Nonlinear stiffness simplified numerical simulation method for flange bolted joint structure

Abstract

This invention provides a simplified numerical simulation method for the nonlinear stiffness of flange bolted connection structures. The method simulates the stiffness of the stop portion using COMBIN40 elements in ANSYS; it simulates the uniaxial tensile stiffness using COMBIN39 elements in ANSYS, obtaining the axial displacement curve and the stiffness curve of the COMBIN39 elements under harmonic load. A finite element solid parameterized model is established based on ANSYS. An equivalent simplified model is established based on the slip force and stiffness coefficient parameters of the stop portion extracted from the finite element solid parameterized model. A transverse harmonic load is applied to the simplified model, obtaining the rotational load curve of the simplified model under the transverse harmonic load. The obtained data are analyzed and verified. This method significantly reduces the number of degrees of freedom and meshes while ensuring the reliability and realism of the research.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of digital simulation, in particular to a simplified numerical simulation method for the non-linear stiffness of a flange bolt connection structure. BACKGROUND

[0002] With the breakthrough and development of computer application technology, digital simulation technology based on finite element analysis has been widely applied in the equivalent modeling of the dynamic characteristics of a joint. With the help of finite element numerical simulation technology, the contact mechanical characteristics of the joint can be conveniently simulated, and then the deformation and stress distribution law in the contact process can be intuitively analyzed. At present, the finite element model of the joint can be roughly divided into two categories.

[0003] The first category is the joint finite element model based on spring-damper elements. Yoshimura proposed in 1979 that each joint of a machine tool can be equivalent to a spring-damper element model with six degrees of freedom. Subsequently, many scholars have adopted this equivalent model to analyze and study the dynamics of the joint. Xu et al. equivalent the BT50 spindle-tool holder joint in the spindle system to a series of spring-damper elements, and established an equivalent dynamic model thereof. Ahmadian modeled and identified the damping of the bolt joint by using a nonlinear spring-damper element. Hu et al. also modeled the T-shaped specimen connected by bolts by using a nonlinear spring element, and analyzed the mechanical characteristics of the specimen under cyclic loading. Kim et al. also equivalent the joint to a spring element, established a numerical model of the joint under different joint conditions, and verified the correctness of the numerical model through modal experiments. Yang took the cantilever beam joint as the research object, equivalent modeled the joint by using a spring-damper element, and further considered the influence of the coupling of the moving and rotating degrees of freedom on the model accuracy. Li Yuansheng et al. established a spring-damper element model of a single bolt joint according to the stiffness equivalent principle, and verified the accuracy of the model through experiments. Qu Zhongnian et al. considered the influence of the bolt joint on the ANSYS software analysis platform, and established an equivalent parameter model of the joint by using a spring-damper element. The joint finite element model based on the spring-damper element has a relatively simple modeling principle, but due to the complex modeling method, it is difficult to be applied to the dynamic performance research of complex bolt connections or large assembly structures.

[0004] The second type is the joint finite element model based on contact elements. The research on the joint problem is ultimately the research on the contact problem. In the finite element software, the contact problem can be simulated directly by defining the contact elements. Williams et al. established a finite element simulation model of a single bolt joint by using contact elements, and analyzed the contact characteristics of the single bolt joint under the action of external tensile load by using the comprehensive technology of analytical method and experimental test. Kim et al. compared several finite element modeling methods of bolt joints, and pointed out that the consistency with the experimental results is the best when the bolt joint is modeled by using the surface-to-surface contact element. Yang et al. also established an equivalent model of the bolt joint by using the contact element. On the basis of the equivalent model, the stiffness formula of the conical pressure distribution connector is modified. Zhou Delian uses the method of combining contact elements and spring damping elements to model the bolt joint by using the finite element method. Liu Zongshan respectively uses the direct binding method, the contact element method and the spring damping element method to model the joint by using the finite element method, and compares the advantages and disadvantages and simulation accuracy of the three methods. In addition, Zhang Xueling and Zhang Xingchao of Tianjin University also use the contact element method to deal with the joint problem. The joint finite element model based on the contact element has the advantage that the contact state of the joint can be simulated intuitively. However, the contact analysis in the finite element software is a highly nonlinear analysis. The nonlinear factors caused by the contact make the convergence calculation of the contact problem complex. In addition, since the contact analysis needs to consume more computing resources, it is not ideal to deal with the contact for the actual structure with more joints.

