Question

For a cantilever beam of T cross section subjected to uniformly distributed load throughout the length, the maximum bending stress occurs at

• A. mid depth
• B. top of cross section
• C. bottom of cross section
• D. junction of flange and web

Hint:

For any beam section, maximum bending stress always occurs at the extreme fibers (the points farthest from the neutral axis).
For asymmetric sections like a T-beam, you must first determine the location of the neutral axis (centroid) to identify which fiber (top or bottom) is farthest away.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the location of the maximum bending stress in a T-section cantilever beam under a UDL.

Step 2: Key Formula or Approach:
The bending stress is given by $\sigma = My/I$. The maximum bending stress occurs at the point with the maximum value of $y$, i.e., at the fiber farthest from the neutral axis. For a T-section, the neutral axis is not at the geometric center.

Step 3: Detailed Explanation:
1.

Bending Moment: A cantilever beam with a UDL has a hogging (negative) bending moment throughout its length, with the maximum magnitude at the fixed support. A hogging moment causes tension in the top fibers and compression in the bottom fibers.
2.

Neutral Axis of a T-section: For a T-section, the centroid (and thus the neutral axis) is located closer to the top flange because the flange has more area. This means the distance from the neutral axis to the top fiber ($y_{top}$) is smaller than the distance from the neutral axis to the bottom fiber of the web ($y_{bottom}$).
\[ y_{bottom} \gt y_{top} \] 3.

Maximum Bending Stress: The magnitude of the bending stress is $|\sigma| = |My/I|$. Since the bending moment $M$ and moment of inertia $I$ are constant for a given cross-section, the stress is maximum where the distance $y$ is maximum.
Since $y_{bottom} \gt y_{top}$, the maximum magnitude of the bending stress will occur at the

bottom of the cross section.
Specifically, the top fiber will have the maximum tensile stress ($ \sigma_{tensile, max} = |M| y_{top} / I $) and the bottom fiber will have the maximum compressive stress ($ \sigma_{compressive, max} = |M| y_{bottom} / I $). The overall maximum magnitude of stress occurs at the bottom.

Step 4: Final Answer:
For a T-section cantilever beam under a UDL, the maximum bending stress (in magnitude) occurs at the bottom of the cross section.