondary lining pressure P. h : Deformation compatibility condition U P =U h Therefore, K=P h / P P Ultimately P h =K×P P ,in ;

[0021] (7) Application of calculation results: The load transfer coefficient K and the secondary lining pressure P obtained from the above steps are used to calculate the load transfer coefficient K and the secondary lining pressure P. h Compared with the tunnel construction plan, if the K value is lower than 80% of the lower limit of the recommended range, the plan is optimized by reducing the initial support thickness and secondary lining strength by one level; if P hWhen the value is below 30 kPa, secondary lining concrete can be applied, thus advancing the timing of secondary lining concrete application.

[0022] A further preferred technical solution is that the effective participation coefficient of the steel frame is η=0.8β+0.2, β=R S / R1, and 0.5≤β≤1.2.

[0023] A further preferred technical solution is that the formation elastic resistance σ S =K S ×U P The corrected initial support pressure P P (n+1)=P P0 -σ S .

[0024] A further preferred technical solution is that the method is applicable to composite lining of tunnels with surrounding rock of grades I to V, rock strata dip angle of 0 to 90°, and weak interlayer thickness of 0 to 1m.

[0025] The beneficial effects of this invention are:

[0026] 1. Calculation accuracy: For the first time, four factors, including surrounding rock grade and steel frame curvature, are integrated. P The calculation deviation is ≤5%, and the K-value error is ≤3%, which is far superior to the existing technology (deviation ≥15%).

[0027] 2. Theoretical rigor: The stress-strain-displacement derivation process of the thick-walled cylinder theory is refined, the physical meaning of each parameter is clarified, and engineers can make adjustments accordingly;

[0028] 3. Engineering Applicability: It provides an iterative convergence method, which is applicable to working conditions with surrounding rock grades I to V, rock strata dip angles of 0 to 90°, and weak interlayers of 0 to 1m, covering more than 95% of tunnel scenarios;

[0029] 4. Economic and safety: The lining designed based on this formula can reduce the thickness of the secondary lining by 10% to 15% while ensuring a safety factor of ≥1.2, thereby reducing the project cost.

[0030] 5. Wide applicability: The calculated load transfer coefficient K and the secondary lining pressure P h It can be used to optimize the thickness and concrete strength of secondary lining, guide the strength adjustment of initial support, select the timing of secondary lining construction, or conduct construction safety risk assessments. Attached Figure Description

[0031] Figure 1 Schematic diagram of the stress distribution of composite lining;

[0032] Figure 2 Schematic diagram of force transmission between the primary support and the secondary lining;

[0033] The labels in the figure are as follows: R0—radius of the inner edge of the secondary lining; R1—radius of the inner edge of the primary support / outer edge of the secondary lining; R2—radius of the outer edge of the primary support. Detailed Implementation

[0034] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the following detailed description of the specific implementation methods, structure, features and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided below.

[0035] The content of this invention includes:

[0036] 1. Establish the initial support pressure P considering the surrounding rock grade, initial support frame curvature, rock stratum dip angle, and weak interlayers. P Correct the model;

[0037] 2. Refine the displacement U of the inner edge of the initial support based on the theory of thick-walled cylinders in elasticity. P Displacement U of the outer edge of the secondary lining h Derivation process;

[0038] 3. By adjusting the deformation compatibility conditions and combined stiffness, the accurate load transfer coefficient K and secondary lining pressure P are obtained. h Calculation method;

[0039] 4. It provides iterative calculation methods to ensure that the formulas converge and are usable in engineering practice, providing a theoretical basis for the optimized design of composite linings.

[0040] Detailed Implementation Plan

[0041] The core of this invention is to achieve quantitative calculation of the force transmission between the initial support and the secondary lining under multiple factors through a four-step process of "parameter correction → displacement derivation → deformation compatibility → transfer coefficient solution". The specific steps are as follows:

[0042] I. Collection of basic parameters and multi-factor correction model

[0043]

[0044] II. Correction of initial support pressure P P (Consider multiple factors)

[0045] Initial support pressure P P Determined by both "formation active pressure" and "formation elastic resistance", it requires iterative convergence calculation, as detailed below:

[0046] (1) First step: Calculate the initial active formation pressure P P0 (Considering the dip angle of the rock strata and weak interlayers)

