Question

At the centroid of a loaded beam cross section,

• A. bending stress is equal to zero
• B. bending stress is maximum
• C. shear stress is equal to zero
• D. shear stress is minimum

Hint:

Remember the stress distribution across a beam's cross-section:
- Bending Stress ($\sigma$): Zero at the neutral axis (centroid), Maximum at the top and bottom (extreme fibers).
- Shear Stress ($\tau$): Maximum at the neutral axis (centroid), Zero at the top and bottom (extreme fibers).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks about the values of bending stress and shear stress at the centroid (neutral axis) of a beam's cross-section. The provided answer is (A), which suggests a potential error in the question or options as shear stress is maximum at the centroid, not zero, and shear stress is minimum at the extreme fibers, not minimum at the centroid. Let's analyze based on bending stress.

Step 2: Key Formula or Approach:
1.

Bending Stress ($\sigma$): The theory of simple bending gives the flexure formula:
\[ \sigma = \frac{My}{I} \] where $M$ is the bending moment, $I$ is the moment of inertia about the neutral axis, and $y$ is the perpendicular distance from the neutral axis. The neutral axis passes through the centroid of the cross-section.
2.

Shear Stress ($\tau$): The shear stress formula is:
\[ \tau = \frac{VA\bar{y}}{Ib} \] where $V$ is the shear force, $A$ is the area above (or below) the point of interest, $\bar{y}$ is the distance from the neutral axis to the centroid of area $A$, $I$ is the moment of inertia, and $b$ is the width of the section at that point.

Step 3: Detailed Explanation:

• Bending Stress at the Centroid: The centroid of the cross-section lies on the neutral axis. At the neutral axis, the distance $y$ is, by definition, equal to zero. Substituting $y=0$ into the flexure formula gives:
\[ \sigma = \frac{M(0)}{I} = 0 \] Therefore, the bending stress is always zero at the centroid (neutral axis) of a beam. This makes option (A) correct. Option (B) is incorrect; bending stress is maximum at the extreme fibers (farthest from the neutral axis).

• Shear Stress at the Centroid: The term $A\bar{y}$ in the shear stress formula is the first moment of area. This term is maximum at the neutral axis for most common shapes (rectangle, I-section, circle). Therefore, the shear stress is typically

maximum at the centroid, not zero or minimum. This makes options (C) and (D) incorrect.



Step 4: Final Answer:
At the centroid of a loaded beam cross section, the bending stress is equal to zero. The selected answer 'A' is correct, but it is important to note that the shear stress is maximum at this location.