1
A Finite Element Study of the Effect of Geometric Dimensioning
and Tolerancing on Bolt Design Stresses in Torque Carrying Bolted
FlangesbyCarney AndersonAn Engineering Project Submitted to the
GraduateFaculty of Rensselaer Polytechnic Institutein Partial
Fulfillment of theRequirements for the degree ofMASTER OF
ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________Ernesto
Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic InstituteHartford, CTApril, 2012(For
Graduation May, 2012)
CONTENTSLIST OF TABLESiiiLIST OF FIGURESivABSTRACTv1.-
INTRODUCTION12.- METHODOLOGY32.1- GEOMETRY32.2- FEA MODELING62.3-
MATERIALS103. - RESULTS AND DISCUSSION113.1 - ASSEMBLY
BEHAVIOR113.2 - BOLT STRESS/LOADS123.3 - VALIDATION184.-
CONCLUSIONS20REFERENCES22
LIST OF TABLESTableDescriptionPage1Typical Mechanical Properties
for Inco 718102Stress/Load Results Summary123Comparison of Fine vs.
Coarse Mesh Results19LIST OF FIGURESFigureDescriptionPage1Torque
Carrying Bolted Shaft Assembly223D CAD Assembly33Tolerance
Conditions4-54Assembly Cross Section/Boundary Conditions75FEA
Component Mesh76Gap and Penetration in FEA ContactModeling97Load
Step Geometry Progression108Loaded Flange Deflection
Results11-129Bolt Shank Stress Plots14-1510Tolerance/Stress
Sensitivity Plots16-1711Fine/Coarse Mesh Comparison1812Coarse Model
Loaded Flange Deflection Results19
ABSTRACTThis paper reports on a Finite Element (FE) modeling
study investigating the contributions of tolerances towards the
limiting bolt stresses in torque carrying bolted flanges. For a
given flange layout, with set boundary conditions, the peak
stresses experienced by the bolts are directly related to the
dimensional variations permitted during the manufacturing of the
flange components. By comparing stress results from Finite Element
Analysis (FEA) models of flange assemblies both in the nominal
condition, and with dimensional error imposed, the Geometric and
Dimensional Tolerancing (GD&T) controls which stresses are most
dependent on may be identified. Furthermore, by running models with
varying magnitudes of dimensional error, a trend curve may be built
to better understand this sensitivity. This allows a designer to
better predict the actual worst case stresses a given design may
experience, and better balance tight tolerance requirements with
nominal configuration changes. A parts price is directly tied to
the tolerances applied to it, therefore this could reduce a parts
manufacturing cost significantly, while better utilizing the design
materials. A list of key tolerances will be identified, and a
stress vs. tolerance curve for each (based on a given flange
configuration) will be provided. Critical design stresses will be
predicted via 3D FEA models of the bolted flange assemblies.
ii
1.- INTRODUCTIONBolted joints are utilized to transmit torque in
a number of high torque applications like automotive drive trains
and jet engine rotor stacks. In these applications it is important
to be able to predict bolt stresses. Although there are a number of
studies using the Finite Element Method (FEM) on the various
methods available for modeling bolted flanges (see e.g. [1]), few
consider the effects of tolerance on the bolt stresses. In order to
reduce part cost, a designer may decide to increase the Geometric
Dimensioning and Tolerancing (GD&T) allowances for their flange
holes. However, without understanding the GD&T contribution to
stress, the designer may inadvertently inflict a durability
shortfall onto the part. This study outlines a method by which to
quantify the increases in bolt stresses from variations in GD&T
allowances. Improving a designers understanding of GD&T on part
stresses should allow to better balance design constraints like
cost, producibility, and durability.
The objective of the study is to predict the level of
sensitivity of bolt design stresses to the principal tolerances for
torque carrying bolted flanges. This will be accomplished by
producing trend curves of tolerance vs. stress. The intent being
that with this information, flanges may be more accurately designed
to the required strength capability, and without unnecessarily
adding cost. This trend info will be summarized by stress vs.
tolerance curves for a typical flange configuration. Bolt hole
diameter and bolt hole true position are the tolerances that will
be assessed (bolt diameter was considered, but it was dropped as it
essentially has the same effect as hole diameter, and fastener
vendors typically hold tight size tolerances relative to the hole
tolerances held by shaft or disk manufacturers), since they are the
major cost drivers when producing bolt holes. Figure 1 shows a 3D
Computer Aided Drafting (CAD) representation of a torque carrying
bolted flange. There are 2 discreet torque carrying shafts
connected in the middle by a piloted flange, and a plurality of
bolts. The bolts both hold the shafts together axially, and are the
mechanism by which torque is transmitted between the 2 shafts.
