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Numerical Modeling of Bolted Joints. An Applied Finite Element Approach

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Numerical Modeling of Bolted Joints. An Applied Finite Element Approach
NUMERICAL MODELING OF BOLTED JOINTS.
AN APPLIED FINITE ELEMENT ANALYSIS
APPROACH
By
Chet Judson VanGaasbeek
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
Major Subject: MECHANICAL ENGINEERING
Approved:
Norberto Lemcoff, Thesis Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
May 2015
c Copyright 2015
by
Chet Judson VanGaasbeek
All Rights Reserved
ii
Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1
Bolt Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.1
Axial Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.2
Shear Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.1.3
Bolt Rotational Stiffnesses . . . . . . . . . . . . . . . . . . . . 10
2.1.3.1 Bolt Beam Stiffnesses . . . . . . . . . . . . . . . . . 11
2.1.3.2 Bolt Torsional Stiffness . . . . . . . . . . . . . . . . . 11
2.2
Typical Bolting Pattern Subject to Overturning Moment . . . . . . . 13
2.3
Stiffness Comparison to Solid Finite Element Model . . . . . . . . . . 16
2.3.1
Solid FEA Preload Calculation and Model Details . . . . . . . 17
3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1
3.2
Solid Model Results and Stiffness Measurement as a Function of Preload 21
3.1.1
Bolt Stiffness Study with Application of Preload and Axial
External Force . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2
Bolt Stiffness Study with Application of Preload and Shear
External Force . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.3
Characterization of Rotational Stiffnesses and Summary of
Stiffnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.4
Summary of Results . . . . . . . . . . . . . . . . . . . . . . . 31
Hand Calculation Comparison to Plate Model . . . . . . . . . . . . . 32
3.2.1
FEA Comparison against Hand Calculation Solution
3.2.1.1 U1 Direction Bolt Load Comparison . . . .
3.2.1.2 U2 Direction Bolt Load Comparison . . . .
3.2.1.3 U3 Direction Bolt Load Comparison . . . .
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37
37
38
39
4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
iii
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
APPENDICES
A. Python Code: Parameter Study: Modulus of Elasticity . . . . . . . . . . . 44
iv
List of Tables
3.1
Connector displacement measured from bolt head to opposite end on
nut as a function of preload force. . . . . . . . . . . . . . . . . . . . . . 26
3.2
Spring constants used for 0.5” diameter 1” long UNC fastener. . . . . . 31
3.3
Comparison between FEA results for overturning moment load and analytical results for the u1 (axial) direction. . . . . . . . . . . . . . . . . 36
3.4
Comparison between u2 direction FEA results for overturning moment
load and analytical results. . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5
Comparison between FEA results for overturning moment load and analytical results for u3 direction forces. . . . . . . . . . . . . . . . . . . . 36
v
List of Figures
1.1
Typical bolting pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Pictorial representation of coordinate systems used in defining the CONN3D2
elements for Abaqus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2
Relation between γ and ∆x for determining shear stiffness in a simple
bolted connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3
Pictorial representation of shear strain. . . . . . . . . . . . . . . . . . . 10
2.4
Depiction of how clearances are taken up by externally applied rotation
on a bolt pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5
Typical bolting pattern with 6 bolts used in overturning moment analysis. 13
2.6
Layout of the solid model with a through-bolt socket head cap screw. . 17
2.7
Axially loaded plate subjecting the bolt to a tensile load. . . . . . . . . 18
2.8
Example of exponential contact definition. The parameters co and po
are provided to the software to fit an exponential to the contact definition. 20
3.1
Pictorial representation of connectors used in solid model. . . . . . . . . 22
3.2
Section cut with Mises stress contour plotted for bolt subjected to 1000
lbf shear load on plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3
Section cut with Mises stress contour plotted for bolt subjected to 1000
lbf shear load on plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4
Plot of measured displacements from top of bolt head to bottom of nut
in the u1 direction as a function of preload for solid model. . . . . . . . 27
3.5
Plot of measured displacements in the u2 direction from top of bolt head
to bottom of nut as a function of preload for solid model. . . . . . . . . 29
3.6
Plot of measured displacements in the u3 direction from top of bolt head
to bottom of nut as a function of preload for solid model with the shear
step excluded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.7
Plot of measured displacements in the u3 direction from top of bolt head
to bottom of nut as a function of preload for solid model. . . . . . . . . 31
3.8
Approximate conical stress distribution for a fastener subjected to 7,000
lbf preload. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
vi
3
3.9
Approximate conical stress distribution for a fastener subjected to 7,000
lbf preload with limits of 2,000 psi Mises stress set. . . . . . . . . . . . . 33
3.10
CTF resultant on bolting pattern for modulus of elasticity = 3 × 105 psi. 34
3.11
CTF resultant on bolting pattern for modulus of elasticity = 3 × 107 psi. 35
3.12
CTF resultant on bolting pattern for modulus of elasticity = 3 × 1010 psi. 35
3.13
Potential effect of footprint length on bolt load distribution . . . . . . . 37
3.14
Magnitude of % relative difference as a function of modulus of elasticity
for the highest loaded bolt in the u1 direction. . . . . . . . . . . . . . . 38
3.15
Difference between expected load as a function of modulus of elasticity
for the highest loaded bolt in the u2 direction. . . . . . . . . . . . . . . 39
3.16
Magnitude of % relative difference as a function of modulus of elasticity
for the highest loaded bolt in the u3 direction. . . . . . . . . . . . . . . 40
vii
ACKNOWLEDGMENT
I would like to thank Professor Norberto Lemcoff, my project adviser for his immeasurable aid and support in the development of this project. Professor GutierrezMiravete, my academic adviser has provided me programmatic support during my
education with RPI, and has been my professor for a number of classes directly
applicable to the topics in this project.
Additionally, I thank my friends and colleagues, without whom I would not
have had resolve to stubbornly persist through the trying times in my work, education, and life in general.
I would like to thank my supervisor at Electric Boat for his support and the
Electric Boat Corporation for aiding in the funding of my graduate education.
Finally, I would like to thank my father, Chester VanGaasbeek, for all of the
love, support, wisdom, and knowledge that he has provided me with throughout my
life.
viii
ABSTRACT
Bolted connections are typically the cheapest means by which two or more parts can
be joined with sufficient strength for their application. Unfortunately, in large systems consisting of multiple sets of bolting patterns, it is not always straightforward
to determine the load on the bolts in the connection. Finite element analysis provides an alternative means to standard hand calculation methods, but characterizing
the stiffness of the bolts and the resulting load distribution on a bolting pattern,
and then verifying that the load distribution is accurate with respect to bounding
hand calculations is a difficult task. Large systems comprised of hundreds or more
bolts require extensive amounts of labor on the part of the engineer to size connections on assemblies without over-designing the connections. This project seeks
to mitigate this burden by developing correlations between finite element analyses
and hand calculations, and attempts to partially quantify the difference in these two
approaches. For stiff plates, this project determines that the axial component of the
load in the analytical solution may under-predict the load and stress in a bolt by
in excess of 30% compared to the finite element solution. The effect of preload on
measured bolt stiffness through a finite element analysis is characterized, and the
convergence towards the hand calculated solution for load distribution in a common
bolt pattern with increasing plate stiffness is demonstrated.
ix
Nomenclature
ka ≡ ku1
Axial Stiffness in the u1 direction, [lbf/in]
kτ ≡ ku2 , ku3
Shear stiffness in the u2 , u3 directions, [lbf/in]
κ ≡ κu5 , κu6
Bending stiffness in the u5 , u6 directions, [in-lbf/radian]
κr ≡ κu4
Rotational / torsional stiffness in the u4 direction, [in-lbf/radian]
θ
Angle of rotation, [radians]
F
Force applied, typically axial
∆x
Elongation in the x direction, [in]
A
Cross-sectional area, [in2 ]
E
Modulus of Elasticity, [psi]
G
Shear Modulus [psi]
σ
Stress [psi]
σy
Yield stress [psi]
τ
Direct shear stress [psi]
γ
Shear strain [radians]
l
Grip length, bolt length [in]
α
Half-apex cone angle [degrees]
dw
Washer diameter [in]
d
Bolt diameter [in]
λ
Overturning moment load standoff distance [in]
I
Area moment of inertia (second moment of area) [in4 ]
Acronyms
ASME
American Society of Mechanical Engineers
ASTM
American Society for Testing and Materials
FEA
Finite Element Analysis
FEM
Finite Element Method
CSYS
Coordinate System
x
Chapter 1
Introduction
Bolted joints are generally regarded as one of the most cost efficient and robust
methods of joining parts together. For non-permanent connections, the versatility
of bolted connections makes them exceptionally attractive from the standpoint of
maintainability and ease of assembly and disassembly. The process of developing a
bolted connection does not impart the same distortion into the adjoined members as
would welding or machining from larger stock materials, and is much less expensive
and prone to error during fabrication or installation. The properties of bolt materials
are tested rigorously, and material data sheets will accompany any certified fastener
as it makes its way onto a system where the fastener’s strength is critical. ASTM,
ASME, and a variety of other technical specification entities publish vast arrays
of information regarding the fits and strengths of bolts in bolted connections, and
guidelines for their lengths of engagement.
