Method for predicting working condition of bolt connection structure based on representation of pretightening force relaxation

Abstract

The invention relates to the technical field of bolt connection, in particular to a method for predicting the working condition of a bolt connection structure based on representation of pretightening force relaxation. The invention provides a double-exponential attenuation function, parameter identification of different functions is carried out, the double-exponential function which is simple in structure and good in characterization is obtained, the fitting precision of the double-exponential attenuation function reaches 99.3%, and the bolt pretightening force relaxation rule can be accurately characterized. According to the method, a pre-tightening force relaxation rule function is introduced into an Iwan model to represent the Iwan model under the relaxation condition meeting the actual condition, the relation between various parameters and pre-tightening is analyzed, a mathematical expression is constructed by adopting a least square method, the pre-tightening relaxation rule in a period is equivalent to the pre-tightening relaxation rule of tangential displacement and loading time, and the pre-tightening force relaxation rule is obtained. And constructing an Iwan model density function under a relaxation condition based on various variable parameters. According to the method, nonlinear behaviors such as energy dissipation and rigidity degradation can be predicted, and particularly, a relatively accurate method is provided for predicting various indexes under the working condition of bolt pretightening force relaxation.

Description

Technical Field

[0001] The present invention relates to the technical field of bolt connection, and in particular to a method for predicting the working condition of a bolt connection structure based on characterization of preload relaxation. Background Art

[0002] In the prior art, preload relaxation experiments with large external loads have been carried out, such as the literature 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. The existing experimental equipment uses universal testing machines, annular force sensors, and other experimental equipment. The preload relaxation under different tangential external load conditions and different initial torque conditions was nonlinearly characterized, and the Allometric model function and the nine-stage polynomial function of the preload relaxation changing with the number of cycles were obtained, with fitting effects reaching 71.7% and 90.4% respectively. Both existing functions have certain shortcomings. The first function has a low fitting accuracy; the second function has 10 unknown parameters to be identified, resulting in a poor function characterization effect. Meanwhile, the Iwan model (see the article IWAN W D. A distributed-element model for hysteresis and its steady-state dynamic response [J]. 1966.) consists of n ideal elastic-plastic Jenkins elements, a single sliding damper and a stiffness of k t The linear spring of / n forms an elastic-plastic Jenkins unit, whose maximum yield force is f i * In the existing Iwan model, the preload force is assumed to be a constant value. However, experimental studies have shown that the preload force relaxation shows a certain pattern. Therefore, it is necessary to introduce the preload force relaxation function into the Iwan model to make the prediction value more accurate. Summary of the Invention

[0003] To address the shortcomings of existing technologies, the present invention provides a method for predicting the operating conditions of bolted joints based on characterizing preload relaxation. This method proposes a bi-exponential decay function and performs parameter identification of different functions, resulting in a simple and well-characterized bi-exponential function. The bi-exponential function achieves a fitting accuracy of 99.3% with only five unknown parameters, enabling precise characterization of bolt preload relaxation.

[0004] The technical solution of the present invention is: a method for predicting the working condition of a bolt connection structure based on characterizing preload relaxation, characterized by comprising the following steps:

[0005] Step 1: Use a double exponential function to characterize the relaxation of preload force:

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

[0007]

[0008] Where F0+S1+S2=initial bolt preload value, F0 represents the first fitting number, S1 represents the second fitting number, S2 represents the third fitting number, l1 represents the fourth fitting number, l2 represents the fifth fitting number is the number of cycles, F y is the preload force;

[0009] Step 2: Predict the bolt working condition by coupling the Iwan model.

