# TRANSVERSE CIRCULAR HOLE IN A ROUND BAR

Transverse circular hole in round bar. Stress concentration factors (Kt) for tension, bending and torsion loads. INPUT PARAMETERS Parameter Value Diameter of shaft [D] mm cm m inch ft Hollow shaft inner diameter [d] Transverse hole 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 (at Point-A) [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 (at Point-A) [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 (at Point-A) [τmax ] --- Maximum tension stress due to torsion (at Point-A) [σmax] ---

Note 1: Maximum stress is occured at point A.

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

Note 3: ** Geometry rises τnom by a factor of Kt . (Kt = 2*τmaxnom)

Note 4: o σnom = 4P/[π(D2 - d2)] (Nominal tension stress occurred due to tension load)

Note 5: + σnom= (32MD)/[π(D4-d4)] (Nominal tension stress occurred due to bending)

Note 6: x τnom = (16TD)/[π(D4-d4)] (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)

### Formulas: Tension $$d/D\le 0.9\quad ,\quad 2r/D\le 0.45$$ $${ C }_{ 1 }=3.0$$ $${ C }_{ 2 }=0.427-6.770\frac { d }{ D } +22.698{ (\frac { d }{ D } ) }^{ 2 }-16.670{ (\frac { d }{ D } ) }^{ 3 }$$ $${ C }_{ 3 }=11.357+15.665\frac { d }{ D } -60.929{ (\frac { d }{ D } ) }^{ 2 }+41.501{ (\frac { d }{ D } ) }^{ 3 }$$ $${ K }_{ t }={ C }_{ 1 }+{ C }_{ 2 }\frac { 2r }{ D } +{ C }_{ 3 }{ \left( \frac { 2r }{ D } \right) }^{ 2 }$$ $${ \sigma }_{ nom }=\frac { 4P }{ \pi ({ D }^{ 2 }-{ d }^{ 2 }) }$$ $${ \sigma }_{ max }={ \sigma }_{ A }={ K }_{ t }{ \sigma }_{ nom }$$ Bending $$d/D\le 0.9\quad ,\quad 2r/D\le 0.4$$ $${ C }_{ 1 }=3.0$$ $${ C }_{ 2 }=-6.250-0.585\frac { d }{ D } +3.115{ (\frac { d }{ D } ) }^{ 2 }$$ $${ C }_{ 3 }=41.000-1.071\frac { d }{ D } -6.746{ (\frac { d }{ D } ) }^{ 2 }$$ $${ C }_{ 4 }=-45.000+1.389\frac { d }{ D } +13.889{ (\frac { d }{ D } ) }^{ 2 }$$ $${ K }_{ t }={ C }_{ 1 }+{ C }_{ 2 }\frac { 2r }{ D } +{ C }_{ 3 }{ \left( \frac { 2r }{ D } \right) }^{ 2 }+{ C }_{ 4 }{ \left( \frac { 2r }{ D } \right) }^{ 3 }$$ $${ \sigma }_{ nom }=\frac { 32MD }{ \pi ({ D }^{ 4 }-{ d }^{ 4 }) }$$ $${ \sigma }_{ max }={ \sigma }_{ A }={ K }_{ t }{ \sigma }_{ nom }$$ Torsion $$d/D\le 0.8\quad ,\quad 2r/D\le 0.4$$ $${ C }_{ 1 }=4.0$$ $${ C }_{ 2 }=-6.055+3.184\frac { d }{ D } -3.461{ (\frac { d }{ D } ) }^{ 2 }$$ $${ C }_{ 3 }=32.764-30.121\frac { d }{ D } +39.887{ (\frac { d }{ D } ) }^{ 2 }$$ $${ C }_{ 4 }=-38.330+51.542\sqrt { d/D } -27.483\frac { d }{ D }$$ $${ K }_{ t }={ C }_{ 1 }+{ C }_{ 2 }\frac { 2r }{ D } +{ C }_{ 3 }{ \left( \frac { 2r }{ D } \right) }^{ 2 }+{ C }_{ 4 }{ \left( \frac { 2r }{ D } \right) }^{ 3 }$$ $${ \sigma }_{ max }={ \sigma }_{ A }={ K }_{ t }{ \tau }_{ nom }$$ $${ \tau }_{ nom }=\frac { 16TD }{ \pi ({ D }^{ 4 }-{ d }^{ 4 }) }$$ $${ \tau }_{ max }={ \tau }_{ A }={ 0.5K }_{ t }{ \tau }_{ nom }$$

### Reference:

• Pilkey, W. D..(2005). Formulas for Stress, Strain, and Structural Matrices Formulas for Stress, Strain, and Structural Matrices .2nd Edition John Wiley & Sons