In general, the published information available on bolted connections is overwhelming, which is a good thing for the engineering industry. References such as
Shigley’s Mechanical Design [1] and Roark’s Formulas for Stress and Strain [2], which
are staple textbooks in the field of engineering, in general treat the problem of bolted
connections with a sufficient rigor for the common engineer to size bolted connections with an ample factor of safety for conservatism using conventional methods
and assumptions. However, the methods presented by standard engineering literature neglect the stress concentrations evident in bolted connections, and fail to
capture the effects of variables such as:
• Stiffness of the adjoined members
• Preload on the fasteners
• Friction effects between the head of the bolt / nut and the adjoined members
• Contact interaction definitions between touching surfaces.
1
2
• Shear stiffness of the bolts
• Rotational stiffness between the nuts and the heads of the bolts
The finite element analyst is familiar with many of these variables, as when
developing a bolted model in a finite element software (for example, Abaqus), the
analyst must explicitly declare the model used for each of these effects, and will
notice the resulting overall deformation shapes in the model are functions of the
above considerations. The forces and moments generated internal to the fasteners
will also vary greatly with the choice of modeling method used by the analyst.
The problem of bolted connections is addressed in this project, and considerations for the above effects are examined. Numerical studies are performed using
both hand calculation and finite element analysis. Guidelines for the modeling of the
aforementioned variables are developed and presented, and their relative significance
and effects on the analysis are discussed.
Figure 1.1 is a pictorial representation of a bolted connection between two
places by four fasteners and nuts. Research on the flexibilities and their effects
on fatigue life due to a number of parameters has been reported by Huth [3], and
the modeling of fastener-clamped member systems by de Rijck [4]. The primary
variables which affect the system as determined by Huth [3] are:
1. Young’s modulus of the clamped member’s materials
2. thickness of the clamped members
3. fastener diameter
4. fastener material
5. single or double-shear configuration
Huth also suggests that the secondary variables, which are concerned with
installation of the bolted connection are:
1. fastener preload
2. fit of the fasteners
3
Bolt Head
Top Plate
Bottom Plate
Nut
Exposed Threads
Figure 1.1: Typical bolting pattern.
3. type of fastener head
4. condition of the faying surfaces
Several of these variables are explored in greater depth in this project, and
considerations for other variables specific to finite element analysis are made which
may have been overlooked or neglected in prior research. The bolted connection
is analyzed using several methods with finite element analysis and their results are
compared to hand calculations. Overall, two base FEA models are considered in
this project.
The first model is a solid model consisting of a single bolt adjoining two parallel
plates. The analytical stiffnesses are calculated and compared to the measured
stiffness of the bolt in this model. Differences between the analytical and model
stiffnesses are compared and contrasted. The main parameter which is varied in
this analysis is the bolt preload.
The second finite element model is a simplified model with a set of connector
elements used to model a series of bolts. A comparison is made to the analytical
hand calculation of bolt force distributions as a function of joined member stiffness.
Chapter 2
Method
In the simplified analysis where bolts are replaced by simple connector elements, the
paramount aspect of capturing the stress and load distribution in a series of fasteners
is the stiffness of the bolts. The analysis is performed with the assumptions that all
materials behave linearly and elastically, and that all materials are isotropic such
that no inhomogeneities exist which would impact the load path. The accuracy of
these analytical stiffness models will be evaluated against a more complicated finite
element analysis.
In order to capture the bolt behavior in a simplified model, the following
stiffnesses are considered:
1. Axial stiffness, ka , corresponding to the local u1 coordinate system direction
in the element definition.
2. Shear stiffness, kτ , corresponding to the local u2 and u3 coordinate system
directions. The forces in the u2 and u3 directions are added together in a
root-sum-square sense, corresponding to vector addition of forces.
3. Bending stiffness, κ, corresponding to relative rotations between the point
representing the bolt head and the nut. As an example, the relative rotation
between the coordinate system axis a2 for coordinate system ’a’ and the coordinate system axis b2 on coordinate system ’b’ is related by an effective torsion
spring: T = κθ. T represents the torque between the two points, and theta
their relative rotation.
4. Torsional stiffness, κr , which is the rotational stiffness of the bolt preventing
rotation under the head and nut of the fastener of the underlying material. For
a bolt which is not preloaded or has a clearance fit, this stiffness is identically
zero because no individual clearance or shoulder bolt resists rotational motion.
The coordinate system used for the connector model of the bolts is shown in Figure
4
5
2.1. The ellipses represent the couplings between the points at the end of the bolt
to the material which is being joined.
a3
CSYS: a
a2
Bolt Head
a1
b3
CSYS: b
b2
Nut
b1
Figure 2.1: Pictorial representation of coordinate systems used in defining the CONN3D2 elements for Abaqus.
2.1
Bolt Stiffness
A model of a bolted connection consisting of one bolt joining two plates is
developed in the Abaqus finite element software. This model is used for characterizing the effects of bolt preload on joint stiffness. Comparisons are made with
hand calculated methods. Additionally, a finite element model consisting of two
plates adjoined by six fasteners subjected to an overturning moment load is developed. It examines the effect of material stiffness (Young’s Modulus, E) on the load
distribution in the fasteners. For this initial consideration, the bolts are assumed
to be springs in the finite element software, and are assigned stiffness based on the
6
assumption that they are not preloaded, and that the stiffness of the fastener heads
for rotational load transfer between the points defining the springs is neglected.
The elements used in this analysis to represent the fasteners are Abaqus’
CONN3D21 elements, which allows direct specification of the behaviors and stiffnesses of the joined reference points. The bolt stiffness is approximated through the
following method:
2.1.1
Axial Stiffness
Hooke’s law indicates that for a simple linear spring:
F = ka ∆x
(2.1)
To determine ka , the axial stiffness of the bolt (to a first approximation),
Equation 2.1 is solved for k.
ka =
F
∆x
(2.2)
The definition of Young’s Modulus, E, is used:
E=
F Lo
A∆L
(2.3)
where Lo is the initial length of the material being deformed, A is the cross sectional
area of the material undergoing deformation, and ∆L is the change in length of the
material. Calling x = L, and solving Equation 2.3 for ∆x:
∆x =
F Lo
AE
(2.4)
Substituting Equation 2.4 into Equation 2.2:
ka =
F
AE
F
= F Lo =
∆x
Lo
AE
(2.5)
Equation 2.5 expresses the stiffness of an effective spring in terms of the bolt’s area,
the bolt’s modulus of elasticity, and the stressed length of the bolt. This is not
1
Two noded connector elements used for three-dimensional analyses.
7
necessarily an accurate measure of the stiffness of the joint, and applies only to the
bolt. It is important to note that this is an idealization of the stiffness. The details
of this idealization are as follows:
1. Bolts will conform to a material specification, but it is difficult to know the
exact modulus of elasticity of the material used in the specification because
there is an uncertainty associated with it. While the manufacturer of a lot
of material may quote his modulus of elasticity for steel as E = 30 × 106 psi,
however, depending on the quality of the supplier and the sophistication of
their manufacturing process, the actual values may vary, for example, between
29.6 × 106 psi and 30.4 × 106 psi. So, while the fasteners used in a given
application may be made from the same type of material, if they are not
made from the same lot, there may be variations from the originally quoted
properties due to an uncertainty in manufacturing and testing. Additionally,
it is worth noting that even if the fasteners are all cut from the same lot of
material, their properties may vary together as a whole.
2. The choice of the bolt’s cross sectional area is also a point of contention. There
are a several diameters associated with a given screw thread. The primary two
diameters of concern are the major and the minor diameters. At the minor
diameter, the material condition is at a minimum, and at the major diameter,
at a maximum. The area of the threaded region on a bolt is variable and
subject to tolerance, and the effects of this variation in area is likely difficult
to determine without accurate testing. However, in the case of through-bolts,
where the shank of the bolt2 is the primary region being stressed by a clamping
load, this issue is considered negligible.
3. Finally, the length of the effective spring bolt which is used must be carefully
chosen. In a tensile test, where an experimentalist would try to determine a
modulus of elasticity, this length is typically the length of stressed material.
This must be the case for the model of the bolt’s stiffness as well. When a nut
2
The shank of a bolt refers to the unthreaded region where there is an excess of material with
respect to the threads.
