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Finite element modelling and experimental validation of bolt loosening due to thread wear under transverse cyclic loading
Engineering Failure Analysis 104 (2019) 341–353
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Finite element modelling and experimental validation of bolt loosening due to thread wear under transverse cyclic loading
T
Mingyuan Zhanga, Dongfang Zenga, , Liantao Lua, Yuanbin Zhangb, Jing Wangc, Jingmang Xud ⁎
a
State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China CRRC Qingdao Sifang Co., LTD., Qingdao 266111, China c College of Mechanical Engineering, Chengdu Technology University, Chengdu 611730, China d MOE Key Laboratory of High-speed Railway Engineering, Southwest Jiaotong University, Chengdu 610031, China b
ARTICLE INFO
ABSTRACT
Keywords: Finite element model Self-loosening Bolted joint Fretting wear
A finite element model, considered for the wear profile evolution of a threaded surface, is proposed to simulate the self-loosening of bolted joints under transverse loading. The method for wear profile simulation is based on an energy-based approach. To verify this finite element model, interrupted bolt loosening tests were carried out. The finite element model successfully simulated the phenomenon in which fretting wear causes a gradual reduction in clamping force by changing the distribution and magnitude of contact stress between the threads. The predicted results agreed with the experimental results with respect to the evolutionary trend of the wear profile and clamping force. It can be seen from the finite element analysis that the fretting wear causes a change in the distribution and magnitude of contact stress, and an increment in the sliding distance along the radial direction on the thread surface, which can further change the distribution of the wear depth. Because the accumulation of wear debris is not considered in the finite element model, the predicted wear depth and reduction of clamping force is larger than that obtained from the experimental results.
1. Introduction Bolted joints are widely used in mechanical components because of their advantages of simple structure and ease of assembly and disassembly. However, due to the complexities of the working environment, bolted joints often experience self-loosening (gradual loss of clamping force) with increased service time, which causes a decrease in structural stiffness, and in some cases, may even lead to fatal consequences if undetected. Great efforts have been made to investigate the self-loosening phenomenon. It was found that transverse loading (perpendicular to the bolt axis) is the most severe form of loading for self-loosening, and that the loosening is attributed to the reduction in circumferential holding friction as a result of slip at the fastener surfaces caused by the applied load [1–6]. Recently, Liu [7,8], Zhang [9,10], and Zhou [11] investigated the loosening mechanism through loosening tests. The test results showed that fretting wear can be caused by small-amplitude oscillatory slip between threads of the bolt and the nut, which leads to a slow self-loosening of the bolted joint. Many studies have been done to simulate the loosening process using finite element (FE) analysis. Some early research [12–14] used an axisymmetric simplification of the thread to obtain the stress distribution in bolted joints. Such models cannot be used to model loosening because the helical geometry of the threads cannot be represented. Subsequently, three-dimensional FE models
⁎
Corresponding author. E-mail address: [email protected] (D. Zeng).
https://doi.org/10.1016/j.engfailanal.2019.05.001 Received 4 January 2019; Received in revised form 21 March 2019; Accepted 1 May 2019 Available online 02 May 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.
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containing the thread helix were developed [15–21]. Parameters related to loosening, such as contact stress and sliding distance, were calculated based on the FE models, while these parameters cannot be directly measured from loosening tests. Hess [15] introduced the term ‘localised slip’ and divided the loosening process into four different types: (1) localised head slip with localised thread slip; (2) localised head slip with complete thread slip; (3) complete head slip with localised thread slip; (4) complete head slip with complete thread slip. Izumi [18,19] pointed out that loosening is initiated when complete thread slip is achieved prior to bolt head slip. Liu [21] investigated the effect of ramp angle on the loosening of wedge self-locking nuts using FE modelling and obtained the optimised ramp angle to improve the anti-loosening ability. Although fretting wear has been proven to cause self-loosening of bolted joints, the evolution of the thread surface profile due to fretting wear has not been considered in these FE models. McColl [22] proposed an FE method to simulate the evolution of fretting wear with the number of wear cycles in a cylinder with a flat configuration based on a modified Archard equation. Following this work, Madge [23] proposed a more accurate and efficient method to analyse both fretting wear and fretting fatigue using the user subroutine UMESHMOTION in ABAQUS. Fouvry [24] developed an energybased approach with higher stability. Experimental results have shown that this approach is better than the Archard-based model [25,26]. This methodology has been used to predict the fretting wear occurring in some configurations, such as spine coupling [27], thin steels [28], and modular hip implants [29]. In previous studies, this methodology was used to predict the fretting wear profile of a press-fitted shaft by the authors [30,31]. However, this methodology has not been applied for bolted joints so far. In this study, interrupted tests were first conducted to obtain the wear profile evolution and self-loosening curves of bolted joints under transverse loading. Then, an FE model with consideration for thread wear was developed to simulate the self-loosening of bolted joints under transverse loading. The FE model was validated by comparing experimental results and FE results. Based on the model, the roles of thread wear on the self-loosening of bolted joints were analysed. 2. Self-loosening experiments 2.1. Specimens and materials An M12, 1.75 × 100 mm hexagonal bolt and nut were used in the experiment. To ensure the quality and uniformity of test specimens, the bolt and the nut were made from SCM435 steel through milling machining according to standard processing technology. The manufacturing process included rod cutting of steel wire rods, head making, trimming, and thread milling. The mechanical properties of the material are listed in Table 1. 2.2. Experimental method A diagram of the setup for the self-loosening experiment is shown in Fig. 1. Two plates and a compression load cell (resolution 100 N) were clamped with a bolt and a nut. The nut was slowly tightened using a digital display torque wrench to the set preload. The thickness of the upper plate and the bottom plate were 15 mm and 40 mm, respectively. Both of the plates were made of AISI 1045 carbon steel. The upper and bottom plates were connected to the upper and lower grips of a Shimadzu servo-hydraulic fatigue testing machine. A load cell with a thickness of 16.3 mm was used for the measurement of the clamping force. A high-precision rotation sensor (resolution 0.045°) was attached to the bolted joint with the implementation of a rotation sensor support to measure the relative rotation between the nut and the bolt. An extensometer (resolution 0.001 mm) was attached to the two plates to measure the relative displacement between the two plates. During each test, the upper plate was driven to move, and the bottom plate was fixed. To avoid the wear on contact surfaces between the two plates, 3 mm diameter rollers were placed in two 25 × 10 × 1.4 mm slots on the contact surface of each plate. Furthermore, solid grease was applied to the contact surfaces. With this study, we hope to obtain a more distinct thread wear profile from the loosening test to facilitate the comparison between the experimental and FE results. Because a lower clamping force can lead to more severe thread wear [7–10], 10 kN was chosen as the preload, which produces a tensile stress of 95 MPa, corresponding to 13.5% of the material yield stress. Similar test methods were also adopted by other researchers, such as Pai [2], Bhattacharya [32], Saha [33], and Panja [34], who conducted loosening tests at a lower preload to investigate the loosening mechanism or anti-loosening ability of the bolted joint. Fatigue fracture frequently happened when the loosening tests were performed at a displacement amplitude larger than 0.2 mm, while a displacement amplitude lower than 0.2 mm would not produce distinct wear on the thread surface. Therefore, 0.2 mm was chosen as the displacement amplitude in this study. Bolt loosening is independent of the loading frequency [1]. To speed up the test, the loading frequency was set as high as possible. Therefore, the limiting frequency of the tester 10 Hz was chosen in this study. Considering that the self-loosening rate gradually decreases with the increase in loading cycles, the interrupted interval was gradually enlarged. The interrupted tests were terminated at 103, 104, 105, and 2 × 105 cycles, respectively. Three repeated tests were carried out for each test condition to obtain reliable results. The coefficient for the thread friction and under-head friction can be obtained by Eq. (1) [35]. Table 1 Mechanical properties of test material. Young's modulus (GPa)
Yield strength (MPa)
Tensile strength (MPa)
Elongation (%)
208
705
989
4.8
342
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Transverse Load Upper plate Load cell Rotation sensor support Rotation sensor Extensometer
Bolt Nut Bottom plate
Roller Fixed
Fig. 1. Setup for the self-loosening experiment.
