^{1}

^{1}

^{1}

By considering the tightening process, a three-dimensional elastic finite element analysis is conducted to explore the mechanism of bolt self-loosening under transverse cyclic loading. According to the geometrical features of the thread, a hexahedral meshing is implemented by modifying the node coordinates based on cylinder meshes and an ABAQUS plug-in is made for parametric modeling. The accuracy of the finite element model is verified and validated by comparison with the analytical and experimental results on torque-tension relationship. And, then, the fastening states acquired by different means are compared. The results show that the tightening process cannot be replaced by a simplified method because its fastening state is different from the real process. With combining the tightening and self-loosening processes, this paper utilizes the relative rotation angles and velocities to investigate the slip states on contact surfaces instead of the Coulomb friction coefficient method, which is used in most previous researches. By contrast, this method can describe the slip states in greater detail. In addition, the simulation result reveals that there exists a creep slip phenomenon at contact surface, which causes the bolt self-loosening to occur even when some contact facets are stuck.

The bolt joint, as a very common component in engineering, is widely used in a variety of industrial machines because of its simple configuration, convenient operation, and low cost [

A typical bolt joint diagram.

In 1969, Junker [

Through the same method, Jiang et al. [

With the development of computer technology, the finite element method is accepted as the most useful numerical method for solving the bolt self-loosening problem [

In this paper, we implement the hexahedral mesh generation of the thread structure by modifying the node coordinates. Besides, an ABAQUS plug-in is made for parametric modeling and further study. Using this model, we study the differences between different fastening means and their effects on bolt self-loosening. Additionally, the mechanism of bolt self-loosening is analyzed using the relative motion of nodes and a creep slip phenomenon is illustrated.

According to the geometry features of the thread, shown in Figure

The cross-section profile of external thread.

The cross-section profile along the bolt axis.

Assume that the diagram given by Figure

In addition, it can be known that the mathematical expression of the outer surface of thread is periodic.

In summary, the complete expression of the thread cross-section profile can be expressed as follows:

Similarly, the profile of the internal thread can be expressed in the same manner and it also has the periodical piecewise function form.

The mesh generation method Fukuoka proposed requires many cyclic operations on the two-dimensional elements such as “copy”; “translate”; “rotate”; and “merge,” which easily cause the problem of expensive computation. In this study, the thread structure is partitioned properly before meshing. The hexahedral mesh generation is implemented by modifying the node coordinates based on cylindrical hexahedron meshes. The detailed procedures are introduced below. In addition, a self-developed ABAQUS plug-in is made for parametric modeling (Figure

The parametric modeling interface.

Depending on the size of the bolt and nut, the corresponding cylinders are modeled with hexahedral meshes in ABAQUS. To fit the shape of the thread well and improve the calculation efficiency, the model is divided into two parts: the thread region and the nonthread region. The thread region is discretized with finer meshes to guarantee the simulation accuracy of the contact state, while the other region is meshed with relative coarse elements (Figure

The hexahedral models of cylinders.

The node coordinates of the thread region are extracted from the INP file exported previously, and they are modified by a self-compiled program depending on the mathematical expression of thread cross-section profile. The modified INP file is then imported to ABAQUS again, and the hexahedral models of threads are generated (Figure

The hexahedral models of threads.

The bolt shank and bolt head, which are simple cylinders, are built up and meshed with hexahedral elements. Then they are merged with the thread part to obtain a complete bolt model.

Finally, the clamped components are modeled in ABAQUS, and all parts are assembled into a whole analysis model of the bolt self-loosening problem (Figure

The whole hexahedral model of a typical bolt joint.

Through the method described before, the 3D finite element model, which contains a ^{3}, respectively.

Contact interactions have been set between all sliding surfaces, including the interfaces between the threads, bolt underhead surface and the upper part surface, and nut surface and the lower part surface. Contact modeling is very important to the simulation of the tightening and self-loosening process. According to the work by Dinger and Friedrich [

The preload refers to the elastic restoring force of the bolt when it is serving. In engineering, it is always controlled by tightening torque or rotation angle of the nut. In this paper, the tightening process analysis is performed by applying a ramp circumferential displacement

The loading diagram.

Referring to the general rules of tightening for threaded fasteners given in GB/T 16823.2 [

To a

The contrast diagram of torque-tension relationships.

In addition, a test rig has been developed as shown in Figure

The test rig.

The curves in Figure

Comparison between the finite element method and the experimental method.

The preload is a very important factor that should not be ignored in the study of bolt self-loosening. However, few studies considered both the tightening process and the self-loosening process simultaneously. Most of them simulated the preload by stretching the bolt or using a cooling pretightening algorithm that makes the fastening state of bolt different from the real case. Therefore, the differences of different fastening ways are discussed here, followed by their effects on bolt self-loosening.

To simulate the tightening process, a circumferential displacement

The loading diagram of the simplified way.

The relationships between the torque and the preload.

