Three-dimensional Finite Element Analysis of Threa

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    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 sub-

    jected 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 [35] 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 sub-

    jected to dynamic loads perpendicular to the thread axis (shear loading). Loosening under shear loading

    1350-6307/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.P I I : S 1 3 5 0 - 6 3 0 7 ( 0 1 ) 0 0 0 2 4 - 3

    Engineering Failure Analysis 9 (2002) 383402

    www.elsevier.com/locate/engfailanal

    * Corresponding author. Tel.: +1-813-974-5643; fax: +1-813-974-1447.

    E-mail address: [email protected] (D.P. Hess).

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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 [713] to model the loosening phe-

    nomenon, 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 localizedslip 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 [1520] 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 three-

    dimensional 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 finiteelement 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 ana-

    lysis. 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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    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-

    Fig. 2. Typical finite element mesh: (a) joint model, (b) boundary conditions, and (c) contact regions on the screw.

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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 20-

    node 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 loosen-

    ing results are based on the displacements, which converge with a relatively coarse mesh compared to themesh 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 ten-

    dency 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 cir-

    cumferential friction that resists the loosening moment. However, as a result of an applied shear load, thecircumferential 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 char-

    acterized 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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    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

    Fig. 4. Typical loosening sequence of a screw during a transverse vibration test.

    Fig. 3. Loosening moments from component of thread reaction to preload.

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    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.

    Fig. 5. Loosening process characterized by localized head slip with complete thread slip: (a) hysteresis curve, and (b) contact status.

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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 appliedload 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 slipoccurs, 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 illu-

    strated 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 stickduring 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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    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

    Fig. 7. Loosening process characterized by complete head slip with complete thread slip: (a) hysteresis curve, and (b) contact status.

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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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    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.

    Fig. 9. Loosening process characterized by complete head slip with localized thread slip: (a) hysteresis curve, and (b) contact status.

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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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    4. Comparison with experiments

    Fig. 13 shows hysteresis curves comparing experimental data with FE results for the four different loos-

    ening 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

    Fig. 11. Loosening process characterized by localized head slip with localized thread slip: (a) hysteresis curve, and (b) contact status.

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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 allowablerange 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 con-

    tribute 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 screwthrough 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 compo-

    nent 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-line-

    arly 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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    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 0.513 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 applic-

    able 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.

    Fig. 14. Summary of factors influencing slip and reaction force distribution in a joint subjected to shear load. (FSApplied shear force,

    Lloosening moment, Stangential slip force, Rnormal reaction).

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    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 localizedhead 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. 15. Shear force required to cause complete thread slip for different screw length. (0.5-13 UNC screw with dry contacts).

    Fig. 16. FE results of load distribution at the engaged threads for 0.5-13 UNC screw.

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    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 non-

    linearly, 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.

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