Study on Tightening, Anti-Loosening, and Fatigue Resistance Performances of Bolted Joints with Different Anti-Loosening Washers and Nuts

Abstract

Loosening failure of bolted joints often occurs in a vibration environment. This may induce the separation of clamped components and even result in catastrophic consequences in certain situations. Various anti-loosening bolted connections have been developed and widely applied to prevent loosening. Tightening and fatigue resistance performances are also two important characteristics of anti-loosening bolted connections. However, a comprehensive comparison study of the tightening, anti-loosening, and fatigue resistance performances of typical anti-loosening bolted connections is lacking, meaning that there is a lack of material to assist in the selection of anti-loosening bolted connections in engineering. In this study, accurate three-dimensional finite element models of typical anti-loosening bolted connections with the forms of washers or nuts are established. The processes of tightening, vibration-induced loosening, and fatigue are simulated via finite element analysis (FEA). Based on the results of FEA, the tightening, anti-loosening, and fatigue resistance performances of regular bolted joints and several anti-loosening bolted connections are comprehensively compared. The results reveal that bolted joints installed with both a variable-diameter nut and an eccentric double nut are difficult to tighten. We confirm that the researched structures all have superior anti-loosening capacities compared to regular bolted joints. Additionally, the results show that the regular bolted joint achieved the worst fatigue resistance performance, i.e., the application of anti-loosening bolted connections improves fatigue resistance. Finally, the effects of preload distributions on the two nuts in bolted joints with double nuts are discussed in terms of tightening, anti-loosening, and fatigue resistance performances.

1. Introduction

Bolted joints are widely used in industries such as the mechanical, astronautical, and aeronautical sectors, as well as in ocean engineering [1,2,3]. A bolted joint consists of bolts, nuts, and clamped components, as shown in Figure 1. A torque wrench is typically employed to tighten the nut. In this way, the engagement of internal and external threads plays a significant role in connecting and fastening separated components [4]. However, the loosening of bolted joints occurs frequently in environments with strong vibration or temperature alternations, decreasing the preload and potentially triggering a catastrophic accident. For example, a railway accident due to a high-speed train derailment occurred as a result of loosening in the United Kingdom in 2007 [5]. This accident resulted from nuts detaching from bolts, allowing the switch rail to be struck by the inner faces of passing train wheels. Therefore, the reliability of bolted joints cannot be overemphasized in engineering.

Why are bolted joints loose during the period of service, and what is the loosening mechanism? In recent decades, many researchers have conducted investigations on these issues, and some representative results have been achieved. In 1969, a German engineer named Junker [6] designed a machine to apply cyclic transversal vibration to bolted joints. The results showed that the preloads decreased rapidly and that complete loosening may occur under transversal vibration. With respect to the loosening mechanism of bolted joints under transversal vibration, explanations invoking complete slippage theory and local slippage accumulation theory have been proposed [7,8]. Some theoretical models have been established based on the theory of complete slippage to explore the loosening process and its critical conditions under the influence of different factors [9,10].

In 2002, Pai and Hess [11,12] established a three-dimensional finite element model (FEM) of a bolted joint and simulated cyclic transversal vibration. They proposed the concept of local slippage accumulation by observing the contact states of thread and bearing surfaces during vibration. According to local slippage accumulation theory, local slippage in partial contact areas can accumulate and expand to complete slippage, causing continued loosening. This new loosening theory attracted extensive subsequent attention [13,14,15,16,17]. Recently, Gong et al. [18,19,20] proposed modified Iwan models to represent local slippage behavior and provided new insights into loosening due to local slippage accumulation. The abovementioned research mainly focused on loosening owing to reverse rotation against the tightening direction. This is also called “self-loosening” [21]. Many researchers have also found that embedding loss [22], creep [23,24], stress relaxation [25], cyclic plastic deformation [26], stress redistribution [27], and fretting wear [28,29,30] can induce preload loss (i.e., non-rotation loosening) during the service of bolted joints.

In practical engineering applications, the questions of how to prevent loosening and improve the integrity and reliability of bolted joints are highly significant [31]. Some anti-loosening bolted connections are applied in place of regular bolted joints only consisting of nuts and bolts, as shown in Figure 2. There are three common forms. First, various anti-loosening washers are inserted between a nut or bolt head and a clamped component [32]. Typical products include plain washers and wedge washers, among others. Second, the shape or size of the internal thread in a nut is modified, and the engagement states between external and internal threads are changed [33,34]. Typical products include wedge-locking and variable-diameter nuts. Third, two nuts are employed to be fastened to a bolt to achieve excellent anti-loosening performance [35,36,37], including normal double nuts, eccentric double nuts, and double-thread bolt connections (a double-thread bolt and two nuts). The shapes of internal threads in eccentric double nuts are unchanged, but there is an eccentricity in eccentric double nuts, and the outer structures of the two nuts differ from the properties of normal double nuts. In a double-thread bolt connection, the bolt is machined to display both left-hand and right-hand threads. Two nuts with opposite thread directions are assembled on the bolt. Details of eccentric double nuts, and double-thread bolt connections are introduced in the following section.

