Accurate Finite Element Modeling of Bolted Joints and Modified IWAN Model

Abstract

A three-dimensional finite element model of helical bolted joints is established in this paper, and the validity of the model establishment scheme is verified by a numerical examples of “the stress distribution on the bolted joint”. Then, Hysteresis curves of the resultant torque versus the applied torsion angle are obtained from the finite element analysis. The results indicate that: (1) The contact threads have experienced three states of adhesion, partial slip and macro slip during the tightening and loosening process; (2) In the state of macro sliding, the resultant torque exhibits an upward trend due to the increase of the clamping pressure in tightening process. On the contrary, the resultant torque exhibits a downward trend due to the decrease of the clamping pressure in the loosening process. A modified IWAN model is established for the three-dimensional helical bolted joint under torsional load by adding the residual stiffness and correction of torque based on the results of finite element analysis. The results of the modified IWAN model for the M12 bolted joint present that the modified IWAN model can reproduce the hysteresis curves obtained by finite element analysis accurately, and the modified IWAN model composed of three Jenkins elements can cover the dissipated energy precisely.

1 Introduction

The loosening of bolted joints is a complex nonlinear process and will be affected by many factors such as the friction coefficient between the clamped parts, the friction coefficient between the contact threads of bolted joints, the preload of bolted joints, the amplitude of external load, temperature, and the material of the structures [1]. Some research on the mechanism of loosening behavior of bolted joints has been carried out for a long time, and many valuable research results have been summarized. Nassar et al. [2] reported that when the amplitude of axial load exceeds a critical value, the clamped parts will be separated, and the bolts joints will bear all the axial loads, resulting in severe plastic deformation and eventually loosening of the bolt joints. Junker [3] found that dynamic shear load has a greater impact on the looseness of bolted joints than axial load. When bolted joints are excited by shear load, the contact threads of bolted joints will produce a relative torsional angle. As the torsional angle increases, the clamping pressure decreased when the slippage between the contact surfaces occurred. Jiang et al. [4] reproduced the loosening process of bolted joints under cyclic shear load. They found the loosening process can be divided into two stages. In the first stage, there is no obvious slip between the contact threads, and the clamping pressure decreased slowly due to the plastic deformation of the bolted joints; In the second stage, as the clamping pressure decreased to a critical value, clamping pressure dropped rapidly due to the relative rotation between the contact threads. Therefore, it is of great significance to study the mechanical properties of bolted joints under torsional load. In recent years, the finite element method has been used to investigate the mechanical properties of bolted joints. However, it is difficult to generate high-quality finite element meshes because of the complex geometric shape of the helical bolted joints. To improve the finite element mesh quality, the effect of the helix angle of threads of a bolted joint is neglected in most studies. Verwaerde et al. [5] investigate the stress distribution on the bolted joints by using a three-dimensional bolted joint model without threads. Zhao et al. [6] analysed stress concentration at thread roots using an axisymmetric bolted joint model. The finite element model established by the above two methods of bolted joint modeling can not accurately simulate the mechanical properties of the bolted joints with the helix angle of threads. It is an effective method to accurately predict the stress distribution at the thread of bolted joints by using the model with helix angle to carry out finite element analysis. However, For different sizes of bolted joints, the bolted joints need to be remodeled and the efficiency is low [7, 8]. Therefore, the quest for a finite element modeling method to quickly construct the bolted joints with the helix angle is of great significance. It takes a considerable amount of time to conduct a finite element simulation for obtaining the hysteresis curve of the resultant torque versus the applied rotation angle for bolted joints. Hence, it is of great significance to develop a corresponding mathematical model with clear physical significance for the bolted joints [9]. With the development of contact mechanics and tribology, many theoretical models have been developed to predict the nonlinear mechanical behavior of bolted joints. Gaul et al. [10] examined the dynamic response of bolted joints under torsional load. Subsequently, they established a model for bolted joints by utilizing the Valanis model with several Jenkins elements arranged in parallel to one another. Liu [11] developed the modified Valanis model based on experimental and finite element analysis, and proved the effectiveness of the modified Valanis model. At present, the most representative models include Valanis model [12], Bouc-Wen model [13, 14] and IWAN model [15, 16], etc. Among them, due to the clear and intuitive physical meaning of the IWAN model, it has been widely used in nonlinear model establishment and dynamic behavior analysis of rivet connection and thread connection [17], etc. The remainder of this paper is organized as follows: In Sect. 2, the three-dimensional finite element modeling method of the bolted joints with the helix angle of threads is presented, and the parametric modeling platform based on this method is briefly introduced. In Sect. 3 the axial load on the M12 bolted joints are simulated in ABAQUS. Meanwhile, the stress distribution and initial location of plastic deformation for the M12 bolted joint are given. The dynamic responses of bolted joints under torsional load are studied in 9 different working conditions, and their hysteretic curves are obtained. In Sect. 4, the classical IWAN model is modified to obtain hysteresis curves of the resultant torque versus the applied rotation angle according to the simulation results. The correctness of the modified IWAN model is verified. Concluding remarks are provided in Sect. 5.

