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DoIntroductionThreaded fasteners are the most widely used machine elements

ecause they can repeatedly be assembled and disassembled by anasy operation. Mechanical behaviors of the threaded fasteners,uch as the strength and the stiffness of bolted joints, have beennalyzed by experiment, theoretical analysis based on elasticheory, and numerical method. Finite element method FEM isound to be the most powerful numerical method for solving theroblems of bolted joints. The development of FEM made it pos-ible to evaluate the stress concentration at the thread root withigh accuracy 1,2. The stiffness of bolted joints, which has aominant effect on its fatigue strength, and stress concentrationsf the bolt thread and the bolt head fillet have also been studiedystematically with help of FEM 3,4. In the conventional studiesn the stress analysis of bolted joints, axisymmetric FEM hasainly been used. In the case of three-dimensional analysis,

hreaded portions were modeled by using the threads with axisym-etric geometry, i.e., the effects of lead angle and the helix of

hread profile were neglected.Recently, some researchers have started to use helical threadodels that were constructed with the advanced modeling func-

ions provided by a couple of commercial software 58. Sometudies have tried to elucidate the loosing phenomena of boltedoints using the helical thread models thus obtained 9. However,he aforementioned procedures do not necessarily provide helical

thread models adequate for analyzing the stress concentration atthe thread root and contact pressure distributions at nut loadedsurface, because of the complexity of thread profile and the limi-tation of softwares functions. Meanwhile, a thread cross sectionperpendicular to the bolt axis is identical at any position. Accord-ingly, the thread profile can be defined mathematically using rig-orous expressions by taking the effects of root radius, where thecross section is divided into three portions.

In this paper, an effective modeling scheme for three-dimensional FE analysis, which can accurately construct helicalthread geometry, is proposed using the equations defining thethread cross section perpendicular to the bolt axis. The presentprocedure has such beneficial performances as modeling eachthread with one-pitch height independently and using fine meshesonly around threaded portions. Therefore, it is possible to con-struct finite element models of bolted joints with high accuracyand computation efficiency. Using the FE models thus obtained,the mechanical behavior caused by the helical thread geometryhas been evaluated, such as the distributions of the thread rootstress along the helix and nonsymmetric contact pressure distribu-tions at the nut loaded surface. It is found that the maximum boltstress occurs at the thread root located half a pitch from nut loadedsurface, and the axial load along engaged threads shows a differ-ent distribution pattern from the previous studies by taking thehelical thread geometry into account.

2 Mathematical Expressions of Thread Cross SectionProfile

The specifications of thread profiles are given in ISO 68, 261,262, and 724. The thread root has an appropriate amount of round-ness to avoid an excessive stress concentration. In Japanese Indus-trial Standard JIS, it is recommended that the thread root radius

Contributed by the Pressure Vessel and Piping Division of ASME for publicationn the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received September 1,006; final manuscript received December 8, 2006; published online January 17,008. Review conducted by Sayed Nassar. Paper presented at the 2006 ASME Pres-ure Vessels and Piping Conference PVP2006, Vancouver, British Columbia,anada, July 2327, 2006.

ournal of Pressure Vessel Technology FEBRUARY 2008, Vol. 130 / 011204-1Copyright 2008 by ASMEToshimichi FukuokaProfessor

e-mail: fukuoka@maritime.kobe-u.ac.jp

Masataka NomuraAssociate Professor

e-mail: nomura@maritime.kobe-u.ac.jp

Faculty of Maritime Sciences,Kobe University,

5-1-1 Fukaeminami,Higashinada, Kobe 658-0022, Japan

ProposiModelinGeometAnalysisDistinctive mechanigeometry. Recently,loosening phenomenmost cases, mesh gcommercial softwarequate for analyzingare the primary congeneration schemegeometry to analyzedistributions causedwith accurate threadsure vary along theand why the secondmaximum stress ocsurface, and the axifrom those obtainedsecond peak of Misepattern of z. DOIKeywords: fixing econcentration, contawnloaded 14 Feb 2013 to 129.5.16.227. Redistribution subject to ASMEon of Helical ThreadWith Accurateand Finite Element

behavior of bolted joints is caused by the helical shape of threadumber of papers have been published to elucidate the strength orf bolted joints using three-dimensional finite element analysis. Inrations of the bolted joints are implemented with the help ofhe mesh patterns so obtained are, therefore, not necessarily ad-e stress concentration and contact pressure distributions, whichns when designing bolted joints. In this paper, an effective meshroposed, which can provide helical thread models with accuratecific characteristics of stress concentrations and contact pressurethe helical thread geometry. Using the finite element (FE) modelsometry, it is shown how the thread root stress and contact pres-

