Proceedings of the Institution of Mechanical
Engineers, Part G: Journal of Aerospace
Engineering
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Estimating clamping pressure distribution and stiffness in aircraft bolted joints by finite-element
analysis
R H Oskouei, M Keikhosravy and C Soutis
Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 2009 223: 863
DOI: 10.1243/09544100JAERO596
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863
Estimating clamping pressure distribution and stiffness
in aircraft bolted joints by finite-element analysis
R H Oskouei1∗ , M Keikhosravy2 , and C Soutis3
1
Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Victoria, Australia
2
Department of Mechanical Engineering, Islamic Azad University, Firuzkooh Branch, Firuzkooh, Iran
3
Aerospace Engineering, The University of Sheffield, Sheffield, UK
The manuscript was received on 21 April 2009 and was accepted after revision for publication on 2 June 2009.
DOI: 10.1243/09544100JAERO596
Abstract: In this study, a finite-element (FE) stress analysis of aircraft structural double-lap bolted
joints was performed using the commercially available computational package ANSYS in order
to obtain the clamping pressure distribution and to estimate the stiffness of the joined plates
(members) within the clamped region. The joint consists of three aluminium alloy 7075-T6 plates
clamped by a single bolt, washer, and nut. A three-dimensional (3D) FE model of the joint was
generated, and then subjected to three different simulated clamping forces. 3D surface-to-surface
contact elements were employed to model the contact between the various components of the
bolted joint. The model included friction between all contacting surfaces, and also a clearance
between the bolt shank and the joint hole. FE results revealed an overall crock-shaped pressure
distribution at the joined plates. Moreover, some beneficial longitudinal compressive stresses
were observed around the fastener hole as a result of the clamping compressive effect.
Keywords: bolted joints, clamping pressure, joint stiffness, finite element modelling, aluminium
alloys
1
INTRODUCTION
Threaded fasteners, including overwhelming varieties
of bolts and nuts, are largely used to create bolted
joints for transferring loads among components in the
construction of aircraft structures. Connections of aluminium truss mounts in the aircraft engine support
structure to the trusses’ attached lugs are typical examples of bolted joints in aircraft structures. There are
conventional methods to design bolted joints and to
select the appropriate fasteners under different using
and loading conditions [1–5]. However, because of the
inevitable presence of drilled holes in the joint members, these connections are inherently vulnerable to
failure because of the localized stress concentration
and the bearing stresses at the fastener hole. As the
bolted joints represent such potential weak points in
the structure, where fatigue cracks can grow, the design
∗ Corresponding author: Department of Mechanical and Aerospace
Engineering, Monash University, Clayton, Victoria 3800, Australia.
email: reza.oskouei@eng.monash.edu.au
JAERO596
of the joint can have a large influence over the structural integrity and load-carrying capacity of the overall
structure.
Aluminium alloys and composite materials have
always been the preferred materials for aircraft construction because of their high strength-to-weight
ratio. Therefore, aluminium and composite bolted
joints are important elements in designing safe and
efficient aircraft structures. On account of this importance, many attempts have been conducted to develop
and optimize the design of aircraft structural bolted
joints under both static and dynamic loadings [6–12].
In this respect, finite-element analysis (FEA) is the
most extensively used numerical tool. For instance,
the stress results of a two-dimensional FEA were
used to understand failure modes of a bolted joint
in low-temperature cure carbon fibre reinforced plastic (CFRP) woven laminates loaded in tension and
to predict the bearing strength [12]. The numerical
results compared favourably to experimental measurements but no detailed analysis was performed for
the clamping force.
Tightening (twisting) the nut stretches the bolt
axially to produce the clamping force (called the
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R H Oskouei, M Keikhosravy, and C Soutis
pretension or bolt preload). It exists in the connection
after the nut has been properly tightened. Since the
members are being clamped together, the clamping
force that produces tension in the bolt induces compression in the members [1]. Previous research results
showed that the bolt clamping effect can decrease the
stress concentration at the bolted hole region when
the joint is subjected to an axial tensile load, and
thus increases the tensile strength and fatigue life of
the joint significantly [13–19]. Recent studies on a
single clamped plate showed that a clamping force
introduces some beneficial longitudinal compressive
stresses near the bolt-filled hole especially at the critical edge of the hole. FEA results confirmed that the
magnitude of the compressive stress at the hole region
increases significantly when higher clamping forces
are applied [18, 19].
