# Articulated trunnion in the rod

The trunnion is used to pivot the machine parts that transmit forces perpendicular to the trunnion axis. The trunnions can also be used for short axes such as wheels, pulleys, etc. They are usually mounted with clearance, allowing relative movement. The connecting Trunnions must be secured against axial displacement. Fig.1

Shear stress in the trunnion:

${\tau }_{s\left(t\right)}=\frac{2F}{\pi {d}^{2}}\le {\tau }_{all}$

τs(t) - shear stress in the trunnion - [MPa]

F - force - [N]

d - trunnion diameter - [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 - []

Bending stress in the trunnion:

${\sigma }_{B}=\frac{4F\left(b+2a+4s\right)}{\pi {d}^{3}}\le {\sigma }_{Ball}$

σB - bending stress in the trunnion - [MPa]

F - force - [N]

d - trunnion diameter - [mm]

a - thickness the rod - [mm]

b - thickness the clevis - [mm]

s - gap - [mm]

σBall - allowable bending stress - [MPa]

Allowable bending stress:

${\sigma }_{Ball}=\frac{0,6{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σBall - allowable bending 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 - []

Combined stress in the trunnion:

${\sigma }_{tresca}=\sqrt{{\sigma }_{B}^{2}+4{\tau }_{s\left(t\right)}^{2}}\le {\sigma }_{Call}$

σtresca - combined stress in the trunnion - [MPa]

σB - bending stress in the trunnion - [MPa]

τs(t) - shear stress in the trunnion - [MPa]

σCall - allowable combined stress - [MPa]

Allowable combined stress:

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

σCall - allowable combined 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 rod:

${p}_{r}=\frac{F}{d*a}\le {\sigma }_{all}$

pr - bearing stress in the rod - [MPa]

F - force - [N]

d - trunnion diameter - [mm]

a - thickness the rod - [mm]

σall - allowable bearing stress - [MPa]

Allowable bearing stress:

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

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 - []

Cb - coefficient of use of joints according to trunnion support - []

Coefficient of use of joints according to trunnion support:

 Trunnion support [] fixed fit 1 running fit 0,25

Bearing stress in the clevis:

${p}_{c}=\frac{F}{2d*b}\le {\sigma }_{all}$

pc - bearing stress in the clevis - [MPa]

F - force - [N]

d - trunnion diameter - [mm]

b - thickness the clevis - [mm]

σall - allowable bearing stress - [MPa]

Axial stress in the rod:

${\sigma }_{r}=\frac{{K}_{tr}*F}{\left({l}_{1}-d\right)a}\le {\sigma }_{Aall}$

σr - axial stress in the rod - [MPa]

Ktr - concentration factor in the rod - []

F - force - [N]

d - trunnion diameter - [mm]

a - thickness the rod - [mm]

l1 - width the rod - [mm]

σall - allowable bearing stress - [MPa]

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 - []

Concentration factor in the rod:

${K}_{tr}=12,882-52,714\left(\frac{d}{{l}_{1}}\right)$ $+89,762{\left(\frac{d}{{l}_{1}}\right)}^{2}-51,667{\left(\frac{d}{{l}_{1}}\right)}^{3}$

Ktr - concentration factor in the rod - []

d - trunnion diameter - [mm]

l1 - width the rod - [mm]

Axial stress in the clevis:

${\sigma }_{c}=\frac{{K}_{tc}*F}{\left({l}_{2}-d\right)2b}\le {\sigma }_{Aall}$

σc - axial stress in the clevis - [MPa]

Ktc - concentration factor in the clevis - []

F - force - [N]

d - trunnion diameter - [mm]

b - thickness the clevis - [mm]

l2 - width the clevis - [mm]

σAall - allowable axial stress - [MPa]

Concentration factor in the clevis:

${K}_{tc}=12,882-52,714\left(\frac{d}{{l}_{2}}\right)$ $+89,762{\left(\frac{d}{{l}_{2}}\right)}^{2}-51,667{\left(\frac{d}{{l}_{2}}\right)}^{3}$

Ktc - concentration factor in the clevis - []

d - trunnion diameter - [mm]

l2 - width the clevis - [mm]

Shear stress in the rod:

${\tau }_{s\left(r\right)}=\frac{F}{{h}_{1}*a}\le {\tau }_{all}$

τs(r) - shear stress in the rod - [MPa]

F - force - [N]

a - thickness the rod - [mm]

h1 - length the rod - [mm]

τall - allowable shear stress - [MPa]

Shear stress in the clevis:

${\tau }_{s\left(c\right)}=\frac{F}{{h}_{2}*2b}\le {\tau }_{all}$

τs(c) - shear stress in the clevis - [MPa]

F - force - [N]

b - thickness the clevis - [mm]

h2 - length the clevis - [mm]

τall - allowable shear stress - [MPa]

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

A. Bolek, J. Kochman a kol.: Části a mechanismy strojů I. 1989

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