# Square head for shaft-hub connection

The advantage of this connection is easy assembly and disassembly. The disadvantage is the low manufacturing precision and consequent consequences for limited speeds and small torques.

For a simplified calculation, it is assumed that the joint is without will, and that the torque causes the contact stress to be half of each function area of the square head. It is possible to assume a triangular distribution of this stress.

Load distribution will differ from the assumption due to production inaccuracy due to looseness or prestressing of joints and shaft deformations by from torsion torque. These deviations can include in the calculation a coefficient max. stress Ss=1,3-2 the lower value of which applies to short joints l≤s and for high accuracy of manufacturing.

Fig.1 square head for shaft-hub connection

Bearing stress:

$p=\frac{{M}_{T}*{s}_{s}}{2a*l*b}\le {\sigma }_{all}$

p - bearing stress - [MPa]

MT - torque - [Nm]

ss - coefficient of maximum stress increase - []

l - length square head in the hub - [mm]

b - distance of the resultant of the pressure - [mm]

Distance of the resultant of the pressure:

$b={a}_{1}+\frac{2}{3}a$

b - distance of the resultant of the pressure - [mm]

${a}_{1}=\frac{{d}_{9}}{2}\mathrm{sin}\left({\mathrm{cos}}^{-1}\frac{s}{{d}_{9}}\right)$

d9 - free diameter - [mm]

s - width square head - [mm]

$a=\frac{{d}_{8}}{2}\mathrm{sin}\left({\mathrm{cos}}^{-1}\frac{s}{{d}_{8}}\right)-{a}_{1}$

d8 - diameter square head - [mm]

s - width square head - [mm]

Allowable bearing stress:

${\sigma }_{all}=\frac{0,9{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σall - allowable bearing stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []

Coefficient of use of joints according to load:

Torsion stress in the shaft:

${\tau }_{s}=\frac{{16M}_{T}}{\pi {d}^{3}}\le {\tau }_{all}$

τs - torsion stress in the shaft - [MPa]

MT - torque - [Nm]

d - diameter of the shaft - [mm]

τall - allowable shear stress - [MPa]

Allowable shear stress:

${\tau }_{all}=\frac{0,4{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

τall - allowable shear stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []

Torsion stress in the hub:

${\tau }_{h}=3,962\frac{{16M}_{T}}{\pi \left(\left({D}_{h}^{4}-{4s}^{4}\right)/{D}_{h}\right)}\le {\tau }_{all}$

τh - torsion stress in the hub - [MPa]

MT - torque - [Nm]

Dh - diameter of the hub - [mm]

s - width square head - [mm]

τall - allowable shear stress - [MPa]

If the shaft is loaded with the bending moment in the joint, the bending stress must be checked. If the shaft is loaded with a shear force in the joint, the shear stress must be checked. The shaft may be load in the joint by axial force. The shaft must be checked for axial stresses. When calculating the different load types, it is necessary to calculate the combined stress.

Bending stress in the shaft:

${\sigma }_{B}=\frac{{32M}_{B}}{\pi {d}^{3}}\le {\sigma }_{Ball}$

σB - bending stress in the shaft - [MPa]

MB - bending moment - [Nm]

d - diameter of the shaft - [mm]

σBall - allowable bending stress - [MPa]

Allowable bending stress:

${\sigma }_{Ball}=\frac{0,6{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σBall - allowable bending stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []

Shear stress in the shaft:

${\tau }_{s\left(s\right)}=\frac{4{F}_{R}}{\pi {d}^{2}}\le {\tau }_{all}$

τs(s) - shear stress in the shaft - [MPa]

FR - shear forces - [N]

d - diameter of the shaft - [mm]

τall - allowable shear stress - [MPa]

Axial stress in the shaft:

${\sigma }_{A}=\frac{4{F}_{A}}{\pi {d}^{2}}\le {\sigma }_{Aall}$

σA - axial stress in the shaft - [MPa]

FA - axial forces - [N]

d - diameter of the shaft - [mm]

σAall - allowable axial stress - [MPa]

Allowable axial stress:

${\sigma }_{Aall}=\frac{0,45{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σAall - allowable axial stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []

Combined stress in the shaft:

${\sigma }_{tresca}=$ $\sqrt{{\left({K}_{tB}*{\sigma }_{B}\right)}^{2}+{\left({K}_{tA}*{\sigma }_{A}\right)}^{2}+4\left({\left({K}_{ts}*{\tau }_{s}\right)}^{2}+{\tau }_{s\left(s\right)}^{2}\right)}$ $\le {\sigma }_{Call}$

