For a simply supported rectangular beam with a central load W, the maximum bending stress is equal to the direct tensile stress W/(bd) produced by the same load. The ratio of length to depth of the beam is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 29/01/2018 (Evening)
Detailed Explanation
Compare two stresses produced by the same load W: the bending stress at mid-span, M/Z, and the direct stress W/A on the same rectangular section.
\[ \sigma_b = \frac{WL/4}{bd^2/6} = \frac{3WL}{2bd^2},\qquad \sigma_t = \frac{W}{bd} \]
\[ \frac{3WL}{2bd^2} = \frac{W}{bd} \Rightarrow \frac{L}{d} = \frac{2}{3} \]
Why “2/3” Is Correct
The two stresses are equal only when L/d = 2/3. For any normal beam (L much larger than d) bending stress is far larger than direct stress, which is why bending governs beam design.
| Section | I about centroidal axis | y max | Section modulus Z |
|---|---|---|---|
| Rectangle b × d (axis ∥ b) | bd³/12 | d/2 | bd²/6 |
| Square side a (sides horizontal) | a⁴/12 | a/2 | a³/6 |
| Square side a (diagonal horizontal) | a⁴/12 | a/√2 | a³/(6√2) |
| Solid circle, diameter d | πd⁴/64 | d/2 | πd³/32 |
| Triangle b × h (axis ∥ base) | bh³/36 | 2h/3 | bh²/24 |
| Hollow circle D, d | π(D⁴ − d⁴)/64 | D/2 | π(D⁴ − d⁴)/(32D) |
Why the Other Options Are Wrong
- A. 1/2: 1/2 is the answer for a circular beam (Q82), not a rectangular one.
- C. 1/4: 1/4 does not satisfy 3L/(2d) = 1.
- D. 1/3: 1/3 gives a bending stress of only half the direct stress.
Key Point
Rectangle, central load: σb/σt = 3L/(2d); equal when L/d = 2/3.
Key Concepts for Students
- Bending stress grows with span; direct stress does not.
- Z = bd²/6 for a rectangle.
- Similar logic for a circle gives L/d = 1/2.
Practice more: this is question 15 in Bending and Shear Stress MCQs – Part 1. Try the whole set and check every answer instantly.