Simulation of Bolted Joint
with Frictional Contacts
Robin Nykänen
Mechanical Engineering, master's level
2019
Luleå University of Technology
Department of Engineering Sciences and Mathematics
ABSTRACT
An easy and reliable way to join two or more components is to use a bolted joint. When
torque is applied on the bolt head, a clamp force is achieved. However about 90% of this
torque will be used just to overcome the friction in the interfaces. To be able to fasten
these in an efficient and precise manner the friction in the bolt threads and underhead
area is important to understand. This is currently investigated at Atlas Copco by using a
friction test rig, FTR, which measures the clamp force, the total torque and the shank
torque.
To evaluate the test rig and also to be able to evaluate different friction models, a
parameterised simulation model of a bolted joint is built. This is a 3D-model of the joint
containing a bolt, a nut and a test specimen. The stiffness and frictional behaviour of the
FTR is used in the model to get a good comparison between these two. Different contact
formulations and settings are tested to achieve a good model. Also, mesh size, step size
and material models are evaluated to see the effect of these.
The results show a good correlation between the FTR data and the simulation model.
For the total torque to clamp force ratio a difference of about 1% is achieved. The average
difference in shank torque and underhead torque is at 3.1% and -1.6% respectively.
The pressure distribution for this model is shown to be unevenly distributed along the
threads. This is minimized by softening the contact, but this instead increases the error
between the FTR data and the simulation model. For the evaluation of friction models
that are based on the contact pressure this needs to be investigated further to find a good
compromise between the contact pressure and the frictional behaviour.
ii
PREFACE
This is the master thesis for the Master of Science in Engineering degree at Lulea University
of Technology. It is a part of the course E7011T ans is done together with Atlas Copco.
I would like to thank my supervisors at Atlas Copco, Erik Persson and Mayank Kumar
and my supervisor at LTU, Jan-Olov Aidanpaa, for the guidance throughout this thesis
work. Also a thank you to Andreas Rydin for the help with FEM simulations in ANSYS.
Robin Nykanen
iii
CONTENTS
Chapter 1 – Introduction 1
1.1 Atlas Copco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Similar Work By Others . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 2 – Theory 4
2.1 Tightening Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Geometry of the Bolted Joint . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Torque and Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4.1 Coulomb Friction Model . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 FEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5.1 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5.2 Non-linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5.3 Newton-Raphson Process . . . . . . . . . . . . . . . . . . . . . . . 9
2.5.4 Contact Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5.5 Friction in ANSYS . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.6 ANSYS Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter 3 – Method 14
3.1 BLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 Stiffness of the BLM . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.2 Test Data BLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.3 Evaluation of Test Results . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 Parametrisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Geometry Preparations . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.3 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.4 Loads and Constrains . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.5 General Simulation Settings . . . . . . . . . . . . . . . . . . . . . 22
3.3 Constant Coefficient of Friction Model . . . . . . . . . . . . . . . . . . . 24
3.4 Test Data FTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.1 Snug Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Varying Coefficient of Friction Model . . . . . . . . . . . . . . . . . . . . 27
3.6 Contact Pressure Distribution . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 4 – Results 29
4.1 BLM Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 BLM Tightening Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Constant COF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.1 Mesh Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.2 Step Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3.3 Symmetrical versus Asymmetrical . . . . . . . . . . . . . . . . . . 35
4.3.4 Linear versus Non-linear Material . . . . . . . . . . . . . . . . . . 36
4.3.5 Gauss Integration Points versus Surface Projection . . . . . . . . 37
4.3.6 Comparison With Calculations and Test Data . . . . . . . . . . . 39
4.4 FTR Tightening Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5 Varying COF Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.6 Stress in Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.7 Contact Pressure Distribution . . . . . . . . . . . . . . . . . . . . . . . . 47
4.7.1 Contact Stiffness Factor . . . . . . . . . . . . . . . . . . . . . . . 49
Chapter 5 – Discussion 51
5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1.1 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1.2 Contact Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Snug Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 53
Nomenclature
FTR - Friction Test Rig
FEA - Finite Element Analysis
COF - Coefficient of friction
MTot Total torque [Nm]
MT Shank torque [Nm]
MH Underhead torque [Nm]
F Clamp force [kN]
p Pitch [mm]
µ Coefficient of friction
µt Thread coefficient of friction
µb Underhead coefficient of friction
α Thread profile angle [°]d Nominal thread diameter [mm]
dm Pitch diameter [mm]
db Equivalent diameter of underhead area [mm]
dw Outer diameter of underhead area [mm]
dh Clearance hole diameter [mm]
Kb Stiffness of bolt [kN/mm]
Kb Stiffness of clamped parts [kN/mm]
[K] Stiffness matrix [kN/mm]
F Force vector [kN]
u Displacement vector [mm]
Fn Contact force [kN]
kn Contact stiffness [kN/mm]
xp Contact penetration [mm]
λ Additional term in Augmented Lagrange formulation
A Area [mm2]
P Pressure [MPa]
τy Shear strength [MPa]
vi
CHAPTER 1
Introduction
1.1 Atlas CopcoAtlas Copco was founded in 1873 in Stockholm, Sweden. They are a leading supplier of
compressors, vacuum solutions, generators, pumps, power tools and assembly systems [1].
Tightening Technique
One of the most common ways to assemble components is to use a threaded fastener, a
bolted joint. It is an easy and reliable way to clamp parts together. However, sometimes
these joints need some extra attention to make them serve their purpose in an efficient
manner.
Bolted joints are preloaded by for example using a torque wrench, so that when the
joint is affected by a force it will not lose the clamp force. If the clamp force could be
measured in the joints with high precision then the optimal joint dimensions and materials
could be chosen for every application. However, when tightening the bolted joint there
are a large number of factors that will influence the clamp force. One that will greatly
affect this is friction. Friction will arise in the threads and in the interface between the
clamped part and the bolt head or the nut. It is estimated, depending on the coefficient
of friction (COF), that of the torque that is applied on the bolt 50% is caused by the
underhead friction, 40% is caused by the friction in the threads and only 10% goes into
the preloading of the bolt [2]. The joint components and the torque distribution are
shown in figure 1.1.
1.2 PurposeIt is important to understand the friction in the bolted joints. Small variations in friction
will have a large effect on the clamp force that is achieved in the joint for a certain
tightening torque. The purpose of this master thesis is to create a contact simulation
1
1.3. Requirements 2
Figure 1.1: Torque distribution in a bolted joint.
model for bolted joints to compare with and evaluate the friction test rig that is constructed
by Atlas Copco and which they are using today for their measurements. Together with
this test rig, this simulation model will make it possible to evaluate different friction
models. The model should be parameterised so different sizes of bolts and nuts can be
used in the simulation model.
1.3 RequirementsThe FE-model, of the bolted joint, should behave similarly to the friction test rig (FTR)
in a tightening procedure. The clamped parts in this model should have the same stiffness
as the test rig that is being evaluated, and the bolt and nut should have dimensions that
can be tested in the test rig. The model should simulate the tightening procedure and be
able to measure the preload, underhead torque and the shank torque.
