A beam of the cross section shown is extruded from an...

A beam of the cross section shown is extruded from an aluminum alloy for which $\sigma_{y}=250 \mathrm{MPa}$ and $\sigma_{U}=450 \mathrm{MPa}$. Using a factor of safety of $3.00,$ determine the largest couple that can be applied to the beam when it is bent about the $z$ axis.

Key Concepts

Flexure Formula

The flexure formula relates the bending moment applied to a beam to the stress developed in the material. Specifically, it states that the bending stress (?) at a distance (c) from the neutral axis is given by ? = M·c/I, where M is the bending moment and I is the moment of inertia of the cross section. This relationship is fundamental in structural analysis when designing beams to ensure that the stress does not exceed the material’s limit.

Moment of Inertia

The moment of inertia is a geometric property of a cross section that quantifies its resistance to bending. It is calculated based on the shape and distribution of the area relative to an axis. A higher moment of inertia indicates that the beam can resist larger bending moments for a given material and yield stress without experiencing excessive deformation or failure.

Material Strength Properties

Material strength properties, such as yield stress and ultimate stress, describe the limits at which a material deforms plastically or fails under load. The yield stress represents the point where the material begins to undergo permanent deformation, while the ultimate stress is the maximum stress the material can withstand before fracture. These properties are crucial in determining the maximum allowable load for a structural component.

Factor of Safety

The factor of safety is a design criterion used to provide a margin for error in engineering calculations. It is applied by dividing the material’s yield or ultimate stress by the safety factor to obtain an allowable stress level. This ensures that the design remains safe under unpredictable loads or variations in material properties, effectively reducing the risk of failure by operating well within the material’s capacity.

Transcript

Please give Ace some feedback