8
is threaded onto a bolt and torqued, the stressed region falls off as a function of
the fit of the bolt in the nut. There is no sharp transition region which can be
called the ”stressed length”. Additionally, there is some stiffness in the head
of the bolt which might be taken into account. If the analyst were to choose
the total length of the bolt as the spring length of the bolt, overestimating the
stressed length, the effect on Equation 2.5 would result in a bolt model which
was too compliant. In general, a common practice in engineering is to consider
this length to be the thickness of the clamped members, or the ’grip length’;
however this neglects the end effects on the bolt head and near the nut, which
may lead to an overly stiff solution. Further analysis of this problem will be
discussed later, and the thickness of the clamped members, or the ’grip’ is
assumed to be an appropriate length.
This bolt stiffness can be used in addition to the clamped member stiffness to
determine an effective joint stiffness. Shigley’s assumes a conical stress distribution
through the material [1]. The expression for member stiffness is:
km =
πE tan α
tan αdw −d)(dw +d)
2 ln (l(ltan
α+dw +d)(dw −d)
(2.6)
where α is the assumed cone’s half-apex angle, dw is the diameter of the washer,
and d is the diameter of the bolt.
The bolt stiffness model used in this project is a simplified version used primarily to evaluate the effects of preload and joined member stiffness, and to compare to
conventional hand calculation methods. The overall joint stiffness can be considered
as a group of springs in series and parallel. Member stiffnesses may be modeled as
springs in series because the force going through the members is the same. Because
the bolt and the joined plates share the same displacement at the nut and head, bolt
axial stiffness can be modeled as a spring in parallel with the member stiffnesses.
For the purposes of the initial hand calculation of the bolted joint stiffness,
and as a first approximation, Equation 2.6 will be neglected, however it is noted that
this stiffness will likely begin to dominate as the preload is increased. The effect of
bolt stiffness on the resulting stresses calculated in the bolts will also be examined.
9
2.1.2
Shear Stiffness
The shear stiffness of a bolt is a difficult topic to address for several reasons.
The shear stiffness of the bolt is a function of several variables:
1. Shear Modulus of the bolt material
2. Preload of the fastener
3. Stiffness of the clamped plates
4. Number of clamped plates
5. Friction between clamped plates
6. Clamped length
7. Clearances between the fastener and the plates (if the preload is low enough
such that the plates may slip relative to one another and the material may
contact the fastener walls).
Assuming the bolt material is isotropic, the shear modulus is calculated[2,
eq.2.2-7] using the modulus of elasticity of the material, E, and Poisson’s ratio, ν.
G=
E
2(1 + ν)
(2.7)
An approximation of the shear stiffness of the bolts used in this analysis is
calculated by considering the grip length and the shear strain caused in the bolt.
The following simple single-shear plate is analyzed for this analysis with one bolt.
Using the definition of engineering strain, with γ = ∆x/L = tan θ and the
definition of shear modulus:
G=
F/A
τ
=
∆x/L
γ
(2.8)
the shear stiffness, kτ of the material can be found by solving for P/∆x which is an
effective spring constant. Rearranging Equation 2.8:
kτ =
P
GA
=
∆x
L
(2.9)
10
P
∆x
P
90◦ -γ
Figure 2.2: Relation between γ and ∆x for determining shear stiffness in
a simple bolted connection.
∆x
P
θ
L
Figure 2.3: Pictorial representation of shear strain.
In Equation 2.9, A, is the area of the bolt, and L is the stressed length of the fastener.
This approach neglects friction interactions between the touching members, which
will be examined for accuracy in a solid model finite element analysis to determine
if this serves as a reasonable model for the stiffness of the bolt.
2.1.3
Bolt Rotational Stiffnesses
The bolt rotational stiffnesses are characterized by two separate primary com-
ponents:
1. Bolt ’beam bending’ stiffnesses, κu5 , κu6 , which relate the rotation of the top
point modeling the bolt to the bottom point. The moment are added in a
vector sum sense. The subscripts u5 and u6 indicate that they are rotational
about the local u2 and u3 axes, respectively, in Figure 2.1. The beam bending
stiffness in these two directions is assumed to be equal because the bolt and
11
clamped members are isotropic.
2. Bolt ’twisting’ stiffness, κr , which would capture the moment carried by the
connector element for a preloaded connection if the two plates joined by a
single fastener were to be twisted about that single bolt.
2.1.3.1
Bolt Beam Stiffnesses
The first-approximation bolt rotational stiffnesses are calculated by making
some assumptions about the behavior of the bolt under the effect of a tipping load.
The head of the bolt and the joined plates are assumed to be infinitely stiff. Additionally, the flexibility of the bolt is assumed to be such that the head and nut may
not pry off of the underlying material, and neglects preload effects. This system
is approximated as a cantilever beam subjected to an external moment Mo . Using
Reference [2], table 8.1 case 3a, the slope of the beam for a = 0 is:
θ=
−Mo L
EI
Rearranging Equation 2.10 and solving for
κ=
(2.10)
Mo
:
θ
Mo
−EI
=
θ
L
(2.11)
Equation 2.11 is the expression for rotational beam stiffness of this bolt model. It
is important to note that this methodology, based on our assumptions of how the
bolt follows the rotation of the joined plates, is a function of the assumed pivot
point. Also, this stiffness corresponds to prescribing a rotation, θ. The deflections
due to beam stiffness are neglected because of a small angle approximation for this
rotation.
2.1.3.2
Bolt Torsional Stiffness
For analyses without preload, the bolt torsional stiffness is assumed to be zero
because the bolt head and nut will slip relative to the joined plates. Additionally,
where more than one bolt is used in a bolt pattern, there is a limiting amount by
which the bolts can slip before taking up the clearances and going into pure shear.
12
Rotation
c4
c3
c1
c2
Rotation
Figure 2.4: Depiction of how clearances are taken up by externally applied rotation on a bolt pattern
Figure 2.4 depicts how the clearances are quickly resolved by the rotation of
adjoined plates. The hole clearances c1 through c4 are not the same due to tolerances
which are in general on the order of thousandths of an inch for a hole clearance that
may be no more than 10 thousandths of an inch.
The torsional stiffness, κr for a shaft of uniform cross section may be calculated:
T =
JT Gθr
L
(2.12)
where JT is the torsion constant for the section, G is the shear modulus, and L
is the length of the twisted member. Rearranging 2.12 and solving for T /θr yields:
κr =
T
GJT
=
θr
L
(2.13)
13
Y
X
RP−1
Z
X
Z
RP−2
Y
Z
X
Figure 2.5: Typical bolting pattern with 6 bolts used in overturning moment analysis.
This equation would model the torsion of a bolt subjected to simple twist,
however the effect of preload may directly relate to a stiffness in accordance with
Equation 2.13.
2.2
Typical Bolting Pattern Subject to Overturning Moment
The problem of a typical bolting pattern subjected to a shear and an overturn-
ing moment is examined. An example geometry with three bolts in a lap joint is
chosen, as depicted in Figure 2.5. The bolts are represented by green circles which
represent coupling constraints joining nodes between the top white plate and the
base green plate to the connector element modeling each bolt.
The bolting pattern is three equidistant bolts offset from each side by 1.5
inches bolting together two 2 inch wide by 6 inch long by 0.5” thick plates. As a
first estimate of the forces and stresses experienced by the bolts, the standard bolt
pattern tipping approach is employed:
14
Static Equilibrium Equations For a given load P, transverse to the plate and
above the bolting pattern at a distance above the plate, L, we can sum forces in the
vertical direction:
X
Fy = 0 = R0 + Fy1 + Fy2 + Fy3
(2.14)
where Fyi for i ∈ {1, 3} is the vertical force on the ith bolt. Summing forces in the
X direction provides:
X
Fx = 0 = P − Fx1 − Fx2 − Fx3
(2.15)
Finally, moments are summed about the tipping point of the bolting pattern, which
is assumed to be at the edge of the adjoined plate. This particular assumption is
of primary importance for determining the actual load on the bolts, and will be
discussed in Section 4. λ is the standoff distance above the plate.
X
M = 0 = −P λ + Fy1 × d1 + Fy2 × (d1 + d2 ) + Fy3 × (d1 + d2 + d3 )
(2.16)
Geometric Assumptions The problem as defined is not solvable without the introduction of some geometric considerations. An assumption about the relationship
of the strains of the bolts is made, which is not entirely valid for preloaded bolts
or flexible joined plates. The strains are assumed to be linearly proportional to the
tipping point.
1 ×
d1 + d2
d1 + d2 + d3
d1
= 2 ×
= 3 ×
d1 + d2 + d3
d1 + d2 + d3
d1 + d2 + d3
(2.17)
Equation 2.17 indicates that the furthest bolt from the tipping pattern will
experience the highest strain, and consequently it will experience the highest stress
as well. Using Equation 2.17, we can substitute the definition for strain as = σ/E,
and substitute σ = F/A, where F is the load in the fastener and A is the cross
sectional area of the fastener. These substitutions into Equation 2.17 allow us to
solve equations 2.14, 2.15, and 2.16 for the force on each bolt in the pattern.
