Proposition of Helical Thread Modeling With Accurate Geometry and Finite Element Analysis Toshimichi Fukuoka Professor e-mail:
[email protected]
Masataka Nomura Associate Professor e-mail:
[email protected] Faculty of Maritime Sciences, Kobe University, 5-1-1 Fukaeminami, Higashinada, Kobe 658-0022, Japan
Distinctive mechanical behavior of bolted joints is caused by the helical shape of thread geometry. Recently, a number of papers have been published to elucidate the strength or loosening phenomena of bolted joints using three-dimensional finite element analysis. In most cases, mesh generations of the bolted joints are implemented with the help of commercial software. The mesh patterns so obtained are, therefore, not necessarily adequate for analyzing the stress concentration and contact pressure distributions, which are the primary concerns when designing bolted joints. In this paper, an effective mesh generation scheme is proposed, which can provide helical thread models with accurate geometry to analyze specific characteristics of stress concentrations and contact pressure distributions caused by the helical thread geometry. Using the finite element (FE) models with accurate thread geometry, it is shown how the thread root stress and contact pressure vary along the helix and at the nut loaded surface in the circumferential direction and why the second peak appears in the distribution of Mises stress at thread root. The maximum stress occurs at the bolt thread root located half a pitch from nut loaded surface, and the axial load along engaged threads shows a different distribution pattern from those obtained by axisymmetric FE analysis and elastic theory. It is found that the second peak of Mises stress around the top face of nut is due to the distinctive distribution pattern of z. 关DOI: 10.1115/1.2826433兴 Keywords: fixing element, bolted joint, FEM, helical thread modeling, stress concentration, contact pressure distribution
1
Introduction
Threaded fasteners are the most widely used machine elements because they can repeatedly be assembled and disassembled by an easy operation. Mechanical behaviors of the threaded fasteners, such as the strength and the stiffness of bolted joints, have been analyzed by experiment, theoretical analysis based on elastic theory, and numerical method. Finite element method 共FEM兲 is found to be the most powerful numerical method for solving the problems of bolted joints. The development of FEM made it possible to evaluate the stress concentration at the thread root with high accuracy 关1,2兴. The stiffness of bolted joints, which has a dominant effect on its fatigue strength, and stress concentrations of the bolt thread and the bolt head fillet have also been studied systematically with help of FEM 关3,4兴. In the conventional studies on the stress analysis of bolted joints, axisymmetric FEM has mainly been used. In the case of three-dimensional analysis, threaded portions were modeled by using the threads with axisymmetric geometry, i.e., the effects of lead angle and the helix of thread profile were neglected. Recently, some researchers have started to use helical thread models that were constructed with the advanced modeling functions provided by a couple of commercial software 关5–8兴. Some studies have tried to elucidate the loosing phenomena of bolted joints using the helical thread models thus obtained 关9兴. However, the aforementioned procedures do not necessarily provide helical Contributed by the Pressure Vessel and Piping Division of ASME for publication in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received September 1, 2006; final manuscript received December 8, 2006; published online January 17, 2008. Review conducted by Sayed Nassar. Paper presented at the 2006 ASME Pressure Vessels and Piping Conference 共PVP2006兲, Vancouver, British Columbia, Canada, July 23–27, 2006.
Journal of Pressure Vessel Technology
thread models adequate for analyzing the stress concentration at the thread root and contact pressure distributions at nut loaded surface, because of the complexity of thread profile and the limitation of software’s functions. Meanwhile, a thread cross section perpendicular to the bolt axis is identical at any position. Accordingly, the thread profile can be defined mathematically using rigorous expressions by taking the effects of root radius, where the cross section is divided into three portions. In this paper, an effective modeling scheme for threedimensional FE analysis, which can accurately construct helical thread geometry, is proposed using the equations defining the thread cross section perpendicular to the bolt axis. The present procedure has such beneficial performances as modeling each thread with one-pitch height independently and using fine meshes only around threaded portions. Therefore, it is possible to construct finite element models of bolted joints with high accuracy and computation efficiency. Using the FE models thus obtained, the mechanical behavior caused by the helical thread geometry has been evaluated, such as the distributions of the thread root stress along the helix and nonsymmetric contact pressure distributions at the nut loaded surface. It is found that the maximum bolt stress occurs at the thread root located half a pitch from nut loaded surface, and the axial load along engaged threads shows a different distribution pattern from the previous studies by taking the helical thread geometry into account.
