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Three-dimensional finite element analysis of threaded fastener loosening due to dynamic shear load
Engineering Failure Analysis 9 (2002) 383–402 www.elsevier.com/locate/engfailanal
Three-dimensional finite element analysis of threaded fastener loosening due to dynamic shear load N.G. Pai, D.P. Hess* Department of Mechanical Engineering, University of South Florida, 4202 E. Fowler Avenue, ENB 118, Tampa, FL 33620-5350, USA Received 7 May 2001; accepted 20 May 2001
Abstract The two most widespread causes of failure of threaded fasteners subjected to dynamic loads are fatigue and vibration induced loosening. This paper presents results of a study on failure of threaded fasteners by vibration induced loosening caused due to dynamic shear loads. Previous experimental work has revealed that fastener loosening occurs as a result of complete or localized slip at the thread and head contact surfaces. A three-dimensional finite element (FE) model is used to study details of four different loosening processes that are characterized by either complete or localized slip at the head and thread contacts. The FE model is found to be capable of adequately modeling factors that influence slip and predicting the different loosening processes. Primary factors that influence slip at fastener contacts are discussed. The results show that loosening can occur at relatively low shear loads due to the process of localized slip. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Threaded fasteners; Vibration induced loosening; Fastener failures; Joint failures
1. Introduction and background Threaded fasteners are widely used in assemblies because of their ability to develop a clamping force and ease of disassembly for maintenance. The two most common modes of failure of threaded fasteners subjected to dynamic loads are fatigue and vibration induced loosening. This paper studies failure of threaded fasteners by vibration induced loosening caused due to dynamic shear loads. Such failures can be avoided by proper joint design using guidelines based on the understanding of loosening caused by dynamic loads. The work presented in this paper is a step towards development of such design guidelines. Research on loosening of threaded fasteners due to vibration spans nearly six decades. The reader is referred to Hess [1] and Bickford [2] for a comprehensive review of the literature. Early work [3–5] focused on loosening due to dynamic loads acting along the fastener axis (axial loading). However, experimental studies in the late 1960s by Junker [6] demonstrated that loosening is more severe when the joint is subjected to dynamic loads perpendicular to the thread axis (shear loading). Loosening under shear loading * Corresponding author. Tel.: +1-813-974-5643; fax: +1-813-974-1447. E-mail address: [email protected] (D.P. Hess). 1350-6307/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S1350-6307(01)00024-3
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was attributed to reduction of circumferential holding friction as a result of slip at the fastener surfaces caused by the applied shear load. There have been numerous attempts [7–13] to model the loosening phenomenon, however, all have had limited success in adequately predicting loosening. It was recently shown [14] that a fastener could turn loose under dynamic shear loading as a result of accumulation of localized slip in the form of strain at the fastener contacts surfaces. Adequate modeling of such loosening requires inclusion of fastener geometry, stiffness, as well as contact with friction. In this work a three-dimensional finite element (FE) model that includes these features is used to study fastener loosening. Most of the previous work relating to FE analysis [15–20] of threaded fastener have been restricted to axisymmetric models aimed primarily at stress analysis. Such models cannot be used to simulate loosening because they do not include the helical thread geometry. Zadoks and Kokatam [21] have presented a threedimensional FE model of a screw that includes the thread helix. The screw in their model is not meshed as a single continuous body; instead the screw body (a cylinder) and the threads (helix) are meshed separately and then joined using fixed contact elements. Although this approach makes the modeling process less complex, it significantly increases the solution computational cost due to the fixed contacts elements used to bond the threads to the body. In addition, for a given mesh density, the fixed contact approach is less accurate than using a continuous mesh for the entire screw. As a result of these considerations, the finite element model in the present work utilizes a continuous mesh for the screw. This work is aimed at improving the understanding of fastener loosening under dynamic shear loads as a step towards development of quantitative guidelines for joint design based on fastener loosening. Previous experimental work [14] has identified four different loosening processes that cause fasteners to turn loose under shear loading. Details of the different loosening process are illustrated using results from the FE analysis. Factors that influence slip, and the resulting loosening are discussed. The finite element model is used to illustrate the effect of some significant factors that influence loosening.
2. Finite element model The most widely used apparatus for experimental study of loosening under dynamic shear load is the transverse vibration test apparatus developed by Junker [6]. The finite element model developed in the present study models the joint in such a test system. The test specimen (Fig. 1) clamps the top movable plate to the rigid fixed base through a threaded insert. Roller bearings are placed between the top plate and the fixed base to prevent galling. The top plate is subjected to a cyclic shear load through an arm connected to an eccentric. The fastener preload, and the applied shear load are measured through load cells. In addition, the displacement of the top plate is measured using an LVDT. The FE model of the test joint was developed using ANSYS, which is a general-purpose finite element analysis software. A typical finite element mesh of the model used for the study is shown in Fig. 2. It
Fig. 1. Transverse vibration test apparatus.
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Fig. 2. Typical finite element mesh: (a) joint model, (b) boundary conditions, and (c) contact regions on the screw.
