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:

τ_{s (t)} = \frac{2 F}{π d^{2}} \leq τ_{a l l}

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

F - force - [N]

d - trunnion diameter - [mm]

τ_all - allowable shear stress - [MPa]

Allowable shear stress:

τ_{a l l} = \frac{0,4 R_{p 0,2 T}}{S_{F}} * C_{c}

τ_all - allowable shear stress - [MPa]

R_p0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

S_F - safety factor - []

C_c - coefficient according to load - []

Coefficient according to load:

load	[]
Unidirectional load, non-impact load	0,8
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:

σ_{B} = \frac{4 F (b + 2 a + 4 s)}{π d^{3}} \leq σ_{B a l l}

σ_B - 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]

σ_Ball - allowable bending stress - [MPa]

Allowable bending stress:

σ_{B a l l} = \frac{0,6 R_{p 0,2 T}}{S_{F}} * C_{c}

σ_Ball - allowable bending stress - [MPa]

R_p0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

S_F - safety factor - []

C_c - coefficient according to load - []

Combined stress in the trunnion:

σ_{t r e s c a} = \sqrt{σ_{B}^{2} + 4 τ_{s (t)}^{2}} \leq σ_{C a l l}

σ_tresca - combined stress in the trunnion - [MPa]

σ_B - bending stress in the trunnion - [MPa]

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

σ_Call - allowable combined stress - [MPa]

Allowable combined stress:

σ_{C a l l} = \frac{R_{p 0,2 T}}{S_{F}} * C_{c}

σ_Call - allowable combined stress - [MPa]

R_p0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

S_F - safety factor - []

C_c - coefficient according to load - []

Bearing stress in the rod:

p_{r} = \frac{F}{d * a} \leq σ_{a l l}

p_r - bearing stress in the rod - [MPa]

F - force - [N]

d - trunnion diameter - [mm]

a - thickness the rod - [mm]

σ_all - allowable bearing stress - [MPa]

Allowable bearing stress:

σ_{a l l} = \frac{0,9 R_{p 0,2 T}}{S_{F}} * C_{c} * C_{b}

σ_all - allowable bearing stress - [MPa]

R_p0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

S_F - safety factor - []

C_c - coefficient according to load - []

C_b - 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}{2 d * b} \leq σ_{a l l}

p_c - bearing stress in the clevis - [MPa]

F - force - [N]

d - trunnion diameter - [mm]

b - thickness the clevis - [mm]

σ_all - allowable bearing stress - [MPa]

Axial stress in the rod:

σ_{r} = \frac{K_{t r} * F}{(l_{1} - d) a} \leq σ_{A a l l}

σ_r - axial stress in the rod - [MPa]

K_tr - concentration factor in the rod - []

F - force - [N]

d - trunnion diameter - [mm]

a - thickness the rod - [mm]

l₁ - width the rod - [mm]

σ_all - allowable bearing stress - [MPa]

Allowable axial stress:

σ_{A a l l} = \frac{0,45 R_{p 0,2 T}}{S_{F}} * C_{c}

σ_Aall - allowable axial stress - [MPa]

R_p0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

S_F - safety factor - []

C_c - coefficient according to load - []

Concentration factor in the rod:

K_{t r} = 12,882 - 52,714 (\frac{d}{l_{1}})

+ 89,762 {(\frac{d}{l_{1}})}^{2} - 51,667 {(\frac{d}{l_{1}})}^{3}

K_tr - concentration factor in the rod - []

d - trunnion diameter - [mm]

l₁ - width the rod - [mm]

Axial stress in the clevis:

σ_{c} = \frac{K_{t c} * F}{(l_{2} - d) 2 b} \leq σ_{A a l l}

σ_c - axial stress in the clevis - [MPa]

K_tc - concentration factor in the clevis - []

F - force - [N]

d - trunnion diameter - [mm]

b - thickness the clevis - [mm]

l₂ - width the clevis - [mm]

σ_Aall - allowable axial stress - [MPa]

Concentration factor in the clevis:

K_{t c} = 12,882 - 52,714 (\frac{d}{l_{2}})

+ 89,762 {(\frac{d}{l_{2}})}^{2} - 51,667 {(\frac{d}{l_{2}})}^{3}

K_tc - concentration factor in the clevis - []

d - trunnion diameter - [mm]

l₂ - width the clevis - [mm]

Shear stress in the rod:

τ_{s (r)} = \frac{F}{h_{1} * a} \leq τ_{a l l}

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

F - force - [N]

a - thickness the rod - [mm]

h₁ - length the rod - [mm]

τ_all - allowable shear stress - [MPa]

Shear stress in the clevis:

τ_{s (c)} = \frac{F}{h_{2} * 2 b} \leq τ_{a l l}

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

F - force - [N]

b - thickness the clevis - [mm]

h₂ - length the clevis - [mm]

τ_all - 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

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