[0010] Step 21 Initial stiffness K T Characterization

[0011]

[0012] p1, p2, and p3 are the parameters to be identified;

[0013] Step 22: fitting the slope k1 of the peak pressure and the slope k2 of the maximum radius into a linear function

[0014] The slope formulas of the peak pressure and the maximum radius are:

[0015] k1(F y )=q1+q2F y (32)

[0016]

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

[0018] Step 23: Determine the preload correction factor ω

[0019] The formula for the preload correction factor ω is:

[0020]

[0021] N is the theoretical preload value;

[0022] Step 24: Determine the relationship between the initial pressure value e1, the initial linear pressure distribution function slope e2 and the preload force

[0023] The expressions for the initial pressure value and the slope of the initial linear pressure distribution function are:

[0024]

[0025] Where z1 and z2 are the initial pressure value e1 and the preload force function F, respectively. y The intercept and slope of

[0026] Step 25: Determining the relationship between the vertical and horizontal multiplication coefficients and the slope

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

[0028]

[0029]

[0030] T represents the magnitude of the tangential tension;

[0031] Step 26: Determine the relationship between preload relaxation and displacement

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

[0033] x=0.1×t (39)

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

[0035] Step 27: Steps for establishing and predicting the coupled Iwan model

[0036] Let A, B, C, D, and E replace complex parameters:

[0037]

[0038] Then the yield displacement density function of the Iwan model under relaxation conditions is:

[0039] ρ(φ)=A(B-Cφ)(D-3Eφ) (46)

[0040] μ is the friction coefficient, τ is the friction shear stress, λ is the mapping parameter, p max is the maximum pressure, b ymax is the maximum value of the major semi-axis of the ellipse contact boundary.

[0041] According to the method for predicting the working condition of a bolt connection structure based on characterizing preload relaxation as described above, it is characterized in that: in step 1, a bolt preload relaxation experiment under a small load condition is used for verification.

[0042] According to the method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation, the characteristic is that in step 26, when n=10, the relationship between preload relaxation and displacement is:

[0043]

[0044] According to the method for predicting the working condition of a bolt connection structure based on characterizing the relaxation of preload force, it is characterized in that: in step 27, k t =K T (F y )-k ∞ (F y ).

[0045] According to the method for predicting the working condition of bolt connection structure based on characterizing preload relaxation, it is characterized by: the friction shear stress τ and the yield force f * There is a mapping relationship, and the relationship is:

[0046] τ=λf * (43).

[0047] According to the method for predicting the working condition of a bolted connection structure based on characterizing the relaxation of the preload force, the method is characterized in that the expression of the mapping parameter λ is:

[0048] BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 Improvements to the preload relaxation function.

[0050] Figure 2 Preload relaxation experiments: (a) Bolt loosening experimental setup, (b) relaxation at different initial preloads under a 0.1 mm tangential external load, and (c) relaxation at different displacements under an initial preload of 11,567 N.

[0051] Figure 3 The relationship between multiple parameters: (a) Flowchart of the relationship between the four, (b) initial stiffness curve, (c) coefficient change,

[0052] (d) Initial pressure change, (e) preload discretization analysis, and (f) displacement change.

[0053] Figure 4 Identification of backbone curve parameters.

[0054] Figure 5Variation patterns of the minor and major semi-axes of the contact boundary.

[0055] Figure 6 Two pressure correction methods.

[0056] Figure 7 Discretized pressure distribution.

[0057] Figure 8 Diagram of tangential force change process. DETAILED DESCRIPTION

[0058] The technical solution of the present invention is further described below with reference to the accompanying drawings.

[0059] A method for predicting the working condition of a bolt connection structure based on characterizing preload relaxation of the present invention comprises the following steps:

[0060] Step 1: Use a double exponential function to characterize the relaxation of preload force:

[0061] The original data (the original data are preload relaxation data under different cyclic displacements and different initial torques in the published paper: Prediction of Pre-Loading Relaxation of Bolt Structure of Complex Equipment under Tangential CyclicLoad. The inventor first discovered that the Allometricl model function and the nine-stage polynomial function can characterize the preload relaxation of the bolt. However, in subsequent calculations, it was found that both functions are not suitable for the Iwan model. Therefore, this patent proposes another function: a double exponential decay function) were used to identify the parameters of different functions (single exponential decay function, double exponential decay function, Gaussian function, and linear rational function) (parameter identification is performed in the Origin software, and the data is fitted with one click to obtain the specific parameter values of the fitting formula), and a double exponential function with a simple structure and good representation was obtained. The fitting accuracy of the double exponential function reaches 99.3%, such as Figure 1 As shown. Figure 1 It can be seen that under the same initial preload, from 0.5mm to 2mm tangential displacement load, the bolt preload changes as the number of cycles increases. The double exponential function in this step has only five unknown parameters and can accurately characterize the bolt preload relaxation law. The formula of the double exponential function is:

