# Key(s) for shaft hub connection

## Values for calculation

$M_T$ $\mathrm{Nm}$
$D$ $\mathrm{mm}$
$D_h$ $\mathrm{mm}$
$S_{y-shaft}$ $\mathrm{MPa}$
$S_{y-hub}$ $\mathrm{MPa}$
$S_{y-key}$ $\mathrm{MPa}$
$l$ $\mathrm{mm}$
$i$
$C_c$
$S_F$
$M_B$ $\mathrm{Nm}$
$F_R$ $\mathrm{kN}$
$F_A$ $\mathrm{kN}$
$b$ $\mathrm{mm}$
$h$ $\mathrm{mm}$
$t$ $\mathrm{mm}$
$t_t$ $\mathrm{mm}$
$t_1$ $\mathrm{mm}$
$t_{1t}$ $\mathrm{mm}$
$r_1$ $\mathrm{mm}$
$r_2$ $\mathrm{mm}$
$l_t$ $\mathrm{mm}$

## 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 bearing stress the hub

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

### Allowable combined stress the shaft

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

### Allowable shear stress the hub

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

### Allowable shear stress the key

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

### Allowable bearing stress the key

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

### Coefficient $B_T$

$$B_T=1.953+0.1434\cdot\left(\cfrac{0.1}{r_2/D}\right)-0.0021\cdot\left(\cfrac{0.1}{r_2/D}\right)^2$$

### 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 key

$$τ_{S-key}=\cfrac{2\cdot M_T\cdot 10^3}{D\cdot i\cdot\left(\left(l-b-l_t\right)\cdot b+\cfrac{π\cdot b^2}{4}\right)}$$

$$τ_{S-key}\le τ_{all-S-key}$$

### The height of the key in the shaft

$$h_s=t-\cfrac{D}{2}+\cfrac{D}{2}\cdot\cos\left(\sin^{-1}{\cfrac{b}{D}}\right)$$

### Bearing stress in the key and shaft

$$P_{B-key-shaft}=\cfrac{2\cdot M_T\cdot 10^3}{D\cdot i\cdot\left(\left(l-b-l_t\right)\cdot\left(h_s-t_{1t}-\left(t+t_1-h\right)-r_1\right)\right)}$$

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

### The height of the key in the hub

$$h_h=t+t_1-h_s$$

### Bearing stress in the key and hub

$$P_{B-key-hub}=\cfrac{2\cdot M_T\cdot 10^3}{D\cdot i\cdot\left(\left(l-b-l_t\right)\cdot\left(h_h-t_{1t}-\left(t+t_1-h\right)-r_1\right)\right)}$$

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

### Torsion stress in the hub

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

$$τ_{T-hub}\le τ_{all-S-hub}$$

### Coefficient $B_B$

$$B_B=1.426+0.1643\cdot\left(\cfrac{0.1}{r_2/D}\right)-0.0019\cdot\left(\cfrac{0.1}{r_2/D}\right)^2$$

### 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

$$τ_{S-shaft}=\cfrac{10^3 \cdot F_R}{\cfrac{π \cdot D^2}{4}-b\cdot t\cdot i}$$

$$τ_{S-shaft}\le τ_{all-S-shaft}$$

### Coefficient $B_A$

$$B_A=1.6$$

### Axial stress in the shaft

$$σ_{A-shaft}=\cfrac{4\cdot 10^3 \cdot F_A \cdot B_A}{π \cdot D^2}$$

$$σ_{A-shaft}\le σ_{all-A-shaft}$$

### Combined stress in the shaft

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

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

## Requirements

$$\sqrt{\left(D+2\cdot t_1\right)^2+b^2}< D_h$$