Shoulder-Fillet Stresses Finesse
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 stress-concentration
factor ( Kt) 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
Peterson’s solutions take the form of separate graphs for the three
loading modes: bending, tension, and torsion. They have been published
in mechanical-design 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 shoulder-fillet
stress-concentration factors (SCFs) were actually rough engineering
estimates. The analytical tools available when Peterson compiled his
stress-concentration factors forced him to base his calculations for
shoulder fillets on solutions for similar geometries, instead of
directly computing the stress state.
generated for Kt
at various values
of r/d and D/d
were based on
filleted shafts in
1997 and earlier
Others had found formulas for the stress concentrations in
hemispherical notches (or grooves, g) in two-dimensional
plates, Kt,g,2D, as well as for fillets ( f) in
two-dimensional plates, Kt,f,2D. The hemispherical-notch
solution had also been expanded to three dimensions to yield a SCF for
circumferential grooves in round bars, Kt,g,3D. But the
flat-plate fillet solution had not been translated into a 3D round shaft
to get Kt,f,3D.
Peterson combined the three existing solutions to estimate the
unknown Kt values for a filleted bar in three dimensions.
(See graphic). His estimate relied on the ratio:
where each Kt term stands in for its corresponding equation. The result is cumbersome to calculate, so Peterson generated a series of curves
showing Kt 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.
Modern analytical techniques can provide a more accurate,
easier-to-handle solution. In fact, the 2007 edition of Peterson’s
Stress Concentration Factors cites 1996 work, also from the
University of Tulsa, that used finite-element 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
more-reliable 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 σ1, and
the stress in the circumferential direction was σ2.
Equivalent stress σe was computed by:
The SCFs, Kt, in terms of the maximum
principal stress can be calculated by:
where Snom = 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, Kq, since
fillet stresses are multiaxial in nature. To do so,
replace σ1 in the above equation with σeq.
In the torsional loading case, the analysis returned
τmax. The resulting Kt is given
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 FEA-derived solutions can be represented by
graphs like Peterson’s or distilled into a single design
The cx in the equation above are numerical
constants chosen based on the loading regime to yield
either Kt or Kq. 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,
combined-loading 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, σ2 using the
principle and equivalent stresses from the bending or
tension loading, σ1 and σeq .
Use the earlier equation for σeq to compute
the maximum equivalent stress of the system from σ1
and the new σ2.
Article edited by Leslie Gordon,
Sr Editor, Machine Design
||Article reprinted by permission of Penton Media,
publisher of Machine Design