σtresca - combined stress in the shaft - [MPa]

KtB - concentration factor in bending stress - []

σB - bending stress in the shaft - [MPa]

KtA - concentration factor in axial stress - []

σA - axial stress in the shaft - [MPa]

Kts - concentration factor in torsion stress - []

τs - torsion stress in the shaft - [MPa]

τs(s) - shear stress in the shaft - [MPa]

σCall - allowable combined stress - [MPa]

Concentration factor in bending stress:

${K}_{tB}={C}_{1B}+{C}_{2B}\left(\frac{D-d}{D}\right)$ $+{C}_{3B}{\left(\frac{D-d}{D}\right)}^{2}+{C}_{4B}{\left(\frac{D-d}{D}\right)}^{3}$
${C}_{1B}=0,947+1,206\sqrt{\frac{D-d}{2r}}$ $-0,131\frac{D-d}{2r}$
${C}_{2B}=0,022-3,405\sqrt{\frac{D-d}{2r}}$ $+0,915\frac{D-d}{2r}$
${C}_{3B}=0,869+1,777\sqrt{\frac{D-d}{2r}}$ $-0,555\frac{D-d}{2r}$
${C}_{4B}=-0,810+0,422\sqrt{\frac{D-d}{2r}}$ $-0,260\frac{D-d}{2r}$

KtB - concentration factor in bending stress - []

C1B - coefficient - []

C2B - coefficient - []

C3B - coefficient - []

C4B - coefficient - []

D - diameter of the shaft - [mm]

d - diameter of the shaft - [mm]

Concentration factor in axial stress:

${K}_{tA}={C}_{1A}+{C}_{2A}\left(\frac{D-d}{D}\right)$ $+{C}_{3A}{\left(\frac{D-d}{D}\right)}^{2}+{C}_{4A}{\left(\frac{D-d}{D}\right)}^{3}$
${C}_{1A}=0,926+1,157\sqrt{\frac{D-d}{2r}}$ $-0,099\frac{D-d}{2r}$
${C}_{2A}=0,012-3,036\sqrt{\frac{D-d}{2r}}$ $+0,961\frac{D-d}{2r}$
${C}_{3A}=-0,302+3,977\sqrt{\frac{D-d}{2r}}$ $-1,744\frac{D-d}{2r}$
${C}_{4A}=0,365-2,098\sqrt{\frac{D-d}{2r}}$ $+0,878\frac{D-d}{2r}$

KtA - concentration factor in axial stress - []

C1A - coefficient - []

C2A - coefficient - []

C3A - coefficient - []

C4A - coefficient - []

D - diameter of the shaft - [mm]

d - diameter of the shaft - [mm]

Concentration factor in torsion stress:

${K}_{ts}={C}_{1s}+{C}_{2s}\left(\frac{D-d}{D}\right)$ $+{C}_{3s}{\left(\frac{D-d}{D}\right)}^{2}+{C}_{4s}{\left(\frac{D-d}{D}\right)}^{3}$
${C}_{1s}=0,905+0,783\sqrt{\frac{D-d}{2r}}$ $-0,075\frac{D-d}{2r}$
${C}_{2s}=-0,437-1,969\sqrt{\frac{D-d}{2r}}$ $+0,553\frac{D-d}{2r}$
${C}_{3s}=1,557+1,073\sqrt{\frac{D-d}{2r}}$ $-0,578\frac{D-d}{2r}$
${C}_{4s}=-1,061+0,171\sqrt{\frac{D-d}{2r}}$ $+0,086\frac{D-d}{2r}$

Kts - concentration factor in torsion stress - []

C1s - coefficient - []

C2s - coefficient - []

C3s - coefficient - []

C4s - coefficient - []

D - diameter of the shaft - [mm]

d - diameter of the shaft - [mm]

Allowable combined stress:

${\sigma }_{Call}=\frac{{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σCall - allowable combined stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []

Literature:

AISC: Specification for structural steel buildings: Allowable Stress design and plastic design 1989

Walter D. Pilkey, Deborah F. Pilkey: Peterson’s stress concentration factors. 2008

Joseph E. Shigley, Charles R. Mischke, Richard G. Budynas: Konstruování strojních součástí 2010.

MET-Calc: Allowable stress

A. Bolek, J. Kochman a kol.: Části a mechanismy strojů I. 1989.

K. Kříž a kol.: Strojní součásti 1. 1984.