Computing the stress concentration factor in bolted joint using FEM.
It is becoming increasingly important to accurately predict the behavior of bolted joints such as the shown in fig 1. This research work is focused particularly on the stress concentration that develops at the root of the loaded threads, when the bolts are loaded statically with uniaxial external loads.
Historically, the value of the bolt as a fastener is very great. Bolted joints have been used in a wide range of engineering structures for hundreds of years. Bolted joints are widely used in industries e.g. pressure vessels, automobiles, machine tools, home appliances and so on. It is also likely that bolted connections will continue to grow in usefulness in the future.
Bolted joints are most commonly used components in machines and structures. The fasteners generally represents the largest single cause of the warranty claims faced by the automobile manufactures  often potential durability problems in the machine or structures are proportional to the number of bolted joints used.
The design of a "nuts and bolts" might seem to be one of the least interesting aspects, but in fact is one most fascinating. Moreover, the design and manufacture of fastener is very big business and is a significant part of economy. Literally thousands of different types of fasteners are offered by vendors and thousands to million of fasteners are used in a single complex assembly such as an automobile or aircraft. The Boeing 747 uses about 2.5 million fasteners, some of which cost several dollars each.
[FIGURE 1 OMITTED]
Mechanical joints can be classified into two categories 1) those fastened permanently 2) those held with removable fasteners. Permanently fastened joints are produced by Riveting, Removable joints are usually held with threaded fasteners. However, threaded fasteners are used as permanent fasteners in many cases. Rivets must be destroyed if removed. They are usually less expensive than threaded fasteners, but their strength in shear or tension may be lower, particularly when compared with heat treated bolts. The increased strength of heat treated bolts generally makes it possible to use fewer or smaller fasteners. In addition, threaded fasteners offer the advantage of high clamping forces, and when tooling and assembly costs are considered their in-place cost may be less than that of rivets.
The strength of a joint fastened with threaded members depends on the overall design and the amount of pretension in the bolt. A bolted joint is designed to resist axial or eccentric axial tension as well as shear under external loads. Most commercially available fasteners today are produced with cold-rolled threads in which usually shear and tensile strength is small, but the fatigue strength is considerably high. This occurs because of residual compressive stresses in the thread root, work hardening and improvement in grain growth.
The formation of threads is of two types i.e. (i) rolled (ii) cut threads. The method of formation of thread profile of threaded members has an effect on thread strength. In thread rolling the amount of cold work strain strengthening is unknown; the endurance strength is higher than in cut threads. For example, the bolt body diameter can be reduced to absorb impact energy, and then the body is stretched at a rate as the threaded portion. In this case the reduced body should be as long as possible, and body area should be about equal to the tensile stress area of the bolt threads.
Tightening of a bolted joint can be divided into basic categories, where screw is utilized either in its elastic or plastic region. The most commonly used method of tightening in the elastic region of a screw is the torque control method, while tightening a screw into its plastic region is often done by using the torque and angle method or the torque gradient method . Tightening over the yield point generates higher levels of preload than elastic tightening, and the scatter in preload is greatly reduced. The influence of friction is also reduced and the main parameter affecting the scatter in preload is the yield characteristic of the bolt. The methods of tightening a bolt into its plastic region either require expensive equipment, which can often only be acquired by large industries or profitable small workshops or else are so unreliable that there is a risk of over tightening the bolt or of failing to reach the yield point altogether. In addition, the reusability of yield tightened fasteners is often said to be very poor.
In general, if preload is high, the relative amount of external load carried by a screw in an axially loaded joint is small, and external load is almost entirely carried by the clamped parts. While tightening bolted joints, high stress concentration usually occurs around the thread root because of the complex geometry. It causes the stress concentration factor. The thread stress concentration factor is highest in the first engaged threads and decreases in each successive thread moving towards the end of the bolt .
Stress Concentration Factor
A shaft may have grooves for snap-rings or O-rings or have keyway and holes for the attachment of other parts. Bolts are threaded and have head bigger than their shank. Any one these changes in cross sectional geometry will cause the localized stress concentrations.
