Finite Element Based Member Stiffness Evaluation of Axisymmetric Bolted Joint

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Finite Element Based Member Stiffness Evaluation of Axisymmetric Bolted
Joint
Mr. Chetan Chirmurkar
Department Of Mechanical Engg. (Design Engg.), Savitribai Phule Pune University, D.Y.P.S.O.E. Lohegaon Pune, India
chetanchirmurkar@gmail.com, chirmurkarchetan@yahoo.co.in
Prof. Mr. A. P. Deshmukh,
Department Of Mechanical Engg. (Design Engg.), Savitribai Phule Pune University, D.Y.P.S.O.E. Lohegaon Pune, India
Abstract:
For a reliable design of bolted joints, it is necessary to evaluate the actual fraction of the external load
transmitted through the bolt. The stiffness of the bolt and the member of the joint decide the fractions of external
load shared by the bolt and the member. Bolt stiffness can be evaluated simply by assuming the load flow to be
uniform across the thickness and the deformation is homogeneous. Then, bolt may be modeled as a tension
member and the stiffness can be easily evaluated. But, the evaluation of the member stiffness is difficult because
of the heterogeneous deformation. In the present work, joint materials are assumed to be isotropic and
homogeneous, and linear elastic axisymmetric finite element analysis was performed to evaluate the member
stiffness. Uniform displacement and uniform pressure assumptions are employed in idealizing the boundary
conditions. Wide ranges of bolt sizes, joint thicknesses, and material properties are considered in the analysis to
evaluate characteristic behavior of member stiffness. Empirical formulas for the member stiffness evaluation are
proposed using dimensionless parameters. The results obtained are compared with the results available in the
literature.
Keywords- bolted joints, member stiffness, finite element analysis, axisymmetric model.
INTRODUCTION:
In engineering and day to day life, many cases arise when we have to join the two members. There are many
joining methods used in engineering, as welding, brazing, soldering, and bolting. Each method has its own
characteristics and use. For a particular case one of method is good and reliable, but for another cases it may
worst or impossible. Bolted joints are used in abundance within the mechanical design of machinery. Most of
these joints are noncritical to the overall success of a machine’s function; nevertheless, a certain number of these
joints are extremely critical and require in depth analysis for determining their acceptability. In automobile field
and machineries, the bolted joints are famous. In bolted joints screw threads are used for tightening of two
members. For tightening of members in bolted joints it require a bolt, it may be hexagonal or square but mostly
hexagonal is preferred, and nut. It is as shown in fig.1
Fig.1 Bolted joint of a member
PRESTRESSING:
When it is necessary that the bolted member is joined enough strong to external force. So to tight the member
prestressing is must. At the time of tightening extra torque is applied to the nut so that the member is compressed

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and due to which compression force (Fm) is acting on the member and tensile force (Fb) is generated in the bolt.
The main two advantages of prestressing are it increase the fatigue life of the joints and it create a locking effect
to the joints, it means the joint will not lose easily. The resultant loads acting on bolt and member is
FB= Fi (1)
Fm= Fi (2)
Km and Kb are stiffness of the member and bolt material which is
Km= (3)
Km= (4)
STIFFNESS:
Stiffness is load per unit displacement of any elastic member. In the previous article we have discussed about the
calculation of the stiffness of the member as
Km= (5)
If there are more than one member in joined by nut-bolt it assumed to be spring structure in series as shown in
fig.2.
Fig .2 Stiffness model of member
The overall stiffness can be calculated,
(6)
This formula is used when there is use of gasket in between the members and gasket is having stiffness less
than other members, then other can be neglected for all practical purposes and only gasket stiffness is used. But if
there is no gasket and if only a two member are joined by nut-bolt then it is difficult to calculate the joint
stiffness, except by actual experimentation. Because deflection and force, area acting on the member is non
uniform, it may be more at near the head and nut and low at the interface of two members.
In such cases where area under the compression are not known Prof. Charles Michke suggest the concept of
pressure cone. Many analytical methods were proposed to calculate the area under the calculation but the pressure
cone method is dominated. To model the actual system the conical shaped system is proposed. Prof. Michke
propose the cone angle of 450
Fig.3 Cone geometry of bolted member

