A calculation method for pipe-roof support of a two-parameter foundation beam

Abstract

The present invention discloses a calculation method for pipe-roof support of a double-parameter foundation beam. According to the flexural differential control equation of a beam on the Pasternak foundation model, considering the three-dimensional influence of the beam on the elastic foundation, the flexural differential equation is solved to obtain the analytical solution of the flexural deformation under any load on the foundation model; the entire length of the pipe-roof is divided into different sections for theoretical analysis and calculation; an appropriate beam model is selected to simplify each section; appropriate constraint conditions are selected to simplify the pipe-roof end; according to the actual tunnel burial depth, an appropriate calculation theory for surrounding rock pressure is selected, and the load distribution form of the entire length of the pipe-roof is appropriately assumed; the analytical solutions of the flexural deformations of each section are obtained by solving through the continuity conditions of the deflection w, rotation angle θ, bending moment M, and generalized shear force Q at the sectional nodes and the boundary conditions of the pipe-roof end.

Description

Technical Field

[0001] The present invention relates to the field of tunnel and underground engineering, and particularly to a calculation method for pipe shed support of a two-parameter foundation beam. Background Technique

[0002] In the construction of tunnel projects with poor surrounding rock conditions or large excavation sections, advanced support measures are often required to prevent ground collapse, control advanced deformation, and gain time for the initial support to take effect. Among the existing advanced support measures, pipe sheds are widely used in the construction of tunnel entrances and exits, poor geological sections, or super-large cross-section tunnels due to their advantages such as simple technology, convenient construction, and relatively low cost. The wide application of pipe sheds has promoted the development of pipe shed design and calculation theories.

[0003] The existing pipe shed calculation formulas based on the elastic foundation beam theory generally regard the pipe shed as an Euler beam resting on an elastic foundation. Based on the assumption that the load is uniformly distributed along the entire length of the pipe shed and ignoring the three-dimensional influence of the beam on the elastic foundation, a calculation method for the pipe shed on the elastic foundation is established. These formulas can guide the pipe shed design calculation to a certain extent and reflect the stress state of the pipe shed. However, due to the lack of consideration of actual engineering conditions, there is a large deviation between the theoretical calculation results and the actual test results. In the theoretical analysis of similar problems, the vast majority do not comprehensively consider the three-dimensional influence of the beam on the elastic foundation. Compared with the undisturbed rock and soil mass in the deep part, the subgrade reaction coefficient and shear modulus of the disturbed rock and soil mass in front of the tunnel face have decreased to a certain extent. Using the existing pipe shed support calculation theory will inevitably overestimate the bearing capacity of the pipe shed support.

[0004] The design parameters of the pipe shed are related to the control effect of the surrounding rock stability and the construction safety. Therefore, it is necessary to improve the pipe shed support calculation theory and establish a more complete pipe shed support calculation method to guide the pipe shed construction and ensure the safety of tunnel project construction. Summary of the Invention

[0005] The purpose of the present invention is to provide a calculation method for pipe shed support of a two-parameter foundation beam. The specific technical solution is as follows: A calculation method for pipe shed support of a two-parameter foundation beam, characterized by including the following steps:

[0006] S1. According to the deflection differential control equation of the beam on the Pasternak foundation model, considering the three-dimensional influence of the beam on the elastic foundation, use the initial parameter solution method to solve the deflection differential equation to obtain the analytical solution of the deflection deformation under any load on the Pasternak foundation model;

[0007] S2. According to the actual tunnel construction steps, divide the entire length of the pipe shed into different sections for theoretical analysis and calculation;

[0008] S3. Select an appropriate beam model to simplify each section according to the actual support situation;

[0009] S4. Select appropriate boundary conditions to simplify the end of the pipe shed according to the actual excavation conditions and the stress and deformation of the pipe shed end;

[0010] S5. Select an appropriate theory for calculating the surrounding rock pressure and appropriately assume the load distribution form along the entire length of the pipe shed according to the actual tunnel burial depth;

[0011] S6. Obtain the analytical solution of the flexural deformation of each section by solving through the continuity conditions of the deflection w, rotation angle θ, bending moment M, and generalized shear force Q at the segmented nodes and the boundary conditions at the end of the pipe shed.