22
Figure 1: Torque carrying bolted shaft assembly (part
configuration and loading scenario to be analyzed via 3D FEA in
this study)2.- METHODOLOGYIn order to run this study, a number of
3D FEA models were built for a series of slightly varying bolted
flange/shaft assemblies. The 3D geometry was built in the Computer
Aided Drafting (CAD) program Unigraphics, and the FEA modeling and
post processing was done in Ansys. The models were all run at room
temperature, and all the parts were modeled as nickel alloy. The
following subchapters provide the specifics of, and the reasoning
behind, the geometry of the different cases, the mesh and boundary
conditions applied, and the material properties that were used.
2.1- GEOMETRY
The geometry was modeled using the FEM, with the interfaces
modeled as contacts (see figure 4 for an axisymmetric
representation). All the models consist of two short shafts with
their flanges snapped together, and bound axially by pre-loaded
bolt members.
Figure 2: 3D Cad model of the bolted flange assembly.
In order to populate the trend curves, 9 separate FE models were
constructed, a baseline case, 4 cases with varying hole size
tolerance (with geometric deviations ranging from 0.001 to 0.004),
and 4 cases with varying hole true position tolerance (with
geometric deviations ranging from 0.001 to 0.004). For all
conditions, the bolts were modeled with nominal geometry. For the
baseline condition, both flanges were modeled in a nominal
condition (the hole diameters were consistent at 0.251, and were
equally spaced circumferentially). For the tolerance cases, shaft 1
was left at nominal, and the dimensional deviations from nominal
geometry were applied to the holes of shaft 2. The application of
tolerances was done in a worst case fashion; for bolt hole diameter
(figure 3a), a single hole was modeled at the minimum diameter,
while the remainder of the holes in that flange had maximum
diameter holes; for true position (figure 3b), one hole in the
shaft 2 flange was offset the full tolerance circumferentially (see
figure 3 for a picture of the tolerance conditions).
Figure 3a: Hole size tolerance condition
Figure 3b: True position tolerance condition
Both of these tolerance conditions cause one bolts shank to
contact its bolt hole prior to the others, therefore forcing that
bolt to bear the torque load until it deflects enough that the
remaining bolts shanks come into contact with their associated
flange holes. As a result, the toleranced holes bolt carries more
torque than the remaining bolts, which is what drives the
relatively high stress into said bolt (these trends can be seen in
the load/stress results in table 2). The sliding contacts allow for
the hole edges to contact the bolt shank sections, and accurately
represent the compressive contact that occurs in real parts.
Maximum principal stress, and maximum bearing stress (via minimum
principal stress on the surface of the bolt shank) were checked, as
they are the typical stresses that limit part life. Von-Mises
stress was checked, in order to understand the likely extent of
plasticity in the bolt. The nominal flange model was built to
baseline the stresses, and to verify that the model functions
properly and produces reasonable results. Next, subsequent models
with discrete geometry variations were run in order to evaluate
tolerance sensitivity. A range of models (see table 2) were run in
order to gather data to populate the trend curves. Since the stress
results from the models are being compared, it is important to make
sure that the mesh is approximately the same for all. Specifically,
the mesh density at the contact locations and high stress locations
needs to match. Variations in mesh density will drive variations in
stress, which can be seen in the results shown in table 3.
2.2- FEA MODELINGThe FEA analysis was performed in Ansys. All
parts are meshed entirely with 8 noded bricks (solid 45 in Ansys).