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Fig.3 shows cone geometry, it has washer diameter dw, on both sides and joint of equal thickness member, this
is assumed to simplify the case. But in actual there may be member of unequal thickness and may be more than
one member of different modulus of elasticity. Shiglay [1] in his book assume α=450
.Later the same author
proposed the angle as α=300
.Verein Deutscher Ingenieur procedure consider it as α=300
.
The stiffness of frustum is given by Shiglay as
Km =
π
α ! !
α ! !
(7)
Assuming washer diameter dw=1.5d, & α=300
as best value
Km=
".$%%π
$&
'.()) '.( !
'.()) *.( !
(8)
The distinguishing characteristics of this system are cone angle’α’, washer diameter ‘dw’, bolt diameter ‘d’ and
clamped material thickness ‘L’. Perhaps the most unsettled topic concerning this model has been the selection for
the cone angle ‘α’, as the different author use different angle, no one has discussed relative important of choosing
the proper value for this parameter. Verein Deutscher Ingenieur (VDI) procedure assumes α to be 30°. Juvinall
and Marshek proposed the following expression to estimate the effective clamping area, Am, of the member.
Am =
+
,
-.
/ /0
1 2 345 (9)
Where dh=d for small clearance, d1=1.5d, d2= d1+Ltanα, and
α=300
. Above equation is
Am=d2
+0.68dl+0.065L (10)
The stiffness of the member can be calculated by substituting above eq. in the equation of stiffness.
Apart from analytical models, many researchers used numerical methods such as finite element method to
estimate member stiffness. Wileman et al performed axisymmetric finite element analysis (FEA) and proposed an
exponential expression for member stiffness. In the analysis, the washer diameter dw was assumed to be 1.5 times
the diameter of the bolt. They considered the displacement of the uppermost node located on the center line of the
washer for stiffness calculation. Finite element analyses for various aspect ratios (dh L) were carried out and
finally an exponential relation for member stiffness evaluation was proposed. This is as below,
Km = Em dh A67 8/:
(11)
Where,
A, B = material constants.
A = 0.78952 and B=0.62914
Which valid for engineering materials such as steel, copper, aluminum, and cast iron. This formula is mostly
applicable for the cases with dh /L≤ 2 and shown to be quite simple and efficient method to calculate the member
stiffness. Later many researchers used and developed the same technique to evaluate the member stiffness. The
work is proposed an exponential relationship using no dimensional geometric and material parameters for
member stiffness evaluation. The analysis was done using finite elements considering various bolt sizes,
geometries, and different member materials. Most of the methods presented in the literature review have
differences in the results. The differences in the results are mainly due to the assumptions made during the model
development. This is value of pressure cone angle and confined uniform stress field in a particular region.
Manring [3] showed the member stiffness dependents on the choice of cone angle. The differences caused by
various assumptions need higher safety factors for reliable design. Wileman formula, given in Eq. (11), is an
attempt to handle the inaccuracies but the formula itself is restricted to limited variation of parameters. In this, we
are doing an attempt to carry out systematic study and evaluation of member stiffness for various geometric and

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material properties of the joint members. An ideal condition are considered where effects of member diameter,
thread friction, contact friction between bolt head/washer and members, interface slip between members, and
external shear loads are not considered in the present analysis; hence all such effects if present should be
compensated by proper safety factor. The proposed formulas should not be applied for joints in which members
may subject to very high external loads and tend to separate from each other.
METHOD OF APPROACH:
Our main aim is to propose a formula for member stiffness evaluation based on the results of linear finite element
analysis. The member stiffness is calculated using finite element method and it is divided by a stiffness measure
Ko to get a dimensionless quantity R, called “correction factor,” as given by
R =
; <!
;"
(11)
(12)
Where,
Ko=the stiffness of hollow cylinder with internal and external diameters the same as that of the washer and length
equal to the joint thickness.
If, Ko = KFEM , then R=1
So we are comparing the KFEM to the stiffness of the hollow cylinder with internal and external diameters the
same as that of the washer. Correction factors for various joint geometries and material constants are obtained by
detailed parametric finite element analysis and empirical relations to calculate the correction factor are proposed
by least squares curve fitting.
Finite Element Modeling and Analysis:
Fig. 4 Axisymmetric half model of the bolted joint
For the finite element modeling of the bolted joint, the assumptions made are
1) The geometry of the joint is assumed to be perfectly Axisymmetric.
2) The two members of the joint are assumed to be made of the same material and have the same thickness,
3) Also assumed to be in perfect contact without slippage.
4) The effect of screw threads is ignored.
5) The member material is assumed to be homogeneous, isotropic, and linearly elastic.
Due to the second assumption, the two members of the joint can be treated as a single continuous member and
planar symmetry about the joint midplane is introduced in the model. Both the symmetries are incorporated in the
model by applying proper symmetry constraints, as shown in Fig. 4. Commercial finite element software ANSYS
is used for modeling and analysis. To avoid the edge effects, the outer diameter of the member is taken to be five
times the diameter of the bolt hole. The model is meshed by eight noded axisymmetric quadrilateral elements. A
schematic finite element discretization is shown in Fig.5. The member material is assumed to be homogeneous,
isotropic, and linearly elastic with Young’s modulus, Em, equal to 210 GPa, for all the analyses.