Although some research on the anti-loosening mechanisms of the abovementioned structures has been preliminarily conducted, comparison studies of their anti-loosening performance under vibration are still lacking. In fact, tightening and fatigue resistance performances are also two important characteristics of these anti-loosening bolted connections. For example, both insufficient tightening preload and material fatigue aggravate the vibration-induced loosening behavior of anti-loosening bolted connections. However, to the best of our knowledge, there have been no investigations of the tightening and fatigue resistance performances of the abovementioned anti-loosening bolted connections in previous publications. As such, there is no comprehensive evidence that can be used to provide guidance for the selection of proper anti-loosening bolted connections in engineering. This is the motivation of the present study.

In this paper, a comprehensive comparison study on the tightening, anti-loosening, and fatigue resistance performances of several typical anti-loosening bolted connections (shown in Figure 2) is conducted for the first time. In this way, people can understand the performances of these anti-loosening bolted connections; therefore, this study can provide technical assistance and guidance for engineers in selecting appropriate anti-loosening bolted connections in engineering applications. Moreover, we discuss the effects of the different preload distributions of the two nuts on tightening, anti-loosening, and fatigue resistance performances of three bolted joints installed with double nuts. We found that it is necessary to increase the proportion of preload assigned to the upper nut in order to achieve better anti-loosening performance and extend the fatigue life. This work can guide engineers in tightening double nuts with appropriate technologies and processes.

2. Details of Anti-Loosening Bolted Connections

A plain washer is a kind of gasket with an annular shape that has become an international standard due to its widespread use in engineering. In the standard, the sizes, materials, surface treatment technologies and other parameters are provided. Plain washers were originally designed to protect the surfaces of clamped components instead of preventing bolted joints from loosening. Whether plain washers contribute to anti-loosening performance is controversial. A wedge washer consists of a pair of locking washers with a cam on one side and radial teeth on the other side, as shown in Figure 3. The anti-loosening mechanism is based on negative feedback adjustment, i.e., loosening leads to relative misalignment of upper and lower washers and, in turn, causes tensile elongation of the bolt and an increase in preload.

In the second type of anti-loosening bolted connections, a wedge-locking nut has a wedge ramp at the root of the thread, as shown in Figure 4a. During tightening of the wedge-locking nut, the bolt tips are pushed against the wedge ramps, and linear contact is formed between the internal and external threads, which gives a wedge-locking nut a greater locking force than a regular nut. The tips of external threads are easily deformed, resulting in a large frictional force and uniform load distribution on the wedge thread. A variable-diameter nut is another type of anti-loosening bolted connection, as shown in Figure 4b. This nut is similar to conventional prevailing torque nuts, which essentially change the size of the internal thread in a normal nut to generate interference between the internal thread in the nut and the standard external thread in a bolt.

Double nuts consist of two identical regular nuts, as shown in Figure 2e, where the lower nut is called the power nut and the upper nut is called the locking nut. The power nut is first tightened, followed by the locking nut. After the tightening process of regular double nuts, the thread of the lower nut may come into contact with the surface opposite that it is in contact with during the standard tightening process. A locking force is generated to prevent loosening. An eccentric double nut is composed of an eccentric convex nut and a concentric concave nut. When the convex nut and the concentric nut are combined, a powerful locking effect can be produced via the wedge principle, as shown in Figure 5a. The interference between convex and concave nuts generates a strong frictional force after tightening, achieving the aim of loosening resistance. Double-thread bolts, which engage with two nuts with different rotational thread directions, were invented for anti-loosening. The nut with right-hand thread is the power nut, and the one with left-hand thread is the locking nut, as shown in Figure 5b. The power nut is tightened first, followed by the locking nut. In vibration or impact environments, the power nut tends to loosen first, causing the locking nut to rotate in the tightening direction; thus, the loosening of the power nut is prevented.

3. Finite Element Models of Anti-Loosening Bolted Connections

The FEM built up in this paper is based on a typical bolted joint structure. The nominal diameter and pitch of bolt and nut were M10 and 1.5 mm, respectively. Other dimensions of the regular bolted joint are shown in Figure 6, where A = 6 mm, B = 33 mm, C = 12 mm, D₁ = 10 mm, D₂ = 11.2 mm, E = 15.7 mm, F = 18.1 mm, G = 7.5 mm, H₁ = 9 mm, and H₂ = 15 mm.

Subsequently, a three-dimensional FEM of a regular bolted joint was built using HyperMesh 12.0 commercial preprocessor software, as shown in Figure 7. Using the method of Fukuoka and Nomura [38], the meshes of internal and external threads were generated accurately. This method takes the geometric structures of a bolted joint into account and generates three-dimensional hexahedral meshes. Compared with tetrahedral elements, hexahedral elements have higher calculation accuracy and better deformation resistance. In the process of modeling, the bolt head and hexagonal nut were both simplified as cylinders with a diameter of 15.7 mm. In ISO 16130-2015 [39] and German national standard DIN 65151 [40], the cyclic transversal vibration acts on the upper plate, and the lower plate is applied with a fixed constraint. Therefore, the upper plate is called a movable plate, and the lower plate is called a fixed plate. Since the friction effect between the two plates could be ignored, the fixed plate was omitted to simplify the calculation, and only the FEM of the movable plate was built.