The membrane section of one flange is fixed from moving
circumferentially, axially and radially, while the second flanges
membrane section has tangentially oriented loads applied to it in
order to apply the torque. A full 360 degree model was built
instead of opting for a simpler partial model with symmetry
constraints applied for two main reasons: a) The symmetry
constraints would artificially hold the 2 flange parts on center,
instead of allowing the 2 parts to re-center themselves to shifted
bolt circle as it would in reality b) It would force multiple holes
to be out of tolerance based on number of sectors specified (i.e.
if a sector was used, the symmetric geometry assumption would be
that 4 bolts were at the tolerance condition), and this is an
attempt to look at a worst case condition. The geometry was modeled
in Unigraphics (see figure 2), and then imported into Ansys. A
cross section of the assembly, and the applied boundary conditions
and interfaces can be seen in figure 4. Figure 5 shows the mesh of
the various components; in the assembly view, you can see that the
mesh density in the bolt is much finer than that in the most of the
flange body, since the study is primarily concerned with the bolt
results. The picture of the hole mesh shows the relatively fine and
uniform mesh around the bolt holes. This was created by dividing
out the volume around the hole, and sweeping the mesh from front to
back. It is important to have good mesh density in this portion of
the flange in order to ensure good results for the contact
solutions. The bolt mesh view shows the uniform bolt mesh that was
created by sweeping axially down the center of the bolt, and
circumferentially around the outside of the bolt.
Figure 4: Diagram showing where boundary conditions and contacts
are applied on a cross sectional view of the bolted flange
assembly.
Figure 5: FEA MeshThese FEA models are complex and difficult to
solve for a number of reasons. The first being the large quantity
of moving parts included in the model. After the assembly load case
is solved, in which the applied interference between the bolt head
(see figure 7a) and one of the flange faces is resolved (stretching
the bolt, and compressing the flange material under the bolt head,
see figure 7b) a bolt pre-load is achieved; then a new time point
is initiated where the torque is applied. As stated earlier, the
free end of shaft 1 is locked in position, and a circumferential
load is applied to the other end. In order to resolve the torque
step, shaft 2 (the loaded shaft) must spin about the diametral snap
fit until the bolt hole barrel faces contact the individual bolt
shanks (see figures 7c, and 7d). Next, the loaded flange forces the
bolts to bend within the fixed flange until the shank contacts the
barrels of those holes. Finally, once these contacts are all
established, and somewhat settled, the system can start resolving
the load balance in a semi-linear manner. The contacts themselves
significantly increase the difficulty of solving the model. Contact
models are inherently non-linear since they essentially change
boundary conditions depending on the amount of deflection within
the system. Once contact is established, there is all of a sudden
another constraint on the model which didnt exist in the previous
solve iteration step. In this analysis, this complexity is
exacerbated by the sheer number of contacts that need to be
resolved (each bolt shank with 2 separate flange holes, each bolt
and nut washer face with the flange faces, and finally the flanges
to each other). The original intent was to use surface to surface
contact at all of the interfaces, but the model was too unstable,
and would not solve. For the sake of solving, the interface between
shaft 1 (the fixed shaft) and the bolt heads on that side were
modeled as bonded contacts. Bonded contact is created first by
creating node pairs (one from each part associated with the
interface) where said nodes are aligned within a specified
tolerance, then by coupling the deflections of the two nodes in
each set, in all degrees of freedom (DOFs). Its somewhat
unrealistic, since it does not allow any sliding between the two
surfaces. In order to keep this constraint from affecting the
results, the stress values and load summations were all pulled from
the free half (shaft #2 side) of the bolts.
Modeling contacts starts by identifying likely contact locations
between parts (see figure 6). During the solving algorithm, at each
step, the gap between the established locations is quantified. If
at any point this gap comes back as negative, contact has been
established, and the resultant forces between the two contacting
bodies are added to the equilibrium equations. The local contact
forces are driven by the deformation of the mesh required to
resolve the identified penetration. The new applied load
(essentially an added boundary condition) complicates equilibrium
resolution, especially because the level of penetration and
contacting area change with each solution iteration. Contact may be
modeled node to node, node to surface, or surface to surface. In
this case we are using surface to surface due to the relatively
large displacements of the bolts. In the case of surface to surface
contacts, 2D surface elements are modeled on each side of the areas
identified as possible contact locations. The Gauss points (located
along the element edges) are used to calculate the gap between the
two surfaces.
Figure 6: Gap and Penetration measurements of meshed bodies in
near-contact and contact, respectively, for surface to surface
contact (references 1 and 2).
Figure 7: a) Shows the initial step, where the bolt head
interferes with the flange b) shows the geometry after said initial
fit has been resolved; c) shows a bolt shank relative to a
corresponding hole in flange 2 prior to torque application, while
d) shows where the bolt shank ends up within the bolt hole after
the torque load has been resolved
2.3- MATERIALS
Inconel 718 (see table 1 for properties) was used for all parts,
since it is an extremely common material in moderate temperature,
torque loaded applications.