2 Accurate Finite Element Model of Bolted Joints with the Helix Angle of Threads

2.1 Mathematical Expressions of Thread Profile

Here, we will briefly introduce the mathematical model of the thread profile, which has been widely described in the literature [18, 19]. Figure 1a illustrates the profile of the external thread of a bolt along the central axis of the bolt. The specification of the thread profile is given in ISO 68, 261, 262, and 724. To prevent excessive stress concentration at the thread root, the thread root radius $\rho$ is required to be greater than 0.125. The symbol r is the external radial coordinate; is the circumferential coordinate; $d_{1}$ is the minor diameter of thread; d is the nominal diameter of thread, H is the thread overlap, P is the thread pitch, ABCD and A′B′C′D′ are symmetrical points about $\theta = 0$ on the thread profile. The profile of the external thread of the bolt starts at point A and can be divided into three parts from $\theta = 0$ to $\theta = \pi$, A-B (thread flank), B-C (thread tooth side) and C-D (thread crest).

Fig. 1

The profiles of helical thread of the bolt

The thread profile of these three parts is projected onto a plane perpendicular to the bolt axis. Thus, the cross-section of the external thread profile can be obtained, as shown in Fig. 1b. The mathematical expression of the external thread profile can be obtained:

$$r = \left\{ {\begin{array}{*{20}l} {\frac{d}{2} - \frac{7}{8}H + 2\rho - \sqrt {\rho^{2} - \frac{{P^{2} }}{{4\pi^{2} }}\theta^{2} } } \hfill & {(0 \le \theta \le \theta_{1} )} \hfill \\ {\frac{H}{\pi }\theta + \frac{d}{2} - \frac{7}{8}H} \hfill & {(\theta_{1} \le \theta \le \theta_{2} )} \hfill \\ \frac{d}{2} \hfill & {(\theta_{2} \le \theta \le \pi )} \hfill \\ \end{array} } \right.$$

(1)

where $\theta_{1} = \frac{\sqrt 3 \pi }{P}\rho$, $\theta_{2} = \frac{7}{8}\pi$, $\rho \le \frac{\sqrt 3 }{{12}}P$, $H = \frac{\sqrt 3 }{2}P$. In this way, the mathematical expression of the internal thread profile of the nut can be obtained:

$$r = \left\{ {\begin{array}{*{20}l} {\frac{{d_{1} }}{2}} \hfill & {{(}0 \le \theta \le \theta_{1} )} \hfill \\ {\frac{H}{\pi }\theta + \frac{d}{2} - \frac{7}{8}H} \hfill & {{(}\theta_{1} \le \theta \le \theta_{2} )} \hfill \\ {\frac{d}{2} + \frac{1}{8}H - 2\rho_{n} { + }\sqrt {\rho_{n}^{2} - \frac{{P^{2} }}{{4\pi^{2} }}(\pi - \theta )^{2} } } \hfill & {{(}\theta_{2} \le \theta \le \pi )} \hfill \\ \end{array} } \right.$$

(2)

where $\theta_{1} = \frac{\pi }{4}$, $\theta_{2} = \pi \left( {1 - \frac{{\sqrt 3 \rho_{{\text{n}}} }}{P}} \right)$, $\rho_{{\text{n}}} \le \frac{\sqrt 3 }{{24}}P$, $\rho$ and $\rho_{n}$ are the upper limits of the root radius of external thread and internal thread respectively.

