Ratio of Strain Energy for Solid and Hollow Shafts

To determine the ratio of strain energy for solid and hollow shafts subjected to the same torque, we need to consider the formulas for strain energy and the properties of the shafts.

Strain Energy:
The strain energy of a shaft is the energy stored within it due to the applied torque. It can be calculated using the formula:

E = (T^2 * L) / (2 * G)

Where:
E is the strain energy
T is the applied torque
L is the length of the shaft
G is the shear modulus of the material

Properties of Solid and Hollow Shafts:
In this scenario, we are comparing a solid shaft and a hollow shaft with the same outside diameter. Let's denote the outside diameter as D, and the inside diameter of the hollow shaft as d.

1. Area:
The cross-sectional area of a solid shaft is given by:
A_solid = π * (D^2) / 4

The cross-sectional area of a hollow shaft is given by the difference between the outer and inner circles:
A_hollow = π * (D^2 - d^2) / 4

2. Torsional Rigidity:
The torsional rigidity of a solid shaft is given by:
J_solid = π * (D^4) / 32

The torsional rigidity of a hollow shaft is given by:
J_hollow = π * (D^4 - d^4) / 32

3. Shear Modulus:
The shear modulus, G, represents the material's resistance to shearing deformation. It is a material property and remains constant for both the solid and hollow shafts.

Ratio of Strain Energy:
To find the ratio of strain energy between the solid and hollow shafts, we can divide the strain energy equations:

E_solid / E_hollow = [(T_solid^2 * L) / (2 * G)] / [(T_hollow^2 * L) / (2 * G)]

As the lengths and shear modulus are the same for both shafts, they cancel out:

E_solid / E_hollow = (T_solid^2) / (T_hollow^2)

From this equation, we can see that the ratio of strain energy between the solid and hollow shafts is determined solely by the square of the applied torque. The geometry and properties of the shafts do not affect this ratio.

Therefore, for shafts with the same outside diameter, the ratio of strain energy between a solid and a hollow shaft subjected to the same torque is equal to the square of the ratio of the applied torques.