A displacement calculation method of assembled underground structure timoshenko-pasternak foundation beam considering internal force continuity of ring segment

Abstract

This invention discloses a method for calculating the displacement of a Timoshenko-Pasternak foundation beam in a prefabricated underground structure considering the continuity of internal forces within ring segments. The method includes the following steps: Step S1: Simplifying the prefabricated underground structure into a Timoshenko beam model placed on the Timoshenko-Pasternak foundation, and setting that bending moment, shear force, and external loads are continuous within the ring segments, while deformation is discontinuous at the joints of the ring segments; Step S2: Based on the single-variable Timoshenko beam theory, establishing the longitudinal force control equations of the structure, including bending and shear displacements; Step S3: Introducing continuous constraints between ring segments, upgrading the traditional 4th-order differential equations to 6th-order differential control equations; Step S4: Discretizing the 6th-order differential control equations using the finite difference method, separating and independently solving for bending and shear displacements; Step S5: Based on the solution results, outputting the distribution curves of the total vertical displacement, bending displacement, and shear displacement of the prefabricated underground structure along the longitudinal direction.

Description

Technical Field

[0001] This invention belongs to the field of underground structure engineering technology, specifically relating to a method for calculating the displacement of the foundation beam of the Timoshenko-Pasternak prefabricated underground structure that considers the continuity of internal forces in the ring segment. Background Technology

[0002] Prefabricated underground structures (such as shield tunnels, underground utility tunnels, and prefabricated culverts) have been widely used in modern urban underground space development due to their advantages of fast construction speed, minimal environmental impact, and high degree of industrialization. These structures are typically assembled longitudinally from several prefabricated segments or rings connected by bolts. In terms of longitudinal non-uniform stress, prefabricated underground structures exhibit significant discontinuity and discontinuity due to the presence of numerous weak ring joints. Under the additional external forces caused by adjacent underground construction (such as adjacent excavation and unloading of foundation pits, close-range crossings of new railway lines, and sudden large-area surface loading), the assembled ring segments, due to their high rigidity, often exhibit rigid body displacement and overall bending. Meanwhile, the ring joints, with their weak resistance to deformation, are prone to rotation (leading to joint opening) and misalignment (leading to inter-ring slippage). Excessive longitudinal non-uniform deformation can directly cause serious safety accidents such as cracking of the lining segments, water leakage from the ring joints, and even train derailment. Therefore, accurate prediction and detailed analysis of the longitudinal non-uniform deformation of prefabricated underground structures under external loads are of vital engineering significance.

[0003] Currently, elastic foundation beam models are commonly used both domestically and internationally to assess the longitudinal deformation of underground structures. The most common approach is to simplify the tunnel into an equivalent continuous foundation beam (such as the Euler-Bernoulli beam or Timoshenko beam model). However, existing Timoshenko-Pasternak foundation models have the following significant drawbacks when dealing with prefabricated underground structures:

[0004] The inability to quantitatively separate displacement components: Existing model systems are mostly based on fourth-order differential equations, which can only solve for the overall vertical displacement of the structure. They cannot quantitatively calculate the total displacement by breaking it down into bending displacement and shear displacement. It is difficult to accurately distinguish the contribution of the two types of displacement to the total deformation of the structure, which restricts the rationality of the refined design of prefabricated structures.

[0005] The mechanical constraints and discontinuous deformation characteristics between rings are ignored: Existing models often blindly simplify the structure into a homogeneous continuous beam, failing to fully consider the continuous constraints of bending moment, shear force, and external load between adjacent ring segments. They also ignore the actual stress characteristics of "smooth and continuous deformation of the ring segments themselves, while the deformation at the ring joints is discontinuous", resulting in a large deviation between the calculation results and the actual engineering situation.

[0006] The model boundary and joint weakening treatment is too rough: The existing calculation method is difficult to take into account the dual characteristics of continuous internal force and discontinuous deformation in the ring segment. When facing the opening of the ring joint and the misalignment, it lacks a refined mathematical framework and solution algorithm, resulting in insufficient accuracy of longitudinal deformation calculation.