15
2 =
d1 + d2
3
d1 + d2 + d3
1 =
d1
3
d1 + d2 + d3
Using the definition:
σ
E=
=
P
σ
=
AE
Substituting the equations for 2 and 1 into these expressions and solving for P2 , P1 :
P2 = 2 A2 E2 =
d1 + d2
A2 E2 3
d1 + d2 + d3
P1 = 1 A2 E2 =
d1
A1 E1 3
d1 + d2 + d3
Now substituting the above expressions into Equation 2.16:
0 = −P λ +
...
d1
A1 E1 3 d1 + ...
d1 + d2 + d3
(2.18)
d1 + d2
A2 E2 3 (d1 + d2 ) + A3 E3 3 (d1 + d2 + d3 )
d1 + d2 + d3
Solving Equation 2.18 for 3 yields:
3 =
λP
A3 E3 (d1 + d2 + d3 ) +
A2 E2 (d1 +d2 )2
d1 +d2 +d3
+
A1 +E1 +d21
d1 +d2 +d3
The force in bolt 3 may be determined:
P3 =
!
λP
A3 E3 (d1 + d2 + d3 ) +
A2 E2 (d1 +d2 )2
d1 +d2 +d3
+
A1 E1 d21
d1 +d2 +d3
A3 E3
(2.19)
Likewise the force in the second bolt is:
d1 + d2
P2 =
A2 E2
d1 + d2 + d3
!
λP
A3 E3 (d1 + d2 + d3 ) +
A2 E2 (d1 +d2 )2
d1 +d2 +d3
+
A1 E1 d21
d1 +d2 +d3
(2.20)
16
and for the first bolt in the bolting pattern closest to the pivot point:
d1
A2 E2
P1 =
d1 + d2 + d3
!
λP
A3 E3 (d1 + d2 + d3 ) +
A2 E2 (d1 +d2 )2
d1 +d2 +d3
+
A1 E1 d21
d1 +d2 +d3
(2.21)
This method can be extended to any number of fasteners in a rectangular
bolting configuration, and any number of layers of bolts in that rectangular pattern.
If, for example, there were n rows of three bolts instead of one row of three bolts,
the force in the bolts would be divided by n in each location.
2.3
Stiffness Comparison to Solid Finite Element Model
A solid model is constructed consisting of two plates and several load cases
which are intended to provide an assessment of the accuracy of the stiffness of
Equations 2.5, 2.9, 2.11, and 2.13 as they relate to using connector elements to
model fasteners. For the purposes of this analysis, a bolt of nominal 1/2 inch
diameter is used. A 1/2”-13UNC-2A x 1” long fastener has a minor diameter area
of dm = 0.1257 in2 per Reference [5]. An effective reduced radius which captures
this minimum material condition is calculated:
A = πr2
0.1257 in2 = π × r2
r
0.1257 in2
= 0.200 in
r=
π
Note that the radius calculated is less than the nominal radius of the bolt (0.20
inches versus 0.25 inches). While the minor diameter is the smallest diameter to use
for the analysis of bolted connections, for stress purposes it is the most conservative.
For purposes of evaluating stiffness, the calculated result will likely be slightly more
compliant than an actual bolt, however the difference is considered small.
Reference [5] lists the head diameter of a 1/2” socket head cap screw as 0.75”
max 0.735” min, with a head height of 0.500” max to 0.492” minimum. The tolerances on the head height and head diameter are considered negligible, and, for
17
this analysis, their mean values (0.75”+0.735”)/2 = 0.742” and (0.500”+0.492”)/2
= 0.496” are used.
Z
Y
X
Z
Y
X
Figure 2.6: Layout of the solid model with a through-bolt socket head
cap screw.
A hole size of 0.520” is used through the two plates. The large plate will be
referred to as the ”bottom” or ”fixed” plate, and the smaller of the two plates will
be the ”top” or ”loaded” plate. Contact between the two plates is defined using
Abaqus’ surface-to-surface contact feature, assuming that the two plates can slide
relative and onto one another. Assuming the two plates are non-lubricated steel, a
Coulomb friction coefficient, µ, of 0.6 is used in this analysis.
The analysis takes place over two steps per load case. The first step is a
preload step which establishes a preload in the fastener by externally reducing the
length of the bolt using a load in Abaqus. The second step in the analysis is a step
which applies load to the plates such that the effective stiffness of the bolt can be
calculated.
2.3.1
Solid FEA Preload Calculation and Model Details
The axial stiffness of the bolt subjected to a preload from this model will be
calculated by applying a load to the top plate in the vertical direction along the axis
of the bolt, as per Figure 2.7
18
Total distributed load: P Distributed pressure load on top of face
Fixed Base
Top plate
Bottom Plate
Figure 2.7: Axially loaded plate subjecting the bolt to a tensile load.
The preload applied to the bolt is calculated as a force applied to the bolt.
This force is calculated by assuming the bolt is brought to approximately 60% of
yield. Assuming a bolt yield stress of σy = 90 ksi, the required preload is calculated:
0.6σy =
P
P
=
A
0.1257 in2
Solving for P:
P = 6.78 kip
The effects of interaction property on the results must also be evaluated. For
convergence purposes, often times the analyst will use a ’softened’ contact definition
in the finite element software, which allows for penetration of nodes during contact
to allow a more stable result to converge quicker; however this may influence the
observed deflections in the material. An example of this exponential contact definition is provided in Figure 2.8. The parameters co and po are provided by the
analyst. co refers to the clearance between nodes at which the contact pressure is
zero, and as the clearance is reduced, penetration will begin to occur. At the point
where clearance is zero, the contact pressure is defined as po .
This is not the only method of ’softening’ contact in finite element analysis,
19
however it is one of the more robust means by which contact softening is applied
because the analyst can allow for some small penetration on the order of the surface
roughness or tolerance of the material without too significantly affecting the model.
For the purposes of determining the stiffness in the solid model FEA, every effort is
made to avoid artificially softening the contact and having an adverse effect on our
results. A stiff exponential contact definition is used in the plate model with bolts
represented as simple CONN3D2 elements to improve the analysis convergence.
Because the stiffness of the connection is determined using the standard ’hard
contact’ definition, which does not allow penetration between the master and slave
surfaces in the analysis, the use of a softened contact is not expected to have a
significant effect on the results of the simplified analysis, as long as the contact
stiffness is sufficiently high. The contact stiffness clearance is set to a value of
1 × 10−7 inches with a contact pressure at zero clearance of 1 × 107 psi. The stiffness
and clearances were set by decreasing the clearance and increasing the stiffness until
the results show that the bolt pattern has a tipping point nearly about the edge of
the bolted flange, which is similar to what would be expected with a hand calculation
assumption. Depending on the width of the plate/flange, the tipping point may be
adjusted in a hand calculation, however, this parameterization is left as subject for
potential future research. The primary goal of this overturning moment plate model
is to demonstrate how the finite element model results approach the hand calculated
values as the plate stiffness is increased.
20
Contact
pressure
po
Clearance
co
Figure 2.8: Example of exponential contact definition. The parameters
co and po are provided to the software to fit an exponential to
the contact definition.
Chapter 3
Results
3.1
Solid Model Results and Stiffness Measurement as a
Function of Preload
A solid model is developed with a single bolt in order to assess the effects of
preload and determine the accuracy of Equations 2.5, 2.9, 2.11, and 2.13 as they
relate to using connector elements to model fasteners.
The model is developed in Abaqus/CAE which is the graphical user interface
version of Abaqus. Specifically version 6.13-2 was used to create the model, and
Abaqus/Standard 6.13-2 was used to solve the input deck. The solution is calculated over 4 analysis steps. The first step is the PRELOAD step, which applies
a bolt preload which was varied from 10 lbf to 7,000 lbf. The second step is the
APPLIEDLOAD step, which applies a pulling pressure load of 1,000 total lbf to the
top plate upwards, which attempts to pull the plates apart. The third step is the
REMOVEAXLOAD step, which disables the vertical pressure-pull load. Finally,
the plates are sheared with 1000 lbf at the top plate reference point.
The reference points on the sides of the bottom plate are fixed in all degrees
of freedom and kinematically coupled to the bottom plate.
Hard contact is defined with a Coulomb friction coefficient, µ, is assumed to be
equal to 0.6 to model un-lubricated steel on steel. This is an arbitrary choice consistent with a reasonably high friction. Friction coefficient is not varied and therefore,
the change in stiffness with respect to preload is modeled. For this analysis, because
the primary concern is the relationship between bolt stiffness and preload, the materials are assumed to be the same. The plates and the bolt are assigned a modulus of
elasticity of 30 × 106 psi, consistent with standard alloy steel. The material stiffness
used in this analysis for the bolt is not the same as the material stiffness used in
the 6 bolt overturning moment model, as the 6 bolt overturning moment model uses
the analytical stiffnesses calculated in Equations 2.5, 2.9, 2.11, and 2.13.
