The shaft is supported by a smooth thrust bearing at A and a...

The shaft is supported by a smooth thrust bearing at $A$ and a smooth journal bearing at $B .$ If $P=5 \mathrm{kN}$ and the shaft is made from steel having an allowable normal stress of $\sigma_{\text {allow }}=150 \mathrm{MPa}$ and an allowable shear stress of $\tau_{\text {allow }}=85$ MPa, determine the required minimum wall thickness $t$ of the shaft to the nearest millimeter to safely support the load.

Key Concepts

Material Allowable Stress

This concept refers to the maximum stress that a material can safely endure under normal (tensile/compressive) or shear loading without failing or undergoing permanent deformation. In design, the calculated stresses imposed on a component are compared to these allowable limits to ensure safety and reliability. The allowable values are derived from the material’s properties and include safety factors to account for uncertainties in loading and material behavior.

Normal (Bending) Stress

When a shaft or structural member is subjected to bending loads, one side experiences tensile stress while the opposite side undergoes compressive stress. The magnitude of this normal bending stress is a function of the applied moment and the geometry of the cross-section. Ensuring that the maximum bending stress is below the allowable normal stress is critical to prevent material failure in components like shafts.

Shear Stress

Shear stress arises from forces that tend to cause sliding between adjacent sections of a material. In shaft design, shear forces can be imposed through torsion or transverse loading, and the resulting shear stresses must be carefully calculated. These stresses must not exceed the allowable shear stress of the material in order to avoid issues such as torsional failure or excessive deformation.

Shaft Geometry and Wall Thickness Design

The design of a shaft involves selecting an appropriate cross-sectional geometry—often a circular or hollow section—to resist applied loads. The wall thickness, in particular, is a critical parameter as it directly influences both bending and shear stress distributions. Adequate thickness must be determined through calculations to ensure that the shaft can safely carry the applied loads while remaining efficient in terms of weight and material usage.

Bearing Support Conditions

Different types of bearings, such as thrust bearings and journal bearings, provide support to the shaft and influence how loads are transmitted and distributed along its length. Thrust bearings typically deal with axial loads, while journal bearings support radial loads. Understanding these support conditions is crucial in determining the load paths, reaction forces, and resulting stresses in the shaft, thereby influencing its overall design and required dimensions.

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