[0062]

[0063] Where, F0+S1+S2=initial bolt preload value, F0 represents the first fitting number, S1 represents the second fitting number, S2 represents the third fitting number, l1 represents the fourth fitting number, and l2 represents the fifth fitting number (the values of the five parameters F0, S1, S2, l1, and l2 are obtained by fitting the preload relaxation data obtained through experiments using the least squares method and performing nonlinear data fitting). is the number of cycles, F y For preload.

[0064] In order to verify the claim that the preload relaxation is large in the first stage and the first loading, the present invention adopts the bolt preload relaxation test under small load conditions. The 50KN electro-hydraulic fatigue testing machine, annular force sensor, oscilloscope and other equipment are used for control and measurement. Figure 2 As shown in (a). Nickel steel has the advantages of corrosion resistance, weather resistance and seismic resistance to enhance structural safety. It is often used in marine engineering and shipbuilding, construction and bridges, etc., as special structural supports and bridge components in corrosive environments. Therefore, the flat plate is made of nickel steel. The complex equipment is simplified to the interaction between two flat plates. The complexity of the 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 is only assembled and unassembled once. Before the experiment begins, align the screw holes of the two nickel steel plates and place them on the experimental machine. Tighten the bolts with a torque wrench and place the force sensor between the nut and the plate. The experiment can be carried out through the computer control system.

[0065] Several studies on the relaxation law of the first stage preload under different working conditions have been carried out, such as Figure 2 As shown in (bc). When the tangential displacement is 0.1mm and the load is loaded once, the preload relaxes by 3.12%, 1.84%, 4.30% and 4.21% respectively. At the 10th cycle, the preload relaxes by 0.07%, 0.05%, 0.07% and 0.02% respectively. When the displacement is 0.1mm for the first time, the bolt preload relaxes the fastest. When the displacement reaches 10 cycles, the preload remains stable. When the displacement increases, the preload relaxes faster at the first time. When the displacement is 0.25mm, the preload relaxes by 17.12% and 20.51% at the first and tenth time respectively. Based on formula (31), a double exponential decay function is used to describe the first stage preload relaxation law under small displacement load. The average fitting accuracy of the fitting function reaches 99.7%, which can well characterize the law of preload relaxation in multiple loading cycles. The applicability of the mathematical function under small load conditions is proved.

[0066] Therefore, under small load displacement, the preload reaches its maximum relaxation in the first stage, and the preload relaxation is small, but the relaxation during the initial loading cannot be ignored. In the second stage, the bolt basically does not loosen. Large load displacement will cause the bolt to continue loosening throughout the entire vibration cycle.

[0067] Step 2: Predict the bolt working condition by coupling the Iwan model.

[0068] Under tangential cyclic loading, the bolt preload gradually relaxes. How to introduce the function of preload-cycle number into the density function of yield displacement in the modified Iwan model has become a difficult problem in the research.

[0069] Preload relaxation will lead to the parameters of the Iwan model: initial stiffness K T , the slope k1 of the pressure peak, the slope k2 of the maximum radius, the initial pressure value e1, the preload correction factor ω, and the slope e2 of the initial linear pressure distribution function change nonlinearly.

[0070] like Figure 8 As shown in the figure, when the preload is relaxed and there is no tangential load, the contact area is circular. Under the action of tangential force, the contact area between the plates can be approximated as an ellipse. The direction of the tangential tension is defined as x and the vertical direction as y. Therefore, the contact boundary shape is approximated as an ellipse, and the formula is:

[0071]

[0072] With the increase of tangential displacement load, the contact ellipse boundary gradually changes, and the minor semi-axis a x Shortened, semi-major axis b y Increase and exceed the maximum width of the plate. According to the finite element model, x 、b y Evolution values, such as Figure 5 shown.

