Finite Element Analysis of a Three-dimensional Threaded
Structural Fastener
by
Ramin M. Rafatpanah
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the Degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
May 2013
© Copyright 2013
by
Ramin M. Rafatpanah
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS ........................................................................................................ vi
ABSTRACT .................................................................................................................... vii
1 - INTRODUCTION / BACKGROUND ........................................................................ 1
2 - THEORY / METHODOLOGY.................................................................................... 5
3 - RESULTS AND DISCUSSION ................................................................................ 15
4 - MESH ADEQUACY ................................................................................................. 22
5 - CONCLUSION .......................................................................................................... 24
REFERENCES ................................................................................................................ 25
iii
LIST OF TABLES
Table 1: General Model Dimensions ................................................................................ 5
Table 2: Run Information ............................................................................................... 10
Table 3: Description of Contact Pairs ............................................................................. 12
Table 4: Mesh Density Comparison ............................................................................... 22
Table 5: Mesh Density Run Information ........................................................................ 23
iv
LIST OF FIGURES
Figure 1: Example of a Bolted Joint ................................................................................. 1
Figure 2: Bolted Joint Clamping Forces ........................................................................... 2
Figure 3: Joint Resiliency Ratio [1] .................................................................................. 3
Figure 4: Bolted Joint Force Distribution (No Separation) .............................................. 4
Figure 5: Bolted Joint Force Distribution (Joint Separation) ........................................... 4
Figure 6: Threaded Fastener Thread Profiles ................................................................... 7
Figure 7: Cross-section of External Thread Perpendicular to Bolt Axis [2] .................... 8
Figure 8: Node Generation Pattern ................................................................................... 8
Figure 9: Overall View of Threads in Finite Element Model ........................................... 9
Figure 10: Cross-section of Finite Element Model......................................................... 10
Figure 11: End of Threads on Nut and Bolt ................................................................... 11
Figure 12: Cross-section View of Finite Element Model ............................................... 11
Figure 13: Pure Penalty Contact Formulation ................................................................ 12
Figure 14: Tangential Displacement (mm) ...................................................................... 15
Figure 15: Vertical Displacement (mm) ......................................................................... 16
Figure 16: Equivalent Stress Cross-section (ksi) ............................................................ 17
Figure 17: Equivalent Stress (ksi)................................................................................... 17
Figure 18: Stress in Root of Bolt versus Axial Height ................................................... 18
Figure 19: Equivalent Stress of Middle Elevation of Bolt Shank (ksi) .......................... 19
Figure 20: Contact Pressure of Threads - Top Engagement (ksi) .................................. 20
Figure 21: Contact Pressure of Threads - Bottom Engagement (ksi) ............................. 20
Figure 22: Contact Pressure of Top of Nut (ksi) ............................................................ 21
Figure 23: Mesh Convergence Plot ................................................................................ 22
v
LIST OF SYMBOLS
Angle at Minor Diameter
θ1
radians
Angle at Major Diameter
θ2
radians
Length of Engagement
Le
inches
Membrane Stress
σm
psi
Nominal Bolt Diameter
d
inches
Pitch
P
inches
Tensile Area
At
square inches
Thread Overlap
H
inches
Thread Root Radius (bolt)
Pb
inches
Thread Root Radius (nut)
Pn
inches
Thread Shear Area
AS
square inches
Thread Shear Stress
τ
psi
Torque
T
inch-pounds
Torque Coefficient
K
dimensionless
vi
ABSTRACT
Threaded structural fasteners have been analyzed for years by hand calculations to
determine the average stress in the shank and the average thread shear stress. A threedimensional finite element model of a bolted joint is developed in ANSYS®1 to compare
to the typical hand-calculated stresses. A method to generate the three-dimensional
thread geometry without discontinuities is developed and evaluated. The model consists
of a threaded structural fastener, nut, and flange assembly. A torque is applied to the nut
to tension the fastener and simulate the effects of frictional surfaces. The results of the
