Nominal Stress for Sizing a Rectangular Bar with a Hole

When designing mechanical components, especially those with changes in geometry like holes, fillets, or notches, the stress at these specific points can become significantly higher than the average stress in other parts of the component. This phenomenon is called stress concentration. To ensure the component does not fail under load, engineers must account for this increased stress by using a stress concentration factor (\(K_t\)).

The stress concentration factor is a dimensionless ratio that relates the maximum stress at a discontinuity to the nominal (or average) stress in the cross-section if the discontinuity were not present. It is defined by the formula:

\[K_t = \frac{\sigma_{max}}{\sigma_{nom}}\]

Where:

• \(\sigma_{max}\) is the actual maximum stress that occurs at the point of concentration (e.g., at the edge of the hole).
• \(\sigma_{nom}\) is the nominal average stress calculated for the cross-section without considering the stress concentration effect. This is the stress value often used for the overall sizing of the component.

In this problem, we are given the maximum permissible stress the material can withstand and the stress concentration factor for the specific configuration (a rectangular bar with a circular hole). We need to determine the nominal average stress that should be used for sizing the section.

Given Parameters for Stress Calculation

Let's list the information provided in the question:

• Actual maximum permissible stress (\(\sigma_{max}\)) = 150 MPa
• Stress concentration factor (\(K_t\)) for the configuration = 3.0

Calculating the Nominal Average Stress for Sizing

Our goal is to find the nominal average stress (\(\sigma_{nom}\)). We can rearrange the formula for the stress concentration factor to solve for \(\sigma_{nom}\):

\[K_t = \frac{\sigma_{max}}{\sigma_{nom}}\]

Rearranging the equation to solve for \(\sigma_{nom}\):

\[\sigma_{nom} = \frac{\sigma_{max}}{K_t}\]

Now, we substitute the given values into this rearranged formula:

\[\sigma_{nom} = \frac{150 \text{ MPa}}{3.0}\]

Performing the division:

\[\sigma_{nom} = 50 \text{ MPa}\]

This result indicates that for the actual maximum stress at the hole to not exceed 150 MPa, the average stress across the section (the nominal average stress) should be limited to 50 MPa. This is the stress value that would be used in general design equations to determine the required cross-sectional area of the bar.

Conclusion on Nominal Stress

Based on our calculation, the nominal average stress for sizing the rectangular bar section with the circular hole is 50 MPa. This corresponds to the correct option.