Generally, the determination of local stress distribution at a bolted joint is a three-dimensional (3D)
problem because of the bending effects and clamping of the fastener. The stress state in the vicinity of a
bolted hole depends on many complex factors such as
friction properties of the members, contact problem,
geometry and stiffness of the joined members, joint
configuration, clamping force, and loading conditions.
Inclusion of all these factors in a stress analysis of a
joint based on the conventional analytical methods is
extremely cumbersome.
In the selection of a bolt preload to resist the tensile
joint separation load, the bolted joint is traditionally
characterized by the joint stiffness constant, which
depends on both the stiffness of the bolt and the
effective stiffness of the clamped material. The stiffness that is difficult to quantify is that of the clamped
members, because the stress distribution in the members must be estimated or found using FEA [20]. In
order to compute the member stiffness, some wellaccepted assumptions of pressure distribution are
used within the clamped zone such as an equivalent hollow cylinder and a pair of frustum hollow
cones. Meyer and Strelow [21] used the hollow cylinder assumption to study the member stiffness. In this
method, an equivalent hollow cylinder cross-sectional
area with a member diameter greater than three times
the diameter of the bolt is assumed to estimate the
member stiffness. The assumption of a conical distribution (with a cylindrical through hole) is most
commonly used to represent the pressure distribution at a bolted joint. Rötscher [22] proposed that the
stresses are contained within two conical frusta symmetric about the mid-plane of the joint, each having
a vertex angle of 2α. Then a half-apex angle of α = 45◦
was selected to compute the stiffness [22]. Ito et al. [23]
used ultrasonic techniques to determine the pressure
distribution, and the results showed that the pressure
remains high out to about 1.5 bolt radii. However, the
pressure falls off farther away from the bolt. Thus Ito
suggested the use of Rötscher’s pressure-cone method
for stiffness calculations with a variable cone angle.
This method is quite complicated; therefore, a simpler
approach using a fixed cone angle is suggested to use
for design [1].
A review of current literature confirmed that there
is no clear available method to determine the stress
state in bolted joints. How all important factors fully
influence the stress distribution in a joint is complex and has still not been thoroughly investigated.
It seems that the FE method is a convenient and efficient way to determine and analyse the stresses and
strains at the bolted joints. In previous research work
[18], an FE model was generated for a simplified bolted
joint (a single plate with a bolt-filled hole) to achieve
the clamping stress distribution around the hole in
order to investigate isolated effect of clamping force
on the fatigue life of the plate. Focusing on the single
clamped plate could provide the stress state because
of the only clamping of the bolt head and nut in the
absence of other joint plates. Obtained experimental
results of the fatigue life and experimental observations of initial cracks location verified the FEA results
very well. This agreement validated the FE modelling
of the bolted plate and FE simulation of the clamping
force.
This study was carried out to develop an FE simulation approach for the bolted joints in order to
analyse stresses and strains because of the clamping force. An aluminium alloy double-lap bolted joint,
which is extensively used in aircraft structures, was
selected to determine clamping pressure distribution
and stiffness of the members in the clamped zone. An
appropriate 3D FE model was generated and developed to simulate the bolt clamping force based on the
previously verified simulation approach. Stress results
developed in the clamped zone are discussed with
the aim to improve design of aircraft structural bolted
joints.
2
FE MODELLING DETAILS
In view of an FEA, the two primary characteristics
of a bolted joint that need to be considered are the
bolt preload and contact of mating surfaces. Preload
and contact capabilities are not available in all FE
codes. Therefore, workarounds are sometimes necessary. No single method has become the industry
standard, but ANSYS preload and contact elements
have helped in modelling the above characteristics
[24]. ANSYS includes a full complement of linear and
non-linear elements, material laws ranging from metal
to rubber, and the most comprehensive set of solvers.
It can handle relatively complex assemblies, especially
those involving non-linear contact problems, and is a
good choice for determining stresses, temperatures,
displacements, and contact pressure distributions on
all component and assembly designs.
