# Square head for shaft-hub connection

## Values for calculation

$M_T$ $\mathrm{Nm}$
$d$ $\mathrm{mm}$
$D$ $\mathrm{mm}$
$D_h$ $\mathrm{mm}$
$s$ $\mathrm{mm}$
$d_8$ $\mathrm{mm}$
$d_9$ $\mathrm{mm}$
$r$ $\mathrm{mm}$
$S_{y-shaft}$ $\mathrm{MPa}$
$S_{y-hub}$ $\mathrm{MPa}$
$l$ $\mathrm{mm}$
$S_s$
$C_c$
$S_F$
$M_B$ $\mathrm{Nm}$
$F_R$ $\mathrm{kN}$
$F_A$ $\mathrm{kN}$

## 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 hub

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

### Allowable bearing stress the hub

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

$\text{if }\ d_9=0$
$$a_1=0$$
$\text{else}$
$$a_1=\cfrac{d_9}{2}\cdot\sin{\left(\cos^{-1}{\cfrac{s}{d_9}}\right)}$$

$\text{if }\ d_8=0$
$$a=0$$
$\text{else}$
$$a=\cfrac{d_8}{2}\cdot\sin{\left(\cos^{-1}{\cfrac{s}{d_8}}\right)}-a_1$$

### Distance of the resultant of the pressure

$$b=a_1+\cfrac{2}{3}\cdot a$$

### Bearing stress

$$P_B=\cfrac{M_T\cdot 10^3 \cdot S_s}{2\cdot a\cdot l\cdot b}$$

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

### Coefficient $B_{1T}$

$$B_{1T}=0.905+0.783\cdot\sqrt{\cfrac{D-d}{2\cdot r}}-0.075\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_{2T}$

$$B_{2T}=-0.437-1.969\cdot\sqrt{\cfrac{D-d}{2\cdot r}}+0.553\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_{3T}$

$$B_{3T}=1.557+1.073\cdot\sqrt{\cfrac{D-d}{2\cdot r}}-0.578\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_{4T}$

$$B_{4T}=-1.061+0.171\cdot\sqrt{\cfrac{D-d}{2\cdot r}}+0.086\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_T$

$$B_T=B_{1T}+B_{2T}\cdot\left(\cfrac{D-d}{D}\right)+B_{3T}\cdot\left(\cfrac{D-d}{D}\right)^2+B_{4T}\cdot\left(\cfrac{D-d}{D}\right)^3$$

### Torsion stress in the shaft

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

$$τ_{T-shaft}\le τ_{all-S-shaft}$$

### Torsion stress in the hub

$$τ_{T-hub}=3.962\cdot\cfrac{16\cdot 10^3 \cdot M_T}{π\cdot\left(\left(D_h^4-4\cdot s^4\right)/D_h\right)}$$

$$τ_{T-hub}\le τ_{all-S-hub}$$

### Coefficient $B_{1B}$

$$B_{1B}=0.947+1.206\cdot\sqrt{\cfrac{D-d}{2\cdot r}}-0.131\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_{2B}$

$$B_{2B}=0.022-3.405\cdot\sqrt{\cfrac{D-d}{2\cdot r}}+0.915\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_{3B}$

$$B_{3B}=0.869+1.777\cdot\sqrt{\cfrac{D-d}{2\cdot r}}-0.555\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_{4B}$

$$B_{4B}=-0.810+0.422\cdot\sqrt{\cfrac{D-d}{2\cdot r}}-0.260\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_B$

$$B_B=B_{1B}+B_{2B}\cdot\left(\cfrac{D-d}{D}\right)+B_{3B}\cdot\left(\cfrac{D-d}{D}\right)^2+B_{4B}\cdot\left(\cfrac{D-d}{D}\right)^3$$

### 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

$$τ_{S-shaft}=\cfrac{4\cdot 10^3 \cdot F_R}{π \cdot d^2}$$

$$τ_{S-shaft}\le τ_{all-S-shaft}$$

### Coefficient $B_{1A}$

$$B_{1A}=0.926+1.157\cdot\sqrt{\cfrac{D-d}{2\cdot r}}-0.099\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_{2A}$

$$B_{2A}=0.012-3.036\cdot\sqrt{\cfrac{D-d}{2\cdot r}}+0.961\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_{3A}$

$$B_{3A}=-0.302+3.977\cdot\sqrt{\cfrac{D-d}{2\cdot r}}-1.744\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_{4A}$

$$B_{4A}=0.365-2.098\cdot\sqrt{\cfrac{D-d}{2\cdot r}}+0.878\cdot\cfrac{D-d}{2\cdot r}$$

### Coefficient $B_A$

$$B_A=B_{1A}+B_{2A}\cdot\left(\cfrac{D-d}{D}\right)+B_{3A}\cdot\left(\cfrac{D-d}{D}\right)^2+B_{4A}\cdot\left(\cfrac{D-d}{D}\right)^3$$

### 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

$$0.25< \left(D-d\right)/\left(2\cdot r\right)\le2$$$$s< d_8$$$$d_8/s \le \cfrac{1}{\sin{45°}}$$$$d_9< d_8$$