Tension-compression coupled gap beam unit and iterative calculation method thereof

Abstract

The application provides a tension-compression coupled gap beam unit and an iterative calculation method thereof, and belongs to the field of bolted structure design and safety evaluation. A specific degree of freedom is released in an existing unit stiffness matrix, and an axial gap control parameter is introduced, so as to realize coupled simulation of tension-compression bidirectional gap and bending moment transmission. Meanwhile, an adaptive iterative calculation method embedded with dichotomy is provided. The tension-compression coupled gap beam unit can realize coupled simulation of tension-compression bidirectional gap and bending moment transmission, can solve the problem that the prior art gap rod unit cannot consider horizontal bending moment, and provides a reliable means for accurate analysis of internal force and deformation of a three-dimensional bolt (pin) joint structure and has high precision. The calculation method of the gap beam unit comprises the following steps: (1) degree of freedom condensation and release; (2) matrix element supplement; (3) supplement of axial gap, specific in-plane bending stiffness and rod torsional stiffness elements.

Description

Technical Field

[0001] This application relates to the field of bolted structure design and safety assessment, and specifically proposes a tension-compression coupled gap beam element and its iterative calculation method. Background Technology

[0002] Bolted structures, with their advantages of convenient construction, reliable quality, and strong adaptability, are widely used in engineering fields such as bridges, steel structure buildings, machinery and equipment, and aerospace. In beam and arch truss structures composed of members connected by ordinary bolts or pins, the inelastic deformation caused by the gap between the bolts (pins) and the bolt holes will affect the overall deformation of the structure. If the influence of these gaps is ignored in the structural calculation, the calculated deflection result will differ significantly from the actual structural deflection.

[0003] Currently, some large-scale general-purpose commercial finite element software and bridge-specific commercial finite element software disclose related technical solutions for tension-only or compression-only gap bar elements to simulate the influence of gaps on members. However, existing such elements cannot directly simulate the mechanical behavior and deformation process of gap members that can be subjected to both compression and tension. Some published technical documents provide examples of combining tension-only gap bar elements, that is, proposing tension-compression gap bar elements that simultaneously consider tension and compression, and correspondingly proposing iterative algorithms to match them. The stiffness matrix and state control parameters of the tension-compression bar element are shown in the following formulas (1) and (2), and are attached later. Figure 1 to Figure 3 The diagrams for the three states of tension, criticality, and compression shown in formula (2) are as follows: In the formula: E The elastic modulus of the rod, A The cross-sectional area of ​​the rod is α These are state control parameters. l Change the initial length of the theoretical unit. D This is the unit gap value. F i It is a rod i Concentrated force at the end along the axial direction, F j It is a rod j Concentrated force at the end along the axial direction, u i It is a rod i End node displacement, u j It is a rod j The displacement of the end node.

[0004] The tension / compression bar gap element disclosed in the prior art exhibits stiffness values ​​that depend on the actual change in element length under different conditions. Therefore, its calculation process is non-linear and requires an iterative algorithm. Existing methods matching this type of algorithm are iterative algorithms based on the residual load increment theory. This method adjusts the overall stiffness matrix and determines the load step size of the iterative algorithm based on the change in the "life" or "death" state of the gap element during loading. This method assumes no additional open or closed gaps at the beginning of each load step. If gaps open or close during loading, the overall stiffness matrix needs to be reconstructed. For each load increment step, the structural deformation needs to be corrected, and the above process needs to be repeated multiple times to complete the full load application. This results in significant limitations of existing tension / compression bar gap elements, which can be summarized as follows: On the one hand, although existing gap elements can simulate the effect of axial gap and reduce displacement and internal force errors caused by not considering gap, the structure in actual engineering can only be regarded as a hinged joint with gap in the plane perpendicular to the direction of bolt (pin) axis, and bending moment can be transmitted in other directions. It is obviously not realistic to use existing gap rod elements or beam elements to simulate the mechanical behavior of the actual structure here.

[0005] On the other hand, when such gap element structures are subjected to loads not perpendicular to the pin axis direction for various reasons, calculations using existing beam elements or axial gap bar elements will inevitably lead to deviations in the calculation of structural displacements and internal forces. Directly using such calculation results to determine the safety of the members is obviously imprecise and unreliable. Therefore, in the field of bolted structure design and safety assessment, there is an urgent need to implement a tension-compression coupled gap beam element that considers the bending stiffness in the plane determined by the bolt hole axis and the member axis, as well as an iterative algorithm suitable for this element.

[0006] Therefore, this patent application is hereby filed. Summary of the Invention

[0007] The tension-compression coupled gap beam element and its iterative calculation method described in this application aim to solve the shortcomings of the existing technology that cannot simultaneously consider the in-plane bending stiffness determined by the axial gap, the bolt hole axis, and the rod axis. A novel beam element is proposed, which releases specific degrees of freedom in the stiffness matrix of the classical beam element and introduces axial gap control parameters to achieve coupled simulation of tension-compression bidirectional gap and bending moment transmission. At the same time, an adaptive iterative calculation method with embedded bisection method is proposed to balance computational efficiency and stability.

[0008] To achieve the above design objectives, this application proposes a construction method based on tension-compression coupled gap beam elements. These tension-compression coupled gap beam elements are used to simulate the mechanical behavior of bolted structures in finite element analysis. The construction method includes the following steps: (1) Degrees of freedom condensation; The stiffness matrix of the two-node Euler beam element was condensed using the degree-of-freedom condensation method, releasing the two rotational degrees of freedom about the Z-axis of the element's local coordinate system. i iz , i jz This makes the unit obtained by releasing a specific degree of freedom appear as a hinge in a plane perpendicular to the bolt axis; (2) Matrix order supplementation; The element stiffness matrix obtained after the cohesion process in step (1) is supplemented with matrix order to restore it to the same 12×12 order as the stiffness matrix of the two-node Euler beam element, so as to facilitate the integration of the overall stiffness matrix; (3) Introduction of gap characteristics; Based on the actual physical behavior to be simulated by the unit, the matrix elements representing axial stiffness in the stiffness matrix obtained in step (2) are modified to control parameters that can characterize the state of the unit. β The controlled variable also adjusts other matrix elements in the stiffness matrix that involve element length parameters. In this process, the axial stiffness element in the element stiffness matrix obtained in step (3) can be assigned a value depending on whether the element is under tension, compression, or a critical state. In the critical state, the axial stiffness is set to a minimum value that contributes approximately zero to the overall stiffness of the structure to simulate the gap effect. For example, its magnitude can be set to 10. -5 ~10 -7 The axial stiffness of the element when it is under tension or compression.

[0009] The local coordinate system of the tension-compression coupled gap beam element is a right-handed rectangular coordinate system, in which the X-axis is connected to the nodes at both ends of the element. The lines are aligned, and the Y-axis and Z-axis lie in a plane perpendicular to the X-axis and are perpendicular to each other.

[0010] The theoretical initial length of the tension-compression coupled gap beam element The length of the element when it is not under any load. The element's length is the change in length when it is subjected to a load. Based on this, the element state is determined by the change in length of the element. With critical gap parameter x The comparison confirms: like > x If the element is in tension, the axial stiffness of the element is a non-zero positive value. like < - x If the element is under compression, the axial stiffness of the element is a non-zero positive value. like ≤ x If the element is in a critical state, the axial stiffness of the element is set to a minimum value that is approximately zero.

[0011] Using the above construction method, this application proposes a novel tension-compression coupled gap beam element, whose element stiffness matrix is ​​as follows: In the matrix, in, E The elastic modulus of the rod; A The cross-sectional area of ​​the rod; l This is the initial length of the element when it is not under stress. D This refers to the unit gap value; I y For the local coordinate system of the element y Moment of inertia of the axis; I z For the local coordinate system of the element z Moment of inertia of the axis; G Shear modulus; J The torsional moment of inertia of the unit section; β These are the unit state control parameters; l Parameters are used to calibrate the critical state; x This is a critical state member length control parameter, which only functions independently in the critical state, under tension and compression. x and β The values ​​are equal.

[0012] Based on the aforementioned tension-compression coupled gap beam element, this application proposes a novel iterative calculation method, which includes the following: step: (1) Initialization; Set the total load vector, initial load step size, convergence tolerance, maximum number of iterations, and bisection tolerance; set the initial state of all gap beam elements to the critical state, and integrate the initial global stiffness matrix. During integration, the axial stiffness of all gap beam elements is not included in the integration. (2) Load step cycle; For each load step, perform a balance iteration, including: assembling the current global stiffness matrix, solving for displacement increments, updating element states and internal forces, and performing convergence checks; (3) Binary backtracking; If the equilibrium iteration fails to converge within the maximum number of iterations, the bisection method is used to backtrack to the previous convergent state and search for a convergent solution within a smaller load interval. (4) Termination; When the cumulative applied load reaches the total load set in the model and the last load step converges, the entire nonlinear analysis is successfully completed, and the final displacement and internal force results are output.

[0013] Furthermore, by adaptively adjusting the load increment step size, the algorithm uses a larger step size in solving the smooth region to improve efficiency. In regions of sudden state change, a smaller step size is used to ensure convergence.

[0014] Furthermore, in the load step iteration step, after completing a convergent load step, the iteration count of that load step is used as a basis for determining the load step iteration count.