Subsequently, the three-dimensional FEMs of seven anti-loosening bolted connections were built, as shown in Figure 8. The meshes of a plain washer were generated via sweep, and all the meshes were hexahedrons. The mesh density was consistent with that of the moving plate. The inner diameter of the plain washer was consistent with that of the movable plate, and the outer diameter was slightly larger than that of the regular nut, as shown in Figure 8a. The meshes of the wedge washer were divided carefully, especially the radial tooth surfaces on both sides. The mesh density of radial teeth was set to a relatively high value in order to ensure the calculation accuracy of the contact analysis, while the other areas were meshed with low density, as shown in Figure 8b. The plain washer and wedge washer were both installed under a nut; the FEMs of the bolt, nut, and moving plate were the same as in the model of a regular bolted joint shown in Figure 7.

The shapes or sizes and internal threads in a wedge-locking nut and variable-diameter nut were changed. Therefore, the mathematical expression of each portion on the thread profile along the axis within a pitch should be established first. The mathematical expression of the thread profile in a wedge-locking nut is a little complicated, as discussed in [34], while that in a variable-diameter nut is modified compared with a normal internal thread [38]. Next, the meshes of wedge-locking and variable-diameter nuts were both built using Fukuoka and Nomura’s meshing method. In the model of the wedge-locking nut, the wedge angle was 30°. In the model of the variable-diameter nut, the internal and external threads were interfered. Figure 8c,d show the entire bolted joints with a wedge self-locking nut and a variable-diameter nut, respectively. The FEMs of the bolt and moving plate remained consistent with those of the bolted joint shown in Figure 7.

In the finite element modeling process of bolted joints with regular double nuts, the length of the bolt and the number of thread teeth were increased, and the modeling method with two nuts was the same as that for a regular nut. Figure 8e presents the FEM result. The thread meshes of an eccentric double nut with high density were built based on Fukuoka and Nomura’s method. The meshes in the interference area between convex and concave nuts required careful generation. All the meshes were hexahedral in shape, and the density was set to a value as small as possible in order to improve calculation efficiency. The inclined surfaces in the FEMs of convex and concave nuts had similar meshes in order to attain a solid match when in contact. It should be noted that there was an eccentricity in the FEM of the convex nut. The entire bolted joint with an eccentric double nut was established as shown in Figure 8f, where the bolt and moving plate were unchanged. The external thread shape of the double-thread bolt was complex, combining both left-hand and right-hand threads. The eccentricity of the eccentric double nut was set to be 0.1 mm. The FEMs of normal bolts with left-hand and right-hand threads were first established. Next, a Boolean operation was used to retain the common parts of the two models to generate the FEM of a double-thread bolt, as shown in Figure 8g. The FEMs of nuts with opposite thread directions were generated based on Fukuoka and Nomura’s method, and the moving plate remained unchanged.

The material of all the parts was 1045 steel, and the parameters of FEMs are shown in

Table 1. Parameters of FEMs.

Item	Value
Element type	Solid 185
Young’s modulus	210 GPa
Poisson’s ratio	0.3
Friction coefficient	0.15

. Considering that a wedge-locking nuts avoid loosening via plastic deformation of threads, elastic–plastic parameters were added in the finite element model. The bilinear yielding model was applied to model material plastic deformation, and the yield strength of a wedge-locking nut was set to 355 MPa. Each pair of contact interfaces consisted of a contact component and a target component. The element types of the contact component and the target component were set as CONTA173 and TARGE170, respectively. A friction coefficient value of 0.15 was assigned to the contact interfaces, which is a typical value adopted by other researchers [13]. In the modeling progress, the boundary conditions were as follows: (1) movements in all directions for the bottom of the bolt head were fully constrained, and (2) movement in the direction parallel to the bolt axis for the bottom of the movable plate was constrained.

4. Simulations of Tightening, Vibration-Induced Loosening, and Fatigue

4.1. Tightening

There are many methods to generate preloads of bolted joints in FEA. In this study, the preload of a bolted joint was generated by applying torque to the nut, which conformed to engineering practice. In this way, the torsion–tension relationship of bolted joints could be obtained, which helped us to evaluate and compare the tightening characteristics of different types of anti-loosening bolted connections. For a bolted joint with a single nut, the target tightening torque was generated by applying clockwise shear stress distributed evenly on each node around the outer cylindrical surface of the nut [41], as shown in Figure 9. The letter r denotes the equivalent radius of the outer cylindrical nut surface. In terms of a bolted joint with double nuts, the target tightening torque was applied to the outer cylindrical surface of the lower and upper nuts successively using the same method. The rotation angle of the nut during the tightening process was controlled by changing the value of shear stress. The FEA was performed using ANSYS 19.2 software [19,42]. In this study, the generated preload of different threaded fasteners was maintained at 30 ± 1 kN by controlling the rotation angle of the nut.

