U-SHAPED CIRCUMFERENTIAL GROOVE ON A SHAFT

U-shaped circumferential groove on a shaft. Theoretical stress concentration factors (Kt) calculator for tension, bending and torsion loads.

Stress concentration factors for U-shaped groove
INPUT PARAMETERS
Parameter Value
Diameter of larger shaft section [D]
Diameter of smaller shaft section [d]
Radius [r]
Tension force [P]
Bending moment [M]
Torque [T]


Note:Use dot "." as decimal separator.


 RESULTS
LOADING TYPE - TENSION
Stress concentration factors for U-shaped groove under tension
Parameter Value
Stress concentration factor [Kt] * --- ---
Nominal tension stress at shaft [σnom ] o ---
Maximum tension stress due to tension load (at Point-A) [σmax ] ---
LOADING TYPE - BENDING
Stress concentration factors for U-shaped groove under bending
Parameter Value
Stress concentration factor [Kt] * --- ---
Nominal tension stress at shaft [σnom ] + ---
Maximum tension stress due to bending (at Point-A) [σmax ] ---
LOADING TYPE - TORSION
Stress concentration factors for U-shaped groove under torsion
Parameter Value
Stress concentration factor [Kt] ** --- ---
Nominal shear stress at shaft [τnom ] x ---
Maximum shear stress due to torsion (at Point-A) [τmax ] ---

Note 1: Maximum stress is occurred at point A.

Note 2: * Geometry rises σnom by a factor of Kt .  (Kt = σnom/σmax)

Note 3: ** Geometry rises τnom by a factor of Kt . (Kt = τnom/τmax)

Note 4: o σnom = 4P/(πd2) (Nominal tension stress occurred due to tension load)

Note 5: + σnom = 32M/(πd3) (Nominal tension stress occurred due to bending)

Note 6: x   τnom = 16T/(πd3) (Nominal shear stress occurred due to torsion)


Definitions:

Stress Concentration Factor: Dimensional changes and discontinuities of a member in a loaded structure causes variations of stress and high stresses concentrate near these dimensional changes. This situation of high stresses near dimensional changes and discontinuities of a member (holes, sharp corners, cracks etc.) is called stress concentration. The ratio of peak stress near stress riser to average stress over the member is called stress concentration factor.

Kt: Theoretical stress concentration factor in elastic range = (σmax/σnom)

Formulas:

Stress concentration factors for U-shaped groove
Tension
Stress concentration factors for U-shaped groove under tension
  0.1 ≤ h/r ≤ 2.0 2.0 ≤ h/r ≤ 50.0
C1  $$0.89+2.208\sqrt { h/r } -0.094h/r$$ $$1.037+1.967\sqrt { h/r } +0.002h/r$$
C2 $$-0.923-6.678\sqrt { h/r } +1.638h/r$$ $$-2.679-2.980\sqrt { h/r } -0.053h/r$$
C3 $$2.893+6.448\sqrt { h/r } -2.516h/r$$ $$3.090+2.124\sqrt { h/r } +0.165h/r$$
C4 $$-1.912-1.944\sqrt { h/r } +0.963h/r$$ $$-0.424-1.153\sqrt { h/r } -0.106h/r$$
Kt=C1+C2(2h/D)+C3(2h/D)2+C4(2h/D)3

σnom=4P/πd2

σmax=σA=Ktσnom
Bending
Stress concentration factors for U-shaped groove under bending
  0.25 ≤ h/r ≤ 2.0 2.0 ≤ h/r ≤ 50.0
C1  $$0.594+2.958\sqrt { h/r } -0.520h/r$$ $$0.965+1.926\sqrt { h/r }$$
C2 $$0.422-10.545\sqrt { h/r } +2.692h/r$$ $$-2.773-4.414\sqrt { h/r } -0.017h/r$$
C3 $$0.501+14.375\sqrt { h/r } -4.486h/r$$ $$4.785+4.681\sqrt { h/r } +0.096h/r$$
C4 $$-0.613-6.573\sqrt { h/r } +2.177h/r$$ $$-1.995-2.241\sqrt { h/r } -0.074h/r$$
Kt=C1+C2(2h/D)+C3(2h/D)2+C4(2h/D)3

σnom=32M/πd3

σmax=σA=Ktσnom
Torsion
Stress concentration factors for U-shaped groove under torsion
  0.25 ≤ h/r ≤ 2.0 2.0 ≤ h/r ≤ 50.0
C1  $$0.966+1.056\sqrt { h/r } -0.022h/r$$ $$1.089+0.924\sqrt { h/r } +0.018h/r$$
C2 $$-0.192 - 4.037\sqrt { h/r } +0.674h/r$$ $$-1.504 - 2.141\sqrt { h/r } -0.047h/r$$
C3 $$0.808 +5.321\sqrt { h/r } -1.231h/r$$ $$2.486 +2.289\sqrt { h/r } +0.091h/r$$
C4 $$-0.567-2.364\sqrt { h/r } +0.566h/r$$ $$-1.056 -1.104\sqrt { h/r } -0.059h/r$$
Kt=C1+C2(2h/D)+C3(2h/D)2+C4(2h/D)3

τnom=16T/πd3

τmax=τA=Ktτnom

Reference: