A method for predicting working conditions of a bolted joint structure based on representing pre-tightening force relaxation

Abstract

The present application relates to the technical field of bolt connection, in particular to a method for predicting the working condition of bolt connection structure based on the representation of pre-tightening force relaxation. The present application proposes a double exponential decay function, and the parameters of different functions are identified. The double exponential function is simple in structure and good in representation, and the fitting accuracy reaches 99.3%, which can accurately represent the bolt pre-tightening force relaxation law. The pre-tightening force relaxation law function is introduced into the Iwan model to represent the Iwan model under the relaxation condition conforming to the actual situation, the relationship between various parameters and pre-tightening is analyzed, the least square method is used to construct the mathematical expression, the pre-tightening relaxation law in the cycle is equivalent to the pre-tightening relaxation law of the tangential displacement and the loading time, and the Iwan model density function under the relaxation condition is constructed based on various changing parameters. The method of the present application can predict the nonlinear behaviors such as energy dissipation and stiffness degradation, and especially provides a more accurate method for predicting various indicators under the pre-tightening relaxation condition of the bolt.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of bolt connection, in particular to a method for predicting working conditions of bolt connection structure based on characterization of pre-tightening relaxation. BACKGROUND

[0002] In the prior art, pre-tightening relaxation experiments under large external loads have been carried out, such as the document LU X, ZHU M, LI C, et al. Prediction of Pre-Loading Relaxation of Bolt Structure of Complex Equipment under Tangential Cyclic Load [J]. Sensors, 2024, 24(11): 3306. Existing experimental equipment such as universal testing machines and ring-shaped force sensors are used. The pre-tightening relaxation under different tangential external load conditions and different initial torque conditions is nonlinearly characterized, and the Allometricl model function and the nine-stage polynomial function of the pre-tightening relaxation with the number of cycles are obtained, with fitting effects of 71.7% and 90.4%, respectively. The existing two functions have certain shortcomings. The first function has low fitting accuracy; the second function has 10 unknown parameters to be identified, resulting in poor function characterization effect. At the same time, the Iwan model (see the article IWAN W D. A distributed-element model for hysteresis and its steady-state dynamic response [J]. 1966.) is composed of a perfect elastic-plastic Jenkins unit, while a single sliding damper and a linear spring with a stiffness of form an elastic-plastic Jenkins unit, and the maximum yield force is . In the existing Iwan model, the pre-tightening force is assumed to be a constant value, but experimental research shows that the pre-tightening relaxation presents a certain rule, so it is necessary to introduce the function of pre-tightening relaxation into the Iwan model to make the predicted value more accurate. SUMMARY

[0003] In view of the deficiencies of the prior art, the present application provides a method for predicting working conditions of bolt connection structure based on characterization of pre-tightening relaxation. The present application proposes a double exponential decay function, and the parameters of different functions are identified to obtain a double exponential function with simple structure and good characterization. The fitting accuracy of the double exponential function of the present application is 99.3%, and there are only five unknown parameters, which can accurately characterize the pre-tightening relaxation rule of the bolt.

[0004] The technical scheme of the present application is: a method for predicting working conditions of a bolted joint structure based on a representation of pre-tightening force relaxation, characterized by comprising the following steps:

[0005] Step one, a step of representing pre-tightening force relaxation by a double exponential function:

[0006] The formula of the double exponential function is:

[0007] (31)

[0008] In the formula, F0+S1+S2=the pre-tightening force value of the initial bolt, F0 represents the first fitting number, S1 represents the second fitting number, S2 represents the third fitting number, 1 represents the fourth fitting number, 2 represents the fifth fitting number is the number of cycles, is the pre-tightening force;

[0009] Step two, a step of predicting the working conditions of the bolt by coupling an Iwan model,

[0010] Step 21 representation of initial stiffness

[0011]

[0012] is the parameter to be identified;

[0013] Step 22, a step of fitting the slope of the pressure peak , the slope of the maximum radius as a linear function

[0014] The formula of the slope of the pressure peak and the slope of the maximum radius is:

[0015] (32)

[0016] (33)

[0017] q1, q2, q 11 , q 22 , q 33 are all parameters to be identified,

[0018] Step 23, a step of determining the pre-tightening force correction factor

[0019] The formula of the pre-tightening force correction factor is:

[0020] (35)

[0021] ​​ The initial pressure value;

[0022] Step 24, initial pressure value , initial linear pressure distribution function slope Step of determining the relationship with the preload

[0023] The expression of the initial pressure value and the initial linear pressure distribution function slope is:

[0024] (36)

[0025] In the formula, , are the intercept and slope of the initial pressure value and the preload function F y , respectively.

