91Ó°ÊÓ

Strain is a measure of deformation representing the elongation or compression of a material. It is crucial in the mechanics of materials to determine if the material can withstand the applied load without failing. Strain occurs when a material is subjected to a load, causing it to change its length.
In this case, the original length of the aluminum rod is 250 mm, and after applying a load, the length becomes 250.28 mm. The formula to calculate strain, denoted by \( \epsilon \), is:

• Change in length \( \Delta L = 250.28 \, \text{mm} - 250 \, \text{mm} = 0.28 \, \text{mm}\)
• Original length \( L_0 = 250 \, \text{mm} \)
• Strain \( \epsilon = \frac{ \Delta L }{ L_0 } = \frac{ 0.28 }{ 250 } \approx 0.00112 \)

This value indicates how much the rod has stretched per unit length. It's important to note that strain is a dimensionless quantity and provides insight into the ductile or brittle nature of a material.

Stress is a measure of internal forces within a material when it is subjected to an external load. For engineers, performing stress analysis is essential to ensure structures can handle everyday stresses without failure.
Stress \( \sigma \) is calculated using Hooke’s Law, which states that the stress is equal to the product of the modulus of elasticity \( E \) and the strain \( \epsilon \). Given that the modulus of elasticity for aluminum, \( E \), is \( 73 \, \text{GPa} \), we calculate stress as follows:

• \( \sigma = E \times \epsilon = 73 \, \text{GPa} \times 0.00112 \)
• This gives \( \sigma = 81.76 \, \text{MPa} \)

Understanding stress values helps determine how close a material is to its ultimate strength. In this instance, the aluminum rod experiences a stress of 81.76 MPa, which needs to be compared against its ultimate strength to assess its safety.

The factor of safety (FoS) is a critical concept in engineering design, providing a margin between the applied stress and the material's strength to account for uncertainties in material properties, loading conditions, and potential flaws.
The factor of safety is defined as the ratio of the material's ultimate strength (the maximum stress it can withstand before failure) to the actual applied stress:

• Ultimate strength for aluminum is \( 140 \, \text{MPa} \)
• Calculated stress is \( 81.76 \, \text{MPa} \)
• \( \text{FoS} = \frac{140}{81.76} \approx 1.71 \)

A factor of safety of 1.71 implies that the aluminum rod can sustain loads 1.71 times greater than the current stress before reaching its ultimate strength. Engineering applications often aim for specific FoS values depending on the criticality of the use; a higher FoS indicates a more conservative design.