2.2 Finite Element Modeling Platform for Bolts and Nuts

To construct the finite element model of bolted joints quickly with different sizes and types in practical engineering, we developed the bolts and nuts finite element modeling platform with C++ language in the Qt development framework. The main interface of the “modeling platform” is shown in Fig. 2.

Fig. 2

The main interface of the “modeling platform”

The flowchart of constructing the finite element model of bolts and nuts is described in Fig. 3. At the beginning of the program, we need to input the basic parameters of bolts and nuts, i.e., thread pitch P, number of pitches N, nominal diameter d, and the number of elements in single turn thread n_e, etc. Then, the node coordinates of elements are calculated by Eqs. (1) and (2). The obtained node coordinates data and the index of nodes are stored in the predefined VCORG matrix space. Finally, the hexahedral element is constructed by the index of each element's nodes, and the index of the elements and nodes are all stored in the predefined NELEM matrix space.

Fig. 3

Flowchart of constructing the model

3 The Finite Element Model

In this section, the finite element model for the ISO M12 bolted joint with a performance grade of 5.8 is selected for numerical analysis in commercial finite element software ABAQUS. The results are compared with the experimental results to verify the effectiveness of the finite element modeling scheme for bolted joints. The finite element model for the M12 bolted joint is shown in Fig. 4, the model consists of four components: upper plate, lower plate, bolt, and nut. These components are all meshed by hexahedral reduced integral elements, with a total of 87,245 nodes and 77,712 elements. Before finite element analysis, we define the following four contact pairs: (Cont1) between the bottom surface of the bolt head and the top surface of the upper plate, (Cont2) between the bottom surface of the upper plate and the top surface of the lower plate, (Cont3) between the bottom surface of the lower plate and the top surface of the nut, (Cont4) between the contact threads. The above four contact pairs are all defined as surface-surface constraints (Standard) in ABAQUS. The contact behavior is set as tangential and normal, where the normal behavior of the bolt and nut surface is set as an exponential soft contact constraint, and the normal behavior of the other contact surfaces is set as a hard contact constraint. The tangential behavior is constrained by the penalty formula. A torsional load is applied to the nut to rotate it upwards to clamp the two plates.

Fig. 4

Overall finite element model of bolted joints

In the model, the bolt material is 35CrMn with an elastic modulus of about 213,000 MPa and Poisson ratio of about 0.286; The nut is made of 45 Steel with an elastic modulus of about 209,000 MPa and Poisson ratio of about 0.269. The Plane material is Carbon Steel with an elastic modulus of about 210,000 MPa and Poisson ratio of about 0.280; The density of the above materials is 7.87e−9 t/mm³. The plastic strain behavior of the bolt and nut are given in Table 1. The general static analysis step is established, and the loading time is set as 1 s. The number of maximum time steps is set to 100, the initial time step is 0.01, and the influence of geometric nonlinearity is considered. The field output selects stress, strain, etc.

Table 1 Plastic strain behavior of bolt and nut

Regarding the application of preload to bolt joints, ABAQUS provides a method to simulate bolt preload by imposing BOLT LOAD over the cross-section of the bolted joints. However, this method is unable to simulate the friction torque applied to the bolt joints during the pre-tightening process. Therefore, this paper proposes the imposition of a rotation angle on the outside of the nut to simulate the pre-tightening process of the bolted joints. We get the equivalent stress distribution and maximum plastic strain distribution on the bolt, as shown in Fig. 5 and Fig. 6 respectively. It can be found from Fig. 5 that the first three turns of threads bear the largest load, and the larger value of stress is mainly concentrated at the root of the thread, which is consistent with the previous experimental results [20]. It can be seen from Fig. 6 that the plastic deformation first occurs at the root of the first turn of thread of the bolt, and the fatigue fracture of the bolt often occurs at the root of the first turn of thread in practical engineering [21].