[0026] Step 25, step of determining the relationship between the longitudinal and transverse multiplication coefficients and the slope

[0027] The longitudinal and transverse equal ratio multiplication coefficients of the linear pressure distribution function are:

[0028] (37)

[0029] (38)

[0030] Represents the size of the tangential tension;

[0031] Step 26, step of determining the relationship between the preload relaxation and displacement

[0032] In the ABAQUS finite element software, the time of each analysis step is 1s, so the tangential displacement and time present a mapping relationship distribution function:

[0033] (39)

[0034] is the tangential displacement, is the time;

[0035] Step 27, step of establishing and predicting the coupled Iwan model

[0036] Let , , , , replace the complex parameters:

[0037] (41)

[0038] (42)

[0039] (43)

[0040] (44)

[0041] (45)

[0042]

[0043] (46)

[0044] is the friction coefficient, is the friction shear stress, is the mapping parameter, is the maximum pressure, is the maximum value of the long semi-axis of the elliptical contact boundary.

[0045] According to the method for predicting the working condition of a bolted joint structure based on the relaxation of the pre-tightening force, characterized in that the bolt pre-tightening force relaxation experiment under a small load working condition is used for verification in step one.

[0046] According to the method for predicting the working condition of a bolted joint structure based on the relaxation of the pre-tightening force, characterized in that in step 26, when , the relationship between the pre-tightening force relaxation and the displacement is:

[0047] (40).

[0048] According to the method for predicting the working condition of a bolted joint structure based on the relaxation of the pre-tightening force, characterized in that in step 27, .

[0049] According to the method for predicting the working condition of a bolted joint structure based on the relaxation of the pre-tightening force, characterized in that the friction shear stress and the yield force are in a mapping relationship, and the relationship is:

[0050] .

[0051] According to the method for predicting the working condition of a bolted joint structure based on the relaxation of the pre-tightening force, characterized in that the mapping parameter is expressed as:

[0052] (44).

[0053] BRIEF DESCRIPTION OF THE DRAWINGS​

[0054] Figure 1 Improvement of the pre-tightening force relaxation function.

[0055] Figure 2 Pre-tightening force relaxation experiment: (a) bolt loosening experimental device, (b) relaxation of different initial pre-tightening forces under 0.1mm tangential external load, (c) relaxation of different displacements under 11567N initial pre-tightening force.

[0056] Figure 3 Relationship between various parameters: (a) flow chart of relationship between the four, (b) initial stiffness curve, (c) coefficient variation, (d) initial pressure variation, (e) pre-tightening force discretization analysis, (f) displacement variation.

[0057] Figure 4 Identification of backbone curve parameters.

[0058] Figure 5 Variation law of short semi-axis and long semi-axis of contact boundary.

[0059] Figure 6 Two pressure correction methods.

[0060] Figure 7 Discretized pressure distribution.

[0061] Figure 8 Tangential force variation process diagram. DETAILED DESCRIPTION

[0062] The technical solutions of the present application are further described below in combination with the drawings.

[0063] A method for predicting the working condition of a bolted joint structure based on representing pre-tightening force relaxation according to the present application comprises the following steps:

[0064] Step one, using a double exponential function to represent the pre-tightening force relaxation:

[0065] The original data (the original data is the pre-tightening force relaxation data of different initial torques under different cyclic displacements in the published paper: Prediction of Pre-Loading Relaxation of Bolt Structure of Complex Equipment under Tangential Cyclic Load. The inventor first found that the allometricl model function and the nine-stage polynomial function can represent the pre-tightening force relaxation of the bolt, but in subsequent calculations, it was found that the two functions are not suitable for the Iwan model, therefore, another function: double exponential decay function is proposed. The parameters of different functions (single exponential decay function, double exponential decay function, Gaussian function, and first-order rational function) are identified (parameter identification is carried out in Origin software, one-key fitting is performed on the data, and the specific parameter values of the fitting formula are obtained), and the double exponential function with simple structure and good representation is obtained. The fitting accuracy of the double exponential function reaches 99.3%, as shown in Figure 1 From Figure 1 , it can be known that under the condition of the same initial pre-tightening force, the bolt pre-tightening force changes with the increase of the cyclic number under the tangential displacement load from 0.5mm to 2mm. The double exponential function in this step has only five unknown parameters, which can accurately represent the bolt pre-tightening force relaxation law, and the formula of the double exponential function is:

[0066] (31)

[0067] In the formula, F0+S1+S2=the initial bolt pre-tightening force value, F0 represents the first fitting number, S1 represents the second fitting number, S2 represents the third fitting number, 1 represents the fourth fitting number, 2 represents the fifth fitting number (the values of F0, S1, S2, 1, 2 five parameters are obtained through the pre-tightening force relaxation data obtained through experiments, and the least square method is used for nonlinear data fitting), is the cyclic number, is the pre-tightening force.

[0068] In order to verify the first stage and the first loading pre-tightening force relaxation is larger. The bolt pre-tightening force relaxation experiment under small load working condition is adopted. The 50KN electro-hydraulic fatigue testing machine, ring-shaped force sensor, oscilloscope and other equipment are used for control and measurement, as shown in Figure 2(a) shown. Nickel steel has corrosion resistance, weather resistance and anti-vibration performance, and the advantages of enhancing the safety of the structure. Often used in marine engineering and shipbuilding, construction and bridge, etc. As a special structure support, bridge components in corrosive environment. Therefore, the flat plate is made of nickel steel material. The complex equipment is simplified to two plates interacting with each other. The complexity of model calculation can be reduced, and the influence of tangential torque can be prevented. A large number of standard M8 bolts are used in the experiment. In order to control a single variable, each bolt only completes one assembly and disassembly. Before the experiment, the screw holes of the two fast nickel steel plates are aligned and placed on the experimental machine. The torque wrench is tightened and the load sensor is placed between the nut and the plate. Through the computer terminal control system, the experiment can be carried out.

[0069] The first stage of pre-tightening force relaxation law research under several different working conditions is carried out, such as Figure 2 (b-c) shown. When the tangential displacement is 0.1mm, the load is loaded once, and the pre-tightening force is relaxed: 3.12%, 1.84%, 4.30%, 4.21%. At the 10th cycle, the pre-tightening force is relaxed: 0.07%, 0.05%, 0.07%, 0.02%. When the displacement is 0.1mm, the pre-tightening force is relaxed the fastest when it is loaded for the first time, and when it reaches the 10th cycle, the pre-tightening force remains stable. When the displacement increases, the pre-tightening force relaxes faster when it is loaded for the first time. When the displacement is 0.25mm, the pre-tightening force is relaxed: 17.12%, 20.51% when it is loaded for the first time and the tenth time. Based on equation (31), the double exponential decay function is used to describe the first stage pre-tightening relaxation law under small displacement load. The average fitting accuracy of the fitting function reaches 99.7%, which can better characterize the pre-tightening relaxation law of multiple loading cycles. It is proved that the mathematical function is applicable under small load conditions.

[0070] Therefore, under small load displacement, the pre-tightening force basically reaches the maximum relaxation degree in the first stage, and the pre-tightening relaxation is small, but the relaxation situation when it is loaded for the first time cannot be ignored. In the second stage, the bolt is basically not loose. Large displacement will cause the bolt to continue to loosen throughout the vibration cycle.

[0071] Step two, predict the working condition of the bolt by coupling the Iwan model,

[0072] Under the action of tangential cyclic load, the pre-tightening force of the bolt gradually relaxes. How to introduce the pre-tightening force-cycle function into the density function of the yield displacement of the modified Iwan model becomes a research difficulty.

[0073] Pre-tightening relaxation will cause the parameters of the Iwan model: initial stiffness , slope of pressure peak , slope of maximum radius , initial pressure value , pre-tightening correction factor initial linear pressure distribution function slope Nonlinear change occurs.