Figure 3.1 shows the connector elements used to measure the elongation as a
21
22
Reference Point coupled to bolt head
RP coupled to bottom plate
RP−5
RP−1
RP−4
RP coupled to top plate
Cartesian + Rotation
Z
z
Cartesian + Rotation
Cartesian + Rotation
Cartesian + Rotation
Y
y
X
x
RP−3
X
x
Z
Z
Zz
Z
Y
y
Cartesian
X+ Rotation
Y
y
X
X
x
Y
X
RP−6
Z
z
RP−2
RP coupled to bottom plate
Cartesian + Rotation Connectors
used to measure displacements
Z
Y
X
Reference Point coupled to nut
Figure 3.1: Pictorial representation of connectors used in solid model.
function of applied load. A 1,000 pound load is applied after the preload is set, and
the difference in elongation between the load step and the preload step is measured
by requesting elongation output on these connectors. Because the connectors do
not apply external forces to the model, and are coupled in a general sense to the
extreme ends of the bolt, the elongation can be measured off the model without
affecting the result. Additional measurement connectors are used to determine the
effects of clamped plate elongation and assess whether these deformations will affect
the measurement of the bolt length, however, their magnitudes with respect to the
bolt elongations were small and neglected.
A typical example of a highly preloaded bolt’s Mises stress output is shown
in Figure 3.2. The top plate in Figure 3.2 appears to experience a bending effect,
causing a high stress at the top and bottom of the plate. The bolt’s stress was set
to be approximately 60% of the material’s yield stress.
23
S, Mises
(Avg: 100%)
93.19E+03
85.44E+03
77.68E+03
69.93E+03
62.18E+03
54.42E+03
46.67E+03
38.92E+03
31.17E+03
23.41E+03
15.66E+03
7.91E+03
153.53E+00
x
z
z
y
x
y
z
Cartesian + Rotation
Cartesian + Rotation
Cartesian + Rotation
Cartesian + Rotation
x
y
Z
X
Y
Step: APPLIEDLOAD
Increment
1: Step Time = 1.000
Primary Var: S, Mises
Deformed Var: U Deformation Scale Factor: +1.00e+00
Figure 3.2: Section cut with Mises stress contour plotted for bolt subjected to 1000 lbf shear load on plates.
S, Mises
(Avg: 100%)
93.81E+03
86.01E+03
78.21E+03
70.40E+03
62.60E+03
54.79E+03
46.99E+03
39.18E+03
31.38E+03
23.58E+03
15.77E+03
7.97E+03
163.47E+00
x
z
z
y
x
z
y
Cartesian + Rotation
Cartesian + Rotation
x
Cartesian + Rotation
Cartesian + Rotation
y
Z
X
Y
Step: APPLYSHEARLOAD
Increment
1: Step Time = 1.000
Primary Var: S, Mises
Deformed Var: U Deformation Scale Factor: +1.00e+00
Figure 3.3: Section cut with Mises stress contour plotted for bolt subjected to 1000 lbf shear load on plates.
24
3.1.1
Bolt Stiffness Study with Application of Preload and Axial External Force
The stiffness of the member is studied as a function of preload with an applied
external force of 1000 lbf or 1 kip. When the bolt is preloaded, the material which
is clamped by the bolt acts as a series of springs with stiffness in accordance with
Equation 2.6. This reduces the load on the fastener as external load is applied.
The analytical predicted elongation of the fastener initially is calculated (neglecting the stiffness of the adjoined members):
F = k∆x
∆x = F/k
∆x =
where ka =
EA
L
=
6.78 kip
EA
L
=
6.78 kip
26×106 psi×0.1257 in2
1.0 in
26×106 psi×0.1257 in2
1.0 in
= 2.075 × 10−3 in
= 3.268 × 106 lbf/in.
This predicted increase in length is due to the preload placing the bolt into
tension. To obtain a better approximation, the stiffness of the plates is characterized.
The stiffness of the adjoined plates is calculated using Equation 2.6, assuming α =
30◦ , l = 1.0 in, dw = 0.742 in, d = 0.400 in, and E = 30 × 106 psi is:
km =
km =
2 ln
πE × d × tan α
tan α+dw −d)(dw +d)
2 ln (l(l tan
α+dw +d)(dw −d)
π × 30 × 106 psi × 0.400 in × tan 30◦
◦
(1.0 in tan 30 +0.742 in−0.400 in)(0.742 in+0.400 in)
(1.0 in tan 30◦ +0.742 in+0.400 in)(0.742 in−0.400 in)
= 1.877 × 107
lbf
in
If advancement of the thread did not partially elongate the fastener, the bolt
would elongate by a distance of 2.075×10−3 in. In reality the threads would advance
through the nut based on the friction present in the threaded connection and the
torque applied to achieve this preload. This analysis neglects the advancement of
the thread for simplicity, and applies the preload directly to the bolt cross section.
The measured displacement from the model is therefore primarily attributed to the
stiffness of the joined members.
25
For a 6.78 kip preload, the expected reduction in length of the members is
361 µin. When compared to the finite element model, the deformed distance between
two nodes under the head of the bolt and the underside of the nut is measured to
be 0.999620 in. Subtracting the deformed length from the initial distance:
∆um = 1.00 in − 0.999620 in = 380 µin
The difference between the measured and expected value for material deformation is on the order of 5%, which is in good agreement with the theory. In order to
determine the cone angle corresponding to a minimum difference between measured
and calculated, Equation 2.6 is substituted for km in the following expression, and
α is solved for numerically.
P
= ∆um
km
P
∆um =
πE×d×tan α
2 ln
= 380 µin
(l tan α+dw −d)(dw +d)
(l tan α+dw +d)(dw −d)
The difference between measured and expected goes to zero for α ≈ 27.46◦ ,
which corresponds well with prior research[1] which provides a range of 25◦ ≤ α ≤
33◦ .
The measured deflection between the top of the bolt head and the bottom of
the nut is tabulated in the u1, u2, and u3 directions as a function of preload for
various loading cases in Table 3.1. Load case LC1 corresponds to the model with no
load other than the specified preload on the plate. Load case LC2 corresponds to the
model with the specified preloaded bolt held at a fixed length through modification
of the Abaqus bolt load, while a 1,000 lbf upward pressure load pulls the top plate
vertically. Load case LC3 corresponds to the top plate being pulled in the shear
direction. All dimensions are in microinches, and all preloads are listed in pounds
force. The preload is varied from 10 lbf, or effectively no preload, to 7000 lbf. An
additional data point is added at 6780 pounds preload which corresponds to 60%
of the yield stress of this fictional bolt. Smaller steps between preload increases are
taken at lower preloads to capture effects near in the vicinity of the applied external
26
load, and to search for a tendency for the plates to want to separate.
The deflections themselves are loosely correlated to the stiffness of the adjoined
plates, however take into account some flexibility of the bolt heads and deformation
in the bolt as well. The primary focus of this model is to qualitatively determine
the effect of preload on the change in length of the bolt-member pair so that future
research may be able to develop better informed approaches to joint stiffnesses.
Table 3.1: Connector displacement measured from bolt head to opposite
end on nut as a function of preload force.
Preload
LC1
LC1
LC1
LC2
LC2
LC2
LC3
LC3
lbf cu1 µin cu2 µin cu3 µin cu1 µin cu2 µin cu3 µin cu1 µin cu2 µin
10
-1.3 0.003 0.000 337.6 0.076 0.019 284.8 1.270
500
-64.6 0.061 -0.022 108.5 0.076 0.018
65.8 0.615
750
-95.9 0.079 -0.036
-7.5 0.076 0.018
-38.8 0.322
950 -120.7 0.093 -0.050
-74.1 -0.005 -0.157 -111.3 0.390
1000 -126.8 0.096 -0.054
-84.0 0.101 -0.070 -125.7 0.377
1050 -133.0 0.099 -0.058
-93.2 0.180 -0.123 -137.6 0.398
1250 -157.4 0.110 -0.076 -127.5 0.328 0.611 -163.0 0.292
1500 -187.7 0.124 -0.100 -165.6 -0.005 -0.244 -191.0 0.048
2000 -247.5 0.148 -0.156 -235.5 0.196 0.415 -250.6 0.164
3000 -365.1 0.192 -0.287 -363.7 0.233 0.018 -368.2 0.112
4000 -480.9 0.232 -0.432 -485.2 0.240 -0.447 -484.1 0.145
5000 -595.8 0.268 -0.583 -603.7 0.554 -0.199 -598.9 0.290
6000 -709.9 0.304 -0.738 -719.8 0.372 -0.539 -713.0 0.496
6780 -800.6 0.360 -0.783 -811.9 0.309 -0.882 -803.6 0.381
7000 -823.5 0.340 -0.897 -835.2 0.203 -0.859 -826.5 0.328
LC3
cu3 µin
-2100.0
-2020.0
-1930.0
-1770.0
-1710.0
-1620.0
-1110.0
-487.3
-375.5
-263.5
-203.9
-166.5
-140.8
-84.3
-122.3
It is evident from Table 3.1 that as preload is increased, the measured displacement in the u1 direction decreases in the preload step. This is because the
joint tends to shrink, as predicted by Equation 2.6. The relative difference between
the displacement in the u1 direction in the preload step versus the axially applied
load step is an important quantity for joint life’s sake. Joints are typically preloads
to prevent separation, and to prevent bolts from taking on cyclic loads which would
reduce their fatigue life. At preloads of 10 lbf to nearly 750 lbf, the joint fails to
stay together and separates under the 1000 pound load.