[0005] The stiffness characteristics and simulation methods of bolt connection structures and flange connection structures have been studied a lot and fully, but most of the researches do not consider the influence of the joint on the stiffness characteristics of the flange, and do not pay attention to the hysteresis phenomenon caused by the slip of the joint contact surface. Therefore, the stiffness characteristics and simulation methods of the flange connection structure with the joint are key problems to be studied. Therefore, a simplified numerical simulation method of the nonlinear stiffness of the flange bolt connection structure is needed to solve the above problems. SUMMARY

[0006] The purpose of the present application is to provide a simplified numerical simulation method of the nonlinear stiffness of the flange bolt connection structure to solve the problem that most of the researches on the stiffness characteristics and simulation methods of bolt connection structures and flange connection structures do not consider the influence of the joint on the stiffness characteristics of the flange, and do not pay attention to the hysteresis phenomenon caused by the slip of the joint contact surface.

[0007] The present application provides a simplified numerical simulation method of the nonlinear stiffness of the flange bolt connection structure, comprising:

[0008] Simulate the stiffness of the stopper part based on COMBIN40 unit in ANSYS, obtain the COMBIN40 unit stiffness curve when constant load is applied and the COMBIN40 unit stiffness curve when harmonic load is applied;

[0009] Simulate the one-way tensile stiffness based on COMBIN39 unit in ANSYS, obtain the axial displacement curve when harmonic load is applied and the COMBIN39 unit stiffness curve when harmonic load is applied;

[0010] Establish a finite element entity parametric model based on ANSYS;

[0011] According to the slip force of the stopper part extracted from the finite element entity parametric model, an equivalent simplified model is established according to the stiffness coefficient parameter;

[0012] Apply a transverse harmonic load to the simplified model to obtain the rotation load curve of the simplified model when a transverse harmonic load is applied;

[0013] The obtained data are analyzed and verified.

[0014] Further, based on COMBIN40 unit in ANSYS, the COMBIN40 unit stiffness curve when constant load is applied and the COMBIN40 unit stiffness curve when harmonic load is applied are obtained, including:

[0015] The COMBIN40 is a combination of mutually parallel spring sliders and dampers, and a gap controller is connected in series; the COMBIN40 simulates the nonlinear hysteresis characteristics of the bolt connection part stopper part, according to the unit characteristics and geometric characteristics of the COMBIN40 unit in ANSYS, a real constant is set to make K1+K2 parallel stiffness simulate the stiffness characteristics before the stopper generates slip, and K2 represents the stiffness characteristics after the stopper generates slip, load is applied, the hysteresis characteristics of the nonlinear stiffness of the stopper part are simulated; a single COMBIN40 unit is established to fix one end, and load is applied to the other end to obtain the COMBIN40 unit stiffness curve when constant load is applied and the COMBIN40 unit stiffness curve when harmonic load is applied.

[0016] Further, based on COMBIN39 unit in ANSYS, the one-way tensile stiffness is simulated, the axial displacement curve when harmonic load is applied and the COMBIN39 unit stiffness curve when harmonic load is applied are obtained, including:

[0017] Based on COMBIN39 unit in ANSYS, the one-way tensile characteristic of the bolt connection part mounting edge is simulated, according to the unit characteristic, a real constant is set, and the stiffness of the COMBIN39 unit is not manually input by the user, but is automatically solved according to the F-D curve; the stiffness K=dF / dD; F=the shear stress*the bonding area; with the development of the slip, the stiffness is constantly degenerated; the COMBIN39 unit simulates the function of the COMBIN39 unit which can be stretched and cannot be compressed by setting the real constant curve to set the normal stiffness and the infinite reverse stiffness, a simple harmonic load is applied, and the hysteresis characteristic of the non-linear stiffness of the part of the mounting edge of the engine case is simulated.

[0018] Further, the finite element entity parametric model is established based on ANSYS, including:

[0019] The finite element entity parametric model is established based on ANSYS, the actual bolt connection contact characteristic is set, one end is constrained, and one end is applied with a transverse simple harmonic load as a boundary condition, the slip force and the stiffness curve of the part of the stop are extracted. Data support and comparison reference are provided for the subsequent simplified model, and the rotation angle load curve is obtained.

