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Key(s) for shaft hub connection

Key(s) for shaft hub connection D h D b h t
Key(s) for shaft hub connection
Key l b h r 1 r 1
Key

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

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

Allowable bending stress the shaft

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

Allowable shear stress the shaft

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

Allowable bearing stress the shaft

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

Allowable bearing stress the hub

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

Allowable combined stress the shaft

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

Allowable shear stress the hub

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

Allowable shear stress the key

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

Allowable bearing stress the key

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

Coefficient $ B_T $

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

$$τ_{T-shaft}=\cfrac{16\cdot 10^3 \cdot M_T\cdot B_T}{π \cdot D^3}$$
$τ_{T-shaft}=\cfrac{16\cdot 10^3 \cdot M_T\cdot B_T}{π \cdot D^3}$
Equations in LaTeX code
τ_{T-shaft}=\cfrac{16\cdot 10^3 \cdot M_T\cdot B_T}{π \cdot D^3}
$$τ_{T-shaft}\le τ_{all-S-shaft}$$
$τ_{T-shaft}\le τ_{all-S-shaft}$
Equations in LaTeX code
τ_{T-shaft}\le τ_{all-S-shaft}

Shear stress key

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

The height of the key in the shaft

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

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

The height of the key in the hub

$$h_h=t+t_1-h_s$$
$h_h=t+t_1-h_s$
Equations in LaTeX code
h_h=t+t_1-h_s

Bearing stress in the key and hub

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

Torsion stress in the hub

$$τ_{T-hub}=\cfrac{16\cdot 10^3 \cdot M_T \cdot B_T}{π\cdot\left(D_h^4-D^4\right)}\cdot D$$
$τ_{T-hub}=\cfrac{16\cdot 10^3 \cdot M_T \cdot B_T}{π\cdot\left(D_h^4-D^4\right)}\cdot D$
Equations in LaTeX code
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
$$τ_{T-hub}\le τ_{all-S-hub}$$
$τ_{T-hub}\le τ_{all-S-hub}$
Equations in LaTeX code
τ_{T-hub}\le τ_{all-S-hub}

Coefficient $ B_B $

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

$$σ_{B-shaft}=\cfrac{32\cdot 10^3 \cdot M_B \cdot B_B}{π \cdot D^3}$$
$σ_{B-shaft}=\cfrac{32\cdot 10^3 \cdot M_B \cdot B_B}{π \cdot D^3}$
Equations in LaTeX code
σ_{B-shaft}=\cfrac{32\cdot 10^3 \cdot M_B \cdot B_B}{π \cdot D^3}
$$σ_{B-shaft}\le σ_{all-B-shaft}$$
$σ_{B-shaft}\le σ_{all-B-shaft}$
Equations in LaTeX code
σ_{B-shaft}\le σ_{all-B-shaft}

Shear stress in the shaft

$$τ_{S-shaft}=\cfrac{10^3 \cdot F_R}{\cfrac{π \cdot D^2}{4}-b\cdot t\cdot i}$$
$τ_{S-shaft}=\cfrac{10^3 \cdot F_R}{\cfrac{π \cdot D^2}{4}-b\cdot t\cdot i}$
Equations in LaTeX code
τ_{S-shaft}=\cfrac{10^3 \cdot F_R}{\cfrac{π \cdot D^2}{4}-b\cdot t\cdot i}
$$τ_{S-shaft}\le τ_{all-S-shaft}$$
$τ_{S-shaft}\le τ_{all-S-shaft}$
Equations in LaTeX code
τ_{S-shaft}\le τ_{all-S-shaft}

Coefficient $ B_A $

$$B_A=1.6$$
$B_A=1.6$
Equations in LaTeX code
B_A=1.6

Axial stress in the shaft

$$σ_{A-shaft}=\cfrac{4\cdot 10^3 \cdot F_A \cdot B_A}{π \cdot D^2}$$
$σ_{A-shaft}=\cfrac{4\cdot 10^3 \cdot F_A \cdot B_A}{π \cdot D^2}$
Equations in LaTeX code
σ_{A-shaft}=\cfrac{4\cdot 10^3 \cdot F_A \cdot B_A}{π \cdot D^2}
$$σ_{A-shaft}\le σ_{all-A-shaft}$$
$σ_{A-shaft}\le σ_{all-A-shaft}$
Equations in LaTeX code
σ_{A-shaft}\le σ_{all-A-shaft}

Combined stress in the shaft

$$σ_{tresca-shaft}=\sqrt{σ_{B-shaft}^2+σ_{A-shaft}^2+4\cdot\left(τ_{T-shaft}^2+τ_{S-shaft}^2\right)}$$
$σ_{tresca-shaft}=\sqrt{σ_{B-shaft}^2+σ_{A-shaft}^2+4\cdot\left(τ_{T-shaft}^2+τ_{S-shaft}^2\right)}$
Equations in LaTeX code
σ_{tresca-shaft}=\sqrt{σ_{B-shaft}^2+σ_{A-shaft}^2+4\cdot\left(τ_{T-shaft}^2+τ_{S-shaft}^2\right)}
$$σ_{tresca-shaft}\le σ_{all-C-shaft}$$
$σ_{tresca-shaft}\le σ_{all-C-shaft}$
Equations in LaTeX code
σ_{tresca-shaft}\le σ_{all-C-shaft}

Requirements

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