[0015] Adaptive adjustment of the next load increment Δ P n+1 The adjustment formula is: Δ P n+1 = α( N iter )·Δ P n Where α is a step size scaling factor function based on the number of iterations, Δ P n The load step size is the previous convergent load step.

[0016] The binary backtracking steps include: (3.1) Revert; The overall load, nodal displacements, and element states are rolled back to the state of the previous convergent load step. (3.2) Determine the search interval; Set search range [ P low , P high ],in P low For the convergence load of the previous step, P high For the load that has not converged in the current step; (3.3) Binary iteration; Perform a binary search iteration within the search interval, repeatedly executing the following steps until the interval size is [ P low , P high Less than the preset bisection tolerance (3.3.1) Calculate the midpoint load P mid = ( P low + P high ) / 2; (3.3.2) withP mid Perform iterative equilibrium calculations for the target load; (3.3.3) If the trial calculation converges, then let P low = P mid Otherwise, let P high = P mid ; (3.4) Step size update and continued calculation; With the final P low As the new convergence point, the next load step size is determined by the preset step size update rule, and subsequent calculations continue.

[0017] When updating the element state during the load step cycle, if all gap beam elements are in a non-critical state, i.e., all gaps are... When the beam is closed, the solution mode is switched to the solution mode when the beam end degrees of freedom are released; when any element becomes critical again, the iterative calculation method of the tension-compression coupled gap beam element is activated.

[0018] Based on the above-mentioned iterative calculation method for gap beam elements with tension-compression coupling, this application proposes a non-easy calculation method with a stored computer program. A destructible computer-readable storage medium, wherein the computer program, when executed by a processor, is capable of implementing the above-described iterative calculation method for tension-compression coupled gap beam elements.

[0019] In summary, the tension-compression coupled gap beam element and its iterative calculation method have the following advantages and beneficial effects: This application proposes a novel tension-compression coupled gap beam element. By releasing specific degrees of freedom in the stiffness matrix of a classical beam element and introducing axial gap control parameters, it realizes the coupled simulation of tension-compression bidirectional gap and bending moment transmission. This solves the problem that existing gap rod elements cannot consider horizontal bending moment, and provides a reliable and highly accurate means for the accurate analysis of internal forces and deformations of three-dimensional bolted (pin) connected structures.

[0020] This application proposes an adaptive load step size control iterative algorithm with an embedded bisection method, adapted to the proposed tension-compression coupled gap beam element. This algorithm can dynamically identify changes in element state and intelligently adjust the solution path, thereby ensuring the stability and efficiency of the calculation. This application includes two complementary solution strategies, which can automatically and efficiently realize gap simulation and moment transfer in complex structures, and achieve faster solution speed.

[0021] The iterative calculation algorithm for tension-compression coupled gap beam elements proposed in this application integrates the element mechanical model with the solution algorithm. By embedding an adaptive load step size control strategy based on the bisection method, it is specifically optimized for the physical phenomenon of sudden state changes caused by gap opening and closing, thereby overcoming the limitations of general solvers in such problems, such as low computational efficiency and high dependence on user experience.

[0022] The iterative calculation algorithm for tension-compression coupled gap beam elements proposed in this application exhibits strong robustness and versatility. The introduction of the bisection method acts as a "safety net" for the algorithm, effectively handling the strong nonlinear behavior caused by sudden changes in element states (such as the simultaneous opening and closing of multiple gaps), greatly enhancing the algorithm's convergence ability under complex load conditions. It is not limited to the tension-compression coupled gap beam elements proposed in this application, but can also be widely applied to solving other nonlinear problems involving contact, separation, and slip.

[0023] The iterative calculation method for tension-compression coupled gap beam elements proposed in this application has the outstanding feature of high overall efficiency. The adaptive load step size control strategy adopted avoids the redundancy caused by fixed small step size calculations and also avoids convergence failure caused by fixed large step size calculations. The algorithm can advance rapidly in the computationally flat region and automatically refine in the computationally difficult region, and its overall computational efficiency is significantly better than that of existing publicly available iterative algorithms for beam elements or axial rod gap elements. Attached Figure Description

[0024] Figure 1 This is a schematic diagram of the tension state of a prior art tension-compression bar gap unit; Figure 2 This is a schematic diagram of the critical state of a tension / compression bar gap unit in the prior art; Figure 3 This is a schematic diagram of the compression state of a prior art tension / compression bar gap unit; Figure 4 For elements corresponding to the stiffness matrix of existing beam elements i , j A schematic diagram of the degrees of freedom of a node in each direction; Figure 5 This is a schematic diagram of the iteration of a continuous function; Figure 6 This is a schematic diagram of the iterative calculation method proposed in this application; Detailed Implementation

[0025] The technical solutions in the embodiments will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described solutions are some embodiments of this application, but not all embodiments. Based on the embodiments proposed in this application, those skilled in the art can make improvements or modifications without creative effort, and all such improvements and modifications should fall within the protection scope of the appended claims.