2 Mathematical Expressions of Thread Cross Section Profile The specifications of thread profiles are given in ISO 68, 261, 262, and 724. The thread root has an appropriate amount of roundness to avoid an excessive stress concentration. In Japanese Industrial Standard 共JIS兲, it is recommended that the thread root radius
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1 =
冑3 P
冑3 冑3 7 2 = 艋 P H = P 8 12 2
where d and H represent nominal diameter and thread overlap. The profile of internal thread can be expressed in the same manner.
r=
冦
d1 2
共0 艋 艋 1兲
d 7 H + − H 2 8
共 1 艋 艋 2兲
d H + − 2n + 2 8
1 = 4
should be more than 0.125P 共P: thread pitch兲 for external threads made of high strength steel. Figure 1 shows the cross sectional profile along the bolt axis including the thread root radius. Assuming that the rounded portion of the thread root is a part of a single circle with diameter , the surface of external thread can be divided into three parts such as A-B 共root radius兲, B-C 共thread flank兲, and C-D 共crest兲. The thread profile perpendicular to the bolt axis can be obtained by expanding those three parts into the plane, as shown in Fig. 2. Its shape is naturally identical at any cross section along the bolt axis. In the next section, helical thread models are to be constructed by utilizing the characteristics explained here. The thread profile shown in Fig. 1 is expressed by means of the following equations.
r=
冦
冑
2 −
P2 2 共0 艋 艋 1兲 42
d 7 H + − H 2 8
共 1 艋 艋 2兲
d 2
共2 艋 艋 兲
冧
共1兲
Fig. 2 Profile of the cross section of external thread perpendicular to the bolt axis
011204-2 / Vol. 130, FEBRUARY 2008
2n −
冉
P2 共 − 兲2 共2 艋 艋 兲 42
2 = 1 −
冑3n P
冊
n 艋
冑3 24
冧
共2兲
P
There are upper limits for the root radii of external and internal threads, and n, appeared in Eqs. 共1兲 and 共2兲, in connection with the thread geometry of minor and nominal diameters.
Fig. 1 Thread cross section along the bolt axis
d 7 − H + 2 − 2 8
冑
3 Proposition of Helical Thread Modeling With Accurate Geometry 3.1 Conventional Methods. When analyzing the mechanical behavior of bolted joints with three-dimensional analysis, it has been a common practice that the threaded portion of the FE models has axisymmetric geometry, where the effects of lead angle are neglected because of its small value. That is, external and internal threads are modeled by stacking an appropriate number of threads with axisymmetric geometry. Recently, some researchers start to use helical thread models because of a growing recognition of the importance of helical effects, e.g., loosening phenomena of bolted joints. Their modeling procedures are classified roughly into three categories in the case of external threads 关5–9兴. Type 1. Two-dimensional thread cross section model with onepitch height is rotated helically around the bolt axis 关5–7兴. This procedure inevitably generates a small hole around the bolt axis. Type 2. Helical thread model made in the similar manner to Type 1 is attached around a solid cylinder 关8,9兴. Mesh patterns are not coincident at the interface between helical threads and the cylinder. Type 3. Surface models of bolt and nut are made by means of a sophisticated performance provided by commercial software, and then the inside of the helical-shaped solid models of bolt and nut is divided into three-dimensional elements using its automatic mesh generation function. In the case of Type 1, the effect of the small hole seems insignificant. However, unfavorable meshes are to be generated due to the helical rotation, especially around the far end thread and the area connecting the thread runout and the bolt cylinder. The same problem still remains in the case of Type 2. It is not an easy practice even for Type 3 that highly stressed area is intensively divided using small elements while the overall mesh pattern being well balanced. 3.2 Helical Thread Modeling by Stacking Cross Sections With Accurate Geometry. Helical thread modeling procedure proposed in this paper is based on the fact that the shape of the cross section perpendicular to the bolt axis is identical at any position. The profiles of external and internal threads are expressed mathematically by means of Eqs. 共1兲 and 共2兲. In Fig. 3, illustrated are the real shapes of the cross section of external threads with coarse pitch of P. In the following, it is shown how the helical thread models of external thread with accurate geometry can be constructed, where each ridge with one-pitch height is divided into n thin plates with the same configuration. The procedure consists of six steps. Transactions of the ASME
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Fig. 3 Accurate cross section profile of metric coarse thread