Table 1: Typical Mechanical Properties for Precip. Hardened Inco
718 (ref 4)
3. - RESULTS AND DISCUSSIONThe following section provides an
overview of the loaded flange behavior, detailed bolt stress
results and kts, and the method/results for the validation of the
FEA practices used.
3.1 - ASSEMBLY BEHAVIORThe deflection results are as expected,
with uniform hoop deflection (Uy of RSYS 5, in Ansys), about the
circumference, of the bolts and flanges in the baseline case
(displacement increases with radius since the angular displacement
of the flange is constant), and skewed displacement of the bolt and
flange section near the tolerance hole in the tolerance cases (see
figure 8). The head of the bolt in the toleranced hole is
deflecting significantly more than that of the rest of the bolts,
indicating that it is undergoing greater shear load (as was
intended). Alternatively, the flange section adjacent to the
tolerance hole is deflecting less, since its small hole contacts
the bolt shank prior to that of the other holes.
Figure 8a: Hoop deflection plot, in inches, of the free flange
and the free side bolt heads for the baseline case
Figure 8b: Hoop deflection plot, in inches, of the free flange
and the free side bolt heads for the hole +.002 case3.2 - BOLT
STRESS/LOADSThe resultant total torque loading (checked by querying
the reaction forces on the nodes of shaft 1 that were locked into
position) matches within 0.1% of the intended applied loading. The
baseline case also shows reasonable stress results (see table 2),
some stresses are a bit high, but this is likely due to the
relatively coarse mesh in the bolt head to shank fillet, and the
fact that it is not unlikely to yield bolts slightly from pre-load
alone.
Table 2: Test Case array, and corresponding stress/load
results.
This highlights one of the difficulties in modeling edge of
contact bearing stress with FEA; since the effective contact area
is extremely small when 2 round surfaces contact, elastic stresses
spike significantly (see bearing stress plot in figure 9a). In the
real world, a small area of each part would yield until the contact
area was large enough to reduce the stresses to elastic levels. A
better result may be obtained by using elastic-plastic material
models, and extremely fine mesh at the high stress locations.
However, this would significantly increase the complexity of the
models (due to added non-linearity of the material models, and the
increase in model size associated with mesh refinement). Figure 9a
shows that the max bearing stress occurs at the center of the bolt
shank, where the two flanges contact. The nodal reaction load
vectors, shown in figure 9c, align with the max bearing stress
results, showing the nodal reaction loads peaking at the axial
center of the bolt, and falling off towards the ends. The highest
tensile loads (illustrated in figures 9b and 9d) are seen to occur
in the bolt head to shank fillet, which is not surprising since
there is a significant stress concentration associated with a small
fillet, and the bending that the bolt head induces. For the purpose
of studying the stress trends, it is sufficient to look at the
relative stresses between cases in spite of the fact that they are
over yield, as long as the modeling method remains consistent.
Additionally, it should be understood that an elastically predicted
stress greater than the tensile strength for a given material, does
not necessarily indicate a likely failure. Elastically predicted
bending stresses appear artificially high when they reach the
plastic regime, because they continue to grow linearly, while in
reality the material would relax, and the actual stress would
increase much slower. The shear load summation on the highly loaded
bolt also provides a clear picture of how much additional load the
bolt is taking. This load alone could be used to assess how much
additional stress that bolt is experiencing via hand calculations.
It is important to note that the intent of this study is not to
create stress bounds for design, but instead to elaborate on the
relative effects of component stress and tolerance allowance. These
relationships are illustrated best in the trend curves shown in
figures 10a and 10b, which show the estimated stress increases
relative to varying GD&T error. The horizontal axes represent
deviation from nominal shape in inches, while the vertical axes
show the corresponding percent increase in bolt stress or load over
that of the nominal configuration. The origins of the plots
represent the nominal case, and 4 data points of increasing
dimensional deviation were used to build the trend lines. The trend
lines were created using 5th order polynomials in excel.
Figure 9a: Bearing Stress (Min Principal Stress, in psi) plot of
the bolt shank section 2 surface elements for the baseline stress
case.
Figure 9b: Max Running Tensile Stress (Max Principal Stress, in
psi) plot of the bolt shank section 2 surface elements for the
baseline stress case.
