FEM_Course_Project_2

8/8/2015
I-BEAM
A Study in Stress Vs. Fastener Location
jitesh.bhogal@gmail.com

Table of Contents
Table OF Figures _______________________________________________ 1
INTRODUCTION _______________________________________________ 1
METHODS ___________________________________________________ 2
Tetrahedron Element___________________________________________ 2
Case 1: 2 Fasteners ____________________________________________ 3
Case 2: 3 Fasteners ____________________________________________ 3
Results ______________________________________________________ 4
Case 1 _____________________________________________________ 4
Case 1 Revision _______________________________________________ 4
Case 2:_____________________________________________________ 5
ANALYSIS____________________________________________________ 6
Along the Depth of the Beam______________________________________ 6
Along the Length of the Beam _____________________________________ 6
CONCLUSION _________________________________________________ 7
APPENDIX __________________________________________________ A0

Table OF Figures
Figure 1 : Loading Conditions ___________________________________________________1
Figure 2 : I-Beam Cross Section _________________________________________________ 2
Figure 3 : Tetrahedron Element _________________________________________________ 2
Figure 4 : Coarse Mesh Sample Stress Plot _________________________________________ 2
Figure 5 : Coarse Mesh Stress Peak ______________________________________________ 2
Figure 6 : Mesh Refinement At Critical Features _____________________________________ 3
Figure 7 : Mesh Refinement Max Stress Plot ________________________________________ 3
Figure 8 : Case 1 Start Location _________________________________________________ 3
Figure 9 : Case 1 End Location__________________________________________________ 3
Figure 10 : Case 2 Fasteners ___________________________________________________ 4
Figure 11 : Case 1 Results Plot __________________________________________________ 4
Figure 12 : Case 1 Adjusted Path_________________________________________________ 4
Figure 13: Case 1 Rev Car Load Plot _______________________________________________5
Figure 14 : Case 2 Rev Sample Stress Plot ___________________________________________5
Figure 15 : Case 2 Rev Stress Plot _________________________________________________5
Figure 16 : Stress Reduction Benefits _____________________________________________ 6
Table 1 : I-Beam Dimensions ___________________________________________________ 2
Table 2 : Case 1 Rev. Coarse Data________________________________________________ 4
Table 3 : Case 1 All Load Cases_________________________________________________ A0
Table 4 : Case 1 Rev Car Load Case ______________________________________________A1
Table 5 : Case 1 Rev Car Load Case Benefit Data At Y E.D 2.5 ___________________________ A2
Table 6 : Case 2 Rev All Load Cases Stress Data _____________________________________ A3
Table 7 : Stress Reduction Benefit Case 2 Rev at Y ED 2________________________________ A3

Figure 1 : Loading Conditions
I-BEAM
A Study in Stress Vs. Fastener Location
Consider the I-Beam (Figure 1). The Beam has a
length L = 10 ft and other relevant dimension
shown in Figure 2 and Table 1. For this analysis
we are using a standard industrial Grade-8
1 inch bolt as a reference for the holes sizes. The
Beam will be transversely loaded with limits
corresponding to loads experienced in a bridge
(Figure 1). The stress will be measured at the
fastener locations. For the purpose of this report
only the MAX stress experienced will be
relevant. This report will outline the relationship
between beam stress concentrations and bolt
placement, specifically edge distance. Loading
will be distributed to represent vehicles standing
idle over the beam. We will be using Aluminum
instead of steel for simplicity and to eliminate
the material composition of steel as a defining
factor. The purpose of this report is solely to
analyze the effects of moving fasteners in the
Maximum stress area under distributed loading
conditions.
INTRODUCTION
Bridges are an important part of interconnecting
roads across the world. I-Beams are widely used
in the construction of bridges for their ability to
resist shear forces. We will be analyzing the
effect of fastener location on stress around the
fastener holes on a clamped beam clamped at
the fastener holes. There will be 3 Load cases:
Car weight (approx. 5000 Lbs), Empty Truck
(18000 Lbs), and Max Loaded Truck (30000
Lbs). A single beam will never experience the
entire load from vehicles passing over the
bridge, as such for simplicity we will load the
beam with 50% (Gross over estimation of the
loading but will reduces computational
complexity) of the vehicle weight and constraint
it on the edges at fastener holes. The Isotropic
properties of aluminum are as follows:
E=1.015e+007psi Poisson Ratio = 0.346
Density=0.098lb_in3 Yield Strength =13778psi