(1)
T = P0 × (0.159p + 0.578µt d2 + 0.5De µh )
where T is the tightening torque, P0 is the bolt preload, p is the pitch of the thread (1.75 mm), d2 is the basic pitch diameter of the thread (10.86 mm), De is the effective diameter of the bolt head (15.75 mm), μt is the coefficient of thread friction, μh is the coefficient of under-head friction. Generally, it can be assumed that the two friction coefficients are equal [36,37]. Therefore, the friction coefficient can be calculated according to the measured tightening torque and the preload. 2.3. Measurements of wear profile The greatest amount of load is carried by the first engaged thread [38,39], which leads to the most severe damage occurring on this thread surface. Thus, the wear profile on the surface of the first thread was observed in this study. Due to the helical thread structure involved in the bolted joint, the contact stress and sliding distance vary along the circumferential direction on the threads. This may cause a difference in the degree of wear along the thread. Therefore, wear profiles on the two positions shown in Fig. 2 (Position A, θ = 0° and Position B, θ = 90°) were analysed. After the test, the bolt and the nut were cut by a wire electrical discharge machine and cleaned in an ultrasonic bath. Then, the fretting wear profile on the threads of the bolt and the nut were observed and measured using a confocal laser scanning microscope (Olympus OLS4100). Fig. 3(a) shows the cut parts of the bolt and the nut after a total loading of 2 × 105 cycles. Fig. 3(b) displays the optical microscopy observation of the contact surface at Position A of the bolt. Wear is evident in the contact region. The wear profile was measured from thread crest to root, as shown in Fig. 3(c). The flat part shown in Fig. 3(c) corresponds to the regions near the thread root, where the wear does not occur. Therefore, it can be used as the measuring datum to determine the wear depth from the wear profile. The average value of the wear depth from the three repeated specimens was then calculated to obtain the final wear profile. 2.4. Experimental results Table 2 shows the results of the self-loosening tests. The torque–preload relationship for a bolted joint is highly sensitive to friction variation in the threads and the under-head bearing [7], which causes scatter of the tightening torque for each test condition.
Position B y Transverse Load
Position A x
Thread Fig. 2. Observed positions on the first engaged thread. 343
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(a)
Nut (b) Thread root
Thread crest
Outer edge
Position A 3 mm
Contact region
inner edge
Bolt (c)
Fig. 3. Process of observing and measuring the wear profile on thread surface: (a) cut parts of the bolt and the nut, (b) optical microscopy observation of the contact surface and (c) measurement of wear profile.
The friction coefficient was calculated using Eq. (1) based on the measured tightening torque and preload. A higher value for the ratio of the clamping force to the preload (P/P0) indicates superior anti-loosening ability. In addition, the transverse load did not change greatly during the test and remained at a relatively low value of around 0.55 kN, suggesting that no distinct wear was produced on the plates during the test. The measured rotation angle was zero for each test, indicating that the self-loosening was not caused by the relative rotation between the nut and the bolt. 2.4.1. Loosening curve Fig. 4 shows the self-loosening curves within the given loading cycles. These self-loosening curves show a similar tendency in the variation of clamping force and loading cycles. For each test, the first 200 loading cycles were used to apply the transverse amplitude to the preset value. After the first 200 loading cycles, the self-loosening curves can be divided into two stages: a rapid loosening process from 200 to 500 loading cycles (Stage I), and a gradual loosening process after 500 loading cycles (Stage II). As previously reported [9,10], the rapid loosening is caused by the cyclic plastic deformation that occurs near the roots of the threads, and the fretting wear that occurs on the threads leads to the subsequent gradual loosening. After 2 × 105 loading cycles, about 12% of the initial preload is lost, in which Stage I and Stage II occupy about 6%, respectively. 2.4.2. Wear profile Fig. 5 shows the evolution of surface damage at Position A and B of the bolt. No distinct damage except the ploughing caused by the tightening process could be observed on the thread after 103 loading cycles. The ploughing gradually disappears, and the surface damage Table 2 Results of self-loosening tests. Test number 1 2 3 4 5 6 7 8 9 10 11 12 a
Loading cycles 3
10 103 103 104 104 104 105 105 105 2 × 105 2 × 105 2 × 105
Tightening torque (Nm)
Friction coefficienta
P/P0 (%)
Mean
Standard deviation
36.3 37.1 35.1 38.4 36.6 38.2 37.8 36.6 35.9 36.6 37.0 37.5
0.236 0.242 0.228 0.251 0.238 0.250 0.247 0.238 0.234 0.239 0.241 0.245
93.43 93.46 92.68 91.35 91.02 91.66 89.43 89.05 88.92 88.61 88.42 88.00
93.22
0.4718
91.37
0.3604
89.10
0.2650
88.32
0.3121
The coefficient of thread friction and underhead friction is assumed to be equal. 344
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Fig. 4. Self-loosening curves within given loading cycles.
becomes more and more severe with increased loading cycles. Fig. 6 shows the evolution of wear depth at Position A and B of the bolt and the nut. It can be seen that the wear depth gradually increases with the increase in loading cycles. The fretting wear along the radial direction of the thread surface is nonuniform. For any given number of loading cycles, the wear depth reaches a maximum at the outer edge of the contact region for the bolt and the inner side of the contact inner edge for the nut. In addition, the wear depth at Position A is larger than that at Position B, which indicates that the wear damage along the circumferential direction of the thread is not identical. 3. FE modelling 3.1. FE model According to the structure and dimensions of the designed testing setup shown in Fig. 1, an FE model was developed using HYPERMESH software, as shown in Fig. 7. The model contained two clamped plates, a load cell, a bolt, and a nut. The clamped length
Fig. 5. Evolution of surface damage on Position A and B of the bolt. 345
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Fig. 6. Evolution of measured wear depth on Position A of (a) the bolt and (b) the nut, and Position B of (c) the bolt and (d) the nut.