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Fig. 5 Schematic finite element mesh
The outer diameter of the washer is taken to be 1.5 times the Diameter of the bolt, i.e., same as the diameter of the
bolt head (distance between flats of metric hexagonal bolt head). Dimensions of various bolts and corresponding
washers used in the analysis are given in Table 1.
BOUNDARY CONDITIONS:
1) E1=E2
2) t1=t2=l/2
3) Plane of symmetry is steady i.e. plane of symmetry is remains plane after deformation.
4) Force applied at top
In this work, the load transfer is simulated by two different assumptions, which are explained below and the
results obtained from both the assumptions are compared.
UDA:
This case corresponds to a joint in which the washer is highly rigid (hard washer) compared with the member.
Due to this, the displacement of member under the washer is almost uniform. This condition is realized in the
finite element analysis by applying a predetermined axially downward uniform displacement to the topmost
nodes, which fall within the outer diameter of the washer. The sum of the upward reactions, Fr, of the nodes
below the washer is divided by twice the applied displacement (due to symmetry) to get the stiffness of the
member.
The stiffness is given by
(13)
UPA:
This case corresponds to a joint in which the washer is soft compared with the member material. So in this case
the contact pressure acting on the member surface is nearly uniform. In order to realize this condition, a
predetermined uniform pressure is applied on the topmost nodes, which fall just within the washer outer diameter.
Then the average member displacement is calculated by averaging the individual displacements of these nodes.
The member stiffness is given by
(14)
Where Aw is the area under the washer and is given by
(15)
The average nodal displacement is used for calculation of KFEM only for UPA because the displacement field
under the washer contact area is varying in the radial direction with a maximum value at the bolt hole and a

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minimum value at the outer edge of the washer. Longitudinal stiffness also varies along the radial direction. From
the mean value theorem, for constant pressure boundary conditions, the average displacement gives the member
stiffness in an average sense. Earlier, Lehnhoff et al. also used average displacement of nodes below the bolt head
to calculate the member stiffness.The uniform displacement assumption (UDA) corresponds to the finite element
boundary condition where the washer is assumed to be highly rigid compared with the member and the uniform
pressure assumption (UPA) corresponds to the boundary condition where the washer is assumed to be very soft
compared with the member. Plots of displacement contours of a 40 mm thick joint with M20 bolt with UDA and
UPA are shown in Figs. 6 and 7, respectively.
Fig.6Displacement contours with UDA(M20 joint, L=40 m) Fig.7Displacement contours with UPA (M20 joint,L=40 m)
In the actual joint neither the displacement nor the contact pressure below the washer is uniform. The idea
behind using the assumptions for preload simulation is to calculate two
Fig. 8 Variation in R with dw/L for bolt size M20 (UDA) Fig. 9 Variation in R with dw/L for bolt size M20 (UPA)
different estimates of member stiffness, which will work as upper and lower bounds and the actual value of
member stiffness will be in between the limits when the analysis is carried out with the actual washer stiffness.
The assumptions of UDA and UPA may give (not proved in the present study) the upper and lower bounds of the
member stiffness pertaining to the material rigidity of the washer relative to the member. In practice, the washer
will have stiffness, which may not be very rigid or soft, and the rigidity will be of finite values. Convergence
study is carried out on the initial finite element model by decreasing the element size and converged model with
element edge length of 0.33 mm is used for the rest of the