1.4 DelimitationsThis model does not need to include all components of the actual friction test rig. It
should still achieve a stiffness that leads to the same clamp force for the same tightening
angle. The model does not need to have the hexagonal shape of the bolt and nut head,
but instead a cylindrical shape is used. Further simplifications of the bolt and nut may
be appropriate since the stress distribution in the model is not what will be examined,
and therefore some simplifications of the geometry might be possible.
1.5. Methodology 3
1.5 MethodologyThe first step in this master thesis is to create a parametric CAD model with parameters
for the important dimensions of the bolt and nut. The assembly of the bolted joint will
then be exported to ANSYS where a FE-model will be constructed. Frictional contacts
will be used for the underhead surface and for the threads. In order to achieve a stiffness
of the joint, a spring element will be used in the model. Mesh and load step convergence
will be done to ensure reliable results. The results will then be compared with the test
data from the BLM and the FTR to evaluate the model, but also to evaluate the test rigs
themselves. Different simulation settings will be evaluated such as contact formulations,
contact detection methods and material models.
1.6 Similar Work By OthersPlenty on models for bolted joints have previously been made for analyzing the mechanical
properties of a threaded joint. However, many of these have been simple models. For
example, models where only the nodes in the center of the bolt were translated to achieve
a pretensioning for understanding the behaviour of ultrasonic wave propagation in a
preloaded bolt [3].
Instead of three dimensional models, some axisymmetric models [4][5] have been made
to avoid complicated and heavy models. The problem with axisymmetric threaded models
is that they ignore the helical effect.
In a more complex model an actual 3D-model was built to achieve a preload by rotating
the bolt [6]. The frictional values used in this case were from experimental data to achieve
accurate results. In order to achieve a model with a high accuracy, but as computationally
cheap as possible, the threads had a dense mesh on the contacting flanks and a less-dense
mesh on the flanks that were not in contact. In this case it resulted in a simulation with
only 1% error compared to the experimental data.
A procedure for generating a 3D-model of a bolted joint was proposed to in an accurate
manner be able to study the helical threads [5]. Also, the load distribution in the threads
was thoroughly investigated.
To evaluate the effect of friction coefficients on the tightening torque a 3D-model has
been constructed [7]. Also in this model the FEA corresponded well with the theoretical
results and it showed that friction has a great influence on the tightening torque.
CHAPTER 2
Theory
2.1 Tightening ProcedureThe tightening procedure is done to achieve a desired clamp force in the joint. This
procedure has four zones and these are shown in figure 2.1. The first zone is the rundown
phase. This is when the tightening starts but the clamped parts are not yet in contact with
each other and the tightening torque is relatively low. The second zone is the alignment
zone. Here the surfaces are aligned and snugged. This is affected by for example the
roughness and shape of the surfaces. The third zone is when the elastic clamping starts and
the ratio between the tightening torque and the clamp angle is constant. The fourth and
last zone is when the elastic zone is exceeded and plastic deformation of the components
is starting [2].
Figure 2.1: The four zones of the tightening procedure.
4
2.2. Geometry of the Bolted Joint 5
2.2 Geometry of the Bolted JointThe bolted joint consists mainly of the bolt, the nut and the clamped parts. The geometry
of these joints can vary much, but a schematic figure is seen in figure 2.2.
Figure 2.2: Bolted joint.
In figure 2.2, the two most interesting areas of the bolted joint, with frictional behaviour
in mind, are shown. These two are the underhead region, which is between the bolt
head and the clamped part, and the threaded region, which is between the bolt and the
nut. The geometry of the bolt and nut are according to ISO standards. The geometry
of the threads according to the ISO [8] is shown in figure 2.3. Here H is the height of
the fundamental triangle, the nominal diameter of the thread is denoted as d, the pitch
diameter is denoted as dm and the pitch is denoted as p.
2.3. Torque and Forces 6
Figure 2.3: Thread geometry according to ISO [8].
2.3 Torque and ForcesThe torque needed to tighten the joint depends on two main components; the shank
torque and the underhead torque [9], as shown in equation 2.1,
Mtot = MT +MH . (2.1)
The shank torque is caused by the pitch torque and the friction in the threads, the
thread torque. The underhead torque is caused by the friction between the bolt head and
the clamped part. The forces on the threads are shown in figure 2.4.
Figure 2.4: Schematic figure of the forces along the helix and on the profile of the thread.
2.3. Torque and Forces 7
Here the clamp force is denoted as F . Equation 2.1 can be approximated to equation
2.2, where db is the equivalent diameter of the underhead area [10],
Mtot = F( p
2π+
µtdm2cosα
2
+µbdb
2
). (2.2)
The pitch diameter is given by the nominal diameter and the thread pitch as
dm = d− p3√
3
8, (2.3)
The equivalent diameter of the underhead surface is provided by
db =2
3
d3w − d3hd2w − d2h
, (2.4)
where dw and dh are the outer diameter of the underhead surface and the diameter of
the clearance hole respectively [11].
The load in the threads is distributed in a way where the first engaged threads carry
the majority of the load [12]. This distribution is seen in figure 2.5.
Figure 2.5: Load distribution in threads [12].
The contact pressure under the head of the bolt can be approximated with
P =F
A=
F
π(d2w − d2h)/4. (2.5)
The preload in the joint is dependant on the tightening angle and the equivalent stiffness
of the joint. When turning the bolt and by that increasing the preload, the bolt will
stretch but at the same time the clamped parts will be compressed [2]. The equivalent
stiffness is given by equation 2.6.
2.4. Friction 8
KbKc
Kb +Kc
(2.6)
If the coefficients of friction are assumed to have constant values along the whole
tightening procedure then this should result in a linear relationship between the applied
torque and the achieved preload.
2.4 FrictionIn the contact between the two thread interfaces and between the head of the bolt and the
clamped part there will be friction. Friction is highly non-linear and this causes problems
when trying to simulate the tangential reaction forces between these surfaces. Some of
the parameters that friction is dependant on are contact geometry and topology, surface
properties and surface velocities. Lubrication and contamination will also have a large
effect on the friction. In dry sliding contacts the friction can be modeled by the elastic
and plastic deformation forces on the surface asperities that are in contact [13]. Here
it is assumed that every asperity is plastically or elastically deformed until the area in
contact is enough to carry the load that is distributed to each asperity. The tangential
deformations of these asperities are elastic until the shear strength is exceeded. The
sliding friction is then dependent on the shear strength and the hardness of the material.
2.4.1 Coulomb Friction Model
The main function of friction is that it is preventing relative motion and that it is
independent of velocity and contact area. The frictional force is proportional to the
normal load and is referred to as Coulomb friction, equation 2.7. This model does not
result in any specific friction force at zero velocity.