finite element model analysis contained herein are compared to hand calculations. Based
these results, a finite element analysis should be used if a greater level of detail is
required for the analysis of a threaded structural fastener.
1
ANSYS, ANSYS Workbench, Ansoft, AUTODYN, CFX, EKM, Engineering Knowledge Manager,
FLUENT, HFSS and any and all ANSYS, Inc. brand, product, service and feature names, logos and
slogans are trademarks or registered trademarks of ANSYS, Inc. or its subsidiaries located in the United
States or other countries. ICEM CFD is a trademark used by ANSYS, Inc. under license. CFX is a
trademark of Sony Corporation in Japan.
All other brand, product, service and feature names or
trademarks are the property of their respective owners.
vii
1 - INTRODUCTION / BACKGROUND
Threaded structural fasteners are one of the most common methods used to join
assemblies in mechanical components. They allow components to be disassembled and
reassembled with greater ease, as compared to other methods like welding. However,
there are several difficulties associated with using threaded structural fasteners (i.e., the
ability to determine the preload applied to a bolt and the non-linear behavior of a bolted
assembly). An example of a bolted assembly is shown in Figure 1.
Flange
Nut
Bolt
Figure 1: Example of a Bolted Joint
The bolt preload is the clamping force that holds bolted assemblies together.
Therefore, the bolt preload is an important factor used to determine the acceptability of a
given joint. The amount of bolt preload at installation can be estimated by Equation 1
from [5]:
T
Preload
K d
(1)
However, there can be significant scatter when determining the torque coefficient
(K) and the applied torque (T). These values can be affected by the friction of the threads
or bearing surfaces, thread geometry, and how the torque is applied to the bolt or nut.
As the bolt is tensioned, the applied load is divided between compressing the flange
and tensioning the bolt; see Figure 2. The ratio of flange stiffness to bolt stiffness
1
determines how much of the applied load is divided between compressing the flange and
tensioning the bolt.
The same load sharing behavior exists after the desired preload is reached in the
bolt. As an external load is applied to the joint, part of the load is reacted by the bolt and
part is reacted by the flange. The joint resiliency ratio can determine the percentage of
the applied external load that the bolt will resist; see Figure 3. Figure 3 shows a plot of
how the joint resiliency ratio changes depending on the clamping ratio. Figure 3 also
shows that there is a maximum joint resiliency ratio that is dependent on the clamping
material properties.
Bolt
Flange
Joint
Nut
Figure 2: Bolted Joint Clamping Forces
2
Figure 3: Joint Resiliency Ratio [1]
Figure 4 and Figure 5 are charts that explain the loading scenarios in a bolted joint.
For both figures, the vertical axes represent force and the horizontal axes represent
deflection. The left sides of the figures show the bolt extension and the right sides show
the joint compression. Therefore, if there is no external applied load, then the force
carried by the bolt and flange are equal, but the displacements are not equal. These
figures help explain how an external load is reacted by both the bolt and flange, as well
as when the bolted flange will separate due to an external load.
Figure 4 shows a scenario when the external load is less than the required joint
separation load. The difference in stiffness between the bolt and flange determines how
much of the external load is reacted by the bolt. To separate the joint, the external load
needs to be greater than preload over one minus the joint resiliency ratio. Therefore, the
3
separation load is larger than the preload in the bolt, as shown in Figure 5. This is
because the external load concurrently reduces the clamping force while increasing the
bolt force.
External Load
dF (flange)
Preload
Force
dF (bolt)
Bolt Extension
Joint Compression
Bolt Extension
External Load
dF (bolt)
Preload
Force
Figure 4: Bolted Joint Force Distribution (No Separation)
Joint Compression
Figure 5: Bolted Joint Force Distribution (Joint Separation)
4
2 - THEORY / METHODOLOGY
A typical M16 bolt was utilized for this analysis. An M16 bolt is a standard
metric size bolt, as described in [3]. The analysis in [2] defines a method to develop a
finite element mesh for a bolted joint. Furthermore, the analysis in [2] establishes results
for the stress in the bolt threads. The same dimensions presented in [2] are used for this
analysis; the results from the two analyses are compared here. The thread profiles were
taken from [2] for a typical M16 bolt. General dimensions of the bolted assembly are
given in Table 1. Typical material properties for stainless steel are used for all parts in
this simulation. All of the geometry is developed in millimeters; therefore, the modulus
of elasticity is input as 200E3 MPa (N/mm2) with a Poisson’s ratio of 0.3.
Table 1: General Model Dimensions
Description
Dimension (mm)
Flange Inner Diameter
17.5
Flange Outer Diameter
128
Flange Thickness
48
Nut Thickness
10
Nut Outer Diameter
24
Bolt Head Diameter
24
Bolt Head Thickness
8
Bolt Threaded Length
14
The following equations develop the thread profile for the bolt and nut using a
typical M16 bolt with a pitch of 2 mm. Figure 6 plots the thread radius as a function of
circumferential position for both the external and internal threads.
5
) 
3
d  16 mm P  2 mm pb 
 P  0.289 mm 1 
12
H

