2007-01-0258
Dynamic Load and Stress Analysis of a Crankshaft
Farzin H. Montazersadgh and Ali Fatemi
The University of Toledo
Copyright © 2007 SAE International
in which FEM and BEM (Boundary Element Method)
were used. Obtained stresses were verified by
experimental results on a 1.9 liter turbocharged diesel
engine with Ricardo type combustion chamber
configuration. The crankshaft durability assessment tool
used in this study was developed by RENAULT. The
software used took into account torsional vibrations and
internal centrifugal loads. Fatigue life predictions were
made using the multiaxial Dang Van criterion. The
procedure developed is such it that could be used for
conceptual design and geometry optimization of
crankshaft.
ABSTRACT
In this study a dynamic simulation was conducted on a
crankshaft from a single cylinder four stroke engine.
Finite element analysis was performed to obtain the
variation of stress magnitude at critical locations. The
pressure-volume diagram was used to calculate the load
boundary condition in dynamic simulation model, and
other simulation inputs were taken from the engine
specification chart. The dynamic analysis was done
analytically and was verified by simulation in ADAMS
which resulted in the load spectrum applied to crank pin
bearing. This load was applied to the FE model in
ABAQUS, and boundary conditions were applied
according to the engine mounting conditions. The
analysis was done for different engine speeds and as a
result critical engine speed and critical region on the
crankshaft were obtained. Stress variation over the
engine cycle and the effect of torsional load in the
analysis were investigated. Results from FE analysis
were verified by strain gages attached to several
locations on the crankshaft. Results achieved from
aforementioned analysis can be used in fatigue life
calculation and optimization of this component.
Guagliano et al. [2] conducted a study on a marine
diesel engine crankshaft, in which two different FE
models were investigated. Due to memory limitations in
meshing a three dimensional model was difficult and
costly. Therefore, they used a bi-dimensional model to
obtain the stress concentration factor which resulted in
an accuracy of less than 6.9 percent error for a centered
load and 8.6 percent error for an eccentric load. This
numerical model was satisfactory since it was very fast
and had good agreement with experimental results.
Payer et al. [3] developed a two-step technique to
perform nonlinear transient analysis of crankshafts
combining a beam-mass model and a solid element
model. Using FEA, two major steps were used to
calculate the transient stress behavior of the crankshaft;
the first step calculated time dependent deformations by
a step-by-step integration using the newmark-betamethod. Using a rotating beam-mass-model of the
crankshaft, a time dependent nonlinear oil film model
and a model of the main bearing wall structure, the
mass, damping and stiffness matrices were built at each
time step and the equation system was solved by an
iterative method. In the second step those transient
deformations were enforced to a solid-element-model of
the crankshaft to determine its time dependent stress
behavior. The major advantage of using the two steps
was reduction of CPU time for calculations. This is
because the number of degrees of freedom for
performing step one was low and therefore enabled an
efficient solution. Furthermore, the stiffness matrix of the
INTRODUCTION
Crankshaft is a large component with a complex
geometry in the engine, which converts the reciprocating
displacement of the piston to a rotary motion with a four
link mechanism. This study was conducted on a single
cylinder four stroke cycle engine.
Rotation output of an engine is a practical and applicable
input to other devices since the linear displacement of
an engine is not a smooth output as the displacement is
caused by the combustion of gas in the combustion
chamber. A crankshaft changes these sudden
displacements to a smooth rotary output which is the
input to many devices such as generators, pumps,
compressors.
A detailed procedure of obtaining stresses in the fillet
area of a crankshaft was introduced by Henry et al. [1],
1
materials and manufacturing process technologies were
compared and durability assessment procedure, bench
testing, and experimental techniques used for
crankshafts were discussed. Their review also included
cost analysis and potential geometry optimizations of
crankshaft.
solid element model for step two needed only to be built
up once.
In order to estimate fatigue life of crankshafts, Prakash
et al. [4] performed stress and fatigue analysis on three
example parts belonging to three different classes of
engines. The classical method of crankshaft stress
analysis (by representing crankshaft as a series of rigid
disks separated by stiff weightless shafts) and an FEMbased approach using ANSYS code were employed to
obtain natural frequencies, critical modes and speeds,
and stress amplitudes in the critical modes. A fatigue
analysis was also performed and the effect of variation
of fatigue properties of the material on failure of the parts
was investigated. This was achieved by increasing each
strain-life parameter (σf′, εf′, b and c) by 10% and
estimating life. It was shown that strength and ductility
exponents have a large impact on life, e.g. a 10%
increase of b leads to 93% decrease in estimated life.