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Estimating clamping pressure distribution and stiffness
2.1
Geometric modelling and meshing
Figure 1 presents the double-lap bolted joint that was
selected to be modelled and analysed using the ANSYS
FE package. The joint includes three identical aluminium alloy 7075-T6 plates with a thickness of 3 mm
and a 5 mm-diameter hole. A standard aerospace bolt
fastener (AN3-6A) with a diameter of 3/16 in was
selected to clamp the plates together.
Fig. 1
Joint geometry and dimensions (in mm)
Fig. 2
A 3D solid model was generated under preprocessing capabilities of ANSYS, according to the dimensions
of the plates and fastener. Owing to the joint geometry
and loading symmetry with respect to two Cartesian
planes, only one-fourth of the full model was numerically analysed (Fig. 2). Therefore, symmetric displacement boundary conditions were defined for the nodes
on these planes of symmetry. In order to model a bolt
fastener in the structure with a bolted joint, several different kinds of bolt models such as solid bolt model,
coupled bolt model, spider bolt model, and no-bolt
model have been introduced [24, 25]. Each model
has some advantages and disadvantages; however, the
solid bolt model is the most realistic FE model with
the best simulation approach for accuracy in which
tensile, bending, and thermal loads can be transferred
through the bolt. Kim et al. [25] reported that among all
four kinds of the bolt models, the solid bolt model for
a structure with a bolted joint could most accurately
predict the physical behaviour of the structure.
Since the solid bolt is the closest simulation of the
actual bolt, this approach was employed in this study
for modelling the single bolt fastener and the simulation of the clamping force in the joint. To simplify
the bolt head geometry, a circular shape was assumed
for the bolt head instead of its hexagonal shape. As
the bolt and its washer have similar tensile properties,
geometric model of the washer was added to the bolt
head (Fig. 3) in order to minimize the contact element
use (with ignoring contact elements between the bolt
head and washer). This simplification considerably
reduces computation processing time with a very good
approximation to the experimental observations [18].
Geometric modelling: (a) full model and Cartesian coordinate axes and (b) model of
one-fourth for FE analysis
Fig. 3
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Developed solid model of double-lap bolted joint
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R H Oskouei, M Keikhosravy, and C Soutis
In bolted joints, the length of the unthreaded section of
the bolt shank (grip length) should be approximately
equal to the total thickness of the fastened components [26]. According to the standard dimensions of
the selected AN3-6A bolt, the grip length is 3/8 in,
which is approximately equal to the total thickness of
the joint. Therefore, there is no need to consider the
bolt threads in this model. Moreover, a radial clearance
of 0.1 mm is between the bolt shank and the fastener
hole, as shown in Fig. 3.
The model consists of one set of solid elements for
the plates and another set for the bolt. The 3D structural brick elements, as called SOLID45 in ANSYS, were
used for the 3D modelling of the bolt and plate sections. This cubic-shaped element has eight nodes (one
on each vertex of a cube), each having three degrees of
freedom (translations in the x, y, and z directions). The
use of these elements provides the same accuracy in
plasticity (2 × 2 × 2 integration points) as the higherorder elements (20-node element), but requires much
less computational power to converge the numerical
solutions especially in non-linear problems such as
contact analysis [27]. Figure 3 illustrates the required
dummy divisions for the plates and bolt to achieve the
optimal mesh density based on the mesh refinement
around the hole, identification of contact zones, and
also obtaining a converged solution. Size of the mesh
regions and thus the elements were modified several
times in order to achieve element-size-independent
results. The final refined FE model is shown in Fig. 4.
2.2
subjected to the selected maximum wrenching torque
of 5.5 Nm [9]. The friction effect between all potential contacting surfaces was included in the analysis
using the elastic Coulomb friction model with a friction coefficient of 0.33 between the steel bolt (head)
and top aluminium plate [28], and 0.32 between the
top and middle plates [13].