Figure 9c: Bearing stress (psi) plot with nodal reaction loads
on the bolt shank sections 1 and 2 surface elements for the
baseline stress case.
Figure 9d: Max Von-Mises stress (psi) plot of the bolt shank
section 2 surface elements for the baseline stress case.
Figure 10a: Trend curves for critical stresses/loads vs. hole
tolerance
Figure 10b: Trend curves for critical stresses vs. true position
tolerance
3.3 - VALIDATIONIn order to investigate the sensitivity of the
results to the mesh density, a side study was performed. The bolt
mesh was coarsened significantly (0.05 bolt shank elements in the
coarse case vs. 0.025 elements in the standard case, see figure 11)
in the baseline case and the 0.004 true position case, and the
models were run and post-processed as before.
Figure 11: Mesh comparison of the finely meshed bolt (used for
the primary models) and the coarsely meshed bolt (used for the mesh
sensitivity study models).
The deflection results of the coarse models were slightly
greater than that of the fine models. This was to be expected
because contact penetration sensitivity goes down with contact
surface mesh density, since theres significantly fewer Gauss
points. However, the deflection contours are very similar (see
figure 12). The stress results of the coarse models show a
relatively small change in the results relative to the large change
in mesh density (see table 3); only -10% to -14% delta stress
changes when element size doubled.
Figure 12: Hoop deflection plot, in inches, of the free flange
and the free side bolt heads for the baseline-coarse case.
Table 3: Results for the Mesh Density Sensitivity Testing.
4.- CONCLUSIONSThe trend curves (see figures 10a and 10b) show
that both hole size and true position tolerance have significant
influences on the bolt limiting stresses. The results shown are
worst case, and it should be understood that the large majority of
parts machined to a given tolerance will be well within that
tolerance, except in the case that the applied tolerance is far
tighter than the manufacturing process capability. It is even more
unlikely that the distribution of tolerances within a feature set
will end up as were modeled in this study (worst case as described
in the method section). However, this does yield insight into the
effect of tolerances on a given parts stresses, and it is
reasonable to assume that in a mass production situation, a certain
fraction of parts produced will be similar to those modeled. These
curves can be used to estimate the likely worst case stresses that
may be seen in a design with a given nominal stress and bolt hole
tolerance allowances.
Bolt hole tolerances are shown to have a slightly larger effect
on stress than true position (see relative kts between trend curves
in figure 10a vs 10b). Fortunately, hole size is likely more easily
controlled than hole position; as hole size is largely a function
of the cutting/drilling tool size, and position is a function of
many smaller components. Based on these two factors it could be
concluded that a relatively cheap and robust design may be achieved
by holding hole size to a relatively tight tolerance, and true
position to something less so.
Some factors that affect the results of a study like this are
material properties, flange geometry, and flange loading. A
material with a low Youngs modulus will be softer, and will likely
result in better load sharing between bolts with all else being
equal. Nominal bolt hole size relative to the bolt will likely have
a large affect on both bearing stress (based on Hertzian contact
theory) and tensile stresses in the bolt (since a larger bolt hole
will put more bending into the head of the bolt for a given bolt
size and pre-load). If the flange torque loading is not significant
enough to deflect the bolts until they share load, then the stress
spikes will be much higher (the entire torque load may be taken by
one bolt alone). It is therefore important to understand the
differences between an actual design, and a study model like the
one used here.REFERENCES1. Finite element analysis and modeling of
structure with bolted joints by Jeong Kim, Joo-Cheol Yoon, and
Beom-Soo Kang; Applied Mathematical Modelling 31 (2007) 8959112.
The Finite Element Method for Solid and Structural Mechanics: Sixth
edition, O.C. Zienkiewicz, R.L. Taylor; First published in 1967 by
McGraw-Hill, Fifth edition published by Butterworth-Heinemann 2000,
Reprinted 2002, Sixth edition 20053. Ansys Users Manual: Contact
Technology Guide, Ansys, Inc; Release 12.0 April 2009:
http://www1.ansys.com/customer/content/documentation/120/ans_ctec.pdf4.
Inconel Alloy 718 General Information provided by Special Metals
http://www.specialmetals.com/documents/Inconel%20alloy%20718.pdf
Hole MeshBolt Mesh
Assembly Cut-Away View
Fine Bolt Mesh
Coarse Bolt Mesh