A theoretical, or geometrical, stress concentration factor Kt or Kts is used to relate the actual maximum stress at the discontinuity to the nominal stress. Where Kt is used for nominal stresses and [Kts for shear stresses . The maximum stress at a local stress raiser is the defined as
[[sigma].sub.max] = [K.sub.t] [[sigma].sub.nom] (1)
[[tau].sub.max] = [K.sub.ts] [[tau].sub.nom] (2)
Where, [[sigma].sub.nom] and [[tau].sub.nom] are the nominal stresses calculated for the particular applied loading and net cross section, assuming a stress distribution across the section that would for a uniform geometry. The nominal stress distribution is linear and stress on the outer fiber is [[sigma].sub.nom]=Mc/I. The stress at the notches would then be [[sigma].sub.max]=[K.sub.t] Mc/I. In an axial tension case the nominal stress distribution would be as shown in Fig. 2. Where p is applied load
[FIGURE 2 OMITTED]
Note that the nominal stress is calculated using the net cross which is reduced by the notch geometry. The factor [K.sub.t] and [K.sub.t]s takes into the effect of the part geometry into account and does not consider how the material behaves in the face of stress concentration. The ductility and brittleness of the material and the type of loading, whether static or dynamic, also effect how it responds to stress concentration. The stress concentration factor is the ratio of highest value of Von mises stress recorded for each of the engaged bolt thread and the nominal stress.
The value for nominal stress used in calculation of stress concentration factor for the threads can be calculated by the fallowing relation.
The geometric experimental stress concentration factors are based on fatigue stress concentration factor. The data presented by source  gives the reduced fatigue stress concentration factor [K.sub.f] based on the type of the thread (rolled or cut) and bolt material. Since we are not modeling the residual stresses, results will be compared to the value given for cut threads only.
Since the experimental values of fatigue stress concentration factor are available, it is necessary to use following relation to obtain the geometric experimental stress concentration factor value [K.sub.t].
[K.sub.t] = ([Ks.ub.f] - 1/q) + 1 (4)
Where q, [K.sub.f] are notch sensitivity factor which depends on the materiel strength and fatigue stress concentration factor.
Tension Joint-dynamic Loading
Tension loaded bolted joints subjected to fatigue action can be analyzed directly by listing strength reduction at beginning of the threads on the bolt shank. Its values are shown in table 1 which is already corrected for notch sensitivity and for square finish. Designer should be aware that a situation may arise in which it would be advisable to investigate these factor more closely. Since they are only average value, peterson observed  that the distribution of typical bolt failure is about 15% under the head, 20% at the end of the thread and 65% in the thread at the nut face.
Most of the time, the type of fatigue loading encountered in the analysis of bolted joints is one in which the externally applied load fluctuates between zero and some maximum force P. This would be the situation in a pressure cylinder, for example where a pressure either exits or does not exit.
Mechanical behavior of bolted joints and thread load distribution and thread stresses have been studied widely for about a century. Of the many examples in the literature on this topic, a few are mentioned here to give the flavour of what prior work has been performed. Sopwith  carried out a detailed analytical analysis, primarily on thread deformations and stresses, by theory of elasticity methods; he modeled threads as a series and parallel network of springs. The analysis is based on the following assumptions,
Errors in pitch have been neglected; these errors will not be zero, nor will they in general bear any simple relation to the position along thread.
Errors in angle of thread have been neglected.
Limiting friction is assumed. Loads applied axially and not as torque on the nut.
No account has been taken of the stress concentration effects at the roots of the thread; these are unlikely to affect the load distribution, since the high stresses are local and will have little effect on the displacements.
Many finite element analysis of threaded connections have been performed and results are published [2, 3, 4, 5, 6]. A FEM analysis has been carried out by Johnson and Englund  using ANSYS software. To study the interaction and stresses developed in the threads of a bolted connection 2-D axisymmtric is used. In case of 2-D, both nut and bolt are meshed with axisymmetric PLANE82 element.
A smaller mesh size was used around thread roots and flanks because this was the region of most interest. The areas away from roots and flanks were meshed with a coarser mesh. At the interface of the bolt and nut flanks contact elements were included. This model was then solved in a nonlinear, static analysis with axial loads applied gradually. In 3-D, helical threads were generated in both nut and bolt with great difficulty. Similar conditions were used in 3-D, as used in 2-D. From both analyses, stress distribution at the engaged threads are quite similar.