2.3
Contact definition
Contact between components of a bolted joint is a
main feature that transfers the applied load in the
joint. Therefore, it is essential to accurately model contact conditions in bolted joints in order to achieve
much more reliable results. Contact problems are
Material properties
Fig. 4 Typical FE model used
An elastic–plastic multilinear kinematic hardening
material model was used to represent the aluminium
alloy 7075-T6 stress–strain behaviour. This material
model was selected to analyse plastic as well as elastic stresses and strains on condition that the applied
bolt preloads cause the material to be plastically
deformed. To do this, a true stress–strain diagram for
Al-alloy 7075-T6 was obtained from a simple tensile
test (Fig. 5); then, seven data points from this diagram were used as input data for the material model,
as given in Table 1. Furthermore, the elastic modulus
and Poisson’s ratio were measured to be E = 71.0 GPa
and ν = 0.33, respectively. However, for the steel bolt
and its steel washer, a linear elastic material relationship was assumed with a Young’s modulus of 210 GPa
and a Poisson’s ratio of 0.30. This is based on the
tested fact that the bolt material remained in the elastic
region (without any plastic deformation) when it was
Fig. 5 True stress–strain diagram for Al-alloy 7075-T6
Table 1 Test data points of stress–strain diagram for 7075-T6
Strain
0
7 × 10−3
8.2 × 10−3
9.4 × 10−3
12 × 10−3
16 × 10−3
1 × 10−1
Stress (MPa)
0
497
524
538
552
565
647
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Estimating clamping pressure distribution and stiffness
generally classified into two classes: rigid-to-flexible
and flexible-to-flexible. Bolted joints are an example of flexible-to-flexible contact problems. ANSYS
can model contact problems with contact elements
based on the present Lagrange multiplier, penalty
function, and direct constraint approach. When meshing a model, the nodes on potential contacting surfaces comprise the layer of contact elements whose
four Gauss integration points are used as contacting checkpoints. ANSYS provides three contact
models: node-to-node, node-to-surface, and surfaceto-surface. Each type of contact model uses a different
set of ANSYS contact elements and is appropriate for
specific types of problems. Contact in the bolted joints
is addressed using these contact types and their elements depending on the model being used. For the
solid 3D modelling, the surface-to-surface contact is
mostly used.
For the FE model of the bolted joint, a 3D fournode surface-to-surface contact element CONTA173
was used to represent the contact between contacting
surfaces in the joint model. This element was preferred
to the eight-node element CONTAC174 (with mid-side
nodes) because of the use of structural SOLID45 elements for the solid model of the joint which have no
mid-side nodes [29]. A 3D target segment element
TARGE170 was also used to associate with CONTA173
via a shared real constant set.
2.4
867
the nut is tightened by applying a torque, thus causing
the bolt to axially elongate. Since the bolt head and nut
react to this behaviour of the bolt, a preload is created
in the bolt, and consequently causes the joint members to clamp together (called a joint clamping force).
Based on this fact, a combined approach was developed for simulation of the clamping force in the joint
model. In this method, the clamping effect is considered by directly applying an axial displacement at the
bottom surface of the bolt shank mid-section in the
solid bolt model.
To apply the clamping force, the problem was
numerically solved by applying an initial negative displacement at the bottom surface of the bolt shank
in the Y direction (see Fig. 3). Then the corresponding clamping force because of the axial displacement
was quantified by obtaining the total reaction force
in the solid bolt model. As the bolt model includes
half of the bolt shank, the bolt clamping force was
finally determined by multiplying the obtained quantity by 2. This approach was repeated several times
to accurately achieve three previously selected clamping forces of 1000, 3000, and 6000 N. This method of
simulation for clamping force in bolted joints was validated in earlier studies where the comparison between
the experiments and simulation results showed a very
good agreement and verified the validation and accuracy of the bolt modelling approach in a single bolted
plate [10, 18].
Clamping simulation
In FE modelling, applying a clamping force can physically simulate fastening bolt at the joint. It was
reported that bolt clamping can generally be modelled
with different methods such as thermal deformation,
constraint equation, and initial strain [25, 30]. In the
thermal deformation method, the preload is generated by assigning virtually different temperatures and
thermal expansion coefficients to the bolt and components. In the case of the constraint equation method,
the preload is a special form of coupling, with which
equations can be applied to govern the behaviour of
the associated nodes. The initial strain method is a
more direct approach, in which the initial displacement is considered as a portion of the preload on the
structure with a bolted joint [25]. In solid bolt model, to
apply the clamping force over the bolt, the virtual thermal deformation method is employed. The thermal
expansion coefficient is assumed to be unit and the
temperature difference T is regarded as the following
relation
T =
4Fcl
Eπd 2
FE STRESS RESULTS
3.1
Pressure distribution due to clamping force
The contour of transverse normal stress σy , which
presents the clamping pressure in the clamped region,
resulting after application of a 6 kN clamping force is
shown in Fig. 6. An overall crock-shaped pressure distribution is observed at the joint considering all plates.