2 
The beam cross-Section is shown in Figure 2 and
corresponding values can be found in Table 1.
This is a typical I-Beam as per ATSM A6
standard
METHODS
Using finite element analysis we capture stress
data for various loading cases and are able to
capture the behavior of our model over a wide
array of varying Edge Distances.
Tetrahedron Element
Below is the element which will be used to model
the I-Beam. The I-Beam will be broken down by
the Meshing tool into several Tet-Elements
(Figure 3). This is done to approximate results
within a certain confidence interval of the actual
solution. Reducing the element volume close to
ZERO will produce more accurate results
however; doing so will also significantly increase
compute time and restrict large or very fine
parameter sweeps.
Figure 3 : Tetrahedron Element
Figure 4 shows the Von Misses stress for a
coarse mesh at Initial Conditions under Max
loading. Here the Mesh is restricted to an
element length of 200 mm globally. As it can be
seen the coarse mesh yields in high spikes in
stress concentration around the Clamped
Fastener holes as shown in Figure 5. To yield
better results we have made local refinements to
the mesh scheme around the fastener holes and
the fillet Radius. Here we only compare the
effects of meshing on results from one case
which will determine the meshing scheme that
will be used for the remainder of the Report.
Figure 6 and 7 show the Stress using the refined
Mesh.
Figure 4 : Coarse Mesh Sample Stress Plot
Figure 5 : Coarse Mesh Stress Peak
D (in) Tw (in) Bf (in) Tf (in) R (in)
12 0.35 5 .35 .25
Table 1 : I-Beam Dimensions
Figure 2 : I-Beam Cross Section

Figure 6 : Mesh Refinement at Critical Features
Figure 7 : Mesh Refinement Max Stress Plot
The mesh refinement constitutes subjecting all
fastener holes and the fillet radiuses to a max
element length of 5mm. Note that Max Von
Misses stress contour is now much smoother
and contains no highly concentrated stress area
spikes; particularly around the fastener holes.
Further refinement did not yield more accurate
results and increased the computing time of the
solutions.
Edge Distance (ED) is defined as the distance
from the edge of a feature (Flat or Plane surface)
to the center of the faster hole. An ED of 1
corresponds to the center of the fastener hole
being 1 Diameter length away from the Edge of
the feature.
In this Report we will analyze the effects of
varying ED on stress concentration around the
Clamping holes on the I-Beam. The resulting
max stress will be plotted as a function of ED to
determine if a relationship exists between ED
and Max stress experienced. Further we will also
analyze the change in Max stress with each step
to determine if an optimal range exists beyond
which increasing ED yields negligible gains.
Case 1: 2 Fasteners
Using a single column of two fasteners we will
step the ED by constant increments. I
hypothesize that the max stress should reduce as
ED increases; however the beam should fail
under Truck loading conditions. Figures 8 & 9
show the start and stop locations of Case 1.
Figure 8 : Case 1 Start Location
Figure 9 : Case 1 End Location
Case 2: 3 Fasteners
Using a single row of three fasteners we will step
the ED by constant increments. The max value is
derived from the physical limitations of
installing fasteners; the bolt head size and the
tool size must be accounted for during
installation. As such fasteners closer than 1.5D
cannot physically be installed. I hypothesize that
the beam will be able to take a greater load than
in Case 1. Start location for case 2 is similar to
that shown in Figure 8 with the addition of a
third fastener hole in the center of the two
existing holes. Figure 10 shows the final position
of case 2.

