Designing Shafts to Withstand Shear Stress

[loststudent123]

Homework Statement

A rule often used in shaft design states that the angle of twist shall not exceed 1° on a length equal to 20 diameters. What stress in the material does this imply if the modulus of rigidity is equal to 80 × 103 N/mm2 ?

Homework Equations

T/J=Gθ/L=τ/r
J=π?32(D^4-d^4)[/B]
G=shear stress/shear strain
shear stress=F/A
F=Wx9.81

The Attempt at a Solution

so i found that the length of the beam is D(20) letting D equal its diameter
so i got the equation
(80x10^3)/(D(20)=Wx9.81Xπ(D/2)/(D/2)
but I am not sure if that's right or how to solve when there is two unknowns?

 

[SteamKing]

Homework Statement

A rule often used in shaft design states that the angle of twist shall not exceed 1 on a length equal to 20 diameters. What stress in the material does this imply if the modulus of rigidity is equal to 80 × 103 N/mm2 ?

Homework Equations

T/J=Gθ/L=τ/r
J=π?32(D^4-d^4)[/B]
G=shear stress/shear strain
shear stress=F/A
F=Wx9.81

The Attempt at a Solution

so i found that the length of the beam is D(20) letting D equal its diameter
so i got the equation
(80x10^3)/(D(20)=Wx9.81Xπ(D/2)/(D/2)
but I am not sure if that's right or how to solve when there is two unknowns?

It's not clear what criterion you are intended to use here. The text of your post doesn't correctly display the words, 'the angle of twist shall not exceed 1 [squiggle] on a length equal to 20 diameters.' What are the units of 1 [squiggle]?

If you know anything about calculating the shear stress of circular shafts, it should be that shear stress ≠ F / A.
What is the correct formula for calculating the shear stress for a circular shaft in torsion?

 

[loststudent123]

my lecture gave only those equations so I am not sure of anyothers and i fixed the angle of twist however.

 

[SteamKing]

my lecture gave only those equations so I am not sure of anyothers and i fixed the angle of twist however.

Well, I must say that was a poor lecture then.

For a circular shaft, the angle of twist θ = TL / G J, where

T = applied torque
L = length of the shaft
G = modulus of rigidity of the shaft material
J = polar moment of inertia for the shaft, J = π D⁴ / 32 for solid shafts; J = π (D_o⁴ - D_i⁴) for hollow shafts,
D_o = outer diameter of shaft
D_i = inner diameter of shaft

The torsional shear stress, τ = T ⋅ r / J, where

T = applied torque
r = location of the point where shear stress is calculated, measured from the center of the shaft cross section
J = polar moment of inertia for the shaft cross section (see above)

You still haven't answered the question about the units of the limiting twist in the shaft (1 [squiggle] in 20 diameters).

 

[loststudent123]

its degree so its one degree is the max angle of twist and its over the length of 20 diameters

 

[SteamKing]

its degree so its one degree is the max angle of twist and its over the length of 20 diameters

You should be all set.

 

[loststudent123]

what should i put in for applied torque?

 

[SteamKing]

what should i put in for applied torque?

You know the angle of twist produced by a certain value for T. The max. shear stress in the outer fiber of the shaft also depends on this value of T.

What you want is to find the shear stress in the shaft so that the angle of twist is no more than 1 degree in 20 diameters.

Use algebra to eliminate T from the two formulas.

 

[loststudent123]

thank you :)