However, the clamping pressure at the outer plates,
which are closer to the bolt head and nut, is represented by a frustum of a hollow cone, according to the
(1)
where E is the Young’s modulus of the material, d is the
bolt nominal diameter, and Fcl is the clamping force.
However, when a bolt and nut are used to fasten a joint,
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3
Fig. 6
Distribution of transverse normal stress σy in MPa
at clamped plates because of a 6 kN clamping
force
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R H Oskouei, M Keikhosravy, and C Soutis
front view of the joint thickness shown in Fig. 7. Moreover, the pressure-cone angles α1 and α2 due to the
applied clamping forces were obtained for each case of
loading by drawing a tangent line to the pressure band,
which indicates a compressive stress of approximately
−10 MPa in the contour of stress σy .
As can be observed in Fig. 8, compressive transverse normal stresses due to the bolt clamping are
symmetrically distributed around the hole of the middle plate like circular bands for all three differently
clamped models. The most compressive transverse
stress occurs at the hole edge whose magnitude
increases significantly when a higher clamping force is
applied. The results showed that the transverse stress
σy at the hole edge of the middle plate increased
from −12 to −64 MPa when the clamping force was
enhanced from 1 to 6 kN, respectively. However, the
stress faded further from the hole edge even in the
firmly clamped model. Moreover, the contour reveals
that the stress distribution is quite uniform along the
thickness of the middle plate.
The observation of strain results confirmed that
all components of the strain remained in the elastic
region of the plate material, and no plastic deformation occurred in the joint plates even because of the
maximum applied clamping force (6 kN).
3.2
Longitudinal stress in middle plate due to
clamping
It is obvious that the final fracture in double-lap bolted
joints loaded in a longitudinal tension occurs in the
middle plate at or in the vicinity of the reduced section
at the fastener hole. In fact, the nature of the load transfer mechanism in the joint causes the middle plate
to bear the applied longitudinal tensile load by itself
rather than two outer plates. Therefore, it is important
to investigate whether the clamping effect can introduce some beneficial compressive stresses around the
fastener hole in the middle plate. Figure 9 illustrates
the longitudinal normal stress σx contour in the middle plate after applying a 6 kN clamping force to the
joint model. Some desirable compressive stresses are
observed near the hole especially at the critical edge
of the hole. This beneficial effect of the clamping is
presented in Table 2 where a higher clamping force
introduces a higher magnitude of the longitudinal
compressive stress throughout the thickness of the
middle plate at the critical edge of the hole. According
to the obtained numerical results, although the magnitude of the longitudinal compressive stresses even
for the highest clamping force is not considerable, this
stress component can play a key role in reducing the
resultant in-plane stress when the joint is subjected to
a remotely applied longitudinal tensile loading.
3.3
Stiffness of clamped members
A general view of the clamping compression geometry at the joined members with the half-apex angle α
is shown in Fig. 10. In order to compute the stiffness
of the double-lap bolted plates in the clamped zone,
a pressure distribution in the form of a frustum of a
Fig. 7
Pressure distribution with pressure-cone angles
because of the applied set of clamping forces
Fig. 9
Distribution of longitudinal normal stress σx in
MPa in middle plate because of a 6 kN clamping
force
Table 2
Fig. 8
Distribution of transverse normal stress σy in MPa
in middle plate because of a 6 kN clamping force
Effect of clamping to increase longitudinal compressive stress magnitude (averaged
throughout the thickness of middle plate at
critical edge of the hole)
Applied clamping force (kN)
1
3
6
Longitudinal compressive stress
(MPa)
−3.10
−6.66
−7.92
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Estimating clamping pressure distribution and stiffness
Fig. 10
Clamping compression at double-lap bolted plates represented by a pair of frustum hollow