Effect of the contact condition at the first ridge on mechanical behaviors of threaded connections were analyzed by Fukuoka et al.  using an axisymmetric model, which is appropriate for the three body contact problem including the effects of friction on two contact surfaces between threads of bolt and nut. The cross sectional shape of nut at the first ridge changes circumferentially due to the lead angle of thread, which causes nonsymmetrical mechanical behaviors concerning the ratio of flank load and stress concentration at the root of bolt. Four types of models were used to estimate these non symmetric behaviors approximately. Each of them represents the cross section rotated one fourth revolution around the axis as shown in Fig. 3. The analysis has been carried out using the axisymmetric model without considering the effects of lead angle. The analysis is based on the assumption that, axial load was supposed to be transmitted only through the pressure flank and shear stress on contact surfaces is assumed to be subject to Coulomb's friction law.
[FIGURE 3 OMITTED]
Fukuoka and Takaki  study, the mechanical behaviour of a bolted joint during tightening, such as variations of axial tension and torque are investigated both experimentally. The friction coefficients on pressure flank of screw thread and the nut loaded surface are estimated by measuring the total torque applied to nut, axial tension and thread friction torque and compared with variations of axial tension and torque. Numerical models are consisting of three elastic bodies, such as bolt, nut and fastened plate, with three contact surfaces of pressure flank of thread, nut loaded surface and bearing surface of bolt head.
Fukuoka et. al .,  propose a numerical method based on Okamoto's method, which is intended for solving two bodies with one contact surface and it is extended to be appropriate three contact surfaces. Taking into account the effect of lead angle into consideration only for the nodes on pressure flank, three dimensional analysis involved can be executed using two dimensional mesh pattern, with each node having three degrees of freedom.
Stress concentration factors for the threads and bolt head fillet in a bolted connection have been suggested by Lehnhoff and Bunyard  using linear finite element analysis. There were ten models studied. The models included bolted connections where two 20mm thick circular steel plates were bolted together by a single bolt. The bolt diameters used were 8, 12, 16, 20 and 24mm. Two models were made for each bolt diameters. The model used the maximum allowable thread depth based on tolerances for the matric thread profile. The second model used the minimum allowable thread depth. The head fillet was modeled at its minimum radius. The FEA models consisted of parabolic axisymmetric solid elements. Gap elements were used to separate the threads in the bolt/nut region and in the areas of contact between the nut, bolt and members. Element size for engaged threads and head fillet areas were determined by obtaining several solutions for the model with the element size being reduced each time. Friction was included between all surfaces that came into contact. From analysis, it is found that thread stress concentration factors were highest in the first engaged threads and decreased in each successive thread moving towards the end of the bolt. Stress concentration factors in the head fillet were 3.18, 3.23, 3.63, 3.58 and 3.90 for the 8, 12, 16, 20 and 24mm bolts respectively.
An investigation is carried by Hobbs et al.,  to study the effect of nut thread run out on stress distribution in a bolt using FEM. In this investigation, both 2-D and 3-D analysis are carried. The bolted connection includes threaded bolt, nut and washer. Analysis has been performed using ANSYS software. For the two dimensional models, axisymmetric plane four node elements were used to model the bolt, nut and washer. The contact surfaces between the nut and washer and between the mating threads were modeled using point-to-surface contact element. Model is meshed with coarser quadrilateral shaped element and thread helix is not included for 2-D. From both 2-D and 3-D analysis results found to be quite similar. It shows that higher stress concentration occurs at the nut thread run out.
Tightening of bolted joints can be divided into two basic categories (i) elastic tightening (ii) plastic tightening. The elastic tightening is common method. In plastic tightening, tightening is done over the yield point, so that it develops plastic region. Tightening over the yield point generates higher levels of preload than elastic tightening, and the scatter in preload is greatly reduced. Toth  presented a technique, based on Monte-Carlo simulation of utilizing a screw over its yield point. This technique predicts permanent elongation, maximum tightening angle, final torque and preload tightening.
