# Longitudinal pin for shaft-hub connection

The easiest and oldest way joints. It is a joint with a shape contact. The pin serves primarily to ensure the mutual positioning of the two parts. They are cylindrical or conical. The pins are dimensioned under simplified assumptions without will and without the pressing effect. When calculating the pin, the length of the pin should not be considered, which is different from the nominal cross-section see, for example, the thread in the pin etc.

When calculating, see below, the hub is simplified to transfer only torsion moments. In practice, it may not be true. If the hub is carrying another load e.g. axial the charge must be assessed on the individual stress components plus the combined stress.

Fig.1 Longitudinal pin for shaft-hub connection

Torsion stress in the shaft:

${\tau }_{s}=\frac{{16M}_{T}}{\pi {\left(D-d\right)}^{3}}\le {\tau }_{all}$

τs - torsion stress in the shaft - [MPa]

MT - torque - [Nm]

D - diameter of the shaft - [mm]

d - diameter of the pin - [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 - []

Coefficient of use of joints according to load:

Shear stress in the pin:

${\tau }_{p}=\frac{{2M}_{T}}{D*d*l*i}\le {\tau }_{all}$

τp - shear stress in the pin - [MPa]

MT - torque - [Nm]

D - diameter of the shaft - [mm]

d - diameter of the pin - [mm]

l - length pin (without threads, etc.) - [mm]

i - number of pins - []

τall - allowable shear stress - [MPa]

Bearing stress in the pin, shaft and hub:

$p=\frac{{4M}_{T}}{D*d*l*i}\le {\sigma }_{all}$

p - bearing stress in the pin, shaft and hub - [MPa]

MT - torque - [Nm]

D - diameter of the shaft - [mm]

d - diameter of the pin - [mm]

l - length pin (without threads, etc.) - [mm]

i - number of pins - []

σall - allowable bearing stress - [MPa]

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 - []

Torsion stress in the hub:

${\tau }_{h}={K}_{t}\frac{{16M}_{T}}{\pi \left(\left({D}_{h}^{4}-{\left(D+d\right)}^{4}\right)/{D}_{h}\right)}\le {\tau }_{all}$

τh - torsion stress in the hub - [MPa]

Kt - concentration factor - []

MT - torque - [Nm]

D - diameter of the shaft - [mm]

Dh - diameter of the hub - [mm]

d - diameter of the pin - [mm]

τall - allowable shear stress - [MPa]

Concentration factor:

${K}_{t}=1,953+0,1434\left(\frac{0,2}{d/D}\right)$ $-0,0021{\left(\frac{0,2}{d/D}\right)}^{2}$

Kt - concentration factor - []

D - diameter of the shaft - [mm]

d - diameter of the pin - [mm]

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:

${\sigma }_{B}=\frac{{32M}_{B}}{\pi {\left(D-d\right)}^{3}}\le {\sigma }_{Ball}$

σB - bending stress in the shaft - [MPa]

MB - bending moment - [Nm]

D - diameter of the shaft - [mm]

d - diameter of the pin - [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:

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

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

FR - shear forces - [N]

D - diameter of the shaft - [mm]

d - diameter of the pin - [mm]

τall - allowable shear stress - [MPa]

Axial stress in the shaft:

${\sigma }_{A}=\frac{4{F}_{A}}{\pi {\left(D-d\right)}^{2}}\le {\sigma }_{Aall}$

σA - axial stress in the shaft - [MPa]

FA - axial force - [N]

D - diameter of the shaft - [mm]

d - diameter of the pin - [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:

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

σtresca - combined stress in the shaft - [MPa]

σB - bending stress in the shaft - [MPa]

σA - axial stress in the shaft - [MPa]

Kt - concentration factor - []

τs - torsion stress in the shaft - [MPa]

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

σCall - allowable combined stress - [MPa]

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 - []

When i≥4 the weakened part of the shaft and the hub must be seen see Fig. 2 to control the bending stress and the shear according to the Grashof formula.

Fig.2 weakened part of shaft and hub

Angle between pins:

$\alpha =\frac{2\pi -\left({2\mathrm{i*sin}}^{-1}\frac{d}{D}\right)}{i}$

α - angle between pins - [rad]

D - diameter of the shaft - [mm]

d - diameter of the pin - [mm]

i - number of pins - []

Width between pins:

$t=D*\mathrm{cos}\left({\mathrm{sin}}^{-1}\frac{d}{D}\right)\mathrm{*sin}\frac{\alpha }{2}$

t - width between pins - [mm]

d - diameter of the pin - [mm]

D - diameter of the shaft - [mm]

α - angle between pins - [rad]

Bending stress in the weakened part of the shaft-hub:

${\sigma }_{B\left(s-h\right)}=\frac{{3d*M}_{T}}{D*l*{t}^{2}*i}\le {\sigma }_{Ball}$

σB(s-h) - bending stress in the weakened part of the shaft-hub - [MPa]

d - diameter of the pin - [mm]

MT - torque - [Nm]

D - diameter of the shaft - [mm]

l - length pin (without threads, etc.) - [mm]

t - width between pins - [mm]

i - number of pins - []

σBall - allowable bending stress - [MPa]

Shear stress in the weakened part of the shaft-hub

${\tau }_{s-h}=\frac{{3M}_{T}}{D*t*l*i}\le {\tau }_{all}$

τs-h - shear stress in the weakened part of the shaft-hub - [MPa]

MT - torque - [Nm]

D - diameter of the shaft - [mm]

t - width between pins - [mm]

l - length pin (without threads, etc.) - [mm]

i - number of pins - []

τall - allowable shear stress - [MPa]

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.