In this paper, first dynamic load analysis of the
crankshaft investigated in this study is presented. This
includes a discussion of the loading sources, as well as
importance of torsion load produced relative to bending
load. FE modeling of the crankshaft is presented next,
including a discussion of static versus dynamic load
analysis, as well as the boundary conditions used.
Results from the FE model are then presented which
includes identification of the critically stressed location,
variation of stresses over an entire cycle, and a
discussion of the effects of engine speed as well as
torsional load on stresses. A comparison of FEA
stresses with those obtained from strain gages of a
crankshaft in a bench test is also presented. Finally,
conclusions are drawn based on the analysis preformed
and results presented.
A geometrically restricted model of a light automotive
crankshaft was studied by Borges et al. [5]. The
geometry of the crankshaft was geometrically restricted
due to limitations in the computer resources available to
the authors. The FEM analysis was performed in
ANSYS software and a three dimensional model made
of Photoelastic material with the same boundary
conditions was used to verify the results. This study was
based on static load analysis and investigated loading at
a specific crank angle. The FE model results showed
uniform stress distribution over the crank, and the only
region with high stress concentration was the fillet
between the crank-pin bearing and the crank web.
LOAD ANALYSIS
The crankshaft investigated in this study is shown in
Figure 1 and belongs to an engine with the configuration
shown in Table 1 and piston pressure versus crankshaft
angle shown in Figure 2. Although the pressure plot
changes for different engine speeds, the maximum
pressure which is much of our concern does not change
and the same graph could be used for different speeds
[9]. The geometries of the crankshaft and connecting rod
from the same engine were measured with the accuracy
of 0.0025 mm (0.0001 in) and were drawn in the I-DEAS
software, which provided the solid properties of the
connecting rod such as moment of inertia and center of
gravity (CG). These data were used in ADAMS software
to simulate the slider-crank mechanism. The dynamic
analysis resulted in angular velocity and angular
acceleration of the connecting rod and forces between
the crankshaft and the connecting rod.
Shenoy and Fatemi [6] conducted dynamic analysis of
loads and stresses in the connecting rod component,
which is in contact with the crankshaft. Dynamic analysis
of the connecting rod is similar to dynamics of the
crankshaft, since these components form a slide-crank
mechanism and the connecting rod motion applies
dynamic load on the crank-pin bearing. Their analysis
was compared with commonly used static FEA and
considerable differences were obtained between the two
sets of analysis. Shenoy and Fatemi [7] optimized the
connecting rod considering dynamic service load on the
component. It was shown that dynamic analysis is the
proper basis for fatigue performance calculation and
optimization of dynamically loaded components. Since a
crankshaft experiences similar loading conditions as a
connecting rod, optimization potentials of a crankshaft
could also be obtained by performing an analytical
dynamic analysis of the component.
Fz
Fx Fy
A literature survey by Zoroufi and Fatemi [8] focused on
durability performance evaluation and comparisons of
forged steel and cast iron crankshafts. In this study
operating conditions of crankshaft and various failure
sources were reviewed, and effect of parameters such
as residual stress and manufacturing procedure on the
fatigue performance of crankshaft were discussed. In
addition, durability performance of common crankshaft
Figure 1: Crankshaft geometry and bending (Fx),
torsional (Fy), and longitudinal (Fz) force directions
2
100
40000
Crankshaft radius
Piston Diameter
Mass of the connecting rod
Mass of the piston assembly
Connecting rod length
Izz of connecting rod about the
center of gravity
Distance of C.G. of connecting
rod from crank end center
Maximum gas pressure
Angular Velocity (rad/s)
80
37 mm
89 mm
0.283 kg
0.417 kg
120.78 mm
35 Bar
Cylinder Pressure (bar)
180
15
10
5
0
500
0
720-10000
540
-40
-20000
-60
-30000
Acceleration
-40000
Forces applied to the crankshaft cause bending and
torsion. Figure 1 demonstrates the positive directions
and local axis on the contact surface with the connecting
rod. Figure 4 shows the variations of bending and
torsion loads and the magnitude of the total force
applied to the crankshaft as a function of crankshaft
angle for the engine speed of 3600 rpm. The maximum
load which happens at 355 degrees is where
combustion takes place, at this moment the acting force
on the crankshaft is just bending load since the direction
of the force is exactly toward the center of the crank
radius (i.e. Fy = 0 in Figure 1). This maximum load
situation happens in all types of engines with a slight
difference in the crank angle. In addition, most analysis
done on engines with more cylinders (e.g. 4, 6, and 8) is
on a portion of the crankshaft that consists of two main
journal bearings, two crank webs, and a connecting rod
pin journal. Therefore, analysis done for this single
cylinder engine can be extended to larger engines.