A new analytical model of bolted joints was suggested by Zhang and Poirier . The conventional theory of bolted joints has been developed long ago. The theory basically is analogous to the preloaded joint to combination of two springs in-parallel, one from the bolt and the other from the compression members. However, it is obviously over simplified. The theory does not distinguish whether the external load is at the bolt head directly or from some distance to the bolt head. Finite element analysis and experiments also showed that, the joint stiffness and bolt load are strongly dependent on the magnitude of the external load. It has been known that actual behavior of a joint is much more complicated than that the conventional theory described. In this paper , actual responses of axisymmetric bolted joint in tension are reinvestigated and the major control parameters are identified, based on which a new analytical method is represented. Finite element analysis is performed to confirm the new model and observations. The new model of bolted joint would help to understand the joint behavior and serve as a base for the future research, analysis and design.
In the present work, a study is carried out to understand the mechanical behavior of a bolted joint during tightening using FEM. It is found that the deformation of the models is maximum at bolt shank near head side.
The simulated results are found to be reasonably accurate when compared with the experimental values, since the error between simulated and experimental results is as below as 3.9 percent.
Problem is analyzed as static elastic (material nonlinear) contact problem. A bolted joint consists of bolt, nut and fastened plate. Bolt, nut and fastened plate meshed with axisymmetric PLANE42 element. Two pair's of contact12 elements were generated between at i) engaged threads of nut and bolt. ii) Nut bearing area. The Figure 4 shows
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The stress distribution in the bolted joint and he load distribution in Attention: the bolted joints
Tresca yield criteria (Max shear stress yield criteria)
 Tresca postulated that a material under a multi axis state of stress would yield when the max shear stress reaches some critical value. Using this assumption the yield envelope for a bi-axial state of stress as shown in figure 7
Why the yield envelope takes this shape can be illustrated by inspecting Mohr's circle for various states of stress. First, consider the uniaxial tensile test at the moment yield occurs. For this one of the principal stresses will be equal to the yield stress and the other principal stress will be zero. This condition defines the critical value of shear stress at which yield occurs.
[FIGURE 5 OMITTED]
According to Tresca,
1/2 ([[sigma].sub.1] - [[sigma].sub.1]) = k (5)
k is yield stress in shear
For tensile test, [[sigma].sub.1] - [[sigma].sub.2] and [[sigma].sub.2] =[[sigma].sub.3] = [sigma]. Maximum shear in tensile is, [[sigma].sub.y]/2 shown in Fig 6. (Mohr's circle)
[[sigma].sub.1] - [[sigma].sub.2] = [[sigma].sub.y] (6)
Comparing 5 and 6,
[[sigma].sub.y] = 2k k = [[sigma].sub.y]/2 (7)
That is, the yield stress in simple shear is half the yield stress in simple tension according to Tresca's yield criterion.
[FIGURE 6 OMITTED]
If either of the principal stresses exceeds the yield stress, the out of plane shear will exceed the max shear value (Fig.8). This is because for 2D problem the out of plane principal stress ([[sigma].sub.3]) is assumed to be zero. Therefore, neither [[sigma].sub.1] nor [[sigma].sub.2] may exceed [[sigma].sub.y].
[FIGURE 7 OMITTED]
In the case where one of the principal stresses has the opposite sign of the other (i.e. one in compression the in tension. Quadrant 2 and 4 of the Tresca yield Envelope) yield will occur before either of the principal stresses reaches the yield stress. As illustrated in figure 10. If [[sigma].sub.2] becomes more negative [[sigma].sub.1] must also decrease (other wise the shear will increase beyond the max value).
Essentially you may slide a constant size circle along the axis between positive and negative [[sigma].sub.y].
[FIGURE 8 OMITTED]
Von Mises yield criteria
Von Mises postulated that a material would yield when the distortional energy at the point reaches a critical value. The distortional energy written in terms of the 2D principal stresses and the yield stress is as follows.
[([[sigma].sub.1] - [[sigma].sub.2]).sup.2] + [[sigma].sup.2.sub.1] + [[sigma].sup.2.sub.2] = 2[[sigma].sup.2.sub.y] (8)
Figure 9 shows the Von Mises yield envelope supposed over the Tresca yield envelope. For hand calculation Tresca is easy to implement; for a computer program Von Mises is convenient because the entire envelope can be represented with a single equation.