[0026] Example 1: As described in the background section above, while the prior art's gap bar element can simulate the effect of axial clearance and reduce displacement and internal force errors caused by not considering clearance, this element cannot directly transmit bending moment and torque at the nodes. In actual engineering, the actual mechanical behavior of many bolted structures can only be considered as a hinged connection with clearance in a plane perpendicular to the bolt (pin) axis; bending moment or torque can be transmitted in other directions. Therefore, when a structure is subjected to loads not located in a plane perpendicular to the pin axis for various reasons, using existing beam elements or gap bar elements for calculation will inevitably lead to deviations between the calculated structural displacement and internal force and the actual situation. The root cause of this defect is that the elements used in modeling cannot completely describe the actual mechanical behavior of the structure. The stiffness matrix of a two-node Euler beam element in the prior art is shown below: The unit corresponding to the above matrix i , j The degree of freedom identifiers for each direction of a node, such as Figure 4 As shown.

[0027] In the formula, E The elastic modulus of the rod; A The cross-sectional area of ​​the rod; l The initial length of the theoretical unit; I y Let be the moment of inertia about the y-axis of the local coordinate system; I z For around the local coordinate system z Moment of inertia of the axis; G Shear modulus; J For torsional moment of inertia.

[0028] In response, this embodiment proposes a tension-compression coupled gap beam element, which can be derived according to the following construction method and is capable of simulating the coupling of tension-compression bidirectional gaps and bending moment transmission.

[0029] Specifically, the method for constructing the tension-compression coupled gap beam element includes the following steps: (1) Degrees of freedom cohesion; The finite element equations of the gap beam element in the local coordinate system can be expressed as: In the formula, These are the degrees of freedom that need to be eliminated within the unit; These are the degrees of freedom that need to be retained within the unit; It consists of the corresponding elements of the element stiffness matrix before condensation that need to retain the degree of freedom terms; It is composed of the corresponding elements of the degree of freedom to be condensed in the element stiffness matrix before condensation; and They are transposes of each other and consist of coupling terms in the element stiffness matrix before condensation that describe the relationship between the degrees of freedom to be retained and the degrees of freedom to be condensed. It consists of the corresponding elements of the degree of freedom terms that need to be retained in the nodal load array before condensation; It consists of the corresponding elements of the degrees of freedom to be condensed in the nodal load array before condensation; From the above formula, we can see that The resulting matrix form is: In the formula, K * The stiffness matrix of the condensed element; This represents the nodal load vector after condensation. The process of condensation and release of the above degrees of freedom is as follows: Because it is necessary to incorporate the existing technology's beam element stiffness matrix... i iz , i jz Since the two degrees of freedom are being condensed and released, the stiffness matrix is ​​adjusted as follows: In the matrix above, the element in the top left corner (10×10) is... K 0 The element in the upper right corner that is 10×2 is... K 0c The element in the bottom left 2×10 is... K c0 The 2x2 element in the bottom right corner is... K cc ; The displacement vector and nodal load vector of the stiffness matrix after swapping the element positions are as follows:

[0030]

[0031] Based on the above formula, it can be derived that the unit i , j The element stiffness matrix after the rotational degree of freedom of the nodes along the z-axis is condensed is as follows: At this point, the displacement vector and nodal load vector that match the element stiffness matrix after condensation and release of degrees of freedom are respectively: (2) Matrix order supplementation; Before condensation, the element stiffness matrix is ​​a 12×12 matrix. After condensation, the order of the element stiffness matrix is ​​reduced. To ensure consistency in the program's expression, The element stiffness matrix retains its original order. In the current situation, zero elements can be added to the relevant rows and columns to maintain the element stiffness matrix as a 12×12 matrix. The specific form of the element stiffness matrix after addition is as follows: In the above-mentioned element stiffness matrix, the matrix elements have been rearranged so that the positions of the stiffness matrix elements are consistent with the order of the nodal load vectors and displacement vectors of the existing beam element. (3) Introduction of gap characteristics; The element stiffness matrix centrally characterizes the mechanical properties of the element. In engineering practice, adjusting the corresponding parameters in the stiffness matrix to accurately reflect the true mechanical behavior of the structure is a commonly used and reliable analytical method.