4 
Figure 10 : Case 2 Fasteners
Results
Case 1
As expected the Beam experiences failure under
the truck load cases. However contrary to
expected results, the max stress increased with
increasing ED. Figure 11 shows the Max stress
for all the Load cases.
Figure 11 : Case 1 Results Plot
Case 1 Revision
To investigate this behavior a secondary set of
measurements were made using the following
changes. The initial fastener locations are the
same as Figure 8 however with each step along
the length of the beam we now measure the
stress and deflection stepwise along the depth of
the beam. In other words the fastener location
will NOT change diagonally each step but will be
varied in one direction at a time as shows in
Figure 12.
Figure 12 : Case 1 Adjusted Path
Table 2 shows the coarse results from this new
step method. NOTE all following measurements
will now be taken according to the revised
methodology. The Columns represent the ED in
the Y (along the Depth) direction while the Rows
represent the ED in the X (along the Length)
direction. The Stress values are in psi.
ED Y direction
X 2 2.5 3 3.5
2.5 5.74E+03 6.34E+03 7.50E+03 8.31E+03
3 5.62E+03 6.09E+03 6.84E+03 8.84E+03
3.5 5.21E+03 5.92E+03 6.52E+03 7.83E+03
4.2 4.94E+03 6.45E+03 6.80E+03 7.30E+03
Table 2 : Case 1 Rev. Coarse Data
Here it can be seen that the stress behaves as
predicted. Keeping the ED along the beam depth
constant and increasing the ED along the beam
length reduces the overall strain experienced by
the fastener locations.
More importantly the stress seems to Dip
around the highlighted area in the table before
rising again. To investigate this nonlinear
behavior a more in-depth analysis is performed
and detailed in Figure 13.
From the trend of the gathered data it can be
seen that there seem to be points of nonlinear
behavior. These data points are taken using the
Car loading case. The cause of these sudden
peaks is not clear and requires an in depth
analysis of the system architecture to determine
whether the peaks are true stress concentration
points or an artifact inherent within the
software. Such an analysis is out of the scope of
this report.

Overall it can be seen from figure 3 that moving
fastener holes towards the center of the web
increases the stress concentration at the fastener
location. Additionally the data seems to indicate
that stresses behave linearly in relation of Edge
Distance. This behavior is parallel to reducing
the effective length of the beam to increase its
ability to take on more loading. In industrial
design the beam dimensions would be altered
more significantly instead of increasing the Edge
Distance.
Case 2:
As can be seen from Figure 15 the third hole
alleviates some of the stress and allows for a
stronger fixture. Table 6 in the appendix shows
the data plotted in Figure 15. From the graph it
can be seen that an increase of ED from the
Flanges of the I-Beam continue to result in a net
increase of MAX stress experience around the
fastener locations. Similarly an increase in ED
along the length of the beam reduces the max
stress experience.
Note that the beam appears to behave non
linearly under Load Case 2 and Load Case 3. If
we were to continue accumulating data points
the limit of the minimum stress should
correspond to an equivalent of clamping the
beam at the midpoint length. Thus this provides
support to the assumption that beyond a certain
ED along the beam gains are negligible.
Figure 14 : Case 2 Rev Sample Stress Plot
Figure 15 : Case 2 Rev Stress Plot
Figure 13: Case 1 Rev Car Load Plot

6 
ANALYSIS
The primary purpose of this report is to
determine the effects of fastener location on an
I-Beam. Doing so allows us to quantify the
benefits of increasing ED. In Case 2 we saw an
exaggerated non-linear behavior of the beam
when experiencing failure under loading
conditions of Case 2 and Case 3.
Along the Depth of the Beam
As we learned, increasing ED in the vertical
direction, along the depth of the beam, results in
a significantly positive and seemingly linear
increase in stress concentrations. Therefore the
optimal position along the depth of the beam for
fastener locations is at the minimum required
edge distance from features which will not
restrict installation or assembly of the beam
onto the final structure. In the case of multiple
fasteners such as in Case 2 the designer should
maintain symmetry along the fastener axis for
optimum results.
Along the Length of the Beam
Increasing ED in along the length of the I-Beam
results in a somewhat linear reduction of the
stress. Although there is some interesting non-
linear behavior (stress spikes) which occurs at
varying locations this behavior is out of scope for
the purpose of the report. The overall linear
behavior is parallel to reducing the effective
length of the beam. From a design point of view
choosing an ED beyond 3d should trigger a re-
design of the beam and result in a shortening of
the length or other dimensional alterations to
accommodate for the stress concentrations.
Figure 16 and Figure 17 show graphs which show
the benefit (a reduction of MAX stress
concentrations on the fasteners) of increasing
the ED along the beam with two fasteners
(Figure 16) and three fasteners (Figure 17).
Tabulated data for these graphs can be found in
the Appendix. The important takeaway from
these graphs is that the largest reduction occurs
between 1ED and 2 ED after which the data
seems to hover around zero. The increases near
3.5D are most likely due to inherent factors of
the software which can be classified as outliers
temporarily. A more detailed parameter sweep
was not done due to time constraints.
Figure 16 : Stress Reduction Benefits Case 1
As expected the stress at 0.75D are significantly
higher than at higher ED’s. However with each
step the reduction in Max stress experienced is
reduced and seems to approach the zero. As such
when designing the connections for an I-Beam
or any similar beam which experiences
transverse loading the designer should choose
an ED between 1.5D-2.5D.
Figure 17 : Stress Reduction Benefit Case 2

