The circular hollow shaft is supported by a smooth thrust...

The circular hollow shaft is supported by a smooth thrust bearing at $A$ and smooth journal bearing at $B .$ If the shaft is made from steel having an allowable normal stress of $\sigma_{\text {allow }}=150 \mathrm{MPa}$ and allowable shear stress of $\tau_{\text {allow }}=85 \mathrm{MPa}$, determine the maximum allowable magnitude of the two forces $\mathbf{P}$ that can be applied to the shaft. The thickness of the shaft's wall is $t=5 \mathrm{mm}$.

Key Concepts

Hollow Circular Shaft Properties

Hollow circular shafts are structural elements that use a tubular cross?section to balance strength with reduced weight. Their performance under load depends largely on geometric parameters such as outer diameter, inner diameter, and wall thickness. These parameters determine the moment of inertia and polar moment of inertia, which are essential for analyzing their bending and torsional responses.

Allowable Stress in Materials

Allowable stress represents the maximum limit of stress that a material can safely withstand under loading without failure. It is divided into normal (tensile/compressive) and shear components. Designers use these limits to ensure that the stresses from applied loads do not exceed what the material can sustain, providing a margin of safety in engineering applications.

Bending Stress Analysis

Bending stress analysis involves calculating the stress distribution that develops when a shaft or beam is subjected to bending moments. The maximum bending stress is typically determined using the relationship between the bending moment, the distance from the neutral axis, and the moment of inertia of the cross-section. This analysis is key to ensuring that the material does not yield under bending loads.

Torsional Stress Analysis

Torsional stress analysis deals with forces applied in a twisting action, leading to shear stresses within a shaft. The maximum torsional shear stress is calculated using the applied torque, the radius of the section, and the polar moment of inertia. This concept is critical to prevent failure from excessive twisting in shafts transmitting torque.

Sectional Properties of Thin-Walled Structures

For thin-walled circular sections, specific approximations simplify the computation of the moment of inertia and polar moment of inertia. In such cases, the wall thickness is much smaller than the overall dimensions, allowing for formulas that streamline the design and analysis process when assessing both bending and torsional stresses. This simplification is particularly useful in early design and quick-check calculations.

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