Some scholars [43] have accurately established a model of the mathematical relationship between tightening torque (T) and preload (F) through theoretical analysis, as shown in Equation (1).

$$T=F\left({\mu }_b{r}_b+\frac{p}{2\pi }+\frac{{\mu }_t{r}_t}{\cos \alpha }\right)$$

(1)

where u_b is the friction coefficient between the turning nut and its surface, u_t is the friction coefficient between threads, r_b is the effective radius of the bearing surface, r_t is the effective thread contact radius, p is the pitch of the thread, and α is the helix angle. The accuracy of FEA results was validated by comparing the torque–tension relationship obtained by FEA and that obtained using Equation (1). The results are presented in Figure 10. It is obvious that the discrete points of torque and preload obtained by FEA almost lay in a straight line based on Equation (1), meaning that the simulation of nut tightening using the torque method is reliable.

4.2. Vibration-Induced Loosening

It has been validated that bolted joints tend to loosen under cyclic transversal vibration and that loosening results in the loss of preload. In this study, the process of the preload decay under cyclic transversal vibration was simulated via FEA with the purpose of evaluating the anti-loosening ability of different kinds of bolted joints. Gong et al. [1] investigated the change in preload with an increase in vibration cycles under different vibration frequencies through FEA and experimentation. It was proven that the vibration frequency had no effect on the loosening behavior. Since the experimental vibration frequency was less than 12.5 Hz [44], the cyclic transversal vibration of the movable plate was assumed to be a quasi-static process. Based on the quasi-static assumption, cyclic transversal vibration was applied to all nodes on the outer cylindrical surface of the movable plate, as shown in Figure 11. The vibration-induced loosening simulation process of all bolted joints was carried out under an initial preload of 30 ± 1 kN. The vibration amplitude was set to 0.5 mm [39] along the x+ and x− directions, and vibration was performed with a total of 30 cycles.

After the simulation of tightening and cyclic vibration was completed, the stress distributions of the bolt thread and bearing surface for a regular bolted joint were extracted, as shown in Figure 12. It can be seen that the position of maximum stress in the bolt thread root moved upward after cyclic transversal vibration, which indicates that rotation loosening occurred. Additionally, the stress on the bearing surface of the movable plate decreased obviously after vibration.

In order to verify the influence of FE mesh density on the result, the anti-loosening bolted connections were meshed with different densities (the element size was small), as shown in Figure 13. The preload variations under cyclic transversal vibration in different mesh densities are depicted in Figure 14. It can be seen that with the improvement in mesh refinement, the FEA results tend to be consistent. However, it is evident that an increase in mesh density significantly reduces the solution efficiency of FE simulation. Thus, considering the achieved accuracy and efficiency, the choice of the FEA elements (type 1) reported in this manuscript was reasonable.

4.3. Vibration-Induced Fatigue

In engineering applications, another failure form of bolted joints is the fatigue fracture of bolts under long-term cyclic vibration [45,46]. Fatigue life is denoted by the number of loading cycles before specimen failure. The presence of some anti-loosening bolted connections, such as washers, can increase the fatigue lives of bolted joints [47]. In order to study the fatigue characteristics of different anti-loosening bolted connections, the cyclic transversal vibration applied to movable plates was simulated using the same method deployed for the simulation of vibration-induced loosening. Due to the large calculations and small changes in stress amplitude under the same simulation condition required for FE models, only one vibration period of simulation was performed, and the vibration amplitude was set to 0.5 mm.

It has been validated that the bolt thread root located within one pitch of the bearing surface has the maximum stress level, which is where fatigue cracks tend to occur most easily [20]. Therefore, the root point in the bolt was extracted as shown in Figure 15 in order to analyze the fatigue resistance performance of different bolted joints. Von Mises stress, which takes the first, second, and third principal stresses into account, is often used to evaluate fatigue resistance performance.

After the simulations are completed, the von Mises stresses of root points in all bolts within a complete vibration cycle are extracted. The stress time series obtained via FEA is a continuous random process. The issue of how to account for complex loading history is a basic problem in fatigue life estimation [48]. Cycle counting is an effective way to simplify complicated loading history into simple load sequences with a cyclic concept for the purpose of estimating fatigue life using existing knowledge [49]. The rainflow counting method is one of the typical methods developed to calculate the number of cycles in complex loading history, and this technique has been successfully applied in fatigue life prediction for the automotive, railway, and aircraft industries [50]. This counting method uses all the information that impacts the fatigue damage of materials [51]. In order to carry out statistical processing for subsequent fatigue analysis, a MATLAB program based on the rainflow counting method was developed. Figure 16 shows the flow chart of the program to explain it in detail. For the purpose of cyclic counting, the irregular stress fluctuation is converted into a series of peaks and valleys. Then, a stress range histogram can be obtained via the rainflow counting method.