[0020] Further, the equivalent simplified model is established according to the slip force and the stiffness coefficient parameter of the part of the stop extracted from the finite element entity parametric model, including:

[0021] Two tempered planes are used to simulate the flange mounting edge, a beam unit is used to replace the engine case barrel, and a bolt connection structure is simulated by a spring 39 unit, a spring 40 unit and two 14 units.

[0022] Further, the transverse simple harmonic load is applied to the simplified model, and the rotation angle load curve of the simplified model under the transverse simple harmonic load is obtained, including:

[0023] The same boundary condition is set for the simplified model, the transverse simple harmonic load is applied, the non-linear hysteresis characteristic of the bolt is simulated, the rotation angle displacement of the load point is taken as the abscissa, and the applied simple harmonic load is taken as the ordinate to obtain the rotation angle load curve of the simplified model under the transverse simple harmonic load.

[0024] Further, the obtained data is analyzed and verified, including:

[0025] Analysis of the obtained data reveals that for bolted connections with stop flanges, hysteresis occurs when the connection is subjected to a transverse load parallel to the contact surface. Based on the different bending stiffnesses, the entire curve can be divided into several approximately linear stages: the first stage is the initial loading stage, where the rotation angle increases linearly with the load. At the boundary between stages one and two, the bending stiffness of the structure undergoes a sudden change, remaining linear before and after the change; at the end of the second stage, the flange rotation angle reaches its maximum, and then in the third stage, the load enters the unloading process, at which point the flange's bending stiffness is the same as in the first stage; the fourth stage is the continued unloading process, and the fifth stage is the reverse loading process. The bending stiffness in these two stages is approximately linear, with only slight fluctuations near the zero rotation angle; the bending stiffness is largely consistent with the second stage; the sixth stage, like the third stage, is the unloading process. The entire curve, after removing the first stage, is symmetrical about the origin. The entire curve exhibits two stiffnesses: stages one, three, and six have one stiffness, while stages two, four, and five have approximately the same stiffness. Furthermore, it is verified that the stiffness curve of the solid model can be simulated using ANSYS combination elements.

[0026] The beneficial effects of this invention are as follows: This invention provides a simplified numerical simulation method for the nonlinear stiffness of flange bolt connection structures. For bolt connection structures with stops, it simulates the hysteresis phenomenon that occurs when the connection structure is subjected to a transverse load parallel to the contact surface using ANSYS combination elements. It also verifies that the stiffness curve of the solid model can be simulated using ANSYS combination elements, which greatly reduces the number of degrees of freedom and meshes while ensuring the reliability and authenticity of the research. Attached Figure Description

[0027] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0028] Figure 1 A flowchart illustrating a simplified numerical simulation method for the nonlinear stiffness of a flange bolt connection structure provided in an embodiment of the present invention;

[0029] Figure 2 A COMBIN40 geometric model for a simplified numerical simulation method of nonlinear stiffness of a flange bolt connection structure provided in an embodiment of the present invention;

[0030] Figure 3The stiffness curve of the COMBIN40 element under constant load is provided for a simplified numerical simulation method of nonlinear stiffness of flange bolt connection structure provided in an embodiment of the present invention.

[0031] Figure 4 The stiffness curve of the COMBIN40 element under harmonic load is provided for a simplified numerical simulation method of nonlinear stiffness of flange bolt connection structure in an embodiment of the present invention.

[0032] Figure 5 The axial displacement curve under harmonic load is provided by a simplified numerical simulation method for the nonlinear stiffness of a flange bolt connection structure according to an embodiment of the present invention.

[0033] Figure 6 The stiffness curve of the COMBIN39 element under harmonic load is provided for a simplified numerical simulation method of nonlinear stiffness of flange bolt connection structure in an embodiment of the present invention.

[0034] Figure 7 A schematic diagram of a solid model of a simplified numerical simulation method for the nonlinear stiffness of a flange bolt connection structure provided in an embodiment of the present invention;

[0035] Figure 8 The solid model of the simplified numerical simulation method for nonlinear stiffness of a flange bolt connection structure provided in this embodiment of the invention includes the transverse load and angular load curves.

