A hollow cylindrical shaft of length L, mean radius cm, and...

A hollow cylindrical shaft of length $L,$ mean radius $c_{m},$ and uniform thickness $t$ is subjected to a torque of magnitude $T .$ Consider, on the one hand, the values of the average shearing stress $\tau_{\text {ave }}$ and the angle of twist $\phi$ obtained from the elastic torsion formulas developed in Secs. $3.1 \mathrm{C}$ and 3.2 and, on the other hand, the corresponding values obtained from the formulas developed in Sec. 3.10 for thin-walled shafts. $(a)$ Show that the relative error introduced by using the thin-walled-shaft formulas rather than the elastic torsion formulas is the same for $\tau_{\text {ave }}$ and $\phi$ and that the relative error is positive and proportional to the ratio $t / c_{m \cdot}(b)$ Compare the percent error corresponding to values of the ratio $t / c_{m}$ of 0.1,0.2 and 0.4.

Key Concepts

Elastic Torsion Theory

This concept involves the fundamental equations and principles that govern how solid cylindrical shafts behave under applied torsional loads. It utilizes the relationships derived from elasticity theory to determine key response parameters such as shear stress distribution and the angle of twist, considering the material properties and geometric dimensions of the shaft.

Thin-Walled Shaft Approximation

This approximation simplifies torsion analysis by assuming that the wall thickness of a cylindrical shaft is small compared to its mean radius. Under this assumption, the stress distribution and deformation can be expressed with simpler formulas, which are easier to use in engineering calculations while still capturing the essential behavior of the structure.

Shear Stress and Angle of Twist in Torsion

These are primary response quantities in torsion problems. The shearing stress in a shaft under torque develops in a predictable manner, and the resultant angle of twist represents the rotation of the shaft due to the applied load. Both quantities are determined using the elastic torsion theory for a general case and can be approximated using simplified formulas in the thin-walled limit.

Relative Error Analysis

This concept focuses on quantifying the discrepancy between values obtained from the exact elastic torsion formulas and those from the thin-walled approximations. It involves comparing how the approximated average shearing stress and angle of twist deviate from their exact values, typically showing that the error is positive and linearly proportional to the ratio of the wall thickness to the mean radius. This analysis is essential for understanding the limits and applicability of simplified formulas.

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