[0007] Therefore, developing a refined 6th-order differential control displacement calculation method that can fully consider the continuity of internal forces in the ring segment, accurately characterize the abrupt deformation characteristics of the joint, and achieve quantitative separation of bending and shear displacements has become an urgent technical problem to be solved in this field. Summary of the Invention

[0008] The purpose of this invention is to disclose a displacement calculation method for Timoshenko-Pasternak foundation beams in prefabricated underground structures that considers the continuity of internal forces within ring segments. This method aims to solve the technical problems of existing Timoshenko-Pasternak foundation beam models, which only use fourth-order differential equations, cannot separate bending and shear displacements, do not consider the continuity constraint of internal forces between ring segments, and have insufficient calculation accuracy. This invention achieves the construction of sixth-order differential equations, separates bending and shear displacements and calculates them quantitatively, which is consistent with the actual stress characteristics of continuous deformation of ring segments and discontinuous deformation between rings in prefabricated underground structures, thereby improving the accuracy of displacement calculation.

[0009] To achieve the above objectives, the present invention provides the following technical solution:

[0010] This invention proposes a method for calculating the displacement of foundation beams in the Timoshenko-Pasternak prefabricated underground structure, considering the continuity of internal forces within the ring segment, comprising the following steps:

[0011] Step S1: Simplify the prefabricated underground structure into a Timoshenko beam model placed on the Timoshenko-Pasternak foundation. Assume that the foundation is an elastic foundation and satisfies the dual parameter control of foundation shear stiffness and subgrade coefficient. Set that the bending moment, shear force and external load are continuous within the ring segment, and the deformation at the joint of the ring segment is discontinuous.

[0012] Step S2: Based on the single-variable Timoshenko beam theory, establish the total vertical displacement w and bending displacement w b With shear displacement w s The decoupling relationship was determined, and the relationship regarding the bending displacement w was derived. b The 6th-order differential governing equation:

[0013]

[0014] Where EI is the flexural stiffness of the ring segment, GA is the shear stiffness of the ring segment, κ is the shear coefficient of the Timoshenko beam, kp and ks are the shear layer stiffness coefficient and foundation reaction coefficient of the Pasternak foundation, respectively, and q(y) is the external load acting on the structure.

[0015] Step S3: Define the boundary constraints at the joint of the ring segment, using the joint rotational stiffness K. θ and joint shear stiffness K s Establish the deformation jump boundary conditions at the joint of the ring segment;

[0016] Boundary constraints are satisfied:

[0017]

[0018]

[0019] Where △θ is the joint rotation angle, δ is the joint shear displacement, M is the bending moment between adjacent ring segments, and Q is the shear force between adjacent ring segments;

[0020] Step S4: Discretize the 6th-order differential control equation using the finite difference method. Use the standard central difference scheme inside the loop segment and introduce virtual nodes at the loop segment joints to satisfy the boundary constraints and construct a linear equation system matrix.

[0021] Step S5: Solve the system of linear equations to obtain the bending displacement w at each node. b Shear displacement w s The total vertical displacement w is output, along with the distribution curves of the total vertical displacement, bending displacement, and shear displacement of the prefabricated underground structure along the longitudinal direction.

[0022] Furthermore, in step S2, the total vertical displacement w and the bending displacement w b The conversion formula is as follows:

[0023]

[0024] Shear displacement w s With bending displacement w b The conversion formula is as follows:

[0025]

[0026] Furthermore, in step S3, the bending moment M at the ring joint j j and shear force Q j The continuity and discontinuous deformation characteristics satisfy:

[0027] Internal force continuity condition:

[0028]

[0029]

[0030] Deformation jump relationship:

[0031]

[0032]

[0033] Among them, M j - and M j + The bending moments of the ring sections on the left and right sides of joint j are respectively; Q j - and Q j + These represent the shear forces of the ring sections on the left and right sides of joint j, respectively; M j Qj and θ represent the transmitted bending moment and transmitted shear force at the joint, respectively; j - and θ j + These are the cross-sectional rotation angles at the left and right annular ends of the joint, respectively; w b,j - and w b,j + These represent the bending displacements of the cross-sections at the left and right ends of the joint, respectively; Δθ j Δw represents the relative angular jump value between the two sides of the joint, used to characterize the opening deformation of the circumferential joint; b,j The relative bending displacement jump value on both sides of the joint is used to characterize the shear misalignment deformation of the circumferential joint.