consists of a screw, which fastens the top plate through a threaded insert. The geometry is simplified to include only the essential features of the system. Since the base is assumed to be rigid, only a small region around the threaded insert is modeled, and the nodes on the external surface of this region are constrained (see Fig. 2b). Also, only a small region of the top plate around the screw is modeled and its end surfaces are constrained to remain plane to model the behavior of a longer member (see Fig. 2b). Since the friction at the interface between the top plate and the fixed base is negligible due to the roller bearings, the bottom and side contacts are simply modeled by nodal constraints at the bottom of the top plate in the z and y directions. Fastener preload is simulated by an initial interference between the screw head and the upper surface of the top plate. The transverse load in the x direction is applied to the clamped component through a spring element (see Fig. 2a). The stiffness of the spring element is based on the experimentally determined stiffness of the load transmitting members of the transverse test apparatus. Due to the presence of contact and significant rotation from the screw turn, the problem requires a non-linear solution that utilizes the New-
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ton-Raphson method [22]. Loosening is modeled as a quasi-static process based on previous results that show loosening in the transverse test apparatus to be independent of the loading frequency [6]. Each loading cycle is divided into at least thirty two load increments with some cases utilizing higher number of divisions to enhance convergence of the non-linear solution. The finite element mesh used in this study utilizes a predominantly hexahedral mesh, which provides good results within a reasonable runtime. A typical model with twelve screw threads and five and a half internal threads consist of 6126 nodes, and 4584 elements. The model mainly comprises of 8-node and 20node brick elements, with a small portion of the fastener meshed with 10 node tetrahedral elements. Higher order elements (20-node brick) are used in the contact regions of the components (see Fig. 2c). These contact regions are overlaid with high order general-purpose contact elements capable of modeling friction. The model includes contact regions between the screw head and the clamped component (see Fig. 2c), the clamped component hole and the screw body, and between the internal and external threads. Preliminary studies were conducted to determine the effect of the mesh density on the loosening results. It was found that a relatively coarse mesh (such as shown in Fig. 2) utilizing high order elements at the contact regions provided good results in a reasonable amount of time. This is not surprising since loosening results are based on the displacements, which converge with a relatively coarse mesh compared to the mesh required for accurate determination of stresses. Typical run time for a loading cycle is approximately 6 h on a 700 MHz Pentium III PC with 512MB RAM on the Windows NT platform. 3. Loosening processes The earliest explanation for loosening under shear loading was provided by Junker [6] and recently extended based on an experimental study by the authors [14]. Threaded fasteners have an inherent tendency to loosen due to the helical slope at the threads. Fig. 3 shows the thread reactions, RPn, n=1. . .4, to the preload, FP, at four points around a thread. The loosening moment is developed from the circumferential components RPLn,n=1. . .4. In absence of external loads, the fastener remains tight because of the circumferential friction that resists the loosening moment. However, as a result of an applied shear load, the circumferential friction force reduces as the direction of contact forces changes from circumferential to the direction of the applied shear force. Consequently the fastener turns loose as contacts at the head and threads slip. For additional details of the loosening process, the reader is referred to Pai and Hess [14]. Slip at the contacts under the applied load can be classified as complete slip or localized slip. Complete slip occurs when the entire contact surface (at the head or threads) slips, while localized slip occurs when only parts of the contact regions slip. Complete slip requires that the loads acting on the fastener are sufficiently large to overcome friction over the entire contact, while localized slip occurs only in parts of the contact where the friction force has been overcome. Significant loosening occurs only when the entire fastener turns, which requires that the two fastener contact surfaces, i.e. at the head and threads, either undergo complete slip, or localized slip that accumulates over loading cycles. Loosening processes can be divided into four different types depending on the nature of slip at the head and thread contacts. These are characterized by (1) localized head slip with localized thread slip, (2) localized head slip with complete thread slip, (3) complete head slip with localized thread slip, and (4) complete head slip with complete thread slip. The influence of the type of slip on loosening is illustrated in Fig. 4, which shows a typical preload versus cycles plot of loosening obtained using the transverse vibration test apparatus. The loosening rate shows a drastic increase as soon as head slip changes from localized to complete slip. All the FE results presented in this paper model a 63.5 mm long, Grade 8, Class 2A/2B, 0.5-13 UNC screw (12.7 mm diameter) screw subjected to 1 mm displacement through the spring element of stiffness 1.2 kN/mm. The modulus of elasticity of the fastener materials used for the simulations is 200 GPa. The coefficient of friction is 0.261 for dry contact, 0.158 for contacts with machine oil lubrication, and 0.086 for MoS2 grease lubricated contacts based on previously obtained experimental data [14].
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Fig. 3. Loosening moments from component of thread reaction to preload.
Fig. 4. Typical loosening sequence of a screw during a transverse vibration test.
Figs. 5 and 6 show loosening resulting from localized head slip and complete thread slip for a fastener with head and thread contacts lubricated with machine oil and a preload of nearly 11 kN. This loosening process is the same as the initial loosening process occurring in Fig. 4. Fig. 5a shows a hysteresis curve, which is a plot of the shear force acting on the top plate versus its displacement. The slope of the hysteresis
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Fig. 5. Loosening process characterized by localized head slip with complete thread slip: (a) hysteresis curve, and (b) contact status.
curve provides an indication of the joint stiffness in the transverse direction. The reduction in slope of the hysteresis curve is a sign of slip at the contacts. Fig. 5b shows the state of the fastener at three points during the cycle. Black regions indicate contact regions that stick, while the gray regions indicate slip. Segments ab and cd in Fig. 5a indicate parts of the cycle where the threads undergo complete slip while the head contact undergoes localized slip (see Fig. 5bi and biii). Both thread and head undergo localized slip during parts of the cycle indicated by the steeper segments bc and da (Fig. 5bii). Note that head contact regions that stick during the first half of cyclic loading, slip during the second half (see Fig. 5bi and biii). This enables the entire head contact to slip over a complete cycle.
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Fig. 6. Loosening process characterized by localized head slip with complete thread slip: (a) thread turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . – . – ., at 270 ; — —, average, (b) head turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . — . — ., at 270 ; — —, average, and (c) preload.
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Fig. 6a and b shows the turning angle at four points around the thread and head respectively. The thread turn angle shown is the average from all engaged threads. Thread loosening angles show a rapid increase due to complete thread slip, while the head turning rate is lower due to localized slip. This results in the screw body being under torsion. The plots indicate that points at 90 and 270 oscillate with the applied load with a net loosening at the end of each half cycle as reflected by the angles at 0 and 180 . Since the screw body is in a state of torsion, the corresponding oscillating points at the head and thread move in opposite directions. Note that the thread turn angle is far larger than the head angle, and that the preload loss per cycle (Fig. 6c) is fairly low. Figs. 7 and 8 illustrate loosening caused by complete head slip and complete thread slip for the thread and head contacts lubricated with machine oil and at a preload of approximately 5.5 kN. The hysteresis curve shown in Fig. 7a indicates regions with three distinct slopes. Region i reflect parts of the loading where complete thread slip occurs and the head undergoes localized slip (Fig. 7bi). At region ii the entire head and thread slip as shown in Fig. 7bii. The initial stage of the unloading portion of the cycle is seen to have a higher slope because of localized slip at both the head and the thread (see Fig. 7biii). Fig. 8a and b shows the loosening at four points at the thread and head. The loosening angles at all four points are nearly identical. The head loosening angle shows nearly a step increase at the point where complete slip occurs, while the thread angle increases more gradually with a small step reflecting the occurrence of complete head slip. The loosening angles at the head and thread are comparable with the thread angle leading the head angle. This is expected since the entire loosening moment is developed at the threads. Also, the rate of preload drop (Fig. 8c) is significantly larger than that found with localized head slip illustrated in Fig. 6c. This loosening process corresponds to the loosening occurring at a rapid rate in Fig. 4. Hysteresis curves and loosening angles of the loosening process characterized by complete head slip and localized thread slip are shown in Figs. 9 and 10 for a fastener with MoS2 grease at the head and dry threads with a preload of about 11.9 kN. The hysteresis curve is characterized by two distinct slopes. The stiffer segments indicate parts of loading where the head and thread undergo localized slip (see Fig. 9bi). Regions ii and iii reflect parts of the cycle where the entire head slips, while the threads continue to undergo localized slip (see Fig. 9bii and biii). From Fig. 9bii and biii it is seen that different parts of the threads stick during different parts of the cycle (note that orientation of the ii is opposite of that of iii). The thread angles at four points (Fig. 10a) show that points at 90 and 270 oscillate with the applied input with a net increase in the loosening angle per half a loading cycle. The head angles (Fig. 10b) display a step increase at the point where the complete head slips, with the points at 90 and 270 reflecting the oscillations at the threads. As in the earlier cases, the thread angle leads the head angle. The loosening rate (Fig. 10c) is not as severe as in the case of complete thread and head slip (Fig. 8c). Figs. 11 and 12 illustrate the loosening process resulting from localized head and localized thread slip with dry threads and MoS2 grease at the head and at preload of nearly13.2 kN. The hysteresis curve shows nearly the same stiffness at all regions with a very slight reduction in the slope as the thread slip region increases. The regions of localized slip (Fig. 11b) are seen to vary at different parts of the cycle, however the lower thread contact stick throughout the entire cycle. Fig. 12a and 12b show the loosening angles at the thread and head, which essentially display an oscillatory character at the 90 and 270 points with a slight increase in the loosening angle per cycle. As with the other loosening processes, the thread loosening angle leads the head angle. The rate of preload loss (Fig. 12c) is seen to be quite small. Although the lower threads stick throughout the entire cycle in the results presented here, the preload loss resulting due to the localized slip at the first few threads can accumulate and cause progressively larger slip and eventually lead to complete thread slip. The above FE results illustrate the four possible loosening processes that can occur in a threaded fastener subjected to dynamic shear loads. Of these four processes, only three are widespread in practice. In the typical case of fastener loosening, the initial stage of loosening is characterized by localized slip at both the head and the threads. This progresses to loosening by localized head slip with complete thread slip, and
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Fig. 7. Loosening process characterized by complete head slip with complete thread slip: (a) hysteresis curve, and (b) contact status.