There are several fatigue damage cumulative theories. Among them, linear damage cumulative theory is often used to estimate fatigue life due to the simplification of the fatigue damage mechanism and the convenience of its calculation. This theory assumes that the fatigue damage generated under various levels of stress is independent of other forms, meaning that the cumulative total damage can be obtained using linear superposition theory. Miner’s linear damage cumulative rule [52,53] given in Equation (2) is used in this study to evaluate fatigue damage of a threaded fastener.

$$D={\displaystyle \sum _i\frac{n_i}{N_i}}$$

(2)

where D is the fatigue damage index, n_i is the number of the existing stress cycles in each stress range, and N_i is the number of stress cycles in each range that could lead to fatigue failure.

The data needed in the rule can be extracted from the histograms by referring to S-N diagrams, which indicate the relationship between the stress ranges and fatigue life within set stress ranges. S-N curves for bolts with different specifications are available in DNVGL-RP-C203 [54], including the M10 × 1.5 bolts used in the bolted joints in this study, as shown in Equation (3).

$$\log N=16.301-5.0\log \Delta \sigma$$

(3)

According to Equation (2), the S-N curve of M10 × 1.5 bolts was produced to calculate fatigue damage in this study, as shown in Figure 17, which shows data obtained from constant-amplitude experiments. Finally, the fatigue lives of different types of bolted joints can be estimated according to the linear damage cumulative theory. For ease of understanding, the flow chart of fatigue damage evaluation is shown in Figure 18.

5. Results

5.1. Tightening Performance Analysis

The tightening process simulations of all bolted joints mentioned above were accomplished via FEA. First, the discrete data points and curves of tightening torques and the corresponding preloads in five different bolted joints installed with a single nut were obtained, as shown in Figure 19, which shows that it is difficult to differentiate between the discrete data points of regular bolted joints and those of the bolted joints installed with plain and wedge washers due to coincidence. The curve drawn based on discrete data points obtained from the bolted joint installed with a variable-diameter nut is nonlinear and runs above those of the other bolted joints, which are almost linear.

To quantitatively evaluate the degrees of tightening difficulties, linear fits were performed for the data points. The slopes of fitted lines for the torsion–tension relationship reflect the tightening performances of different bolted joints. The larger the slope of the fitted line, the more difficult it is to tighten the corresponding bolted joints. The slopes of fitted lines are listed in

Table 2. Slopes of fitted lines for the torsion–tension relationship in the bolted joints installed with a single nut.

Bolted Joint	Slopes (mm)
Regular bolted joint	1.983
Plain washer	2.004
Variable-diameter nut	5.498
Wedge-locking nut	2.365
Wedge washer	1.952

. By comparing the slopes shown in Table 2, it is obvious that the slope of variable-diameter nut is as large as 5.498 mm, while the wedge-locking nut has a slope of 2.365 mm. They are increased by 125% and 20%, respectively, compared with the values of a regular bolted joint. However, the slopes of the bolted joints installed with plain and wedge washers are almost the same as those of the regular bolted joint. This indicates that the variable-diameter nut and wedge-locking nut are more difficult to tighten than the normal nut and that the applications of plain and wedge washers do not affect the tightening performance.

Gong et al. [18] found that the torque–tension relationships of a regular bolted joint, plain washer, and wedge-locking nut obtained by test machine were similar to the FEA results shown in Figure 19. Specifically, when tightening the wedge-locking nut, a higher torque was required to achieve the same expected preload. Therefore, the results reported in our paper and the available literature prove that a wedge-locking nut is more difficult to tighten than a regular nut and that the application of a plain washer does not affect the tightening performance.

Secondly, the discrete data points and curves of tightening torques and the corresponding preloads in three bolted joints installed with double nuts were also obtained, as shown in Figure 20. In this study, the total preload was set to approximately 30 kN, and the upper and lower nuts were both assigned values of 15 kN. It can be observed from the figure that there are two distinct stages in these relationship curves. In the first stage, the lower nut was tightened to obtain the discrete data and curves of torques and preloads. When tightening the second nut, we found that the preload existed but that the torque increased from 0. Consequently, there was a sudden change in the relationship curves. Subsequently, the preload and torque increased together as tightening proceeded. In the first stage, we found that the curve slope of the eccentric double nut was the largest, while the other two bolted joints had a similar curve slope. In the second stage, the curve slope of the double-thread bolt connection was the smallest, while those of the other two bolted joints were close. The final torques in the first and second stages were applied in order to quantitatively evaluate the degrees of tightening difficulty, and they were named the first final torque (FFT) and the second final torque (SFT), respectively. The values of FFTs and SFTs were obtained, as shown in

Table 3. Final torques obtained in the two stages of tightening.

Bolted Joint	Lower Nut (Nmm)	Upper Nut (Nmm)
Regular double nuts	58,972	38,791
Double-thread bolt connection	55,362	31,560
Eccentric double nut	81,928	96,540

. The FFT and SFT of the eccentric double nut were both the largest, followed by regular double nuts and the double-thread bolt connection. This demonstrates that the interference and eccentricity in an eccentric double nut increase the difficulty of tightening. It is interesting to discover that it is more convenient to tighten a double-thread bolt connection than regular double nuts.