[0032] For release i iz and i jz The stiffness matrix of the beam element with degrees of freedom is obtained by incorporating the axial clearance into the above element stiffness matrix and considering the effect of changes in the axial clearance on the stiffness in other directions. The stiffness matrix of the tension-compression coupled gap beam element considering the axial clearance, the bending stiffness in a specific plane, and the torsional stiffness of the member is as follows: in, E The tensile and compressive modulus of elasticity of the rod; A The cross-sectional area of ​​the rod; l Change the initial length of the theoretical unit; D This refers to the unit gap value; I y For the local coordinate system y Moment of inertia of the axis; I z For around the local coordinate system z Moment of inertia of the axis; G Shear modulus; J For torsional moment of inertia; β These are the unit state control parameters; l These are parameters for calibrating the critical state; their default value range is 10. -5 Up to 10 -7 between; xThis is a member length control parameter under critical conditions, and it only functions independently under critical conditions, specifically under tension and compression. x = β ; It is important to note that the value of ξ determines the element length in the current element stiffness matrix. This value is determined through calculation, specifically, the actual spacing between nodes at the convergence point of a given iteration equals... In actual iteration, the spacing between unit nodes determines whether the unit is under tension, compression, or a critical state.

[0033] Furthermore, when using the stiffness matrix of the aforementioned tension-compression coupled gap beam element, the element in xoy When the element only bears end node loads in the plane (a more common case), the corresponding element node load vector should be modified to the following form to ensure that the axial force at the rod end is consistent with the actual load: It should be noted that, although the displacement in the shear direction and the surrounding direction in the above matrix... z The displacements of the axes are not independent, but for clarity, the load vectors are... The relevant items are still displayed in the text.

[0034] Thus, a new tension-compression coupled gap beam element with a specific expression has been formed, which releases the degree of freedom in the plane ( xoy The behavior of the plane is consistent with that of the rod, and in another plane perpendicular to the plane of released degrees of freedom ( xoz The behavior of the plane is consistent with that of the beam, and in the plane perpendicular to the axis of the member ( yoz The plane can withstand torque and can take into account the effect of clearance on the unit behavior.

[0035] As described above, in the method for constructing the stiffness matrix of a tension-compression coupled gap beam element, the axial stiffness of the element will only take effect when the change in the length of the element nodes (node ​​spacing) meets specific conditions.

[0036] Therefore, it is uncertain whether the axial stiffness of the beam elements participates in the calculation. Since it is impossible to predict whether the axial stiffness of the elements will participate in the stress before loading the model, the calculation of the entire finite element model is a nonlinear process. Furthermore, it should be noted that the stiffness in other directions will continuously change with the change in node spacing during the solution process, and the stress state of the elements will change with this change in node spacing. Regarding the aforementioned nonlinear solution process, traditional linear solvers or fixed-step-size nonlinear iterative algorithms are difficult to handle effectively, easily leading to non-convergence or low computational efficiency. Furthermore, while the solvers built into many large-scale general-purpose finite element algorithms can solve the problem, their focus on the universality and stability of the computational method inevitably results in low solution efficiency and a high dependence on user experience.

[0037] To address this issue, this application proposes an iterative calculation method for gap beam elements with tension-compression coupling, based on the aforementioned construction method and adaptive load step size control using a bisection method. This iterative calculation method can dynamically identify changes in element state and intelligently adjust the solution path, thereby significantly improving the iterative solution efficiency of gap beam elements with tension-compression coupling and nonlinear finite element models while ensuring the accuracy and stability of the calculation.

[0038] Applying this method, it is easy to see that when a structure is in a stable state under a certain load, applying a small external disturbance (such as a nodal load or temperature load) usually does not significantly change the state of the internal elements. Only when the disturbance increases to a certain critical value will the structural state change significantly. Therefore, how to quickly and stably determine this critical load value is the core objective of this calculation method. To achieve this objective, the calculation method proposed in this paper sets all gap beam elements to a critical state (i.e., neither under compression nor tension) at the beginning of each load step, integrating only the stiffness of non-axial elements. In subsequent iterations, if the state of an element changes, the program automatically integrates the updated element stiffness matrix into the overall stiffness matrix according to the actual situation. In the specific solution process, this calculation method includes two calculation modes: the conventional mode and the bisection mode. The conventional mode is characterized by the fact that, unlike traditional methods that mechanically determine the next load step size, this algorithm can preset a discrete or continuous step size control function. The program automatically determines the corresponding increment step coefficient according to the number of iterations and multiplies this coefficient by the step size of the previous step as the actual load increment for the next step.

