A short column carries an eccentric load. Which stress distribution at the base AB represents the case where the intensity of direct stress is less than the intensity of bending stress? (σmin = minimum resultant stress, σmax = maximum resultant stress)
Try it first: pick an option, then read the explanation.
Asked in SSC JE 06/06/2024 (4 pm–6 pm)
Detailed Explanation
An eccentric load P at eccentricity e is equivalent to an axial load P (direct stress σ0 = P/A, uniform) plus a moment Pe (bending stress σb = M/Z, linear across the base). The base stress is the sum of the two, so the shape of the diagram depends on which one is bigger.
\[ \sigma_{max} = \sigma_0 + \sigma_b = \frac{P}{A}\left(1 + \frac{6e}{b}\right),\qquad \sigma_{min} = \sigma_0 - \sigma_b = \frac{P}{A}\left(1 - \frac{6e}{b}\right) \]
\(\sigma_0 > \sigma_b\): \(\sigma_{min} > 0\) (trapezium) \(\sigma_0 = \sigma_b\): \(\sigma_{min} = 0\) (triangle) \(\sigma_0 < \sigma_b\): \(\sigma_{min} < 0\) (tension appears)
Why “Two triangles crossing AB: σmin tensile on one side, σmax compressive on the other” Is Correct
With σ0 < σb the minimum stress becomes negative. One edge of the base goes into tension while the other is in compression, so the diagram crosses the base line as two opposite triangles. For a rectangular base this happens when e > b/6, i.e. the load lies outside the middle-third (core) of the section.
Why the Other Options Are Wrong
- A. Trapezium: σmin and σmax both compressive: A trapezium with both ends compressive means σ0 > σb (load inside the core).
- B. Rectangle: uniform stress, σmin = σmax: A uniform rectangle means there is no bending at all (axial load only).
- D. Triangle: σmin = 0 at A, σmax at B: σmin = 0 at one edge is the limiting case σ0 = σb (load exactly at the core boundary, e = b/6).
Key Point
σmin = σ0 − σb: positive → trapezium, zero → triangle, negative → tension on one side (σ0 < σb).
Key Concepts for Students
- Middle-third rule: no tension in a rectangular base if e ≤ b/6.
- Core (kern) of a circular section of diameter d is a circle of diameter d/4.
- Masonry dams and chimneys are designed so that the resultant stays inside the middle third.
Practice more: this is question 1 in Bending and Shear Stress MCQs – Part 1. Try the whole set and check every answer instantly.