[0012] Step S1 is specifically as follows:

[0013] S1.1. Establish the differential control equation of the flexure of the beam on the Pasternak foundation, and its expression is Equation (1):

[0014]

[0015] where k and G p are the subgrade reaction coefficient and the subgrade shear modulus respectively, EI is the equivalent flexural rigidity of the pipe shed cross-section, EI = E c I c + E s I s , E c , I c , E s and I s are the elastic modulus of the filled concrete, the elastic modulus of the steel for the pipe shed, the moment of inertia of the cross-section of the filled concrete, and the moment of inertia of the cross-section of the steel for the pipe shed respectively; b is the calculated width of the load on the beam, b = 0.5πD, D is the outer diameter of the pipe shed; q(x) is the load on the beam, w(x) is the deflection at the position x on the beam; b * is the calculated width of the subgrade reaction,

[0016] S1.2. Considering the three-dimensional influence of the beam on the elastic foundation, use the initial parameter method to solve the differential control equation of the flexure in Step S1.1, and obtain the analytical solution of the flexural deformation under any load on the Pasternak foundation model, specifically:

[0017] (1) When,

[0018]

[0019]

[0020]

[0021]

[0022] (2) When

[0023]

[0024]

[0025]

[0026]

[0027] (3) When

[0028]

[0029]

[0030]

[0031]

[0032] (4)G p When = 0, the model degenerates into the Winkler foundation beam model,

[0033]

[0034]

[0035]

[0036]

[0037] (5)k = G p When = , the model degenerates into the ordinary beam model,

[0038]

[0039] [[ID=*66]]

[0040]

[0041]

[0042] In Equations (2) - (21): w0, θ0, M0, and Q0 are the deflection, rotation angle, bending moment, and generalized shear force of the initial cross-section, and z is the integration variable,

[0043] It should be noted that there seems to be a small error in the original text where "When = " in ID=58 should probably be "When = 0" or some other correct value. This translation is based on the provided text as accurately as possible.

[0044]

[0045]

[0046]

[0047] J1(x) = cos(λx)sinh(λx), J2(x) = cos(λx)cosh(λx),

[0048] J3(x) = sin(λx)cosh(λx), J4(x) = sin(λx)sinh(λx).

[0049] Step S2 is specifically as follows: According to the typical tunnel construction steps, the full length of the pipe shed is divided into the initial support closed section, the initial support unclosed section, the excavation unprotected section, the loosened section of the soil mass in front of the heading face, and the unloosened section; among them, the lengths of the initial support closed section, the initial support unclosed section, and the excavation unprotected section are determined according to the actual support situation; the length of the loosened section of the soil mass in front of the heading face is: where h is the tunnel excavation height, is the internal friction angle of the soil mass in front; the unloosened section is determined according to the distance between the end of the pipe shed and the heading face: where ΔL is the distance between the end of the pipe shed and the heading face.

[0050] Step S3 is specifically as follows: Considering the supporting effect of the initial support, the initial support closed section and the initial support unclosed section are simplified into a Pasternak foundation beam model; considering the hysteresis effect of the initial support, the formation parameters of the initial support unclosed section are obtained by reducing the formation parameters of the initial support closed section, and the reduction coefficient is taken as ξ; the excavation unprotected section is simplified into a general beam model and into a Winkler foundation beam model when considering the pipe shed grouting reinforcement shell effect; the loosened section and the unloosened section of the soil mass are simplified into a Pasternak foundation beam model, and the formation parameters of the loosened section are obtained by reducing the formation parameters of the unloosened section, and the reduction coefficient is taken as η.

[0051] Step S4 is specifically as follows: The end of the pipe shed close to the initial support is simplified into a fixed constraint with a certain initial deflection and rotation angle or a spring constraint with a certain stiffness; the end far from the initial support is simplified into a fixed constraint or a free end.