[0039] When the conventional mode becomes difficult to solve (e.g., it fails to converge due to excessive iterations), the program automatically switches to the bisection mode. This mode determines a reasonable load interval based on the incremental step information calculated in the conventional mode, and then uses the bisection method to search for a suitable load step within this interval. By successively halving the search interval, the bisection method significantly improves numerical stability while maintaining solution accuracy. Therefore, the core of the iterative calculation method for tension-compression coupled gap beam elements described in this application lies in the following: at the beginning of each load step, it is assumed that all gap beam elements are in a critical state and trial calculations are performed; the element state and overall stiffness matrix are updated based on the trial calculation results, and residual forces are eliminated iteratively until convergence; if convergence is difficult, the bisection method is automatically activated for backtracking to find a convergent solution within a smaller load interval.

[0040] Specifically, the implementation steps include the following: Step (1), Initialization; Set the total load vector Initial load step size Convergence tolerance (usually taken) Maximum number of iterations Dichotomy tolerance ; Initialize current load displacement vector Internal force vector ; Initial state of all gap beam elements Set to a critical state; Integrated initial global element stiffness matrix During integration, the axial clearance stiffness of the unit is not included in the integration of the overall stiffness matrix; only the bending and shear stiffness matrices are integrated. Integrated into the initial middle; Step (2), load cycle; For each load step Perform the following procedure; Step (2.1): Apply the load increment; Calculate the total load currently applied. .

[0041] Step (2.2), balance iterative loop; Set the number of iterations Initialize the displacement within this load step. ; Step (2.2.1): Assemble the stiffness matrix; Based on the current state of all units Assemble the overall stiffness matrix ; Step (2.2.2): Solve for the displacement increment; Calculate the residual force (unbalanced force) using the following formula: in, This represents the overall nodal force vector assembled from the internal forces of the elements; The equation to be solved is: The displacement increment is calculated from the above formula. ; Step (2.2.3): Update the displacement using the following formula;

[0042] Step (2.2.4): Update the unit state and internal forces; according to Calculate the length variation of each gap beam element. Update the cell status according to the following rules and internal energy: According to the new status Calculate the internal forces of each element and reassemble. ; Step (2.2.5), convergence check; Calculate the new residual force using the following formula: If satisfied Then the load converges stepwise, and let Exit the iteration and proceed to step (4) below; if the current load step does not converge, and Then let Return to step (2.2.1) and continue iterating; if the desired result is reached... If the load step still does not converge, then terminate the iteration of the current load step and proceed to the following step (3). In this step, after completing a convergent load step, the iteration count of that load step is used as the basis for determining the final result. Adaptive adjustment of the next load increment Δ P n+1 The adjustment formula is: Δ P n+1 = α( N iter )·Δ P n Where α is a step size scaling factor function based on the number of iterations, Δ P n The load step size is the previous convergent load step. The adaptive load step size control process can be divided into the following two cases: The first type is the discrete step size scaling factor; The following formula is merely an example; a sufficiently large number of function intervals can be defined during the iteration process, and a coefficient can be set for each function interval: The second type is the continuous step scaling factor: Many kinds of continuous functions are automatically defined. This serves as the basis for determining the step size scaling factor, but attention should be paid to the parameters. The value of is limited to positive integers; this strategy aims to increase the step size when iteration is easy and decrease the step size when iteration is difficult. When the regular iteration mode fails to converge within the maximum number of iterations, the algorithm automatically switches to the following binary search mode; When updating the element state, the solution will be obtained when all gap beam elements are in a non-critical state, i.e., when all gaps are closed. The solution mode is switched to the solution mode when the degrees of freedom at the beam ends are released; when any element becomes critical again, the iterative calculation method for the gap beam element with tension-compression coupling is activated. Step (3): Binary search backtracking; Step (3.1), rollback; All total loads, displacements, and element states are rolled back to the previous convergent load step (i.e., ...). The state of ); Step (3.2): Determine the search interval; Set the current load range Alternatively, a smaller load range can be selected based on the results of the previous iteration, which can be controlled by the analyst. Step (3.3), binary search iteration; Step (3.3.1): Calculate the load at the midpoint of the interval. ; Step (3.3.2), with P mid Perform iterative equilibrium calculations for the target load; Attempt to apply load vector Perform the balancing iteration process in step (2); If the calculation converges, then let Record the current convergence state, exit the iteration, and continue to the next load step; if it does not converge: then let ; After completing a convergent load step, the iteration count of that load step is used as a basis. Adaptive adjustment of the next load increment Δ P n+1 The adjustment formula is: Δ P n+1 = α( N iter )·Δ P n Where α is a step size scaling factor function based on the number of iterations, Δ P n The load step size is the previous convergent load step. Step (3.3.3): Check the interval size; If the trial calculation converges, then let P low =P mid Otherwise, let P high = P mid ; Step (3.4), step size update and continued calculation; With the final P low As the new convergence point, the next load step size is determined by the preset step size update rule and the subsequent calculation continues; Step (4), termination; When the cumulative applied load achieve When the last load step converges, the entire nonlinear analysis is successfully completed, and the final displacement and internal force results are output.