Fig. 5

The contour of the equivalent stress

Fig. 6

The contour of the plastic strain

3.1 Dynamic Behavior of a Bolted Joint Subjected to Torsional Excitation

The influence of different friction coefficients of Cont2 and Cont4 on the dynamic response of bolted joints is being studied using three levels of friction coefficients, 0.05, 0.1, and 0.15 respectively. It is ensured that other variables remain constant. In order to investigate the influence of different preloads, denoted as $F_{P}$, on the dynamic response of a bolted joint, the preloads of 5 kN, 10 kN, and 20 kN were applied to the bolted joints, respectively. The numerical calculations were carried out for each preload value of $F_{P}$. To study the influence of different torsional load amplitudes on the dynamic response of bolted joint. A reference point, denoted as $p_{reference}$, is set directly below the nut along the bolt axis. Nodes on the outer surface of the nut are bound to Preference using distributive coupling. Torsional amplitudes of 0.002, 0.0035, and 0.005 rad are then externally imposed on the reference point $p_{reference}$, to evaluate the numerical response of the bolted joints to torsional excitation from the nut. The different cases of finite element simulation are shown in Table 2.

Table 2 Summary of finite element simulation cases

3.2 The Hysteretic Curves

The hysteretic curves of torque T versus torsion angle $\theta$ at the reference point $p_{reference}$ under the control of various factors are shown in Fig. 7. As shown in the hysteretic curve, there are three stages that could occur throughout the tightening and loosening process adhesion, partial slip, and macroscopic slip. The gradual alteration of the sliding contact area is indicated in the tightening process by the smooth transition from the adhesive stage to the partial slip stage. Sliding begins in the region of low contact pressure. The torque steadily grows as the torsional angle increases and moves to the region of high contact pressure at the mating hole. On the other hand, the change from sliding to adhesion is not as seamless when the speed direction is opposite. This can be explained as follows: Sliding gradually starts on the contact surface as the torsional Angle rises and comes to an end when the velocity direction is reversed. The friction coefficient between the clamped parts $\mu_{c}$ and the friction coefficient between the contact threads $\mu_{t}$, as shown in Fig. 7c, d, have a significant impact on the friction dissipation energy of the bolted joints. Cont2 has a more pronounced effect on the friction dissipation of the bolted joints than Cont4, which causes the friction dissipation energy of the bolted joints to change more significantly and is the primary cause of the loosening of the bolted joints collectively.

Fig. 7

The influence of different variables on hysteresis curve

4 Modified IWAN Model

According to the finite element analysis, the hysteresis curves provided by the classical IWAN model are insufficient to describe the hysteresis characteristics of bolted joints the helix angle of threads under torsional load. The classical IWAN model needs to be modified to account for these differences.

4.1 Residual Stiffness in Macro Slip Stage

The classical IWAN model is shown in Fig. 8a. It is composed of N Jenkins elements in parallel. Each Jenkins element is composed of a linear spring with stiffness $k_{i}$ and a coulomb element with yield force $R_{i}$ in series. In its adhesive stage, the Coulomb element provides resistance that is both equal to and in opposition to the external load. The Jenkins element reaches the sliding stage when its friction capacity is achieved, which occurs when the external load is larger than the friction coefficient times the normal force. The load that imposed on the sliding element is equal to the friction coefficient multiplied by the sliding element's natural resistance. When only some Jenkins elements slip during the periodic harmonic motion of the model, it is considered partial slip. If all Jenkins elements slip, the model has reached the stage of macro slip. The classical IWAN model incorporates a spring with an elastic coefficient $k_{s}$, that is coupled in parallel to take into account the residual stiffness to accurately replicate the macro slip process. In Fig. 8b presents the IWAN model with residual stiffness.

Fig. 8

The IWAN model

Figure 9a illustrates the hysteresis curve of a-b-c-d-e-f-a, which is bounded by a small area and drawn using the classical IWAN model. In the loosening part of a-b-c-d and the tightening part of d-e-f-a, the model is in a state of adhesion. Figure 9b illustrates the hysteretic curve enclosed by a′-b′-c′-d′-e′-f′-a′ drawn using the IWAN model that incorporates residual stiffness. In this instance, as the external load increases, all Jenkins elements in the model will eventually slide, resulting in macro slip phenomena. The model is in a state of adhesion at the loosening part a′-b′ and tightening part d′-e′. At this stage, all the Jenkins elements in the IWAN model do not yield, making the model equivalent to a linear spring. The middle part, b′-d′ of loosening part and e′-f′ of tightening part, are the segments where the model experiences partial slip. At this stage, some Jenkins elements in the IWAN model yield. On the other hand, in the loosening part c′-d′ and tightening part f′-a′, the model is in a state of macro slip. The slope that corresponds to e′-d′ and f′-a′ parts specifies the residual stiffness. The IWAN model's Jenkins elements yield at this point, and only the residual stress acts, resulting in macro slip of the model. The relationship between torque and torsion angle of the bolted joints can be determined after accounting for the residual stiffness $k_{s}$:

$$T_{j} = k_{s} \theta + \sum\limits_{1}^{m} {\left\{ {\begin{array}{*{20}l} {r_{i} (t)} \hfill & {{\text{abs}}(r_{i} (t)) < R_{i} } \hfill \\ {R_{i} {\text{sgn}} (\dot{\theta })} \hfill & {{\text{else}}} \hfill \\ \end{array} } \right.}$$