20
400
360
Figure 3: Variation of angular velocity and angular
acceleration of the connecting rod over one complete
engine cycle at a crankshaft speed of 2800 rpm
25
300
-20 0
28.6 mm
30
200
0
Crankshaft Angle (Deg)
35
100
10000
20
-100
40
0
20000
40
-80
0.663×10-3 kg-m2
30000
Velocity
60
Angular Acceleration (rad/s^2)
Table 1: Configuration of the engine to which the
crankshaft belongs
600
700
Crankshaft Angle (Deg)
Figure 2: Piston pressure versus crankshaft angle
diagram used to calculate forces at the connecting rod
ends
There are two different load sources acting on the
crankshaft. Inertia of rotating components (e.g.
connecting rod) applies forces to the crankshaft and this
force increases with the increase of engine speed. This
force is directly related to the rotating speed and
acceleration of rotating components. Variation of angular
acceleration and angular velocity of the connecting rod
for the engine speed of 3600 rpm is shown in Figure 3.
The second load source is the force applied to the
crankshaft due to gas combustion in the cylinder. The
slider-crank mechanism transports the pressure applied
to the upper part of the slider to the joint between
crankshaft and connecting rod. This transmitted load
depends on the dimensions of the mechanism.
20
15
Force (kN)
10
Total
5
0
0
100
200
300
400
500
600
700
-5
Bending
Torsional
-10
Crankshaft Angle (Deg)
Figure 4: Bending, torsional, and the resultant force at
the connecting rod bearing at the engine speed of 3600
rpm
3
at the fillets where the stresses are higher due to stress
concentrations. As a crankshaft is designed for very long
life, stresses must be in the linear elastic range of the
material. Therefore, all carried analysis are based on the
linear properties of the crankshaft material. The meshed
crankshaft with 122,441 elements is shown in Figure 6.
In many studies the torsional load is neglected for the
load analysis of the crankshaft, and this is because
torsional load is less than 10 percent of the bending load
[10]. In this specific engine with its dynamic loading, it is
shown in the next sections that torsional load has no
effect on the range of von Mises stress at the critical
location. The main reason of torsional load not having
much effect on the stress range is that the maxima of
bending and torsional loading happen at different times
(see Figure 4). In addition, when the peak of the bending
load takes place the magnitude of torsional load is zero.
The dynamic loading of the crankshaft is complicated
because the magnitude and direction of the load
changes during a cycle. There are two ways to find the
stresses in dynamic loading. One method is running the
FE model as many times as possible with the direction
and magnitude of the dynamic force. An alternative and
simpler way of obtaining stress components is
superposition of static loading. The main idea of
superposition is finding the basic loading positions, then
applying unit load on each position according to dynamic
loading of the crankshaft, and scaling and combining the
stresses from each unit load. In this study both methods
were used with 13 points over 720 degrees of crankshaft
angle. The results from 6 different locations on the
crankshaft showed identical stress components from the
two methods.
Figure 5 compares the magnitude of maximum torsional
and bending loads at different engine speeds. As can be
seen in this figure, the maximum of total load magnitude,
which is equal to the maximum of bending load
decreases as the engine speed increases. The reason
for this situation refers to the load sources that exist in
the engine at 355 degree crank angle. At this crank
angle these two forces act in opposite directions. The
force caused by combustion which is greater than the
inertia load does not change at different engine speeds
since the same pressure versus crankshaft angle is
used for all engine speeds. The load caused by inertia
increases in magnitude as the engine speed increases.
Therefore, as the engine speed increases, a larger
magnitude of inertia force is deducted from the
combustion load, resulting in a decrease of the total load
magnitude.
Max Bending
Max Torsion
Range of Bending
Range of Torsion
25
Force Magnitude (kN)
20
15
Figure 6: FEA model of the crankshaft with fine mesh in
fillet areas
10
It should be noted that the analysis is based on dynamic
loading, though each finite element analysis step is done
in static equilibrium. The main advantage of this kind of
analysis is more accurate estimation of the maximum
and minimum loads. Design and analyzes of the
crankshaft based on static loading can lead to very
conservative results. In addition, as was shown in this
section, the minimum load could be achieved only if the
analysis of loading is carried out during the entire cycle.
The minimum value of von Mises stress which is
obtained at the minimum load is needed for the stress
range calculation and considering it zero will lead to
smaller values for the stress range.
5
0
2000
2800
3600
Engine Speed (RPM)
Figure 5: Comparison of maximum and range of bending
and torsional loads at different engine speeds
FE MODELING OF THE CRANKSHAFT
As the dynamic loading condition is analyzed, only two
main loading conditions are applied to the surface of the
crankpin bearing. These two loads are perpendicular to
each other and their directions are shown in Figure 1 as
The FE model of the crankshaft geometry has about 105
quadratic tetrahedral elements, with the global element
length of 5.08 mm and local element length of 0.762 mm
4
element. Figure 10 shows the maximum stress, mean
stress, and stress range at the engine speed of 2000
rpm at different locations. It can be seen that element
number 2 not only has the maximum von Mises stress,
but it also carries the largest stress range and mean
stress among other locations. This is important in fatigue
analysis since the range and mean stress have more
influence than the maximum stress. This is another
reason for why having the stress history of critical
elements are more useful than static analysis of the
crankshaft.
Fx and Fy. Since the contact surface between connecting
rod and crankpin bearing does not carry tension, Fx and
Fy can also act in the opposite direction to those shown
in Figure 1. Any loading condition during the service life
of the crankshaft can be obtained by scaling and
combining the magnitude and direction of these two
loads.
Boundary conditions in the FE model were based on the
engine configuration. The mounting of this specific
crankshaft is on two different bearings which results in
different constraints in the boundary conditions. One
side of the crankshaft is fixed to the engine block by a
ball bearing and the other side is rolling over a journal
bearing. When under load, only 180 degrees of the
bearing surfaces facing the load direction constraint the
motion of the crankshaft. Therefore, a fixed semicircular
surface as wide as the ball bearing width was used to
model that section. This indicates that the surface can
not move in either direction and can not rotate. The
other side was modeled as a fixed thin semicircular ring
which only holds the crankshaft centerline in its original
position and acts as a pivot joint. In other words, the
journal bearing is modeled in a way that allows the
crankshaft to rotate about axis 1 as well as slide in
direction 3 as occurs in a journal bearing. These defined
boundary conditions are shown in Figure 7. Boundary
conditions rotate with the direction of the load applied.
a
5
7
A
d
c
1
6
A-A
b
A
3
2
Figure 8: Locations on the crankshaft where the stress
variation was traced over one complete cycle of the
engine, and locations where strain gages were mounted
1
Applied load; constant
pressure over 120°
2
3
4
5
6
200
2
3
1
Fixed ring in
directions 1 & 2
o
over 180
Stress Magnitude (MPa)
150
Fixed surface
in all degrees
of freedom
o
over 180
100
50
0
Figure 7: Boundary conditions used in the FEA model
0
180
360
540
720
-50
RESULTS AND DISCUSSION OF STRESS
ANALYSIS
Crankshaft Angle (Deg)
Some locations on the geometry were considered for
depicting the stress history. These locations were
selected according to the results of FE analysis, and as
expected, all the selected elements are located on
different parts of the fillet areas due to the high stress
concentrations at these locations. Selected locations are
labeled in Figure 8 and the von Mises stresses with sign
for these elements are plotted in Figure 9. The critical
loading situation is at the crank angle of 355 where the
combustion exerts a large impact on the piston. At this
time all stresses are at their highest level during stress
time history in a cycle. As can be seen, location number
2 experiences the highest stress at this moment.
Therefore, element number 2 was selected as the critical
Figure 9: von Mises stress history (considering sign of
principal stress) at different locations at the engine
speed of 2000 rpm
5
Maximum
Minimum
Range
remains the same with and without considering torsional
load. This is due to the location of the critical point which
is not influenced by torsion since it is located on the
crankpin bearing. Other locations such as 1, 6, and 7 in
Figure 8 experience the torsional load. Figure 12 shows
changes in minimum, maximum, mean, and range of
von Mises stress at location 7 with considering torsion
and without considering it during service life at two
different engine speeds. It can be seen that the
minimum von Mises stress does not change since the
minimum happens at a time when the torsional load is
zero. The effect of torsion is about 16 percent increase
in the stress range at this location.
