ShoulderFillet Stresses Finesse
1/8/2009
by
Machine Design Staff Finite element analysis of shoulder fillets
reveals stress concentrations up to 24% higher than anticipated by
industry standard guide  and are located not where previously thought. Take a look at any nearby shaft or axle and
chances are you’ll see a shoulder fillet. Although it’s better than a
sharp corner, the shoulder fillet still has a stressconcentration
factor ( K_{t}) designers must take into account. For years,
designers relied on the formulas and graphs published in Peterson’s
Stress Concentration Factors. But what if they weren’t as accurate as we
thought?
Peterson’s solutions take the form of separate graphs for the three
loading modes: bending, tension, and torsion. They have been published
in mechanicaldesign textbooks and handbooks since their original
appearance in 1953.
The original stress concentration solution was a ratio of 2D notched plate, 3D notched shaft, and 2D filleted plate solutions.
Peterson’s original equations for shoulderfillet
stressconcentration factors (SCFs) were actually rough engineering
estimates. The analytical tools available when Peterson compiled his
stressconcentration factors forced him to base his calculations for
shoulder fillets on solutions for similar geometries, instead of
directly computing the stress state.
The curves
Peterson
generated for Kt
at various values
of r/d and D/d
were based on
an estimated
solution for
filleted shafts in
1997 and earlier
versions of
Peterson’s Stress
Concentration
Factors
Others had found formulas for the stress concentrations in
hemispherical notches (or grooves, g) in twodimensional
plates, K_{t,g,2D}, as well as for fillets ( f) in
twodimensional plates, K_{t,f,2D}. The hemisphericalnotch
solution had also been expanded to three dimensions to yield a SCF for
circumferential grooves in round bars, K_{t,g,3D}. But the
flatplate fillet solution had not been translated into a 3D round shaft
to get K_{t,f,3D}.
Peterson combined the three existing solutions to estimate the
unknown K_{t} values for a filleted bar in three dimensions.
(See graphic). His estimate relied on the ratio:
where each K_{t} term stands in for its corresponding equation. The result is cumbersome to calculate, so Peterson generated a series of curves
showing K_{t} as a function of fillet radius,
r, over the smaller shaft diameter, d.
Each curve represents a different ratio of the larger
shaft diameter, D, to d. The solution
is only applicable over limited ranges of r/d.
Peterson developed a separate set of curves for each
loading scenario: bending, tension, and torsion.
FEA Fidelity
Modern analytical techniques can provide a more accurate,
easiertohandle solution. In fact, the 2007 edition of Peterson’s
Stress Concentration Factors cites 1996 work, also from the
University of Tulsa, that used finiteelement analysis (FEA) to overcome
many of the inaccuracies of the previous method.
Researchers finely meshed thousands of shaft and fillet geometries like this one with r/d = 0.01 and D/d = 3. Applied bending, tension, and torsion stresses revealed the magnitude and location of the stress concentration.
With the advent of FEA, designers could use numerical equations to
calculate SCFs instead of relying on printed curves and tables. They
still needed a separate design equation for each loading mode, however.
Researchers strove to develop a simpler design equation that would cover
all loading regimes. With advances in computing speed and power, they
were able to mesh the filleted shaft model more finely and cover more
combinations of r, d, and D.
The fine mesh ensured the FEA would converge. It also produced a
morereliable location for the maximum stress in the fillet.
The analysis returned the maximum principal stress at each node as a
result of axial loading, F, bending moment, M, and
applied torsion, T. For the bending and tension cases, maximum
principal stress in the axial direction was denoted σ_{1}, and
the stress in the circumferential direction was σ_{2}.
Equivalent stress σ_{e} was computed by:
The SCFs, K_{t}, in terms of the maximum
principal stress can be calculated by:
where S_{nom} = nominal engineering stress on
the system. In bending this is given as:
In tension use:
FEA also permitted calculation of the multiaxial von
Mises equivalent stress for the same r/d and D/d values
Designers may also wish to compute the SCF in terms
of von Mises equivalent stress, K_{q}, since
fillet stresses are multiaxial in nature. To do so,
replace σ_{1} in the above equation with σ_{eq}.
In the torsional loading case, the analysis returned
τ_{max}. The resulting K_{t} is given
by:
where:
The von Mises SCF in torsion is related to the maximum
principal SCF in torsion by:
The resulting maximum SCFs reveal that Peterson’s
original equations can underestimate the stress
concentration at a shoulder fillet by as much as 24%.
The FEAderived solutions can be represented by
graphs like Peterson’s or distilled into a single design
equation:
The c_{x} in the equation above are numerical
constants chosen based on the loading regime to yield
either K_{t} or K_{q}. The constants are
listed in the accompanying tables.
FEA analysis also returned the angle Φ from the smaller
shaft up the fillet to the point of maximum stress
concentration for each type of loading
Knowing the location of the maximum stress is just as
critical as knowing its value. The information can be
invaluable for experimental stress analysis or failure
analysis. The angle from the smaller portion of the
shaft up the fillet to the point of maximum stress is
denoted as Φ.
Replace the ax terms with the
numerical constants listed in the accompanying table for
bending, tension, and torsion loading.
FEA results plotted curves similar to Peterson’s but with more accurate results over a wider range of r/d and D/d values.
Although the equations and graphs depicted here focus
on loading in bending, tension, or torsion alone,
combinedloading cases are the real world norm. To arrive
at the maximum equivalent stress when bending or tension
is accompanied by torsion, design engineers can figure
circumferential stress, σ_{2} using the
principle and equivalent stresses from the bending or
tension loading, σ_{1} and σ_{eq} .
Use the earlier equation for σ_{eq} to compute
the maximum equivalent stress of the system from σ_{1}
and the new σ_{2}.
Article edited by Leslie Gordon,
Sr Editor, Machine Design

Article reprinted by permission of Penton Media,
publisher of Machine Design 
