July 24, 2017 | Author: Abhijith Madabhushi | Category: Screw, Nut (Hardware), Stress (Mechanics), Helix, Finite Element Method

#### Description

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

1

Introduction

Journal of Pressure Vessel Technology

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

FEBRUARY 2008, Vol. 130 / 011204-1

␪1 =

where d and H represent nominal diameter and thread overlap. The profile of internal thread can be expressed in the same manner.

r=

d1 2

d 7 H ␪+ − H ␲ 2 8

d H + − 2␳n + 2 8

␲ ␪1 = 4

r=

␳2 −

P2 2 ␪ 共0 艋 ␪ 艋 ␪1兲 4␲2

d 7 H ␪+ − H ␲ 2 8

d 2

Fig. 2 Profile of the cross section of external thread perpendicular to the bolt axis

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␳2n −

P2 共␲ − ␪兲2 共␪2 艋 ␪ 艋 ␲兲 4␲2

␪2 = ␲ 1 −

␳n 艋

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

Fig. 3 Accurate cross section profile of metric coarse thread

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

Fig. 8 Normalized maximum Mises stress occurred at the bolt thread root

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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

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

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.

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