[0034] Furthermore, the cross-sectional rotation angle θ and the bending displacement w b The relation is:

[0035]

[0036] Furthermore, in step S4, when discretizing the 6th-order differential control equation, a seven-point central difference scheme is used, and the difference expression at any node i is:

[0037]

[0038] in, Let δ be the difference step size. 2 δ 4 δ 6 These are the second-order, fourth-order, and sixth-order central difference operators, respectively.

[0039] The second-order, fourth-order, and sixth-order central difference operators correspond to the second derivatives of the governing equations with respect to the spatial coordinate y. Fourth derivative and sixth derivative The discrete approximation scheme;

[0040] The second-order central difference operator satisfies the following formula, which uses the bending displacement values ​​of the current node i and its left and right adjacent nodes (a total of 3 nodes) to approximate the second-order spatial rate of change of the bending displacement.

[0041]

[0042] The fourth-order central difference operator satisfies the following equation, which uses the displacement values ​​of the current node i and the two extrapolated nodes to its left and right (a total of 5 nodes) to approximate the fourth-order spatial rate of change of the bending displacement.

[0043]

[0044] The sixth-order central difference operator satisfies the following formula, which uses the displacement values ​​of the current node i and the three extrapolated nodes to its left and right (a total of seven nodes, i.e., the seven-point format) to approximate the sixth-order spatial rate of change of bending displacement.

[0045]

[0046] Furthermore, in step S4, a virtual node is introduced at the joint between ring segment j and ring segment j+1. and It is used to solve the internal force balance and deformation compatibility equations of the nodes on both sides of the joint simultaneously.

[0047] Furthermore, in step S4, the constructed linear equation system matrix is ​​in the following form:

[0048]

[0049] Where [K] is the total stiffness matrix composed of the difference coefficients of the 6th order governing equations and the boundary constraint equations, {Wb} is the bending displacement vector of all nodes along the line, and {Q} is the load vector.

[0050] Furthermore, in step S5, after obtaining the bending displacement w... b Then, the total vertical displacement w of any node i is calculated using the following formula:

[0051]

[0052] Based on the above-mentioned method for calculating the displacement of Timoshenko-Pasternak foundation beams of prefabricated underground structures considering the continuity of internal forces in the ring segment, this invention also proposes an electronic device, including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the steps of any of the above-mentioned methods for calculating the displacement of Timoshenko-Pasternak foundation beams of prefabricated underground structures considering the continuity of internal forces in the ring segment.

[0053] Furthermore, the present invention also proposes a computer-readable storage medium storing a computer program, wherein the computer program, when executed by a processor, performs the steps of any of the above-mentioned methods for calculating the displacement of the Timoshenko-Pasternak foundation beam of a prefabricated underground structure considering the continuity of internal forces within a ring segment.

[0054] Compared with the prior art, the beneficial effects of the present invention are:

[0055] (1) A scientific 6th-order force control system was constructed: Based on the single-variable Timoshenko beam and Pasternak dual-parameter foundation model, this invention successfully constructed a high-order (6th-order) differential control difference equation by integrating the internal force continuity and deformation discontinuity constraints of the circumferential joint at the discrete level. This fundamentally overcomes the mechanical defect that the traditional 4th-order continuous beam model cannot accurately reflect the longitudinal discontinuity characteristics of the assembled structure.

[0056] (2) Achieved high-precision independent quantitative separation of bending and shear displacement: This method can clearly and rigorously decompose the total longitudinal deformation into bending displacement and shear displacement, accurately extract the contribution value and proportion of the deformation of the ring segment itself and the opening of the joint and the misalignment to the overall uneven deformation, and provide quantitative guidance indicators for the longitudinal seepage prevention design of prefabricated underground structures and the selection of bolt shear stiffness.