Step 1. Cross section of external thread with accurate geometry, shown in Fig. 3, is properly modeled using two-dimensional elements. This is a basic mesh model. Step 2. The basic model is placed at the reference position, z = 0. Step 3. Rotating the basic model anticlockwise by an amount of 2 / n, it is placed at the position of z = P / n. Step 4. Connecting the corresponding nodes of the two basic models placed at z = 0 and P / n, a three-dimensional model with P / n thickness is obtained. Step 5. One-pitch helical thread model is completed by repeating Steps 3 and 4 n times. Step 6. An appropriate number of one-pitch model obtained in Step 5 is stacked, according to the number of threads of the objective bolted joint. If the one-pitch model is constructed by simply stacking the basic model according to the aforementioned procedure, the elements around the threaded portion might have a large aspect ratio, which causes low accuracy of the numerical analysis. In addition, the mesh pattern around the bolt axis becomes finer than is necessary. From the numerical accuracy and computation efficiency points of view, therefore, the finite element meshes for thread area and bolt core portion should be constructed separately. Figure 4 shows an example of the mesh patterns of the cross sections perpendicular to and along the bolt axis. The circular area inside the four arrows, shown in Fig. 4共a兲, is divided by rather coarse meshes, and the outside area is modeled by fairly fine meshes. Therefore, the bolt core portion is simply modeled as a cylinder and only the thread area is modeled following Steps 1–6. Figure 5共a兲 shows a one-pitch helical thread model thus obtained. The mesh patterns of the two separate models are completely coincident at the interface. It follows that the helical model constructed here is expected to attain both high accuracy and computation efficiency. Thread runout is modeled by gradually varying the depth of the groove along the helix so as to be smoothly connected with bolt cylinder. Following the above procedure, it is possible to construct an entire bolted joint model only by eightnode brick elements. Internal threads can be modeled in the same manner. In this case, the outside area of threaded portion is modeled as a hollow cylinder. The outer surface of the nut is modeled as a cylindrical shape for simplicity, although it is possible to construct a hexagonal nut. Figure 5共b兲 shows the cross section of the nut model with helical geometry. Figure 6 is an example of the entire bolted joint model, which is tightened by a single bolt with coarse thread of M16. The total numbers of nodes and elements are 78,520 and 86,504, respectively. Numerical analysis with FE models constructed here can be implemented by standard FE analysis 共FEA兲 software packages. Journal of Pressure Vessel Technology
Fig. 4 Mesh patterns of cross sections of bolt model
4 Stress Analysis of Bolted Joints Using Helical Thread Model 4.1 Numerical Models and Boundary Conditions. The mesh generation scheme proposed here can be executed without any help of commercial software. However, it is favorable to use some sophisticated functions of commercial software for an effec-
Fig. 5 One-pitch model of external thread and cross section of nut model
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Fig. 6 Fienit element model of entire bolted joint
Fig. 7 Mises stress distributions at the bolt thread root along the helix
tive modeling. In this study, Hyper Works is used for supporting the mesh generation and the numerical calculations are conducted as elastic problems by ABAQUS. Referring to the FE model shown in Fig. 6, the axial bolt force is applied as a uniform displacement at the lower end of the bolt cylinder. At the lower surface of the fastened plate, axial displacements are completely restrained and the circumferential ones are restrained at four nodes located 90 deg apart. The analytical objects are bolted joints tightened by a single bolt of M16 or M12 with coarse thread. Bolt, nut, and plate are supposed to be made of carbon steel whose Young’s modulus and Poisson’s ratio are 200 GPa and 0.3, respectively. In the case of M16, bolt hole diameters are changed as 17 mm, 17.5 mm, and 18.5 mm, which correspond to the first, second, and third classes specified in JIS, respectively. Coefficients of friction are varied from 0.05 to 0.20 with an increment of 0.05 and assumed to be identical at pressure flank of screw thread and nut loaded surface. For a parametric study, a standard analytical condition is defined as follows. Standard condition: M16, coefficient of friction= 0.15, bolt hole diameter= 17.5 mm 共second class兲 4.2 Stress Distributions Along Thread Root. Many previous studies have reported that the maximum stress occurs at the bolt thread root located within one pitch of the nut loaded surface. Most of the research on the stress concentration at the thread root have been conducted using axisymmetric FE models. Even if introducing such helical thread models explained in Sec. 3.1, it seems difficult to evaluate the stress concentrations around the thread root with practical accuracy. In this section, stress concentrations around the thread root are analyzed using the FE models obtained in Sec. 3.2. Numerical calculations were performed using a single computer equipped with Pentium 4 of 3.4 GHz with 2 Gbyte RAM. CPU time changes from 3 h to 5 h as coefficient of friction increases. It is shown in Fig. 7 how the maximum stress, which occurred at the thread root, varies along the helix. Mises stress at the thread root eq is normalized with respect to the mean tensile stress b defined at the bolt cylinder. The abscissa represents the distance from the nut loaded surface. The maximum Mises stress eqmax occurs at half a pitch from the nut loaded surface, as in the case of the previous studies 关10兴, where larger coefficient of friction produces higher peak stress. Then, the stresses at the thread root gradually decrease toward the top face of the nut, and they show a second peak. It is considered that this phenomenon is caused by the low stiffness of the last engaged thread for its bending deformation. In Fig. 8, shown are the effects of friction coefficient and bolt hole diameter on the maximum normalized stress eqmax / b. It increases slightly and almost linearly as coefficient of friction increases, and it almost decreases 011204-4 / Vol. 130, FEBRUARY 2008
