A pile of length L with weight W per metre is lifted by two suspension points placed so that the bending moment at the centre equals that at the points of suspension. The maximum bending moment is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 24/01/2018 (Evening)
Detailed Explanation
A pile being lifted is a beam loaded by its own weight (a UDL) and supported at the lifting points. To keep the moment small, the lifting points are placed so that the hogging moment at them equals the sagging moment at the centre.
Let each end overhang be a. Hogging at a support: \( \dfrac{Wa^2}{2} \). Sagging at centre: \( \dfrac{W(L-2a)^2}{8} - \dfrac{Wa^2}{2} \).
Equating: \( \dfrac{W(L-2a)^2}{8} = Wa^2 \Rightarrow L - 2a = 2\sqrt2\,a \Rightarrow a = \dfrac{L}{2(1+\sqrt2)} = 0.207L \)
\[ M_{max} = \frac{W(0.207L)^2}{2} = 0.0214\,WL^2 \approx \frac{WL^2}{46.6} \approx \frac{WL^2}{47} \]
Why “WL²/47” Is Correct
The maximum moment is about WL²/47 — option (c). Lifting at one point (centre) would give WL²/8, almost six times larger.
Why the Other Options Are Wrong
- A. WL/8: WL/8 has the wrong units and corresponds to single-point lifting (WL²/8).
- B. WL²/24: WL²/24 is not the two-point lifting value.
- D. WL²/26: WL²/26 is too large — it does not equalise the moments.
Key Point
Two-point lifting at 0.207L from each end ⇒ M_max ≈ WL²/47.
Key Concepts for Students
- The same 0.207 ratio minimises the moment in a beam with equal overhangs under UDL.
- Three-point lifting reduces it further, to about WL²/90.
- Precast piles are designed for handling stresses as well as driving stresses.
Practice more: this is question 33 in Bending Moment and Shear Force MCQs – Part 3. Try the whole set and check every answer instantly.