Ff = µN (2.7)
2.5 FEAIt is important to understand the basics in finite element analysis (FEA) to use it in
a useful and powerful way. In a static FEA the stiffness and deformation of a body is
considered to calculate the reaction forces following equation 2.8,
F = [K]u. (2.8)
This means that the inertia and damping effects are not considered as they are in
a dynamic FEA. For the static analyses you can have either a linear or a non-linear
simulation. The difference between linear and non-linear analyses is the stiffness matrix
2.5. FEA 9
K of the model. This can be affected by the shape, the material and the supports of the
model.
2.5.1 Linear Analysis
When the model is deformed the stiffness will change due to some of these factors
mentioned above, but in some cases the change in stiffness might be so small that the
stiffness can be assumed as constant. When a FEA is conducted, where the deformations
are small and stresses are kept in the linear elastic range, a linear static analysis is effective
and accurate. The stiffness matrix in this case is kept constant and the time for an
analysis is short compared to doing a non-linear analysis with the same model. If the
elements are going to be greatly translated or deformed a linear analysis will probably
result in a higher error.
2.5.2 Non-linear Analysis
If the model is subjected to large deformations, contact, or has non-linear material
properties, as for example visko-elastic material or if the behaviour of a material after the
yield point is of interest, then a non-linear analysis is needed. Here the stiffness matrix will
change during the load case. The stiffness matrix will be constantly updated as the solver
is iterating for the solution. Material non-linearity is the behavior of a material based on
many parameters, as for example deformation history, rate of deformation, temperatures
and pressures. Contact non-linearity is present when two or more surfaces get in contact
and therefore changes the geometric boundaries. The stiffness of this contact is of course
unknown before the solution and this is what results in the non-linearity.
2.5.3 Newton-Raphson Process
For non-linear analysis ANSYS is using the Newton-Raphson process for iteration until
the solution has converged [14]. This method takes small linear steps by evaluating a
tangent stiffness matrix for the model and takes a step with that stiffness. Then the
results are evaluated by calculating the residual, which is as shown in the equations below.
[KTi ]∆ui = F a − F nr
i (2.9)
ui+1 = ui + ∆ui (2.10)
If the residual is within the tolerance specified in the analysis then the solution has
converged for that load step. If not, a new tangent stiffness matrix will be calculated for
the new position and the iteration process starts again, see figure 2.6.
2.5. FEA 10
Figure 2.6: Illustration of the Newton Raphson process [14].
2.5.4 Contact Modelling
The main objective of this master thesis is to get a simulation model where friction in the
contacts can be evaluated. Contacts in FE models are often very advanced and demand
a lot of computational effort. Together with frictional forces in a contact it will create
an unsymmetrical stiffness matrix which also increases the complexity. Therefore, it is
important to understand how contacts are calculated in FE-models.
Contact Formulations
The contact formulations in ANSYS are based on a contact body and a target body [15].
The contact will be based on points while the target is based on surfaces, see figure 2.7.
Figure 2.7: Contact detection with Gauss points [16].
To evaluate the contact forces between the contact and the target there are different
formulations, some of them will be explained here. The simplest one is the penalty
2.5. FEA 11
formulation. As all contacts, this uses a stiffness between the contact and the target to
determine how much penetration will occur for a specific contact force, see figure 2.8. A
higher value of the stiffness will result in less penetration but might give convergence
problems, while a lower stiffness will result in better convergence and an averaged results
over an area of the surface. However, if the penetration is too high, a worse representation
of the reality will be given. This stiffness can be modified in ANSYS by a Contact Stiffness
Factor, but for frictional contacts the default value is set to 1.
Figure 2.8: Scematic explenation of the Pure Penalty formulation [15].
The contact force for a penalty formulation is calculated as following,
Fn = knxp. (2.11)
For this formulation only an infinite contact stiffness would result in zero penetration.
A modification of this is the Augmented Lagrange that uses an extra term to make it
more robust and efficient, see equation 2.12. This is the default formulation for contacts
in ANSYS Mechanical.
Fn = knxp + λ (2.12)
Another formulation that is frequently used is the Normal Lagrange formulation. This
avoids penetration by solving for contact pressure. Because penetration is preferably
avoided this will be a very stiff contact, computationally heavier and might lead to
divergence problems.
Detection Methods
The detection method determines which nodes or points on the contact should be used in
the contact formulation [16]. These are then constrained against the penetration of the
target surface. This means that between these points there can actually be penetration
without it being recognized, as shown in figure 2.10. As default in ANSYS, Gauss
integration points are used, as seen in Figure 2.7. However, for cases of corner contact
2.5. FEA 12
problems the nodal-normal method, where the nodes of the elements are used, will work
better. Another formulation is the surface projection based contact where the contact
forces are computed with the average penetration of an area, which is based on the target
elements projected on the contact elements, see figure 2.9. The advantages of this method
are generally more accurate contact stresses but it is also less sensitive to high penetrations
at edges of the target body. The disadvantage is mainly a higher computational cost
because of the amount of nodes in each contact constraint.
Figure 2.9: Surface projection based detection method.
In asymmetric contacts one of the bodies needs to be defined as the ”contact part” and
the other as the ”target part”. This puts some demands on the properties of the bodies
because the target part can actually penetrate the contact as long as it does not get in
contact with a detection point on the contact part. To avoid this, it is recommended
that the contact part is the most flexible part, has the finest mesh, or if the surfaces are
curved then the most convex surface should be the contact part.
Figure 2.10: Effect of chosen contact and target sides for an asymmetric contact [17].
2.5. FEA 13
In ANSYS there is the possibility to use a symmetric contact behaviour which constrains
the contact side from penetrating the target, but also the target side is now constrained
from penetrating the contact. This will therefore result in a model that is more compu-
tationally expensive and the results shown on the contact elements will be an average
between the target and the contact.
2.5.5 Friction in ANSYS
For contact elements CONTA174 the coulomb friction is used in ANSYS. It defines the
point of sliding with an equivalent shear stress, τ , that is dependent on the coefficient
of friction and the contact pressure. When this shear stress is reached, the surfaces will
start sliding relative to each other. As default it uses a constant value of the coefficient of
friction. However, this can be modified in different ways. With the command TBFIELD
together with TB and FRIC, the user is able to get a coefficient that is dependent on
parameters such as temperature, time, normal pressure and sliding distance.
2.5.6 ANSYS Solvers
For static structural simulations in ANSYS there are two available solvers to use. The
default solver is an iterative, also called Preconditioned Conjugate Gradient iterative
equation solver (PCG). The other solver is the direct solver, which is called the Sparse
direct equation solver. For non-linear analyses involving contact problems the direct solver
is usually to prefer. However, this will use more memory compared to the iterative solver.
For the purpose of this thesis the direct solver is therefore recommended. The computer
that was used had a large amount of memory, which does not make this a problem.
CHAPTER 3
Method
3.1 BLMThe BLM Testing Machine, shown in figure 3.1, is the one that is currently used by Atlas
Copco to investigate the tightening procedure of a threaded joint. In order to evaluate
the results from the contact model the experimental data from the BLM will be used.
The contact model should therefore have the same stiffness as achieved with the BLM.