was 73 mm. The bolt and the nut respectively had 5 and 4 pitches of thread, and 4 pitches were engaged. The mesh generation for the thread referred to the method proposed by Fukuoka [40], which provided accurate helical thread geometry and continuous mesh to obtain an accurate contact stress and sliding distance during the simulation. The total structure was discretised using C3D8 eightnode elements. There were 432,529 nodes and 390,624 elements in the FE model. The material of the bolt and nut was modelled as an elastic–plastic kinematic hardening material. The properties of the material are listed in Table 1, in which the yield stress is 705 MPa and the hardening modulus is 0.45 MPa. Other components were assumed to be elastic. The Young's modulus and Poisson's ratio are 210 GPa and 0.3, respectively. In the FE model, 5 contact pairs were defined: the contact between the threads (Contact 1), the contact between the nut and the load cell (Contact 2), the contact between the load cell and the bottom plate (Contact 3), the contact between the upper and the bottom plates (Contact 4), and the contact between the upper plate and the bolt head (Contact 5). Finite sliding was applied to Contact 1 and Contact 4 because of the occurrence of large sliding between threads and between plates. To improve computation efficiency, a small sliding value was used for the other contact pairs. For Contact 1, the exponential softened pressure–overclosure relationship and the penalty method with friction were respectively used to model the normal and tangential behaviour. For other contacts, the penalty method with hard contact and friction were used, respectively, in the normal and tangential direction. The friction coefficient between threads and between the upper plate and the bolt head was set to 0.241, which is the mean value of the friction coefficients listed in Table 2. The friction coefficient between plates was set to 0.0005 because the rollers and solid grease were applied between the plates to reduce friction during the test. The friction coefficient was
Fig. 7. FE model. 346
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taken to be 0.2 for other contact surfaces. The preload was applied via thermal loading. The thermal expansion coefficient of the load cell was set to 1.5 × 10−5/°C in the z direction, and that of other components was zero. After the application of temperature, the load cell only elongated in the z direction, but other components did not elongate. A thermal load was applied in Step 1 to achieve a preload. To avoid the generation of large displacement which would cause a convergence problem, all external nodes of the nut were constrained in all directions in this step. In the following steps, the constraint on the external nodes of the nut was removed. To simulate the transverse load, all nodes on the end surface of the bottom plate were constrained in all directions, and a transverse displacement that varied with computational time t, S=S0sin(2πt), was applied to the right end of the upper plate in the x direction. The FE analysis of loosening was performed as a quasi-static process based on the previous results that loosening is independent of the loading frequency [1]. Fig. 4 shows that the reduction of clamping force during Stage I (before 500 loading cycles) was not caused by thread wear. Therefore, this simulation directly takes the clamping force at 500 loading cycles as the initial preload. In this study, a preload of 9400 N, a transverse displacement amplitude of 0.2 mm, and a total loading cycles of 200,000 were used. 3.2. Wear model The energy-based wear approach was used in the FE simulation to simulate fretting wear occurring at thread surface. The global energy wear approach is normally expressed as: N
V=
N
Ei = i=1
Qi dSi
(2)
i=1
where V is the total wear volume, α is the energy wear coefficient, and Ei, Qi and Si are the dissipated energy, tangential force and sliding distance at the ith cycle, respectively. For each cycle, the local wear volume can be described as: (3)
dV (x ) = dh (x ) A (x )
where dh(x) is the local wear depth, A(x) is the infinitesimally small contact area, and x is the horizontal contact position. Then, the local dissipated energy is as follows:
E (x ) =
q (x ) ds (x )
(4)
where q(x) is the local shear stress, and ds(x) is the increment in sliding distance. Therefore, the following equation can be obtained from Eq. (2):
dh (x ) A (x ) =
q (x ) A (x ) ds (x )
(5)
Then, diving both sides by A(x), the following equation is obtained:
dh (x ) =
q (x ) ds (x ) = E (x )
(6)
The wear model described above was implemented into ABAQUS software through the adaptive meshing UMESHMOTION user
Generate FE model ABAQUS Calculate shear stress q(x) and sliding distance s(x) for one increment
Calculate nodal wear depth n h( x ) N q ( x, t ) s ( x, t ) t 1
N N
Yes Yes
Update nodal coordinates and revise FE model
No No
UMESHMOTION
Stop
Fig. 8. Flow chart of the numerical procedure to model thread wear. 347
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Fig. 9. The variation of (a) contact stress and (b) wear depth with the selected mesh size.
subroutine [41]. This subroutine has been used by many researchers to implement wear simulation [23–26]. Fig. 8 shows the flowchart of the numerical procedure to model thread wear. One loading cycle is discretised into n increments. The total wear depth at node x during a full cycle of loading is: n
hstep (x ) =
q (x , t ) s (x , t )
(7)
t=1
where q(x,t) and Δs(x,t) are the shear stress and incremental sliding distance, respectively, at node x over an increment. The cycle jump technique was used to reduce the computational time. This technique was proposed by different authors like McColl [22] and Mary [27], where it is assumed that the contact stress and sliding distance remains constant in the ΔN loading cycles that follow. Therefore, the wear depth Δh(x) for ΔN loading cycles can be calculated as: n
h (x ) = N
q (x , t ) (x , t )
(8)
t=1
The wear coefficient α is a material parameter, which can be measured from conventional fretting tests with a round-flat or ballflat configuration [27,29]. In this study, the energy wear coefficient was set as α = 4.17 × 10−8 based on the fretting wear tests carried out for SCM435 steel using a ball-flat configuration [42]. 3.3. Optimisation of the parameters The mesh size on the thread surface, increments per loading cycle n, and cycle jumping-ΔN are important modelling parameters related to computational accuracy and speed. Optimisation of these parameters was conducted in this section. The mesh size on the thread surface was optimised first. For this analysis, the increments per loading cycle n, cycle jumping-ΔN, and total loading cycles were set to 40, 4000, and 20,000, respectively. Fig. 9(a) shows the variation in contact stress with the selected mesh size without transverse loading. It can be seen that the contact stress increases with decreasing mesh size until the mesh size is less than 73 μm. Fig. 9(b) shows the variation in wear depth with the selected mesh size after 20,000 loading cycles. It can be seen that the predicted wear depth increases with decreasing mesh size. However, when the mesh size is less than 73 μm, the wear depth remains unchanged. It should be pointed out that, for this method to generate thread mesh [40], the total number of nodes and elements is significantly increased when reducing the mesh size. When the mesh size is reduced from 73 μm to 61 μm, over 110,000 nodes and 120,000 elements are generated, which causes an almost four-fold increase in computational time. Therefore, to balance computational accuracy and speed, the mesh size on the thread surface was set to 73 μm.
Fig. 10. The variation of wear depth with the selected (a) increments per loading cycle n and (b) cycle jumping-ΔN. 348
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Fig. 11. The variation of slip range with the selected maximum elastic slip tolerance.
Then the increments per loading cycle n were optimised. For this analysis, the mesh size on the thread surface, cycle jumping-ΔN, and total loading cycles were set to 73 μm, 4000, and 20,000, respectively. As seen from Fig. 10(a), the wear depth slightly increases with an increase in the increments. As the increments are increased to 50, the predicted wear depth is almost unchanged. Because small increments easily cause convergence problems, the increment was chosen as 50 in this study. For the analysis of the cycle jumping-ΔN, the mesh size on the thread surface, increments per loading cycle n, and total loading cycles were set to 73 μm, 50, and 20,000, respectively. Fig. 10(b) shows the variation in wear depth with the selected cycle jumping-ΔN. When the cycle jump is larger than 5000, the wear depth rapidly increases with an increase in cycle jump. But when the cycle jump is less than 5000, the wear profiles remain almost unchanged. Because a smaller cycle jump increases the total computational time by increasing the total calculation steps, it was optimised to be 5000 in this study. Because the convergence problem occurred during the simulation when a Lagrange multiplier contact algorithm was used, the penalty approach was used to model the sticking and slipping behaviour in this study. It was found that the penalty approach could offer satisfactory results if the contact parameters were well-selected [43]. Fig. 11 shows the variation in slip range with the selected maximum elastic slip tolerance. It can be seen that the distribution of slip range remains unchanged with a maximum elastic slip tolerance less than 0.0004. Thus, the maximum elastic slip tolerance was taken to be 0.0004. In this section, the data shown in Figs. 9–11 were obtained from the results of Position A for the bolt.
Fig. 12. Evolution of predicted wear depth on Position A of (a) the bolt and (b) the nut, and Position B of (c) the bolt and (d) the nut. 349
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Fig. 13. Comparison of the predicted and experimental wear profile on Position A after (a) 104 loading cycles and (b) 2 × 105 loading cycles, and Position B after (c) 104 loading cycles and (d) 2 × 105 loading cycles.