eventually to complete slip at the head and the threads. In some cases, the initial loosening starts with localized head slip and complete thread slip, and progresses to complete slip at the head and the threads. The loosening rate at the initial stages characterized by localized slip is fairly low and increases as the loosening transitions from localized to complete slip (see Fig. 4). A contribution of the present work is the identification of loosening caused by localized slip. Loosening at the initial stages resulting from localized slip is critical since it can occur at a significantly lower shear load than that required for complete slip. For
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Fig. 8. Loosening process characterized by complete head slip with complete thread slip: (a) thread turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . – . – ., at 270 ; — —, average, (b) head turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . — . — ., at 270 ; — —, average, and (c) preload.
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Fig. 9. Loosening process characterized by complete head slip with localized thread slip: (a) hysteresis curve, and (b) contact status.
example, in the data shown earlier, loosening by localized slip (Fig. 5a) occurs when the magnitude of the shear force acting on the joint is approximately 9% of the preload, while loosening by complete slip (Fig. 7a) occurs when the shear force is 16% of the preload. In this case, loosening by localized slip occurs at about half of the load required to cause complete slip. Loosening by localized slip is therefore the critical loosening process from a perspective of joint design. Loosening failure in a joint can be avoided by ensuring that the dynamic loads acting on the fastener are lower than the loads required to cause loosening by localized slip.
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Fig. 10. Loosening process characterized by complete head slip with localized thread slip: (a) thread turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . – . – ., at 270 ; — —, average, (b) head turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . — . — ., at 270 ; — —, average, and (c) preload.
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Fig. 11. Loosening process characterized by localized head slip with localized thread slip: (a) hysteresis curve, and (b) contact status.
4. Comparison with experiments Fig. 13 shows hysteresis curves comparing experimental data with FE results for the four different loosening processes. The FE results agree reasonably well with experimental results as far as displaying the four loosening processes reflected by the slopes of the hysteresis curves. Fig. 13a shows the hysteresis curves for the screw lubricated with oil at the threads and head at preloads of 11 and 5.5 kN. The hysteresis curves at the higher preload representing loosening by localized head slip and complete thread slip show a very good match. The comparison of loosening process of complete head and thread slip at the preload of 5.5 kN
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Fig. 12. Loosening process characterized by localized head slip with localized thread slip: (a) thread turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . – . – ., at 270 ; — —, average, (b) head turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . — . — ., at 270 ; — —, average, and (c) preload.
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Fig. 13. Comparison of experimental hysteresis curves and FE results: (a) ——, experiment 11 kN preload with oil lubrication; , FE 11 kN; – – –, experiment 5.5 kN preload with oil lubrication; &, FE 5.5 kN preload. (b) ——, experiment 11.9 kN preload with dry threads and MoS2 grease at head; , FE, and (c) ——, experiment 13.2 kN preload with dry threads and MoS2 grease at head; , FE. (63.5 mm long, Grade 8, 0.5-UNC 13 screw, 1 mm eccentric setting).
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capture the trends, however the net slip occurring during complete head slip is smaller in the FE results than in the experiments. The amount of slip reflected in the hysteresis curve is a function of the fastener dimensions, and is largely influenced by the clearance between the internal and external threads. The difference in results is most likely because the thread dimensions utilized for the FE simulations were defined in the middle of the allowable range for Class 2A/2B threads, and the thread dimensions in the test specimens may be slightly different. The hysteresis curves shown in Fig. 13b show loosening by complete head slip and localized thread slip for a screw with MoS2 grease at head and dry threads at 11.9 kN preload. The FE result is seen to capture the experimental data quite well, and as with the previous case the slip is slightly smaller. A similar trend is observed for loosening caused by localized thread and head slip shown in Fig. 13c for the same lubrication condition at 13.2 kN preload. In all four cases, especially in Fig. 13b and c, the FE results are seen to be slightly stiffer than the experimental data. This is because the FE model includes six threads before the first engaged thread instead of 12 threads found in the test specimens. The additional threads in the screws used in the experiments contribute to the lower bending stiffness observed in the experimental data. A FE model with 12 exposed threads could not be used because of the significant additional computational cost. The influence of this omission is clearly not very significant as indicated by the reasonably good comparisons between the results.
5. Fastener slip The four different loosening processes described earlier are a result of the slip occurring at the fastener contact surfaces. Slip at the fastener surfaces is a function of the distribution of loads acting tangential to the contact surface (i.e. the forces that cause slip), and the contact normal force distribution (since it influences the friction). Fig. 14 summarizes the main factors that influence slip, S, and normal reactions, R, in a joint subjected to shear load [14]. The fastener has an initial loosening moment L due to the thread reaction to preload (see Fig. 3), which is balanced by the friction moments at the threads and the head. During the loading, the clamped component is subjected to a shear load FS, which is transferred to the screw through the head friction. The shear force can also be transferred to the screw through side contact between the clamped component and the screw body, which is caused by head slip or fastener bending. The component of the shear force acting tangential to the thread flank contributes to thread slip, while the remaining part changes the thread normal reaction distribution. The bending moment developed from the shear force changes the head and thread reaction distributions. The bending moment also contributes to slip along the thread flank due to the turning action about the moment axis. As a result of the shear load and the bending moment, the fastener undergoes elastic deformation at the head and the threads, which also contributes to slip. The resulting slip from all of the above factors is also influenced by the state of side contact at the head and the thread (not shown). From the above description it can be seen that several factors influence the loosening process. The main factors have been separated for the purpose of illustration. However, all these factors are coupled non-linearly and together determine the final state and process of loosening. Some important aspects of these factors that influence loosening are discussed below.