[0052] Step S5 is specifically as follows: The load is obtained by any one of the methods of the railway tunnel design code method, the Terzaghi theory, the Prandtl theory, or the measured results; the distribution form of the load is assumed to be any one of the uniform distribution, the triangular distribution, and the normal distribution.

[0053] Applying the technical solution of the present invention has the following beneficial effects:

[0054] The present invention belongs to an analytical calculation method, which is characterized by being fast, simple, reliable, and having a low calculation cost; it can consider the three-dimensional influence on the double-parameter foundation beam and the situation of any load acting on the pipe shed, so it can more reasonably reflect the actual stress state of the pipe shed, calculate the deformation and internal force of the pipe shed more accurately and reliably, and be more in line with the actual engineering conditions.

[0055] In addition to the purposes, features, and advantages described above, the present invention has other purposes, features, and advantages. The following will refer to the drawings to further elaborate on the present invention in detail. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] The drawings forming a part of this application are used to provide a further understanding of the present invention. The schematic embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation to the present invention. In the drawings:

[0057] Figure 1 is a schematic diagram for calculating a short beam on a Pasternak foundation;

[0058] Where: 1 represents the short beam, 2 represents the load on the beam, 3 represents the generalized shear force on the beam, and 4 represents the bending moment on the beam.

[0059] Figure 2 is a diagram for dividing the pipe shed section;

[0060] Where: 5 represents the unclosed section of the primary support, 6 represents the closed section of the primary support, 7 represents the unexcavated and unprotected section, 8 represents the loosened section of the soil in front of the heading face, 9 represents the unloosened section, 10 represents the rupture surface, 11 represents the rupture angle, 12 represents the pipe shed, 13 represents the primary support, and 14 represents the load;

[0061] Figure 3 is a schematic curve diagram generated based on the deflection values of each section in this embodiment. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0062] The following will elaborate on the embodiments of the present invention in detail with reference to the drawings, but the present invention can be implemented in many different ways defined and covered by the claims.

[0063] Embodiment

[0064] This embodiment provides a calculation method for pipe shed support of a double-parameter foundation beam, including the following steps:

[0065] S1. According to the flexural differential control equation of the beam on the Pasternak foundation model, considering the three-dimensional influence of the beam on the elastic foundation, use the initial parameter method to solve the flexural differential equation to obtain the analytical solution of the flexural deformation under any load on the Pasternak foundation model;

[0066] S1.1. Establish the differential control equation for the flexure of a beam on a Pasternak foundation, whose expression is Equation (1):

[0067]

[0068] where k and G p are the subgrade reaction coefficient and the subgrade shear modulus respectively, and EI is the equivalent flexural rigidity of the pipe shed cross-section, EI = E c I c + E s I s E c , I c E s and I s are the elastic modulus of the filled concrete, the elastic modulus of the steel for the pipe shed, the moment of inertia of the cross-section of the filled concrete, and the moment of inertia of the cross-section of the steel for the pipe shed respectively; b is the calculated width of the load on the beam, b = 0.5πD, where D is the outer diameter of the pipe shed; q(x) is the load on the beam, and w(x) is the deflection at the position x on the beam; b * is the calculated width of the subgrade reaction,

[0069] S1.2. Considering the three-dimensional influence of the beam on the elastic foundation, use the initial parameter method to solve the differential control equation for flexure in Step S1.1, and obtain the analytical solution for the flexural deformation under any load on the Pasternak foundation model, specifically:

[0070] (1) When

[0071]

[0072]

[0073]

[0074]

[0075] (2) When

[0076]

[0077]

[0078]

[0079]

[0080] (3) When

[0081]

[0082]

[0083]

[0084]

[0085] (4)G p When = 0, the model degenerates into the Winkler foundation beam model.

[0086]

[0087]

[0088]

[0089]

[0090] (5)k = G p When = 0, the model degenerates into the ordinary beam model.