3.1.1 Stiffness of the BLM
To approximate the stiffness of the BLM a FE model was created in ANSYS. Only the
components that are affected by the clamp force were used for the model, see figure 3.2,
and this was done by following the route of the force. The surfaces of the nut were fixed
and a load of 12.5 kN was applied on the bearing spacer. Because the model was split in
half, a rolling constraint was applied on the split surfaces. The stiffness was calculated by
dividing the force by the maximum nodal displacement.
14
3.1. BLM 15
Figure 3.1: BLM test rig
Figure 3.2: BLM model setup
3.1.2 Test Data BLM
The data used for the model was from tightening tests with the BLM friction test rig. For
these tests a nutrunner was used, which was attached to a support. The 2-step tightening
procedure has a rundown phase where the speed of the nutrunner is at 100 RPM until
a total torque of 5 Nm is achieved. After this the speed is lowered to 20 RPM and is
3.2. Simulation Model 16
Figure 3.3: The components used in the tests. M10 bolt and nut with test specimen.
tightened with clamp force control until a clamp force of 25 kN is reached. After that
there is a relaxation phase and a loosening stage. The signals from the sensors of the
BLM and external transducers were received by a Dewe-43A DAQ and then processed in
Dewesoft X3 software.
These tests were done with partially threaded zinc-iron coated M10 bolts with 8.8
classification, see figure 3.3. These bolts were also waxed to get a smooth frictional
behaviour during the tightenings. The clamp length in these tests was approximately
58mm. The tests were done with ten bolts and nuts but with five test specimens, where
both sides of the test specimens were used. For each bolt and nut five tightening procedures
were done. After these five tightenings the test specimen was flipped and a new bolt and
nut was used.
3.1.3 Evaluation of Test Results
Matlab was used when evaluating the test results. For the evaluation of test data from
the BLM, the angle of tightening was calculated. This was from the point where a clamp
force of 2 kN was reached, so that the rundown phase could be assumed to be complete,
until a clamp force of 25 kN was achieved. An average of the maximum total torque,
shank torque, underhead torque and tightening angle was then calculated for all the
tests. When comparing with the simulation model, average value plots were made for the
test data. These were conducted by extrapolation some of the data up to the average
tightening angle and then interpolation of the data in MATLAB for calculation of the
average values.
3.2 Simulation ModelThe model is built by three components, which are the bolt, the nut and a test specimen.
This is shown in figure 3.4. The CAD-geometry was created in CREO and parameterised
to allow for easy changes between different sizes of bolts and nuts.
3.2. Simulation Model 17
Figure 3.4: Assembly of the parameterised model.
(a) Parametric model of the bolt. (b) Parametric model of
the nut
Figure 3.5: Parameterised geometry of bolt and nut in CREO.
The bolt and nut are simplified by having a circular head shapes instead of the hexagonal
shape, see figure 3.5.
In order to get better convergence possibilities for the simulation model the threads in
the nut were simplified. Consequently, the problem of very thin elements at the contacting
areas where the forces are high is avoided. This will therefore result in a more stable
simulation model and easier convergence.
The threads in the nut compared with the bolt threads have a small offset, both because
of the tolerances in real components but also to help avoid initial penetration in the
FE-model. The sketch used for the helical cut is shown in figure 3.6 where an offset of
3.2. Simulation Model 18
0.01mm was used.
Figure 3.6: Sketch used for the helical cut of nut threads in CREO.
3.2.1 Parametrisation
The bolt is parameterised by choosing the bolt head diameter, nominal diameter, pitch,
length of bolt and the free length, see table 3.1.
Table 3.1: Parameters for the bolt.
Name Value [mm] Description
Bolt Head 14.8 Bolt head diameter
Bolt Dia. 10.0 Nominal bolt diameter
Head Length 6.4 Length of bolt head
Bolt Length 80.0 Length of bolt
Unthreaded 52.0 Unthreaded length of bolt
Pitch 1.5 Thread pitch
The nut is parameterised by its head diameter, nominal bolt diameter, pitch, length
of nut and the offset between bolt and nut threads, see table 3.2. The bolt head has a
diameter that corresponds to the actual area of contact, while the outer diameter of the
nut is corresponds to the real nut size so that the stiffness of the bolt will be accurate.
The threads are constructed according to ISO standards [8], [18], [19], [20], [21].
The parameters are shown in figure 3.7.
3.2. Simulation Model 19
Table 3.2: Parameters for the nut.
Name Value [mm] Description
Nut Size 16.0 Nut outer diameter
Bolt Dia. 10.0 Nominal bolt diameter
Nut Length 8.4 Length of nut
Pitch 1.5 Thread pitch
Offset 0.01 Offset between threads
Figure 3.7: Parameters for the bolt and nut.
3.2.2 Geometry Preparations
The CREO-geometry was then imported as a STEP format to ANSYS. In order to prepare
the geometry SpaceClaim was used to split the surfaces. This was done to in an easier
way be able to put constrains and loads on the model but also to in a better way get a
controlled mesh.
The top surface of the test specimen got a circular split for the contact surface. Both
the upper and bottom surface were then also spit in eight segments and the side surfaces
were split in half along the long sides, see figure 3.8.
3.2. Simulation Model 20
Figure 3.8: Geometry preparation of the test specimen in SpaceClaim.
The bolt was split as seen in figure 3.9 and the nut was split as seen in figure 3.10.
The threads of both the nut and bolt were split to make it possible to measure the load
distribution on the threads for each revolution.
Figure 3.9: Geometry preparation of the bolt in SpaceClaim.
Figure 3.10: Geometry preparation of the nut in SpaceClaim.
3.2. Simulation Model 21
(a) Meshed bolt.
(b) Meshed nut. (c) Meshed test specimen.
Figure 3.11: Meshed components for simulation.
3.2.3 Meshing
For meshing in ANSYS quadratic tetrahedron elements were used with an general mesh
sizing of 1.5mm. The underhead and thread contact areas were then refined for mesh
convergence. ANSYS non-linear shape checking setting, which should be more appropriate
for contact simulations, was used for the mesh in this model. The meshed components
are seen in figure 3.11.
3.2. Simulation Model 22
3.2.4 Loads and Constrains
The FE model was created in ANSYS Mechanical. The setup with the components, loads
and constrains is shown in figure 3.12. As seen, a fixed constraint is set to a part of the
upper surface of the nut. This will still allow some bending of the top surface closest to
the threads.
The test specimen has roller constrains on all four side surfaces to keep the body from
moving in any other direction than along the axis of the bolt and it will also prevent it
from any rotation. The bottom surface of the test specimen is constrained as rigid to
keep it from bending around the head of the bolt. In order to avoid problems between
multiple constrains along the same edges the side surfaces of the test specimen were split
in half. The roller constraint was set to the upper part of these surfaces and will therefore
not interfere with the rigid constraint of the bottom surface.