6. Fastener bending It is seen in Fig. 14 that the bending moment contributes to slip and loosening in several ways. The bending moment contributes to localized slip by changing the reaction distribution which causes slip in regions with lower reaction force (for example see slip regions at the head in Figs. 5b and 11b). The bending moment also causes slip at the thread flanks, and contributes to slip by elastic deformation.
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Fig. 14. Summary of factors influencing slip and reaction force distribution in a joint subjected to shear load. (FS–Applied shear force, L–loosening moment, S–tangential slip force, R–normal reaction).
The influence of bending moment on slip is illustrated in Fig. 15. It shows the shear force required to cause complete slip at the threads for a 0.5–13 UNC screw of different lengths at a preload of 13.3 kN. The shear force required to cause complete thread slip in a 63.5 mm screw is nearly half that for a 31.8 mm screw. This indicates that the additional bending moment developed in the longer screw is a significant factor responsible for thread slip. The bending moment also contributes to head slip by elastic deformation of the head. This is because the bending moment causes a change in the head normal reaction distribution, which changes the head surfaces deformation and contributes to slip as shown by the illustration at the bottom right corner in Fig. 14. The above data indicate that long fasteners require lower shear force to loosen. However, this is applicable only to force loaded systems, such as fasteners used to mount a pillow block bearing subjected to dynamic loads from an unbalanced rotating mass. In the case of displacement-loaded systems such as the transverse vibration test apparatus (Fig. 1) longer fasteners can reduce loosening. This is because the shear force acting at the contacts in such systems is a function of the applied displacement and the fastener stiffness. Since longer fasteners have a lower bending stiffness, they are subjected to lower shear force, and therefore are less vulnerable to loosening. Experience with displacement loaded systems support the notion that longer bolts inhibit vibration induced loosening [2,23]. However, the opposite is true (i.e. longer bolts are more prone to loosening) for force-loaded systems, and this is not widely appreciated.
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Fig. 15. Shear force required to cause complete thread slip for different screw length. (0.5-13 UNC screw with dry contacts).
7. Other factors Another aspect of the loosening process is the influence of the thread load distribution on thread slip. In the results presented earlier, it is found that the first few threads slip before the last ones (see Fig. 11b for example). This is mainly because of the normal load distribution at the threads as shown in Fig. 16. The data show that the first three threads in a FE model with five and a half engaged threads carry about 73% of the preload. This means that the first three threads have significantly higher frictional resistance, and only once these are overcome can the lower threads begin to slip. This is reflected in Fig. 17, which shows the turn angle for nodes at the 90 location on the first five threads for the loosening process caused by localized head and thread slip (Figs. 11 and 12). The magnitude of the turn angle is largest at the first thread and reduces in subsequent threads. The turn angles of the first few threads reflect slip as well as elastic deformation, while those at the last few threads are mainly a result of elastic deformation. These results indicate that the load distribution at the threads influence the loosening process.
Fig. 16. FE results of load distribution at the engaged threads for 0.5-13 UNC screw.
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Fig. 17. Thread turn angle at 90 node for loosening process characterized by localized head slip with localized thread slip: ——, first engaged thread; – – –, second thread; , third thread; . — . —, fourth thread; — , fifth thread.
In all the cases presented in this work, the fastener preload has been fairly low so has to minimize the influence of localized thread yielding, which is likely to influence the thread load distribution. Inclusion of thread yielding in the FE model requires a denser mesh to ensure accurate determination of stresses. This was not attempted at this stage of the study due to the extremely high computational cost. The effect of thread yielding on loosening will be addressed in future work. In addition to fastener length, and thread load distribution, there are various other parameters that influence loosening, including the fastener material, and dimensional tolerances. Since all these factors are coupled nonlinearly, it is important to study their individual effect as well as the influence of their interaction on screw loosening. Results from a parameter study investigating these effects will be reported in a future paper.
8. Conclusion The finite element model presented is capable of predicting the four different loosening processes observed experimentally. The FE results capture the essential features displayed by the experimental data. Several factors influence loosening at different stages of loading, and the final outcome is a result of their non-linear interactions. The FE model includes the primary factors that cause loosening, and provides a powerful tool for evaluation of the details of fastener loosening. The loosening process caused by localized slip can occur at significantly lower shear force than loosening caused by complete slip, and therefore is critical in joint design. Acknowledgements The authors gratefully acknowledge the support from the National Science Foundation under Grant No. CMS-9629217. References [1] Hess DP. Vibration- and shock- induced loosening. In: Bickford JH, Nasser S, editors. Handbook of bolts and bolted joints. New York: Marcel Dekker, 1998. p. 757–824.
