# Longitudinal pin for shaft-hub connection

## Values for calculation

$M_T$ $\mathrm{Nm}$
$D$ $\mathrm{mm}$
$d$ $\mathrm{mm}$
$D_h$ $\mathrm{mm}$
$S_{y-shaft}$ $\mathrm{MPa}$
$S_{y-pin}$ $\mathrm{MPa}$
$S_{y-hub}$ $\mathrm{MPa}$
$l$ $\mathrm{mm}$
$i$
$C_c$
$S_F$
$M_B$ $\mathrm{Nm}$
$F_R$ $\mathrm{kN}$
$F_A$ $\mathrm{kN}$

## Calculation

### Allowable axial stress the shaft

$$σ_{all-A-shaft}=\cfrac{0.45 \cdot S_{y-shaft}}{S_F}\cdot C_c$$

### Allowable bending stress the shaft

$$σ_{all-B-shaft}=\cfrac{0.6 \cdot S_{y-shaft}}{S_F}\cdot C_c$$

### Allowable shear stress the shaft

$$τ_{all-S-shaft}=\cfrac{0.4 \cdot S_{y-shaft}}{S_F}\cdot C_c$$

### Allowable bearing stress the shaft

$$P_{all-B-shaft}=\cfrac{0.9 \cdot S_{y-shaft}}{S_F}\cdot C_c$$

### Allowable combined stress the shaft

$$σ_{all-C-shaft}=\cfrac{S_{y-shaft}}{S_F}\cdot C_c$$

### Allowable shear stress the pin

$$τ_{all-S-pin}=\cfrac{0.4 \cdot S_{y-pin}}{S_F}\cdot C_c$$

### Allowable bearing stress the pin

$$P_{all-B-pin}=\cfrac{0.9 \cdot S_{y-pin}}{S_F}\cdot C_c$$

### Allowable shear stress the hub

$$τ_{all-S-hub}=\cfrac{0.4 \cdot S_{y-hub}}{S_F}\cdot C_c$$

### Allowable bending stress the hub

$$σ_{all-B-hub}=\cfrac{0.6 \cdot S_{y-hub}}{S_F}\cdot C_c$$

### Allowable bearing stress the hub

$$P_{all-B-hub}=\cfrac{0.9 \cdot S_{y-hub}}{S_F}\cdot C_c$$

### Coefficient $B_T$

$\text{if }\ i=1$
$$B_T=1.61148+1.85385\cdot\cfrac{d}{D}-4.38205\cdot\left(\cfrac{d}{D}\right)^2+5.8264\cdot\left(\cfrac{d}{D}\right)^3$$
$\text{else if }\ i=2$
$$B_T=1.57944+2.42327\cdot\cfrac{d}{D}-4.634\cdot\left(\cfrac{d}{D}\right)^2+11.07945\cdot\left(\cfrac{d}{D}\right)^3$$
$\text{else}$
$$B_T=1.90616-4.66479\cdot\cfrac{d}{D}+52.95626\cdot\left(\cfrac{d}{D}\right)^2-156.30366\cdot\left(\cfrac{d}{D}\right)^3+204.97321\cdot\left(\cfrac{d}{D}\right)^4$$

### Coefficient $B_B$

$\text{if }\ i=1$
$$B_B=0.797+7.794\cdot\cfrac{d}{D}-43.549\cdot\left(\cfrac{d}{D}\right)^2+122.372\cdot\left(\cfrac{d}{D}\right)^3-112.269\cdot\left(\cfrac{d}{D}\right)^4$$
$\text{else if }\ i=2$
$$B_B=0.997+4.127\cdot\cfrac{d}{D}-23.179\cdot\left(\cfrac{d}{D}\right)^2+81.608\cdot\left(\cfrac{d}{D}\right)^3-77.795\cdot\left(\cfrac{d}{D}\right)^4$$
$\text{else}$
$$B_B=0.683+9.809\cdot\cfrac{d}{D}-52.437\cdot\left(\cfrac{d}{D}\right)^2+139.668\cdot\left(\cfrac{d}{D}\right)^3-114.395\cdot\left(\cfrac{d}{D}\right)^4$$

