# STRESS CONCENTRATION FACTORS FOR SHOULDER FILLET IN A STEPPED CIRCULAR SHAFT

Theoretical stress concentration factors (Kt) of shoulder fillet in a stepped circular shaft can be calculated by this calculator for tension, bending and torsion loads. Maximum stress at the fillet of the shaft is also calculated.

There exist some validity conditions for the equations which are used in the calculations. If input parameters don't satisfy validity conditions of equations, a warning message is given by the calculator.

### Shoulder Fillet in a Stepped Circular Shaft:

 INPUT PARAMETERS Parameter Value Diameter of larger shaft section [D] mm cm m inch ft Diameter of smaller shaft section [d] Radius [r] Tension force [P] N kN lbf Bending moment [M] N*m lbf*in lbf*ft Torque [T]

Note: Use dot "." as decimal separator.

 RESULTS LOADING TYPE - TENSION Parameter Value Stress concentration factor [Kt] * --- --- Nominal tension stress at shaft [σnom ] o --- MPa psi ksi Maximum tension stress due to tension load [σmax ] --- LOADING TYPE - BENDING Parameter Value Stress concentration factor [Kt] * --- --- Nominal tension stress at shaft [σnom ] + --- MPa psi ksi Maximum tension stress due to bending [σmax ] --- LOADING TYPE - TORSION Parameter Value Stress concentration factor [Kt] ** --- --- Nominal shear stress at shaft [τnom ] x --- MPa psi ksi Maximum shear stress due to torsion [τmax ] ---

Note 1: * Geometry rises σnom by a factor of Kt . (Kt = σmaxnom)

Note 2: ** Geometry rises τnom by a factor of Kt . (Kt = τmaxnom)

Note 3: o σnom = 4P/(πd2) (Nominal tension stress occurred due to tension load)

Note 4: + σnom = 32M/(πd3) (Nominal tension stress occurred due to bending)

Note 5: 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 = (σmaxnom)

### List of Equations:

 Tension $$0.1\le h/r\le 2.0$$ $$2.0\le h/r\le 20.0$$ $${ C }_{ 1 }=0.926+1.157\sqrt { h/r } -0.099h/r$$ $${ C }_{ 1 }=1.200+0.860\sqrt { h/r } -0.022h/r$$ $${ C }_{ 2 }=0.012-3.036\sqrt { h/r } +0.961h/r$$ $${ C }_{ 2 }=-1.805-0.346\sqrt { h/r } -0.038h/r$$ $${ C }_{ 3 }=-0.302+3.977\sqrt { h/r } -1.744h/r$$ $${ C }_{ 3 }=2.198-0.486\sqrt { h/r } +0.165h/r$$ $${ C }_{ 4 }=0.365-2.098\sqrt { h/r } +0.878h/r$$ $${ C }_{ 4 }=-0.593-0.028\sqrt { h/r } -0.106h/r$$ $${ K }_{ t }={ C }_{ 1 }+{ C }_{ 2 }(2h/D)+{ { C }_{ 3 }(2h/D) }^{ 2 }+{ { C }_{ 4 }(2h/D) }^{ 3 }$$ $${ \sigma }_{ nom }={ 4P }/{ \pi { d }^{ 2 } }$$ $${ \sigma }_{ max }={ K }_{ t }{ \sigma }_{ nom }$$ Bending $$0.1\le h/r\le 2.0$$ $$2.0\le h/r\le 20.0$$ $${ C }_{ 1 }=0.947+1.206\sqrt { h/r } -0.131h/r$$ $${ C }_{ 1 }=1.232+0.832\sqrt { h/r } -0.008h/r$$ $${ C }_{ 2 }=0.022-3.405\sqrt { h/r } +0.915h/r$$ $${ C }_{ 2 }=-3.813+0.968\sqrt { h/r } -0.260h/r$$ $${ C }_{ 3 }=0.869+1.777\sqrt { h/r } -0.555h/r$$ $${ C }_{ 3 }=7.423-4.868\sqrt { h/r } +0.869h/r$$ $${ C }_{ 4 }=-0.810+0.422\sqrt { h/r } -0.260h/r$$ $${ C }_{ 4 }=-3.839+3.070\sqrt { h/r } -0.600h/r$$ $${ K }_{ t }={ C }_{ 1 }+{ C }_{ 2 }(2h/D)+{ { C }_{ 3 }(2h/D) }^{ 2 }+{ { C }_{ 4 }(2h/D) }^{ 3 }$$ $${ \sigma }_{ nom }={ 32M }/{ \pi { d }^{ 3 } }$$ $${ \sigma }_{ max }={ K }_{ t }{ \sigma }_{ nom }$$ Torsion $$0.25\le h/r\le 4.0$$ $${ C }_{ 1 }=0.905+0.783\sqrt { h/r } -0.075h/r$$ $${ C }_{ 2 }=-0.437-1.969\sqrt { h/r } +0.553h/r$$ $${ C }_{ 3 }=1.557+1.073\sqrt { h/r } -0.578h/r$$ $${ C }_{ 4 }=-1.061+0.171\sqrt { h/r } +0.086h/r$$ $${ K }_{ t }={ C }_{ 1 }+{ C }_{ 2 }(2h/D)+{ { C }_{ 3 }(2h/D) }^{ 2 }+{ { C }_{ 4 }(2h/D) }^{ 3 }$$ $${ \tau }_{ nom }={ 16T }/{ \pi { d }^{ 3 } }$$ $${ \tau }_{ max }={ K }_{ t }{ \tau }_{ nom }$$

### Examples:

 Link Usage Torsion Of Solid Shaft Example An example about the usage of calculator.