lix and at the nut loaded surface in the circumferential directionak appears in the distribution of Mises stress at thread root. Thes at the bolt thread root located half a pitch from nut loadedoad along engaged threads shows a different distribution patternaxisymmetric FE analysis and elastic theory. It is found that the

ress around the top face of nut is due to the distinctive distribution.1115/1.2826433

ent, bolted joint, FEM, helical thread modeling, stresspressure distribution license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

smpicvflbpcmpm

1 =3

P 2 =

78

312

P H =32

P

Fd

0

Dohould be more than 0.125P P: thread pitch for external threadsade of high strength steel. Figure 1 shows the cross sectional

rofile along the bolt axis including the thread root radius. Assum-ng that the rounded portion of the thread root is a part of a singleircle with diameter , the surface of external thread can be di-ided into three parts such as A-B root radius, B-C threadank, and C-D crest. The thread profile perpendicular to theolt axis can be obtained by expanding those three parts into thelane, as shown in Fig. 2. Its shape is naturally identical at anyross section along the bolt axis. In the next section, helical threadodels are to be constructed by utilizing the characteristics ex-

lained here. The thread profile shown in Fig. 1 is expressed byeans of the following equations.

r =d2

78

H + 2 2 P242

2 0 1

H +

d2

78

H 1 2

d2

2

1

Fig. 1 Thread cross section along the bolt axis

ig. 2 Profile of the cross section of external thread perpen-icular to the bolt axis

11204-2 / Vol. 130, FEBRUARY 2008wnloaded 14 Feb 2013 to 129.5.16.227. Redistribution subject to ASMEwhere d and H represent nominal diameter and thread overlap.The profile of internal thread can be expressed in the samemanner.

r =d12

0 1

H +

d2

78

H 1 2

d2

+H8

2n +n2 P242 2 2 2

1 =

42 = 1 3nP n 324 P

There are upper limits for the root radii of external and internalthreads, and n, appeared in Eqs. 1 and 2, in connection withthe thread geometry of minor and nominal diameters.

3 Proposition of Helical Thread Modeling With Accu-rate Geometry

3.1 Conventional Methods. When analyzing the mechanicalbehavior of bolted joints with three-dimensional analysis, it hasbeen a common practice that the threaded portion of the FE mod-els has axisymmetric geometry, where the effects of lead angle areneglected because of its small value. That is, external and internalthreads are modeled by stacking an appropriate number of threadswith axisymmetric geometry. Recently, some researchers start touse helical thread models because of a growing recognition of theimportance of helical effects, e.g., loosening phenomena of boltedjoints. Their modeling procedures are classified roughly into threecategories in the case of external threads 59.

Type 1. Two-dimensional thread cross section model with one-pitch height is rotated helically around the bolt axis 57. Thisprocedure inevitably generates a small hole around the bolt axis.

Type 2. Helical thread model made in the similar manner toType 1 is attached around a solid cylinder 8,9. Mesh patterns arenot coincident at the interface between helical threads and thecylinder.

Type 3. Surface models of bolt and nut are made by means of asophisticated performance provided by commercial software, andthen the inside of the helical-shaped solid models of bolt and nutis divided into three-dimensional elements using its automaticmesh generation function.

In the case of Type 1, the effect of the small hole seems insig-nificant. However, unfavorable meshes are to be generated due tothe helical rotation, especially around the far end thread and thearea connecting the thread runout and the bolt cylinder. The sameproblem still remains in the case of Type 2. It is not an easypractice even for Type 3 that highly stressed area is intensivelydivided using small elements while the overall mesh pattern beingwell balanced.

3.2 Helical Thread Modeling by Stacking Cross SectionsWith Accurate Geometry. Helical thread modeling procedureproposed in this paper is based on the fact that the shape of thecross section perpendicular to the bolt axis is identical at anyposition. The profiles of external and internal threads are ex-pressed mathematically by means of Eqs. 1 and 2. In Fig. 3,illustrated are the real shapes of the cross section of externalthreads with coarse pitch of P. In the following, it is shown howthe helical thread models of external thread with accurate geom-etry can be constructed, where each ridge with one-pitch height isdivided into n thin plates with the same configuration. The proce-dure consists of six steps.

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sm

=

2

mP

i

SjbmwtepaspfmTa5mdhednpn

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DoStep 1. Cross section of external thread with accurate geometry,hown in Fig. 3, is properly modeled using two-dimensional ele-ents. 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 /n, it is placed at the position of z= P /n.Step 4. Connecting the corresponding nodes of the two basicodels placed at z=0 and P /n, a three-dimensional model with/n thickness is obtained.Step 5. One-pitch helical thread model is completed by repeat-

ng Steps 3 and 4 n times.Step 6. An appropriate number of one-pitch model obtained in

tep 5 is stacked, according to the number of threads of the ob-ective bolted joint.