[FIGURE 9 OMITTED]
FEA Model Two-Dimensional model Assumptions Bolt, nut and fastened plate considered as axisymmetric models. Run out of nut neglected. Loads applied axially and not as a torque on the nut. The effect of lead angle is very small. Therefore lead angle not considered. Model
The geometry of the bolted joint modeled is based on ISO M12 threads. The cross section shown in Figure 5 The two dimensional model includes a bolt, a cylindrical nut and a fastened plate.
The axisymmetric, two dimensional, bolted joint was modeled using a radius root thread profile. This enables us to represent three dimensional objects into two dimensional objects. This is obviously a simplification as certain aspects of the geometry are neglected, most notably the thread helix angle.
All the two dimensional finite element analyses have been preferred using ANSYS software. The model was meshed with two dimensional quadrilateral PLANE42 elements. In order to achieve geometry in the thread region that could be completely meshed with quadrilateral elements. The construction was carefully planned, making sure that all the areas were four sided. A smaller mesh size was used around the thread roots and flanks because this was the region of the most interest. The areas away from the roots and flanks were left with a coarser mesh to cut down on total element count and run time.
[FIGURE 10 OMITTED]
Total number of elements-7742, Total number of nodes-8320. The mesh is Shown in Fig. 11
[FIGURE 11 OMITTED]
Two pairs of contact elements have been generated. The first pair is generated in between upper surface of the nut and the lower surface of fastened plate and the second pair is generated in between threads of the bolt and the nut.
Since the problem is complex i.e. compression in one component and tension in another component occurs at same time. Therefore, problem is solved in two stages. In first stage, lower part of fastened plate is fixed and axial load  is applied on the bolt shank. In second stage, lower part of fastened plate fixed and axial load (in opposite direction) at nut loaded surface. Number of unengaged threads set to be 10. In both stages, axial load of over yield point load 52458.075N applied. Force based convergence criteria was set. Number of iteration set in each load step is 25. This model solved in a linear, static analysis.
Results and Discussions
Two dimensional numerical analysis has been carried out as discussed below.
Analytical results (M12)
[delta] = PL/AE
[delta] = 52458.075 x 58/84.27 x 2 x [10.sup.5] = 0.179121mm (9)
Analytically the deformation of bolted joint of size M12 can determined by the equation 9, where [delta] is deformation or deflection, P is Preload (Clamping load), A Cross sectional area, L is Length, E is Young's modulus. The deformation which has been calculated is 0 .18mm and the deformation which has been found out from FEM analysis is .21mm which is very nearer to the analytical results. Hence by meshing the FEA model with exact number elements and selecting suitale type of element we can get the accurate results.
[FIGURE 12 OMITTED]
Table 2 and 3 shows the comparison between experimentally obtained results and present results. It is obvious from these tables that the accuracy of the present method is quite good for the case of M12.
Comparison results of size M12 by using the von mises & tresca criteria with the experimental results.
The stress concentration by using the Von Mises and the Tresca criteria along the length of the bolt for the size M12 has been depicted in the figure 13 and in figure 16, left side is the head of bolt and right is the nut. The most highly stressed portion is the threaded area of the bolt. This area therefore determines the strength of the bolt. When the Tresca criteria has been consider the results are very nearer to the experimental results  than the Von mises criteria which are shown in table 2 and 3. Secondly the error has been minimized between simulated results and experimental results by choosing suitable yield criteria, proper type of meshing and by using the optimum number of elements.
Stress concentration factor in the threads
Figure 13 shows the amount of stress concentration (Von mises) in the bolted joint assembly. There is an uniform stress distribution accept in the area where plate has engaged with nut. Green colour in the figure depicts the area where plastic zone has formed.
Figure 14 shows the zoomed view of engaged threads. It is observed from this figure that a high amount of stress is developed in the first engaged thread and stress concentration goes on decreasing as we move towards last engaged thread. The maximum stress concentration has developed at below the deepest point of the thread root as depicted in figure 10 the stress in this area is found 2589 N/[mm.sup.2].
Figure 12 shows deformation of the bolted joint assembly under the applied loading conditions as expected, the part of bolt shank near head side has gone under maximum deformation and the deformation goes on decreasing until other end of the bolt shank.