[0057] (3) It truly represents the deformation and abrupt changes of the prefabricated structure: With the help of the local virtual node elimination algorithm, the displacement curve output by this method can truly reproduce the service state of the prefabricated structure of "smooth inside the ring and jumping between the rings". The calculation results are highly consistent with the field measured data, which greatly improves the economy and safety of the design calculation. Attached Figure Description

[0058] Figure 1 For the present invention Figure 1 This is a schematic diagram of the stress and deformation of the Timoshenko-Pasternak foundation beam of the prefabricated underground structure considering the continuity of internal forces in the ring segment in this invention.

[0059] Figure 2 This is a schematic diagram of the bending deformation of the prefabricated underground structure under positive and negative bending moments in this invention, a) negative bending moment action, b) positive bending moment action;

[0060] Figure 3 This is a discrete schematic diagram of the prefabricated underground structure of the present invention;

[0061] Figure 4 This is the virtual node diagram between rings in this invention. Detailed Implementation

[0062] The present invention will now be described in detail with reference to the embodiments shown in the accompanying drawings. However, it should be noted that these embodiments are not intended to limit the present invention. Equivalent changes or substitutions in function, method, or structure made by those skilled in the art based on these embodiments are all within the scope of protection of the present invention.

[0063] This invention proposes a method for calculating the displacement of foundation beams in the Timoshenko-Pasternak prefabricated underground structure, considering the continuity of internal forces within the ring segment. The method includes the following steps:

[0064] Step 1: Determine the computational model and basic assumptions

[0065] The prefabricated underground structure is simplified as a Timoshenko beam, placed on a Timoshenko-Pasternak two-parameter elastic foundation model. The relationship between the subgrade coefficient k and the shear layer stiffness G is then analyzed. p Two parameters control the stratum reaction force; it is assumed that the prefabricated structure satisfies the continuity of bending moment M, shear force Q, and external load q within the ring segment, while exhibiting deformation discontinuity at the joint of adjacent ring segments; based on the single-variable Timoshenko beam theory, the bending deformation and shear deformation effects of the beam are fully considered.

[0066] In step 1, based on the single-variable Timoshenko beam theory, the displacement, internal force, and geometric equations within each ring segment of the prefabricated underground structure satisfy the following expressions:

[0067]

[0068]

[0069]

[0070] In the formula: y is the axial coordinate along the longitudinal direction of the underground structure; θ is the rotation angle of the ring section; w is the total vertical displacement of the ring section; wb and ws are the bending displacement and shear displacement of the ring section, respectively; γ is the shear strain; M and Q are the bending moment and shear force of the ring section, respectively; EI is the longitudinal bending stiffness of the ring section itself; GA is the shear stiffness of the ring section; κ is the Timoshenko beam shear coefficient, which is taken as 0.53 for circular tunnel sections; q is the generalized external load acting on the ring section.

[0071] =At the joint j of the adjacent ring segment, the rotational stiffness K of the joint is introduced. θ The boundary constraints, the bending moment M, the shear force Q between adjacent ring segments, and the relationship between rotational jump and displacement jump, along with the joint shear stiffness Ks, satisfy the following equations:

[0072] Internal force continuity condition:

[0073]

[0074]

[0075] Deformation jump relationship:

[0076]

[0077]

[0078] Where: M j - and M j + The bending moments of the ring sections on the left and right sides of joint j are respectively; Q j - and Q j + These represent the shear forces of the ring sections on the left and right sides of joint j, respectively; M j Qj and θ represent the transmitted bending moment and transmitted shear force at the joint, respectively; j - and θ j + These are the cross-sectional rotation angles at the left and right annular ends of the joint, respectively; w b,j - and w b,j + These represent the bending displacements of the cross-sections at the left and right ends of the joint, respectively; Δθ j Δw represents the relative angular jump value between the two sides of the joint, used to characterize the opening deformation of the circumferential joint; b,j The relative bending displacement jump value on both sides of the joint is used to characterize the shear misalignment deformation of the circumferential joint.

[0079] Step 2: Construct the 6th-order differential governing equation

[0080] Based on the traditional fourth-order differential equations of the Timoshenko-Pasternak foundation, continuous constraints on bending moment M, shear force Q, and distributed load q between adjacent ring segments are introduced. Combined with the force balance equations within the ring segments, the deformation compatibility relationship at the joints, and the constitutive relation of the Timoshenko beam, higher-order derivations are performed. In the difference system, a sixth-order differential governing equation for longitudinal forces that can uniformly characterize the continuous characteristics within the rings and the discontinuous characteristics between the rings is constructed, abandoning the traditional simplified assumption of homogeneous continuous beams.