[0043] By adaptively adjusting the load increment step size, the algorithm uses a larger step size in the solution of the smooth region to improve efficiency, and a smaller step size in the state change region to ensure convergence.

[0044] When all gap beam elements are in a non-critical state, the algorithm switches to the classic solution mode where there are no gap beam elements; when any element state changes and a critical state reappears, the tension-compression coupled gap beam element iterative calculation method is reactivated.

[0045] The following key control nodes exist in the above iterative calculation algorithm for tension-compression coupled gap beam elements: Unit axial stiffness integration: When forming the overall stiffness matrix, the axial stiffness of the unit does not truly "disappear" when the unit is in a critical state. Instead, the influence of the gap is simulated by making the contribution of the axial stiffness to the overall stiffness matrix "approximately 0" through the critical state control parameters in the gap beam unit. This method can ensure the continuity of calculation while ensuring that the results meet the requirements of engineering accuracy.

[0046] Transition between critical and non-critical states: During the iteration process, when an element transitions from a critical state to a non-critical state, the axial stiffness of the element will change significantly (e.g., increase significantly). Due to the limitation of axial stiffness, the load vector will become just enough to close or open the gap. In each load increment step, the deformation of the structure needs to be corrected.

[0047] Solution strategy when all gaps are closed: When all gaps in the structure are closed (i.e., all gap beam elements are either under tension or under load), the structure's behavior under new loads is, in principle, the same as that of beam elements with some degrees of freedom released. In this case, unless the state of the gap beam elements changes further, the element behavior will be consistent with that of beam elements with only some degrees of freedom released. At this time, the overall configuration of the structure is relatively stable, and continuing to load the structure within the same load step will often not result in the stiffness deficiency that may exist when the gaps are not closed. Therefore, to further improve solution efficiency, when all gaps are closed, the iterative algorithm will automatically switch the solution mode to the solution mode when "beam end degrees of freedom are released" (note that there are no unclosed gaps at this time). Only when the element state changes and gaps reappear will the iterative algorithm mentioned above be reactivated. Since the solution mode when "beam end degrees of freedom are released" is basically the same as the traditional solution mode (only with slight differences in element length), this method will not be elaborated upon.

[0048] This embodiment, based on the aforementioned iterative calculation method for gap beam elements with tension-compression coupling, proposes a method storing a computer program. A non-volatile computer-readable storage medium. Specifically, when the computer program is executed by a processor, it can implement the above-described iterative calculation method for tension-compression coupled gap beam elements.

[0049] Although the present application has been presented and described with the above embodiments, it will be understood by those skilled in the art that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the present application, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for constructing a tension-compression coupled gap beam element, wherein the tension-compression coupled gap beam element is used to simulate the mechanical behavior of bolted structures in finite element analysis, characterized in that, Build it in the following way: (1) Degrees of freedom condensation; The stiffness matrix of the two-node Euler beam element was condensed using the degree-of-freedom condensation method, releasing the two rotational degrees of freedom θ about the Z-axis of the element's local coordinate system. iz θ jz This makes the unit obtained by releasing a specific degree of freedom appear as a hinge in a plane perpendicular to the bolt axis; (2) Matrix order supplementation; The element stiffness matrix obtained after the cohesion process in step (1) is supplemented with matrix order to restore it to the same 12×12 order as the stiffness matrix of the two-node Euler beam element, so as to facilitate the integration of the overall stiffness matrix; (3) Introduction of gap characteristics; According to the actual physical behavior to be simulated by the unit, the matrix elements representing axial stiffness in the stiffness matrix obtained in step (2) are modified to variables controlled by the control parameter β that can characterize the state of the unit, and other matrix elements in the stiffness matrix that involve the unit length parameter are adjusted at the same time. Among them, the axial stiffness element in the element stiffness matrix obtained by step (3) can be taken according to whether the element is in tension, compression or critical state. In the critical state, the axial stiffness is set to a minimum value that contributes approximately zero to the overall stiffness of the structure to simulate the gap effect. The element stiffness matrix is ​​as follows: In the matrix, Where E is the elastic modulus of the rod; A is the cross-sectional area of ​​the rod; l is the initial length of the element when unloaded; Δ is the element gap value; I y Let I be the moment of inertia about the y-axis of the element's local coordinate system; z G is the moment of inertia about the z-axis of the element's local coordinate system; J is the shear modulus; β is the torsional moment of inertia of the element's cross section; λ is the element's state control parameter; λ is the critical state calibration parameter; ξ is the critical state member length control parameter, which only plays an independent role in the critical state, and ξ is equal to β in the tension and compression states.