The bolt would only need two free degrees of freedom in this model, rotation around
its axis and translation along it. Instead of using a torque input on the bolt, a rotation
is used on the bolt head. The reaction torque is then achieved after solving. The outer
surface of the bolt head was split in two for applying the rotational constraint on the
upper surface. The splitting of this surface will avoid interference between the rotational
constraint and the contact surface of the underhead. The other rotational degrees of
freedom were fixed for the bolt head. The translational degrees of freedom were only
constrained by weak springs. In order to make calculations cheaper the surface for the
rotational constraint was set as a rigid property at the applied surface. It could be argued
that this is also a more realistic approach because while applying torque to the joint with
a socket wrench the socket will keep the outer surface of the bolt head rather free from
the torsional deformation.
The stiffness of the bolted joint is set by adding a spring element between the test
specimen and a fixed point along the axis of the bolt, see figure 3.13. In order to achieve
a stiffness corresponding to the BLM a first stiffness was chosen and then modified until
a clamp force of 25 kN was achieved for the mean tightening angle of the BLM test data.
To hold the bolt in place in the start a weak spring was added to its bottom surface.
This had a stiffness of 1 N/mm and does not make a noticeable difference in the results
because the translation of the bolt is under half a millimeter.
3.2.5 General Simulation Settings
For the contact in the underhead area a frictional type was chosen with the Augmented
Lagrange formulation. Also, a ”Adjust to touch” setting was set for the contact interface
so that all penetration or gap is removed in the start of the simulation. The contact
in the threads was also set as a frictional type with augmented Lagrange formulation,
but because an offset was created between the thread interfaces in CREO the ”Adjust
to touch” setting would not be ideal in this case. Instead, a stabilization damping was
inserted for this contact, which will result in a more stable model before the surfaces get
3.2. Simulation Model 23
Figure 3.12: Model setup in ANSYS.
Figure 3.13: Spring element with joint stiffness.
3.3. Constant Coefficient of Friction Model 24
in contact. This will keep the threads from penetrating into each other too much in the
first load steps, but this stabilization will consume energy when it is active. Even if only
a low amount of energy is needed for this damping, a ”KEYOPT” function was used to
only have it active during the first load step. This was done by the command which is
shown in figure 3.14.
Figure 3.14: Contact damping command in ANSYS.
This command will use KEYOPT(15), which is for determining the effect of contact
stabilization damping. The ”cid” is for using this command for the contact elements in
the threads while the value of 0 means that it is activated only in the first load step.
For the analysis settings a direct solver was used together with large deflections activated.
”Weak spring” option was also activated to avoid free body motion. The convergence
criterion for the non-linear simulation was as shown in table 3.3
Table 3.3: Convergence criterion for non-linear analysis.
Convergence Tolerance Minimum tolerance
Force [N] 0.5% 0.01
Moment [Nmm] 0.5% 10
Displacement [mm] 0.5% 0
Rotation [°] 0.5% 0
3.3 Constant Coefficient of Friction ModelFor the model with data from the BLM with a 2-step tightening, mesh and step convergence
were done. For this model a constant coefficient of friction was set for the contacts, both in
the threads and between the test specimen and the bolt head. ANSYS uses the Coulomb
friction model as default. The values chosen for this were taken from the BLM test
data, where the coefficient of friction at a clamp force of 25 kN was calculated by solving
equation 2.2 for µb and µt and using the torque mean values of the test data.
The simulation model was evaluated with three different mesh sizes and three different
step sizes. The mesh convergence was done with 113·103, 163·103 and 366·103 nodes. The
mesh with 113e3 nodes had a general mesh sizing for the whole model while the finer
3.3. Constant Coefficient of Friction Model 25
meshes had refinements at the contact areas and their surroundings. Step convergence
was then also done, this with 30, 45 and 60 steps.
In addition of the mesh and step convergence also the affect of non-linear material
models, the difference between symmetrical or asymmetrical contacts and the difference
between Gauss integration points or surface projection based contact detection were
compared with this model.
Gauss integration points as detection method is the default setting in ANSYS for static
structural analyses but to use surface projection based detection the command which is
shown in figure 3.15 was inserted.
Figure 3.15: Surface projection based detection command in ANSYS.
In order to evaluate if the high stress that will occur in the model at some elements
will affect the results a non-linear material model was used. This was a bi-linear model
with yield strength of 640 MPa and a tangent modulus of 1333 MPa, which is the slope
of the plastic region as shown in figure 3.16.The percentage of elongation until fracture,
the yield strength and the tensile strength for the bolt, from ISO 898-1 [22], were used to
calculate the tangent modulus. The result and the time of solving a simulation with a
bi-linear material model was then compared with the same model with a linear material.
Figure 3.16: Explanation of the tangent modulus in a bi-linear material model.
3.4. Test Data FTR 26
Figure 3.17: Atlas Copco Friction Test Rig
3.4 Test Data FTRThe test data used in the previous sections were from a 1-step tightening procedure in the
BLM that were already available at the start of this master thesis. This data was used
to evaluate the constant coefficient of friction model. Because of the change of speed in
the tightening, where the first part is done with high RPM and the second part is done
with a lower RPM, this seems to give a highly non-linear behaviour of the forces in the
beginning of the tightening.
In order to get data that will correspond better to the simulation model along a larger
part of the tightening, a 1-step tightening was conducted. These tests were done with the
friction test rig (FTR) developed at Atlas Copco, see figure 3.17. Instead of changing the
speed of the tightening it was now kept constant through the whole procedure. A total of
100 tightenings were done with the same bolts, nuts and test specimens as in previous
tests. For these tests, 20 bolts and nuts were used together with one test specimen. Every
bolt and nut was used for five tightenings in a row. After the tenth sample of bolt and
nut the test specimen was flipped, so that the side of the test specimen in contact was
changed. Otherwise the testing equipment was the same as for the BLM tests. However,
the first six tightenings were not used for evaluation and comparison with this simulation
model due to errors in the measurement procedure.
For achieving the right stiffness for this model the first assumption was the stiffness
calculated by the constructors of the FTR which was 1.625·106 N/mm [23]. This was
3.5. Varying Coefficient of Friction Model 27
however, in the same manner as for the constant friction model, modified until a clamp
force of 25 kN was achieved for the corresponding mean angle calculated from the test
data.
3.4.1 Snug Point
For this simulation model it was important to find the snug point in an accurate manner.
This point is where the simulation model will assume the starting point of the tightening
and from there an linear force-to-angle ratio will be achieved until the final clamp force
is achieved. The snug point was found by following the tangent of the last part of the
force-angle curve in the BLM data which should correspond to the linear elastic path of
the tightening. The point where this tangent reaches 0kN is chosen as the snug point.
3.5 Varying Coefficient of Friction ModelThe test data from both the BLM and the FTR shows that the coefficient of friction varies
during the tightening. Previously in the simulation model the COF had one constant
value through the whole simulation. In order to better evaluate the FTR, the COF was
set to vary with the rotational angle of the bolt in the simulation according to what was
calculated for the BLM test data. Though, in ANSYS this angle dependency was set as a
time dependency, see figure 3.18. Because the rotational angle of the bolt in ANSYS is a
linear function of time this will make the COF linearly dependant of the time.