[0081]

[0082] Where B is the width of the beam, in mm. One factor contributing to cross-sectional warping; equivalent shear stiffness GAeq; Equivalent bending stiffness EIeq; Let GAeq=T, EIeq=H, Substituting into the above formula, we get:

[0083]

[0084] To simplify the equation expression, four intermediate control coefficients A0, A1, A2, and A3 are introduced, defined as follows:

[0085] (In the formula, B) T = B H = )

[0086] Step 3: Quantitative separation and solution of bending and shear displacement

[0087] Based on the constructed 6th-order differential control equation, the total vertical displacement w of the cross section is decomposed into bending displacement wb and shear displacement ws, which satisfy w = wb + ws. By solving the control matrix equation containing boundary constraints, the independent numerical solutions and proportions of bending displacement and shear displacement are independently calculated, clarifying their contribution ratio to the total deformation of the structure.

[0088] Step 4: Deformation Characterization and Calculation Output

[0089] Based on the physical characteristics of continuous internal forces between adjacent ring segments and discontinuous deformation between rings, a rotational stiffness K is applied at the joint of the ring segments. θ and joint shear stiffness K sControlled deformation jump boundary conditions are applied at the beginning and end of the structure. The finite difference method is used and local virtual nodes are introduced for elimination. The overall stiffness matrix equation is established and solved. The distribution curves of the total vertical displacement, bending displacement and shear displacement along the longitudinal direction of the structure are output to characterize the rotation jump and displacement change characteristics at the joint.

[0090] Step 5: Calculation Verification and Application

[0091] The displacement and component results calculated by this method were compared and verified with the results of the traditional 4th order Timoshenko-Pasternak continuous beam model, three-dimensional refined numerical simulation data, or field engineering measured data. After confirming the high accuracy, this method was directly applied to the longitudinal refined stress and deformation control design of ring-shaped underground structures such as prefabricated underground utility tunnels, shield tunnels, or prefabricated culverts.

[0092] As a further improvement of the present invention, in step 4, the underground structure is discretized using the finite difference method. It is assumed that the structure contains n² ring segments, and each ring segment is equally spaced into n¹ segments of length. The internal differential nodes;

[0093] Two global virtual nodes are added at each end of the underground structure to apply free or fixed boundary conditions at the ends. At the joint j of each adjacent ring segment, an additional local virtual node is introduced to transform the internal force continuity condition and deformation jump relationship into a difference equation and substitute it into the control matrix for elimination and reconstruction. Thus, at the boundary node of the global stiffness matrix equation, a system of linear algebraic equations with difference operator coefficients corresponding to the 6th order differential characteristics is formed. The expansion coefficient terms of the algebraic equation system contain node weights of -20, 15, -6, and 1 with the 6th order central difference characteristics.

[0094] As a further improvement of the present invention, in step 1, the expression for the foundation reaction force p(y) applied to the beam by the Timoshenko-Pasternak two-parameter elastic foundation model is as follows:

[0095]

[0096] In the formula: k is the subgrade reaction coefficient; G p Let w(y) be the shear stiffness of the foundation soil. After obtaining the bending displacement of each node, the total displacement w(y) in the foundation reaction force is eliminated by the displacement separation constitutive relation in equation (1a), and then substituted into the total force equilibrium equation to achieve matrix assembly.

[0097] This invention proposes a method for calculating the displacement of foundation beams in the Timoshenko-Pasternak prefabricated underground structure, considering the continuity of internal forces within the ring segment. The mechanical analysis and numerical reconstruction logic is as follows:

[0098] like Figure 1 As shown, Figure 1 This is a schematic diagram of the stress and deformation of a prefabricated underground structure. The prefabricated underground structure is divided into n1 calculation units along the longitudinal direction, where b represents the semi-minor axis length of the cross-section. The strains of the top and bottom plates at the i-th node are denoted as... and Q is the shear force at the section, and M is the bending moment at the section.