5.2. Vibration-Induced Loosening Performance

The changes in preloads with increasing vibration cycles for eight different bolted joints are shown in Figure 21. For the bolted joints installed with double nuts, the preloads assigned to the upper and lower nuts were both 15 kN. It can be observed from the figure that the preload in the regular bolted joint gradually decreased during the vibration process, showing poor anti-loosening performance. The preloads in the bolted joints installed with a plain washer, wedge washer, variable-diameter nut, eccentric double nut, and double-thread bolt were almost constant after 30 cycles. The preload in the bolted joint installed with a wedge-locking nut decreased gradually during the first five cycles and remained constant subsequently. This phenomenon indicates that a wedge-locking nut can provide anti-loosening ability for the bolted joint, but its anti-loosening performance is not as strong as that of other structures with stable preload under vibration. For the regular double nuts, there was a small drop in the preload following the initial vibration cycle, possibly because of the effect of stress redistribution [27]. Overall, the seven bolted joints installed with different washers or nuts all show a certain anti-loosening performance under transversal vibration.

It is worth mentioning that Gong et al. [18] used a Junker test machine to experimentally investigate the anti-loosening abilities of different bolted joints. They found that the anti-loosening performance of a wedge-locking nut was better than that of a regular bolted joint. Thus, the results of our study and the available literature prove that a wedge-locking nut is more capable of loosening resistance than a regular nut.

5.3. Vibration-Induced Fatigue Performance

After the finite element simulations were completed, the von Mises stresses of root points in all bolts within a complete vibration cycle were extracted. The obtained stress time series of different bolted joints are shown in Figure 22.

It is worth mentioning that the three bolted joints with double nuts had the same preload distribution, that is, the upper and lower nuts were both assigned 15 kN. It can be seen from the figure that the stresses in all bolted joints fluctuate within a vibration cycle. The locations of the peaks and valleys in the curves are almost the same in the six threaded connection structures, i.e., in the regular bolted joint, as well as in the bolted joints installed with a plain washer, variable-diameter nut, wedge-locking nut, regular double nuts, and a double-thread bolt. However, the magnitudes of stresses of the peaks and valleys in the curves are different for the six abovementioned types of bolted joints. We can also see that the locations of the peaks and valleys in the curve for the bolt installed with an eccentric double nut are almost the opposite of those of the six abovementioned bolted joints. In addition, the change in the curves of the bolted joint installed with a wedge washer considerably differs from that of other bolted joints, causing the locations and stresses of the peaks and valleys in the curve to significantly differ.

With the rainflow counting method, the stress time series of threaded connection structures were counted cyclically; the stress range histograms are shown in Figure 23. Under the same preload level, the majority of stress cycles extracted via the rainflow algorithm are small-amplitude stress cycles, while there are few stress cycles with amplitudes higher than 100 MPa.

According to Miner’s cumulative damage criterion, the larger the fatigue damage index under the same condition, the shorter the fatigue life. Therefore, the fatigue resistance performance of different bolted joints can be estimated according to the fatigue damage index accumulated within one vibration cycle. Using the stress range histograms and the S-N curve used for M10 × 1.5 bolts shown in Figure 17, the fatigue damage indexes accumulated within a complete cycle were calculated, as shown in

Table 4. Fatigue damage indexes of bolted joints.

Bolted Joint	Index
Regular bolted joint	0.1101
Plain washer	0.0935
Variable-diameter nut	0.0222
Wedge-locking nut	0.0223
Wedge washer	0.0408
Regular double nuts	0.0676
Double-thread bolt connection	0.0292
Eccentric double nut	0.0572

. The calculation results demonstrate that the fatigue index of regular bolted joint is the highest, representing the worst fatigue resistance performance under the conditions of the same vibration cycles. Moreover, the fatigue resistance performance of the bolted joints installed with a variable-diameter nut and wedge-locking nut is the best among all the threaded connection structures, i.e., these bolts have the longest fatigue life under cyclic vibration. Additionally, among the three bolted joints with double nuts, the double-thread bolt connection shows the best fatigue resistance performance, whereas that of the eccentric double nut is the worst.

6. Discussion

During the assembly of double nuts, the preload distribution applied to the upper and lower nuts may be diverse under different tightening processes, although the total preload remains the same. In this section, the effects of preload distribution on the two nuts on tightening, anti-loosening, and fatigue resistance performances are discussed.

6.1. Effect of Preload Distribution on Tightening Performance

In order to explore the influence of preload distributions on the tightening performance of three bolted joints with double nuts, the preload distributions of the upper and lower nuts were changed by changing the rotational angles of the nuts. The total preload was approximately 30 kN, and the preloads of the upper nut were changed to 5, 10, 15, 20, 25, and 30 kN. The discrete data points and relationship curves of tightening torques and preloads in the bolted joint installed with regular double nuts are shown in Figure 24. From the figure, it can be observed that the curve slopes in the two stages are almost the same for different preload distributions of upper and lower nuts. However, the values of preload when the tightening in the first stage is finished are different; the preload displayed here is called the transition preload (TR). The assignment of a larger preload to the upper nut requires a smaller TR to generate the same total preload during the process of tightening. This results in final tightening torques (FFTs and SFTs) in the upper and lower nuts being different, as shown in

Table 5. Final torques of regular double nuts under different preload distributions.

