## The cross pin connects 2 flanges (torque load)

The easiest and oldest way joints. It is a joint with a shape contact. The pin serves primarily to ensure the mutual positioning of the two parts. They are cylindrical or conical. The pins are dimensioned under simplified assumptions without will and without the pressing effect. When calculating the pin, the length of the pin should not be considered, which is different from the nominal cross-section see, for example, the thread in the pin etc.

Fig. 1 - The cross pin connects 2 flanges (torque load)

Shear stress in the pin:

$\tau_p=\cfrac{8M_T}{i\cdot\pi\cdot d^2\cdot D}\le\tau_{all}$
where:
 $τ_p$ shear stress in the pin $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ Pitch diameter for pins $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $d$ diameter of the pin $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

Allowable shear stress:

$\tau_{all}=\cfrac{0,4R_{p0,2/T}}{S_F}\cdot C_c$
where:
 $τ_{all}$ allowable shear stress $\mathrm{MPa}$ $R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$ $S_F$ safety factor $\mathrm{-}$ $C_c$ coefficient according to load $\mathrm{-}$

load $\mathrm{-}$

Bearing stress in the pin and flange 1:

$p_1=\cfrac{2\cdot M_T}{i\cdot s_1\cdot d\cdot D}\le\sigma_{all}$
where:
 $p_1$ bearing stress in the pin and flange 1 $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ Pitch diameter for pins $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $s_1$ pin length in flange 1 $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $\sigma_{all}$ allowable bearing stress $\mathrm{MPa}$

Allowable bearing stress:

$\sigma_{all}=\cfrac{0,9R_{p0,2/T}}{S_F}\cdot C_c$
where:
 $\sigma_{all}$ allowable bearing stress $\mathrm{MPa}$ $R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$ $S_F$ safety factor $\mathrm{-}$ $C_c$ coefficient according to load $\mathrm{-}$

Bearing stress in the pin and flange 2:

$p_2=\cfrac{{2\cdot M}_T}{i\cdot s_2\cdot d\cdot D}\le\sigma_{all}$
where:
 $p_2$ bearing stress in the pin and flange 2 $\mathrm{MPa}$ $M_T$ torque $\mathrm{Nm}$ $D$ pitch diameter for pins $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $s_2$ pin length in flange 2 $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $\sigma_{all}$ allowable bearing stress $\mathrm{MPa}$

Axial stress in the flange 1:

$\sigma_{A1}=\cfrac{K_{t1}\cdot4\cdot M_T}{i\cdot D\cdot\left(D_{1B}-D_{1A}-2d\right)\cdot s_1}\le\sigma_{Aall}$
where:
 $\sigma_{A1}$ axial stress in the flange 1 $\mathrm{MPa}$ $K_{t1}$ concentration factor in flange 11 $\mathrm{-}$ $M_T$ torque $\mathrm{Nm}$ $D$ pitch diameter for pins $\mathrm{mm}$ $s_1$ pin length in flange 1 $\mathrm{mm}$ $D_{1B}$ outer diameter of flange 1 $\mathrm{mm}$ $D_{1A}$ inner diameter of flange 1 $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $d$ diameter of the pin $\mathrm{mm}$ $\sigma_{Aall}$ allowable axial stress $\mathrm{MPa}$

Concentration factor in flange 1:

$K_{t1}=C_{1A}+C_{1B}\left(\cfrac{d}{{2c}_1}\right)+C_{1C}\left(\cfrac{d}{{2c}_1}\right)^2+C_{1D}\left(\cfrac{d}{{2c}_1}\right)^3$
$C_{1A}=2,9969-0,009\left(\cfrac{c_1}{e_1}\right)+0,01338\left(\cfrac{c_1}{e_1}\right)^2$
$C_{1B}=0,1217+0,518\left(\cfrac{c_1}{e_1}\right)-0,5297\left(\cfrac{c_1}{e_1}\right)^2$
$C_{1C}=0,5565+0,7215\left(\cfrac{c_1}{e_1}\right)+0,6153\left(\cfrac{c_1}{e_1}\right)^2$
$C_{1D}=4,082+6,0146\left(\cfrac{c_1}{e_1}\right)-3,9815\left(\cfrac{c_1}{e_1}\right)^2$
$c_1=\min\left\{\cfrac{D-D_{1A}}{2};\cfrac{D_{1B}-D}{2}\right\}$
$e_1=\max\left\{\cfrac{D-D_{1A}}{2};\cfrac{D_{1B}-D}{2}\right\}$
where:
 $K_{t1}$ concentration factor in flange 1 $\mathrm{-}$ $C_{1A}$ coefficient $\mathrm{-}$ $C_{1B}$ coefficient $\mathrm{-}$ $C_{1C}$ coefficient $\mathrm{-}$ $C_{1D}$ coefficient $\mathrm{-}$ $d$ diameter of the pin $\mathrm{mm}$ $c_1$ dimension $\mathrm{mm}$ $e_1$ dimension $\mathrm{mm}$ $D$ pitch diameter for pins $\mathrm{mm}$ $D_{1B}$ outer diameter of flange 1 $\mathrm{mm}$ $D_{1A}$ inner diameter of flange 1 $\mathrm{mm}$