If the one-pitch model is constructed by simply stacking theasic model according to the aforementioned procedure, the ele-ents around the threaded portion might have a large aspect ratio,hich causes low accuracy of the numerical analysis. In addition,

he mesh pattern around the bolt axis becomes finer than is nec-ssary. From the numerical accuracy and computation efficiencyoints of view, therefore, the finite element meshes for thread areand bolt core portion should be constructed separately. Figure 4hows an example of the mesh patterns of the cross sections per-endicular to and along the bolt axis. The circular area inside theour arrows, shown in Fig. 4a, is divided by rather coarseeshes, and the outside area is modeled by fairly fine meshes.herefore, the bolt core portion is simply modeled as a cylindernd only the thread area is modeled following Steps 16. Figurea shows a one-pitch helical thread model thus obtained. Theesh patterns of the two separate models are completely coinci-

ent at the interface. It follows that the helical model constructedere is expected to attain both high accuracy and computationfficiency. Thread runout is modeled by gradually varying theepth of the groove along the helix so as to be smoothly con-ected with bolt cylinder. Following the above procedure, it isossible to construct an entire bolted joint model only by eight-ode brick elements.

Internal threads can be modeled in the same manner. In thisase, the outside area of threaded portion is modeled as a hollowylinder. The outer surface of the nut is modeled as a cylindricalhape for simplicity, although it is possible to construct a hexago-al nut. Figure 5b shows the cross section of the nut model withelical geometry. Figure 6 is an example of the entire bolted jointodel, which is tightened by a single bolt with coarse thread of16. The total numbers of nodes and elements are 78,520 and

6,504, respectively. Numerical analysis with FE models con-tructed here can be implemented by standard FE analysis FEAoftware packages.

Fig. 3 Accurate cross section profile of metric coarse thread

ournal of Pressure Vessel Technologywnloaded 14 Feb 2013 to 129.5.16.227. Redistribution subject to ASME4 Stress Analysis of Bolted Joints Using HelicalThread Model

4.1 Numerical Models and Boundary Conditions. Themesh generation scheme proposed here can be executed withoutany help of commercial software. However, it is favorable to usesome sophisticated functions of commercial software for an effec-

Fig. 4 Mesh patterns of cross sections of bolt model

Fig. 5 One-pitch model of external thread and cross section ofnut model

FEBRUARY 2008, Vol. 130 / 011204-3 license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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stMhtsttoi2osMtsmllsoptone

0

Doive modeling. In this study, Hyper Works is used for supportinghe mesh generation and the numerical calculations are conducteds elastic problems by ABAQUS. Referring to the FE model shownn Fig. 6, the axial bolt force is applied as a uniform displacementt the lower end of the bolt cylinder. At the lower surface of theastened plate, axial displacements are completely restrained andhe circumferential ones are restrained at four nodes located0 deg apart. The analytical objects are bolted joints tightened bysingle bolt of M16 or M12 with coarse thread. Bolt, nut, and

late are supposed to be made of carbon steel whose Youngsodulus and Poissons ratio are 200 GPa and 0.3, respectively. In

he case of M16, bolt hole diameters are changed as 17 mm,7.5 mm, and 18.5 mm, which correspond to the first, second, andhird classes specified in JIS, respectively. Coefficients of friction

are varied from 0.05 to 0.20 with an increment of 0.05 andssumed to be identical at pressure flank of screw thread and nutoaded surface. For a parametric study, a standard analytical con-ition is defined as follows. Standard condition: M16, coefficientf friction=0.15, bolt hole diameter=17.5 mm second class

4.2 Stress Distributions Along Thread Root. Many previoustudies have reported that the maximum stress occurs at the bolthread root located within one pitch of the nut loaded surface.

ost of the research on the stress concentration at the thread rootave been conducted using axisymmetric FE models. Even if in-roducing such helical thread models explained in Sec. 3.1, iteems difficult to evaluate the stress concentrations around thehread root with practical accuracy. In this section, stress concen-rations around the thread root are analyzed using the FE modelsbtained in Sec. 3.2. Numerical calculations were performed us-ng a single computer equipped with Pentium 4 of 3.4 GHz with

Gbyte RAM. CPU time changes from 3 h to 5 h as coefficientf friction increases. It is shown in Fig. 7 how the maximumtress, which occurred at the thread root, varies along the helix.