402 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
N.G. Pai, D.P. Hess / Engineering Failure Analysis 9 (2002) 383–402 Bickford JH. An introduction to the design and behavior of bolted joints (third edition). New York: Marcel Dekker, 1995. Goodier JN, Sweeney RJ. Loosening by vibration of threaded fastenings. Mechanical Engineering 1945;67:798–802. Sauer JA, Lemmon DC, Lynn EK. Bolts: how to prevent their loosening. Machine Design 1950;22:133–9. Gambrell SC. Why bolts loosen. Machine Design 1968;40:163–7. Junker GH. New criteria for self-loosening of fasteners under vibration. SAE Transactions 1969;78:314–35. Sakai T. Investigations of bolt loosening mechanisms (1st report, on bolts of transversely loaded joints). Bulletin of JSME 1978; 21(159):1385–90. Yamamoto A, Kasei S. A solution for self-loosening mechanism of threaded fasteners under transverse vibration. Bulletin Japan Society of Precision Engineering 1984;18(3):261–6. Vinogradov O, Huang X. On a high frequency mechanism of self-loosening of fasteners. In: Proceedings of 12th ASME conference on mechanical vibration and noise, Montreal, Quebec, 1989. p. 131–7. Daabin A, Chow YM. A theoretical model to study thread loosening. Mechanism and Machine Theory 1992;27(1):69–74. Zadoks RI, Yu X. A preliminary study of self-loosening in bolted connections. ASME Nonlinear Vibrations 1993;DE-54:79–88. Zadoks RI, Yu X. An investigation of the self-loosening behavior of bolts under transverse vibration. Journal of Sound and Vibration 1997;208(2):189–209. Kasei S, Matsuoka H. Considerations of thread loosening by transverse impacts. ASME Pressure Vessels and Piping Division (Publication) PVP 1998;367:117–23. Pai NG, Hess DP. Experimental study of loosening of threaded fasteners due to dynamic shear loads. Submitted to Journal of Sound and Vibration. Maruyama K. Stress analysis of a bolt-nut joint by the finite element method and the copper-electroplating method. Bulletin of the JSME 1976;19(130):360–8. Fukuoka T, Yamasaki N, Kitagawa H, Hamada M. A stress analysis of threaded portions in fastening. Bulletin of JSME 1985; 28(244):2247–53. Tanaka M, Yamada A. The behavior of fasteners under external loading. Bulletin of JSME 1986;29(248):617–24. Dragoni E. Effect of thread pitch and frictional coefficient on the stress concentration in metric nut-bolt connections. Journal of Offshore Mechanics and Arctic Engineering 1994;116:21–6. Englund RB, Johnson DH. Finite element analysis of a threaded connection compared to experimental and theoretical research. Journal of Engineering Technology 1997;14(2):42–7. Lehnhoff TF, Bunyard BA. Bolt thread and head fillet stress concentration factors. Journal of Pressure Vessel Technology 2000; 122:180–5. Zadoks RI, Kokatam DPR. Three-dimensional finite element model of a threaded connection. Computer Modeling and Simulation in Engineering 1999;4(4):274–81. ANSYS Inc. ANSYS user manuals for Release 5.6. Cannonsburg. Pennsylvania: ANSYS, Inc. Heston AW. VDI joint design procedures. In: Bickford JH, Nasser S, editors. Handbook of bolts and bolted joints. New York: Marcel Dekker, 1998. p. 317–40.
Three-dimensional finite element analysis of threaded fastener loosening due to dynamic shear load N.G. Pai, D.P. Hess* Department of Mechanical Engineering, University of South Florida, 4202 E. Fowler Avenue, ENB 118, Tampa, FL 33620-5350, USA Received 7 May 2001; accepted 20 May 2001
Abstract The two most widespread causes of failure of threaded fasteners subjected to dynamic loads are fatigue and vibration induced loosening. This paper presents results of a study on failure of threaded fasteners by vibration induced loosening caused due to dynamic shear loads. Previous experimental work has revealed that fastener loosening occurs as a result of complete or localized slip at the thread and head contact surfaces. A three-dimensional finite element (FE) model is used to study details of four different loosening processes that are characterized by either complete or localized slip at the head and thread contacts. The FE model is found to be capable of adequately modeling factors that influence slip and predicting the different loosening processes. Primary factors that influence slip at fastener contacts are discussed. The results show that loosening can occur at relatively low shear loads due to the process of localized slip. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Threaded fasteners; Vibration induced loosening; Fastener failures; Joint failures
1. Introduction and background Threaded fasteners are widely used in assemblies because of their ability to develop a clamping force and ease of disassembly for maintenance. The two most common modes of failure of threaded fasteners subjected to dynamic loads are fatigue and vibration induced loosening. This paper studies failure of threaded fasteners by vibration induced loosening caused due to dynamic shear loads. Such failures can be avoided by proper joint design using guidelines based on the understanding of loosening caused by dynamic loads. The work presented in this paper is a step towards development of such design guidelines. Research on loosening of threaded fasteners due to vibration spans nearly six decades. The reader is referred to Hess [1] and Bickford [2] for a comprehensive review of the literature. Early work [3–5] focused on loosening due to dynamic loads acting along the fastener axis (axial loading). However, experimental studies in the late 1960s by Junker [6] demonstrated that loosening is more severe when the joint is subjected to dynamic loads perpendicular to the thread axis (shear loading). Loosening under shear loading * Corresponding author. Tel.: +1-813-974-5643; fax: +1-813-974-1447. E-mail address: [email protected] (D.P. Hess). 1350-6307/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S1350-6307(01)00024-3
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was attributed to reduction of circumferential holding friction as a result of slip at the fastener surfaces caused by the applied shear load. There have been numerous attempts [7–13] to model the loosening phenomenon, however, all have had limited success in adequately predicting loosening. It was recently shown [14] that a fastener could turn loose under dynamic shear loading as a result of accumulation of localized slip in the form of strain at the fastener contacts surfaces. Adequate modeling of such loosening requires inclusion of fastener geometry, stiffness, as well as contact with friction. In this work a three-dimensional finite element (FE) model that includes these features is used to study fastener loosening. Most of the previous work relating to FE analysis [15–20] of threaded fastener have been restricted to axisymmetric models aimed primarily at stress analysis. Such models cannot be used to simulate loosening because they do not include the helical thread geometry. Zadoks and Kokatam [21] have presented a threedimensional FE model of a screw that includes the thread helix. The screw in their model is not meshed as a single continuous body; instead the screw body (a cylinder) and the threads (helix) are meshed separately and then joined using fixed contact elements. Although this approach makes the modeling process less complex, it significantly increases the solution computational cost due to the fixed contacts elements used to bond the threads to the body. In addition, for a given mesh density, the fixed contact approach is less accurate than using a continuous mesh for the entire screw. As a result of these considerations, the finite element model in the present work utilizes a continuous mesh for the screw. This work is aimed at improving the understanding of fastener loosening under dynamic shear loads as a step towards development of quantitative guidelines for joint design based on fastener loosening. Previous experimental work [14] has identified four different loosening processes that cause fasteners to turn loose under shear loading. Details of the different loosening process are illustrated using results from the FE analysis. Factors that influence slip, and the resulting loosening are discussed. The finite element model is used to illustrate the effect of some significant factors that influence loosening.
2. Finite element model The most widely used apparatus for experimental study of loosening under dynamic shear load is the transverse vibration test apparatus developed by Junker [6]. The finite element model developed in the present study models the joint in such a test system. The test specimen (Fig. 1) clamps the top movable plate to the rigid fixed base through a threaded insert. Roller bearings are placed between the top plate and the fixed base to prevent galling. The top plate is subjected to a cyclic shear load through an arm connected to an eccentric. The fastener preload, and the applied shear load are measured through load cells. In addition, the displacement of the top plate is measured using an LVDT. The FE model of the test joint was developed using ANSYS, which is a general-purpose finite element analysis software. A typical finite element mesh of the model used for the study is shown in Fig. 2. It
Fig. 1. Transverse vibration test apparatus.