This variation of the friction with the angle was conducted by creating a matrix of
commands in MATLAB with values of time and the corresponding COF.
Figure 3.18: A part of the command in ANSYS to achieve friction variations with time.
3.6. Contact Pressure Distribution 28
3.6 Contact Pressure DistributionIf a friction model as a function of contact pressure is to be used with this model, a
reliable representation of the contact pressure is needed.
By varying the contact stiffness factor for the threads, a softer contact can be achieved.
This will result in more penetration, but also an averaged out pressure distribution. If
the factor is too low, too much penetration will occur and a worse representation of the
reality will be achieved. Simulations were done with an interval from 0.05 to 1.0 of contact
stiffness factors. The pressure distribution along the center of the flank was evaluated
together with the clamp force and the shank torque. The distribution of load in the
threads was also considered.
The contact stiffness of the underhead area was also considered and the stiffness factor
was varied between 0.1-1.0 to evaluate if this would have an effect on the contact pressure,
the clamp force and the underhead torque.
CHAPTER 4
Results
4.1 BLM StiffnessTo calculate the stiffness a simplified FE-model of the BLM was done. The results of this
model are shown in figure 4.1, where the force applied was 12.5 kN.
The maximum nodal displacement was used to determine the stiffness of the model, see
table 4.1
Table 4.1: Results from FE model for calculating the BLM stiffness.
Force [kN] Deformation [mm] Stiffness [N/mm]
12.5 0.0945 1.32·105
This value of the stiffness was used as an first assumption for the simulation model.
4.2 BLM Tightening TestsThe ,BLM test data for the five tests with five tightening procedures each is shown in
figure 4.2 below.
29
4.2. BLM Tightening Tests 30
Figure 4.1: Deformation along the axis of the BLM FE-model.
(a) Total torque for all tightenings. (b) Clamp force for all tightenings.
(c) Shank torque for all tightenings. (d) Underhead torque for all tightenings.
Figure 4.2: Plots of the BLM test data for a 2-step tightening.
4.3. Constant COF Model 31
The mean values that were calculated for these tests at a clamp force of 25 kN are
shown in table 4.2.
Table 4.2: Mean values from the 2-step BLM test data.
Angle [°] Total Torque [Nm]
90 41.3
Shank Torque [Nm] Underhead Torque [Nm]
23.8 17.5
COF Thread COF Underhead
0.139 0.109
4.3 Constant COF ModelFor the model with a constant coefficient of friction the results in table 4.2 from the 2-step
tightening were used. Mesh and step convergence was conducted with this data but also
the evaluation of contact detection methods and material models.
4.3.1 Mesh Convergence
The total torque to clamp force ratio for the three different mesh sizes is shown in the
graph below. The coarse mesh with 113·103 nodes, a middle mesh with 163·103 nodes
and the finest meshing was with 366·103 nodes. The graphs are starting from the point
when the pre-tightening of 2 kN is complete and the final tightening starts. For all of
these simulations the final tightening load step was divided into 60 sub steps.
4.3. Constant COF Model 32
Figure 4.3: Total torque to clamp force ratio comparison for mesh convergence.
Because these simulations were very close to each other when comparing the shank and
underhead torque, difference plots were made for these to in a more accurate manner be
able to compare them. The simulation with 311·103 nodes was used as a reference, see
figure 4.4.
(a) Shank torque, difference plot. (b) Underhead torque, difference plot.
Figure 4.4: Mesh convergence comparison, difference plots with finest mesh as reference.
A slight convergence is noticed when comparing the simulations with each other. The
times for solving and the results at the end of the simulations are shown in table 4.3,
4.3. Constant COF Model 33
where the total torque to clamp force ratio is denoted as TF-ratio.
Table 4.3: Time and results for simulations with 113·103, 163·103 and 366·103 nodes.
NodesTime
[min]
Clamp Force
[kN]
TF-Ratio
[Nm/kN]
113·103 68 24.9 1.66
163·103 103 25.0 1.67
366·103 381 25.1 1.67
NodesTotal Torque
[Nm]
Shank Torque
[Nm]
Underhead Torque
[Nm]
113·103 41.5 24.4 17.1
163·103 41.8 24.6 17.2
366·103 41.9 24.6 17.3
4.3.2 Step Convergence
The total torque to clamp force ratio for the three different substep sizes is shown in the
graph below. For all these simulations the middle mesh size of 163·103 nodes was used.
Figure 4.5: Total torque to clamp force ratio comparison for step convergence.
As for the mesh convergence, these simulations were also very similar to each other
4.3. Constant COF Model 34
when comparing the shank and underhead torque. Difference plots were therefore made
for these to be able to compare them. As reference the simulation with 60 substeps was
used.
(a) Shank torque, difference plot. (b) underhead torque, difference plot.
Figure 4.6: Step convergence comparison, difference plots with 60 substeps as reference.
In this case, a clear case of convergence was not achieved and the difference in the
underhead torque is only due to the tolerance in the simulations. However, with the 60
substep simulation a more stable result was obtained. The times for solving and the
results at the end of the simulations are shown in table 4.4.
Table 4.4: Time and results for simulations with 30, 45 and 60 substeps.
SubstepsTime
[min]
Clamp Force
[kN]
TF-Ratio
[Nm/kN]
30 86 25.0 1.67
45 97 25.0 1.66
60 103 25.0 1.67
SubstepsTotal Torque
[Nm]
Shank Torque
[Nm]
Underhead Torque
[Nm]
30 41.7 24.5 17.2
45 41.6 24.4 17.2
60 41.8 24.6 17.2
4.3. Constant COF Model 35
4.3.3 Symmetrical versus Asymmetrical
The total torque to clamp force ratio and penetration for a symmetrical and an asymmet-
rical contact are shown in the graphs below. These simulations were done with a model
with 163·103 nodes and 60 substeps.
(a) Total torque to clamp force ratio. (b) Maximum penetration of thread contact.
Figure 4.7: Mesh convergence comparison.
There was no obvious difference in the results between these two simulations except for
a slight time difference. The times for solving and the results at the end of the simulations
are shown in table 4.5.
Table 4.5: Time and results for simulations with symmetrical and asymmetrical contacts.
ContactTime
[min]
Clamp Force
[kN]
TF-Ratio
[Nm/kN]
Symmetrical 119 25.0 1.67
Asymmetrical 103 25.0 1.67
ContactTotal Torque
[Nm]
Shank Torque
[Nm]
Underhead Torque
[Nm]
Symmetrical 41.7 24.5 17.2
Asymmetrical 41.8 24.6 17.2
4.3. Constant COF Model 36
4.3.4 Linear versus Non-linear Material
To compare if the material non-linearities would have any effect on the results, a bi-linear
material model was used. The relative difference between the shank and underhead torque
is shown in the graphs below.
