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.)?

Understanding how materials behave under different forces is essential in mechanical engineering, and stress and strain analysis provides the tools to do just that.

Stress, denoted by the symbol \( \sigma \), is the internal force experienced by a material per unit area, typically measured in pascals (Pa) or megapascals (MPa). It's calculated by dividing the applied force \( F \) by the cross-sectional area \( A \) over which the force is distributed, as \( \sigma = \frac{F}{A} \).

Strain, on the other hand, is a measure of deformation representing the elongation or compression of a material. It is dimensionless and calculated as the change in length \( \Delta L \) divided by the original length \( L_0 \) as \( \epsilon = \frac{\Delta L}{L_0} \).

In our cylindrical rod example, applying the provided force could cause deformation. To ensure the rod deforms only within an allowable range, the stress must not exceed the material’s yield strength, hence the stress calculation becomes the foundational step to determine the rod's dimensions that will comply with the performance criteria.

Yield strength is a material property that signifies the maximum stress a material can withstand without permanent deformation. Once a material is stressed beyond its yield strength, it will no longer return to its original shape after unloading. This is why it's crucial for engineers to design components such that they operate safely below their yield point.

The yield strength is determined through material testing and is typically provided in megapascals (MPa) or pounds per square inch (psi). In the given exercise, our steel rod has a yield strength of \( 310 MPa \), which means the designed stress in the rod due to the applied load should not exceed this value.

Evaluating the yield strength ensures that our rod will experience elastic deformation, which is reversible, rather than plastic deformation, which is permanent, when subjected to the specified load. This consideration is fundamental for the rod to be both safe and functional in its application, as the question specifies an acceptable elongation limit.

Calculating the cross-sectional area is necessary when analyzing the structural integrity of components. For cylindrical objects, like rods or pipes, the cross-sectional area depends on the diameter of the cylinder.

To find the area \( A \) of a circle (which is the cross-sectional shape of our rod), we use the formula \( A = \frac{\pi D^2}{4} \), where \( D \) is the diameter. This relationship is central to converting the knowledge of the force applied to the rod and the material's yield strength into a practical dimension—the rod's diameter—in our exercise.

We can solve this formula for diameter, \( D \) to get \( D = \sqrt{\frac{4A}{\pi}} \). From this equation, we see that the diameter is directly related to the cross-sectional area that we calculated in the previous steps. This is how we can use the load and the material's yield strength to determine the needed diameter for the rod, ensuring that it will experience the specified allowable elongation without reaching its elastic limit.