## Radial 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 designing a pin, the cross section of the pin in the shear area must be the nominal cross-section of the pin.

Fig. 1 - Radial pin for shaft-hub connection

Torsion stress in the shaft:

$τ_s=\cfrac{{16M}_T}{\pi D^3}\cdot\cfrac{J_{tube(s)}}{J_{net(s)}}\leτ_{all}$
$\cfrac{J_{tube(s)}}{J_{net(s)}}=1/\cfrac{J_{net(s)}}{J_{tube(s)}}$
where:
 $τ_s$ torsion stress in the shaft $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $τ_{all}$ 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:

$τ_p=\cfrac{{4M}_T}{\pi\cdot d^2\cdot D}+\cfrac{{2F}_A}{\pi\cdot d^2}\leτ_{all}$
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 $\mathrm{mm}$ $F_A$ axial force $\mathrm{N}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

Bearing stress in the pin and shaft:

$p_1=\cfrac{{6M}_T}{D^2\cdot d}+\cfrac{F_A}{D\cdot d}\le\sigma_{all}$
where:
 $p_1$ bearing stress in the pin and shaft $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $F_A$ axial force $\mathrm{N}$ $\sigma_{all}$ 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 $\mathrm{-}$

Bearing stress in the pin and hub:

$p_2=\cfrac{{4M}_T}{d\cdot(D_h^2-D^2)}+\cfrac{F_A}{d\cdot(D_h-D)}\le\sigma_{all}$
where:
 $p_2$ bearing stress in the pin and hub $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$ $F_A$ axial force $\mathrm{N}$ $\sigma_{all}$ allowable bearing stress $\mathrm{MPa}$

Torsion stress in the hub:

$τ_h=\cfrac{{16M}_TD_h}{\pi(D_h^4-D^4)}\cdot\cfrac{J_{tube(h)}}{J_{net(h)}}\leτ_{all}$
$\cfrac{J_{tube(h)}}{J_{net(h)}}=1/\cfrac{J_{net(h)}}{J_{tube(h)}}$
where:
 $τ_h$ torsion stress in the hub $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

If the shaft and hub is loaded with the bending moment in the joint, the bending stress must be checked. If the shaft and hub is loaded with a shear force in the joint, the shear stress must be checked. The shaft and hub may be load in the joint by axial force. The shaft and hub 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:

$\sigma_{B(s)}=\cfrac{{32M}_B}{\pi D^3}\cdot\cfrac{Z_{tube(s)}}{Z_{net(s)}}\le\sigma_{Ball}$
$\cfrac{Z_{tube(s)}}{Z_{net(s)}}=1/\cfrac{Z_{net(s)}}{Z_{tube(s)}}$
where:
 $\sigma_{B(s)}$ bending stress in the shaft $\mathrm{MPa}$ $M_B$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $\sigma_{Ball}$ allowable bending stress $\mathrm{MPa}$

Allowable bending stress:

$\sigma_{Ball}=\cfrac{0,6R_{p0,2/T}}{S_F}\cdot C_c$
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 $\mathrm{-}$

Bending stress in the hub:

$\sigma_{B(h)}=\cfrac{{32M}_BD_h}{\pi(D_h^4-D^4)}\cdot\cfrac{Z_{tube(h)}}{Z_{net(h)}}\le\sigma_{Ball}$
$\cfrac{Z_{tube(h)}}{Z_{net(h)}}=1/\cfrac{Z_{net(h)}}{Z_{tube(h)}}$
where:
 $\sigma_{B(h)}$ bending stress in the hub $\mathrm{MPa}$ $M_B$ torque $\mathrm{Nm}$ $D$ diameter of the shaft $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$ $\sigma_{Ball}$ allowable bending stress $\mathrm{MPa}$

Shear stress in the shaft:

$τ_{s(s)}=\cfrac{{4F}_R}{\pi D^2}\cdot\cfrac{A_{tube(s)}}{A_{net(s)}}\leτ_{all}$
$\cfrac{A_{tube(s)}}{A_{net(s)}}=1/\cfrac{A_{net(s)}}{A_{tube(s)}}$
where:
 $τ_{s(s)}$ shear stress in the shaft $\mathrm{MPa}$ $F_R$ shear force $\mathrm{N}$ $D$ diameter of the shaft $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

Shear stress in the hub:

$τ_{s(h)}=\cfrac{{4F}_R}{\pi(D_h^2-D^2)}\cdot\cfrac{A_{tube(h)}}{A_{net(h)}}\leτ_{all}$
$\cfrac{A_{tube(h)}}{A_{net(h)}}=1/\cfrac{A_{net(h)}}{A_{tube(h)}}$
where:
 $τ_{s(h)}$ shear stress in the hub $\mathrm{MPa}$ $F_R$ shear force $\mathrm{N}$ $D$ diameter of the shaft $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