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Fig. 2. Typical finite element mesh: (a) joint model, (b) boundary conditions, and (c) contact regions on the screw.
consists of a screw, which fastens the top plate through a threaded insert. The geometry is simplified to include only the essential features of the system. Since the base is assumed to be rigid, only a small region around the threaded insert is modeled, and the nodes on the external surface of this region are constrained (see Fig. 2b). Also, only a small region of the top plate around the screw is modeled and its end surfaces are constrained to remain plane to model the behavior of a longer member (see Fig. 2b). Since the friction at the interface between the top plate and the fixed base is negligible due to the roller bearings, the bottom and side contacts are simply modeled by nodal constraints at the bottom of the top plate in the z and y directions. Fastener preload is simulated by an initial interference between the screw head and the upper surface of the top plate. The transverse load in the x direction is applied to the clamped component through a spring element (see Fig. 2a). The stiffness of the spring element is based on the experimentally determined stiffness of the load transmitting members of the transverse test apparatus. Due to the presence of contact and significant rotation from the screw turn, the problem requires a non-linear solution that utilizes the New-
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ton-Raphson method [22]. Loosening is modeled as a quasi-static process based on previous results that show loosening in the transverse test apparatus to be independent of the loading frequency [6]. Each loading cycle is divided into at least thirty two load increments with some cases utilizing higher number of divisions to enhance convergence of the non-linear solution. The finite element mesh used in this study utilizes a predominantly hexahedral mesh, which provides good results within a reasonable runtime. A typical model with twelve screw threads and five and a half internal threads consist of 6126 nodes, and 4584 elements. The model mainly comprises of 8-node and 20node brick elements, with a small portion of the fastener meshed with 10 node tetrahedral elements. Higher order elements (20-node brick) are used in the contact regions of the components (see Fig. 2c). These contact regions are overlaid with high order general-purpose contact elements capable of modeling friction. The model includes contact regions between the screw head and the clamped component (see Fig. 2c), the clamped component hole and the screw body, and between the internal and external threads. Preliminary studies were conducted to determine the effect of the mesh density on the loosening results. It was found that a relatively coarse mesh (such as shown in Fig. 2) utilizing high order elements at the contact regions provided good results in a reasonable amount of time. This is not surprising since loosening results are based on the displacements, which converge with a relatively coarse mesh compared to the mesh required for accurate determination of stresses. Typical run time for a loading cycle is approximately 6 h on a 700 MHz Pentium III PC with 512MB RAM on the Windows NT platform. 3. Loosening processes The earliest explanation for loosening under shear loading was provided by Junker [6] and recently extended based on an experimental study by the authors [14]. Threaded fasteners have an inherent tendency to loosen due to the helical slope at the threads. Fig. 3 shows the thread reactions, RPn, n=1. . .4, to the preload, FP, at four points around a thread. The loosening moment is developed from the circumferential components RPLn,n=1. . .4. In absence of external loads, the fastener remains tight because of the circumferential friction that resists the loosening moment. However, as a result of an applied shear load, the circumferential friction force reduces as the direction of contact forces changes from circumferential to the direction of the applied shear force. Consequently the fastener turns loose as contacts at the head and threads slip. For additional details of the loosening process, the reader is referred to Pai and Hess [14]. Slip at the contacts under the applied load can be classified as complete slip or localized slip. Complete slip occurs when the entire contact surface (at the head or threads) slips, while localized slip occurs when only parts of the contact regions slip. Complete slip requires that the loads acting on the fastener are sufficiently large to overcome friction over the entire contact, while localized slip occurs only in parts of the contact where the friction force has been overcome. Significant loosening occurs only when the entire fastener turns, which requires that the two fastener contact surfaces, i.e. at the head and threads, either undergo complete slip, or localized slip that accumulates over loading cycles. Loosening processes can be divided into four different types depending on the nature of slip at the head and thread contacts. These are characterized by (1) localized head slip with localized thread slip, (2) localized head slip with complete thread slip, (3) complete head slip with localized thread slip, and (4) complete head slip with complete thread slip. The influence of the type of slip on loosening is illustrated in Fig. 4, which shows a typical preload versus cycles plot of loosening obtained using the transverse vibration test apparatus. The loosening rate shows a drastic increase as soon as head slip changes from localized to complete slip. All the FE results presented in this paper model a 63.5 mm long, Grade 8, Class 2A/2B, 0.5-13 UNC screw (12.7 mm diameter) screw subjected to 1 mm displacement through the spring element of stiffness 1.2 kN/mm. The modulus of elasticity of the fastener materials used for the simulations is 200 GPa. The coefficient of friction is 0.261 for dry contact, 0.158 for contacts with machine oil lubrication, and 0.086 for MoS2 grease lubricated contacts based on previously obtained experimental data [14].
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Fig. 3. Loosening moments from component of thread reaction to preload.
Fig. 4. Typical loosening sequence of a screw during a transverse vibration test.
Figs. 5 and 6 show loosening resulting from localized head slip and complete thread slip for a fastener with head and thread contacts lubricated with machine oil and a preload of nearly 11 kN. This loosening process is the same as the initial loosening process occurring in Fig. 4. Fig. 5a shows a hysteresis curve, which is a plot of the shear force acting on the top plate versus its displacement. The slope of the hysteresis
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Fig. 5. Loosening process characterized by localized head slip with complete thread slip: (a) hysteresis curve, and (b) contact status.
curve provides an indication of the joint stiffness in the transverse direction. The reduction in slope of the hysteresis curve is a sign of slip at the contacts. Fig. 5b shows the state of the fastener at three points during the cycle. Black regions indicate contact regions that stick, while the gray regions indicate slip. Segments ab and cd in Fig. 5a indicate parts of the cycle where the threads undergo complete slip while the head contact undergoes localized slip (see Fig. 5bi and biii). Both thread and head undergo localized slip during parts of the cycle indicated by the steeper segments bc and da (Fig. 5bii). Note that head contact regions that stick during the first half of cyclic loading, slip during the second half (see Fig. 5bi and biii). This enables the entire head contact to slip over a complete cycle.
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Fig. 6. Loosening process characterized by localized head slip with complete thread slip: (a) thread turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . – . – ., at 270 ; — —, average, (b) head turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . — . — ., at 270 ; — —, average, and (c) preload.