The results of the u1 direction displacements as a function of preload are
plotted in Figure 3.4
27
400
Measured displacement (µ in)
200
Measured Displacements as a Function of Preload (U1)
Preload Step
Axial Pull: 1000 lbf
Shear Pull: 1000 lbf
0
−200
−400
−600
−800
−1000
0
1000
2000
3000
4000
Preload (lbf)
5000
6000
7000
Figure 3.4: Plot of measured displacements from top of bolt head to bottom of nut in the u1 direction as a function of preload for
solid model.
3.1.2
Bolt Stiffness Study with Application of Preload and Shear External Force
Characterization of the shear stiffness due to tangential contact force between
the joined plates and the bolt/nut, combined with the shear stiffness of the bolt
itself is outside the scope of this project, however, is recommended as a subject for
future research. The axial stiffness formula from Shigley [1] is applicable to axial
loads, however additional work should be performed to turn these loads and stress
distribution assumptions into transverse shear stiffnesses.
Bolt shear stiffness is calculated here for later use in the overturning moment
shell model. The shear stiffnesses, kτ 2 , kτ 3 in the u2 and u3 direction are calculated
28
through the use of Equation 2.9:
kτ 2 = kτ 3
GA
=
=
L
E
A
2(1+ν)
L
=
26×106 psi
2(1+0.3)
0.1257 in2
1.0 in2
= 1.257 × 106 lbf/in
The magnitudes of the u3 (local shear) direction displacement are small in
each step with the exception of the shear pull step. Figure 3.7 exhibits a definitive
pattern wherein the relative displacement between the head of the bolt and the nut
in the u3 direction drops off significantly around a preload of 1000 to 1500 lbf. At low
preloads, the applied transverse load dominates the behavior of the bolt, whereas
at higher preloads (above the transverse load), there is a reduced displacement and
the bolt behavior is more predictable. This dropoff is likely a function of the friction
coefficient, µ, between the two surfaces, their tolerances and surface finishes, and
material combination.
Local u2 direction (orthogonal to the applied shear force direction) displacements are plotted on Figure 3.5, however the magnitudes of these displacements
are on the order of tenths of microinches, and are not considered significant. No
discernible useful pattern appears to exist with these data, and they are presented
for information only.
Figure 3.6 is a plot of the same data contained in Figure 3.7, with the exception that the data from the shear step is ignored and the axes are scaled to more
intricately examine the trends with the other steps. The axial pull step appears to
experience a large amount of ’noise’, and is likely indicative of numerical difficulties
arising from the discretization of the problem with finite element analysis. For an
axial pull, there does not seem to be any reason to include or predict effects on the
order of microinches for most practical applications, and the data is likely not useful
for the purposes of this project.
The magnitude of shear displacement appears to increase steadily during the
preload step. This effect is likely due to the discretization and slight asymmetry of
the mesh.
29
1.4
Measured displacement (µ in)
1.2
Measured Displacements as a Function of Preload (U2)
Preload Step
Axial Pull: 1000 lbf
Shear Pull: 1000 lbf
1.0
0.8
0.6
0.4
0.2
0.0
−0.2
0
1000
2000
3000
4000
Preload (lbf)
5000
6000
7000
Figure 3.5: Plot of measured displacements in the u2 direction from top
of bolt head to bottom of nut as a function of preload for
solid model.
3.1.3
Characterization of Rotational Stiffnesses and Summary of Stiffnesses
The rotational beam stiffnesses κ4 , κ5 are calculated using Equation 2.11:
κ5 = κ6 =
−EI
L
where I is the second moment of area in bending of the cross section, which is assumed to be at the minor diameter of the external threads, d3 = 0.4084 in. Assuming
the bolt has a solid circular cross section:
π
π
Ix = Iy = R4 =
4
4
0.4084 in
2
4
= 1.366 × 10−3 in4
30
0.8
Measured displacement (µ in)
0.6
Measured Displacements as a Function of Preload (U3)
Preload Step
Axial Pull: 1000 lbf
0.4
0.2
0.0
−0.2
−0.4
−0.6
−0.8
−1.0
0
1000
2000
3000
4000
Preload (lbf)
5000
6000
7000
Figure 3.6: Plot of measured displacements in the u3 direction from top
of bolt head to bottom of nut as a function of preload for
solid model with the shear step excluded.
Calculating the bending stiffness:
κ5 = κ6 =
in × kip
−EI
= −35.5
L
rad
The torsional stiffness is calculated using Equation 2.13 as follows:
κ4 =
T
GJ
=
θr
L
where J is the polar moment of inertia: J =
πd4
32
=
π(0.4084 in)4
32
10−3 in4 .
2.731 × 10−3 in4 ×
T
GJ
κ4 =
=
=
θr
L
1.0 in
E
2(1+ν)
= 27.3
in × kip
rad
= 2.731 ×
31
500
Measured Displacements as a Function of Preload (U3)
Measured displacement (µ in)
0
−500
−1000
−1500
Preload Step
Axial Pull: 1000 lbf
Shear Pull: 1000 lbf
−2000
−2500
0
1000
2000
3000
4000
Preload (lbf)
5000
6000
7000
Figure 3.7: Plot of measured displacements in the u3 direction from top
of bolt head to bottom of nut as a function of preload for
solid model.
Spring constants in all 6 degrees of freedom have been defined with hand
calculations and can be compared to the values obtained from finite element analysis
as a function of preload.
Table 3.2: Spring constants used for 0.5” diameter 1” long UNC fastener.
ku1 [kip/in] ku2 [kip/in] ku3 [kip/in] ku4 [in-kip /
radian]
3268
1257
1257
27.3
3.1.4
ku5 [in-kip /
radian]
-35.5
ku6 [in-kip /
radian]
-35.5
Summary of Results
The conical approximation of the stress distribution under the head of the
fastener and nut per Equation 2.6, which in general assumes an angle of 25◦ ≤
32
α ≤ 33◦ , is difficult to verify because the criteria for this pressure falloff is not well
defined. As shown in Figure 3.8, the stress distribution falls off gradually in an
approximate conical shape, however the choice of stress limits on the plot has a
large effect on the graphical angle of the stress cone observed. Figure 3.9 has a limit
of 2000 psi set, where any Mises stress above 2,000 psi is truncated and colored
gray. From this plot, it is clearly not a trivial task to simply measure the stress
cone half-apex angle from a finite element analysis without having some criteria for
determining a stress cutoff. This is a reasonable area for future research.
S, Mises
(Avg: 100%)
34.58E+03
31.70E+03
28.82E+03
25.94E+03
23.06E+03
20.18E+03
17.29E+03
14.41E+03
11.53E+03
8.65E+03
5.77E+03
2.89E+03
8.00E+00
Z
Y
X
Step: PRELOAD
Increment
1: Step Time = 1.000
Primary Var: S, Mises
Deformed Var: U Deformation Scale Factor: +1.00e+00
Figure 3.8: Approximate conical stress distribution for a fastener subjected to 7,000 lbf preload.
3.2
Hand Calculation Comparison to Plate Model
A finite element model is constructed of two plates adjoined by six fasten-
ers. The fasteners are modeled with CONN3D2 elements which have properties in
accordance with Table 3.2.
The values for connector forces are compared to the hand calculated values,
which are based on application of Equation 2.18. Individual forces in the bolts
are calculated using Equations 2.19, 2.20, and 2.21 with the change that the forces
33
S, Mises
(Avg: 100%)
34.58E+03
2.00E+03
1.83E+03
1.67E+03
1.50E+03
1.34E+03
1.17E+03
1.00E+03
838.00E+00
672.00E+00
506.00E+00
340.00E+00
174.00E+00
8.00E+00
Z
Y
X
Step: PRELOAD
Increment
1: Step Time = 1.000
Primary Var: S, Mises
Deformed Var: U Deformation Scale Factor: +1.00e+00
Figure 3.9: Approximate conical stress distribution for a fastener subjected to 7,000 lbf preload with limits of 2,000 psi Mises stress
set.
calculated must be divided by two because of the arrangement of the six bolts.