(3)

Fig. 9

Comparison of hysteretic curves of IWAN models with or without residual stiffness

and

$$r_{i} (t) = k_{i} (\theta - \theta_{{{\text{rev}}}} ) + R_{i} {\text{sgn}} (\dot{\theta }_{{{\text{rev}}}} )$$

(4)

where $\theta$ and $\theta_{rev}$ denote the torsion angle of the bolted joints during forward rotation and reverse rotation, respectively. $\dot{\theta }_{rev}$ denotes the angular velocity of the bolted joints before the speed reversal. m is the number of Jenkins elements in the modified IWAN model. ${\text{sgn}} (\dot{\theta }_{{{\text{rev}}}} )$ can be written as:

$${\text{sgn}} (\dot{\theta }_{{{\text{rev}}}} ) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{if}}\quad \dot{\theta }_{{{\text{rev}}}} > 0} \hfill \\ 0 \hfill & {{\text{if}}\quad \dot{\theta }_{{{\text{rev}}}} = 0} \hfill \\ { - 1} \hfill & {{\text{if}}\quad \dot{\theta }_{{{\text{rev}}}} < 0} \hfill \\ \end{array} } \right.$$

(5)

4.2 The Overall Offset of Torque

The hysteretic curve in Fig. 9 is symmetrical about the origin because the classical IWAN model is only applied to the case of the same reciprocating friction conditions. In the case of reciprocating friction between thread pairs having helix angles, the friction force would increase upon the ascent of the nut. As a result, the hysteretic curve in Fig. 7 obtained via finite element simulation is non-symmetrical about the origin. This feature causes an overall upward torque offset in the resulting curve. In an IWAN model having residual stiffness, the stress relationship meets Masin's assumption [16]. As a result, the relationship between the loosening torque of the model and the tightening torque can be expressed as follows:

$$T_{r} (\theta ) = - T_{u} ( - \theta )$$

(6)

where the footmark r represents the tightening process and u represents the loosening process.

Obviously, Eq. (6) is inconsistent with the relationship between torque and torsion angle of the bolted joint described in real practice. The value of the torque offset can be estimated by calculating the difference in torque between the tightened nut and the loosened nut. The relationship between forces acting on rectangular threads is studied. Unfold the external thread as shown in Fig. 10a along the pitch diameter $d_{2}$ of the thread to obtain an inclined plane with an angle of inclination $\varphi$ as shown in Fig. 10b. When the sliding block is subjected to axial force is pushed at a horizontal and uniform speed with force $F_{t}$ if the sliding block rises along the inclined plane, its tightening torque $T_{r}$ needs to overcome the thread friction moment $T_{1}$ of the thread pair and the end face friction moment $T_{2}$ between the nut and the supporting surface of the bolted joint at the same time. The tightening torque can be obtained:

$$T_{r} = T_{1} + T_{2} = F\tan \left( {\varphi + \rho_{v} } \right)\frac{{d_{2} }}{2} + \frac{{F\mu_{{\text{c}}} }}{3} \times \frac{{D_{w}^{3} - d_{0}^{3} }}{{D_{w}^{2} - d_{0}^{2} }}$$

(7)

Fig. 10

Force diagram of the bolted joint

where $D_{w}$ and $d_{0}$ denote the outer diameter and inner diameter (hole diameter) of the indirect contact surface between the nut and the connected piece respectively. The equivalent friction angle of the thread is:

$$\rho_{v} = \arctan \frac{f}{\cos \gamma }$$

(8)

where f denotes the friction coefficient between threads and $\gamma$ denotes the half angle of the thread profile. Considering that the rising angle of thread $\varphi$ is quite small, the tightening torque Eq. (8) is simplified as follows:

$$T_{r} = T_{1} + T_{2} = F\left[ {\frac{{d_{2} \mu_{{\text{t}}} }}{2\cos \gamma } + \frac{P}{2\pi } + \frac{{\mu_{{\text{c}}} }}{3} \times \frac{{D_{w}^{3} - d_{0}^{3} }}{{D_{w}^{2} - d_{0}^{2} }}} \right]$$