linearly with increasing bolt hole diameter. As for the effects of nominal diameter, larger bolt produces larger stress concentrations as well as the previous studies 关1兴. 4.3 Asymmetric Contact Pressure Distributions at Nut Loaded Surface. The contact pressure at the nut loaded surface decreases outward in the radial direction. It is predicted that the contact pressure also varies in the circumferential direction, though probably a small amount, because of the circumferential variation of the stiffness of engaged threads adjacent to the nut loaded surface. The latter phenomenon can be analyzed only when introducing a helical thread model. Figure 9共a兲 shows the circumferential contact pressure distributions at the nut loaded surface for varying radial positions. The reference point of = 0 is placed at the plate top surface on which a fully formed nut thread with one-pitch height exists. The magnitude of the contact pressure varies in the circumferential direction, which is rather remarkable along the bolt hole and at the outer end of the nut loaded surface. In the radial direction, though not shown here, the contact pres-
Fig. 8 Normalized maximum Mises stress occurred at the bolt thread root
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Fig. 10 Axial load distributions along engaged threads
tained by the helical thread model is compared to that by Yamamoto’s equation. Numerical result by axisymmetric FE analysis is also shown, where the mesh pattern is the same as the cross section of the helical thread model along the bolt axis. The axisymmetric analysis gives a similar load distribution to that of concaveshaped Yamamoto’s equation, except around the top face of the nut. On the other hand, the numerical result by the helical thread model shows slightly convex distributions both around the nut loaded surface and the top face of the nut. In the cases of Yamamoto’s equation and axisymmetric FE analysis, it is assumed that every set of male and female threads is equally engaged. In the actual engaged threads, however, the contact areas of engaged threads rapidly decrease around the nut loaded surface and the top face of the nut. It is considered that such effects could be represented by the helical thread models introduced here.
5 Fig. 9 Circumferential contact pressure distributions at the nut loaded surface
sure decreases smoothly at any circumferential position. Shown in Fig. 9共b兲 are the contact pressure distributions in the circumferential direction for third class bolt hole. The effect of the helical shape of thread geometry appears similarly in the case of second class. Such circumferential variation of the contact pressure distributions might cause various problems in the bolted joints. 4.4 Evaluation of Load Distributions Along Engaged Threads. In this study, finite element meshes around thread ridges have equal thickness in the axial direction, as shown in Fig. 4共b兲. Accordingly, the load distribution along engaged threads can be evaluated by summing up the axial loads exerted on each thin element with equal thickness. In a common bolted joint, bolt cylindrical portion is subjected to axial bolt force Fb at any position between nut loaded surface and bolt head. Meanwhile, the axial load F along the engaged threads gradually decreases from Fb to zero toward the top surface of the nut. It is well known that such load distribution pattern causes various problems inherent to boltnut connections. Yamamoto derives an equation for F along the engaged threads based on the elastic theory, which shows that the axial load F decreases following a hyperbolic function, sinh共x兲, where x denotes the distance from the nut loaded surface 关11兴. In Fig. 10, the load distribution along the engaged threads obJournal of Pressure Vessel Technology
Discussions
A nut is classified into several kinds according to its shapes around bearing surface and top face. The nut used here has a flat bearing surface that is completely in contact with the plate surface. The threads at the top face of the nut are commonly chamfered, i.e., truncated at some angle, toward the bolt hole. The effect of the chamfering is studied by FE analysis. Figure 11共a兲 illustrates the nut cross section with and without chamfering. All the numerical results presented so far are associated with the chamfered nut models. Figure 11共b兲 represents the effect of the chamfering on the stress concentrations at the thread root. It is observed that for both chamfered and nonchamfered nuts, z shows characteristic stress distribution patterns, which steeply vary between positive and negative values. Accordingly, it seems that the second peak appearing in the Mises stress distribution is caused by the distinctive distribution pattern of z. In the case of nonchamfered nut, the second peak of Mises stress shows an unnatural decrease compared to the case of chamfered nut. This phenomenon can be mitigated by chamfering the top face of the nut.
