Which of the following methods used for finding the slope and deflection of beams is also called the 'method of singularity functions'?
Try it first: pick an option, then read the explanation.
Asked in SSC JE 06/06/2024 (4 pm–6 pm)
Detailed Explanation
With several point loads, the ordinary double integration method needs a separate moment equation for every segment and two constants for each. Macaulay's method writes a single moment equation valid for the whole beam using brackets that are zero until x passes a load point.
Why “Macaulay's method” Is Correct
The brackets ⟨x − a⟩ⁿ are singularity (discontinuity) functions, so Macaulay's method is also called the method of singularity functions. Only two integration constants are needed for the whole beam.
| Method | Used for | Basis |
|---|---|---|
| Double integration | Slope and deflection | Integrate EI y'' = M twice; constants from boundary conditions |
| Macaulay's (singularity function) method | Slope and deflection with discontinuous loads | One BM equation with brackets ⟨x − a⟩ |
| Moment-area (Mohr's theorems) | Slope and deflection | Area and moment of area of the M/EI diagram |
| Conjugate beam | Slope and deflection | M/EI diagram applied as load on a conjugate beam |
| Strain energy / Castigliano / unit load | Deflection at a point | Energy principles |
| Mohr's circle | Stresses on inclined planes — not deflection | Graphical stress transformation |
| Moment distribution / three-moment equation | Support moments in indeterminate beams — not deflection | Stiffness and carry-over / compatibility |
Why the Other Options Are Wrong
- B. Mohr's theorem method: Mohr's theorems use areas and moments of the M/EI diagram.
- C. Conjugate beam method: The conjugate beam method loads an imaginary beam with the M/EI diagram.
- D. Moment area method: The moment area method is another name for Mohr's theorem method.
Key Point
Macaulay's method = method of singularity functions (bracket functions ⟨x − a⟩).
Key Concepts for Students
- ⟨x − a⟩ = 0 for x < a and (x − a) for x ≥ a.
- Brackets are integrated as a whole, never expanded.
- A couple M₀ at x = a enters as M₀⟨x − a⟩⁰.
Practice more: this is question 4 in Slope and Deflection MCQs – Part 1. Try the whole set and check every answer instantly.