Preload Assigned to the Upper Nut	5 kN	10 kN	15 kN	20 kN	25 kN	30 kN
First stage (Nmm)	61,083	59,104	58,972	57,584	58,072	57,104
Second stage (Nmm)	11,857	28,804	38,791	52,986	50,489	64,081
Max torque (Nmm)	61,083	59,104	58,972	57,584	58,072	64,081

. It can be observed that as the preload assigned to the upper nut is increased, the FFT of the lower nut is decreased. Table 5 also shows the maximum final torque (MFT), which represents the degree of tightening difficulty. The MFT is the largest among these different preload distributions for the preload of 30 kN assigned to the upper nut, followed by preloads of 5, 10, 15, and 25 kN assigned to the upper nut. When a preload of 20 kN is assigned to the upper nut, the MFT is at a minimum, with very easy tightening.

Similarly, the discrete data points and relationship curves of tightening torque and preload in the bolted joints installed with a double-thread bolt and an eccentric double nut were obtained, as shown in Figure 25 and Figure 26, respectively. The preloads of the upper nut were also set to 5, 10, 15, 20, 25, and 30 kN in FEA. The changes in the curves during the two stages showed the same tendency as those of the regular bolted joints under different preload distributions of the upper and lower nuts. It was obvious that the slope of the curve at the second stage in the bolted joint installed with an eccentric double nut was very high, indicating more tightening difficulties for the upper nut. The FFT, SFT, and MFT for tightening of the two bolted joints at the two stages were calculated as listed in

Table 6. Torques of the double-thread bolt connection required under different preload distributions.

Preload Assigned to the Upper Nut	5 kN	10 kN	15 kN	20 kN	25 kN	30 kN
First stage (Nmm)	60,065	56,125	55,362	53,734	51,222	46,819
Second stage (Nmm)	11,801	21,708	31,560	40,139	49,516	58,543
Max torque (Nmm)	60,065	56,125	55,362	53,734	51,222	58,543

and

Table 7. Final torques of the eccentric double nut under different preload distributions.

Preload Assigned to the Upper Nut	5 kN	10 kN	15 kN	20 kN	25 kN	30 kN
First stage (Nmm)	89,090	88,270	81,928	80,340	75,498	74,497
Second stage (Nmm)	47,766	81,099	96,540	122,718	160,249	182,401
Max torque (Nmm)	89,090	88,270	96,540	122,718	160,249	182,401

, respectively. We can conclude that the change tendencies of FFT and SFT with increases in the preloads assigned to the upper nut were the same as those of regular bolted joints. Table 6 shows that for the double-thread bolt connection, the MFT was the largest when a preload of 5 kN was assigned to the upper nut, indicating the most difficult tightening. The MFT of 51,222 Nmm obtained when assigning a preload of 25 kN to the upper nut was the smallest. In contrast, with respect to the results of the eccentric double nut presented in Table 7, the largest MFT occurred when a preload of 30 kN was assigned to the upper nut, while the smallest value appeared when a preload of 10 kN was assigned to the upper nut. This demonstrates that for different bolted joints with double nuts, the tightening performance is greatly affected by the preload distribution applied to the upper and lower nuts.

6.2. Effect of Preload Distribution on Anti-Loosening Performance

In order to discuss the influence of preload distributions on the anti-loosening abilities of three bolted joints with double nuts, the preload distributions of the upper nuts were also changed (i.e., 5, 10, 15, 20, 25, and 30 kN) under the same total preload (30 kN), and transversal vibrations of movable plates were simulated via FEA. As shown in Figure 27, the preload decays of the bolted joints installed with regular double nuts vary under different preload distributions. When the preloads assigned to the upper nut are 5 and 10 kN, the total preloads decrease gradually under vibration conditions, indicating that the anti-loosening performance is lost. For the other preloads assigned to the upper nut, the total preloads drop slightly in the initial cycle subsequently remain unchanged with continued vibration. This demonstrates that these regular double nuts can achieve anti-loosening performance. This may be because the thread surface on the upper nut retains a stuck state and that interface slippage is prevented when the proportion of preload assigned to the upper nut is increased. Furthermore, it can be deduced that there is a critical preload assigned to the upper nut below which the anti-loosening ability is lost. The critical value ranges from 10 to 15 kN for a bolted joint installed with regular double nuts.

Figure 28 and Figure 29 show the changes in preload as vibration cycles are increased for a bolted joint with a double-thread bolt and an eccentric double nut, respectively. From Figure 28, we can observe that the decline trend of preload is evident when the preloads assigned to the upper nut are 5 kN and 10 kN, while the preloads are almost constant under vibration for the other preloads assigned to the upper nut. This finding is the same for the regular double nuts shown in Figure 27. This indicates that the large preload distribution of the upper nut can help to improve anti-loosening ability. Therefore, the lower nut should be tightened using a small torque, while the upper nut should be tightened using a large torque in order to achieve better anti-loosening performance.