t2( ) 
  
d
2

7
8
H
r_bolt 1( ) 
d
r_bolt2( ) 
H
r_bolt 3( ) 
d

2

7
2
P
2
 H  2 p b  p b 
8
  
d
2

7
8
4 
2

3 
P
 pb
2 
7
8

H 
3
2
P
2
H
2




r_bolt( )  if 0    1r_bolt1( ) if 1    2r_bolt2( ) if 2     r_bolt3( ) 0
p n 
H

  
d
2

7
8
H
3
24
P
1 
r_nut1( ) 
d
r_nut2( ) 
H
r_nut3( ) 
d
2

2


5
8
3 pn 
2     1 

4
P


H
  



H
8
d
2

7
8
H
 2 p n 
2
pn 
2
P
4 
2
(   )

2


r_nut( )  if 0    1r_nut1( ) if 1    2r_nut2( ) if 2     r_nut3( ) 0
6
Circumferential Position (Radians)
Bolt Major Diameter
2
Bolt Root
0
2
Nut Threads
Bolt Threads
7
7.5
8
Radius (mm)
Figure 6: Threaded Fastener Thread Profiles
A finite element model was developed and used to simulate a bolted assembly with
a focus on the threaded interface. This model was generated through rotation and
translation of a two-dimensional plane of nodes. The locations of these nodes were
adjusted at the beginning and end of the threads to provide smooth transitions at the
boundaries of the threads. Elements were generated between these nodes to create a
finite element model for the threaded locations.
The first two-dimensional plane of nodes was generated from an area of the crosssection of the thread profile perpendicular to the bolt axis; see Figure 7. The bolt radius
varies along the circumferential direction in this cross-section. These nodes were copied
several times to generate all the nodes for the threaded region. Each layer rotates the
nodes about the bolt axis 2π/N radians and translates the nodes P/N along the bolt axis
(where N is the number of divisions chosen and P is the pitch of the threads). The bolt
has seven full threads (seven times P/N layers) and the nut has five full threads (five
times P/N layers). The original area is divided into 2N segments along the circumference
to provide better aspect ratios (when compared to the method used in [2]) of the resulting
elements. This process, depicted in Figure 8, enables the entire model to be created with
7
brick elements.
These brick elements are favorable when compared to tetrahedral
elements with regard to solve time and solution accuracy.
One Full Pitch
Figure 7: Cross-section of External Thread Perpendicular to Bolt Axis [2]
P/N
2π / N
Figure 8: Node Generation Pattern
8
The process previously described was repeated for the cross-section of the nut to
generate the threaded section of the nut. Volumes were generated and meshed for the
remaining sections of the bolt shank, bolt head, and flange. The locations of the nodes
were modified at the beginning and end of the threaded section of the bolt and nut before
elements were generated. A 45-degree chamfer with a length of half the pitch length is
modeled on the top of the nut, the bottom of the nut, and the bottom of the bolt. This
approach provides a smooth transition for engagement between the threaded section of
the nut and bolt. A transition between the top of the bolt threaded section and the bolt
shank was utilized. The transition starts one pitch below the top of the threaded section
of the bolt and blends the threads into the bolt shank. The resulting threaded section of
the finite element model is shown in Figure 9. A side view and an isometric view are
presented on the left and right of Figure 9, respectively. These views illustrate the full
thread profile of the assembly. The chamfer on the bottom threads and the blend on the
top threads can be seen on the left part of Figure 9.
Figure 9: Overall View of Threads in Finite Element Model
The mesh density was chosen to adequately resolve the thread geometry. In
particular, the mesh characteristics need to adequately capture the geometric shape of the
thread root. This is an area of interest because the highest stress values and the highest
stress gradients are expected at the thread root. This model was developed and solved