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Fig. 6a and b shows the turning angle at four points around the thread and head respectively. The thread turn angle shown is the average from all engaged threads. Thread loosening angles show a rapid increase due to complete thread slip, while the head turning rate is lower due to localized slip. This results in the screw body being under torsion. The plots indicate that points at 90 and 270 oscillate with the applied load with a net loosening at the end of each half cycle as reflected by the angles at 0 and 180 . Since the screw body is in a state of torsion, the corresponding oscillating points at the head and thread move in opposite directions. Note that the thread turn angle is far larger than the head angle, and that the preload loss per cycle (Fig. 6c) is fairly low. Figs. 7 and 8 illustrate loosening caused by complete head slip and complete thread slip for the thread and head contacts lubricated with machine oil and at a preload of approximately 5.5 kN. The hysteresis curve shown in Fig. 7a indicates regions with three distinct slopes. Region i reflect parts of the loading where complete thread slip occurs and the head undergoes localized slip (Fig. 7bi). At region ii the entire head and thread slip as shown in Fig. 7bii. The initial stage of the unloading portion of the cycle is seen to have a higher slope because of localized slip at both the head and the thread (see Fig. 7biii). Fig. 8a and b shows the loosening at four points at the thread and head. The loosening angles at all four points are nearly identical. The head loosening angle shows nearly a step increase at the point where complete slip occurs, while the thread angle increases more gradually with a small step reflecting the occurrence of complete head slip. The loosening angles at the head and thread are comparable with the thread angle leading the head angle. This is expected since the entire loosening moment is developed at the threads. Also, the rate of preload drop (Fig. 8c) is significantly larger than that found with localized head slip illustrated in Fig. 6c. This loosening process corresponds to the loosening occurring at a rapid rate in Fig. 4. Hysteresis curves and loosening angles of the loosening process characterized by complete head slip and localized thread slip are shown in Figs. 9 and 10 for a fastener with MoS2 grease at the head and dry threads with a preload of about 11.9 kN. The hysteresis curve is characterized by two distinct slopes. The stiffer segments indicate parts of loading where the head and thread undergo localized slip (see Fig. 9bi). Regions ii and iii reflect parts of the cycle where the entire head slips, while the threads continue to undergo localized slip (see Fig. 9bii and biii). From Fig. 9bii and biii it is seen that different parts of the threads stick during different parts of the cycle (note that orientation of the ii is opposite of that of iii). The thread angles at four points (Fig. 10a) show that points at 90 and 270 oscillate with the applied input with a net increase in the loosening angle per half a loading cycle. The head angles (Fig. 10b) display a step increase at the point where the complete head slips, with the points at 90 and 270 reflecting the oscillations at the threads. As in the earlier cases, the thread angle leads the head angle. The loosening rate (Fig. 10c) is not as severe as in the case of complete thread and head slip (Fig. 8c). Figs. 11 and 12 illustrate the loosening process resulting from localized head and localized thread slip with dry threads and MoS2 grease at the head and at preload of nearly13.2 kN. The hysteresis curve shows nearly the same stiffness at all regions with a very slight reduction in the slope as the thread slip region increases. The regions of localized slip (Fig. 11b) are seen to vary at different parts of the cycle, however the lower thread contact stick throughout the entire cycle. Fig. 12a and 12b show the loosening angles at the thread and head, which essentially display an oscillatory character at the 90 and 270 points with a slight increase in the loosening angle per cycle. As with the other loosening processes, the thread loosening angle leads the head angle. The rate of preload loss (Fig. 12c) is seen to be quite small. Although the lower threads stick throughout the entire cycle in the results presented here, the preload loss resulting due to the localized slip at the first few threads can accumulate and cause progressively larger slip and eventually lead to complete thread slip. The above FE results illustrate the four possible loosening processes that can occur in a threaded fastener subjected to dynamic shear loads. Of these four processes, only three are widespread in practice. In the typical case of fastener loosening, the initial stage of loosening is characterized by localized slip at both the head and the threads. This progresses to loosening by localized head slip with complete thread slip, and
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Fig. 7. Loosening process characterized by complete head slip with complete thread slip: (a) hysteresis curve, and (b) contact status.
eventually to complete slip at the head and the threads. In some cases, the initial loosening starts with localized head slip and complete thread slip, and progresses to complete slip at the head and the threads. The loosening rate at the initial stages characterized by localized slip is fairly low and increases as the loosening transitions from localized to complete slip (see Fig. 4). A contribution of the present work is the identification of loosening caused by localized slip. Loosening at the initial stages resulting from localized slip is critical since it can occur at a significantly lower shear load than that required for complete slip. For
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Fig. 8. Loosening process characterized by complete head slip with complete thread slip: (a) thread turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . – . – ., at 270 ; — —, average, (b) head turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . — . — ., at 270 ; — —, average, and (c) preload.
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Fig. 9. Loosening process characterized by complete head slip with localized thread slip: (a) hysteresis curve, and (b) contact status.
example, in the data shown earlier, loosening by localized slip (Fig. 5a) occurs when the magnitude of the shear force acting on the joint is approximately 9% of the preload, while loosening by complete slip (Fig. 7a) occurs when the shear force is 16% of the preload. In this case, loosening by localized slip occurs at about half of the load required to cause complete slip. Loosening by localized slip is therefore the critical loosening process from a perspective of joint design. Loosening failure in a joint can be avoided by ensuring that the dynamic loads acting on the fastener are lower than the loads required to cause loosening by localized slip.
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Fig. 10. Loosening process characterized by complete head slip with localized thread slip: (a) thread turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . – . – ., at 270 ; — —, average, (b) head turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . — . — ., at 270 ; — —, average, and (c) preload.
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Fig. 11. Loosening process characterized by localized head slip with localized thread slip: (a) hysteresis curve, and (b) contact status.
4. Comparison with experiments Fig. 13 shows hysteresis curves comparing experimental data with FE results for the four different loosening processes. The FE results agree reasonably well with experimental results as far as displaying the four loosening processes reflected by the slopes of the hysteresis curves. Fig. 13a shows the hysteresis curves for the screw lubricated with oil at the threads and head at preloads of 11 and 5.5 kN. The hysteresis curves at the higher preload representing loosening by localized head slip and complete thread slip show a very good match. The comparison of loosening process of complete head and thread slip at the preload of 5.5 kN
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Fig. 12. Loosening process characterized by localized head slip with localized thread slip: (a) thread turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . – . – ., at 270 ; — —, average, (b) head turn angle at four points: ——, at 0 ; – – –, at 90 ; , at 180 ; . — . — ., at 270 ; — —, average, and (c) preload.
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Fig. 13. Comparison of experimental hysteresis curves and FE results: (a) ——, experiment 11 kN preload with oil lubrication; , FE 11 kN; – – –, experiment 5.5 kN preload with oil lubrication; &, FE 5.5 kN preload. (b) ——, experiment 11.9 kN preload with dry threads and MoS2 grease at head; , FE, and (c) ——, experiment 13.2 kN preload with dry threads and MoS2 grease at head; , FE. (63.5 mm long, Grade 8, 0.5-UNC 13 screw, 1 mm eccentric setting).