[0074] As shown in Figure 8 , when the pre-tightening relaxation, the contact area is circular under the condition of no tangential load. Under the action of tangential force, the contact area between the plates can be approximated as an ellipse. The tangential tensile force direction is defined as , and the vertical direction is , so the contact boundary shape is approximated as an ellipse, and its formula is:

[0075] (15)

[0076] With the increase of tangential displacement load, the contact ellipse boundary gradually changes, the short semi-axis shortens, and the long semi-axis increases and exceeds the maximum width of the plate. According to the finite element model, the evolution values are shown in , . Figure 5

[0077] The short semi-axis value decreases with the increase of tangential load, and the long semi-axis value increases with the increase of tangential load. The short semi-axis function conforms to the exponential change, and the long semi-axis function conforms to the linear change, and its general formula is:

[0078] (16)

[0079] (17)

[0080] In the formula, , , , , are all to be identified parameters, is the maximum value of the long semi-axis.

[0081] is the length of the short semi-axis in the joint surface elliptical contact boundary, as shown in formula 16, with the increase and decrease of tangential force, the short semi-axis gradually decreases, and the long semi-axis gradually increases. The pressure value near the screw hole is the largest, the radius is 4.5mm, along the direction of the short semi-axis, the pressure value gradually decreases, and the radius gradually increases. When the pressure decreases to 0, the maximum value of the radius is .

[0082] ​Only related to initial working condition, without considering the influence of both. Mathematical relationship expression of pretension and various parameters can be constructed. Pretension relaxation law can be characterized as the relationship between pretension and tangential displacement. The expression relationship between various parameters and tangential displacement can be obtained. Introduce various parameters into the density distribution function of yield displacement, deduce Iwan model under relaxation condition, and the relationship among the four is as shown in Figure 3 (a).

[0083] Step 21 initial stiffness characterization

[0084] The fitting curve is constructed by an exponential function, as shown in Figure 3 (b). The initial stiffness data under different pretensions are obtained by finite element. Based on the least square method, the exponential function is used to fit the data, and the initial stiffness fitting coefficient is 0.97569, and the fitting degree is good. It shows that under different pretension working conditions, this function can predict the value of initial stiffness. The initial stiffness fitting coefficient is 0.97569, and the fitting degree is good. When the pretension relaxation is small, the initial stiffness changes little, and the influence of relaxation can be ignored. When the pretension relaxation is large, the initial stiffness has a large error range. Existing literature shows that the joint surface of bolted connection structure will have stiffness degradation phenomenon under the influence of external load, which gradually decays from large stiffness to small stiffness, and the stiffness is continuous and will not decay to 0, but there is a small residual stiffness. Affected by pretension. When the single cycle relaxation is large, the residual stiffness changes greatly. The residual stiffness needs to be identified by parameters to obtain more accurate residual stiffness. The linear relationship between residual stiffness and pretension cannot be determined, and the situation needs to be analyzed. Therefore, the exponential function is used to characterize the relationship between stiffness and pretension to ensure the high accuracy of Iwan model prediction. Its formula is:

[0085]

[0086]

[0087] The overall stiffness of the joint surface of the connecting structure is equal to the initial stiffness minus the residual stiffness, as shown in Figure 4 (b) below.

[0088] The initial stiffness data under different pretensions are obtained by finite element. Based on the least square method, the exponential function is used to fit the data, and the initial stiffness fitting coefficient is 0.97569, and the fitting degree is good. It shows that under different pretension working conditions, this function can predict the value of initial stiffness. And is the parameter to be identified, which needs to be identified by specific data. The identification result of the model established in this paper is the result in Figure 3 (b).

[0089] Residual stiffness Identification by finite element.

[0090] Step 22, slope of pressure peak , slope of maximum radius Step of fitting to linear function

[0091] Based on the finite element results under different pre-tightening conditions, the parameters , are fitted to linear function and Expdec1 function, respectively, as shown in Figure 3 (c). The values remain basically unchanged, show a nonlinear decrease, and the fitting accuracies are 0.90345 and 0.99172, respectively. The fitting curves have good representation effects and can be used to represent the values of the pre-tightening force changes. The formula is:

[0092] (32)

[0093] (33)

[0094] q1, q2, q 11 , q 22 , q 33 are parameters to be identified. The data under different pre-tightening forces can be obtained through finite element or experiments, and then the parameters can be identified through the formulas (32) and (33), so that accurate values can be obtained.