2. The method for constructing a tension-compression coupled gap beam element according to claim 1, characterized in that, The local coordinate system of the tension-compression coupled gap beam element is a right-hand rectangular coordinate system, in which the X-axis is aligned with the line connecting the two end nodes of the element, and the Y-axis and Z-axis are located in a plane perpendicular to the X-axis and are perpendicular to each other.

3. The method for constructing a tension-compression coupled gap beam element according to claim 1 or 2, characterized in that, The theoretical initial length of the tension-compression coupled gap beam element The length of the element when it is not under any load. The element's length is the change in length when it is subjected to a load. Based on this, the element state is determined by the change in length of the element. Determined by comparison with the critical gap parameter ξ: like If ξ > ξ, then the element is in tension and the axial stiffness of the element is a non-zero positive value. like If the value is less than -ξ, the element is under compression and the axial stiffness of the element is a non-zero positive value. like If ≤ ξ, then the element is in a critical state, and the axial stiffness of the element is set to a minimum value that is approximately zero.

4. A gap beam element constructed using the tension-compression coupling method as described in any one of claims 1 to 3, characterized in that, Its element stiffness matrix is: In the matrix, Where E is the elastic modulus of the rod; A is the cross-sectional area of ​​the rod; l is the initial length of the element when unloaded; Δ is the element gap value; I y Let I be the moment of inertia about the y-axis of the element's local coordinate system; z G is the moment of inertia about the z-axis of the element's local coordinate system; J is the shear modulus; β is the torsional moment of inertia of the element's cross section; λ is the element's state control parameter; λ is the critical state calibration parameter; ξ is the critical state member length control parameter, which only plays an independent role in the critical state, and ξ is equal to β in the tension and compression states.

5. An iterative calculation method for a gap beam element with tension-compression coupling as described in claim 1, characterized in that, Includes the following steps: (1) Initialization; Set the total load vector, initial load step size, convergence tolerance, maximum number of iterations, and bisection tolerance; set the initial state of all gap beam elements to the critical state, and integrate the initial global stiffness matrix. During integration, the axial stiffness of all gap beam elements is not included in the integration. (2) Load step cycle; For each load step, perform a balance iteration, including: assembling the current global stiffness matrix, solving for displacement increments, updating element states and internal forces, and performing convergence checks; (3) Binary backtracking; If the equilibrium iteration fails to converge within the maximum number of iterations, the bisection method is used to backtrack to the previous convergent state and search for a convergent solution within a smaller load interval. (4) Termination; When the cumulative applied load reaches the total load set in the model and the last load step converges, the entire nonlinear analysis is successfully completed, and the final displacement and internal force results are output.

6. The iterative calculation method for tension-compression coupled gap beam elements according to claim 5, characterized in that, By adaptively adjusting the load increment step size, the algorithm uses a larger step size in the solution of the smooth region to improve efficiency, and a smaller step size in the state change region to ensure convergence.

7. The iterative calculation method for tension-compression coupled gap beam elements according to claim 5, characterized in that, In the load step iterative step, after completing a convergent load step, the iteration count of that load step is used to determine the outcome. Adaptively adjust the next load increment ΔP n+1 The adjustment formula is: ΔP n+1 = α(N iter )·ΔP n Where α is a step size scaling factor function based on the number of iterations, and ΔP n The load step size is the previous convergent load step.

8. The iterative calculation method for tension-compression coupled gap beam elements according to claim 5, characterized in that, The binary backtracking steps include: (3.1) Revert; The overall load, nodal displacements, and element states are rolled back to the state of the previous convergent load step. (3.2) Determine the search interval; Set the search range [P] low , P high ], where P low For the convergence load of the previous step, P high For the load that has not converged in the current step; (3.3) Binary iteration; Perform a binary search iteration within the search interval, repeatedly executing the following steps until the interval size [P] is reached. low , P high Less than the preset bisection tolerance (3.3.1) Calculate the midpoint load P mid = (P low + P high ) / 2; (3.3.2) with P mid Perform iterative equilibrium calculations for the target load; (3.3.3) If the trial calculation converges, then let P low = P mid Otherwise, let P high = P mid ; (3.4) Step size update and continued calculation; With the final P low This serves as the new convergence point, and the next load step size is determined according to the preset step size update rule to continue subsequent calculations.

9. The iterative calculation method for tension-compression coupled gap beam elements according to claim 5, characterized in that, When updating the element state in the load step cycle, when all gap beam elements are in a non-critical state, that is, when all gaps are closed, the solution mode is switched to the solution mode when the beam end degrees of freedom are released. When any element returns to a critical state, the described iterative calculation method for the tension-compression coupled gap beam element is activated.

10. A non-volatile computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the iterative calculation method for tension-compression coupled gap beam elements as described in any one of claims 5 to 9.

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