[0036] Figure 9 A simplified model diagram of a simplified numerical simulation method for the nonlinear stiffness of a flange bolt connection structure provided in an embodiment of the present invention;

[0037] Figure 10 The simplified model of the simplified numerical simulation method for nonlinear stiffness of flange bolt connection structure provided in this embodiment of the invention includes the transverse load and angular load curves. Detailed Implementation

[0038] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention. The technical solutions provided by various embodiments of this invention will be described in detail below with reference to the accompanying drawings.

[0039] Please see Figure 1 A simplified numerical simulation method for the nonlinear stiffness of flange bolted connection structures, comprising:

[0040] S101 simulates the stiffness of the stop section based on the COMBIN40 element in ANSYS, and obtains the stiffness curves of the COMBIN40 element under constant load and under harmonic load.

[0041] In this embodiment, COMBIN40 is a combination of parallel spring sliders and dampers, connected in series with a gap controller. This combined unit is as follows: Figure 2 As shown; the COMBIN40 simulates the nonlinear hysteresis characteristics of the stop portion of the bolt connection. Based on the element characteristics and geometric characteristics of the COMBIN40 element in ANSYS, real constants are set so that the parallel stiffness of K1+K2 simulates the stiffness characteristics of the stop before slippage occurs, while K2 represents the stiffness characteristics after slippage occurs. A load is applied to simulate the hysteresis characteristics of the nonlinear stiffness of the stop portion. A single COMBIN40 element is fixed at one end, and a load is applied to the other end, resulting in the following... Figure 3 The stiffness curves of the COMBIN40 element under constant load are shown below. Figure 4 The stiffness curve of the COMBIN40 element is shown when a harmonic load is applied.

[0042] S102 simulates uniaxial tensile stiffness based on the COMBIN39 element in ANSYS, and obtains the axial displacement curve and the stiffness curve of the COMBIN39 element under simple harmonic load.

[0043] In this embodiment, based on the COMBIN39 element in ANSYS, the uniaxial tensile characteristics of the mounting edge of the bolted connection are simulated. Real constants are set according to the element characteristics. Unlike the COMBIN40 element, the stiffness of the COMBIN39 element is not manually input by the user, but automatically calculated by the program based on the FD curve; stiffness K = dF / dD; F = shear stress (constitutive relation of the bond surface) * bond area; as slip develops, the stiffness continuously degrades. The COMBIN39 element is set with normal positive stiffness and infinite negative stiffness by setting a real constant curve to simulate the tensile but incompressible function of the COMBIN39 element. A harmonic load is applied to simulate the hysteresis characteristics of the nonlinear stiffness of the mounting edge of the casing. The result is as follows: Figure 5 The axial displacement curves under harmonic loads are shown below. Figure 6 The figure shows the stiffness curve of the COMBIN39 element under a harmonic load.

[0044] S103 establishes a finite element solid parameterized model based on ANSYS.

[0045] In this embodiment, a finite element solid parameterized model is established based on ANSYS. Figure 7The diagram shows a schematic of the solid model. By setting the actual bolt connection contact characteristics, with one end constrained and the other end subjected to a transverse harmonic load as boundary conditions, the slip force and stiffness curve of the stop section are extracted. This provides data support and a comparative reference for the subsequent simplified model, as shown below. Figure 8 The lateral load and angular load curves of the solid model shown are derived from... Figure 8 It can be seen that the stiffness curve of the bolted connection structure in the solid model exhibits obvious nonlinear hysteresis.

[0046] S104 establishes an equivalent simplified model based on the slip force and stiffness coefficient parameters of the stop portion extracted from the finite element solid parameterized model.

[0047] In this embodiment, as Figure 9 The diagram shows two tempered planes simulating the flange mounting edge, beam elements replacing the casing, and spring elements 39, 40, and two 14 elements used to simulate an equivalent bolted connection structure.

[0048] S105 applies a lateral harmonic load to the simplified model, resulting in the angular load curve of the simplified model under the lateral harmonic load.

