The shaft is supported by journal bearings at A and B that...

The shaft is supported by journal bearings at $A$ and $B$ that exert force components only in the $x$ and $z$ directions. Determine the smallest diameter of the shaft to the nearest In. Use the maximum shear stress theory of failure with $\tau_{\text {allow }}=6 \mathrm{ksi}$.

Key Concepts

Maximum Shear Stress Theory

This theory, often known as the Tresca criterion, posits that a material fails when its maximum shear stress reaches a critical value. It is used to predict yielding when the object is subjected to a complex state of stress, by comparing the computed maximum shear stress to an allowable value derived from the material properties.

Bending Stress Analysis

Bending stress analysis involves determining the stresses induced in a beam or shaft due to bending moments. For circular shafts, the stress is calculated by relating the bending moment to the section modulus, which depends on the moment of inertia and the distance from the neutral axis, thereby dictating the material's resistance to bending.

Combined Loading Analysis

When structural components such as shafts experience more than one type of load (for example, bending along with torsion), the resulting stresses must be combined. This analysis uses failure theories such as the maximum shear stress theory to find an equivalent stress, ensuring that the design does not exceed the material's allowable stress limits.

Shaft Diameter Design

Designing a shaft involves determining the smallest diameter that can withstand the imposed loads without failure. This requires equating the applied bending and, if present, torsional loads to the stress formulas for a circular cross-section, ensuring that the resulting stresses remain below the defined allowable limit in accordance with the chosen failure criterion.

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