[0095] Step 23, pre-tightening force correction factor determination step

[0096] The pre-tightening force correction factor changes with the pre-tightening force relaxation, and the influence of the pre-tightening force needs to be considered. At the same time, the tangential force also influences the correction parameter, and the influences of the two need to be considered comprehensively.

[0097] (35)

[0098] is the theoretical pre-tightening force value; is the discrete calculated pre-tightening force value between the plates. The discrete pre-tightening force value is described in detail below.

[0099] When =0, the maximum radius of the pressure distribution function is 12.33607 mm. However, the identification result of the contact boundary function through finite element shows that the maximum pressure radius is 11.19226. Therefore, the maximum radius when the pressure is zero needs to be corrected. Common correction methods are constraint method and offset method. The constraint method corrects the maximum radius by adding a constraint to the maximum radius. The offset method corrects the maximum radius by offsetting the maximum radius. =0, the pressure distribution function is constrained so that the pressure zero value is located at the maximum of the contact boundary . The shift method shifts the linear function downward so that the pressure result is consistent with the contact boundary function, as shown in Figure 6 .

[0100] (27)

[0101] =0, the linear pressure distribution function formula is expressed as:

[0102] (28)

[0103] (29)

[0104] Therefore, by substituting formula (28), (29) into formula (27), the dynamic pressure distribution function under different tangential forces can be obtained as:

[0105] (30)

[0106] (31)

[0107] Based on the simulation results of finite element ABAQUS, it is found that the initial contact state is circular, and then under the action of static friction, the contact state gradually evolves into an elliptical shape. Under the interaction of static friction and dynamic friction, with the increase of tangential load, the radius (short semi-axis) that reaches the pressure zero value gradually decreases, the long semi-axis continuously increases, and finally it is truncated at the maximum half-width of the flat plate. During this process, the slope of the ellipse gradually decreases, the pressure around the screw hole is the maximum, and the maximum contact boundary pressure is zero. Discrete analysis is carried out on the linear pressure distribution, and the contact area is divided into discrete ellipses. When , the discretized area can be approximated as the area of the entire region. By discretizing the contact area and modifying the linear pressure distribution, the error between the pre-tightening force after pressure correction and the theoretical pre-tightening force is verified, and the accuracy of the two correction methods is also verified. As shown in Figure 7 .

[0108] Assuming that the length of the short semi-axis of the ellipse remains equidistantly increasing, and the long semi-axis changes with the curvature and short semi-axis, then from 0 to , the ellipticity gradually increases, i.e., from 1 to components. Therefore, the length of the short semi-axis, the length of the long semi-axis, and the ellipticity of the th ellipse are respectively:

[0109] (32)

[0110] The length of the short semi-axis.

[0111] Therefore, the area of the ellipse, the discrete preload is respectively:

[0112] (33)

[0113] (34)

[0114] The total calculated preload is The discrete preload value is accumulated, and when =1000, the discrete degree remains stable.

[0115] The preload correction parameter can be represented as a dynamic equation that changes with the tangential external force:

[0116] (35)

[0117] In the formula, N i is the discrete calculated preload value between the plates, which can be obtained by finite element software.

[0118] After obtaining this formula, the relationship between and the preload is characterized.

[0119] Step 24, the initial pressure value , the initial linear pressure distribution function slope and the preload relationship determination step

[0120] The relationship between the initial pressure value and the preload is shown in Figure 3 (d). The initial pressure value and the preload show a linear relationship, with a fitting correlation coefficient of 0.99967, which has a good characterization effect. Therefore, the initial pressure value can be predicted by the preload value. Under ideal conditions, the greater the preload, the better the bolt fastening effect, which will lead to an increase in the interfacial pressure value between the plates, and the formula is:

[0121] (34)

[0122] The slope of the linear pressure distribution function changes with the initial pressure value and the maximum contact radius. When the preload relaxes, the maximum contact radius remains basically unchanged, and the maximum contact radius is considered as a constant value. Only related to the initial working condition. Therefore, the slope and the preload have a nonlinear relationship:

[0123] (36)

[0124] wherein, , are the initial pressure value and the intercept and slope of the pre-tightening force function Fy, after obtaining the data of the two, the specific value of , can be obtained by linear fitting, is the radius value when the pressure is maximum, is the radius value when the pressure decreases to 0, the origin of the radius value is the center position of the screw hole, at the edge of the screw hole, there is only pressure value. Along the positive direction of the minor axis, the radius value gradually increases, and the pressure value gradually decreases.