cones and a middle hollow cylinder
hollow cone at the outer plates and a hollow cylinder at the middle plate is assumed. Using Shigley’s
analytical solution for member stiffness based on the
work of Lehnhoff et al. [31], the contraction of an element of the hollow cone of thickness dy subjected to a
clamping force Fcl is
Fcl dy
E1 A1
dδ1 =
(2)
The area of this cylinder is
A2 =
A1 =
−
D1
+
2
K2 =
=
Fcl
πE1
3t/2
t/2
(4)
Therefore, the spring rate or stiffness of the frustum as
the top/bottom plate is
K1 =
Fcl
=
δ1
πE1 d tan α
(D1 + d)(D1 − d + 2t tan α)
ln
(D1 − d)(D1 + d + 2t tan α)
(5)
Similarly, for the hollow cylinder
δ2 =
Fcl t
E2 A 2
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2
2 d
−
2
πE2 [(D1 /2 + t tan α)2 − (d/2)2 ]
Fcl
A2 E2
=
=
δ2
t
t
(8)
1
2
1
K1 K 2
=
+
⇒ Km =
Km
K1
K2
K1 + 2K2
(3)
dy
[(D1 + d)/2 + (3t/2) tan α − y tan α]
×[(D1 − d)/2 + (3t/2) tan α − y tan α]
(D1 + d)(D1 − d + 2t tan α)
Fcl
ln
πE1 d tan α
(D1 − d)(D1 + d + 2t tan α)
=π
D1
+ t tan α
2
As all three plates act like compressive springs in
series, the total spring rate (stiffness) of the plates is
The total contraction of the hollow cone is obtained by
integrating equation (2) from y = t/2 to y = 3t/2 as
δ1 =
−
ri22 )
Thus, the stiffness of the hollow cylinder as the middle
plate is
ri12 )
2 2 3t
d
− y tan α −
=π
2
2
D1 + d
3t
=π
+
tan α − y tan α
2
2
D1 − d
3t
×
+
tan α − y tan α
2
2
2
π(ro2
(7)
The area of the element is
2
π(ro1
869
(6)
(9)
As previously mentioned, in order to design bolted
joints more properly and to achieve safer and more
reliable joints particularly in aircraft structures, the
stiffness of the clamped material is required to be computed as precisely as possible. This needs to obtain the
clamping pressure distribution at the joint accurately.
The above analysis, which is using the characteristics
of the obtained pressure geometry (α and D2 ), can be
used to develop the design of aluminium bolted joints,
and thus to optimize the clamping force such that aluminium plates in the double-lap bolted joint are not
failing by through-the-thickness crushing.
4
CONCLUSIONS
In order to determine the clamping pressure distribution and joint stiffness in aircraft metallic double-lap
bolted joints, a solid-bolt-based model was used in
ANSYS FE package to simulate the applied clamping
force in the joint. An overall crock-shaped pressure
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R H Oskouei, M Keikhosravy, and C Soutis
distribution was observed at the clamped plates
including a pair of frustum hollow cones developed
at the outer plates and a hollow cylinder shape at the
middle plate uniformly distributed along the thickness. The bolt clamping force can introduce some
beneficial longitudinal compressive stresses around
the fastener hole with localized maximum magnitudes
at the critical edge of the hole. These stresses are more
compressive in firmly clamped joints and can considerably reduce the damaging effect of in-plane tensile
stresses that usually develop at the edge of the hole
when a tensile load is remotely applied to the joint.
The clamping simulation method used in this study
compares favourably with some recent experimental
work performed by the authors [18], but further analysis is required, especially for multiple bolted joints and
fatigue loading conditions.
12
13
14
15
16
© Authors 2009
17
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APPENDIX
Notation
A1
A2
d
dy
D1
D2
E
E1
E2
element cross-section area of hollow cone
cross-section area of cylinder
bolt nominal diameter, hole diameter
element thickness of hollow cone
outside diameter of washer
outer diameter of cylinder
Young’s modulus
Young’s modulus of top/bottom plate
Young’s modulus of middle plate
JAERO596
871
Fcl
Km
K1
K2
ri1
ri2
ro1
ro2
t
clamping force
total stiffness
stiffness of top/bottom plate
stiffness of middle plate
element inner radius of hollow cone
inner radius of cylinder
element outer radius of hollow cone
outer radius of cylinder
thickness
α
δ1
half-apex angle of pressure cone
total transverse contraction of top/
bottom plate
total transverse contraction of middle plate
temperature difference
Poisson’s ratio
longitudinal normal stress
transverse normal stress
δ2
T
ν
σx
σy
Proc. IMechE Vol. 223 Part G: J. Aerospace Engineering
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