# Thin Cylinder Theory: Determining Torsional Shear Stress

• brd
In summary, to determine the torsional shear stress using the thin cylinder theory for a copper cylinder under pressure and a twisting force, you will need to combine the stresses from the pressure and the torque using Mohr's Circle. The maximum shear stress due to the pressure alone can be calculated using the equation \tau_{max} = \frac{pr}{4t}, while the normal stress is \sigma = \frac{pr}{t}. The shear stress from the torque alone is \tau = \frac{T*c}{J}, where T is the applied torque, c is the outer radius, and J is the polar moment of inertia.
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).

Last edited:

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.

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.

## 1. What is Thin Cylinder Theory and how is it used in determining torsional shear stress?

Thin Cylinder Theory is a method used in engineering and physics to calculate the stresses and strains in a cylindrical object subjected to torsion, or twisting. It assumes that the cylinder is thin-walled and the thickness is much smaller than the radius of the cylinder. This theory is used to determine the distribution of torsional shear stress along the length of the cylinder.

## 2. What factors are considered in Thin Cylinder Theory when calculating torsional shear stress?

Thin Cylinder Theory takes into account the material properties of the cylinder, such as its elasticity and shear modulus, as well as the geometry of the cylinder, including its radius and length. It also considers the applied torque and the resulting shear stress on the cylinder.

## 3. What are the assumptions made in Thin Cylinder Theory?

In addition to assuming that the cylinder is thin-walled, Thin Cylinder Theory also assumes that the material is homogeneous, isotropic, and in a state of pure torsion. It also assumes that the cylinder is subjected to a constant and uniformly distributed torque.

## 4. How is torsional shear stress distributed in a thin cylindrical object?

According to Thin Cylinder Theory, the torsional shear stress is assumed to be distributed in a linear manner along the length of the cylinder. This means that the shear stress is directly proportional to the distance from the axis of the cylinder, with the highest stress occurring at the outer surface of the cylinder.

## 5. What are some practical applications of Thin Cylinder Theory in engineering?

Thin Cylinder Theory is commonly used in the design and analysis of structures and components that are subjected to torsional loads, such as drive shafts, propeller shafts, and turbine blades. It is also useful in predicting the failure of thin-walled pressure vessels and pipes due to torsional stresses.

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