[0125] Step 25, the step of determining the relationship between the longitudinal and transverse multiplication coefficients and the slope;

[0126] The longitudinal and transverse equal multiplication coefficients of the linear pressure distribution function are respectively:

[0127] (37)

[0128] (38)

[0129] represents the size of the tangential tension.

[0130] Step 26, the step of determining the relationship between the pre-tightening force relaxation and displacement

[0131] The first stage of pre-tightening force relaxation and the first loading cycle are studied. The expression relationship between the pre-tightening force relaxation and the cycle is constructed. The first cycle is discretized, and it is divided into portions, as shown in Figure 3 (e). Each discrete interval corresponds to a different pre-tightening force. Through the pre-tightening relaxation function, the unique pre-tightening force of different discrete intervals is obtained.

[0132] The equalized first cycle pre-tightening force relaxation is characterized by ABAQUS software. The finite element software is based on the real characterization of relaxation. In the finite element calculation, the calculation time of one cycle is regarded as 1s. In the first cycle portions of equal interval cycle can correspond to the value of time. Therefore, there is a certain mapping relationship between them. In the finite element calculation, the time and the tangential displacement present a mapping relationship, as shown in Figure 3 (f). The distribution function is:

[0133] (39)

[0134] is the tangential displacement, is the time.

[0135] When At this time, the first cycle of the discrete and time corresponding to the pre-tightening force relaxation and displacement relationship:

[0136] (40)

[0137] At this time, it can be understood that when the connecting structure moves to a certain displacement, it corresponds to a pre-tightening force value (not the initial value, but the relaxed one). And this pre-tightening force value is dynamically corresponding to six parameters. Therefore, a closed loop is formed.

[0138] The tangential load is cyclic, not just stretched once, as mentioned in the previous experiment, this patent mainly considers the first cycle of stretching and the first 10-15 times of stretching. Because, under small load displacement, only in the first cycle stage (before 15 times of tangential cycle), the relaxation of pre-tightening force will have a greater impact on the Iwan model, and the later cycles have less impact on the model. (Because the hysteresis will form an area of energy dissipation, which basically remains unchanged, indicating that the structure's energy is basically not dissipated, so it has less impact on the Iwan model.)

[0139] Steps 21-26 get the nonlinear relationship between pre-tightening force and various parameters in the Iwan model. But after getting this relationship, it is not possible to introduce the pre-tightening force into the Iwan model, so the following steps are to study how to introduce the pre-tightening force into the model.

[0140] Step 27, coupling the Iwan model established and predicted

[0141] Based on the above analysis, the relationship between various parameters and tangential displacement can be characterized by mathematical functions. The Iwan model under relaxation conditions is constructed.

[0142] Under the pre-tightening force relaxation condition, the parameters of the Iwan model change with the pre-tightening load. In this paper, the relationship between various parameters-pre-tightening force, pre-tightening force-tangential displacement is characterized by analytical expressions, and the backbone curve of the Iwan model under relaxation conditions is obtained. Based on the six changing parameters (initial stiffness , slope of pressure peak , slope of maximum radius , initial pressure value , pre-tightening force correction factor , initial linear pressure distribution function slope ), the density function of the yield displacement of the Iwan model under relaxation conditions can be derived. To simplify the yield displacement density function equation, let , , , , replace the complex parameters:

[0143] (41)

[0144] (42)

[0145] (43)

[0146] (44)

[0147] (45)

[0148] where the independent variable is tangential force , pre-tightening force relaxation function , and equivalent to tangential displacement. Therefore, the number of independent variables and dependent variables in the density function of yield displacement does not change, and is still tangential displacement , tangential force .