[0099] The prefabricated underground structure is considered longitudinally as composed of several rigid segment rings, connected by longitudinal joints equipped with elastic bolts. The entire structure is placed on a Pasternak two-parameter elastic foundation, which not only simulates the normal support of the soil through the subgrade coefficient k, but also through the shear layer stiffness G. p The simulation of shear transmission effect between soil springs better reflects the actual working conditions of composite strata.

[0100] Constitutive control of internal forces in the ring segment

[0101] Within each independent ring segment, the beam satisfies deformation continuity, strictly adhering to the single-variable Timoshenko beam theory. To decouple and quantitatively separate the displacements, this invention employs a single-variable bending displacement for system construction. From equation (1a), it can be seen that the total vertical displacement w of the cross-section is equal to the bending displacement w0. b With shear displacement w s Superposition:

[0102] w=w b+ w s

[0103] Shear displacement w due to shear deformation s It is directly related to the cross-sectional shear force Q, and according to the physical constitutive law, Furthermore, based on the shear relationship of the univariate Timoshenko beam, we have Integrating the shear geometry (and eliminating the rigid body displacement term) directly yields the precise form of shear displacement expressed as a higher-order derivative of bending displacement:

[0104]

[0105] Therefore, the total vertical displacement of the entire structure can be entirely determined by the bending displacement w. b Its unified representation of the second derivative:

[0106]

[0107] Characteristic constraints at joints

[0108] When the structure crosses the interface between adjacent ring segments (at joint j), the deformation changes abruptly due to the discontinuity of the assembled structure. For example... Figure 2 As shown, Figure 2 This diagram illustrates the bending deformation of a prefabricated underground structure under positive and negative bending moments. According to Newton's third law, the bending moment and shear force at the joint should be strictly equal on both sides (assuming the joint itself does not bear concentrated external applied moments or shear point loads).

[0109]

[0110]

[0111] However, the deformation on both sides is due to the joint rotational stiffness K θ and shear stiffness K s The existence of no longer satisfies continuity, and its relationship between angular jump and bending displacement jump is explicitly expressed as:

[0112]

[0113]

[0114] Finite difference mesh discretization and construction of 6th order force equations

[0115] like Figure 3 As shown, Figure 3 This is a discretization diagram of a prefabricated underground structure. Three virtual nodes are added at both ends, and the structure is discretized into n1n2+6 nodes along the longitudinal direction, where n2 is the number of ring segments, n1 is the number of blocks within a ring segment, and the distance between adjacent nodes is l. s =L n / n1,L n The entire length of the prefabricated structure.

[0116] like Figure 4 As shown, the tunnel has n2 pipe rings along its longitudinal direction, and each pipe ring is discretized into n1 grid segments of length l. To solve the complex discontinuous boundary value problem described above, this invention fundamentally reconstructs and improves the order of the governing equations by setting virtual nodes at joint j.

[0117] For any standard node i within the ring segment, according to the equilibrium equation of the Timoshenko beam on the Pasternak two-parameter foundation: dQ / dy - p(y) + q ext (y) = 0. Substituting the constitutive equations of internal forces and the equations of foundation reactions, the governing differential equations are mathematically expressed in fourth-order form:

[0118]

[0119] However, at the junction j of adjacent ring segments, in order to forcibly satisfy the conflicting mechanical boundaries of continuous internal forces (equal bending moment and shear force) and discontinuous deformation (rotation and displacement jump), this invention introduces local virtual nodes at the end of the left ring segment and the beginning of the right ring segment, respectively. The derivative is discretized using the finite difference central difference operator:

[0120]

[0121] Substitute the angle jump equation (restricting first-order derivative jump) and displacement jump equation (restricting third-order derivative jump) at the joint into the difference equation at the junction, and simultaneously eliminate the local virtual nodes at the junction.