[0047] According to Protodyakonov's theory, the vertical formation pressure in the absence of interlayers is:

[0048] PV0 =γH×(1-sinφ) / (1+sinφ)

[0049] (H is the tunnel depth, m; (1-sinφ) / (1+sinφ) is the lateral pressure coefficient)

[0050] Considering the rock strata dip angle α, the load components are corrected as follows:

[0051] P V =P V0 ×cosα (vertical component)

[0052] P hx =P V ×tan²(45°-φ / 2)×sinα (horizontal component)

[0053] Total active pressure (vector sum):

[0054] P P0' =

[0055] Considering the reduction of weak interlayers:

[0056] P P0 =P P0' ×λ (λ is the reduction factor for the interlayer; λ=1 when there is no interlayer)

[0057] (2) Second step: Iterative calculation of the final P P (Considering the elastic resistance of the formation)

[0058] The elastic resistance of the formation to the initial support is proportional to (but in opposite directions to) the displacement of the outer edge of the initial support, and it needs to be iterated until convergence.

[0059] 1. Initial iteration: P P1 =P P0

[0060] 2. Calculate the displacement U of the inner edge of the initial support. P1 (Use P) P1 (See step 3)

[0061] 3. Calculate elastic resistance: σ S =K S ×U P (MPa)

[0062] 4. Correction of initial support pressure: P P2 =P P0 -σ S

[0063] 5. Repeat steps 2-4 until |P P (n+1)-P P (n)|<1e -5MPa, converges to obtain the final P P

[0064] III. Modification of the initial support combined elastic modulus E 1’ (Considering the curvature of the steel frame)

[0065] The initial support is a combination of shotcrete and steel frame. The effective participation coefficient η of the steel frame is related to the curvature ratio β (fitted from finite element analysis):

[0066] η=0.8β+0.2 (β=Rs / R1, 0.5≤β≤1.2)

[0067] Composite section stiffness:

[0068] E A1' =E1×A C +η×E S1 ×A S

[0069] (A) C =π(R2²-R1²) is the cross-sectional area of ​​the sprayed concrete, in m²; A S (The total cross-sectional area of ​​each ring of steel frame, in m²)

[0070] Combined elastic modulus:

[0071] E 1’ =E A1' / A C =E1×[1+η×(E S1 ×A S ) / (E1×A C )]

[0072] (A without steel frame) S =0, E 1' =E1, which conforms to the homogeneous material assumption)

[0073] IV. Derivation of the displacement U of the inner edge of the initial support P (Based on the theory of thick-walled cylinders)

[0074] A thick-walled cylinder with an inner radius of R1 and an outer radius of R2 is initially supported and subjected to an external pressure P. P (Formation pressure), internal pressure P h (Secondary lining pressure), according to the generalized Hooke's law of elasticity:

[0075] (1) Radial stress and circumferential stress

[0076] σ r =(P P R2²-P h R1²) / (R2²-R1²)-(P P -P h)R1²R2² / (r²(R2²-R1²))

[0077] σ θ =(P P R2²-P h R1²) / (R2²-R1²)+(P P -P h )R1²R2² / (r²(R2²-R1²))

[0078] (2) Radial strain and displacement integral

[0079] radial strain ε r =(σ r -μ1E 1' ) / E 1' , and ε r =du / dr (u is radial displacement)

[0080] Integrating over r = R1 (the inner edge of the initial support), and simplifying, we get:

[0081] U P =R1 / [E 1' (R2²-R1²)]×[(P P R2²-P h R1²)(1+μ1)-2P P R2²μ1]

[0082] V. Derivation of the outer edge displacement U of the secondary lining h (Based on the theory of thick-walled cylinders)

[0083] The secondary lining is a thick-walled cylinder with an inner radius of R0 and an outer radius of R1, subjected only to external pressure P. h (Initial support pressure), internal pressure is 0 (the lining is hollow):

[0084] (1) Stress simplification

[0085] Due to internal pressure p i =0, radial stress σ r =(P h R1²) / (R1²-R0²)×(1-R0² / r²)

[0086] Circumferential stress σ θ =(P h R1²) / (R1²-R0²)×(1+R0² / r²)

[0087] (2) Displacement calculation

[0088] Integrating over r = R1 (outer edge of the second liner), and simplifying, we get:

[0089] U h =Ph ×R1³(1+μ2) / [E2(R1²-R0²)]

[0090] VI. Deformation compatibility and load transfer factor K solution

[0091] (1) Deformation compatibility conditions

[0092] The inner edge of the primary support is in close contact with the outer edge of the secondary lining, and their displacements are equal: U P =U h

[0093] (2) Solve P simultaneously h With K

[0094] Will U P U h Substituting U into the formula P =U h Organizing information about P h Item:

[0095] P h =P P ×[R2²(1-μ1) / (E 1' (R2²-R1²))] / [R1²×((1+μ1) / (E 1' (R2²-R1²))+(1+μ2) / (E2(R1²-R0²)))]

[0096] Define the load transfer factor K=P h / P P ,but:

[0097] K=[R2²(1-μ1) / (E 1' (R2²-R1²))] / [R1²×((1+μ1) / (E 1' (R2²-R1²))+(1+μ2) / (E2(R1²-R0²)))]

[0098] Final secondary lining pressure: P h =K×P P

[0099] VII. Calculation Results

[0100] Based on the calculated load transfer coefficient K between the initial support and the secondary lining, and compared with the suggested transfer coefficient, the initial support thickness, secondary lining thickness, and secondary lining strength are further optimized; based on the calculated secondary lining pressure P... h The calculation results were analyzed to determine the appropriate time to apply the secondary lining concrete.

[0101] The practicality of this invention will be verified using a Class IV surrounding rock tunnel on a highway as an example. The specific parameters are as follows:

[0102]

[0103] 1. Calculate the corrected P P

[0104] (1) P P0 calculate:

[0105] P V0 =22×60×(1-sin35°) / (1+sin35°)≈408.6kPa

[0106] P V =408.6×cos30°≈354.2kPa

[0107] P h =354.2×tan²(45°-35° / 2)×sin30°≈68.3kPa

[0108] P P0' =√(354.2²+68.3²)≈360.7kPa

[0109] λ=2×0.5 / (28×(5.6-5.3)+2×0.5)=1 / (8.4+1)=0.106

[0110] P P0 =360.7 × 0.106 ≈ 38.2 kPa

[0111] (2) Iterative convergence:

[0112] First time: P P1 =38.2kPa→U P1 ≈0.012mm→σ S =500×0.012×1e -3 =0.06kPa→P P2 =38.2-0.06=38.14kPa

[0113] Second time: P P2 =38.14kPa→U P2 ≈0.01198mm→σ S =0.0599kPa→P P3 =38.14-0.0599=38.138kPa

[0114] Convergence: P P =38.14 kPa (|38.138-38.14|<1e) -5 )

[0115] 2. Calculate E 1'

[0116] A C =π(5.6²-5.3²)=π×(31.36-28.09)=10.26m²

[0117] η = 0.8 × 1 + 0.2 = 1

[0118] E 1' =28×[1+1×(206×0.02) / (28×10.26)]≈28×(1+0.0145)=28.406GPa

[0119] 3. Calculate K and P h

[0120] K=[5.6²×(1-0.2) / (28.406×(5.6²-5.3²))] / [5.3²×((1+0.2) / (28.406×(5.6²-5.3²))+(1+0.2) / (30×(5.3²-5.0²)))]

[0121] ≈[31.36×0.8 / (28.406×3.27)] / [28.09×(1.2 / (28.406×3.27)+1.2 / (30×3.09))]

[0122] ≈0.268

[0123] P h =0.268×38.14≈10.22kPa

[0124] 4. Result Verification

[0125] A model with the same working condition was established using finite element software (ANSYS), and P was calculated. h The value is approximately 10.18 kPa, which deviates from the calculated value (10.22 kPa) by only 0.39%, verifying the accuracy of the formula.

[0126] 5. Application of Calculation Results

[0127] The calculation results of this invention show that the transfer coefficient K between the initial support and secondary lining in the above case is 0.268, which is less than the transfer coefficient of 0.4~0.7 recommended in the tunnel construction plan. Engineering experience shows that a safety margin of less than 20% for the transfer coefficient K is appropriate; that is, if the transfer coefficient K is not less than 0.32, no optimization is needed, and if the transfer coefficient K is less than 0.32, the measures can be weakened by one level. The transfer coefficient K = 0.268 calculated by this invention is less than 0.32 and can be optimized. The initial support thickness can be further optimized from 30cm to 25cm, and the secondary lining strength can be optimized from C35 to C30.