4. Results and discussion 4.1. Wear profile The FE simulation for bolt loosening was conducted using the optimised parameters stated above. Fig. 12 illustrates the evolution of the predicted wear depth at Position A and B of the bolt and the nut. It can be seen that the predicted wear depth at Position A and B gradually increases with increased loading cycles. Fig. 12a and c show that the predicted wear depth at Position A and B of the bolt reaches a maximum at the contact outer edge for each loading cycle, and it decreases away from the contact outer edge as the number of loading cycles reaches 103 and 104. As the loading cycles are further increased, a change in the distribution of wear depth take place, namely, the wear depth first decreases and then increases from the outer edge to the inner edge. The nut shows a similar trend in the change of wear depth distribution (Fig. 12b and d), but it should be noted that the maximum wear depth occurs at the contact inner edge of the nut. In addition, the predicted wear depth at Position A is larger than that at Position B. The wear profiles at 104 and 2 × 105 loading cycles were used as an example to validate the FE results. Fig. 13 displays a comparison of the predicted and experimental wear profile at Position A and B, respectively. It can be seen that the wear profile predicted by the FE model describes a contour similar to the experimental profile. The predicted wear width is in good agreement with the experimental result, while the predicted wear depth is larger than the experimental result. This may be caused by the wear debris piling up on the thread surface. The wear debris covers the contact area, which separate the threads of the bolt and the nut and reduces the wear. Because this model cannot consider the accumulation of wear debris, a larger wear depth is predicted in the FE simulation. In addition, a more distinct difference in wear depth between the FE and experiment results occurs around the contact edges, because the wear debris more easily accumulates around the contact edges under repeated slip motion [44]. 4.2. Contact variables The distribution of contact variables of the bolt at different loading cycles was obtained from the FE result, as shown in Fig. 14. The corresponding cloud diagrams of contact stress are shown in Fig. 15. For the unworn surface (0 cycles), the contact stress gradually decreases from the outer edge to the inner edge of the contact region, while the sliding distance increases from the outer edge to the inner edge. The sliding distance at different locations increases with the increase in loading cycles, as shown in Fig. 14b and d. However, Figs. 14a, c, and 15 show that the contact stress on the side near the outer edge gradually decreases when increasing the loading cycles, while the contact stress on the side near the inner edge gradually increases. In the initial stage, the distribution of wear depth for the bolt, as shown in Fig. 12a and c, is attributed to the difference in contact stress at varied locations. The initial large loss of surface material near the outer edge relieves the stress concentration in this region, 350
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Fig. 14. Evolution of contact variables distribution: (a) contact stress and (b) sliding distance of the bolt on Position A, (c) contact stress and (d) sliding distance of the bolt on Position B.
Fig. 15. Cloud diagrams of contact stress for different stages.
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Fig. 16. Comparison of the predicted and experimental self-loosening curves.
and thereby reduces the wear rate. Meanwhile, because of the low initial wear rate, less loss of surface material occurs on the side near the inner edge in the initial stage. This leads to an increase in contact stress in this region and accelerates the fretting wear. Therefore, as the number of loading cycles is increased, a change in the distribution of wear depth takes place: the wear depth first decreases and then increases from the outer edge to the inner edge for loading cycles from 105 to 2 × 105. In addition, both the contact stress and sliding distance at Position A are larger than that at Position B, thus a larger wear depth is produced at Position A. Because the same material is used for the bolt and the nut, and surface-to-surface contact elements are used to model the contact between the threads, the contact stress, sliding distance and wear depth at the alignment locations for the thread surface of the bolt and the nut are identical. Therefore, the relationship between contact variables and wear depth for the nut is not described in this paper. 4.3. Loosening curve Fig. 16 shows the self-loosening curve obtained from the FE simulation together with the experimental result. It should be pointed out that this experimental curve only contains the portion of Stage II in Fig. 4, where the self-loosening is caused by fretting wear. The two curves show the same trend in the variation of clamping force with loading cycle. It can be seen from Fig. 12 that the wear depth increases with increasing loading cycles; this reduces the contact stress between the threads (Fig. 14), and thereby causes a gradual reduction in clamping force (Fig. 16). However, the reduction in clamping force for the FE simulation is larger than for the experimental result. This may be related to the piling up of wear debris on the thread surface. Because the wear debris partly fills up the gap generated by the fretting wear, it would alleviate the decrease in clamping force. This simulation work improves our understanding of loosening behaviour due to thread wear. It further confirms that the fretting wear that occurs during thread wear causes a slow self-loosening. We developed an FE model that considered the evolution of thread wear. In addition to analysing the experiment results, this model can also be used to optimise the structural parameters involved in the bolted joint, such as thread pith, thread fit, and hole clearance. In particular, it is difficult to optimise the structural parameters through experimental methods for large bolts. This model may play a more important role in the optimisation design for large bolts. 5. Conclusions In this study, an FE model was established to simulate the loosening behaviour of a bolted joint caused by thread wear under transverse loading. The simulation results were compared with the experimental results. The following conclusions can be drawn from the simulation and experimental results: 1. To obtain an appropriate balance between computational accuracy and speed, the mesh size on the thread surface, the increments per loading cycle n, the cycle jumping-ΔN, and the maximum elastic slip tolerance in the FE model were optimised to be 73 μm, 50, 5000, and 0.0004, respectively. 2. The FE model successfully simulated the phenomenon in which fretting wear causes a gradual reduction in clamping force by changing the distribution and magnitude of contact stress between the threads. 3. The predicted results agree with the experimental results with respect to the evolutionary trend of wear profile and clamping force. Because the accumulation of wear debris is not considered in the FE model, the predicted wear depth and reduction in clamping force is larger than in the experimental results. 4. The fretting wear causes a change in the distribution and magnitude of contact stress, and an increment in sliding distance along the radial direction on the thread surface, which can further change the distribution of wear depth in this direction. Acknowledgement This work was supported by the Independent Research Projects of State Key Laboratory of Traction Power (2015TPL_T16 and 352