[0149] Six parameters (initial stiffness , slope of pressure peak , slope of maximum radius , initial pressure value , pre-tightening force correction factor , slope of initial linear pressure distribution function ) are obtained by finite element collection, and parameters under different pre-tightening forces are collected. These parameters are fitted to obtain a function, which is the function of steps 21-25. The parameters are dynamically changed, and the relationship between the parameters and the pre-tightening force constructed by steps 21-25 is a general formula.

[0150] The initial stiffness minus the residual stiffness is equal to the total stiffness of the joint surface of the connecting structure, as shown below Figure 4 , the content in the article is correct

[0151] Friction shear stress and yield force are in a mapping relationship, and the relationship is:

[0152]

[0153] In the formula, according to the relationship between the mapping parameter and the maximum pressure in the literature

Study on Tangential Stiffness Nonlinear Softening of Bolted Joint in Friction-Sliding Process

[0154] (44)

[0155] In the formula, is the friction coefficient, which can be set by finite elements, and the value is 0.6 this time.

[0156] . is the macroscopic displacement point, which is identified by the stiffness degradation curve of the finite element. is the area of the maximum contact boundary, which is identified by the finite element. is the maximum value of the long semi-axis of the elliptical contact boundary, which is identified by the parameter of the finite element.

[0157] The yield displacement density function of the Iwan model under relaxation conditions is:

[0158] (46)

[0159] In the formula, the yield displacement density function is the same as the density function under the condition of constant pre-tightening force. MATLAB software is used to integrate the function to obtain the backbone curve of the micro-slip nonlinear stage. Six parameters (initial stiffness , slope of pressure peak , slope of maximum radius , initial pressure value , pre-tightening force correction factor , and initial linear pressure distribution function slope ) change with the tangential force and the pre-tightening force relaxation. The pre-tightening force also presents a nonlinear relationship with the tangential displacement (the tangential displacement is the independent variable in the Iwan model, and the restoring force (tangential force) is the dependent variable). By discretizing the cycle period, the relationship between displacement and discretization period is constructed. The backbone curve and hysteresis loop in each displacement period can be calculated to obtain the Iwan model under relaxation conditions in multiple cycle loading.

[0160] The existing Iwan model backbone curve expression of the linear spring is:

[0161] (49)

[0162] This formula is only applicable to the micro-slip nonlinear stage, and the Coulomb friction theory cannot be applied to the stick stage, which presents a linear change proportional to the initial stiffness. The macro-slip stage also presents a linear change proportional to the residual stiffness, as shown in Figure 4 . Therefore, the backbone curve solving expressions of the stick stage and the macro-slip stage are respectively:

[0163] (50)

[0164] (51)

[0165] wherein, , are micro-slip and macro-slip displacement points, is the amplitude of the load, , are initial stiffness and residual stiffness, respectively.

[0166] The hysteresis loop expression of the loading stage and unloading stage is:

[0167] (52)

[0168] According to the Masing criterion, the expression of the model hysteresis loop can be derived from the backbone curve, and formula (52) cannot be directly applied to the existing model, so the criterion is obtained by mathematical substitution, and the hysteresis loop of the loading and unloading stages is:

[0169] (53)

[0170] The density distribution function of the yield displacement is brought into the formula to obtain the backbone curve and the hysteresis loop.

[0171] The relaxation condition can better reflect the true backbone curve and hysteresis loop law. Under the action of the tangential cyclic displacement, the curve gradually converges inwards, and the energy dissipation gradually decreases. The hysteresis loop change law obtained by the experimental method. The present application is more in line with the actual working condition, and has the advantages of high precision and high applicability.

[0172] The relaxation of the pre-tightening force will cause the hysteresis loop of the model to gradually converge inwards. The smaller the tangential external load, the lower the degree of bolt loosening, and the smaller the influence on the hysteresis loop. At this time, after several cycles of tangential vibration, the hysteresis loop reaches a stable state. It shows that the bolt pre-tightening force remains basically unchanged. In the case of large tangential external load, it is easy to cause the bolt to loosen completely. When the hysteresis loop converges to a certain extent, it shows that the bolt has loosened, which may affect the safety of the structure. The method of the present application has general significance for mechanical engineering structures, and can predict nonlinear behaviors such as energy dissipation and stiffness degradation. Especially, it provides a more accurate method for predicting various indicators under the condition of bolt pre-tightening force relaxation.