6
Conclusions
An effective three-dimensional thread modeling scheme, which can accurately take account of its helical geometry, is proposed using the equations defining the real configuration of the thread cross section perpendicular to the bolt axis. It is shown how the thread root stress varies along the helix and that the maximum stress occurs at half a pitch from the nut loaded FEBRUARY 2008, Vol. 130 / 011204-5
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Nomenclature ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ eqmax ⫽ z ⫽ Di d d1 F Fb H P r z , n b eq
bolt hole diameter nominal diameter minor diameter axial load along engaged threads axial bolt force thread overlap thread pitch radial coordinate axial coordinate coefficient of friction circumferential coordinate root radii of external and internal threads mean tensile stress defined at bolt cylinder Mises stress at thread root maximum Mises stress at thread root axial stress
References
Fig. 11 Effect of the chamfering of the nut top thread
surface. The stresses at the thread root gradually decrease toward the top face of the nut and they show a second peak because of the low stiffness of the last engaged threads. It is shown how the contact pressure at the nut loaded surface varies in the circumferential direction due to the effect of the helical thread geometry. The axial load distribution along engaged threads analyzed by helical thread models shows a different distribution pattern from those obtained by axisymmetric FE analysis and elastic theory. The second peak appearing in the distributions of Mises stress at the thread root is caused by the distinctive distribution pattern of z.
Acknowledgment The authors would like to acknowledge Mr. Yuuya Morimoto 共DAIHATSU Motor Co.兲 for his contribution to the numerical calculations conducted in this research.
011204-6 / Vol. 130, FEBRUARY 2008
关1兴 Fukuoka, T., 1997, “Evaluation of the Method for Lowering Stress Concentration at the Thread Root of Bolted Joints With Modifications of Nut Shape,” ASME J. Pressure Vessel Technol. 119共1兲, pp. 1–9. 关2兴 Fukuoka, T., and Takaki, T., 2003, “Elastic Plastic Finite Element Analysis of Bolted Joint During Tightening Process,” ASME J. Mech. Des. 125共4兲, pp. 823–830. 关3兴 Lehnhoff, T. F., Ko, K. I., and Mckay, M. L., 1994, “Member Stiffness and Contact Pressure Distribution of Bolted Joints,” ASME J. Mech. Des. 116共2兲, pp. 550–557. 关4兴 Lehnhoff, T. F., and Bunyard, B. A., 2000, “Bolt Thread and Head Fillet Stress Concentration Factors,” ASME J. Pressure Vessel Technol. 122共2兲, pp. 180– 185. 关5兴 Chen, J., and Shih, Y., 1999, “A Study of the Helical Effect on the Thread Connection by Three Dimensional Finite Element Analysis,” Nucl. Eng. Des. 191, pp. 109–116. 关6兴 Bahai, H., and Esat, I. I., 1994, “A Hybrid Model for Analysis of Complex Stress Distribution in Threaded Connectors,” Comput. Struct. 52共1兲, pp. 79– 93. 关7兴 Rhee, H. C., 1990, “Three-Dimensional Finite Element Analysis of Threaded Joint,” Proceedings of the Nintu International Confedence on Offshore Mechanics Arctic and Engineering, Vol. 3, Pt. A, pp. 293–297. 关8兴 Zadoks, R. I., and Kokatam, D. P. R., 1999, “Three-Dimensional Finite Element Model of a Threaded Connection,” Comput. Model. Simul. Eng. 4共4兲, pp. 274–281. 关9兴 Zhang, M., and Jiang, Y., 2004, “Finite Element Modeling of Self-Loosening of Bolted Joints,” PVP 共Am. Soc. Mech. Eng.兲, 478, pp. 19–27. 关10兴 Fukuoka, T., Yamasaki, N., Kitagawa, H., and Hamada, M., 1986, “Stresses in Bolt and Nut—Effects of Contact Conditions at the First Ridge-,” Bull. JSME 29共256兲, pp. 3275–3279. 关11兴 Yamamoto, A., 1970, Theory and Practice of the Tightening Process of Screw Threads, Yokendo, Tokyo, pp. 1–2 共in Japanese兲.
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