## 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 - Articulated trunnion in the rod

Shear stress in the trunnion:

$\tau_{s\left(t\right)}=\cfrac{2F}{\pi d^2}\le\tau_{all}$
where:
 $τ_{s(t)}$ shear stress in the trunnion $\mathrm{MPa}$ $F$ force $\mathrm{N}$ $d$ trunnion diameter $\mathrm{mm}$ $τ_{all}$ 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 $\mathrm{-}$

Coefficient according to load:

load $\mathrm{-}$
Unidirectional load, with a small impact load 0,7
Unidirectional load, with a big impact load 0,6
Alternating load, with a small impact load 0,45
Alternating load, with a big impact load 0,25

Bending stress in the trunnion:

$\sigma_B=\cfrac{4F\left(a+2b+2s\right)}{\pi d^3}\le\sigma_{Ball}$
where:
 $\sigma_B$ bending stress in the trunnion $\mathrm{MPa}$ $F$ force $\mathrm{N}$ $d$ trunnion diameter $\mathrm{mm}$ $a$ thickness the rod $\mathrm{mm}$ $b$ thickness the clevis $\mathrm{mm}$ $s$ gap $\mathrm{mm}$ $\sigma_{Ball}$ allowable bending stress $\mathrm{MPa}$

Allowable bending stress:

$\sigma_{Ball}=\cfrac{0,6R_{p0,2/T}}{S_F}\cdot C_c$
where:
 $\sigma_{Ball}$ allowable bending 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 $\mathrm{-}$

Combined stress in the trunnion:

$\sigma_{tresca}=\sqrt{\sigma_B^2+4\tau_{s\left(t\right)}^2}\le\sigma_{Call}$
where:
 $\sigma_{tresca}$ combined stress in the trunnion $\mathrm{MPa}$ $\sigma_B$ bending stress in the trunnion $\mathrm{MPa}$ $τ_{s(t)}$ shear stress in the trunnion $\mathrm{MPa}$ $\sigma_{Call}$ allowable combined stress $\mathrm{MPa}$

Allowable combined stress:

$\sigma_{Call}=\cfrac{R_{p0,2/T}}{S_F}\cdot C_c$
where:
 $\sigma_{Call}$ allowable combined 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 rod:

$p_r=\cfrac{F}{d\cdot a}\le\sigma_{all}$
where:
 $p_r$ bearing stress in the rod $\mathrm{MPa}$ $F$ force $\mathrm{N}$ $d$ trunnion diameter $\mathrm{mm}$ $a$ thickness the rod $\mathrm{mm}$ $\sigma_{all}$ allowable bearing stress $\mathrm{MPa}$

Allowable bearing stress:

$\sigma_{all}=\cfrac{0,9R_{p0,2/T}}{S_F}\cdot C_c\cdot C_b$
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 $\mathrm{-}$ $C_b$ coefficient of use of joints according to trunnion support $\mathrm{-}$

Coefficient of use of joints according to trunnion support:

Trunnion support $\mathrm{-}$
fixed fit 1
running fit 0,25

Bearing stress in the clevis:

$p_c=\cfrac{F}{2d\cdot b}\le\sigma_{all}$
where:
 $p_c$ bearing stress in the clevis $\mathrm{MPa}$ $F$ force $\mathrm{N}$ $d$ trunnion diameter $\mathrm{mm}$ $b$ thickness the clevis $\mathrm{mm}$ $\sigma_{all}$ allowable bearing stress $\mathrm{MPa}$

Axial stress in the rod:

$\sigma_r=\cfrac{K_{tr}\cdot F}{\left(l_1-d\right)a}\le\sigma_{Aall}$
where:
 $\sigma_r$ axial stress in the rod $\mathrm{MPa}$ $K_{tr}$ concentration factor in the rod $\mathrm{-}$ $F$ force $\mathrm{N}$ $d$ trunnion diameter $\mathrm{mm}$ $a$ thickness the rod $\mathrm{mm}$ $l_1$ width the rod $\mathrm{mm}$ $\sigma_{Aall}$ allowable axial stress $\mathrm{MPa}$

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 $\mathrm{-}$

Concentration factor in the rod:

$K_{tr}=12,882-52,714\left(\cfrac{d}{l_1}\right)+89,762\left(\cfrac{d}{l_1}\right)^2-51,667\left(\cfrac{d}{l_1}\right)^3$
where:
 $K_{tr}$ concentration factor in the rod $\mathrm{-}$ $d$ trunnion diameter $\mathrm{mm}$ $l_1$ width the rod $\mathrm{mm}$

Axial stress in the clevis:

$\sigma_c=\cfrac{K_{tc}\cdot F}{\left(l_2-d\right)2b}\le\sigma_{Aall}$
where:
 $\sigma_c$ axial stress in the clevis $\mathrm{MPa}$ $K_{tc}$ concentration factor in the clevis $\mathrm{-}$ $F$ force $\mathrm{N}$ $d$ trunnion diameter $\mathrm{mm}$ $b$ thickness the clevis $\mathrm{mm}$ $l_2$ width the clevis $\mathrm{mm}$ $\sigma_{Aall}$ allowable axial stress $\mathrm{MPa}$

Concentration factor in the clevis:

$K_{tc}=12,882-52,714\left(\cfrac{d}{l_2}\right)+89,762\left(\cfrac{d}{l_2}\right)^2-51,667\left(\cfrac{d}{l_2}\right)^3$
where:
 $K_{tc}$ concentration factor in the clevis $\mathrm{-}$ $d$ trunnion diameter $\mathrm{mm}$ $l_2$ width the clevis $\mathrm{mm}$

Shear stress in the rod:

$\tau_{s\left(r\right)}=\cfrac{F}{h_1\cdot a}\le\tau_{all}$
where:
 $τ_{s(r)}$ shear stress in the rod $\mathrm{MPa}$ $F$ force $\mathrm{N}$ $a$ thickness the rod $\mathrm{mm}$ $h_1$ length the rod $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

Shear stress in the clevis:

$\tau_{s\left(c\right)}=\cfrac{F}{h_2\cdot2b}\le\tau_{all}$
where:
 $τ_{s(c)}$ shear stress in the clevis $\mathrm{MPa}$ $F$ force $\mathrm{N}$ $b$ thickness the clevis $\mathrm{mm}$ $h_2$ length the clevis $\mathrm{mm}$ $τ_{all}$ allowable shear stress $\mathrm{MPa}$

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.
- A. Bolek, J. Kochman a kol.: Části a mechanismy strojů I. 1989.
- K. Kříž a kol.: Strojní součásti 1. 1984.