ises stress at the thread root eq is normalized with respect tohe mean tensile stress b defined at the bolt cylinder. The ab-cissa represents the distance from the nut loaded surface. Theaximum Mises stress eqmax occurs at half a pitch from the nut

oaded surface, as in the case of the previous studies 10, wherearger coefficient of friction produces higher peak stress. Then, thetresses at the thread root gradually decrease toward the top facef the nut, and they show a second peak. It is considered that thishenomenon is caused by the low stiffness of the last engagedhread for its bending deformation. In Fig. 8, shown are the effectsf friction coefficient and bolt hole diameter on the maximumormalized stress eqmax /b. It increases slightly and almost lin-arly as coefficient of friction increases, and it almost decreases

Fig. 6 Fienit element model of entire bolted joint

11204-4 / Vol. 130, FEBRUARY 2008wnloaded 14 Feb 2013 to 129.5.16.227. Redistribution subject to ASMElinearly with increasing bolt hole diameter. As for the effects ofnominal diameter, larger bolt produces larger stress concentrationsas well as the previous studies 1.

4.3 Asymmetric Contact Pressure Distributions at NutLoaded Surface. The contact pressure at the nut loaded surfacedecreases outward in the radial direction. It is predicted that thecontact pressure also varies in the circumferential direction,though probably a small amount, because of the circumferentialvariation of the stiffness of engaged threads adjacent to the nutloaded surface. The latter phenomenon can be analyzed only whenintroducing a helical thread model. Figure 9a shows the circum-ferential contact pressure distributions at the nut loaded surfacefor varying radial positions. The reference point of =0 is placedat the plate top surface on which a fully formed nut thread withone-pitch height exists. The magnitude of the contact pressurevaries in the circumferential direction, which is rather remarkablealong 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. 7 Mises stress distributions at the bolt thread root alongthe helix

Fig. 8 Normalized maximum Mises stress occurred at the boltthread root

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sFtsct

ThAeelblzlnea

w

Fn

J

Doure decreases smoothly at any circumferential position. Shown inig. 9b are the contact pressure distributions in the circumferen-

ial direction for third class bolt hole. The effect of the helicalhape of thread geometry appears similarly in the case of secondlass. Such circumferential variation of the contact pressure dis-ributions might cause various problems in the bolted joints.

4.4 Evaluation of Load Distributions Along Engagedhreads. In this study, finite element meshes around thread ridgesave equal thickness in the axial direction, as shown in Fig. 4b.ccordingly, the load distribution along engaged threads can be

valuated by summing up the axial loads exerted on each thinlement with equal thickness. In a common bolted joint, bolt cy-indrical portion is subjected to axial bolt force Fb at any positionetween nut loaded surface and bolt head. Meanwhile, the axiaload F along the engaged threads gradually decreases from Fb toero toward the top surface of the nut. It is well known that suchoad distribution pattern causes various problems inherent to bolt-ut connections. Yamamoto derives an equation for F along thengaged threads based on the elastic theory, which shows that thexial load F decreases following a hyperbolic function, sinhx,here x denotes the distance from the nut loaded surface 11.In Fig. 10, the load distribution along the engaged threads ob-

ig. 9 Circumferential contact pressure distributions at theut loaded surface

ournal of Pressure Vessel Technologywnloaded 14 Feb 2013 to 129.5.16.227. Redistribution subject to ASMEtained by the helical thread model is compared to that by Yama-motos equation. Numerical result by axisymmetric FE analysis isalso shown, where the mesh pattern is the same as the cross sec-tion of the helical thread model along the bolt axis. The axisym-metric analysis gives a similar load distribution to that of concave-shaped Yamamotos equation, except around the top face of thenut. On the other hand, the numerical result by the helical threadmodel shows slightly convex distributions both around the nutloaded surface and the top face of the nut. In the cases of Yama-motos equation and axisymmetric FE analysis, it is assumed thatevery set of male and female threads is equally engaged. In theactual engaged threads, however, the contact areas of engagedthreads rapidly decrease around the nut loaded surface and the topface of the nut. It is considered that such effects could be repre-sented by the helical thread models introduced here.

