# 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:

${\tau }_{p}=\frac{8{M}_{k}}{i*\pi *{d}^{2}*D}\le {\tau }_{all}$

τp - shear stress in the pin - [MPa]

MT - torque - [Nm]

D - Pitch diameter for pins - [mm]

i - number of pins - []

d - diameter of the pin - [mm]

τall - allowable shear stress - [MPa]

Allowable shear stress:

${\tau }_{all}=\frac{0,4{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

τall - allowable shear stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient according to load - []

Bearing stress in the pin and flange 1:

${p}_{1}=\frac{2*{M}_{T}}{i*{s}_{1}*d*D}\le {\sigma }_{all}$

P1 - bearing stress in the pin and flange 1 - [MPa]

MT - torque - [Nm]

D - Pitch diameter for pins - [mm]

i - number of pins - []

s1 - pin length in flange 1 - [mm]

d - diameter of the pin - [mm]

σall - allowable bearing stress - [MPa]

Allowable bearing stress:

${\sigma }_{all}=\frac{0,9{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σall - allowable bearing stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient according to load - []

Bearing stress in the pin and flange 2:

${p}_{2}=\frac{2*{M}_{T}}{i*{s}_{2}*d*D}\le {\sigma }_{all}$

P2 - bearing stress in the pin and flange 2 - [MPa]

MT - torque - [Nm]

D - Pitch diameter for pins - [mm]

i - number of pins - []

s2 - pin length in flange 2 - [mm]

d - diameter of the pin - [mm]

σall - allowable bearing stress - [MPa]

Axial stress in the flange 1:

${\sigma }_{A1}=\frac{{K}_{t1}*4*{M}_{T}}{i*D*\left({D}_{1B}-{D}_{1A}-2d\right)*{s}_{1}}\le {\sigma }_{Aall}$

σA1 - Axial stress in the flange 1 - [MPa]

Kt1 - concentration factor in flange 1 - []

MT - torque - [Nm]

D - Pitch diameter for pins - [mm]

s1 - pin length in flange 1 - [mm]

D1B - outer diameter of flange 1 - [mm]

D1A - inner diameter of flange 1 - [mm]

i - number of pins - []

d - diameter of the pin - [mm]

σAall - allowable axial stress - [MPa]

Concentration factor in flange 1:

${K}_{t1}={C}_{1A}+{C}_{1B}\left(\frac{d}{{2c}_{1}}\right)$ $+{C}_{1C}{\left(\frac{d}{{2c}_{1}}\right)}^{2}+{C}_{1D}{\left(\frac{d}{{2c}_{1}}\right)}^{3}$
${C}_{1A}=2,9969-0,0090\left(\frac{{c}_{1}}{{e}_{1}}\right)+0,01338{\left(\frac{{c}_{1}}{{e}_{1}}\right)}^{2}$
${C}_{1B}=0,1217+0,5180\left(\frac{{c}_{1}}{{e}_{1}}\right)-0,5297{\left(\frac{{c}_{1}}{{e}_{1}}\right)}^{2}$
${C}_{1C}=0,5565+0,7215\left(\frac{{c}_{1}}{{e}_{1}}\right)+0,6153{\left(\frac{{c}_{1}}{{e}_{1}}\right)}^{2}$
${C}_{1D}=4,082+6,0146\left(\frac{{c}_{1}}{{e}_{1}}\right)-3,9815{\left(\frac{{c}_{1}}{{e}_{1}}\right)}^{2}$
${c}_{1}=min\left\{\frac{D-{D}_{1A}}{2};\frac{{D}_{1B}-D}{2}\right\}$
${e}_{1}=max\left\{\frac{D-{D}_{1A}}{2};\frac{{D}_{1B}-D}{2}\right\}$

Kt1 - concentration factor in flange 1 - []

C1A - coefficient - []

C1B - coefficient - []

C1C - coefficient - []

C1D - coefficient - []

d - diameter of the pin - [mm]

c1 - dimension - [mm]

e1 - dimension - [mm]

D - Pitch diameter for pins - [mm]

D1B - outer diameter of flange 1 - [mm]

D1A - inner diameter of flange 1 - [mm]

Allowable axial stress:

${\sigma }_{Aall}=\frac{0,45{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σAall - allowable axial stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient according to load - []

Axial stress in the flange 2:

${\sigma }_{A2}=\frac{{K}_{t2}*4*{M}_{T}}{i*D*\left({D}_{2B}-{D}_{2A}-2d\right)*{s}_{2}}\le {\sigma }_{Aall}$

σA2 - Axial stress in the flange 2 - [MPa]

Kt2 - concentration factor in flange 2 - []

MT - torque - [Nm]

D - Pitch diameter for pins - [mm]

s2 - pin length in flange 2 - [mm]

D2B - outer diameter of flange 2 - [mm]

D2A - inner diameter of flange 2 - [mm]

i - number of pins - []

d - diameter of the pin - [mm]

σAall - allowable axial stress - [MPa]

Concentration factor in flange 2:

${K}_{t2}={C}_{2A}+{C}_{2B}\left(\frac{d}{{2c}_{2}}\right)$ $+{C}_{2C}{\left(\frac{d}{{2c}_{2}}\right)}^{2}+{C}_{2D}{\left(\frac{d}{{2c}_{2}}\right)}^{3}$
${C}_{2A}=2,9969-0,0090\left(\frac{{c}_{2}}{{e}_{2}}\right)+0,01338{\left(\frac{{c}_{2}}{{e}_{2}}\right)}^{2}$
${C}_{2B}=0,1217+0,5180\left(\frac{{c}_{2}}{{e}_{2}}\right)-0,5297{\left(\frac{{c}_{2}}{{e}_{2}}\right)}^{2}$
${C}_{2C}=0,5565+0,7215\left(\frac{{c}_{2}}{{e}_{2}}\right)+0,6153{\left(\frac{{c}_{2}}{{e}_{2}}\right)}^{2}$
${C}_{2D}=4,082+6,0146\left(\frac{{c}_{2}}{{e}_{2}}\right)-3,9815{\left(\frac{{c}_{2}}{{e}_{2}}\right)}^{2}$
${c}_{2}=min\left\{\frac{D-{D}_{2A}}{2};\frac{{D}_{2B}-D}{2}\right\}$
${e}_{2}=max\left\{\frac{D-{D}_{2A}}{2};\frac{{D}_{2B}-D}{2}\right\}$

Kt2 - concentration factor in flange 2 - []

C2A - coefficient - []

C2B - coefficient - []

C2C - coefficient - []

C2D - coefficient - []

d - diameter of the pin - [mm]

c2 - dimension - [mm]

e2 - dimension - [mm]

D - Pitch diameter for pins - [mm]

D2B - outer diameter of flange 2 - [mm]

D2A - inner diameter of flange 2 - [mm]

Diameter in which there must be no other debilitating element:

${d}_{1}=\frac{4{M}_{T}}{D*i}*max\left\{\frac{1}{{{\tau }_{all}*s}_{1}};\frac{1}{{{\tau }_{all}*s}_{2}}\right\}+d$

d1 - diameter in which there must be no other debilitating element - [mm]

MT - torque - [Nm]

D - Pitch diameter for pins - [mm]

i - number of pins - []

s1 - pin length in flange 1 - [mm]

s2 - pin length in flange 2 - [mm]

d - diameter of the pin - [mm]

τall - allowable shear stress - [MPa]

In the marked diameter d1 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.

MET-Calc: Allowable stress

K. Kříž a kol.: Strojní součásti 1. 1984.