[0073] The value of the minor axis decreases with the increase of the tangential load, while the value of the major axis increases with the increase of the tangential load. The minor axis function conforms to the exponential change, while the major axis function conforms to the linear change. The general formula is:

[0074]

[0075] In the formula, a1, a2, a3, b1, b2 are all parameters to be identified, b ymax is the maximum value of the semi-major axis.

[0076] a x is the length of the short semi-axis in the elliptical contact boundary of the bonding surface. As shown in Formula 16, with the increase or decrease of the tangential force, the short semi-axis a x Gradually decreases, the major axis b y The pressure value near the screw hole is the largest, with a radius of 4.5 mm. 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 radius reaches its maximum value of r max .

[0077] It is only related to the initial working condition and does not consider the influence of the two. The mathematical relationship expression between preload and various parameters can be constructed. The preload relaxation law can be characterized as the relationship between preload and tangential displacement. The expression relationship between various parameters and tangential displacement can be obtained. Introducing various parameters into the density distribution function of yield displacement, the Iwan model under relaxation conditions is derived. The relationship between the four is as follows Figure 3 As shown in (a).

[0078] Step 21 Initial stiffness K T Characterization

[0079] The fitting curve is constructed by exponential function, such as Figure 3 (b) As shown. The initial stiffness data under different preloads were obtained by finite element analysis. Based on the least squares method, the exponential function was used to fit the data, and the initial stiffness fitting coefficient was 0.97569, which is a good fit. This shows that under different preload conditions, this function can predict the value of the initial stiffness. The initial stiffness fitting coefficient is 0.97569, which is a good fit. When the preload relaxation is small, the initial stiffness changes little, and the influence of relaxation can be ignored. When the preload relaxation is large, the initial stiffness K T The error range is large. Existing literature shows that the stiffness degradation phenomenon will occur on the joint surface of the bolted connection structure under the influence of external load, gradually decaying from a larger stiffness to a smaller stiffness, and the stiffness is continuous and will not decay to 0, but there is a smaller stiffness. Residual stiffness k ∞ The residual stiffness is significantly affected by the preload. During single-cycle relaxation, the residual stiffness varies significantly. Parameter identification is required to obtain a more accurate residual stiffness. A linear relationship with the preload cannot be determined, so a case-by-case analysis is required. Therefore, an exponential function is used to characterize the relationship between stiffness and preload to ensure high accuracy of the Iwan model prediction. The formula is:

[0080]

[0081] k t =K T (F y )-k ∞ (F y )

[0082] The overall stiffness of the joint surface of the connection structure is equal to the initial stiffness minus the residual stiffness, as shown below Figure 4 shown.

[0083] The initial stiffness data under different preloads were obtained by finite element method. Based on the least squares method, the exponential function was used to fit the data, and the initial stiffness fitting coefficient was 0.97569, which is a good fitting degree. This shows that under different preload conditions, this function can predict the value of the initial stiffness. P1, P2, and P3 are parameters to be identified and need to be identified through specific data. The results of the model identification established in this paper are Figure 3 The results in (b).

[0084] Residual stiffness k ∞ Identification was performed using finite element method.

[0085] Step 22: fitting the slope k1 of the peak pressure and the slope k2 of the maximum radius into a linear function

[0086] Based on the finite element results under different preload conditions, the parameters k1 and k2 are fitted as linear functions and Expdec1 functions, respectively, as follows: Figure 3 (c) shows that the k1 value remains basically unchanged, while k2 shows a nonlinear decrease, with fitting accuracies of 0.90345 and 0.99172 respectively. The fitting curve has a good representation effect and can be used to represent the preload force F y The value when it changes is as follows:

[0087] k1(F y )=q1+q2F y (32)

[0088]

[0089] q1, q2, q 11 ,q 22 ,q 33 These are all parameters to be identified. The data under different preload forces can be obtained through finite element analysis or experiments. Then, the parameters can be identified using formulas (32) and (33) to obtain accurate values.