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Fig.10Variation of Rwith λm/Em for bolt size M 20 (UDA) Fig.11Variation in R with m/Em for bolt size M 20 (UPA)
Analysis. The number of elements used in the analysis is not constant due to change in geometry. The number of
elements used in the converged analyses ranges from 1000 to 40,859.
RESULTS AND DISCUSSION:
Parametric study is carried for nine different bolts ranging from M6 to M36 with joint thickness ranging from
16 mm to 60 mm with an increment of 4 mm and with five different Poisson ratio values varying from 0.2 to 0.4
with an increment of 0.05 for both UDA and UPA boundary conditions. The number of converged case studies
conducted for each of UDA and UPA is 540. Correction factors are calculated for all the case studies using
Eq.(11) used above. The finite element stiffness values for various bolt sizes and joint geometries are given in
reference [4], but one of it are given in Tables 3.
Empirical Relation for Correction Factor: It is observed from the results of FEA that the stiffness of the member
isdependent on the dimensions of the bolt hole, washer diameter, and grip length of the joint. Along with these
three factors, the Member stiffness is also dependent on Lamé’s constant of the member material, λm. This is
because when the member is axially compressed by the load, the member below the washer will try to expand in
radial direction due to Poisson’s effect and the material around the washer will restrict this expansion due to
which radial stresses are induced in the member. These stresses, which influence the member stiffness, are
observed to be proportional to Lamé’s constant. Therefore, the following dimensionless parametric group is used
to express the correction factor.
(16)
As metric standard bolt dimensions are used throughout the analysis, dw/dh is nearly constant and retained in the
expression only to represent the effects of bolt hole clearance. Variation in correction factor with dw/L is shown
in Figs. 6 and 7 for UDA and UPA, respectively. In each plot, it can be observed that the curves resemble
exponential decay pattern. Figures 8 and 9 show that the correction factor increases with increase in λm/Em (ratio
is a function of Poisson’s ratio). Finally, an empirical relation for correction factor was proposed after observing
the variation with individual variables of Eq. (16) and using a least squares curve fitting algorithm. During the
initial curve fitting process, the curve was assumed to be of the form, exponential proposed by Wileman et al. In
our modeling, three non dimensional parameters (dw/dh, dw/L, and λm/Em) are introduced to account for the wide
range of geometric and material parameter variations. Later, the correction factor (R) variation is characterized
for different geometrical non dimensional parameter variations for a given material property value. It was found
that use of powers of non dimensional variables 3=/34!>0
3=/?!@
fitted the curve well. The process was
repeated for different material property values (λm/Em). Finally, it was observed that the introduction of inverted
hyperbolic function allowed fitting all the results into a single curve with minimal error (less than 3%). The final
expression for correction factor is given by
R = C5 + C6 exp(S) (17)