Mean
250
Stress Magnitude (MPa)
200
150
100
50
Total
Total
Total
Total
0
1
2
3
4
5
6
min stress
max stress
stress range
mean stress
Min stress without Torsion
Max stress without Torsion
Stress range without Torsion
Mean stress without torsion
100
-50
Location Number
80
60
Stress Magnitude (MPa)
Figure 10: Comparison of maximum, minimum, mean,
and range of stress at the engine speed of 2000 rpm at
different locations on the crankshaft
Figure 11 shows the effect of engine speed on minimum,
maximum, mean and range of stress. This figure
indicates the higher the engine speed, the lower the von
Mises stress. It should, however, be noted that there are
many other factors regarding service life of an engine.
Other important factors when the engine speed
increases are wear and lubrication. As these issues
were not of concern in this study, further discussion is
avoided.
Min
Max
Mean
von Mises Stress Magnitude (MPa)
3600
Engine Speed (RPM)
Figure 12: Effect of considering torsion in stresses at
location 7 at different engine speeds
Range
Stress results obtained from the FE model were verified
by experimental component test. Strain gages were
mounted at four locations on the crankpin bearing.
These locations are labeled as a, b, c, and d in Figure 8.
The FE model boundary conditions were changed
according to the fixture of the test assembly. The fixture
constraints the motion of the shaft on the left side of the
crankshaft in Figure 8 and a load is applied on the right
side of the crankshaft with a moment arm of 44 cm.
Therefore, the crankshaft is experiencing bending as a
cantilever beam. Applying load in the direction of axis 2
in Figure 7 will result in stresses at locations a and b,
and applying load in the direction of axis 1 in the same
figure will result in stresses at locations c and d.
Analytical calculations based on pure bending equation,
Mc/I, show the magnitude of stresses to be the same
and equal to 72 MPa at these locations, for a 890 N
load. The values obtained from experiments are
tabulated in Table 2. FEA results are also shown and
compared with experimental results in this table. As can
be seen, differences between FEA and strain gage
results are less than 7 percent for different loading
100
50
0
3000
2000
-20
-80
150
2500
0
-60
200
2000
20
-40
250
-50
1500
40
3500
4000
Engine Speed (RPM)
Figure 11: Variation of minimum stress, maximum
stress, mean stress, and stress range at location 2 on
the crankshaft as a function of engine speed
The effect of torsional load was discussed in the load
analysis section, and was pointed out that it has no
effect on the stress range of the critical location. The von
Mises stress at location number 2 shown in Figure 9
6
conditions. This is an indication of the accuracy of the
FE model used in this study.
200
von Mises Stress Magnitude (MPa)
2
Table 2: Comparison of stress results from FEA and
strain gages located at positions shown in Figure 8
Load
(N)
-890
890
FEA
(MPa)
Location a
EXP
%
(MPa) Difference
-61.6
-59.3
61.5
65.5
3.8%
6.5%
FEA
(MPa)
86.9
Location b
EXP
%
(MPa)
Difference
81.4
6.4%
-86.7
-90.3
4.2%
FEA
(MPa)
Location d
EXP
%
(MPa)
Difference
FEA
(MPa)
Location c
EXP
%
(MPa) Difference
-890
-76.4
-71.7
6.1%
75.5
71.7
5.0%
890
76.3
75.8
0.5%
-75.6
-76.5
1.3%
Load
(N)
150
100
50
6
0
0
1
2
3
4
5
6
7
8
9
10
10
11
12
-50
Tim e
Figure 13: Rain flow count of the von Mises stress with
consideration of sign at location 2 at engine speed of
2000 rpm
Comparison of stresses at locations c and d resulting
from loading in direction 1 in Figure 7 show symmetric
stress values from FEA, experiment, and analytical
method. The results from these three methods are close
to each other. However, stresses obtained form FEA
results and experiment show different stresses (i.e. nonsymmetric) at locations a and b, resulting from loading in
direction 2 in Figure 7. On the other hand, stresses
calculated from the analytical method are symmetric at
these two locations (+/-72 MPa) and different from the
obtained values from FEA and experiment. Therefore,
the use of FE model in the analysis is necessary due to
geometry complexity.
CONCLUSIONS
The following conclusions could be drawn from this
study:
1. Dynamic loading analysis of the crankshaft results in
more realistic stresses whereas static analysis
provides an overestimate results. Accurate stresses
are critical input to fatigue analysis and optimization
of the crankshaft.
2. There are two different load sources in an engine;
inertia and combustion. These two load source
cause both bending and torsional load on the
crankshaft.