9
using ANSYS. This run was performed on a Linux solving cluster with two processors
for the solution. The parameters of this run are summarized in Table 2.
Table 2: Run Information
FEA Software
ANSYS Version 12.1
Number of Processors
2
Run Time
16 Hours
Maximum Scratch Memory Used
5,989 MB
Number of Nodes
227,377
Number of Elements
243,953
A torque of 40 ft-lbf (54,233 N-mm) was applied to the outer diameter of the nut
to tension the bolt. Figure 10 shows the cross-section of the full finite element model.
The outer diameters of the flange and bolt head are fixed in all three degrees of freedom.
Bolt
Flange
Nut
Figure 10: Cross-section of Finite Element Model
10
Figure 11 shows a zoomed view of the transition of the bolt and nut threads.
Figure 12 shows a cross-section view of the threaded region. Figure 11 shows the
resolution of the thread curvature. Figure 12 shows the same region of the threads with a
cutout view.
Figure 11: End of Threads on Nut and Bolt
Figure 12: Cross-section View of Finite Element Model
Contact pairs are utilized on several locations for mesh connections and nonlinear contact. Table 3 summarizes the contact pairs used in this model. A typical
coefficient of friction of 0.5 was used for stainless steel to stainless steel frictional
11
surfaces. A typical coefficient of friction of 0.2 was used for the stainless steel to
stainless steel thread interface, which is similar to the value used in [2].
Table 3: Description of Contact Pairs
Location
Inner Diameter of Threaded Section of
Bolt to Outer Diameter of Bolt Interior
Top of Threaded Section of Bolt to
Bottom of Bolt Shank
Type
Bonded
Bonded
Bottom of Bolt Head to Top of Flange
Bonded
Top of Nut to Bottom of Flange
Frictional
Bolt Threads to Nut Threads
Frictional
Description
Connection between Dissimilar
Mesh Densities
Connection between Dissimilar
Mesh Densities
Assumed a Bonded Connection
for Simplification
Non-linear Frictional Contact
(μ = 0.5)
Non-linear Frictional Contact
(μ = 0.2)
Figure 13: Pure Penalty Contact Formulation
The pure penalty contact formulation is used for all the contact pairs listed in
Table 3. This contact formulation, defined in [4], utilizes a penalty stiffness to reduce the
amount of penetration in a given contact. Figure 13 shows a contact surface (solid blue
line) moving a given distance in a given equilibrium iteration. The distance past the
12
target surface (solid black line) is the resulting penetration of this contact pair (X).
Therefore, to reduce the penetration of this contact pair, a penalty stiffness is applied that
is similar to a spring force between the contact and the target surface. This spring force
is developed by utilizing a scale factor (FKN) multiplied by the contact stiffness (Kcont)
multiplied by the penetration distance (X). This formulation provides a good balance
between accuracy and run time. However, this method will result in a penetration
distance between the contact and the target surface.
Hand calculations are often used to evaluate bolted joints. These hand calculations
were utilized to validate the finite element model. A value of 0.5 is used for the
coefficient of friction between the top of the nut and the bottom of the flange. Therefore,
it is assumed that half of the applied torque is lost due to friction between these surfaces.
applied_torque  40 ft lbf  54233 N  mm
torque 
applied_torque
2
 27116 N  mm
Using a coefficient of friction of 0.2 for the bolt threads to the nut threads, the axial
force in the bolt can be calculated as:
torque
Ftorque 
 8474 N
d  0.20
The length of engagement is determined by the length of the bolt threads that are in
contact with the nut threads, subtracted by half a pitch for the chamfer on both the top
and bottom of the nut:
P
Le  5 P  2    8 mm
2
 