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Maruyama, Stress analysis of a bolt-nut joint by the finite element method and the copper-electroplating method: 3rd Report, influence of pitch error or flank angle error, Transact. Jpn. Soc. Mech. Eng. 41 (348) (1975) 2292–2302. [14] A. Tafreshi, W. Dover, Stress analysis of drillstring threaded connections using the finite element method, Int. J. Fatigue 15 (5) (1993) 429–438. [15] N. Pai, D. Hess, Three-dimensional finite element analysis of threaded fastener loosening due to dynamic shear load, Eng. Fail. Anal. 9 (4) (2002) 383–402. [16] M. Zhang, Y. Jiang, C. Lee, Finite element modeling of self-loosening of bolted joints. Analysis of bolted joints, J. Mech. Design 129 (2) (2007) 218–226. [17] G. Dinger, C. Friedrich, Avoiding self-loosening failure of bolted joints with numerical assessment of local contact state, Eng. Fail. Anal. 18 (8) (2011) 2188–2200. [18] S. Izumi, T. Yokoyama, A. Iwasaki, et al., Three-dimensional finite element analysis of tightening and loosening mechanism of threaded fastener, Eng. Fail. Anal. 12 (4) (2005) 604–615. [19] S. Izumi, T. Yokoyama, M. Kimura, et al., Loosening-resistance evaluation of double-nut tightening method and spring washer by three-dimensional finite element analysis, Eng. Fail. Anal. 16 (5) (2009) 1510–1519. [20] Y. Chen, Q. Gao, Z. Guan, Self-loosening failure analysis of bolt joints under vibration considering the tightening process, Shock. Vib. (2017) 1–15. [21] J. Liu, H. Gong, X. Ding, Effect of ramp angle on the anti-loosening ability of wedge self-locking nuts under vibration, J. Mech. Des. 140 (7) (2018) 072301. [22] I. Mccoll, J. Ding, S. Leen, Finite element simulation and experimental validation of fretting wear, Wear 256 (11) (2004) 1114–1127. [23] J. Madge, S. Leen, I. Mccoll, et al., Contact-evolution based prediction of fretting fatigue life: Effect of slip amplitude, Wear 262 (9) (2007) 1159–1170. [24] S. Fouvry, T. Liskiewicz, P. Kapsa, et al., An energy description of wear mechanisms and its applications to oscillating sliding contacts, Wear 255 (2003) 287–298. [25] S. Fouvry, P. Duo, P. Perruchaut, A quantitative approach of Ti–6A1-4VFretting damage: friction, wear and crack nucleation, Wear 257 (2004) 916–929. [26] K. Elleuch, S. Fouvry, Experimental and modeling aspects of abrasive wear of a A357 aluminum alloy under gross slip fretting conditions, Wear 258 (2005) 40–49. [27] J. Ding, S. Leen, E. Williams, et al., Finite element simulation of fretting wear-fatigue interaction in spline couplings, Tribol. Mater. Surf. Interf. 2 (1) (2008) 10–24. [28] A. Cruzado, S. Leen, M. Urchegui, et al., Finite element simulation of fretting wear and fatigue in thin steel wires, Int. J. Fatigue 55 (5) (2013) 7–21. [29] T. Zhang, N. Harrison, P. Mcdonnell, et al., A finite element methodology for wear–fatigue analysis for modular hip implants, Tribol. Int. 65 (3) (2013) 113–127. [30] Y. Zhang, L. Lu, L. Zou, et al., Finite element simulation of the influence of fretting wear on fretting crack initiation in press-fitted shaft under rotating bending, Wear 400 (2018) 177–183. [31] D. Zeng, Y. Zhang, L. Lu, et al., Fretting wear and fatigue in press-fitted railway axle: A simulation study of the influence of stress relief groove, Int. J. Fatigue 119 (2019) 225–236. [32] A. Bhattacharya, A. Sen, S. Das, An investigation on the anti-loosening characteristics of threaded fasteners under vibratory conditions, Mech. Mach. Theory 45 (2010) 1215–1225. [33] S. Saha, S. Srimani, S. Hajra, et al., On the anti-loosening property of different fasteners, 13th National Conference on Mechanisms and Machines, 2007, pp. 229–232. [34] B. Panja, S. Das, A study of anti-loosening ability of 5/8 BSW fasteners under vibration with high tension steel and stainless steel bolts, International & National Conference on Machines & Mechanisms, 2013, pp. 873–878. [35] DIN 946, Determination of Coefficient of Friction of Bolt–Nut Assemblies under Specified Conditions, German Institute for Standardisation, German, 1991. [36] D. Croccolo, M. Agostinis, N. Vincenzi, Failure analysis of bolted joints: Effect of friction coefficients in torque–preloading relationship, Eng. Fail. Anal. 18 (1) (2011) 364–373. [37] D. Croccolo, M. Agostinis, N. Vincenzi, Influence of tightening procedures and lubrication conditions on titanium screw joints for lightweight applications, Tribol. Int. 55 (2) (2012) 68–76. [38] Z. Guan, J. Mu, F. Su, et al., Pull-through mechanical behavior of composite fastener threads, Applied Composite Materials 3 (22) (2015) 251–267. [39] Z. Wang, B.Xu.Y. Jiang, A reliable fatigue prediction model for bolts under cyclic axial loading, Proceedings of the 5th ISSAT International Conference on Reliability Quality in Design, 1999, pp. 137–141. [40] T. Fukuoka, M. Nomura, Proposition of helical thread modeling with accurate geometry and finite element analysis, J. Press. Vessel. Technol. 130 (1) (2008) 135–140. [41] Abaqus User Manual: Abaqus Theory Guide, Version 6.14, Dassault Systemes Simulia Corp, 2014. [42] P. Wang, Y. Liang, H. Song, Y. Yi, Fretting wear behavior of 35CrMo steel under gross slip condition, Heat Treat. Metals 39 (3) (2014) 141–144. [43] R. Rajasekaran, D. Nowell, On the finite element analysis of contacting bodies using submodelling, J. Strain Anal. Eng. Des. 40 (2) (2005) 95–106. [44] J. Zheng, J. Luo, J. Mo, et al., Fretting wear behaviors of a railway axle steel, Tribol. Int. 43 (5) (2010) 906–911.
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Finite element modelling and experimental validation of bolt loosening due to thread wear under transverse cyclic loading
T
Mingyuan Zhanga, Dongfang Zenga, , Liantao Lua, Yuanbin Zhangb, Jing Wangc, Jingmang Xud ⁎
a
State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China CRRC Qingdao Sifang Co., LTD., Qingdao 266111, China c College of Mechanical Engineering, Chengdu Technology University, Chengdu 611730, China d MOE Key Laboratory of High-speed Railway Engineering, Southwest Jiaotong University, Chengdu 610031, China b
ARTICLE INFO
ABSTRACT
Keywords: Finite element model Self-loosening Bolted joint Fretting wear
A finite element model, considered for the wear profile evolution of a threaded surface, is proposed to simulate the self-loosening of bolted joints under transverse loading. The method for wear profile simulation is based on an energy-based approach. To verify this finite element model, interrupted bolt loosening tests were carried out. The finite element model successfully simulated the phenomenon in which fretting wear causes a gradual reduction in clamping force by changing the distribution and magnitude of contact stress between the threads. The predicted results agreed with the experimental results with respect to the evolutionary trend of the wear profile and clamping force. It can be seen from the finite element analysis that the fretting wear causes a change in the distribution and magnitude of contact stress, and an increment in the sliding distance along the radial direction on the thread surface, which can further change the distribution of the wear depth. Because the accumulation of wear debris is not considered in the finite element model, the predicted wear depth and reduction of clamping force is larger than that obtained from the experimental results.
1. Introduction Bolted joints are widely used in mechanical components because of their advantages of simple structure and ease of assembly and disassembly. However, due to the complexities of the working environment, bolted joints often experience self-loosening (gradual loss of clamping force) with increased service time, which causes a decrease in structural stiffness, and in some cases, may even lead to fatal consequences if undetected. Great efforts have been made to investigate the self-loosening phenomenon. It was found that transverse loading (perpendicular to the bolt axis) is the most severe form of loading for self-loosening, and that the loosening is attributed to the reduction in circumferential holding friction as a result of slip at the fastener surfaces caused by the applied load [1–6]. Recently, Liu [7,8], Zhang [9,10], and Zhou [11] investigated the loosening mechanism through loosening tests. The test results showed that fretting wear can be caused by small-amplitude oscillatory slip between threads of the bolt and the nut, which leads to a slow self-loosening of the bolted joint. Many studies have been done to simulate the loosening process using finite element (FE) analysis. Some early research [12–14] used an axisymmetric simplification of the thread to obtain the stress distribution in bolted joints. Such models cannot be used to model loosening because the helical geometry of the threads cannot be represented. Subsequently, three-dimensional FE models
⁎
Corresponding author. E-mail address: [email protected] (D. Zeng).
https://doi.org/10.1016/j.engfailanal.2019.05.001 Received 4 January 2019; Received in revised form 21 March 2019; Accepted 1 May 2019 Available online 02 May 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.
