If the allowable shear stress is 1 ×10^8 Pa for a hollow shaft having an outside diameter of .6

step 1 Okay, so from the torsional formula we have tau over r equal to G times 5 over l and this is equ

If the allowable shear stress is 1 ×10^8 Pa for a hollow...

If the allowable shear stress is $1 \times 10^8 \mathrm{~Pa}$ for a hollow shaft having an outside diameter of .6 m and an inside diameter of 3 m , what is the maximum twist over a length of 4.5 m for $G=1 \times 10^{11} \mathrm{~Pa}$ ? What is the rate of twist?

Key Concepts

Torsion in Circular Shafts

This concept involves the study of how circular shafts react when subjected to twisting loads. In engineering mechanics, torsion describes the twisting deformation of structural elements, and understanding it is crucial for designing elements such as drive shafts. The analysis involves determining how applied torque leads to shear stress and angular deformation along the element’s length.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross-section that quantifies its ability to resist torsion. For hollow sections, it is calculated based on the outer and inner diameters. This parameter directly influences the torsional stiffness of the shaft, with higher values indicating a greater resistance to twisting under an applied torque.

Shear Stress under Torsion

Shear stress in torsion is the internal force per unit area that develops within a shaft when it is subjected to a torque. There is a direct relationship between the applied torque, the dimensions of the shaft (through the polar moment of inertia), and the resulting shear stress. The concept is critical in ensuring that the operation stays within the material’s allowable limits to prevent failure.

Angle of Twist

The angle of twist refers to the total angular deformation experienced by a shaft along its length due to an applied torque. It is determined by the magnitude of the torque, the length of the shaft, the polar moment of inertia, and the material’s shear modulus. This measure is key in evaluating the performance of structural members in torsion.

Rate of Twist

The rate of twist is defined as the angle of twist per unit length along the shaft. It provides a uniform measure of torsional deformation along the entire length of the member, which is important for designs that require consistent performance and allowable deformation limits.

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