(a) Shank torque, difference plot. (b) Underhead torque, difference plot.
Figure 4.8: Difference plots between linear and nonlinear material models.
As expected, the results of both simulations are following each other at lower clamp
forces, but as the forces grow higher the difference between them is rapidly increasing.
The times for solving and the results at the end of the simulations are shown in table 4.6.
Table 4.6: Time and results for simulations with a linear and non-linear material model.
MaterialTime
[min]
Clamp Force
[kN]
TF-Ratio
[Nm/kN]
Linear 103 25.0 1.67
Non-linear 562 24.9 1.67
MaterialTotal Torque
[Nm]
Shank Torque
[Nm]
Underhead Torque
[Nm]
Linear 41.8 24.6 17.2
Non-linear 41.7 24.5 17.2
4.3. Constant COF Model 37
4.3.5 Gauss Integration Points versus Surface Projection
When using the Gauss integration points as the detection method, high contact pressure
peaks are found at the first thread revolution of the nut and bolt, see figure 4.9. For the
projection based detection method the contact pressures are more evenly distributed.
(a) Contact pressure distribution with Gauss integration points.
(b) Contact pressure with surface projection.
Figure 4.9: Contact pressure distribution comparison.
This is also noticeable by observing the load distribution of the threads. The average
values of the load distribution ratios for the first three thread revolutions are compared
in table 4.7 with the theoretical values.
4.3. Constant COF Model 38
Table 4.7: Load distribution ratio in threads.
Method 1st 2nd 3rd
Theoretical 34% 23% 16%
Gauss 37.2% 23.9% 15.1%
Surface Projection 36.4% 23.8% 15.4%
There is a slight change in the distribution of load on the first engaged thread. The
times for solving and the results at the end of the simulations are shown in table 4.8.
Table 4.8: Time and results for simulations with different detection methods.
DetectionTime
[min]
Clamp Force
[kN]
TF-Ratio
[Nm/kN]
Gauss 83 25.0 1.67
Surface Projection 103 25.0 1.67
DetectionTotal Torque
[Nm]
Shank Torque
[Nm]
Underhead Torque
[Nm]
Gauss 41.9 24.6 17.2
Surface Projection 41.8 24.6 17.2
4.3. Constant COF Model 39
4.3.6 Comparison With Calculations and Test Data
For the comparison with the BLM data and theoretical calculations, the middle mesh size
of 163·103 elements was set together with 60 sub steps. This model used a asymmetric
contact formulation with surface projection based detection method together with a linear
material model. The results are shown in figure 4.10.
(a) Clamp force comparison. (b) Total torque to clamp force ratio compari-
son.
(c) Shank torque comparison. (d) Underhead torque comparison.
Figure 4.10: Comparison between the simulation model, BLM test data and theoretical
calculations.
Here the end values of the simulation are of most importance and they are corresponding
well to both the theoretical calculations and the BLM test data. The results from the
simulation model, BLM test data and theoretical calculations at 25 kN are shown in table
4.9.
4.3. Constant COF Model 40
Table 4.9: Comparison between the simulation model, BLM test data and theoretical
calculations.
ComparisonClamp Force
[kN]
TF-Ratio
[Nm/kN]
Simulation Model 25.0 1.67
BLM Test Data 25 1.65
Theoretical 25 1.66
ComparisonShank Torque
[Nm]
Underhead Torque
[Nm]
Simulation Model 24.6 17.2
BLM Test Data 23.8 17.5
Theoretical 24.0 17.6
4.4. FTR Tightening Tests 41
4.4 FTR Tightening TestsThe 1-step FTR test data for the eighteen tests with five tightening procedures each is
shown in figure 4.11 below.
(a) Total torque as function of tightening angle.(b) Clamp force as function of tightening angle.
(c) Shank torque as function of tightening angle.(d) Underhead torque as function of tightening
angle.
Figure 4.11: Plots of the FTR test data for a 1-step tightening.
The snug point was approximated by the tangent at the end of the force-to-angle curve,
see figure 4.12
4.4. FTR Tightening Tests 42
Figure 4.12: Snug point approximation of FTR tests.
The mean values that were calculated for these tests at a clamp force of 25 kN are
shown in table 4.10. The angle is from the snug point until the 25 kN of clamp force is
reached.
Table 4.10: Mean values from the 1-step FTR test data.
Angle [°] Total Torque [Nm]
79.5 38.6
Shank Torque [Nm] Underhead Torque [Nm]
21.1 17.5
How the friction for the tests with the FTR varies with the angle of tightening for the
threads and for the underhead area is shown in figure 4.13.
4.5. Varying COF Model 43
Figure 4.13: Mean coefficient of friction in threads and the underhead along the tightening
procedure.
4.5 Varying COF ModelThe stiffness used in this simulation model to achieve a clamp force of approximately 25
kN with 79.5° of rotation was 1.265·105 N/mm. This could be compared with the stiffness
calculated by the constructors of the FTR, which was more than ten times higher at a
value of 1.625·106 N/mm. For the friction variations the FTR data was used with 100
substeps. The torque-plots in figure 4.14 show the comparison between the simulation
model and the BLM data.
4.5. Varying COF Model 44
(a) Clamp Force comparison. (b) Total torque to clamp force ratio compari-
son.
(c) Shank torque comparison. (d) Underhead torque comparison.
Figure 4.14: Comparison between the simulation model, FTR test data and theoretical
calculations.
The simulation model is corresponding well with the FTR data, except for the clamp
force, which in the model is growing linearly while in the tests there are some non-
linearities in the start of the tightening. Figure 4.15 shows the percentual error between
the simulation model and the BLM test data for the shank and underhead torque.
4.5. Varying COF Model 45
(a) Shank torque, error plot. (b) Underhead torque, error plot.
Figure 4.15: Percentual error plots between simulation model and FTR test data.
As observed, the percentual error at low clamp forces is very high but as the force is
rising the error converges to a stable level. The results from the simulation model are
compared with the FTR test data at 5, 15 and 25 kN in table 4.9.
Table 4.11: Comparison between simulation model and FTR test data.
Force [kN]Underhead Torque
[Nm]
Shank Torque
[Nm]
5 (Model/FTR) 3.93/3.81 4.66/4.63
15 (Model/FTR) 11.06/10.88 12.98/13.38
25 (Model/FTR) 17.50/17.20 21.09/21.77
Force [kN]TF-Ratio
[kN/Nm]Error TF-Ratio
5 (Model/FTR) 1.69/1.72 -1.75%
15 (Model/FTR) 1.62/1.60 0.92%
25 (Model/FTR) 1.56/1.54 1.01%
If the whole tightening procedure is taken into account then the average error between
these is 1.0% for the shank torque and -2.6% for the underhead torque. The average total
torque error is then -0.7%. However, from 15 kN until 25 kN of clamp force the average
error is 3.1%, -1.6% and 1.0% respectively.