[0122] During the elimination and reconstruction process, due to the strong coupling between the first and third derivative jumps and higher-order internal forces (second and third derivatives), the elimination directly leads to cross-mesh correlations in the difference equations of nodes near the joint. This internalized algebraic treatment of boundary conditions essentially generates an operator structure with 6th-order central difference characteristics in the intersection row of the global stiffness matrix equation. In the coefficients of its expanded algebraic equations, the unknown bending displacement w corresponds to each adjacent node. b Its weighted coefficients are precisely presented as follows:

[0123]

[0124] The characteristic coefficients (-20, 15, -6, 1) represent the mathematical skeleton of a typical 6th-order governing differential equation under the finite difference method. This fully demonstrates that our method, through refined reconstruction of the matrix boundaries, successfully upgrades the traditional coarse 4th-order continuous beam model into a 6th-order discontinuous force control model that closely matches the actual service conditions, perfectly taking into account the profound impact of weakened stiffness at the ring joints on the overall longitudinal non-uniform deformation.

[0125] Matrix solving and displacement separation output

[0126] By applying longitudinal end free boundary conditions (i.e., bending moment M=0 and shear force Q=0 at the end sections) to both ends of the entire structure, and then eliminating variables through global virtual nodes at both ends, the system of linear algebraic equations for the entire structure is obtained:

[0127]

[0128] In the formula: [K] represents the comprehensive integration of the ring segment bending stiffness EI, shear stiffness GA, foundation bed coefficient k, foundation shear layer stiffness Gp, and ring joint comprehensive rotational stiffness K. θ The overall structural large stiffness matrix with shear stiffness Ks; {W b} =[w b,1 , w b,2, ……, w b,n ] T {Q} is the column vector of bending displacements to be determined at each node of the entire structure; {Q} is the known load vector consisting of external additional loads.

[0129] Solving this system of equations using computer matrix inversion or Gaussian elimination iteration method allows for the direct acquisition of precise numerical values ​​of the bending displacements w at all nodes of the entire structure. b,i .

[0130] In obtaining w b,i Then, using the separation equation from step 3, the shear displacement component w at each node is quantitatively calculated directly using a second-order central difference scheme. s,i :

[0131]

[0132] Then, by directly adding the two together, the overall vertical displacement distribution of the output structure along the longitudinal direction can be extracted:

[0133]

[0134] like Figure 4 As shown, the longitudinal deformation distribution curves output by the plotting show distinct non-smooth and discontinuous characteristics: inside the precast pipe ring segment, the total displacement curve slides very smoothly, and the deformation is mainly bending and rigid body translation; while at the junction of the circumferential joints of adjacent ring segments, the total displacement curve shows a clear abrupt change in slope (precisely corresponding to the opening and relative rotation of the joint) and numerical misalignment steps (precisely corresponding to the shear misalignment between rings).

[0135] Typical Engineering Application Cases

[0136] The method of this invention has been successfully applied in the longitudinal safety assessment of a shield tunnel in soft soil in a certain city. The electric shield tunnel has an outer diameter of 6.2 m, a segment thickness of 0.35 m, a ring width of 1.2 m, and a longitudinal flexural stiffness of the lining concrete EI = 1.15 × 10⁻⁶. 8 kN·m 2 Excavation of a large-scale foundation pit above the tunnel induced significant longitudinal uneven unloading of external loads.

[0137] In the calculation and analysis, each ring is divided into n1 = 4 grid nodes, and the subgrade coefficient is taken as k = 18 MN·m. 3 The shear stiffness of the foundation is taken as G. p = 30 MN. Based on the quantity and preload of the inter-ring bolts, input the overall rotational stiffness K of the joint. θ = 1.5×10 5 kN·m / rad, joint shear stiffness K s= 5.0 × 10 5 kN / m. Solving the 6th-order control matrix equation of this invention, the maximum vertical total uplift displacement of the tunnel is output as 24.5 mm. Comparing this calculated curve with the measured data from the automated total station and laser ranging segment monitoring deployed on-site, the curve shape shows a high degree of consistency, with the calculation error at the maximum total displacement point being only 4.2%. Simultaneously, this method successfully identified that the shear displacement jump caused by joint weakening misalignment accounts for 35.2% of the total uplift deformation, providing extremely accurate and quantitative mechanical basis for the longitudinal reinforcement and misalignment reduction scheme of the tunnel's circumferential joint.