(9)

Similarly, the loosening torque can be deduced:

$$T_{u} = T_{1} + T_{2} = F\left[ {\frac{{d_{2} \mu_{{\text{t}}} }}{2\cos \gamma } - \frac{P}{2\pi } + \frac{{\mu_{{\text{c}}} }}{3} \times \frac{{D_{w}^{3} - d_{0}^{3} }}{{D_{w}^{2} - d_{0}^{2} }}} \right]$$

(10)

When the torsion angle is small, the change of axial force F is very small, so it is regarded as a fixed value. Then the average difference between tightening torque and loosening torque is expressed as the upward correction, and the following can be obtained:

$$U = \frac{{T_{r} - T_{u} }}{2} = F\frac{P}{2\pi }$$

(11)

Add the overall torque offset into Eq. (3), and the equation of IWAN model incorporated into the overall torque offset can be obtained:

$$T_{j} = U + k_{s} \theta + \sum\limits_{1}^{m} {\left\{ {\begin{array}{*{20}l} {r_{i} (t)} \hfill & {{\text{abs}}(r_{i} {(}t)) < R_{i} } \hfill \\ {R_{i} {\text{sgn}} (\dot{\theta })} \hfill & {{\text{else}}} \hfill \\ \end{array} } \right.}$$

(12)

The modified IWAN model after incorporating the overall torque offset U and its hysteretic curve are shown in Fig. 11b.

Fig. 11

The modified IWAN model after incorporating the overall torque offset U

4.3 Modified Residual Stiffness Slope

As the average stiffness increases with the increase of the torsion angle in the macro slip stage, the gradient of torque to the torsional angle in the macro slip tightening stage is higher than that in the loosening stage. The residual stiffness values $K_{r}$ and $K_{u}$ can be obtained by estimating the torque when tightening and loosening the nut and then deriving the torque from the torsion angle. Rotating the nut will stretch the bolt while compressing the clamped part, which is equivalent to two springs in a series. The system stiffness of the series spring is $K_{c} = K_{b} K_{j} /(K_{b} + K_{j} )$. According to Hooke's law, the axial force on the bolt is:

$$F = K_{c} P\theta /2\pi = K_{b} K_{j} P\theta /[2\pi (K_{b} + K_{j} )]$$

(13)

Substitute Eq. (13) into Eq. (9) and Eq. (10) respectively to obtain:

$$T_{r} = K_{c} P\theta /2\pi \left[ {\frac{{d_{2} \mu_{{\text{t}}} }}{2\cos \gamma } + \frac{P}{2\pi } + \frac{{\mu_{{\text{c}}} }}{3} \times \frac{{D_{w}^{3} - d_{0}^{3} }}{{D_{w}^{2} - d_{0}^{2} }}} \right]$$

(14)

$$T_{u} = K_{c} P\theta /2\pi \left[ {\frac{{d_{2} \mu_{{\text{t}}} }}{2\cos \gamma } - \frac{P}{2\pi } + \frac{{\mu_{{\text{c}}} }}{3} \times \frac{{D_{w}^{3} - d_{0}^{3} }}{{D_{w}^{2} - d_{0}^{2} }}} \right]$$

(15)

take the derivative of Eqs. (14) and (15) with respect to torsional angle $\theta$:

$$K_{r} = \frac{{dT_{r} }}{d\theta } = \frac{{K_{c} P}}{2\pi }\left[ {\frac{{d_{2} \mu_{{\text{t}}} }}{2\cos \gamma } + \frac{P}{2\pi } + \frac{{\mu_{{\text{c}}} }}{3} \times \frac{{D_{w}^{3} - d_{0}^{3} }}{{D_{w}^{2} - d_{0}^{2} }}} \right]$$