Figure 13 to 16 shows the stress distribution in the bolted joint assembly, unengaged thread and engaged thread of the bolted joint respectively based on Tresca criteria. It has observed that the stress concentration varies along threads in the region of unengaged threads.
Figure 18 shows the meshed model of bolted joint assembly. It may note that the model is meshed with four nodded rectangular element as to get maximum accuracy. The meshed independent study has been carried before selecting a meshed size of elements 7742 and with number of nodes equal to 8320.
Figure 15 shows the variation of stress concentration factor with respect to number of threads. It has been observed from the figure that the stress concentration factor decreases with increase in number of threads. The stress concentration decrease by 71% when number of threads is increased from 1 to 5.
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[FIGURE 16 OMITTED]
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A simulation work has been done to investigate the stress concentration factor. Ansys has been used as simulation tool which works on finite element method. Some leniency has used while generating the physical model of the bolted joint .The simulation represented the behavior of the bolted joint very well. It is generally accepted that local yielding occurs in the thread roots.The stresses in the engaged threads of bolted joint of M12 are maximum at first engaged thread and minimum at last thread. Thus stress concentration factor is maximum at first thread and minimum at last thread. It is observed that in an individual thread the highest stress occurred below the deepest point of the thread root.
It is found that the deformation of the models is maximum at bolt shank near head side.
The simulated results are found to be reasonably accurate when compared with the experimental values, since the error between simulated and experimental results is as below as 3.9 percent.
Their scope for this work is to use the Neural Network to find out the optimum number of elements to get the accurate results which will minimize the error between the experimental results and the FEM results. Secondly by finding out the optimal aspect ratio of the elements, these both can use as parameters and the stress concentration factos can be used as dependent parameter to achieve the results which will be nearer to experimental results.
A Cross section area
[A.sub.b] Cross sectional area in unthreaded portion of bolt
[A.sub.d] Bolt shank area
[A.sub.t] Tensile stress area
[A.sub.th] Cross sectional of bolt in threaded portion of bolt
E Young's modulus
[F.sub.b] Force in bolt
[F.sub.i] Pre load in bolt
H Tangent modulus
[K.sub.b] Stiffness of bolt
[K.sub.d] Stiffness of shank
[K.sub.f] Fatigue stress concentration factor
[K.sub.m] Members stiffness
[K.sub.s] Stiffness of thread
[K.sub.t] Stress concentration factor
[L.sub.d] Shank length
[L.sub.g] Grip length
[L.sub.h] Bolt head length
[L.sub.t] Thread length
P Preload (Clamping load)
[P.sub.f] Proof load
[R.sub.r] Radius at root
[S.sub.p] Proof strength
d Nominal diameter
[d.sub.p] Pitch diameter
[d.sub.r] Minor diameter
k Torque coefficient
l Clamped length
[delta] Deformation or deflection
[psi] Half thread angle
[phi] Tightening angle
[gamma] Poisons ratio
[lambda] Lead angle
[mu] Coefficient of friction
[pi] = 3.142
[[sigma].sub.e] Equivalent stress
[[sigma].sub.max] Maximum stress
[[sigma].sub.t] Axial stress
[[sigma].sub.y] Yield stress or yield strength
[[sigma].sub.1] [[sigma].sub.2] [[sigma].sub.3] Principal stresses
[tau] Shear stress
[[tau].sub.s] Stripping shear stress
[[sigma].sub.th] Nominal stress in threaded portion
[[sigma].sub.b] Nominal stress applied to the nominal diameter of bolt
 Sopwith, D.G., (1948), The distribution of load in screw thread, Institution of Mechanical Engineers, Proceedings, Vol.159, No. 45, pp 373-383.
 Johnson and Englund "Three dimensional modeling of a bolted connection", www.ansys.com.
 Fukuoka, T.,Yamasaki. N.,Kitagawa. H.,Hamada. M,(1986) Stresses in bolt and nut, Bulletin of JSME, Vol.29, No.256, pp 3275-3279,
 Fukuoka, T., Takaki, T. (1998), "Mechanical behaviours of bolted joint during tightening using torque control", JSME International journal, series A, Vol.41, No.2, pp 185-191.