3. The maximum load occurs at the crank angle of 355
degrees for this specific engine. At this angle only
bending load is applied to the crankshaft.
4. Considering torsional load in the overall dynamic
loading conditions has no effect on von Mises stress
at the critically stressed location. The effect of
torsion on the stress range is also relatively small at
other locations undergoing torsional load. Therefore,
the crankshaft analysis could be simplified to
applying only bending load.
5. Critical locations on the crankshaft geometry are all
located on the fillet areas because of high stress
gradients in these locations which result in high
stress concentration factors.
6. Superposition of FEM analysis results from two
perpendicular loads is an efficient and simple
method of achieving stresses at different loading
conditions according to forces applied to the
crankshaft in dynamic analysis.
7. Experimental and FEA results showed close
agreement, within 7% difference. These results
indicate non-symmetric bending stresses on the
crankpin bearing, whereas using analytical method
predicts bending stresses to be symmetric at this
location. The lack of symmetry is a geometry
deformation effect, indicating the need for FEA
Stress results from FE and analytical results have similar
symmetric values for stresses on the main bearing away
from fillet areas. FE results show different stress values
on the fillet area of main bearing. The reason is the
eccentric cylinders geometry which will result in changes
in Kt value around the fillet area.
Load variation over a cycle results in variation of stress.
For proper calculations of fatigue damage in the
component there is a need for a cycle counting method
over the stress history. Using the rainflow counting
method [11] on the critical stress history plot (i.e.
location 2 in Figure 9) shows that in an entire cycle only
one peak is important and can cause fatigue damage in
the component. The result of the rain count flow over the
stress-time history of location 2 at the engine speed of
2000 rpm is shown in Figure 13. It is shown in this figure
that in the stress history of the critical location only one
cycle of loading is important and the other minor cycles
have low stress amplitudes.
7
modeling due to the relatively complex geometry of
the crankshaft.
8. Using the rainflow cycle counting method on the
critical stress history plot shows that in an entire
cycle only one peak is important and can cause
fatigue damage in the component.
6. Shenoy, P. S. and Fatemi, A., 2006, “Dynamic
analysis of loads and stresses in connecting rods,”
IMechE, Journal of Mechanical Engineering
Science, Vol. 220, No. 5, pp. 615-624
7. Shenoy, P. S. and Fatemi, A., "Connecting Rod
Optimization for Weight and Cost Reduction", SAE
Paper No. 2005-01-0987, SAE 2005 Transactions:
Journal of Materials and Manufacturing
8. Zoroufi, M. and Fatemi, A., "A Literature Review on
Durability Evaluation of Crankshafts Including
Comparisons
of
Competing
Manufacturing
Processes and Cost Analysis", 26th Forging Industry
Technical Conference, Chicago, IL, November 2005
9. Fergusen, C. R., 1986, “Internal Combustion
Engines, Applied Thermo Science,” John Wiley and
Sons, New York, NY, USA
10. Jensen, E. J., 1970, “Crankshaft strength through
laboratory testing,” SAE Technical Paper No.
700526, Society of Automotive Engineers
11. Stephens, R. I., Fatemi, A., Stephens, R. R., and
Fuchs, H. O., 2001, “Metal Fatigue in Engineering,”
2nd edition, John Wiley and Sons, New York, NY,
USA
REFERENCES
1. Henry, J., Topolsky, J., and Abramczuk, M., 1992,
“Crankshaft Durability Prediction – A New 3-D
Approach,” SAE Technical Paper No. 920087,
Society of Automotive Engineers
2. Guagliano, M., Terranova, A., and Vergani, L., 1993,
“Theoretical and Experimental Study of the Stress
Concentration Factor in Diesel Engine Crankshafts,”
Journal of Mechanical Design, Vol. 115, pp. 47-52
3. Payar, E., Kainz, A., and Fiedler, G. A., 1995,
“Fatigue Analysis of Crankshafts Using Nonlinear
Transient Simulation Techniques,” SAE Technical
Paper No. 950709, Society of Automotive Engineers
4. Prakash, V., Aprameyan, K., and Shrinivasa, U.,
1998, “An FEM Based Approach to Crankshaft
Dynamics and Life Estimation,” SAE Technical
Paper No. 980565, Society of Automotive Engineers
5. Borges, A. C. C., Oliveira, L. C., and Neto, P. S.,
2002, “Stress Distribution in a Crankshaft Crank
Using a Geometrucally Restricted Finite Element
Model”, SAE Technical Paper No. 2002-01-2183,
Society of Automotive Engineers
8