[0090] Step 23: Determine the preload correction factor ω

[0091] The preload correction factor ω changes with preload relaxation, so the influence of preload must be considered. At the same time, the tangential force also affects the correction parameter, so the influence of both must be considered comprehensively.

[0092]

[0093] N is the theoretical preload value; N i is the calculated preload value after discretization between the plates. The following is a detailed explanation of the discretized preload value.

[0094] When T=0, the maximum radius of the pressure distribution function is 12.33607mm. However, after the finite element identification of the contact boundary function, it is known that the maximum pressure radius is 11.19226. Therefore, the maximum radius when the pressure is zero needs to be corrected. Commonly used corrections are: constraint method and offset method. The constraint method constrains the pressure distribution function at T=0 so that the zero pressure value is located at the maximum contact boundary value a1+a2-r min The offset method offsets the linear function downward so that the pressure result is consistent with the contact boundary function, such as Figure 6 shown.

[0095]

[0096] When T=0, the linear pressure distribution function formulas are expressed as:

[0097]

[0098] p2(r)=e2(a1+a2-r min )-e2r (29)

[0099] Therefore, by substituting formulas (28) and (29) into formula (27), the dynamic pressure distribution function under different tangential forces can be obtained as:

[0100]

[0101] Based on the finite element ABAQUS simulation results, it was concluded that the initial contact state was circular, and then under the action of static friction, the contact state gradually evolved into an ellipse. Under the interaction of static friction and dynamic friction, as the tangential load increases, the radius (semi-short axis) reaching zero pressure gradually decreases, the semi-long axis continues to increase, and finally cuts off at the maximum half-width of the plate. During this process, the slope of the ellipse gradually decreases, the pressure around the screw hole is the maximum value, and the maximum contact boundary pressure is zero. The linear pressure distribution is discretized and the contact area is divided into n discrete ellipses. When n→∞, the discretized area can be approximated as the area of the entire area. The error between the preload force after pressure correction and the theoretical preload force is verified by the discrete contact area and the corrected linear pressure distribution, and the accuracy of the two correction methods can also be verified. Figure 7 shown.

[0102] Assuming that the length of the minor axis of the ellipse increases equidistantly, and the major axis changes with the curvature and the minor axis, the ellipticity increases gradually from 0 to n, that is, from 1 to b ymax / a x Components are added. Therefore, the length of the minor axis, the length of the major axis, and the ellipticity of the i-th ellipse are:

[0103]

[0104] a x is the length of the semi-minor axis.

[0105] Therefore, the area and discrete preload of the i-th ellipse are:

[0106]

[0107] The total calculated preload force is the sum of n discrete preload force values. After calculation, when n=1000, the discrete degree remains stable.

[0108] The preload correction parameter ω can be expressed as a dynamic equation that changes with the tangential external force:

[0109]

[0110] Where N is the theoretical preload value, which can be obtained through finite element software.

[0111] After getting this formula, we can calculate ω and preload F. y The relationship is characterized.

[0112] Step 24: Determine the relationship between the initial pressure value e1, the initial linear pressure distribution function slope e2 and the preload force

[0113] The relationship between the initial pressure value and the preload force is as follows: Figure 3 (d) shows that the initial pressure value is linearly related to the preload, with a fitting correlation coefficient of 0.99967, indicating a good characterization effect. Therefore, the initial pressure value can be predicted by the preload value. Ideally, the greater the preload, the better the bolt tightening effect, which will lead to an increase in the pressure value of the joint surface between the plates. The formula is:

[0114] p max (F y )=e1(F y )=z1+z2F y (34)

[0115] The slope e2 of the linear pressure distribution function varies with the initial pressure value and the maximum contact radius. When the preload is relaxed, the maximum contact radius remains essentially unchanged and is considered a constant. This is only relevant to the initial operating conditions. Therefore, the slope e2 has a nonlinear relationship with the preload:

[0116]

[0117] Where z1 and z2 are the intercept and slope of the initial pressure value e1 and the preload force function Fy, respectively. After obtaining the data of the two, the specific values of z1 and z2 can be obtained by linear fitting, r min is the radius when the pressure is maximum, r maxThe radius is the value when the pressure decreases to 0. The origin of the radius is the center of the screw hole. The pressure value only exists at the edge of the screw hole. Along the positive direction of the semi-minor axis, the radius value gradually increases and the pressure value gradually decreases.