 Lehnhoff, T.F., Bunyard, B.A. (2000), "Bolt thread and head fillet stress concentration factors", Journal of pressure vessel technology, ASME, Vol.122, pp 180-185.
 Hobbs, J.W., Burguete, R.L., Patterson, E.A.( 2003), Investigation into the effect of the nut thread run out on the stress distribution in a bolt using the finite element method, Journal of Mechanical design, ASME, Vol.125, pp 527-532.
 Toth, G.R. (2004), Torque and angle controlled tightening over the yield point of a screw -Based on Monte-Carlo simulations, Journal of Mechanical design, ASME, Vol.126, pp 729-736.
 Zhang, O., Poirier, J.A. (2004), New analytical model of bolted joints, Journal of Mechanical design, ASME, Vol.126, pp 721-728.
 Shigley, J.E., and Mischke, C.R., Mechanical engineering design, 6th edition, 1st reprint 2003, Tata-McGraw Hill.
 Norton, R.L., Machine design, 2nd edition, Third Indian reprint 2004, Pearson Education.
 ISO Metric Screw Threads, Part 1 Basic and Design Profiles, IS:4218 (Part 1) -1976, 3rd Reprint, August 1998.
 Yanyao Jiang,Ming Zhang, Chu-Hwa Lee.( 2003), A Study of Early Stage Self-Loosening of Bolt, Journal of mechanical design, ASME, Vol.125, pp 218-223,. (http://femci.gsfc.nasa.gov/yield/)
K.H. Muhammad-Tandur (1) and Irfan Anjum Magami (2) School of Mechanical Engineering, University Science Malaysia,
(1) Engineering Campus, Seri Ampangan, 14300 Nibong Tebal, Seberang Perai Selatan, Pulau Pinang, Malaysia.
(2) Department of Mechanical Engineering, University of Malaya, Kuala Lumpur, Malaysia Email: email@example.com
Table 1: Fatigue stress concentration factor [K.sub.f] for threaded elements Grade metric Rolled threads Cut threads 3.6 to 5.8 2.2 2.8 6.6 to 10.9 3.0 3.8 Table 2: Von mises criteria [K.sub.t] % error of Shigley [K.sub.t] previous Present % Bolt dia experimental obtained work  error M12 4.42 4.2 9 4.9 Table 3: Tresca criteria [K.sub.t] % error of Shigley [K.sub.t] previous Present % Bolt dia experimental obtained work  error M12 4.42 4.6 9 3.9 Nomenclature A Cross section area [A.sub.b] Cross sectional area in unthreaded portion of bolt [A.sub.d] Bolt shank area [A.sub.t] Tensile stress area [A.sub.th] Cross sectional of bolt in threaded portion of bolt E Young's modulus [F.sub.b] Force in bolt [F.sub.i] Pre load in bolt F Force H Tangent modulus [K.sub.b] Stiffness of bolt [K.sub.d] Stiffness of shank [K.sub.f] Fatigue stress concentration factor [K.sub.m] Members stiffness [K.sub.s] Stiffness of thread [K.sub.t] Stress concentration factor L Length [L.sub.d] Shank length [L.sub.g] Grip length [L.sub.h] Bolt head length [L.sub.t] Thread length P Preload (Clamping load) [P.sub.f] Proof load [R.sub.r] Radius at root [S.sub.p] Proof strength d Nominal diameter [d.sub.p] Pitch diameter [d.sub.r] Minor diameter k Torque coefficient l Clamped length p Pitch [delta] Deformation or deflection [epsilon] Strain [phi] Half thread angle [psi] Tightening angle [gamma] Poisons ratio [lambda] Lead angle [mu] Coefficient of friction [pi] =3.142 [sigma] Stress [[sigma].sub.e] Equivalent stress [[sigma].sub.max] Maximum stress [[sigma].sub.t] Axial stress [[sigma].sub.y] Yield stress or yield strength [[sigma].sub.1] [[sigma].sub.2] Principal stresses [[sigma].sub.3] [tau] Shear stress [[tau].sub.s] Stripping shear stress [MATHEMATICAL EXPRESSION NOT Nominal stress in threaded portion REPRODUCIBLE IN ASCII] [[sigma].sub.b] Nominal stress applied to the nominal diameter of bolt