Analyzing Maximum Shear Stress in Circular Beams

This explanation clarifies the calculation and concept behind the maximum shear stress within a circular beam, comparing it to the average shear stress.

Understanding Shear Stress in Beams

When a beam experiences a shear force ($V$), internal shear stresses are generated across its cross-section. The way these stresses distribute depends significantly on the shape of the beam's cross-section. For different shapes, the stress is not uniform.

Shear Stress Pattern in Circular Sections

In a beam with a circular cross-section, the shear stress distribution follows a parabolic pattern. Specifically:

• The shear stress is minimal (zero) at the top and bottom points of the circle.
• As you move from the edges towards the center (the neutral axis), the shear stress intensity increases.
• The highest value, or maximum shear stress ($\tau_{max}$), is found precisely at the neutral axis, which is the centroidal axis of the circle.

Defining Average Shear Stress

The average shear stress ($\tau_{avg}$) is a simpler measure. It's calculated by taking the total shear force ($V$) applied to the cross-section and dividing it by the total area ($A$) of that cross-section.

The formula is:

$$ \tau_{avg} = \frac{V}{A} $$

This value represents the mean stress across the entire area, not the peak stress.

Calculating the Maximum Shear Stress for a Circular Beam

For a circular cross-section, the shear stress distribution is not uniform. Engineering mechanics provides a formula to calculate the maximum shear stress ($\tau_{max}$) based on the shear force ($V$) and the cross-sectional area ($A$). This maximum stress occurs at the neutral axis.

The formula is:

$$ \tau_{max} = \frac{4V}{3A} $$

Finding the Ratio: Maximum vs. Average Shear Stress

To determine how the maximum shear stress relates to the average shear stress, we can compute their ratio:

$$ \frac{\tau_{max}}{\tau_{avg}} = \frac{\frac{4V}{3A}}{\frac{V}{A}} $$

By simplifying this fraction, we cancel out $V$ and $A$:

$$ \frac{\tau_{max}}{\tau_{avg}} = \frac{4}{3} $$

When we convert the fraction $\frac{4}{3}$ into a decimal:

$$ \frac{4}{3} \approx 1.333... $$

Therefore, the maximum shear stress in a circular beam is approximately 1.33 times the average shear stress.

Evaluating the Options

Let's compare our result with the provided options:

Our calculation shows that $\tau_{max}$ is approximately 1.33 times $\tau_{avg}$, which directly corresponds to Option 2.

Final Conclusion

The stress distribution in a circular beam leads to a maximum shear stress value that is about 1.33 times the average shear stress calculated simply as shear force divided by area.