Thin Cylinder Theory: Determining Torsional Shear Stress

[brd]

Can anyone tell me what equation I need to use to determine the torsional shear stress using the thin cylinder theory to a copper cylinder which is under pressure and a twisting force (torque).

 

[FredGarvin]

You have a combined loading scenario.

The pressure will create a normal stress. However, when you do a Mohr's circle on the pressure alone, you will develop a shear stress on your location rotated 45° in plane. From the pressure alone, Mohr's theory will show you that the max shear is $$\tau_{max} = \frac{pr}{4t}$$ where:
p = pressure in psi
r = outer radius in in.
t = wall thickness in in.

The normal stress resulting from the pressure is $$\sigma = \frac{pr}{t}$$.

The shear stress due solely to the torque applied will be
$$\tau = \frac{T*c}{J}$$ where:
T = applied torque in in*lbf
c = outer radius in in.
J = polar moment of inertia in in^4

You'll have to combine the stresses in the same directions and do a Mohr's Circle on the total load to determine the principle stresses.

 

[ravenprp]

The equation you will need to use to determine the torsional shear stress using the thin cylinder theory is the Tresca's shear stress equation. This equation takes into account both the pressure and the twisting force (torque) applied to the copper cylinder. It is given by:

τ = (P*r)/(2*t) + (T*r)/(J)

Where:
τ = Torsional shear stress
P = Pressure applied to the cylinder
r = Radius of the cylinder
t = Thickness of the cylinder
T = Torque applied to the cylinder
J = Polar moment of inertia of the cylinder

It is important to note that the thin cylinder theory assumes that the cylinder is long and thin, with a ratio of length to diameter greater than 10. If this is not the case, a different equation may need to be used. Additionally, this equation assumes that the material of the cylinder is homogeneous and isotropic. If the material properties are not uniform, a more complex equation may be required.