[0091]

[0092]

[0093]

[0094]

[0095] In Equations (2) - (21): w0, θ0, M0, and Q0 are the deflection, rotation angle, bending moment, and generalized shear force of the initial section, and z is the integration variable.

[0096]

[0097]

[0098]

[0099]

[0100] J1(x) = cos(λx)sinh(λx), J2(x) = cos(λx)cosh(λx),

[0101] J3(x) = sin(λx)cosh(λx), J4(x) = sin(λx)sinh(λx).

[0102] S2. Divide the full length of the pipe shed into different sections according to the actual tunnel construction steps for theoretical analysis and calculation;

[0103] Specifically, according to the typical tunnel construction steps, divide the full length of the pipe shed into the initial support closed section, the initial support unclosed section, the excavation unprotected section, the loosened soil section in front of the heading face, and the unloosened section; among them, the lengths of the initial support closed section, the initial support unclosed section, and the excavation unprotected section are determined according to the actual support situation; the length of the loosened soil section in front of the heading face is: where h is the tunnel excavation height, is the internal friction angle of the soil in front; the unloosened section is determined according to the distance between the end of the pipe shed and the heading face: where ΔL is the distance between the end of the pipe shed and the heading face.

[0104] S3. Select an appropriate beam model to simplify each section according to the actual support situation;

[0105] Specifically, considering the supporting effect of the initial support, simplify the initial support closed section and the initial support unclosed section into the Pasternak foundation beam model; considering the hysteresis effect of the initial support, the formation parameters of the initial support unclosed section are obtained by reducing the formation parameters of the initial support closed section, and the reduction coefficient is taken as ξ; simplify the excavation unprotected section into an ordinary beam model, and simplify it into a Winkler foundation beam model when considering the grouting reinforcement shell effect of the pipe shed; simplify the loosened soil section and the unloosened section into the Pasternak foundation beam model, and the formation parameters of the loosened section are obtained by reducing the formation parameters of the unloosened section, and the reduction coefficient is taken as η. In this embodiment, take ξ = 0.6 and η = 0.5.

[0106] S4. Select appropriate boundary conditions to simplify the end of the pipe shed according to the actual excavation conditions and the stress and deformation conditions at the end of the pipe shed;

[0107] Specifically, simplify the end of the pipe shed close to the initial support into a fixed constraint with a certain initial deflection and rotation angle or a spring constraint with a certain stiffness; simplify the end far from the initial support into a fixed constraint or a free end. In this embodiment, taking the case where the distance between the heading face and the end of the pipe shed is relatively far as an example, simplify the end of the pipe shed close to the initial support into a fixed constraint with a certain initial deflection and rotation angle; simplify the end far from the initial support into a fixed constraint.

[0108] S5. Select an appropriate surrounding rock pressure calculation theory according to the actual tunnel depth situation, and appropriately assume the load distribution form of the full length of the pipe shed;

[0109] Specifically, the load is obtained by any one of the methods in the railway tunnel design code method, Terzaghi's theory, Prandtl's theory, or the measured results; the distribution form of the load is assumed to be any one of the uniform distribution, triangular distribution, and normal distribution. In this embodiment, the load is calculated according to Terzaghi's theory, and it is assumed that the load on the closed section of the primary support, the unclosed section of the primary support, the unclosed excavation section, and the soil loosening section is uniformly distributed; it is assumed that the unloosened section does not bear the load.

[0110] S6. The analytical solution of the flexural deformation of each section is obtained by solving through the continuity conditions of the deflection w, rotation angle θ, bending moment M, and generalized shear force Q at the segmented nodes and the boundary conditions at the end of the pipe shed.

[0111] For different situations, the analytical solutions obtained above are used, and the deformation and internal force at one end of the beam are used as the initial parameters to represent the deformation and internal force at the other end of the beam (where the deformation is the deflection w and rotation angle θ, and the internal force is the bending moment M and generalized shear force Q). As Figure 1 shown, the deformation and internal force at point O are used as the initial parameters to represent the deformation and internal force at point P. When ...