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containing the thread helix were developed [15–21]. Parameters related to loosening, such as contact stress and sliding distance, were calculated based on the FE models, while these parameters cannot be directly measured from loosening tests. Hess [15] introduced the term ‘localised slip’ and divided the loosening process into four different types: (1) localised head slip with localised thread slip; (2) localised head slip with complete thread slip; (3) complete head slip with localised thread slip; (4) complete head slip with complete thread slip. Izumi [18,19] pointed out that loosening is initiated when complete thread slip is achieved prior to bolt head slip. Liu [21] investigated the effect of ramp angle on the loosening of wedge self-locking nuts using FE modelling and obtained the optimised ramp angle to improve the anti-loosening ability. Although fretting wear has been proven to cause self-loosening of bolted joints, the evolution of the thread surface profile due to fretting wear has not been considered in these FE models. McColl [22] proposed an FE method to simulate the evolution of fretting wear with the number of wear cycles in a cylinder with a flat configuration based on a modified Archard equation. Following this work, Madge [23] proposed a more accurate and efficient method to analyse both fretting wear and fretting fatigue using the user subroutine UMESHMOTION in ABAQUS. Fouvry [24] developed an energybased approach with higher stability. Experimental results have shown that this approach is better than the Archard-based model [25,26]. This methodology has been used to predict the fretting wear occurring in some configurations, such as spine coupling [27], thin steels [28], and modular hip implants [29]. In previous studies, this methodology was used to predict the fretting wear profile of a press-fitted shaft by the authors [30,31]. However, this methodology has not been applied for bolted joints so far. In this study, interrupted tests were first conducted to obtain the wear profile evolution and self-loosening curves of bolted joints under transverse loading. Then, an FE model with consideration for thread wear was developed to simulate the self-loosening of bolted joints under transverse loading. The FE model was validated by comparing experimental results and FE results. Based on the model, the roles of thread wear on the self-loosening of bolted joints were analysed. 2. Self-loosening experiments 2.1. Specimens and materials An M12, 1.75 × 100 mm hexagonal bolt and nut were used in the experiment. To ensure the quality and uniformity of test specimens, the bolt and the nut were made from SCM435 steel through milling machining according to standard processing technology. The manufacturing process included rod cutting of steel wire rods, head making, trimming, and thread milling. The mechanical properties of the material are listed in Table 1. 2.2. Experimental method A diagram of the setup for the self-loosening experiment is shown in Fig. 1. Two plates and a compression load cell (resolution 100 N) were clamped with a bolt and a nut. The nut was slowly tightened using a digital display torque wrench to the set preload. The thickness of the upper plate and the bottom plate were 15 mm and 40 mm, respectively. Both of the plates were made of AISI 1045 carbon steel. The upper and bottom plates were connected to the upper and lower grips of a Shimadzu servo-hydraulic fatigue testing machine. A load cell with a thickness of 16.3 mm was used for the measurement of the clamping force. A high-precision rotation sensor (resolution 0.045°) was attached to the bolted joint with the implementation of a rotation sensor support to measure the relative rotation between the nut and the bolt. An extensometer (resolution 0.001 mm) was attached to the two plates to measure the relative displacement between the two plates. During each test, the upper plate was driven to move, and the bottom plate was fixed. To avoid the wear on contact surfaces between the two plates, 3 mm diameter rollers were placed in two 25 × 10 × 1.4 mm slots on the contact surface of each plate. Furthermore, solid grease was applied to the contact surfaces. With this study, we hope to obtain a more distinct thread wear profile from the loosening test to facilitate the comparison between the experimental and FE results. Because a lower clamping force can lead to more severe thread wear [7–10], 10 kN was chosen as the preload, which produces a tensile stress of 95 MPa, corresponding to 13.5% of the material yield stress. Similar test methods were also adopted by other researchers, such as Pai [2], Bhattacharya [32], Saha [33], and Panja [34], who conducted loosening tests at a lower preload to investigate the loosening mechanism or anti-loosening ability of the bolted joint. Fatigue fracture frequently happened when the loosening tests were performed at a displacement amplitude larger than 0.2 mm, while a displacement amplitude lower than 0.2 mm would not produce distinct wear on the thread surface. Therefore, 0.2 mm was chosen as the displacement amplitude in this study. Bolt loosening is independent of the loading frequency [1]. To speed up the test, the loading frequency was set as high as possible. Therefore, the limiting frequency of the tester 10 Hz was chosen in this study. Considering that the self-loosening rate gradually decreases with the increase in loading cycles, the interrupted interval was gradually enlarged. The interrupted tests were terminated at 103, 104, 105, and 2 × 105 cycles, respectively. Three repeated tests were carried out for each test condition to obtain reliable results. The coefficient for the thread friction and under-head friction can be obtained by Eq. (1) [35]. Table 1 Mechanical properties of test material. Young's modulus (GPa)
Yield strength (MPa)
Tensile strength (MPa)
Elongation (%)
208
705
989
4.8
342
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Transverse Load Upper plate Load cell Rotation sensor support Rotation sensor Extensometer
Bolt Nut Bottom plate
Roller Fixed
Fig. 1. Setup for the self-loosening experiment.
(1)
T = P0 × (0.159p + 0.578µt d2 + 0.5De µh )
where T is the tightening torque, P0 is the bolt preload, p is the pitch of the thread (1.75 mm), d2 is the basic pitch diameter of the thread (10.86 mm), De is the effective diameter of the bolt head (15.75 mm), μt is the coefficient of thread friction, μh is the coefficient of under-head friction. Generally, it can be assumed that the two friction coefficients are equal [36,37]. Therefore, the friction coefficient can be calculated according to the measured tightening torque and the preload. 2.3. Measurements of wear profile The greatest amount of load is carried by the first engaged thread [38,39], which leads to the most severe damage occurring on this thread surface. Thus, the wear profile on the surface of the first thread was observed in this study. Due to the helical thread structure involved in the bolted joint, the contact stress and sliding distance vary along the circumferential direction on the threads. This may cause a difference in the degree of wear along the thread. Therefore, wear profiles on the two positions shown in Fig. 2 (Position A, θ = 0° and Position B, θ = 90°) were analysed. After the test, the bolt and the nut were cut by a wire electrical discharge machine and cleaned in an ultrasonic bath. Then, the fretting wear profile on the threads of the bolt and the nut were observed and measured using a confocal laser scanning microscope (Olympus OLS4100). Fig. 3(a) shows the cut parts of the bolt and the nut after a total loading of 2 × 105 cycles. Fig. 3(b) displays the optical microscopy observation of the contact surface at Position A of the bolt. Wear is evident in the contact region. The wear profile was measured from thread crest to root, as shown in Fig. 3(c). The flat part shown in Fig. 3(c) corresponds to the regions near the thread root, where the wear does not occur. Therefore, it can be used as the measuring datum to determine the wear depth from the wear profile. The average value of the wear depth from the three repeated specimens was then calculated to obtain the final wear profile. 2.4. Experimental results Table 2 shows the results of the self-loosening tests. The torque–preload relationship for a bolted joint is highly sensitive to friction variation in the threads and the under-head bearing [7], which causes scatter of the tightening torque for each test condition.
Position B y Transverse Load
Position A x
Thread Fig. 2. Observed positions on the first engaged thread. 343
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(a)
Nut (b) Thread root
Thread crest
Outer edge
Position A 3 mm
Contact region
inner edge
Bolt (c)
Fig. 3. Process of observing and measuring the wear profile on thread surface: (a) cut parts of the bolt and the nut, (b) optical microscopy observation of the contact surface and (c) measurement of wear profile.
The friction coefficient was calculated using Eq. (1) based on the measured tightening torque and preload. A higher value for the ratio of the clamping force to the preload (P/P0) indicates superior anti-loosening ability. In addition, the transverse load did not change greatly during the test and remained at a relatively low value of around 0.55 kN, suggesting that no distinct wear was produced on the plates during the test. The measured rotation angle was zero for each test, indicating that the self-loosening was not caused by the relative rotation between the nut and the bolt. 2.4.1. Loosening curve Fig. 4 shows the self-loosening curves within the given loading cycles. These self-loosening curves show a similar tendency in the variation of clamping force and loading cycles. For each test, the first 200 loading cycles were used to apply the transverse amplitude to the preset value. After the first 200 loading cycles, the self-loosening curves can be divided into two stages: a rapid loosening process from 200 to 500 loading cycles (Stage I), and a gradual loosening process after 500 loading cycles (Stage II). As previously reported [9,10], the rapid loosening is caused by the cyclic plastic deformation that occurs near the roots of the threads, and the fretting wear that occurs on the threads leads to the subsequent gradual loosening. After 2 × 105 loading cycles, about 12% of the initial preload is lost, in which Stage I and Stage II occupy about 6%, respectively. 2.4.2. Wear profile Fig. 5 shows the evolution of surface damage at Position A and B of the bolt. No distinct damage except the ploughing caused by the tightening process could be observed on the thread after 103 loading cycles. The ploughing gradually disappears, and the surface damage Table 2 Results of self-loosening tests. Test number 1 2 3 4 5 6 7 8 9 10 11 12 a
Loading cycles 3
10 103 103 104 104 104 105 105 105 2 × 105 2 × 105 2 × 105
Tightening torque (Nm)
Friction coefficienta
P/P0 (%)
Mean
Standard deviation
36.3 37.1 35.1 38.4 36.6 38.2 37.8 36.6 35.9 36.6 37.0 37.5
0.236 0.242 0.228 0.251 0.238 0.250 0.247 0.238 0.234 0.239 0.241 0.245
93.43 93.46 92.68 91.35 91.02 91.66 89.43 89.05 88.92 88.61 88.42 88.00
93.22
0.4718
91.37
0.3604
89.10
0.2650
88.32
0.3121
The coefficient of thread friction and underhead friction is assumed to be equal. 344
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Fig. 4. Self-loosening curves within given loading cycles.