[0049] In this embodiment, the simplified model is subjected to the same boundary conditions, and a lateral harmonic load is applied to simulate the nonlinear hysteresis stiffness of the bolt. The rotational displacement at the loading point is used as the abscissa, and the applied harmonic load is used as the ordinate to obtain the following result: Figure 10 The simplified model shown is subjected to angular load curves when a lateral harmonic load is applied. Figure 10 It can be seen that the curve shape is similar to a willow leaf, exhibiting obvious hysteresis characteristics, with most of the curves in the two cycles overlapping. During the initial loading stage, when the load is approximately 600 N*m, the bending stiffness of the flange decreases abruptly, which is consistent with the calculation results in the solid model.

[0050] S106 analyzes and verifies the obtained data.

[0051] In this embodiment, analysis of the obtained data shows that for bolted connections with a stop, when the connection is subjected to a transverse load parallel to the contact surface, a hysteresis phenomenon will occur. Based on the different bending stiffnesses, the entire curve can be divided into several approximately linear stages: the first stage is the initial loading stage, where the rotation angle increases linearly with the load. At the boundary between stages one and two, the bending stiffness of the structure undergoes a sudden change, remaining linear before and after the change; at the end of the second stage, the flange rotation angle reaches its maximum, and then in the third stage, the load enters the unloading process, at which point the flange's bending stiffness is the same as in the first stage; the fourth stage is the continued unloading process; and the fifth stage... The first stage represents the reverse loading process. The bending stiffness in these two stages is approximately linear, with only slight fluctuations near the zero rotation angle. The bending stiffness is largely consistent with that in the second stage. The sixth stage, like the third stage, represents the unloading process. The entire curve is symmetrical about the origin after removing the first stage. The entire curve exhibits two types of stiffness: the first, third, and sixth stages have one type of stiffness, while the second, fourth, and fifth stages have approximately the same stiffness. Furthermore, it has been verified that the stiffness curve of the solid model can be simulated using ANSYS-based combination elements, which significantly reduces the number of degrees of freedom and meshes while ensuring the reliability and realism of the research.

[0052] The working principle of the simplified numerical simulation method for nonlinear stiffness of flange bolt connection structures provided by this invention is as follows: The stiffness of the stop portion is simulated using COMBIN40 elements in ANSYS, obtaining the stiffness curves of COMBIN40 elements under constant load and under harmonic load; the uniaxial tensile stiffness is simulated using COMBIN39 elements in ANSYS, obtaining the axial displacement curve and the stiffness curve of COMBIN39 elements under harmonic load; a finite element solid parameterized model is established based on ANSYS; an equivalent simplified model is established based on the slip force and stiffness coefficient parameters of the stop portion extracted from the finite element solid parameterized model; a transverse harmonic load is applied to the simplified model, obtaining the rotational load curve of the simplified model under the transverse harmonic load; the obtained data are analyzed and verified. Verification proves that the stiffness curve of the solid model can be simulated using combination elements based on ANSYS, greatly reducing the number of degrees of freedom and meshes while ensuring the reliability and authenticity of the research.

[0053] This invention also provides a storage medium storing a computer program. When executed by a processor, the computer program implements some or all of the steps in various embodiments of the simplified numerical simulation method for nonlinear stiffness of a flange bolt connection structure provided by this invention. The storage medium may be a magnetic disk, optical disk, read-only memory (ROM), or random access memory (RAM), etc.

[0054] Those skilled in the art will clearly understand that the techniques in the embodiments of the present invention can be implemented using software plus necessary general-purpose hardware platforms. Based on this understanding, the technical solutions in the embodiments of the present invention, or the parts that contribute to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in various embodiments or certain parts of the embodiments of the present invention.

Claims

1. A simplified numerical simulation method for the nonlinear stiffness of a flange bolt connection structure, characterized in that, include: The stiffness of the stop section is simulated using the COMBIN40 element in ANSYS. Based on the element characteristics and geometric characteristics of the COMBIN40 element in ANSYS, real constants are set so that the parallel stiffness of K1+K2 simulates the stiffness characteristics of the stop before slippage occurs. K2 represents the stiffness characteristics of the stop after slippage occurs. Loads are applied to simulate the hysteresis characteristics of the nonlinear stiffness of the stop section. A single COMBIN40 element is fixed at one end, and a load is applied to the other end to obtain the stiffness curves of the COMBIN40 element under constant load and under harmonic load. Based on the COMBIN39 element in ANSYS, uniaxial tensile stiffness is simulated. According to the element characteristics, real constants are set. The stiffness of the COMBIN39 element is automatically obtained by the program based on the FD curve. The COMBIN39 element is set to have normal positive stiffness and infinite negative stiffness by setting the real constant curve to simulate the tensile but incompressible function of the COMBIN39 element. A simple harmonic load is applied to simulate the hysteresis characteristics of the nonlinear stiffness of the casing mounting edge, and the axial displacement curve and the stiffness curve of the COMBIN39 element under the simple harmonic load are obtained. Establish a finite element solid parameter model based on ANSYS; An equivalent simplified model is established based on the slip force and stiffness coefficient parameters of the stop portion extracted from the finite element solid parameterized model. A lateral harmonic load is applied to the simplified model to obtain the angular load curve of the simplified model under the lateral harmonic load. The obtained data is analyzed and verified.