[0112]

[0113]

[0114]

[0115]

[0116] where, w O , θ O , M O , Q O are the deflection, rotation angle, bending moment, and generalized shear force at point O; w P , θ P , M P , Q P are the deflection, rotation angle, bending moment, and generalized shear force at point P; L is the length of the short beam. The above expressions can be written as:

[0117] w P = w O K 11 + θ O K 12 + M O K 13 + Q O K 14 + S1;

[0118] θ P = w O K 21 + θ O K22 +M O K 23 +Q O K 24 +S2;

[0119] M P = w O K 31 + θ O K 32 +M O K 33 +Q O K 34 +S3;

[0120] Q P = w O K 41 + θ O K 42 +M O K 43 +Q O K 44 +S4;

[0121] where K 11 -K 44 is the calculation coefficient for calculating the other endpoint; S1 - S4 are the integral terms generated by the loads on the beam.

[0122] Establish equations through the deformation and continuity conditions at each node:

[0123] See Figure 2 , taking the deformation w A , θ A and internal forces M A , Q A at point A as initial parameters, calculate the deformation and internal forces w B , θ B , M B , Q B at point B; taking the deformation and internal forces at point B as initial parameters, calculate the deformation and internal forces w C , θ C , M C , Q C at point C; taking the deformation and internal forces at point C as initial parameters, calculate the deformation and internal forces w D , θ D , M D , Q D at point D; taking the deformation and internal forces at point D as initial parameters, calculate the deformation and internal forces w E , θ[[ID=1,01]] E , M E , Q E at point E; taking the deformation and internal forces at point E as initial parameters, calculate the deformation and internal forces wF , θ F , M F , Q F , from which several equations are obtained and written in matrix form as follows:

[0124]

[0125] where K 11 ~K 20,4 represent the calculation coefficients of the initial parameters when calculating the end of the segmented beam, and S 20×1 is the integral term corresponding to the deformation and internal force.

[0126] Combined with the boundary conditions of Node A, the known terms related to the deflection and rotation angle are merged to the end; combined with the boundary conditions of Point F, two rows that do not need to be calculated in the matrix can be deleted. Taking both A and F as fixed constraints as an example, w F = θ F = 0,

[0127]

[0128] By solving the above equations, the deformation and internal force values of each node are obtained. Taking the deformation and internal force at the node as the initial parameters and substituting them into the equations of the initial parameters of each section, the analytical solution of the deflection curve equation with each node as the origin can be obtained. For example, when , the expression of the deflection curve of the BC section is:

[0129]

[0130] The deflection values of each section are obtained in the above way and a curve is generated. For example Figure 3 It can be seen that the deflection curve of the pipe shed calculated by the model of this embodiment is more consistent with the on-site measured results compared with the deflection curve of the pipe shed calculated by the existing theory, indicating that the model of this embodiment can more truly reflect the actual stress state of the pipe shed, and thus effectively guide the design of the pipe shed.

[0131] The above are only the preferred embodiments of the present invention and are not used to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A calculation method for pipe-roof support of dual-parameter foundation beams, characterized in that: The following steps are involved: S1. According to the deflection differential governing equation of the beam on the Pasternak foundation model, its expression is Equation 1): Among them, k and G p are the foundation reaction coefficient and foundation shear modulus respectively, EI is the equivalent bending stiffness of the pipe roof section, EI=E c I c +E s I s , E c , I c 、E s and I s are the elastic modulus of the filling concrete, the elastic modulus of the steel used for the pipe roof, the section moment of inertia of the filling concrete, and the section moment of inertia of the steel used for the pipe roof; b is the calculated width of the load on the beam, b = 0.5πD, D is the outer diameter of the pipe roof; q(x) is the load on the beam, w(x) is the deflection at the x position on the beam; b * is the calculation width of foundation reaction force, Considering the three-dimensional influence of the beam on the elastic foundation, the initial parameter solution method is used to solve the flexural differential equation, and the analytical solution of the flexural deformation under arbitrary loads on the Pasternak foundation model is obtained; S2. Divide the entire length of the pipe roof into different sections based on the actual tunnel construction steps and conduct theoretical analysis and calculations; S3. According to the actual support situation, select appropriate beam models to simplify each section; S4. Select appropriate constraints to simplify the pipe-roof end based on the actual excavation conditions and the stress and deformation of the pipe-roof end. S5. Based on the actual tunnel burial depth, select an appropriate surrounding rock pressure calculation theory and make appropriate assumptions about the load distribution along the entire length of the pipe roof. S6. The analytical solution of the flexural deformation of each section is obtained by solving the continuity conditions of the deflection w, rotation angle θ, bending moment M and generalized shear force Q at the segment nodes and the boundary conditions at the ends of the pipe roof.