5 DiscussionsA nut is classified into several kinds according to its shapes

around bearing surface and top face. The nut used here has a flatbearing surface that is completely in contact with the plate sur-face. The threads at the top face of the nut are commonly cham-fered, i.e., truncated at some angle, toward the bolt hole. Theeffect of the chamfering is studied by FE analysis. Figure 11aillustrates the nut cross section with and without chamfering. Allthe numerical results presented so far are associated with thechamfered nut models. Figure 11b represents the effect of thechamfering on the stress concentrations at the thread root. It isobserved that for both chamfered and nonchamfered nuts, zshows characteristic stress distribution patterns, which steeplyvary between positive and negative values. Accordingly, it seemsthat the second peak appearing in the Mises stress distribution iscaused by the distinctive distribution pattern of z. In the case ofnonchamfered nut, the second peak of Mises stress shows an un-natural decrease compared to the case of chamfered nut. Thisphenomenon can be mitigated by chamfering the top face of thenut.

6 ConclusionsAn effective three-dimensional thread modeling scheme, which

can accurately take account of its helical geometry, is proposedusing the equations defining the real configuration of the threadcross section perpendicular to the bolt axis.

It is shown how the thread root stress varies along the helix andthat the maximum stress occurs at half a pitch from the nut loaded

Fig. 10 Axial load distributions along engaged threads

FEBRUARY 2008, Vol. 130 / 011204-5 license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

stl

vhtta

ao

A

c

NomenclatureDi bolt hole diameterd nominal diameter

d1 minor diameterF axial load along engaged threads

Fb axial bolt forceH thread overlapP thread pitchr radial coordinatez axial coordinate coefficient of friction circumferential coordinate

,n root radii of external and internal threadsb mean tensile stress defined at bolt cylindereq Mises stress at thread root

eqmax maximum Mises stress at thread rootz axial stress

References

0

Dourface. The stresses at the thread root gradually decrease towardhe top face of the nut and they show a second peak because of theow stiffness of the last engaged threads.

It is shown how the contact pressure at the nut loaded surfacearies in the circumferential direction due to the effect of theelical thread geometry. The axial load distribution along engagedhreads analyzed by helical thread models shows a different dis-ribution pattern from those obtained by axisymmetric FE analysisnd elastic theory.

The second peak appearing in the distributions of Mises stresst the thread root is caused by the distinctive distribution patternf z.

cknowledgmentThe authors would like to acknowledge Mr. Yuuya Morimoto

DAIHATSU Motor Co. for his contribution to the numericalalculations conducted in this research.

Fig. 11 Effect of the chamfering of the nut top thread

11204-6 / Vol. 130, FEBRUARY 2008wnloaded 14 Feb 2013 to 129.5.16.227. Redistribution subject to ASME1 Fukuoka, T., 1997, Evaluation of the Method for Lowering Stress Concentra-tion at the Thread Root of Bolted Joints With Modifications of Nut Shape,ASME J. Pressure Vessel Technol. 1191, pp. 19.

2 Fukuoka, T., and Takaki, T., 2003, Elastic Plastic Finite Element Analysis ofBolted Joint During Tightening Process, ASME J. Mech. Des. 1254, pp.823830.

3 Lehnhoff, T. F., Ko, K. I., and Mckay, M. L., 1994, Member Stiffness andContact Pressure Distribution of Bolted Joints, ASME J. Mech. Des. 1162,pp. 550557.

4 Lehnhoff, T. F., and Bunyard, B. A., 2000, Bolt Thread and Head Fillet StressConcentration Factors, ASME J. Pressure Vessel Technol. 1222, pp. 180185.

5 Chen, J., and Shih, Y., 1999, A Study of the Helical Effect on the ThreadConnection by Three Dimensional Finite Element Analysis, Nucl. Eng. Des.191, pp. 109116.

6 Bahai, H., and Esat, I. I., 1994, A Hybrid Model for Analysis of ComplexStress Distribution in Threaded Connectors, Comput. Struct. 521, pp. 7993.

7 Rhee, H. C., 1990, Three-Dimensional Finite Element Analysis of ThreadedJoint, Proceedings of the Nintu International Confedence on Offshore Me-chanics Arctic and Engineering, Vol. 3, Pt. A, pp. 293297.

8 Zadoks, R. I., and Kokatam, D. P. R., 1999, Three-Dimensional Finite Ele-ment Model of a Threaded Connection, Comput. Model. Simul. Eng. 44,pp. 274281.

9 Zhang, M., and Jiang, Y., 2004, Finite Element Modeling of Self-Looseningof Bolted Joints, PVP Am. Soc. Mech. Eng., 478, pp. 1927.

10 Fukuoka, T., Yamasaki, N., Kitagawa, H., and Hamada, M., 1986, Stresses inBolt and NutEffects of Contact Conditions at the First Ridge-, Bull. JSME29256, pp. 32753279.

11 Yamamoto, A., 1970, Theory and Practice of the Tightening Process of ScrewThreads, Yokendo, Tokyo, pp. 12 in Japanese.

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