Axial stress in the shaft:

$\sigma_{A(s)}=\cfrac{{4F}_A}{\pi D^2}\cdot\cfrac{A_{tube(s)}}{A_{net(s)}}\le\sigma_{Aall}$
$\cfrac{A_{tube(s)}}{A_{net(s)}}=1/\cfrac{A_{net(s)}}{A_{tube(s)}}$
where:
 $\sigma_{A(s)}$ axial stress in the shaft $\mathrm{MPa}$ $F_A$ axial force $\mathrm{N}$ $D$ diameter of the shaft $\mathrm{mm}$ $\sigma_{Aall}$ 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 $\mathrm{-}$

Axial stress in the hub:

$\sigma_{A(h)}=\cfrac{{4F}_A}{\pi(D_h^2-D^2)}\cdot\cfrac{A_{tube(h)}}{A_{net(h)}}\le\sigma_{Aall}$
$\cfrac{A_{tube(h)}}{A_{net(h)}}=1/\cfrac{A_{net(h)}}{A_{tube(h)}}$
where:
 $\sigma_{A(h)}$ axial stress in the hub $\mathrm{MPa}$ $F_A$ axial force $\mathrm{N}$ $D$ diameter of the shaft $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$ $\sigma_{Aall}$ allowable axial stress $\mathrm{MPa}$

Combined stress in the shaft:

$\sigma_{tresca(s)}=\sqrt{\left({\cfrac{Z_{net(s)}}{Z_{tube(s)}}\cdot} K_{tB(s)}\cdot\sigma_{B(s)}\right)^2+\left(\cfrac{A_{net(s)}}{A_{tube(s)}}\cdot K_{tA(s)}\cdot\sigma_{A(s)}\right)^2+4\left(\left(\cfrac{J_{net(s)}}{J_{tube(s)}}\cdot K_{ts}\cdotτ_s\right)^2+τ_{s(s)}^2\right)}\le\sigma_{Call}$
$\cfrac{Z_{net(s)}}{Z_{tube(s)}}=1-(16/3/\pi)(d/D)$
$\cfrac{A_{net(s)}}{A_{tube(s)}}=1-4/\pi(d/D)$
$\cfrac{J_{net(s)}}{J_{tube(s)}}=1-(8/3/\pi)(d/D)\{1+(d/D)^2\}$
where:
 $\sigma_{tresca(s)}$ combined stress in the shaft $\mathrm{MPa}$ $K_{tB(s)}$ concentration factor shaft in bending stress $\mathrm{-}$ $\sigma_{B(s)}$ bending stress in the shaft $\mathrm{MPa}$ $K_{tA(s)}$ concentration factor shaft in axial stress $\mathrm{-}$ $\sigma_{A(s)}$ axial stress in the shaft $\mathrm{MPa}$ $K_{ts}$ concentration factor shaft in torsion stress $\mathrm{-}$ $τ_s$ torsion stress in the shaft $\mathrm{MPa}$ $τ_{s(s)}$ shear stress in the shaft $\mathrm{MPa}$ $\sigma_{Call}$ allowable combined stress $\mathrm{MPa}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$

Concentration factor shaft in bending stress:

$K_{tB(s)}=3-6,25\left(\cfrac{d}{D}\right)+41\left(\cfrac{d}{D}\right)^2-45\left(\cfrac{d}{D}\right)^3$
where:
 $K_{tB(s)}$ concentration factor shaft in bending stress $\mathrm{-}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$

Concentration factor shaft in axial stress:

$0\le d/D\le0,7$
$K_{tA(s)}=12,806-42,602\left(\cfrac{d}{D}\right)+58,333\left(\cfrac{d}{D}\right)^2$
where:
 $K_{tA(s)}$ concentration factor shaft in axial stress $\mathrm{-}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$

Concentration factor shaft in torsion stress:

$K_{ts}=4-6,055\left(\cfrac{d}{D}\right)+32,764\left(\cfrac{d}{D}\right)^2-38,33\left(\cfrac{d}{D}\right)^3$
where:
 $K_{ts}$ concentration factor shaft in torsion stress $\mathrm{-}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$

Allowable combined stress:

$\sigma_{Call}=\cfrac{R_{p0,2/T}}{S_F}\cdot C_c$
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 $\mathrm{-}$

Combined stress in the hub:

$\sigma_{tresca(h)}=\sqrt{\left({\cfrac{Z_{net(h)}}{Z_{tube(h)}}\cdot} K_{tB(h)}\cdot\sigma_{B(h)}\right)^2+\left(\cfrac{A_{net(h)}}{A_{tube(h)}}\cdot K_{tA(h)}\cdot\sigma_{A(h)}\right)^2+4\left(\left({\cfrac{J_{net(h)}}{J_{tube(h)}}\cdot} K_{th}\cdotτ_h\right)^2+τ_{s(h)}^2\right)}\le\sigma_{Call}$
$\cfrac{Z_{net(h)}}{Z_{tube(h)}}=1-\cfrac{(16/3/\pi)(d/D_h)[1-(D/D_h)^3]}{1-(D/D_h)^4}$
$\cfrac{A_{net(h)}}{A_{tube(h)}}=1-\cfrac{4/\pi(d/D_h)[1-(D/D_h)]}{1-(D/D_h)^2}$
$\cfrac{J_{net(h)}}{J_{tube(h)}}=1-\cfrac{(8/3/\pi)(d/D_h)\{[1-(D/D_h)^3]+(d/D_h)^2[1-(D/D_h)]\}}{1-(D/D_h)^4}$
where:
 $\sigma_{tresca(h)}$ combined stress in the hub $\mathrm{MPa}$ $K_{tB(h)}$ concentration factor hub in bending stress $\mathrm{-}$ $\sigma_{B(h)}$ bending stress in the hub $\mathrm{MPa}$ $K_{tA(h)}$ concentration factor hub in axial stress $\mathrm{-}$ $\sigma_{A(h)}$ axial stress in the hub $\mathrm{MPa}$ $K_{th}$ concentration factor hub in torsion stress $\mathrm{-}$ $τ_h$ torsion stress in the hub $\mathrm{MPa}$ $τ_{s(h)}$ shear stress in the hub $\mathrm{MPa}$ $\sigma_{Call}$ allowable combined stress $\mathrm{MPa}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$

Concentration factor hub in bending stress:

$d/D_h\le0,4;D/D_h\le0,9$
$K_{tB(h)}=C_{1B(h)}+C_{2B(h)}\left(\cfrac{d}{D_h}\right)+C_{3B(h)}\left(\cfrac{d}{D_h}\right)^2+C_{4B(h)}\left(\cfrac{d}{D_h}\right)^3$
$C_{1B(h)}=3$
$C_{2B(h)}=-6,25-0,585(D/D_h)+3,115(D/D_h)^2$
$C_{3B(h)}=41-1,071(D/D_h)-6,746(D/D_h)^2$
$C_{4B(h)}=-45+1,389(D/D_h)+13,889(D/D_h)^2$
where:
 $K_{tB(h)}$ concentration factor hub in bending stress $\mathrm{-}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$ $C_{1B(h)}$ coefficient $\mathrm{-}$ $C_{2B(h)}$ coefficient $\mathrm{-}$ $C_{3B(h)}$ coefficient $\mathrm{-}$ $C_{4B(h)}$ coefficient $\mathrm{-}$

Concentration factor hub in axial stress:

$0< D/D_h\le0,9;d/D_h\le0,45$
$K_{tA(h)}=C_{1A(h)}+C_{2A(h)}\left(\cfrac{d}{D_h}\right)+C_{3A(h)}\left(\cfrac{d}{D_h}\right)^2$
$C_{1A(h)}=3$
$C_{2A(h)}=0,427-6,77(D/D_h)+22,698(D/D_h)^2-16,67(D/D_h)^3$
$C_{3A(h)}=11,357+15,665(D/D_h)-60,929(D/D_h)^2+41,501(D/D_h)^3$
where:
 $K_{tA(h)}$ concentration factor hub in axial stress $\mathrm{-}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$ $C_{1A(h)}$ coefficient $\mathrm{-}$ $C_{2A(h)}$ coefficient $\mathrm{-}$ $C_{3A(h)}$ coefficient $\mathrm{-}$

Concentration factor hub in torsion stress:

$d/D_h\le0,4;D/D_h\le0,8$
$K_{th}=C_{1(h)}+C_{2(h)}\left(\cfrac{d}{D_h}\right)+C_{3(h)}\left(\cfrac{d}{D_h}\right)^2+C_{4(h)}\left(\cfrac{d}{D_h}\right)^3$
$C_{1(h)}=4$
$C_{2(h)}=-6,055+3,184(D/D_h)-3,461(D/D_h)^2$
$C_{3(h)}=32,764-30,121(D/D_h)+39,887(D/D_h)^2$
$C_{4(h)}=-38,33+51,542\sqrt{D/D_h}-27,483(D/D_h)$
where:
 $K_{th}$ concentration factor hub in torsion stress $\mathrm{-}$ $D$ diameter of the shaft $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $D_h$ diameter of the hub $\mathrm{mm}$ $C_{1(h)}$ coefficient $\mathrm{-}$ $C_{2(h)}$ coefficient $\mathrm{-}$ $C_{3(h)}$ coefficient $\mathrm{-}$ $C_{4(h)}$ coefficient $\mathrm{-}$

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