2. The pipe-roof support calculation method for dual-parameter foundation beams according to claim 1 is characterized in that: Considering the three-dimensional influence of the beam on the elastic foundation, the initial parameter solution method is used to solve the flexural differential governing equations, and the analytical solution of the flexural deformation under arbitrary loads on the Pasternak foundation model is obtained, specifically: (1) hour, (2) hour, (3) hour, (4)G p = 0, the model degenerates into the Winkler foundation beam model. (5) k = G p = 0, the model degenerates into an ordinary beam model. In Equations 2) to 21), w0, θ0, M0, and Q0 are the deflection, rotation, bending moment, and generalized shear force of the initial section, and z is the integral variable. J1(x)=cos(λx)sinh(λx), J2(x)=cos(λx)cosh(λx), J3(x)=sin(λx)cosh(λx), J4(x)=sin(λx)sinh(λx).

3. The pipe-roof support calculation method for dual-parameter foundation beams according to claim 1 is characterized in that: Step S2 specifically includes: according to typical tunnel construction steps, the entire length of the pipe roof is divided into an initial support closed section, an initial support unclosed section, an excavation unsupported section, a loose soil section ahead of the tunnel face, and an unloosened section; wherein the lengths of the initial support closed section, the initial support unclosed section, and the excavation unsupported section are determined according to the actual support conditions; the length of the loose soil section ahead of the tunnel face is: Where h is the tunnel excavation height, is the internal friction angle of the soil in front; the unloosened section is determined by the distance between the end of the pipe roof and the tunnel face: Where ΔL is the distance between the end of the pipe roof and the tunnel face.

4. The pipe-roof support calculation method for dual-parameter foundation beams according to claim 1 is characterized in that: Step S3 is specifically as follows: considering the supporting effect of the initial support, the initial support closed section and the initial support unclosed section are simplified into a Pasternak foundation beam model; considering the hysteresis effect of the initial support, the stratum parameters of the initial support unclosed section are obtained by reducing the stratum parameters of the initial support closed section, and the reduction coefficient is taken as ξ; the excavated unsupported section is simplified into an ordinary beam model, and when the pipe-roof grouting reinforcement shell effect is considered, it is simplified to a Winkler foundation beam model; the loosened soil section and the unloosened soil section are simplified into a Pasternak foundation beam model, and the stratum parameters of the loosened section are obtained by reducing the stratum parameters of the unloosened section, and the reduction coefficient is taken as η.

5. The pipe-roof support calculation method for dual-parameter foundation beams according to claim 1 is characterized in that: Step S4 specifically comprises: simplifying the end of the pipe rack close to the initial support into a fixed constraint with a certain initial deflection and rotation angle or a spring constraint with a certain stiffness; and simplifying the end away from the initial support into a fixed constraint or a free end.

6. The pipe-roof support calculation method for dual-parameter foundation beams according to claim 1 is characterized in that: Step S5 specifically includes: obtaining the load by any one of the railway tunnel design code method, Terzaghi theory, Pusch theory or actual measurement results; and assuming that the load distribution is any one of uniform distribution, triangular distribution and normal distribution.

Images