CONCLUSION
It can be concluded from the benefits graph in
figures 16 and 17 that the stress reduction of the
beam with respect to ED along the length of the
beam is in-fact non-linear. This result is not
entirely surprising as at the mid-point of the
beam an increase in ED would effectively result
in a decrease. The reduction of max stress is
either linear with a minimum value at the mid-
point or parabolic with a minimum at the mid-
point.
Our data supports the latter which suggests that
gains from increasing ED approach as we
approach the mid-point of the beam. As such
fasteners should be placed within approximately
3D of the edge.
To this there are additional constraints such as
the connecting element. This element is usually
a flat plate or an L-Flange. Although outside the
scope of the report these connecting elements
also have an optimal range. For example: the L-
Flange. Using an ED greater than 3D would
require a second column of fasteners to be
placed at a smaller ED to prevent creating an
enormous moment arm on the L-Flange.
In conclusion this report analyzed the effects of
Edge Distance on stress concentrations at
fastener locations on an I-Beam. Through the
analysis presented it can be concluded that there
exists an optimal range of Edge Distances at
which fasteners should be placed. If the Max
stress experienced remains beyond the design
threshold the designer should re-evaluate the
beam design and redesign the Beam to
accommodate the stress concentrations.

8 
This Page is Intentionally Left Blank

APPENDIX
Load 15000 9000 2500
Ed(X=Y) Stress
(psi)
Deflection
(mm)
Stress
(psi)
Deflection
(mm)
Stress
(psi)
Deflection
(mm)
1 3.64E+04 1.74 2.18E+04 1.04 6.06E+03 0.29
1.1
1.2 3.68E+04 1.76 2.21E+04 1.06 6.14E+03 0.294
1.3
1.4 3.71E+04 1.81 2.23E+04 1.09 6.19E+03 0.302
1.5
1.6 3.71E+04 1.84 2.22E+04 1.1 6.18E+03 0.306
1.7
1.8 3.96E+04 1.88 2.40E+04 1.13 6.67E+03 0.313
1.9
2 3.75E+04 1.92 2.25E+04 1.15 6.25E+03 0.32
2.1
2.2 3.83E+04 1.96 2.30E+04 1.18 6.38E+03 0.327
2.3
2.4 4.25E+04 2.01 2.55E+04 1.21 7.09E+03 0.335
2.5
2.6 3.86E+04 2.05 2.32E+04 1.23 6.43E+03 0.342
2.7
2.8 4.14E+04 2.12 2.49E+04 1.27 6.91E+03 3.53
2.9
3 4.10E+04 2.18 2.46E+04 1.31 6.84E+03 0.363
3.1
3.2 4.23E+04 2.23 2.54E+04 1.34 7.04E+03 0.372
3.3
3.4 4.40E+04 2.31 2.64E+04 1.39 7.33E+03 0.385
3.5
3.6 4.28E+04 2.39 2.57E+04 1.43 7.13E+03 0.398
3.7
3.8 4.61E+04 2.48 2.77E+04 1.49 7.68E+03 0.414
3.9
4 5.04E+04 2.59 3.02E+04 1.55 8.40E+03 0.431
4.1
4.2 4.70E+04 2.66 2.82E+04 1.6 7.84E+03 0.444
4.3 4.89E+04 2.74 2.93E+04 1.64 8.15E+03 0.456
Table 3 : Case 1 All Load Cases