becomes more and more severe with increased loading cycles. Fig. 6 shows the evolution of wear depth at Position A and B of the bolt and the nut. It can be seen that the wear depth gradually increases with the increase in loading cycles. The fretting wear along the radial direction of the thread surface is nonuniform. For any given number of loading cycles, the wear depth reaches a maximum at the outer edge of the contact region for the bolt and the inner side of the contact inner edge for the nut. In addition, the wear depth at Position A is larger than that at Position B, which indicates that the wear damage along the circumferential direction of the thread is not identical. 3. FE modelling 3.1. FE model According to the structure and dimensions of the designed testing setup shown in Fig. 1, an FE model was developed using HYPERMESH software, as shown in Fig. 7. The model contained two clamped plates, a load cell, a bolt, and a nut. The clamped length
Fig. 5. Evolution of surface damage on Position A and B of the bolt. 345
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Fig. 6. Evolution of measured wear depth on Position A of (a) the bolt and (b) the nut, and Position B of (c) the bolt and (d) the nut.
was 73 mm. The bolt and the nut respectively had 5 and 4 pitches of thread, and 4 pitches were engaged. The mesh generation for the thread referred to the method proposed by Fukuoka [40], which provided accurate helical thread geometry and continuous mesh to obtain an accurate contact stress and sliding distance during the simulation. The total structure was discretised using C3D8 eightnode elements. There were 432,529 nodes and 390,624 elements in the FE model. The material of the bolt and nut was modelled as an elastic–plastic kinematic hardening material. The properties of the material are listed in Table 1, in which the yield stress is 705 MPa and the hardening modulus is 0.45 MPa. Other components were assumed to be elastic. The Young's modulus and Poisson's ratio are 210 GPa and 0.3, respectively. In the FE model, 5 contact pairs were defined: the contact between the threads (Contact 1), the contact between the nut and the load cell (Contact 2), the contact between the load cell and the bottom plate (Contact 3), the contact between the upper and the bottom plates (Contact 4), and the contact between the upper plate and the bolt head (Contact 5). Finite sliding was applied to Contact 1 and Contact 4 because of the occurrence of large sliding between threads and between plates. To improve computation efficiency, a small sliding value was used for the other contact pairs. For Contact 1, the exponential softened pressure–overclosure relationship and the penalty method with friction were respectively used to model the normal and tangential behaviour. For other contacts, the penalty method with hard contact and friction were used, respectively, in the normal and tangential direction. The friction coefficient between threads and between the upper plate and the bolt head was set to 0.241, which is the mean value of the friction coefficients listed in Table 2. The friction coefficient between plates was set to 0.0005 because the rollers and solid grease were applied between the plates to reduce friction during the test. The friction coefficient was
Fig. 7. FE model. 346
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taken to be 0.2 for other contact surfaces. The preload was applied via thermal loading. The thermal expansion coefficient of the load cell was set to 1.5 × 10−5/°C in the z direction, and that of other components was zero. After the application of temperature, the load cell only elongated in the z direction, but other components did not elongate. A thermal load was applied in Step 1 to achieve a preload. To avoid the generation of large displacement which would cause a convergence problem, all external nodes of the nut were constrained in all directions in this step. In the following steps, the constraint on the external nodes of the nut was removed. To simulate the transverse load, all nodes on the end surface of the bottom plate were constrained in all directions, and a transverse displacement that varied with computational time t, S=S0sin(2πt), was applied to the right end of the upper plate in the x direction. The FE analysis of loosening was performed as a quasi-static process based on the previous results that loosening is independent of the loading frequency [1]. Fig. 4 shows that the reduction of clamping force during Stage I (before 500 loading cycles) was not caused by thread wear. Therefore, this simulation directly takes the clamping force at 500 loading cycles as the initial preload. In this study, a preload of 9400 N, a transverse displacement amplitude of 0.2 mm, and a total loading cycles of 200,000 were used. 3.2. Wear model The energy-based wear approach was used in the FE simulation to simulate fretting wear occurring at thread surface. The global energy wear approach is normally expressed as: N
V=
N
Ei = i=1
Qi dSi
(2)
i=1
where V is the total wear volume, α is the energy wear coefficient, and Ei, Qi and Si are the dissipated energy, tangential force and sliding distance at the ith cycle, respectively. For each cycle, the local wear volume can be described as: (3)
dV (x ) = dh (x ) A (x )
where dh(x) is the local wear depth, A(x) is the infinitesimally small contact area, and x is the horizontal contact position. Then, the local dissipated energy is as follows:
E (x ) =
q (x ) ds (x )
(4)
where q(x) is the local shear stress, and ds(x) is the increment in sliding distance. Therefore, the following equation can be obtained from Eq. (2):
dh (x ) A (x ) =
q (x ) A (x ) ds (x )
(5)
Then, diving both sides by A(x), the following equation is obtained:
dh (x ) =
q (x ) ds (x ) = E (x )
(6)
The wear model described above was implemented into ABAQUS software through the adaptive meshing UMESHMOTION user
Generate FE model ABAQUS Calculate shear stress q(x) and sliding distance s(x) for one increment
Calculate nodal wear depth n h( x ) N q ( x, t ) s ( x, t ) t 1
N N
Yes Yes
Update nodal coordinates and revise FE model
No No
UMESHMOTION
Stop
Fig. 8. Flow chart of the numerical procedure to model thread wear. 347
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Fig. 9. The variation of (a) contact stress and (b) wear depth with the selected mesh size.
subroutine [41]. This subroutine has been used by many researchers to implement wear simulation [23–26]. Fig. 8 shows the flowchart of the numerical procedure to model thread wear. One loading cycle is discretised into n increments. The total wear depth at node x during a full cycle of loading is: n
hstep (x ) =
q (x , t ) s (x , t )
(7)
t=1
where q(x,t) and Δs(x,t) are the shear stress and incremental sliding distance, respectively, at node x over an increment. The cycle jump technique was used to reduce the computational time. This technique was proposed by different authors like McColl [22] and Mary [27], where it is assumed that the contact stress and sliding distance remains constant in the ΔN loading cycles that follow. Therefore, the wear depth Δh(x) for ΔN loading cycles can be calculated as: n
h (x ) = N
q (x , t ) (x , t )
(8)
t=1
The wear coefficient α is a material parameter, which can be measured from conventional fretting tests with a round-flat or ballflat configuration [27,29]. In this study, the energy wear coefficient was set as α = 4.17 × 10−8 based on the fretting wear tests carried out for SCM435 steel using a ball-flat configuration [42]. 3.3. Optimisation of the parameters The mesh size on the thread surface, increments per loading cycle n, and cycle jumping-ΔN are important modelling parameters related to computational accuracy and speed. Optimisation of these parameters was conducted in this section. The mesh size on the thread surface was optimised first. For this analysis, the increments per loading cycle n, cycle jumping-ΔN, and total loading cycles were set to 40, 4000, and 20,000, respectively. Fig. 9(a) shows the variation in contact stress with the selected mesh size without transverse loading. It can be seen that the contact stress increases with decreasing mesh size until the mesh size is less than 73 μm. Fig. 9(b) shows the variation in wear depth with the selected mesh size after 20,000 loading cycles. It can be seen that the predicted wear depth increases with decreasing mesh size. However, when the mesh size is less than 73 μm, the wear depth remains unchanged. It should be pointed out that, for this method to generate thread mesh [40], the total number of nodes and elements is significantly increased when reducing the mesh size. When the mesh size is reduced from 73 μm to 61 μm, over 110,000 nodes and 120,000 elements are generated, which causes an almost four-fold increase in computational time. Therefore, to balance computational accuracy and speed, the mesh size on the thread surface was set to 73 μm.
Fig. 10. The variation of wear depth with the selected (a) increments per loading cycle n and (b) cycle jumping-ΔN. 348
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Fig. 11. The variation of slip range with the selected maximum elastic slip tolerance.