In this example:
P = 100 lbf
λ = 1.635 in
d1 = d2 = d3 = 1.25 in
A1 = A2 = A3 = 0.1257 in2
L1 = L2 = L3 = 1.0 in
E1 = E2 = E3 = 26 × 106 psi
Substituting values into Equation 2.19 to solve for P3 :
P3 = 14.01 lbf
P2 = 9.34 lbf
34
CTF, Resultant
56.23E+00
54.21E+00
52.19E+00
50.17E+00
48.14E+00
46.12E+00
44.10E+00
42.08E+00
40.05E+00
38.03E+00
36.01E+00
33.99E+00
31.97E+00
56.23E+00
56.23E+00
33.97E+00
33.97E+00
31.97E+00
31.97E+00
X
Y
Z
Step: Step−1
Increment
1: Step Time = 1.000
Symbol Var: CTF
Deformed Var: U Deformation Scale Factor: +1.00e+00
Figure 3.10: CTF resultant on bolting pattern for modulus of elasticity
= 3 × 105 psi.
P1 = 4.67 lbf
These values are compared to the FEA result values for P1 , P2 , andP3 for varying
modulus of elasticity of top plate in Tables 3.3, 3.4 and 3.5. Examination of the
reaction moments reveals that the rotation between the nut and the head of the
fastener in an overturning moment configuration is a small effect with respect to the
axial and shear stiffness effects, and to a first approximation may safely be neglected
for bolt analysis using FEA with this approach. The resulting moments induced by
the beam stiffnesses and torsional stiffnesses are orders of magnitude lower than the
forces observed in this problem’s configuration.
Modulus of elasticity is increased by an order of magnitude until the analysis
fails to run due to excessive stiffness. The range of modulus swept through begins
with 3×105 psi and is increased to 3×1010 psi, after which the model fails to converge
to a solution. Typical results for connector force magnitude outputs are plotted in
Figures 3.10, 3.11, and 3.12.
35
CTF, Resultant
33.25E+00
32.10E+00
30.94E+00
29.78E+00
28.62E+00
27.47E+00
26.31E+00
25.15E+00
23.99E+00
22.84E+00
21.68E+00
20.52E+00
19.36E+00
33.25E+00
33.25E+00
19.36E+00
19.36E+00
20.08E+00
20.08E+00
X
Z
Y
Step: Step−1
Increment
1: Step Time = 1.000
Symbol Var: CTF
Deformed Var: U Deformation Scale Factor: +1.00e+00
Figure 3.11: CTF resultant on bolting pattern for modulus of elasticity
= 3 × 107 psi.
CTF, Resultant
24.11E+00
23.62E+00
23.13E+00
22.64E+00
22.15E+00
21.66E+00
21.17E+00
20.69E+00
20.20E+00
19.71E+00
19.22E+00
18.73E+00
18.24E+00
24.11E+00
24.11E+00
20.79E+00
20.79E+00
18.24E+00
18.24E+00
X
Y
Z
Step: Step−1
Increment
1: Step Time = 1.000
Symbol Var: CTF
Deformed Var: U Deformation Scale Factor: +1.00e+00
Figure 3.12: CTF resultant on bolting pattern for modulus of elasticity
= 3 × 1010 psi.
36
Table 3.3: Comparison between FEA results for overturning moment
load and analytical results for the u1 (axial) direction.
E (psi)
N/A
3 × 105
3 × 106
3 × 107
3 × 108
3 × 109
3 × 1010
3 × 1011
B3 u1 lbf B2 u1 lbf B1 u1 lbf ∆ B3CTF1 [%] ∆ B2CTF1 [%] ∆ B1CTF1 [%]
14.01
9.34
4.67
N/A
N/A
N/A
55.01
30.99
28.09
292
232
510
34.09
20.09
18.65
143
115
299
30.37
9.48
5.83
116
1.5
24
22.77
11.65
4.57
63
25
2.1
18.07
12.34
7.14
29
32
52
17.62
12.69
7.82
26
36
68
N/C
N/C
N/C
N/C
N/C
N/C
Table 3.4: Comparison between u2 direction FEA results for overturning
moment load and analytical results.
E (psi)
N/A
3 × 105
3 × 106
3 × 107
3 × 108
3 × 109
3 × 1010
3 × 1011
B3 u2 lbf B2 u2 lbf B1 u2 lbf
0
0
0
6.67
1.60
0.17
7.39
2.14
0.021
3.89
1.96
0.099
0.57
0.53
0.15
0.056
0.065
0.018
0.005
0.006
0.0003
N/C
N/C
N/C
∆ B3CTF2 [%] ∆ B2CTF2 [%] ∆ B1CTF2 [%]
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/C
N/C
N/C
Table 3.5: Comparison between FEA results for overturning moment
load and analytical results for u3 direction forces.
E (psi)
N/A
3 × 105
3 × 106
3 × 107
3 × 108
3 × 109
3 × 1010
3 × 1011
B3 u3 lbf B2 u3 lbf B1 u3 lbf ∆ B3CTF3 [%] ∆ B2CTF3 [%] ∆ B1CTF3 [%]
16.67
16.67
16.67
N/A
N/A
N/A
9.55
13.82
15.25
42.7
17.1
8.5
9.13
17.92
19.64
45.2
7.5
17.8
12.98
16.77
19.21
22.1
0.60
15.2
15.73
16.35
17.23
5.6
1.9
3.4
16.37
16.44
16.58
1.8
1.3
0.5
16.46
16.46
16.48
1.3
1.3
1.1
N/C
N/C
N/C
N/C
N/C
N/C
37
Figure 3.13: Potential effect of footprint length on bolt load distribution
3.2.1
FEA Comparison against Hand Calculation Solution
As shown in Tables 3.3, 3.4, and 3.5, it is evident that increasing the stiffness
of the upper plate subjected to the overturning moment reduces the percentagewise difference between the FEA and the hand calculation. It is also seen that for
plates of lower modulus, the hand calculation may potentially under-predict the
axial force on the highest loaded bolt. This load distribution is a function of the
adjoined member and the stiffnesses of the plates being pulled. The load distribution
among the fasteners is also a function of the footprint length of the loaded plate.
This project is concerned only with the effect of modulus on the convergence toward
the hand calculated values, and so the footprint is modeled to the full height of the
plate. The effect on load distribution is qualitatively shown in Figure 3.13.
3.2.1.1
U1 Direction Bolt Load Comparison
The results for relative error for the highest loaded bolt as compared to the
hand calculated values, as shown on Table 3.3 are plotted against logarithmic modulus of elasticity (Figure 3.14). For steel with a modulus of elasticity of approximately
30 × 106 , the percentage difference between the FEA result and the hand calculated
value for the u1 axial direction force is over 100% for this particular model. As
the modulus is increased to 3 × 109 psi, the result begins to approach the hand
calculation solution, with a percentage difference of 26%. While a delta still exists,
38
300
u1 Direction CTF Relative Error as a Function of Modulus
Relative Error ∆, %
250
200
150
100
50
0
105
106
107
108
Modulus (psi)
109
1010
1011
Figure 3.14: Magnitude of % relative difference as a function of modulus
of elasticity for the highest loaded bolt in the u1 direction.
this comparison appears to demonstrate that the assumption of the bolt ∆ L is
correlated with the stiffness of the plate used in the analysis.
The differences observed in the less highly loaded bolts between the hand
calculation and the FEA results appear to reach a minimum with a plate modulus
of elasticity around that of steel, and then begin to increase with higher stiffness.
This effect may be related to the choice of interaction properties, but because this
project is primarily concerned with the stresses seen in the highest loaded bolt, the
results are simply presented for information.
3.2.1.2
U2 Direction Bolt Load Comparison
The comparison between the expected results and the calculated results for
the u2 direction provide confidence that the stiffer the loaded plate, the closer the
FEA result becomes to the hand calculation. The u2 direction is the lateral shear
direction, for which there is no external applied load. A sum of the forces and sum
of the moments in this direction for an infinitely stiff plate predicts that there will
39
Difference from Hand Calculated Load [lbf]
8
u2 Direction CTF difference as a
Function of Modulus for the Highest Loaded Bolt
7
6
5
4
3
2
1
0
105
106
107
108
Modulus (psi)
109
1010
1011
Figure 3.15: Difference between expected load as a function of modulus
of elasticity for the highest loaded bolt in the u2 direction.
be zero load in this lateral direction, which is what is observed as the plate stiffness
is increased. The magnitude of this extra lateral force predicted by the FEA model
is plotted as a function of plate stiffness in Figure 3.15
For a stiffness consistent with steel, for a 100 lbf external load, there is an
additional 3.89 pounds on the highest loaded bolt in the lateral shear direction,
which is a significant effect for the calculation of stress. As the stiffness is increased
by one or two orders of magnitude, the lateral shear forces quickly approach zero
for all three bolts, which is what the hand calculation predicts.
3.2.1.3
U3 Direction Bolt Load Comparison
In the vertical shear direction, u3 , the hand calculation assumes that because
the bolts have the same stiffness, they will see the same shear load. There are
six bolts, so the predicted shear load is 100.0 lbf/6 = 16.67 lbf. With a very low
plate stiffness (3 × 105 psi), the FEA appears to under-predict this shear load.
