A hollow member having the cross section shown is to be...

A hollow member having the cross section shown is to be formed from sheet metal of $\frac{1}{16}$ -in. thickness. Knowing that a 3 -kip.in. torque will be applied to the member, determine the smallest dimension $d$ that can be used if the shearing stress is not to exceed 500 psi.

Key Concepts

Cross-Sectional Geometry and Properties

The geometry of the cross section directly influences the distribution of stresses under torsion. Parameters such as area, thickness, and shape factors are used to calculate the stress intensity and to determine the section’s resistance. An accurate understanding of these geometrical properties is crucial for optimizing design against failure criteria.

Thin-Walled Member Analysis

Thin-walled members are structural elements whose wall thickness is small compared to other dimensions. Special analysis techniques, often involving simplifications and approximations such as assuming constant shear flow along the thickness, are used to assess their strength. This analysis is vital in predicting the performance of such members under torsional loading.

Torsional Loading

This concept involves the application of a twisting moment to a structural member, causing shear stresses within the material. Understanding torsional loading is essential to predicting the behavior of a component under twisting forces and for ensuring that the design can safely resist these stresses.

Shear Stress

Shear stress in a material arises when forces are applied parallel to a cross-sectional area. In the context of torsion, the shear stress is calculated based on the applied torque and the geometric properties of the cross section. The critical design evaluation is to ensure that the induced shear stress does not exceed the material’s allowable limit.

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