[0138] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for calculating the displacement of foundation beams in a prefabricated underground structure, Timoshenko-Pasternak, considering the continuity of internal forces within a ring segment, characterized in that... Includes the following steps: Step S1: Simplify the prefabricated underground structure into a Timoshenko beam model placed on the Timoshenko-Pasternak foundation. Assume that the foundation is an elastic foundation and satisfy the dual parameter control of foundation shear stiffness and subgrade coefficient. Set that the bending moment, shear force and external load are continuous within the ring segment, and the deformation at the joint of the ring segment is discontinuous. Step S2: Based on the single-variable Timoshenko beam theory, establish the total vertical displacement w and bending displacement w b With shear displacement w s The decoupling relationship was determined, and the relationship regarding the bending displacement w was derived. b The 6th-order differential governing equation: ； Where EI is the flexural stiffness of the ring segment, GA is the shear stiffness of the ring segment, κ is the shear coefficient of the Timoshenko beam, kp and ks are the shear layer stiffness coefficient and foundation reaction coefficient of the Pasternak foundation, respectively, and q(y) is the external load acting on the structure. Step S3: Define the boundary constraints at the joint of the ring segment, using the joint rotational stiffness K. θ and joint shear stiffness K s Establish the deformation jump boundary conditions at the joint of the ring segment; Boundary constraints are satisfied: ； ； Where △θ is the joint rotation angle, δ is the joint shear displacement, M is the bending moment between adjacent ring segments, and Q is the shear force between adjacent ring segments; Step S4: Discretize the 6th-order differential control equation using the finite difference method. Use the standard central difference scheme inside the loop segment and introduce virtual nodes at the loop segment joints to satisfy the boundary constraints and construct a linear equation system matrix. Step S5: Solve the system of linear equations to obtain the bending displacement w at each node. b Shear displacement w s The total vertical displacement w is output, along with the distribution curves of the total vertical displacement, bending displacement, and shear displacement of the prefabricated underground structure along the longitudinal direction.

2. The method according to claim 1, characterized in that, In step S2, the total vertical displacement w and the bending displacement w b The conversion formula is as follows: ； Shear displacement w s With bending displacement w b The conversion formula is as follows: 。 3. The method according to claim 1, characterized in that, In step S3, the bending moment M at the ring joint j j and shear force Q j The continuity and discontinuous deformation characteristics satisfy: Internal force continuity condition: ； ； Deformation jump relationship: ； ； Among them, M j - and M j + The bending moments of the ring sections on the left and right sides of joint j are respectively; Q j - and Q j + These represent the shear forces of the ring sections on the left and right sides of joint j, respectively; M j Qj and θ represent the transmitted bending moment and transmitted shear force at the joint, respectively; j - and θ j + These are the cross-sectional rotation angles at the left and right annular ends of the joint, respectively; w b,j - and w b,j + These represent the bending displacements of the cross-sections at the left and right ends of the joint, respectively; Δθ j Δw represents the relative angular jump value between the two sides of the joint, used to characterize the opening deformation of the circumferential joint; b,j The relative bending displacement jump value on both sides of the joint is used to characterize the shear misalignment deformation of the circumferential joint.

4. The method according to claim 3, characterized in that, The cross-sectional rotation angle θ and bending displacement w b The relation is: 。 5. The method according to claim 1, characterized in that, In step S4, when discretizing the 6th-order differential control equation, a seven-point central difference scheme is used, and the difference expression at any node i is: ; in, Let δ be the difference step size. 2 δ 4 δ 6 These are the second-order, fourth-order, and sixth-order central difference operators, respectively.

6. The method according to claim 5, characterized in that, In step S4, a virtual node is introduced at the joint between ring segment j and ring segment j+1. and It is used to solve the internal force balance and deformation compatibility equations of the nodes on both sides of the joint simultaneously.

7. The method according to claim 1, characterized in that, In step S4, the constructed linear equation system matrix is ​​in the following form: ； Where [K] is the total stiffness matrix composed of the difference coefficients of the 6th order governing equations and the boundary constraint equations, {Wb} is the bending displacement vector of all nodes along the line, and {Q} is the load vector.

8. The method according to claim 1, characterized in that, In step S5, after obtaining the bending displacement w b Then, the total vertical displacement w of any node i is calculated using the following formula: 。 9. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method as described in any one of claims 1 to 8.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the steps of the method as described in any one of claims 1 to 8.

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