### Coefficient $B_A$

$\text{if }\ i=1$
$$B_A=1.482+1.635\cdot\cfrac{d}{D}-17.72\cdot\left(\cfrac{d}{D}\right)^2+64.654\cdot\left(\cfrac{d}{D}\right)^3-58.064\cdot\left(\cfrac{d}{D}\right)^4$$
$\text{else if }\ i=2$
$$B_A=1.491+1.283\cdot\cfrac{d}{D}-12.812\cdot\left(\cfrac{d}{D}\right)^2+40.026\cdot\left(\cfrac{d}{D}\right)^3-33.061\cdot\left(\cfrac{d}{D}\right)^4$$
$\text{else}$
$$B_A=1.463+2.297\cdot\cfrac{d}{D}-23.872\cdot\left(\cfrac{d}{D}\right)^2+81.116\cdot\left(\cfrac{d}{D}\right)^3-72.492\cdot\left(\cfrac{d}{D}\right)^4$$

### Torsion stress in the shaft

$$τ_{T-shaft}=\cfrac{16\cdot 10^3 \cdot M_T\cdot B_T}{π \cdot D^3}$$

$$τ_{T-shaft}\le τ_{all-S-shaft}$$

### Shear stress in the pin

$$τ_{S-pin}=\cfrac{2\cdot 10^3 \cdot M_T}{D\cdot d\cdot l\cdot i}$$

$$τ_{S-pin}\le τ_{all-S-pin}$$

### Bearing stress

$$P_B=\cfrac{4\cdot 10^3 \cdot M_T}{D\cdot d\cdot l\cdot i}$$

$$P_B\le\min\left(P_{all-B-shaft}, P_{all-B-pin}, P_{all-B-hub}\right)$$

### Torsion stress in the hub

$\text{if }\ \cfrac{d}{D_h-D} > 0.25$
$$τ_{T-hub}=\cfrac{16\cdot 10^3 \cdot M_T \cdot B_T}{π\cdot\left(D_h^4-D^4\right)}\cdot D$$
$\text{else}$
$$τ_{T-hub}=\cfrac{16\cdot 10^3 \cdot M_T}{π\cdot\left(D_h^4-D^4\right)}\cdot D_h$$

$$τ_{T-hub}\le τ_{all-S-hub}$$

### Bending stress in the shaft

$$σ_{B-shaft}=\cfrac{32\cdot 10^3 \cdot M_B \cdot B_B}{π \cdot D^3}$$

$$σ_{B-shaft}\le σ_{all-B-shaft}$$

### Shear stress in the shaft

$$τ_{S-shaft}=\cfrac{8\cdot 10^3 \cdot F_R}{2\cdot π \cdot D^2-π \cdot d^2\cdot i}$$

$$τ_{S-shaft}\le τ_{all-S-shaft}$$

### Axial stress in the shaft

$$σ_{A-shaft}=\cfrac{8\cdot 10^3 \cdot F_A \cdot B_A}{2\cdot π \cdot D^2-π \cdot d^2\cdot i}$$

$$σ_{A-shaft}\le σ_{all-A-shaft}$$

### Combined stress in the shaft

$$σ_{tresca-shaft}=\sqrt{σ_{B-shaft}^2+σ_{A-shaft}^2+4\cdot\left(τ_{T-shaft}^2+τ_{S-shaft}^2\right)}$$

$$σ_{tresca-shaft}\le σ_{all-C-shaft}$$

### Angle between pins

$$α=\cfrac{2 \cdot π-\left(2\cdot i \cdot \sin\left(\cfrac{d}{D}\right)\right)}{i}$$

### Width between pins

$$t=D\cdot\cos\left(\sin\left(\cfrac{d}{D}\right)\right)\cdot\sin\left(\cfrac{α}{2}\right)$$

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

$$σ_{B-shaft-hub}=\cfrac{3 \cdot 10^3 \cdot d \cdot M_T}{D\cdot l \cdot t^2 \cdot i}$$

$$σ_{B-shaft-hub}\le \min\left(σ_{all-B-shaft}, σ_{all-B-hub}\right)$$

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

$$τ_{S-shaft-hub}=\cfrac{3 \cdot 10^3 \cdot M_T}{D\cdot l \cdot t \cdot i}$$

$$τ_{S-shaft-hub}\le \min\left(τ_{all-S-shaft}, τ_{all-S-hub}\right)$$

## Requirements

$$\cfrac{d}{D_h-D}\le 0.4$$$$\cfrac{d}{D}\geq 0.1$$$$2\cdot i\cdot \sin^{-1}\left(\cfrac{d}{D}\right)< 2\cdot π$$