As shown before, the value of the torque in the tightening process is obviously higher than that obtained by stretching the bolt under the same preload, which leads to making the self-loosening more likely to occur. The resultant torque on the thread interface consists of two parts: the pitch torque and the thread friction torque. Figure

The relationships between different kinds of torque and preload.

Simulating the tightening process

Stretching the bolt

Considering the influence of the tightening process, the preload is produced by applying a constrained circumferential displacement on the side surface of the nut, followed by its removal. To conduct the FEA of bolt joints self-loosening, a transversal excitation

The diagrams of transversal harmonic load.

To investigate the effects of different fastening means on the self-loosening process of the bolt joint, the same preload is produced by adjusting the circumferential displacement and the separation distance in finite element analysis, and the other attributes of the two models are completely identical. A cyclic transversal displacement is then loaded on the clamped components. The evolutions of the preload of different fastening means during the first 15 load cycles are illustrated in Figure

The preload variations under different fastening ways.

It can be drawn from the above analysis that the approximate pretightening algorithm cannot take the place of the tightening process to study the self-loosening mechanism of bolt joints. To preform further analysis of the evolution of the preload during bolt self-loosening, the number of loading cycles is increased to 150 and the initial preload is set to 8 kN. The curve in Figure

The curve of preload variation during the 150 cycles.

The self-loosening behavior is mainly caused by slip at contact surfaces. Therefore, the dynamics during self-loosening is the main focus of the following analysis. This paper uses the relative motion of nodes to present the slip state, which is different from previous researches, and it is proved to be in greater detail by contrast with the traditional method using the friction relation.

The process of the preload variation mainly consists of two stages (exampled in Section

The diagrams of the contact surface and the calculation points.

According to Figure

The rotation angles of each point in different stages.

The rapid decline stage

The flat stage

As shown before, all the nodes rotate along the loose direction as a whole which causes the preload loss. However, at the beginning of the self-loosening process, not all of the nodes rotate at the same time but one node rotates firstly and drives the rotation of the other nodes. Moreover, in the process of rotating, the rotation angles of some nodes are large and some are small. When the movement direction of clamped component changes, the rotation angles of those whose rotation angles are large previously begin to decrease. Meanwhile, the rotation angles of those whose rotation angles are small increase. This presents a creep slip phenomenon at contact surface under reversed cyclic load. With increase of the loading cycles, the preload continues to decline. And, finally, all nodes present a back and forth rotation at one place, which causes the flat stage.

To analyze the slip state during the initial stage of self-loosening (when preload is 27.2 kN), all nodes along the outer edge are taken into account, and their relative rotation velocities around node A are carried out, as shown in Figure

The node number along the outer edge.

The relative rotation velocity of each node at different moment.

The slip state contours at different time.

In addition, the relative motion between thread interfaces is analyzed in a similar way. Two helical segments are intercepted from the contact location of bolt and nut (Figure

The schematic diagrams of helical segments.

The position-velocity fields of bolt and nut at 0.25 s.

Based on the position-velocity curves of bolt and nut, the relative position-velocity relationship between thread interfaces can be acquired by subtracting them. Figure

The relative rotation velocity of each node at different moment.

The slip state contours between thread interfaces.

To further strengthen the trust in the results summed before, the relation between transverse force (shear force) and transverse displacement during the initial fifteen cycles is shown in Figure

Hysteresis loops of transverse displacement and load.

The self-loosening process of bolt joints is investigated combining the tightening process by a three-dimensional finite element model in this paper. The FE model is meshed with hexahedral elements, and its accuracy is verified and validated compared with the analytical and experimental results. Followed by simulating different fastening means, the differences between them and their effects on bolt self-loosening are discussed. Finally, we utilize the relative motion of nodes to describe the contact states, and the conventional Coulomb friction method is also applied for contrast. Based on the FEA results, the following conclusions are drawn:

Based on the mathematical expression, the threads are meshed with hexahedral elements by modifying the node coordinates of the cylindrical hexahedral meshes, which is proved to be effective. And a self-developed plug-in is made for parametric modeling, and its functions can be expanded in further study.

Through comparing with a simplified pretightening algorithm, it is demonstrated that the tightening process cannot be replaced, because the simplified way may cause a smaller resultant torque due to the opposite direction of the two torque components on the thread interface. For the same reason, it will lead to a greater loss of preload than the value in reality under the same number of load cycles.

By contrast, the relative motion between nodes is found in a greater detail to describe the slip state at contact surfaces than Coulomb’s law of friction. According to the simulation results of bolt self-loosening, it reveals that there exists a creep slip phenomenon on the bolt head bearing surface, which causes the bolt self-loosening to occur even when some contact facets are stuck.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The paper is supported by National Science and Technology Major Project of the Ministry of Science and Technology of China (no. 2011ZX02403), National Natural Science Foundation of China (no. 11302035 and no. 11272074), and the Fundamental Research Funds for the Central Universities.