A cylindrical rod of steel (E=207 GPa, 30 ×10^∘. psi) having a yield strength of 310 MPa(45,000

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A cylindrical rod of steel (E=207 GPa, 30 ×10^∘. psi) having...

A cylindrical rod of steel $\left(E=207 \mathrm{GPa}, 30 \times 10^{\circ}\right.$ psi) having a yield strength of $310 \mathrm{MPa}(45,000$ psi) is to be subjected to a load of $11,100 \mathrm{~N}$ (2500 $\left.\mathrm{Ib}_{i}\right)$. If the length of the rod is $500 \mathrm{~mm}(20.0 \mathrm{in}$.), what must be the diameter to allow an elongation of $0.38 \mathrm{~mm}(0.015$ in.)?

Key Concepts

Circular Cross-Sectional Geometry

For a cylindrical rod, the cross-sectional area is a function of its diameter, typically calculated using the formula A = (?/4)d². This geometric relationship is essential when linking the applied load and the resulting deformation, as the area directly influences the stress level experienced by the material.

Modulus of Elasticity

The modulus of elasticity, or Young’s modulus, is a measure of a material’s stiffness, indicating how much it will deform under a given load. It is defined as the ratio of stress to strain in the elastic region of the material’s stress–strain curve and is a critical parameter when designing components to ensure they deform within acceptable limits under operational loads.

Yield Strength

Yield strength is the stress level at which a material begins to deform plastically, meaning that deformation becomes permanent. Knowing the yield strength is crucial in design to ensure that the material does not exceed its elastic limit under the applied loads, thereby preventing permanent deformation or failure.

Axial Deformation

Axial deformation refers to the change in length of a material or structural member when subjected to a force along its length. This concept forms the basis for calculating elongation in components like rods, where the change in length is directly related to the applied force, the material properties, and the geometry of the cross-section.

Hooke's Law

Hooke's Law is the principle that states the extension or compression of an elastic material is proportional to the applied load, within the elastic limit. In the context of axial deformation, it leads to the relationship between stress and strain, and ultimately to the formula relating force, elongation, modulus of elasticity, and the cross-sectional area of the rod.

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