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capture the trends, however the net slip occurring during complete head slip is smaller in the FE results than in the experiments. The amount of slip reflected in the hysteresis curve is a function of the fastener dimensions, and is largely influenced by the clearance between the internal and external threads. The difference in results is most likely because the thread dimensions utilized for the FE simulations were defined in the middle of the allowable range for Class 2A/2B threads, and the thread dimensions in the test specimens may be slightly different. The hysteresis curves shown in Fig. 13b show loosening by complete head slip and localized thread slip for a screw with MoS2 grease at head and dry threads at 11.9 kN preload. The FE result is seen to capture the experimental data quite well, and as with the previous case the slip is slightly smaller. A similar trend is observed for loosening caused by localized thread and head slip shown in Fig. 13c for the same lubrication condition at 13.2 kN preload. In all four cases, especially in Fig. 13b and c, the FE results are seen to be slightly stiffer than the experimental data. This is because the FE model includes six threads before the first engaged thread instead of 12 threads found in the test specimens. The additional threads in the screws used in the experiments contribute to the lower bending stiffness observed in the experimental data. A FE model with 12 exposed threads could not be used because of the significant additional computational cost. The influence of this omission is clearly not very significant as indicated by the reasonably good comparisons between the results.
5. Fastener slip The four different loosening processes described earlier are a result of the slip occurring at the fastener contact surfaces. Slip at the fastener surfaces is a function of the distribution of loads acting tangential to the contact surface (i.e. the forces that cause slip), and the contact normal force distribution (since it influences the friction). Fig. 14 summarizes the main factors that influence slip, S, and normal reactions, R, in a joint subjected to shear load [14]. The fastener has an initial loosening moment L due to the thread reaction to preload (see Fig. 3), which is balanced by the friction moments at the threads and the head. During the loading, the clamped component is subjected to a shear load FS, which is transferred to the screw through the head friction. The shear force can also be transferred to the screw through side contact between the clamped component and the screw body, which is caused by head slip or fastener bending. The component of the shear force acting tangential to the thread flank contributes to thread slip, while the remaining part changes the thread normal reaction distribution. The bending moment developed from the shear force changes the head and thread reaction distributions. The bending moment also contributes to slip along the thread flank due to the turning action about the moment axis. As a result of the shear load and the bending moment, the fastener undergoes elastic deformation at the head and the threads, which also contributes to slip. The resulting slip from all of the above factors is also influenced by the state of side contact at the head and the thread (not shown). From the above description it can be seen that several factors influence the loosening process. The main factors have been separated for the purpose of illustration. However, all these factors are coupled non-linearly and together determine the final state and process of loosening. Some important aspects of these factors that influence loosening are discussed below.
6. Fastener bending It is seen in Fig. 14 that the bending moment contributes to slip and loosening in several ways. The bending moment contributes to localized slip by changing the reaction distribution which causes slip in regions with lower reaction force (for example see slip regions at the head in Figs. 5b and 11b). The bending moment also causes slip at the thread flanks, and contributes to slip by elastic deformation.
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Fig. 14. Summary of factors influencing slip and reaction force distribution in a joint subjected to shear load. (FS–Applied shear force, L–loosening moment, S–tangential slip force, R–normal reaction).
The influence of bending moment on slip is illustrated in Fig. 15. It shows the shear force required to cause complete slip at the threads for a 0.5–13 UNC screw of different lengths at a preload of 13.3 kN. The shear force required to cause complete thread slip in a 63.5 mm screw is nearly half that for a 31.8 mm screw. This indicates that the additional bending moment developed in the longer screw is a significant factor responsible for thread slip. The bending moment also contributes to head slip by elastic deformation of the head. This is because the bending moment causes a change in the head normal reaction distribution, which changes the head surfaces deformation and contributes to slip as shown by the illustration at the bottom right corner in Fig. 14. The above data indicate that long fasteners require lower shear force to loosen. However, this is applicable only to force loaded systems, such as fasteners used to mount a pillow block bearing subjected to dynamic loads from an unbalanced rotating mass. In the case of displacement-loaded systems such as the transverse vibration test apparatus (Fig. 1) longer fasteners can reduce loosening. This is because the shear force acting at the contacts in such systems is a function of the applied displacement and the fastener stiffness. Since longer fasteners have a lower bending stiffness, they are subjected to lower shear force, and therefore are less vulnerable to loosening. Experience with displacement loaded systems support the notion that longer bolts inhibit vibration induced loosening [2,23]. However, the opposite is true (i.e. longer bolts are more prone to loosening) for force-loaded systems, and this is not widely appreciated.
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Fig. 15. Shear force required to cause complete thread slip for different screw length. (0.5-13 UNC screw with dry contacts).
7. Other factors Another aspect of the loosening process is the influence of the thread load distribution on thread slip. In the results presented earlier, it is found that the first few threads slip before the last ones (see Fig. 11b for example). This is mainly because of the normal load distribution at the threads as shown in Fig. 16. The data show that the first three threads in a FE model with five and a half engaged threads carry about 73% of the preload. This means that the first three threads have significantly higher frictional resistance, and only once these are overcome can the lower threads begin to slip. This is reflected in Fig. 17, which shows the turn angle for nodes at the 90 location on the first five threads for the loosening process caused by localized head and thread slip (Figs. 11 and 12). The magnitude of the turn angle is largest at the first thread and reduces in subsequent threads. The turn angles of the first few threads reflect slip as well as elastic deformation, while those at the last few threads are mainly a result of elastic deformation. These results indicate that the load distribution at the threads influence the loosening process.
Fig. 16. FE results of load distribution at the engaged threads for 0.5-13 UNC screw.
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Fig. 17. Thread turn angle at 90 node for loosening process characterized by localized head slip with localized thread slip: ——, first engaged thread; – – –, second thread; , third thread; . — . —, fourth thread; — , fifth thread.
In all the cases presented in this work, the fastener preload has been fairly low so has to minimize the influence of localized thread yielding, which is likely to influence the thread load distribution. Inclusion of thread yielding in the FE model requires a denser mesh to ensure accurate determination of stresses. This was not attempted at this stage of the study due to the extremely high computational cost. The effect of thread yielding on loosening will be addressed in future work. In addition to fastener length, and thread load distribution, there are various other parameters that influence loosening, including the fastener material, and dimensional tolerances. Since all these factors are coupled nonlinearly, it is important to study their individual effect as well as the influence of their interaction on screw loosening. Results from a parameter study investigating these effects will be reported in a future paper.
8. Conclusion The finite element model presented is capable of predicting the four different loosening processes observed experimentally. The FE results capture the essential features displayed by the experimental data. Several factors influence loosening at different stages of loading, and the final outcome is a result of their non-linear interactions. The FE model includes the primary factors that cause loosening, and provides a powerful tool for evaluation of the details of fastener loosening. The loosening process caused by localized slip can occur at significantly lower shear force than loosening caused by complete slip, and therefore is critical in joint design. Acknowledgements The authors gratefully acknowledge the support from the National Science Foundation under Grant No. CMS-9629217. References [1] Hess DP. Vibration- and shock- induced loosening. In: Bickford JH, Nasser S, editors. Handbook of bolts and bolted joints. New York: Marcel Dekker, 1998. p. 757–824.
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