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Square head for shaft-hub connection

Square head for shaft-hub connection b l M T d 9 s a a 1 d 8 D h d D l d 45° r r b l M T d 9 s a a 1 d 8 D h d D l d 45i r r
Square head for shaft-hub connection

Values for calculation

$ \{M_T\in\mathbb{R}\mid M_T\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ M_T $ $ \mathrm{Nm} $
$ \{d\in\mathbb{R}\mid d>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ d $ $ \mathrm{mm} $
$ \{D\in\mathbb{R}\mid D>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ D $ $ \mathrm{mm} $
$ \{D_h\in\mathbb{R}\mid D_h>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ D_h $ $ \mathrm{mm} $
$ \{s\in\mathbb{R}\mid s>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ s $ $ \mathrm{mm} $
$ \{d_8\in\mathbb{R}\mid d_8>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ d_8 $ $ \mathrm{mm} $
$ \{d_9\in\mathbb{R}\mid d_9\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ d_9 $ $ \mathrm{mm} $
$ \{r\in\mathbb{R}\mid r>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ r $ $ \mathrm{mm} $
$ \{S_{y-shaft}\in\mathbb{R}\mid S_{y-shaft}>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ S_{y-shaft} $ $ \mathrm{MPa} $
$ \{S_{y-hub}\in\mathbb{R}\mid S_{y-hub}>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ S_{y-hub} $ $ \mathrm{MPa} $
$ \{l\in\mathbb{R}\mid l>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ l $ $ \mathrm{mm} $
$ \{S_s\in\mathbb{R}\mid S_s>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ S_s $
$ C_c $
$ \{S_F\in\mathbb{R}\mid S_F>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ S_F $
$ \{M_B\in\mathbb{R}\mid M_B\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ M_B $ $ \mathrm{Nm} $
$ \{F_R\in\mathbb{R}\mid F_R\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ F_R $ $ \mathrm{kN} $
$ \{F_A\in\mathbb{R}\mid F_A\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ F_A $ $ \mathrm{kN} $

Calculation

Allowable axial stress the shaft

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

Length square head without load

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

Length square head with load

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

Distance of the resultant of the pressure

$$b=a_1+\cfrac{2}{3}\cdot a$$
$b=a_1+\cfrac{2}{3}\cdot a$
Equations in LaTeX code
b=a_1+\cfrac{2}{3}\cdot a

Bearing stress

$$P_B=\cfrac{M_T\cdot 10^3 \cdot S_s}{2\cdot a\cdot l\cdot b}$$
$P_B=\cfrac{M_T\cdot 10^3 \cdot S_s}{2\cdot a\cdot l\cdot b}$
Equations in LaTeX code
P_B=\cfrac{M_T\cdot 10^3 \cdot S_s}{2\cdot a\cdot l\cdot b}
$$P_B\le\min\left(P_{all-B-shaft}, P_{all-B-hub}\right)$$
$P_B\le\min\left(P_{all-B-shaft}, P_{all-B-hub}\right)$
Equations in LaTeX code
P_B\le\min\left(P_{all-B-shaft}, P_{all-B-hub}\right)

Coefficient $ B_{1T} $

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

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

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

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

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

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

Torsion stress in the hub

$$τ_{T-hub}=3.962\cdot\cfrac{16\cdot 10^3 \cdot M_T}{π\cdot\left(\left(D_h^4-4\cdot s^4\right)/D_h\right)}$$
$τ_{T-hub}=3.962\cdot\cfrac{16\cdot 10^3 \cdot M_T}{π\cdot\left(\left(D_h^4-4\cdot s^4\right)/D_h\right)}$
Equations in LaTeX code
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)}
$$τ_{T-hub}\le τ_{all-S-hub}$$
$τ_{T-hub}\le τ_{all-S-hub}$
Equations in LaTeX code
τ_{T-hub}\le τ_{all-S-hub}

Coefficient $ B_{1B} $

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

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

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

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

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

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

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

Coefficient $ B_{1A} $

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

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

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

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

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

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

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