4.6. Stress in Joint 46
4.6 Stress in JointThe average stress in the shank of the bolt should be approximately 318 MPa for a M10
bolt at a clamp force of 25 kN. The Von-Mises equivalent stress plot for the simulation
model confirms this, as shown in figure 4.16.
Figure 4.16: Von-Mises equivalent stress plot of joint.
The Von-Mises equivalent stress in more detail for the underhead and thread area is
shown in figure 4.17
(a) Stress plot of underhead area. (b) Stress plot of thread area.
Figure 4.17: Von-Mises equivalent stress plot of underhead and thread area.
4.7. Contact Pressure Distribution 47
4.7 Contact Pressure DistributionThe contact area for the underhead area can be assumed to be affected be a uniform force
distribution [24]. This should give a pressure of about 420 MPa for a M10, according
to equation 2.5. Where the contact surface has an outer diameter 14 mm and an inner
diameter 11 mm. The pressure distribution behaviour from the FEM simulations is shown
in figure 4.18. The pressure in the model corresponds well to the theoretical values.
Figure 4.18: Contact pressure at the underhead area.
This pressure distribution is seen in figure 4.19, where the pressure is plotted from the
inner edge of the contact in the underhead area until the outer edge of contact.
4.7. Contact Pressure Distribution 48
Figure 4.19: Underhead contact pressure with distance from the center of the bolt.
For the contact pressure in the thread area, as seen in figure 4.20, an uneven distribution
is achieved in the simulation models.
Figure 4.20: Contact pressure in the threads.
The contact pressure in the threads is unevenly distributed as shown in figure 4.21,
where the pressure along the center of the flank is plotted with the distance along the
4.7. Contact Pressure Distribution 49
thread. The pressure is oscillating in some areas with 100-200 MPa.
Figure 4.21: Contact pressure along the center of the flank.
4.7.1 Contact Stiffness Factor
The contact stiffness factor for the thread contact was varied between 1 and 0.05. The
contact pressure along the center of the flank for factors 1, 0.1 and 0.05 is shown in figure
4.22
Figure 4.22: Contact pressure along the center of the flank for different stiffness factors.
4.7. Contact Pressure Distribution 50
However, the lower values of contact stiffness will lead to more penetration in the
contacts and it will also affect the shank torque, see figure 4.23.
(a) Total torque to clamp force ratio comparison.(b) Comparison of penetration in the threads.
Figure 4.23: Effects of lowering the contact stiffness factor.
This could be observed by computing the difference between the simulation models
and the FTR test data at a specific load. The load used for this comparison was 23 kN
because this is the force that the softest contact reached when using the same tightening
angle and the same joint stiffness. The results are seen in table 4.12
Table 4.12: Error between simulation and test data when varying the contact stiffness.
Contact Stiffness Factor Error of shank torque at 23 kN
1 2.88%
0.1 3.69%
0.05 5.64%
CHAPTER 5
Discussion
If not comparing the results of the clamp force, but focusing on the shank torque, the
underhead torque and the total torque to force ratio, this model is corresponding very
good to the test data with an average error of only 1% on the total torque from 15
kN until 25 kN. This is better than in many other models that were found during the
background research. The shank torque has an error of 3.1% in this force interval. It is
the more complicated contact area and will need further evaluation.
5.1 ModelRegarding the mesh convergence, a finer mesh could have been tested, but the increase
of the computational effort that was needed for a higher amount of degrees of freedom,
was rapidly increasing. The medium mesh size was chosen in all the later simulations.
This was because the frictional behaviour was not changing much at all when comparing
between the coarse and the fine mesh. For the step convergence, the difference in the
results achieved were mostly due to the convergence tolerances in the simulation. A high
amount of substeps was chosen for the later simulations, partly because the time for the
simulations were not greatly affected by the amount of sub steps while the results got a
bit more steady, but also because the friction variations would then be more accurate.
As the graphs for the difference between the linear and non-linear material models show,
the error, or the difference between the simulations, with linear and bi-linear material
models is starting to grow in the end of the simulation. This is probably because of the
stresses in the model that are growing and eventually reach the yield point. Up to this
point however, these two models should be identical. So if clamp forces higher than 25 kN
are to be evaluated for this bolt and nut, then a non-linear model might be a good choice.
51
5.2. Snug Point 52
5.1.1 Stiffness
The stiffness used for the joint in the simulation model was achieved by varying the
stiffness until the average tightening angle from the test rig data that was used in the
model achieved a clamp force of about 25 kN. This assumes that the stiffness of the
modeled bolt corresponds to the stiffness of the real M10 bolt. If the modeled bolt has a
higher stiffness, this would lead to a lower approximation of the test rig stiffness that is
used in the simulations. If different bolts were modeled and simulated then this could be
evaluated to see if the stiffness of the test rig has been chosen correctly.
5.1.2 Contact Pressure
Because of uneven contact pressure distribution between the threads of the bolt and nut,
different measures were made to find what causes this problem and what could solve it.
A local mesh for the first engaged thread of the bolt and nut was refined to a very fine
mesh with eight elements along the flanks to see if the mesh sizing causes the problem.
Another test was with a hexahedron mesh to see if it would solve the problem. Also a
change to use the nut as contact and the bolt as target was tried. The problem behind
this uneven distribution of pressure might be caused by the discretization of the rather
complicated flank surface.
To solve this problem however, or at least to make it less of a problem, the contact
stiffness in the threads was lowered. The contact stiffness of the thread contact is however,
of big significance in the model. A higher stiffness factor as used for the constant COF
and varying COF models will result in lower errors compared to the BLM and FTR data
when comparing the frictional behaviour. A lower stiffness will instead give a higher
penetration and therefore a more evenly distributed contact pressure.
5.2 Snug PointFor the BLM data the snug point was not correctly evaluated but the start point was
just chosen at 2kN. For the FTR test data however, the snug point was more accurately
chosen. The tolerances of the gauges were initially considered in this step to see if it
would have a large influence of this non-linearity in the tightening curve, and if it would
affect in where the snug point is to be located. However, the band of the tolerances for
the gauges are small enough for this so it should not be a problem.
5.3 TolerancesThe tolerances in the measurements is an important factor to have in mind. Gauges
meant for 100 kN of clamp force and 100 Nm of torque are used in the friction test rigs
but they are used at much lower ranges than this. The force was at a maximum of 25 kN
53
while the shank torque was not exceeding 30 Nm. At these lower ranges the gauges are
showing larger errors because of the non-linearity.
5.4 Conclusion and Future WorkThe model is simplified compared to the actual test conditions, so some differences in
the results should actually be expected. However, the results for the model, when a stiff
contact is used, results in a good correlation with the tests that are done on both the FTR
and the BLM. The parameterised model together with following the work flow in this
thesis work gives an easy way of creating a model of different dimensions, or if another
software is to be used for the FEA.
In order to achieve a good model, that will be used with for example a friction model
dependant on the contact pressure, a compromise might be needed between the frictional
behaviour and the contact pressure distribution. It should also be investigated to see if a
simplification of the thread surface could be done numerically and by that therefore the
contact stiffness could remain high.
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