[0118] Step 25, determining the relationship between the longitudinal and transverse multiplication coefficients and the slope;

[0119] The longitudinal and transverse geometric multiplication coefficients of the linear pressure distribution function are:

[0120]

[0121] T represents the magnitude of the tangential tension.

[0122] Step 26: Determine the relationship between preload relaxation and displacement

[0123] The first stage and the first loading cycle where the preload relaxation is fast are studied. The expression of the preload relaxation with the cycle period is constructed. The first cycle is discretized and divided into n equal parts, such as Figure 3 (e) Each discrete interval corresponds to a different preload force. Using the preload relaxation function, a unique preload force is obtained for each discrete interval.

[0124] The relaxation of the preload force in the first cycle is characterized by ABAQUS software. The relaxation is truly characterized based on the finite element software. In the finite element calculation, the calculation time of one cycle is considered to be 1s. The n equal-difference cycles in the first cycle can correspond to the value of time. Therefore, there is a certain mapping relationship between the two. In the finite element calculation, time and tangential displacement show a mapping relationship, such as Figure 3 (f). The distribution function is:

[0125] x=0.1×t (39)

[0126] x is the tangential displacement and t is the time.

[0127] When n=10, the first cycle’s discreteness corresponds to time, and the relationship between preload relaxation and displacement is:

[0128]

[0129] At this point, it can be understood that when the connection structure moves to a certain displacement, a corresponding preload value (not the initial value, but the relaxation value) is generated. This preload value is dynamically correlated with the six parameters, thus forming a closed loop.

[0130] Tangential loading is cyclical, not just a single stretch. As mentioned in the previous experiments, this patent primarily considers the first stretching cycle and the first 10-15 stretching cycles. This is because, under small load displacements, preload relaxation only significantly affects the Iwan model during the first cycle (before 15 tangential cycles); subsequent cycles have less of an impact. (Because the area enclosed by the hysteresis line represents energy dissipation, the fact that the area enclosed by it remains essentially unchanged indicates that the structure dissipates little energy, thus minimizing the impact on the Iwan model.)

[0131] Steps 21-26 yielded a nonlinear relationship between preload and various parameters in the Iwan model. However, even with this relationship, it was not possible to incorporate preload into the Iwan model. Therefore, the next steps involved investigating how to incorporate preload into the model.

[0132] Step 27: Steps for establishing and predicting the coupled Iwan model

[0133] 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.

[0134] Under the condition of preload relaxation, the parameters of the Iwan model change with the preload. This paper characterizes the relationship between multiple parameters-preload and preload-tangential displacement through analytical expressions, and obtains the backbone curve of the Iwan model under relaxation conditions. T , the slope of the peak pressure k1, the slope of the maximum radius k2, the initial pressure value e1, the preload correction factor ω, and the slope of the initial linear pressure distribution function e2) can be used to derive the density function of the yield displacement of the Iwan model under relaxation conditions. To simplify the yield displacement density function equation, let A, B, C, D, and E replace the complex parameters:

[0135]

[0136] In the formula, the independent variables are the tangential force T and the preload relaxation function F y , F y Therefore, the number of independent and dependent variables in the density function of yield displacement does not change, and it is still the tangential displacement x and tangential force T.

[0137] Six parameters (initial stiffness K TThe following parameters (k1, k2, k ...