Claims

1. A method for predicting the working conditions of bolted connections based on characterizing preload relaxation, characterized in that: Includes the following steps: Step 1: Characterizing the preload relaxation process using a double exponential function: The formula for the double exponential function is: (31) In the formula, F0 + S1 + S2 = the initial bolt preload value, F0 represents the first fitted number, S1 represents the second fitted number, and S2 represents the third fitted number. 1 represents the fourth fitted number, 2 represents the fifth fitted number. The number of loops. For preload; Step 2: Predicting bolt conditions using the coupled Iwan model. Step 21 Initial Stiffness The representation ； These are the parameters to be identified; Step 22, the slope of the peak pressure The slope of the maximum radius Steps to fit a function to a linear function The formulas for the slope of the peak pressure and the slope of the maximum radius are: (32) (33) q1, q2, q 11 q 22 q 33 All of these are parameters to be identified. Step 23, Preload Correction Factor Determined steps Preload Correction Factor The formula is: (35) The calculated preload value is the discretized value between the plates; Step 24, Initial pressure value The slope of the initial linear pressure distribution function Steps for determining the relationship with preload The expressions for the initial pressure value and the slope of the initial linear pressure distribution function are: (36) In the formula, , These are the initial pressure values. and preload function F y The intercept and slope, r max This is the radius value when the pressure decreases to 0; Step 25: Determining the relationship between the longitudinal and lateral multiplication factors and the slope. The longitudinal and transverse proportional multiplication factors of the linear pressure distribution function are respectively: (37) (38) This represents the magnitude of the tangential tensile force; the multiplication factor contains two independent variables, tangential force and tensile force. Preload relaxation function ; This is the maximum pressure. Step 26: Determining the relationship between preload relaxation and displacement; Construct an expression for the preload relaxation as a function of the cycle period; discretize the first cycle and divide it into arithmetic progressions. Share; when At that time, the relationship between preload relaxation and displacement is: (40) Step 27, the steps for coupling the Iwan model to establish and predict: Let , , , , Replace complex parameters: (41) (42) (43) (44) (45) The yield displacement density function of the Iwan model under relaxation conditions is: (46) The coefficient of friction, For mapping parameters, The maximum value of the semi-major axis of the elliptical contact boundary. The area of ​​the maximum contact boundary. This is the radius value when the pressure is at its maximum. The length of the minor semi-axis, The overall stiffness of the joint surface of the connecting structure.

2. The method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation according to claim 1, characterized in that: Step 1 uses a bolt preload relaxation test under small load conditions for verification.

3. The method for predicting the working condition of bolted connections based on characterizing preload relaxation according to claim 1, characterized in that: In step 27, , This represents the residual stiffness.

4. The method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation according to claim 1, characterized in that: It also includes the calculation formulas for the minor and major semi-axles: (15) (16) (17) In the formula, , , , , All are parameters to be identified. This is the maximum value of the major semi-axis. Defined for the direction of tangential tension. Vertical direction For the short half-axis, It is the long half-axis.

5. The method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation according to claim 4, characterized in that: Mapping parameters The expression is: (44)。 6. The method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation according to claim 1, characterized in that: Step 26 also includes the analysis of tangential displacement versus time; the analysis of tangential displacement versus time is as follows: In the ABAQUS finite element software, the time for each analysis step is 1 second, therefore the tangential displacement and time exhibit a mapping relationship distribution function as follows: (39) For tangential displacement, For time.

7. A method for characterizing preload relaxation using a double exponential function, characterized in that: The formula for the double exponential function is: (31) In the formula, F0 + S1 + S2 = the initial bolt preload value, F0 represents the first fitted number, S1 represents the second fitted number, and S2 represents the third fitted number. 1 represents the fourth fitted number, 2 represents the fifth fitted number. The number of loops. For preload.

8. The method for characterizing preload relaxation using a double exponential function according to claim 7, characterized in that: The verification was conducted using a bolt preload relaxation test under low load conditions.

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