(16)

$$K_{u} = \frac{{dT_{u} }}{d\theta } = \frac{{K_{c} P}}{2\pi }\left[ {\frac{{d_{2} \mu_{{\text{t}}} }}{2\cos \gamma } - \frac{P}{2\pi } + \frac{{\mu_{{\text{c}}} }}{3} \times \frac{{D_{w}^{3} - d_{0}^{3} }}{{D_{w}^{2} - d_{0}^{2} }}} \right]$$

(17)

according to the above equation, the residual stiffness value of the classical IWAN model is:

$$k_{s} = \frac{{K_{c} P}}{2\pi }\left[ {\frac{{d_{2} \mu_{{\text{t}}} }}{2\cos \gamma } + \frac{{\mu_{{\text{c}}} }}{3} \times \frac{{D_{w}^{3} - d_{0}^{3} }}{{D_{w}^{2} - d_{0}^{2} }}} \right]$$

(18)

there is a constant stiffness difference between the tightening process and the loosening process, i.e.,

$$K_{{\text{T}}} - K_{{\text{L}}} = \frac{{K_{c} P^{2} }}{{2\pi^{2} }}$$

(19)

In the process of tightening and loosening the nut, because the angular displacement is opposite, the stiffness symbol reflected on the hysteretic curve in the tightening and loosening process is opposite. That is, based on the residual stiffness, add the stiffness value of $k_{p} = - \frac{{K_{s} P^{2} }}{{4\pi^{2} }}$ in the unloading stage and the stiffness value of $k_{p} = \frac{{K_{s} P^{2} }}{{4\pi^{2} }}$ in the loading section. Add the corrected residual stiffness slope into the residual stiffness term in Eq. (12), and the model expression can be rewritten as:

$$T_{j} = (k_{s} + {\text{sgn}} (\dot{\theta })k_{p} )\theta + U + \sum\limits_{1}^{m} {\left\{ {\begin{array}{*{20}l} {r_{i} (t)} \hfill & {{\text{abs}}(r_{i} (t)) < R_{i} } \hfill \\ {R_{i} {\text{sgn}} (\dot{\theta })} \hfill & {{\text{else}}} \hfill \\ \end{array} } \right.}$$

(20)

where $k_{p}$ is the corrected value of residual stiffness. The modified IWAN model after corrected residual stiffness slope and its hysteretic curve are shown in Fig. 12.

Fig. 12

The modified IWAN model after corrected residual stiffness slope

4.4 Solution Process of IWAN Modified Model

By modifying the classical IWAN model, the new model (hereinafter referred to as the modified IWAN model) is able to describe the force of the bolted joints with helix angle under torsional load. The expression of the modified IWAN model is Eq. (20). By solving the following parameters: $k_{s}$, $k_{i}$, $k_{p}$, U and $R_{i}$. Finally, the relationship between the reaction torque and the torsion angle of the modified IWAN model can be obtained. Parameter extraction results of the modified IWAN model with different number Jenkins and their dissipated energy are shown in Table 3.

Table 3 Parameter extraction results of IWAN model with different number Jenkins elements

According to the hysteresis curve fitted by the modified IWAN model in Fig. 13, it is not difficult to find that the approximation of fitting increases gradually with the increase of Jenkins units in the IWAN model. When the classical IWAN model is used, the calculation results of dissipated energy match well with the finite element simulation results. However, the expression of mechanical relation can not reflect the mechanical behavior of the bolted joints under torsional load. By using the modified IWAN model, the force relation of the bolted joints with helix Angle can be more truly described, the equivalent dissipation behavior of the bolted joints can be simulated without distortion, and the hysteresis curve of the bolted structure can be obtained. Compared with the finite element model, the modified IWAN model can greatly reduce the computational cost without reducing the accuracy. The modified IWAN model consisting of only three Jenkins unit models can reproduce the hysteretic curve obtained in the finite element analysis accurately enough, and further save the processing time.

Fig. 13

Hysteresis curves of the modified models with different Jenkins elements

5 Conclusion

Based on the mathematical expressions of thread profile, the effective finite element model for bolted joints with helix angle is obtained. Using the finite element model thus obtained, the hysteresis curves on the thread interface of the resultant torque versus the applied rotation angle are reproduced under different variables. The modified IWAN model is obtained by analyzing the hysteresis curve and the mechanical properties of the bolted joints. The hysteresis curves of the bolted joint can be well reproduced by using a third order modified IWAN model, and good agreement was achieved for the energy dissipation from the finite element model and the third order modified model.