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Where S is an intermediate variable used for convenience of expression and it is given by
S = C4 sinh 0
3=/34!>0
3=/?!@
! E FG/HG!@I
(18)
The constants used in Eqs(17) and (18) for UDA and UPA are given in Table 2. The maximum error in curve
fitting is below 3% for both UDA and UPA.The member stiffness for any arbitrary joint geometry can now be
calculated using the proposed empirical formula. However, it is very important to carefully review the inherent
assumptions used to obtain this formula before using it, particularly the preload simulation methods using UDA
and UPA. As explained earlier, both the assumptions are extreme idealizations of the actual phenomenon.
Accordingly, the stiffness obtained by UDA always predicts the higher value for the member stiffness than that
obtained by UPA. The selection of the appropriate assumption should be made depending on the problem at hand.
The diameter of the washer is taken to be equal to the bolt head diameter (1.5 times the diameter of bolt) in the
present work. As the diameter of the washer influences the member stiffness, these formulas should not be used if
the difference between the washer diameter and the bolt head diameter is considerable. In the case of a joint
without washer, formula can be used by replacing washer diameter by bolt head diameter.It is observed in the
present work that the member stiffness is increased by 8–13% approximately with the increase in (λm/Em)( as
(λm/Em) is a function of Poisson ratio) alone) from 0.28 (µ =0.2) to 1.43(µ =0.4), see Figs. 10 and 11. This
variation was accounted for in the proposed empirical formulas.In this work, the outer diameter of the member,
dm, is
taken to be five times the diameter of bolt hole, dm, to avoid the
edge effects. Except for very small values of dw/L, increase in
member diameter beyond 3.5 times dh results in a negligible increase in stiffness, see Fig. 10. It is also evident
from Fig. 10 that the member stiffness decreases steeply with a decrease in the member diameter beyond three
times dh. Hence the proposed formula can be safely used for member diameters starting from 3.5 times dh .
CONCLUSIONS:
In the present work, the member stiffness of bolted joints is calculated using the axisymmetric finite
element analysis and an empirical formula for the computation of member stiffness is proposed. The external load
on the joint member is applied using UDA and UPA. Parametric study has been carried out for different bolts
ranging from M6 to M36, joint thickness ranging from 16 mm to 60 mm, and Poisson ratio varying from 0.2 to
0.4. It is observed that the member stiffness varies by 8–13% with an increase in Poisson’s ratio from 0.2 to 0.4.
The proposed empirical formulas fit very well with the finite element simulations and the maximum deviation
observed was less than 3%. Finally, the results are compared with the results obtained from previous methods in
the literature and it is observed that the reported results overestimate the stiffness. The overestimation of the
member stiffness for the conical clamp analytical model is mainly due to the cone angle selection. Selecting a
lower cone angle may give better estimates of the stiffness.
NOMENCLATURE:
A, B = material constants
C = radial clearance
D = nominal diameter of the bolt
L = grip length/thickness of the joint
Pext = external load
p = uniform pressure applied on the nodes under washer
R = correction factor

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S = intermediate variable
r, z = radial and axial coordinates
α = semivertex angle of the frustum of pressure cone
δ = boundary displacement applied on the nodes under washer
µ = Poisson ratio
Ab = nominal cross sectional area of the bolt shank
Am = effective clamping area
Aw = area under the washer/bolt head in the case of no washer
Ci = constants used in correction factor expression, the subscript i vary from 1 to 6
dh = diameter of bolt hole or washer inner diameter
(dh=d+2c)
dm = outer diameter of the member
dw = washer outer diameter/bolt head diameter in the case of no washer
d1, d2 = minor and major diameters of the conical frusta
Eb = elastic modulus of bolt material
Em = elastic modulus of member material
Fb, = fraction of external load acting on bolt
Fi = preload applied on the joint
Fm = fraction of external load acting on members
Fr = sum of reaction forces of nodes under washer in UDA
Kb = bolt stiffness
KFEM = member stiffness, calculated by finite element analysis
Km = member stiffness
Ko = stiffness of hollow cylinder with inner and outer diameters as that of washer and length equal to the thickness of
the joint.
δa = average displacement of nodes under washer
δb = effective elongation of bolt due to preload
δm = effective compression of member due to preload
REFERENCES:
[1] Shigley, J. E., and Mitchell, L. D., 1983, Mechanical Engineering Design, 4thMcGraw-Hill, New York.
[2] Norton, R. L., 2000, Machine Design, An Integrated Approach, 2nd Edition, Prentice Hall, Upper Saddle River, New
Jersey.
[3] Manring, N. D., 2003,Sensitivity Analysis of the Conical Shaped Equivalent Model of a Bolted Joint,” ASME J. Mech.
Des., 125, pp. 642–646.
[4] Sethuraman R.,Sasi Kumar T.,2009,” Finite Element Based Member Stiffness Evaluation of Axisymmetric Bolted
Joints”ASME J. MECH. Des.131,
[3] Abdel-Hakim Bouzid, Hichem Galai, “A New Approach to Model Bolted Flange Joints With Full Face Gaskets”, ASME
J. Mech. Des
[4] Abdel-Hakim Bouzid, Hichem Galai, “A New Approach to Model Bolted Flange Joints With Full Face Gaskets”, ASME
J. Mech. Des
[5] Sayed A. Nassar, Antoine Abboud , “An Improved Stiffness Model for Bolted Joints”, ASME J. Mech. Des

Finite Element Based Member Stiffness Evaluation of Axisymmetric Bolted Joint

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