Then the increments per loading cycle n were optimised. For this analysis, the mesh size on the thread surface, cycle jumping-ΔN, and total loading cycles were set to 73 μm, 4000, and 20,000, respectively. As seen from Fig. 10(a), the wear depth slightly increases with an increase in the increments. As the increments are increased to 50, the predicted wear depth is almost unchanged. Because small increments easily cause convergence problems, the increment was chosen as 50 in this study. For the analysis of the cycle jumping-ΔN, the mesh size on the thread surface, increments per loading cycle n, and total loading cycles were set to 73 μm, 50, and 20,000, respectively. Fig. 10(b) shows the variation in wear depth with the selected cycle jumping-ΔN. When the cycle jump is larger than 5000, the wear depth rapidly increases with an increase in cycle jump. But when the cycle jump is less than 5000, the wear profiles remain almost unchanged. Because a smaller cycle jump increases the total computational time by increasing the total calculation steps, it was optimised to be 5000 in this study. Because the convergence problem occurred during the simulation when a Lagrange multiplier contact algorithm was used, the penalty approach was used to model the sticking and slipping behaviour in this study. It was found that the penalty approach could offer satisfactory results if the contact parameters were well-selected [43]. Fig. 11 shows the variation in slip range with the selected maximum elastic slip tolerance. It can be seen that the distribution of slip range remains unchanged with a maximum elastic slip tolerance less than 0.0004. Thus, the maximum elastic slip tolerance was taken to be 0.0004. In this section, the data shown in Figs. 9–11 were obtained from the results of Position A for the bolt.
Fig. 12. Evolution of predicted wear depth on Position A of (a) the bolt and (b) the nut, and Position B of (c) the bolt and (d) the nut. 349
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Fig. 13. Comparison of the predicted and experimental wear profile on Position A after (a) 104 loading cycles and (b) 2 × 105 loading cycles, and Position B after (c) 104 loading cycles and (d) 2 × 105 loading cycles.
4. Results and discussion 4.1. Wear profile The FE simulation for bolt loosening was conducted using the optimised parameters stated above. Fig. 12 illustrates the evolution of the predicted wear depth at Position A and B of the bolt and the nut. It can be seen that the predicted wear depth at Position A and B gradually increases with increased loading cycles. Fig. 12a and c show that the predicted wear depth at Position A and B of the bolt reaches a maximum at the contact outer edge for each loading cycle, and it decreases away from the contact outer edge as the number of loading cycles reaches 103 and 104. As the loading cycles are further increased, a change in the distribution of wear depth take place, namely, the wear depth first decreases and then increases from the outer edge to the inner edge. The nut shows a similar trend in the change of wear depth distribution (Fig. 12b and d), but it should be noted that the maximum wear depth occurs at the contact inner edge of the nut. In addition, the predicted wear depth at Position A is larger than that at Position B. The wear profiles at 104 and 2 × 105 loading cycles were used as an example to validate the FE results. Fig. 13 displays a comparison of the predicted and experimental wear profile at Position A and B, respectively. It can be seen that the wear profile predicted by the FE model describes a contour similar to the experimental profile. The predicted wear width is in good agreement with the experimental result, while the predicted wear depth is larger than the experimental result. This may be caused by the wear debris piling up on the thread surface. The wear debris covers the contact area, which separate the threads of the bolt and the nut and reduces the wear. Because this model cannot consider the accumulation of wear debris, a larger wear depth is predicted in the FE simulation. In addition, a more distinct difference in wear depth between the FE and experiment results occurs around the contact edges, because the wear debris more easily accumulates around the contact edges under repeated slip motion [44]. 4.2. Contact variables The distribution of contact variables of the bolt at different loading cycles was obtained from the FE result, as shown in Fig. 14. The corresponding cloud diagrams of contact stress are shown in Fig. 15. For the unworn surface (0 cycles), the contact stress gradually decreases from the outer edge to the inner edge of the contact region, while the sliding distance increases from the outer edge to the inner edge. The sliding distance at different locations increases with the increase in loading cycles, as shown in Fig. 14b and d. However, Figs. 14a, c, and 15 show that the contact stress on the side near the outer edge gradually decreases when increasing the loading cycles, while the contact stress on the side near the inner edge gradually increases. In the initial stage, the distribution of wear depth for the bolt, as shown in Fig. 12a and c, is attributed to the difference in contact stress at varied locations. The initial large loss of surface material near the outer edge relieves the stress concentration in this region, 350
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Fig. 14. Evolution of contact variables distribution: (a) contact stress and (b) sliding distance of the bolt on Position A, (c) contact stress and (d) sliding distance of the bolt on Position B.
Fig. 15. Cloud diagrams of contact stress for different stages.
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Fig. 16. Comparison of the predicted and experimental self-loosening curves.
and thereby reduces the wear rate. Meanwhile, because of the low initial wear rate, less loss of surface material occurs on the side near the inner edge in the initial stage. This leads to an increase in contact stress in this region and accelerates the fretting wear. Therefore, as the number of loading cycles is increased, a change in the distribution of wear depth takes place: the wear depth first decreases and then increases from the outer edge to the inner edge for loading cycles from 105 to 2 × 105. In addition, both the contact stress and sliding distance at Position A are larger than that at Position B, thus a larger wear depth is produced at Position A. Because the same material is used for the bolt and the nut, and surface-to-surface contact elements are used to model the contact between the threads, the contact stress, sliding distance and wear depth at the alignment locations for the thread surface of the bolt and the nut are identical. Therefore, the relationship between contact variables and wear depth for the nut is not described in this paper. 4.3. Loosening curve Fig. 16 shows the self-loosening curve obtained from the FE simulation together with the experimental result. It should be pointed out that this experimental curve only contains the portion of Stage II in Fig. 4, where the self-loosening is caused by fretting wear. The two curves show the same trend in the variation of clamping force with loading cycle. It can be seen from Fig. 12 that the wear depth increases with increasing loading cycles; this reduces the contact stress between the threads (Fig. 14), and thereby causes a gradual reduction in clamping force (Fig. 16). However, the reduction in clamping force for the FE simulation is larger than for the experimental result. This may be related to the piling up of wear debris on the thread surface. Because the wear debris partly fills up the gap generated by the fretting wear, it would alleviate the decrease in clamping force. This simulation work improves our understanding of loosening behaviour due to thread wear. It further confirms that the fretting wear that occurs during thread wear causes a slow self-loosening. We developed an FE model that considered the evolution of thread wear. In addition to analysing the experiment results, this model can also be used to optimise the structural parameters involved in the bolted joint, such as thread pith, thread fit, and hole clearance. In particular, it is difficult to optimise the structural parameters through experimental methods for large bolts. This model may play a more important role in the optimisation design for large bolts. 5. Conclusions In this study, an FE model was established to simulate the loosening behaviour of a bolted joint caused by thread wear under transverse loading. The simulation results were compared with the experimental results. The following conclusions can be drawn from the simulation and experimental results: 1. To obtain an appropriate balance between computational accuracy and speed, the mesh size on the thread surface, the increments per loading cycle n, the cycle jumping-ΔN, and the maximum elastic slip tolerance in the FE model were optimised to be 73 μm, 50, 5000, and 0.0004, respectively. 2. The FE model successfully simulated the phenomenon in which fretting wear causes a gradual reduction in clamping force by changing the distribution and magnitude of contact stress between the threads. 3. The predicted results agree with the experimental results with respect to the evolutionary trend of wear profile and clamping force. Because the accumulation of wear debris is not considered in the FE model, the predicted wear depth and reduction in clamping force is larger than in the experimental results. 4. The fretting wear causes a change in the distribution and magnitude of contact stress, and an increment in sliding distance along the radial direction on the thread surface, which can further change the distribution of wear depth in this direction. Acknowledgement This work was supported by the Independent Research Projects of State Key Laboratory of Traction Power (2015TPL_T16 and 352
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