For the stiffness of a steel plate, the highest loaded bolt’s percent difference is
40
50
u3 Direction CTF Relative Error as a Function of Modulus
Relative Error ∆, %
40
30
20
10
0
105
106
107
108
Modulus (psi)
109
1010
1011
Figure 3.16: Magnitude of % relative difference as a function of modulus
of elasticity for the highest loaded bolt in the u3 direction.
calculated to be 22.1% under the hand-calculated value, with the lowest loaded bolt
experiencing a higher load than hand-calculated. The complicated interaction due
to friction between the plates and the curvature of the plate due to the overturning
moment load makes this effect difficult to qualitatively understand at low modulus
of elasticity. As the stiffness of the plate is increased, the load quickly begins to
approach the hand calculated value with a 1.3% difference at a modulus of 3 × 1010
psi. The magnitude of the difference between force predicted by the hand calculation
and the value calculated by the FEA model is plotted as a function of plate stiffness
in Figure 3.16
Chapter 4
Conclusions
A solid finite element model with a single bolt is developed to aid in characterizing
the variation of bolt stiffness with respect to preload. The joined member stiffness
is measured for the plates under a bolt subjected to a 60% σy preload, and good
agreement with reference textbook results is obtained for approximation of the halfapex cone angle for material stiffness. The cone angle bounds provided by the
reference textbook is found to be applicable to the steel plate and bolt model used.
Bolt displacement as a function of preload in all three principle directions is
measured for preloads ranging from 10 lbf to 7000 lbf. For the preload step, the
measured displacement is almost identically linear with respect to the magnitude of
preload applied. For the axial and shear pull steps, the displacements are approximately linear until the magnitude of the applied force approaches the magnitude
of the preload, at which point a second linear slope begins and follows closely the
trend of the preload step. The orthogonal direction displacements which are likely
primarily functions of numerical approximation and the Poisson’s ratio are found
not to have significant effects on the stiffness of the bolt. The shear direction displacement is found not to vary in the preload or axial pull steps of the solid model.
However, the shear pull step appears to be significantly less stiff in the regime where
the preload is less than the applied load, and then stiffens over the range of about
1000-1500 lbf preload significantly for a preload of 1000 lbf. The measured shear
displacements for the preload and axial pull steps are found to generally increase in
magnitude with increasing preload, although this is considered a second order effect
due to discretization of the problem and slight asymmetries.
A plate finite element model is developed consisting of six bolts. The modulus
of elasticity of the plate subjected to an external load is varied over the range
of 3 × 105 psi to 3 × 1011 psi. The bolt load distributions are determined from
the results of the model and compared with the analytical solution, assuming that
the bolt pattern tips about the end of the plate. The location about which the
41
42
bolt pattern tips is recommended for future research as a topic of a parametric
analysis of overturning moment loads on bolt patterns. The assumption of contact
definition between the two surfaces in the plate model is found to affect the resulting
load distribution significantly at high modulus of elasticity, indicating that the stiff
system is sensitive to the definition of the ”bolt tipping point”. Additionally, the
effect of footprint length of the loaded plate bolt distribution is considered another
interesting topic for future research into the analysis of bolt patterns.
This project has determined that bolt patterns may be analyzed reasonably
using finite element analysis, however, reasonable factors of safety must always be
used to ensure the safe construction and operation of a bolted system. In analyzing
the load distribution for steel plates, no comparison with test data was made in
this project. However, it is noted that for a stiff plate, an additional factor of
safety of on the order of 25-30 % may be reasonable to apply to the tensile force
component on the highest stressed bolt. For steel, this additional factor of safety
could be considered an effective ”double” in the load on the highest loaded bolt. The
orthogonal shear direction is considered approximately negligible for steel plates of
moderate stiffness. The shear load for the highest loaded bolt appears to decrease
marginally for steel plates, indicating that hand calculation would over-predict this
stress slightly.
Finite element analysis is a reasonable approach to solving problems concerning load distributions in bolted connections. In sizing these connections, it is important to exercise a reasonable design factor of safety to ensure that the tipping point
in the bolting pattern is accurately identified. No substitution can be made for direct
testing of bolts used in design, however the finite element analysis tools available
today allow for fast and cheap design iteration and sizing of bolted connections.
REFERENCES
[1] R.G. Budynas and J.K. Nisbett. Shigley’s Mechanical Engineering Design.
Number v. 10 in McGraw-Hill series in mechanical engineering. McGraw-Hill,
2008.
[2] W. Young, R. Budynas, and A. Sadegh. Roark’s Formulas for Stress and
Strain, 8th Edition. McGraw Hill professional. McGraw-Hill Education, 2011.
[3] H. Huth. Influence of fastener flexibility on the prediction of load transfer and
fatigue life for multiple-row joints. ASTM STP 927, 1986:221–250, 1986.
[4] J. Schijve R. Benedictus J.J. Homan J.J.M de Rijck, S.A. Fawaz. Stress
analyses of mechanically fastened joints in aircraft fuselages. 24th ICAF
Symposium - Naples, 24, 2007.
[5] E. Oberg, F.D. Jones, H.L. Horton, H.H. Ryffel, C.J. McCauley, R.M. Heald,
and M.I. Hussain. Machinery’s Handbook: A Reference Book for the
Mechanical Engineer, Designer, Manufacturing Engineer, Draftsman,
Toolmaker, and Machinist. Machinery’s Handbook: A Reference Book for the
Mechanical Engineer, Designer, Manufacturing Engineer, Draftsman,
Toolmaker, and Machinist. Industrial Press, 2004.
43
Appendix A
Python Code: Parameter Study: Modulus of Elasticity
# −∗− c o d i n g : u t f −8 −∗−
"""
Created on Tue Apr 09 13:07:10 2015
This program uses a base input file and iterates over the modulus
of elasticity over a range of values for a parametric study.
This work is published free of use as part of a
master ’s project in mechanical engineering at RPI.
@author : Chet VanGaasbeek
"""
import re , subprocess
import numpy as np
import time
from odbAccess import ∗
# Try t o r e m o v e t h e
iterated
file ,
if
it
exists .
try:
os. remove (’t2.inp ’)
print ’file removed ’
except :
print ’no file removed ’
# Number o f d a t a p o i n t s / j o b s
# Vary m o d u l u s o f
elasticity
t o run
from 3 e3 t o 3 e12
#m o d u l u s = np . l i n s p a c e ( 3 0 0 0 0 0 , 3 0 0 0 0 0 0 0 0 0 0 0 , n )
modulus = [3e5 , 3e6 , 3e7 , 3e8 , 3e9 , 3e10 , 3e11]
n = range (len( modulus ))
44
45
for i in range(len( modulus )):
with open(’t.inp ’) as fi:
with open(’TRUN ’+str(i)+’.inp ’,’w+’) as fo:
fileinp = fi.read ()
val = modulus [i]
seval = str(val)
fileout = re.sub(’Elastic \ n . ∗ ’,’Elastic \ n ’+ \
seval+’, 0.3 ’,fileinp )
time.sleep (1)
fo.write( fileout )
time.sleep (1)
rstring = " abaqus job=OF"+str(i)+" interactive inp = \
TRUN"+str(i)+".inp cpus =2"
print ’Running Job # ’+str(i)
print ’with code: ’+ rstring
rc = subprocess .Popen(rstring ,shell=True)
rc.wait ()
allctfs , allctms , ctfs , ctms , compn , i, j = [], [], [], \
[], [], [], []
dat = [0 for x in range(n)]
for dp in range(n):
odbstr = ’OF’+str(dp)+’.odb ’
odb = openOdb (path= odbstr )
step1 = odb.steps[’Step −1’]
frame = step1. frames[ −1]
allctfs = frame. fieldOutputs [’CTF ’]
allctms = frame. fieldOutputs [’CTM ’]
setnames = [’BOLT1 ’,’BOLT2 ’,’BOLT3 ’]
cf = [[0 for x in range (6)] for x in range(len( setnames ))]
for i, setname in enumerate ( setnames ):
setcall = odb. rootAssembly . elementSets [ setname ]
46
ctfs = allctfs . getSubset ( region = setcall )
ctms = allctms . getSubset ( region = setcall )
ctfs = [[ float (ctfs. values [ctf ]. data[compn ]) for \
compn in range (3)] \
for ctf in range(len(ctfs. values ))]
ctms = [[ float (ctms. values [ctf ]. data[compn ]) for \
compn in range (3)] \
for ctm in range(len(ctms. values ))]
for j in range (6):
if j < 3:
cf[i][j] = ctfs [0][j]
else:
cf[i][j] = ctms [0][j −3]
dat[dp] = cf
with open(’output .txt ’,’w+’) as f:
for a in range(n):
for b in range (3):
print dat[a][b]
f.write(str(dat[a][b])+ ’ \ n’)
f.write(’Modulus : ’+str( modulus )+’ \ n’)
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