[0138] k t =K T (F y )-k ∞ (F y ) The initial stiffness minus the residual stiffness equals the overall stiffness of the joint surface of the connection structure, as shown below Figure 4 As shown, the content in the article is correct: friction shear stress τ and yield force f * There is a mapping relationship, and the relationship is:

[0139] τ=λf * (43)

[0140] Where, according to the literature [Study on Tangential Stiffness Nonlinear Softening of Bolted Joint in Friction-Sliding Process], the mapping parameter λ and the maximum pressure p max The relationship between the tangential force T and the preload force F can be derived. y The dynamic relationship is:

[0141]

[0142] Where μ is the friction coefficient, which can be set by finite element method and is taken as 0.6 in this case.

[0143] k t =K T (F y )-k ∞ (F y ). φ2 is the macro sliding displacement point, which is identified by the stiffness degradation curve of the finite element. max is the area of the maximum contact boundary, identified by finite element method. b ymax is the maximum value of the major semi-axis of the elliptical contact boundary, and the parameters are identified by finite element method.

[0144] Then the yield displacement density function of the Iwan model under relaxation conditions is:

[0145] ρ(φ)=A(B-Cφ)(D-3Eφ) (46)

[0146] In the formula, the yield displacement density function ρ(φ) has the same form as the density function under the condition of constant preload. The function is integrated using MATLAB software. The backbone curve of the micro-slip nonlinear stage is obtained. The six parameters (initial stiffness K T (F y ), the slope of the peak pressure k1(F y ), the slope of the maximum radius k2(F y )、initial pressure value e1(F y ), preload correction factor ω(F y ), the slope of the initial linear pressure distribution function e2(F y )) varies with the tangential force T and preload relaxation. Preload, in turn, exhibits a nonlinear relationship with tangential displacement (tangential displacement is the independent variable in the Iwan model, while the restoring force (tangential force) is the dependent variable). By discretizing the cyclic period, a relationship between displacement and discretization period is constructed. The backbone curve and hysteresis loop within each displacement period can be calculated, yielding the Iwan model under relaxation conditions during multiple cyclic loading.

[0147] The backbone curve expression of the existing Iwan model with the introduction of linear springs is:

[0148]

[0149] This formula is only applicable to the micro-slip nonlinear stage. Coulomb friction theory cannot be applied to the sticky stage, which shows a linear change and is proportional to the initial stiffness. The macro-slip stage also shows a linear change and is proportional to the residual stiffness, such as Figure 4 Therefore, the backbone curve solutions for the sticking stage and the macro-slip stage are:

[0150] T(x)=K T (F y )x 0≤x≤φ1 (50)

[0151] T(x)=T2+k ∞ (F y )(x-φ2) φ2≤x≤A (51)

[0152] Where φ1 and φ2 are the micro-slip and macro-slip displacement points, A is the loading amplitude, and K is the T 、k ∞ are the initial stiffness and residual stiffness, respectively.

[0153] The hysteresis loop expressions of the loading and unloading stages are:

[0154]

[0155] According to the Masing criterion, the expression of the model hysteresis loop can be derived from the backbone curve. However, formula (52) cannot be directly applied to the existing model. Therefore, the criterion is mathematically replaced to obtain the hysteresis loops of the loading and unloading stages:

[0156]

[0157] Substituting the density distribution function of yield displacement into the formula, the backbone curve and hysteresis loop can be obtained.

[0158] Relaxation conditions better reflect the true backbone curve and hysteresis loop patterns. Under the influence of tangential cyclic displacement, the hysteresis loop gradually converges, and energy dissipation gradually decreases. The present invention experimentally obtains the hysteresis loop variation patterns. This method better reflects actual operating conditions and offers the advantages of high precision and applicability.

[0159] Preload relaxation will cause the hysteresis loop of the model to gradually retract. The smaller the tangential external load, the lower the degree of bolt loosening and the smaller the impact on the hysteresis loop. At this time, after several cycles of tangential vibration, the hysteresis loop reaches a stable state. This shows that the bolt preload remains basically unchanged. When the tangential external load is large, it is easy to cause the bolt to completely loosen. When the hysteresis loop is retracted to a certain extent, it indicates that the bolt has loosened, which may affect the safety of the structure. The method of the present invention has general significance for mechanical engineering structures, and can predict nonlinear behaviors such as energy dissipation and stiffness degradation. In particular, it provides a more accurate method for predicting various indicators under the condition of bolt preload relaxation.