Y direction
CAR LC 2 2.25 2.5 2.75 3 3.25 3.5 3.75
x Dir Stress
(psi)
Stress
(psi)
Stress
(psi)
Stress
(psi)
Stress
(psi)
Stress
(psi)
Stress
(psi)
Stress
(psi)
0.75 8.43E+03
1 8.42E+03
1.25 8.24E+03
1.5 7.70E+03
1.75 7.06E+03
2 6.81E+03
2.25 6.53E+03
2.5 5.74E+03 6.34E+03 7.50E+03 8.31E+03
2.75 5.65E+03 6.14E+03 6.16E+03 6.70E+03 7.47E+03 7.57E+03 8.14E+03 8.62E+03
3 5.62E+03 6.09E+03 6.84E+03 8.84E+03
3.25 6.07E+03 6.70E+03 7.96E+03
3.5 5.21E+03 5.92E+03 6.52E+03 7.83E+03
3.65 6.01E+03 6.50E+03
3.75 5.96E+03 6.51E+03
3.85 5.88E+03 6.66E+03
4 5.70E+03 6.31E+03 7.50E+03
4.1 5.87E+03 6.70E+03
4.15 5.94E+03 6.22E+03
4.2 4.94E+03 6.45E+03 6.80E+03 7.30E+03
4.5 5.15E+03 5.42E+03 6.10E+03
5 5.11E+03 5.36E+03 5.94E+03 6.66E+03
5.5 5.39E+03
6 5.04E+03 5.28E+03 5.88E+03 6.32E+03
7 4.61E+03 5.19E+03 5.71E+03 6.26E+03
8 4.48E+03 4.99E+03 5.42E+03 5.91E+03
Table 4 : Case 1 Rev Car Load Case

2 
X start Delta Stress (psi) % Reduction
0.75 1.20E+01 0.19%
1 1.75E+02 2.76%
1.25 5.39E+02 8.50%
1.5 6.38E+02 10.06%
1.75 2.53E+02 3.99%
2 2.77E+02 4.37%
2.25 1.91E+02 3.01%
2.5 1.81E+02 2.85%
2.75 7.30E+01 1.15%
3 1.40E+01 0.22%
3.25 1.50E+02 2.37%
3.5 -8.70E+01 -1.37%
3.65 5.30E+01 0.84%
3.75 8.00E+01 1.26%
3.85 1.83E+02 2.89%
4 -1.77E+02 -2.79%
4.1 -7.00E+01 -1.10%
Table 5 : Case 1 Rev Car Load Case Benefit Data At Y E.D 2.5

3 holes Loaded Truck Truck Car
EdY EdY EdY
Stress (psi)
Edx 2 2.5 3 2 2.5 3 2 2.5 3
1 4.29E+04 4.86E+04 5.20E+04 2.57E+04 2.92E+04 3.12E+04 7.15E+03 8.10E+03 8.66E+03
1.5
2 3.72E+04 3.99E+04 4.74E+04 2.23E+04 2.39E+04 2.84E+04 6.20E+03 6.65E+03 7.90E+03
2.5
3 3.41E+04 3.70E+04 4.07E+04 2.05E+04 2.22E+04 2.44E+04 5.69E+03 6.17E+03 6.78E+03
3.5
4 3.10E+04 3.43E+04 3.78E+04 1.86E+04 2.06E+04 2.27E+04 5.17E+03 5.71E+03 6.30E+03
Table 6 : Case 2 Rev All Load Cases Stress Data
Edx car Truck loaded
Tuck
1 20.41% 13.18% 13.20%
1.5 6.69% 7.19% 7.18%
2 6.41% 7.27% 7.28%
Table 7 : Stress Reduction Benefit Case 2 Rev at Y ED 2

FEM_Course_Project_2

Recomendados

Recomendados

Más contenido relacionado

La actualidad más candente

La actualidad más candente (17)

Destacado

Destacado (8)

Similar a FEM_Course_Project_2

Similar a FEM_Course_Project_2 (20)

FEM_Course_Project_2