Similarly, it can be said that the critical preload assigned to the upper nut is between 10 kN and 15 kN for double-thread bolt connection, below which the anti-loosening ability is lost. Figure 29 shows that only when the upper nut is assigned a value of 5 kN is the preload decreased continuously. Otherwise, the preloads are unchanged with an increase in vibration cycles. Therefore, when the preload assigned to the upper nut is equal to or less than 5 kN, the anti-loosening ability of the eccentric double is lost. The critical preload assigned to the upper nut ranges from 5 kN and 10 kN for an eccentric double nut. In summary, increasing the preload assigned to the upper nut results in better anti-loosening performance for different bolted joints with double nuts.

6.3. Effect of Preload Distribution on Fatigue Resistance Performance

In order to explore the influence of preload distribution on the fatigue resistance performance of three bolted joints with double nuts, the preload distributions assigned to the upper nut were changed. After the tightening simulations were completed, one-cycle vibration simulation under the same conditions was performed, and the von Mises stresses of root points were extracted. The obtained stress time series and stress range histograms of the three thread connection structures are shown in Figure 30, Figure 31 and Figure 32, respectively. From these figures, it can be found that the von Mises stresses of root points were larger when the preloads assigned to the upper nuts were smaller (5 kN). The minimum stresses at root points were almost the same for different preload distributions on the upper and lower nuts. From the figures of stress range histograms, it can be observed that the stress cycles extracted via the rainflow counting method are mostly low-amplitude, while only a few high-amplitude stress cycles are present.

The fatigue damage indexes accumulated within one vibration cycle were calculated, as listed in

Table 8. Fatigue damage indexes of three bolted joints installed with double nuts.

Preload Assigned to the Upper Nut	5 kN	10 kN	15 kN	20 kN	25 kN	30 kN
Regular double nuts	0.0812	0.0674	0.0676	0.0585	0.0370	0.0298
Double-thread bolt connection	0.0438	0.0378	0.0292	0.0248	0.0243	0.0099
Eccentric double nut	0.1041	0.0878	0.0752	0.0655	0.0651	0.0416

. The damage indexes change regularly as the preload assigned to the upper nut is increased. The larger the preload assigned to the upper nut, the smaller the damage index, indicating better fatigue resistance performance. Therefore, it is recommended to increase the proportion of preload assigned to the upper nut in engineering applications with the purpose of extending the fatigue life of thread connection structures.

7. Conclusions

In this study, three-dimensional FEMs of a regular bolted joint and seven anti-loosening bolted connections were established, and the processes of tightening, vibration-induced loosening, and fatigue were simulated via FEA. A MATLAB program based on a rainflow counting method was developed to perform cycle counting and accumulate the fatigue damage of bolted joints. According to the results of the simulation, the tightening, anti-loosening, and fatigue resistance characteristics of a typical bolted joint and anti-loosening bolted connections were evaluated comprehensively. The main conclusions are summarized as follows.

(1)

The results of the tightening simulation indicate that variable-diameter nuts and wedge-locking nuts are more difficult tighten than regular nuts and that the application of plain and wedge washers does not affect the tightening characteristics. The simulation results are the same as the experimental results reported in [18]. In terms of bolted joints with double nuts, it was found that the interference and eccentricity in an eccentric double nut make it more difficult to tighten than regular and double-thread bolted joints.

(2)

The results of vibration-induced loosening simulation verify that the researched bolted connections installed with different washers or nuts all possess anti-loosening capacities under transversal vibration compared to a regular bolted joint. Specifically, the anti-loosening performance of a wedge-locking nut is not as strong as that of other bolted connections. Additionally, a small drop in the preload of regular double nuts was observed after the initial vibration cycle. This phenomenon might be caused by the effect of stress redistribution.

(3)

The results of fatigue simulation demonstrate that the application of anti-loosening bolted connections improves fatigue resistance to varying degrees. Moreover, the fatigue resistance performances of bolted joints installed with a variable-diameter nut and wedge-locking nut are the best among all the anti-loosening bolted connections. It is worth mentioning that although the bolted joint installed with eccentric double nut showed superior anti-loosening ability, its fatigue resistance performance under transversal vibration was the worst among the three bolted joints with double nuts.

(4)

The effects of preload distribution on the two nuts on tightening, anti-loosening, and fatigue resistance performances were discussed. The results show that, for different bolted joints with double nuts, the tightening performance is greatly influenced by the preload distribution applied to the upper and lower nuts. We recommend increasing the proportion of preload assigned to the upper nut in order to achieve better anti-loosening performance and extend the fatigue life of bolted connections. The importance of reasonably distributing the preload of the two nuts should be emphasized in engineering applications.

Overall, the results obtained in this work can provide technical assistance and guidance for engineers in terms of selecting appropriate anti-loosening bolted connections in critical applications.