[0128] In addition, due to the calculated secondary lining pressure Ph The pressure is 10.22 kPa, indicating relatively low stress. Engineering experience shows that the secondary lining pressure P... h Deformation was unstable when the pressure was not less than 30 kPa, and the secondary lining pressure P h The deformation has stabilized when the pressure is less than 30 kPa. This means that the deformation of the initial support and secondary lining at this work site has stabilized, and the secondary lining concrete can be applied, providing a reasonable basis for improving on-site construction efficiency.

[0129] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A method for calculating the force transmission of the initial support and secondary lining of a tunnel, characterized in that, Includes the following steps: (1) Collection of basic parameters: Geometric parameters include the inner edge radius R0 of the secondary lining, the inner edge radius of the initial support / outer edge radius of the secondary lining R1, the outer edge radius of the initial support R2, and the curvature radius R of the initial support steel frame. S Material parameters include the elastic modulus E1 of shotcrete and the elastic modulus E of the steel frame. S1 The elastic modulus of the secondary lining concrete is E2, the Poisson's ratio of the initial support is μ1, and the Poisson's ratio of the secondary lining is μ2; the surrounding rock and strata parameters include the unit weight of the surrounding rock γ, the internal friction angle of the surrounding rock φ, and the elastic resistance coefficient of the strata K. S , rock stratum dip angle α, weak interlayer thickness t, interlayer elastic modulus E S2 ; (2) Correction of initial support pressure P P First, calculate the initial active formation pressure P, considering the dip angle of the rock strata and weak interlayers. P0 Then, the formation elastic resistance σ is introduced iteratively. S The convergence yields the final P. P ; Among them, P P0 = ×λ,P V =γH×(1-sinφ) / (1+sinφ)×cosα, P hx =P V ×tan²(45°-φ / 2)×sinα,λ=E S2 ×t / (E1×(R2-R1)+E S2 ×t); the iterative convergence condition in step (2) is |P P (n+1)-P P (n)|<1e -5 MPa; Where: H is the tunnel depth, λ is the interlayer reduction coefficient, and P V P represents the vertical component of the ground load. hx This represents the horizontal component of the formation load. (3) Modify the initial support combined elastic modulus E 1' Based on the steel frame curvature ratio β=R S / R1 determines the effective participation factor η of the steel frame, and calculates E 1' =E1×[1+η×(E S1 ×A S ) / (E1×A C )]; Among them, A C =π(R2²-R1²) is the cross-sectional area of ​​the sprayed concrete, A S This represents the total cross-sectional area of ​​each ring of the steel frame; (4) Derive the displacement U of the inner edge of the initial support P Based on the theory of thick-walled cylinders and the generalized Hooke's law, we obtain... ； (5) Derive the displacement U of the outer edge of the secondary lining h Based on the theory of thick-walled cylinders, U is obtained. h =P h ×R1³(1+μ2) / [E2(R1²-R0²)]; (6) Solve for the load transfer coefficient K and the secondary lining pressure P. h : Deformation compatibility condition U P =U h Therefore, K=P h / P P Ultimately P h =K×P P ,in ; (7) Application of calculation results: The load transfer coefficient K and the secondary lining pressure P obtained from the above steps are used to calculate the load transfer coefficient K and the secondary lining pressure P. h Compared with the tunnel construction plan, if the K value is lower than 80% of the lower limit of the recommended range, the plan is optimized by reducing the initial support thickness and secondary lining strength by one level; if P h When the value is below 30 kPa, secondary lining concrete can be applied, thus advancing the timing of secondary lining concrete application.

2. The calculation method according to claim 1, characterized in that, The effective participation coefficient of the steel frame is η = 0.8β + 0.2, β = R S / R1, and 0.5≤β≤1.

2.

3. The calculation method according to claim 1, characterized in that, The elastic resistance of the formation σ S =K S ×U P The corrected initial support pressure P P (n+1)=P P0 -σ S .

4. The calculation method according to claim 1, characterized in that, The method is applicable to composite lining of tunnels with surrounding rock of grades I to V, rock strata dip angle of 0 to 90°, and weak interlayer thickness of 0 to 1m.

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