## 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:

$τ_s=\cfrac{{16M}_T}{\pi(D-d)^3}\leτ_{all}$
where:
 $τ_s$ torsion stress in the shaft $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

Allowable shear stress:

$τ_{all}=\cfrac{0,4R_{p0,2/T}}{S_F}\cdot C_c$
where:
 $τ_{all}$ allowable shear stress $\mathrm{MPa}$ $R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$ $S_F$ safety factor $\mathrm{-}$ $C_c$ coefficient of use of joints according to load $\mathrm{-}$

Coefficient according to load:

load $\mathrm{-}$
Unidirectional load, with a small impact load 0,7
Unidirectional load, with a big impact load 0,6
Alternating load, with a small impact load 0,45
Alternating load, with a big impact load 0,25

Shear stress in the pin:

$τ_p=\cfrac{{2M}_T}{D\cdot d\cdot l\cdot i}\leτ_{all}$
where:
 $τ_p$ shear stress in the pin $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $l$ length pin (without threads, etc.) $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

Bearing stress in the pin, shaft and hub:

$p=\cfrac{{4M}_T}{D\cdot d\cdot l\cdot i}\le\sigma_{all}$
where:
 $p$ bearing stress in the pin, shaft and hub $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $l$ length pin (without threads, etc.) $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $\sigma_{all}$ allowable bearing stress $\mathrm{MPa}$

Allowable bearing stress:

$\sigma_{all}=\cfrac{0,9R_{p0,2/T}}{S_F}\cdot C_c$
where:
 $\sigma_{all}$ allowable bearing stress $\mathrm{MPa}$ $R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$ $S_F$ safety factor $\mathrm{-}$ $C_c$ coefficient of use of joints according to load $\mathrm{-}$

Torsion stress in the hub:

$τ_h=K_t\cfrac{{16M}_T}{\pi((D_h^4-(D+d)^4)/D_h)}\leτ_{all}$
where:
 $τ_h$ torsion stress in the hub $\mathrm{MPa}$ $K_t$ concentration factor $\mathrm{-}$ $M_T$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$ $d$ diameter of the hub $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

Concentration factor:

$K_t=1,953+0,1434\left(\cfrac{0,2}{d/D}\right)-0,0021\left(\cfrac{0,2}{d/D}\right)^2$
where:
 $K_t$ concentration factor $\mathrm{-}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$

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:

$\sigma_B=\cfrac{{32M}_B}{\pi(D-d)^3}\le\sigma_{Ball}$
where:
 $\sigma_B$ bending stress in the shaft $\mathrm{MPa}$ $M_B$ bending moment $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $\sigma_{Ball}$ allowable bending stress $\mathrm{MPa}$

Allowable bending stress:

$\sigma_{Ball}=\cfrac{0,6R_{p/T}}{S_F}\cdot C_c$
where:
 $\sigma_{Ball}$ allowable bending stress $\mathrm{MPa}$ $R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$ $S_F$ safety factor $\mathrm{-}$ $C_c$ coefficient of use of joints according to load $\mathrm{-}$

Shear stress in the shaft:

$τ_{s(s)}=\cfrac{4F_R}{\pi(D-d)^2}\leτ_{all}$
where:
 $τ_{s(s)}$ shear stress in the shaft $\mathrm{MPa}$ $F_R$ shear force $\mathrm{N}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

Axial stress in the shaft:

$\sigma_A=\cfrac{4F_A}{\pi(D-d)^2}\le\sigma_{Aall}$
where:
 $\sigma_A$ axial stress in the shaft $\mathrm{MPa}$ $F_A$ axial force $\mathrm{N}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $\sigma_{Aall}$ allowable axial stress $\mathrm{MPa}$

Allowable axial stress:

$\sigma_{Aall}=\cfrac{0,45R_{p0,2/T}}{S_F}\cdot C_c$
where:
 $\sigma_{Aall}$ allowable axial stress $\mathrm{MPa}$ $R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$ $S_F$ safety factor $\mathrm{-}$ $C_c$ coefficient of use of joints according to load $\mathrm{-}$

Allowable axial stress:

$\sigma_{tresca}=\sqrt{\sigma_B^2+\sigma_A^2+4((K_t\cdotτ_s)^2+τ_{s(s)}^2)}\le\sigma_{Call}$
where:
 $\sigma_{tresca}$ combined stress in the shaft $\mathrm{MPa}$ $\sigma_B$ bending stress in the shaft $\mathrm{MPa}$ $\sigma_A$ axial stress in the shaft $\mathrm{MPa}$ $K_t$ concentration factor $\mathrm{-}$ $τ_s$ torsion stress in the shaft $\mathrm{MPa}$ $τ_{s(s)}$ shear stress in the shaft $\mathrm{MPa}$ $\sigma_{Call}$ allowable combined stress $\mathrm{MPa}$

Allowable combined stress:

$\sigma_{Call}=\cfrac{R_{p0,2/T}}{S_F}\cdot C_c$
where:
 $\sigma_{Call}$ allowable combined stress $\mathrm{MPa}$ $R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$ $S_F$ safety factor $\mathrm{-}$ $C_c$ coefficient of use of joints according to load $\mathrm{-}$

When $i\geq4$ 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:

$\propto=\cfrac{2\pi-\left({2i\cdot\sin}^{-1}{\cfrac{d}{D}}\right)}{i}$
where:
 $\propto$ angle between pins $\mathrm{rad}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$

Width between pins:

$t=D\cdot\cos{\left(\sin^{-1}{\cfrac{d}{D}}\right)}\cdot\sin{\cfrac{\propto}{2}}$
where:
 $t$ width between pins $\mathrm{mm}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $\propto$ angle between pins $\mathrm{rad}$

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

$\sigma_{B(s-h)}=\cfrac{{3d\cdot M}_T}{D\cdot l\cdot t^2\cdot i}\le\sigma_{Ball}$
where:
 $\sigma_{B(s-h)}$ bending stress in the weakened part of the shaft-hub $\mathrm{MPa}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $M_T$ torque $\mathrm{Nm}$ $l$ length pin (without threads, etc.) $\mathrm{mm}$ $t$ width between pins $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $\sigma_{Ball}$ allowable bending stress $\mathrm{MPa}$

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

$τ_{s-h}=\cfrac{{3M}_T}{D\cdot t\cdot l\cdot i}\leτ_{all}$
where:
 $τ_{s-h}$ shear stress in the weakened part of the shaft-hub $\mathrm{MPa}$ $D$ diameter of the shaft $\mathrm{mm}$ $M_T$ torque $\mathrm{Nm}$ $l$ length pin (without threads, etc.) $\mathrm{mm}$ $t$ width between pins $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

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.
- A. Bolek, J. Kochman a kol.: Části a mechanismy strojů I. 1989.
- R. Kříž: Strojní součásti I pro SPŠ strojnické 1984.