Allowable axial stress

$\sigma_{Aall}=\cfrac{0,45R_{p0,2/T}}{S_F}\cdot C_c$
where:
 $\sigma_{Aall}$ allowable axial stress $\mathrm{MPa}$ $R_{p 0,2 /T}$ the minimum yield strength or 0,2% proof strength at calculation temperature $\mathrm{MPa}$ $S_F$ safety factor $\mathrm{-}$ $C_c$ coefficient according to load $\mathrm{-}$

Axial stress in the flange 2:

$\sigma_{A2}=\cfrac{K_{t2}\cdot4{\cdot M}_T}{i\cdot D\cdot\left(D_{2B}-D_{2A}-2d\right)\cdot s_2}\le\sigma_{Aall}$
where:
 $\sigma_{A2}$ axial stress in the flange 2 $\mathrm{MPa}$ $K_{t2}$ concentration factor in flange 2 $\mathrm{-}$ $M_T$ torque $\mathrm{Nm}$ $D$ pitch diameter for pins $\mathrm{mm}$ $s_2$ pin length in flange 2 $\mathrm{mm}$ $D_{2B}$ outer diameter of flange 2 $\mathrm{mm}$ $D_{2A}$ inner diameter of flange 2 $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $d$ diameter of the pin $\mathrm{mm}$ $\sigma_{Aall}$ allowable axial stress $\mathrm{MPa}$

Concentration factor in flange 2:

$K_{t2}=C_{2A}+C_{2B}\left(\cfrac{d}{{2c}_2}\right)+C_{2C}\left(\cfrac{d}{{2c}_2}\right)^2+C_{2D}\left(\cfrac{d}{{2c}_2}\right)^3$
$C_{2A}=2,9969-0,009\left(\cfrac{c_2}{e_2}\right)+0,01338\left(\cfrac{c_2}{e_2}\right)^2$
$C_{2B}=0,1217+0,518\left(\cfrac{c_2}{e_2}\right)-0,5297\left(\cfrac{c_2}{e_2}\right)^2$
$C_{2C}=0,5565+0,7215\left(\cfrac{c_2}{e_2}\right)+0,6153\left(\cfrac{c_2}{e_1}\right)^2$
$C_{2D}=4,082+6,0146\left(\cfrac{c_2}{e_2}\right)-3,9815\left(\cfrac{c_2}{e_2}\right)^2$
$c_2=\min\left\{\cfrac{D-D_{2A}}{2};\cfrac{D_{2B}-D}{2}\right\}$
$e_2=\max\left\{\cfrac{D-D_{2A}}{2};\cfrac{D_{2B}-D}{2}\right\}$
where:
 $K_{t2}$ concentration factor in flange 2 $\mathrm{-}$ $C_{2A}$ coefficient $\mathrm{-}$ $C_{2B}$ coefficient $\mathrm{-}$ $C_{2C}$ coefficient $\mathrm{-}$ $C_{2D}$ coefficient $\mathrm{-}$ $d$ diameter of the pin $\mathrm{mm}$ $c_2$ dimension $\mathrm{mm}$ $e_2$ dimension $\mathrm{mm}$ $D$ pitch diameter for pins $\mathrm{mm}$ $D_{2B}$ outer diameter of flange 2 $\mathrm{mm}$ $D_{2A}$ inner diameter of flange 2 $\mathrm{mm}$

Diameter in which there must be no other debilitating element:

$d_1=\cfrac{4M_T}{D\cdot i}\cdot \max\left\{\cfrac{1}{{\tau_{all}\cdot s}_1};\cfrac{1}{{\tau_{all}\cdot s}_2}\right\}+d$
where:
 $d_1$ diameter in which there must be no other debilitating element $\mathrm{mm}$ $M_T$ torque $\mathrm{Nm}$ $D$ pitch diameter for pins $\mathrm{mm}$ $i$ number of pins $\mathrm{-}$ $s_1$ pin length in flange 1 $\mathrm{mm}$ $s_2$ pin length in flange 2 $\mathrm{mm}$ $d$ diameter of the pin $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

In the marked diameter $d_1$ and there must be no other weakening element in the cross-section in which the pin is located.

Literature:
- AISC: Specification for structural steel buildings: Allowable Stress design and plastic design 1989.
- Walter D. Pilkey, Deborah F. Pilkey: Peterson’s stress concentration factors. 2008.
- Joseph E. Shigley, Charles R. Mischke, Richard G. Budynas: Konstruování strojních součástí 2010.
- K. Kříž a kol.: Strojní součásti 1. 1984.