The bolt dimensions are used to determine the tensile area and the thread shear area:
rminor_bolt  r_bolt( 0)  6.773 mm
rpitch_bolt 
rmajor_bolt  rminor_bolt
2
2
2
A t    rmajor_bolt  201.062 mm
 
rmajor_bolt  r_bolt 2  8mm
 7.387 mm
Le
2
AS    2 rpitch_bolt 
 185.645 mm
2

13

Then, the membrane stress in the shank of the bolt and the thread shear stress can be
calculated:
m 
Ftorque
At
 thread 
7
 4.215  10 Pa
Ftorque
AS
m  6.113 ksi
7
 4.565  10 Pa
 thread  6.62 ksi
These stresses are the average stresses used to compare to allowable stress values
for design criteria. The average stress values will also be obtained from this finite
element model and compared to the analytical values.
14
3 - RESULTS AND DISCUSSION
In this analysis, a torque is applied to the outer diameter of the nut. This torque
drives the nut in a circumferential direction and causes the nut to travel a large distance
when sliding along the bolt threads. Therefore, this analysis utilizes large deflections.
The tangential displacement of the nut due to the applied torque is shown in Figure 14.
Figure 14: Tangential Displacement (mm)
As the nut rotates, the threads on the nut tension the bolt. This creates a preload
in the bolt shank. This preload is reacted by the top of the nut onto the bottom of the
flange and by the bottom of the bolt head on the top of the flange. The vertical
displacement of the bolt and nut is shown in Figure 15. Figure 15 shows that the bolt is
moving downward and that the nut is moving upward. This shows how the flange is
clamped by the nut and bolt head.
15
Figure 15: Vertical Displacement (mm)
The equivalent stress of the threaded region is shown in Figure 16 and Figure 17.
The maximum stress occurs at the root of the threads on the bolt. The maximum stress
occurs at half pitch from the top of the nut because the first half pitch is chamfered.
Therefore, the thread on the nut is a partial thread until half pitch from the top of the nut.
Figure 16 shows the stress profile through the cross-section of a given thread. The stress
decreases in the threads when moving outward in the radial direction along a given
thread.
16
Figure 16: Equivalent Stress Cross-section (ksi)
Figure 17 shows how the stress intensity in the root of the threads on the bolt
changes along the height of the bolt threads. The root of the threads on the bolt has the
highest stress.
Figure 17: Equivalent Stress (ksi)
17
Figure 18 plots the scaled stress (thread root stress divided by the average stress
on the bolt shank) along the root of the thread along the bolt. The thread stress profile
presented in Figure 18 agrees well with the data presented in Figure 7 of [2]. However,
Figure 7 of [2] has a second peak labeled “Run Out of Thread”. This does not occur in
Figure 18 because the threads are blended into the bolt shank.
Stress in the Root of the Bolt vs Axial Height
6.0
Thread Root
Top of Nut
Normalized Stress (Thread/Shank)
5.0
Bottom of Nut
Top of Bolt Threads
Bottom of Bolt Threads
4.0
3.0
2.0
1.0
0.0
-1
0
1
2
3
4
5
6
Axial Height (Pitch)
Figure 18: Stress in Root of Bolt versus Axial Height
The average stress in the bolt shank is determined by selecting a cross-section of
nodes in the bolt shank at half of the height of the bolt shank. The nodal stress is
averaged across all of the nodes to produce the bolt shank stress. Figure 19 shows the
stress distribution of the bolt shank cross-section.
18
Figure 19: Equivalent Stress of Middle Elevation of Bolt Shank (ksi)
All of the bolt threads are selected and the nodal stresses are averaged to determine
the average thread shear stress. These two average stresses are compared to the hand
calculations previously performed:
m_ansys  6.51020303 ksi
 thread_ansys  6.71734516 ksi
m_ansys  m
m
 6.5 %
 thread_ansys   thread
 thread
 1.5 %
The contact pressures for the non-linear frictional contacts are plotted in Figure 20,
Figure 21, and Figure 22.
19
Figure 20: Contact Pressure of Threads - Top Engagement (ksi)
Figure 20 and Figure 21 show how the threads on the nut engage on the threads
on the bolt. This engagement profile is tapered and gradual because of the chamfer to
create partial first and last threads on the nut threads. The engagements of the first and
last thread are shown by Figure 20 and Figure 21, respectively.
Figure 21: Contact Pressure of Threads - Bottom Engagement (ksi)
20
Figure 22 shows the distribution of contact pressure between the top surface of
the nut and the flange. A higher contact pressure is developed closer to the bolt threads.
Figure 22: Contact Pressure of Top of Nut (ksi)
21
4 - MESH ADEQUACY
The finite element method uses several approximations to develop a resulting
solution. One of these approximations is the discretization of the domain. A
