A centric load 𝐏 must be supported by the steel bar A B....

A centric load $\mathbf{P}$ must be supported by the steel bar $A B$. Using allowable stress design, determine the smallest dimension $d$ of the cross section that can be used when $(a) P=108 \mathrm{kN},(b) P=$ $166 \mathrm{kN} .$ Use $\sigma_{Y}=250 \mathrm{MPa}$ and $E=200 \mathrm{GPa}$.

Key Concepts

Allowable Stress Design

This design approach is based on ensuring that the stresses induced in a member under load do not exceed a specified allowable value, typically derived from the material's yield strength. It simplifies the design process by directly comparing the applied load-induced stresses with the material’s capacity, incorporating a factor of safety through the allowable stress limit.

Axial Loading

When a load is applied along the axis of a member, it creates a uniform normal stress over the cross-sectional area. The relationship between the load and the cross-sectional area determines whether the member can sustain the load without exceeding the material’s yield strength, which is crucial for ensuring the structural integrity of the component.

Material Properties

Understanding the material properties, particularly the yield strength and the modulus of elasticity, is essential in design. The yield strength defines the maximum stress that can be applied without causing permanent deformation, while the modulus of elasticity relates the stress to the strain in the elastic region. Both properties guide the selection of the cross-sectional dimensions to safely support the applied load.

Cross-Sectional Geometry

The geometry of the cross-section, such as a circular shape with a defined diameter, directly influences the area available to resist applied loads. By relating the cross-sectional area to the desired dimensions, designers can calculate the minimum required dimension that will ensure the stress does not exceed the allowable limit. This geometric relationship is fundamental in optimizing material usage while maintaining structural safety.

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