Claims

1. A method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation, characterized by: The following steps are involved: Step 1: Use a double exponential function to characterize the relaxation of preload force: The formula for the double exponential function is: Where F0+S1+S2=initial bolt preload value, F0 represents the first fitting number, S1 represents the second fitting number, S2 represents the third fitting number, l1 represents the fourth fitting number, l2 represents the fifth fitting number is the number of cycles, F y is the preload force; Step 2: Predict the bolt working condition by coupling the Iwan model. Step 21 Initial stiffness K T Characterization p1, p2, and p3 are the parameters to be identified; Step 22: fitting the slope k1 of the peak pressure and the slope k2 of the maximum radius into a linear function The slope formulas of the peak pressure and the maximum radius are: k1(F y )=q1+q2F y (32) q1, q2, q 11 ,q 22 ,q 33 are parameters to be identified. Step 23: Determine the preload correction factor ω The formula for the preload correction factor ω is: N i is the calculated preload value after discretization between the plates; Step 24: Determine the relationship between the initial pressure value e1, the initial linear pressure distribution function slope e2 and the preload force The expressions for the initial pressure value and the slope of the initial linear pressure distribution function are: Where z1 and z2 are the initial pressure value e1 and the preload force function F, respectively. y The intercept and slope of Step 25: Determining the relationship between the vertical and horizontal multiplication coefficients and the slope The longitudinal and transverse geometric multiplication coefficients of the linear pressure distribution function are: T represents the magnitude of the tangential tension; there are two independent variables in the multiplication coefficient, the tangential force T and the preload relaxation function F y ; Step 26, determining the relationship between preload relaxation and displacement; Step 27: The step of establishing and predicting the coupled Iwan model.

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: The process of step 27 is: let A, B, C, D, and E replace the complex parameters: Then the yield displacement density function of the Iwan model under relaxation conditions is: ρ(φ)=A(B-Cφ)(D-3Eφ) (46) μ is the friction coefficient, τ is the friction shear stress, λ is the mapping parameter, p max is the maximum pressure, b ymax is the maximum value of the major semi-axis of the ellipse contact boundary.

3. The method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation according to claim 1, characterized in that: In step 1, a bolt preload relaxation test under a small load condition is used for verification.

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: In step 26, when n=10, the relationship between preload relaxation and displacement is:

5. The method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation according to claim 1, characterized in that: In step 27, k t =K T (F y )-k ∞ (F y ).

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: It also includes the shortening of the minor axis and the calculation formula of the major axis: In the formula, a1, a2, a3, b1, b2 are all parameters to be identified, b ymax is the maximum value of the semi-major axis, x is defined as the direction of tangential tension, y is the vertical direction, and a x is the minor axis, b y is the semi-major axis.

7. The method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation according to claim 6, characterized in that: The mapping parameter λ is expressed as:

8. The method for predicting the working condition of a bolted connection structure based on characterizing preload relaxation according to claim 1, characterized in that: The analysis of tangential displacement and time in step 26 is as follows: In the ABAQUS finite element software, the time of each analysis step is 1 second, so the tangential displacement and time present a mapping relationship distribution function: x=0.1×t (39) x is the tangential displacement and t is the time.

9. A method for characterizing preload relaxation using a double exponential function, characterized in that: The formula for the double exponential function is: Where F0+S1+S2=initial bolt preload value, F0 represents the first fitting number, S1 represents the second fitting number, S2 represents the third fitting number, l1 represents the fourth fitting number, l2 represents the fifth fitting number is the number of cycles, F y For preload.

10. The method of characterizing preload relaxation using a double exponential function according to claim 1, characterized in that: The bolt preload relaxation test under small load conditions is used for verification.

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