discretization that is too coarse can impact the accuracy of the results. Various mesh
densities are produced to evaluate the discretization of the domain’s affect on the
solution. Similar results with a much finer mesh show that the solution is not
significantly affected by the discretization error inherent in finite element
approximations. Table 4 and Figure 23 compare the mesh density for the models used in
this comparison. The mesh density is changed by changing the number of divisions both
circumferentially and axially along the bolt and nut threads. Based on the mesh
sensitivity study, the number of vertical divisions per pitch selected for this analysis was
28.
Table 4: Mesh Density Comparison
Number of
Vertical Divisions
per Pitch
16
28
40
48
8.5
Stress (ksi)
8
Number
of Nodes
Number of
Elements
Average Shank
Stress (ksi)
106,914
227,377
441,570
612,340
108,861
243,953
475,983
664,239
6.83
6.51
6.38
6.37
Average Thread
Shear Stress
(ksi)
8.26
6.72
6.87
6.86
Mesh Convergence
Average Shank Stress
Average Thread Shear Stress
7.5
7
6.5
6
1E+05
1E+06
Number of Nodes
Figure 23: Mesh Convergence Plot
22
Table 5 describes various run properties for the mesh adequacy runs performed.
The number of processors used for each run was based on the availability of resources at
the time of the run. The maximum scratch memory used is the amount of random access
memory taken for the given run. This is also dependent on available shared resources on
the Linux solving cluster. The page file size is a file written to the hard disk, which is
used as additional scratch space. Very large page files are used for this analysis during
the model generation, as opposed to during the solution of the model which is more
typical. This is because large matrices are stored to create the thread profile.
The major contributor to increased solve time on the larger models is due to the
contact surfaces. The more vertical divisions per pitch on the threads creates a larger
number of smaller contact surfaces where each of the contact surfaces can contribute to
chatter. The computational difficulty on the contact pairs in the threads requires the
solution to be solved in several smaller increments of the total applied load.
Table 5: Mesh Density Run Information
Number of Vertical
16
28
40
48
Finite Element
ANSYS
ANSYS
ANSYS
ANSYS
Analysis Software
Version 12.1
Version 12.1
Version 12.1
Version 12.1
Run Type
Distributed
Distributed
Distributed
Distributed
8
2
8
16
1.4 Hours
16 Hours
20.7 Hours
25 Hours
568 MB
5,989 MB
3,682 MB
3,660 MB
32,988 MB
58,997 MB
58,997 MB
58,987 MB
Divisions per Pitch
Number of
Processors
Run Time
Maximum Scratch
Memory Used
Page File Size
23
5 - CONCLUSION
Hand calculations are used extensively in engineering analyses because they provide
a good estimate with minimal analytical effort. The typical equations used to analyze
threaded structural fasteners compare very well to a finite element model of a threaded
structural fastener. However, there is a difference in the results obtained from hand
calculations compared to those obtained from a finite element model. Hand calculations
provide average stresses. Typically stress concentration factors are used to determine
peak stresses from the average stresses. However, the threads do not all carry an equal
load. The first half of the engaged threads carries a much higher load as compared to that
carried by the second half of the engaged threads. Therefore, a stress concentration
factor multiplied by the average stress value will not yield the peak stress for all of the
threads.
This discrepancy is mitigated by engineering design criteria. Safety factors are used
on allowable stress values to determine an acceptable design. This often leads to overengineering a part to ensure that it will meet the design criteria in operation. However,
more sophisticated methods, such as finite element analyses, can yield better designs by
obtaining more accurate results when compared to hand calculations. Therefore a finite
element analysis should be used if a greater level of detail is required for the analysis of
a threaded structural fastener.
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REFERENCES
1. Aaronson, Stephen F, “Analyzing Critical Joints,” Machine Design, January 21,
1982.
2. Fukuoka, T. and Nomura, M., “Proposition of Helical Thread Modeling With
Accurate Geometry and Finite Element Analysis,” ASME J. Pressure Vessel
Technol., 130(1), 011204, 2008.
3. ISO 724, “ISO general-purpose metric screw threads – Basic dimensions,” 1993.
4. ANSYS User Manual, Version 11.0
5. Juvinall, Robert C. and Kurt M. Marshek, “Fundamentals of Machine
Component Design,” Third Edition, John Wiley & Sons, Inc., United States,
2002.
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