The relation 'slope = area of BMD / EI' is given by:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 30/10/2020 (Evening)
Detailed Explanation
The moment-area method has two theorems. The first deals with slope, the second with deflection (tangential deviation).
Mohr I: \[ \theta_B - \theta_A = \int_A^B \frac{M}{EI}\,dx = \frac{\text{Area of BMD between A and B}}{EI} \]
Mohr II: \[ t_{B/A} = \frac{\text{Moment of that area about B}}{EI} = \frac{A\bar{x}}{EI} \]
Why “Mohr's first theorem” Is Correct
Slope (change of slope) from the area of the M/EI diagram is Mohr's first theorem. For a cantilever, measuring from the fixed end where the slope is zero, the area gives the slope directly.
| Theorem | Statement | Gives |
|---|---|---|
| Mohr's first | θ_B − θ_A = area of M/EI diagram between A and B | Slope |
| Mohr's second | Deviation of B from tangent at A = moment of that area about B ÷ EI | Deflection |
Why the Other Options Are Wrong
- A. Mohr's second theorem: Mohr's second theorem gives deflection (moment of the area), not slope.
- B. Macaulay's theorem: Macaulay's method is an integration technique, not this theorem.
- D. Castigliano's theorem: Castigliano's theorem uses strain energy: δ = ∂U/∂P.
Key Point
Mohr I: slope = area of M/EI; Mohr II: deflection = moment of area of M/EI.
Key Concepts for Students
- Cantilever tip load: area = (1/2)(WL)(L) = WL²/2 ⇒ θ = WL²/2EI.
- Moment of that area about the tip = (WL²/2)(2L/3) ⇒ δ = WL³/3EI.
- Moment-area and conjugate beam methods use the same M/EI diagram.
Practice more: this is question 17 in Slope and Deflection MCQs – Part 2. Try the whole set and check every answer instantly.