2. The simplified numerical simulation method for nonlinear stiffness of flange bolt connection structures according to claim 1, characterized in that, The COMBIN40 is a combination of parallel spring sliders and dampers, connected in series with a clearance controller; the COMBIN40 simulates the nonlinear hysteresis characteristics of the bolt connection stop portion.

3. The simplified numerical simulation method for nonlinear stiffness of flange bolt connection structures according to claim 1, characterized in that, Based on the COMBIN39 element in ANSYS, the uniaxial tensile characteristics of the mounting edge of the bolted connection are simulated. The stiffness of the COMBIN39 element is K=dF / dD; F=shear stress*bonded area; as slip develops, the stiffness continuously degrades.

4. The simplified numerical simulation method for nonlinear stiffness of flange bolt connection structures according to claim 1, characterized in that, A finite element solid parametric model was established based on ANSYS, including: A finite element solid parameterized model was established based on ANSYS. By setting the actual bolt connection contact characteristics, setting one end constraint, and applying a transverse harmonic load to the other end as boundary conditions, the slip force and stiffness curve of the stop part were extracted. This provides data support and comparison reference for the subsequent simplified model, and the rotational load curve was obtained.

5. The simplified numerical simulation method for the nonlinear stiffness of a flange bolt connection structure according to claim 1, characterized in that, Based on the slip force and stiffness coefficient parameters extracted from the parametric finite element model of the stop portion, an equivalent simplified model is established, including: Two tempered planes are used to simulate the flange mounting edge, beam elements are used to replace the casing, and spring elements 39, spring elements 40, and two elements 14 are used to simulate a bolted connection structure.

6. The simplified numerical simulation method for nonlinear stiffness of a flange bolt connection structure according to claim 1, characterized in that, Applying a lateral harmonic load to the simplified model yields the angular load curves of the simplified model under the applied lateral harmonic load, including: The same boundary conditions are set for the simplified model, and a lateral harmonic load is applied to simulate the hysteresis characteristics of the bolt's nonlinear stiffness. The angular load curve of the simplified model under the applied lateral harmonic load is obtained by using the rotational displacement of the loading point as the abscissa and the applied harmonic load as the ordinate.

7. The simplified numerical simulation method for nonlinear stiffness of flange bolt connection structures according to claim 1, characterized in that, The obtained data was analyzed and verified, including: Analysis of the obtained data reveals that for bolted connections with stop flanges, hysteresis occurs when the connection is subjected to a transverse load parallel to the contact surface. Based on the different bending stiffnesses, the entire curve can be divided into six approximately linear stages: the first stage is the initial loading stage, where the rotation angle increases linearly with the load. At the boundary between stages one and two, the bending stiffness of the structure undergoes a sudden change, remaining linear before and after the change; at the end of the second stage, the flange rotation angle reaches its maximum, and then in the third stage, the load enters the unloading process, at which point the flange's bending stiffness is the same as in the first stage; the fourth stage is the continued unloading process, and the fifth stage is the reverse loading process. The bending stiffness in these two stages is approximately linear, with only slight fluctuations near the zero rotation angle; the bending stiffness is largely consistent with the second stage; the sixth stage, like the third stage, is the unloading process. The entire curve, excluding the first stage, is symmetrical about the origin. The entire curve exhibits two stiffnesses: stages one, three, and six have one stiffness, while stages two, four, and five have approximately the same stiffness. Furthermore, it is verified that the stiffness curve of the solid model can be simulated using ANSYS combination elements.

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