The slope at the fixed support of a cantilever of length L and flexural rigidity EI, carrying a point load P at mid-span and another point load P at the free end, is:
Try it first: pick an option, then read the explanation.
Correct answer: D. zero
Asked in SSC JE 11/10/2023 (9 am–11 am)
Detailed Explanation
The question asks for the slope at the fixed support, not at the free end. A fixed support is defined by zero rotation and zero displacement.
Why “zero” Is Correct
Whatever loads act on the cantilever, the slope at the built-in end stays zero. The loads only change the slope and deflection elsewhere along the beam.
| Beam and loading | Maximum slope | Maximum deflection | Where |
|---|---|---|---|
| Cantilever, point load W at free end | WL²/2EI | WL³/3EI | Free end |
| Cantilever, UDL w over full length | wL³/6EI | wL⁴/8EI | Free end |
| Cantilever, triangular load (w at fixed end, 0 at free end) | wL³/24EI | wL⁴/30EI | Free end |
| Cantilever, couple M at free end | ML/EI | ML²/2EI | Free end |
| Simply supported, central point load W | WL²/16EI | WL³/48EI | Slope at supports, deflection at centre |
| Simply supported, UDL w over full span | wL³/24EI | 5wL⁴/384EI | Slope at supports, deflection at centre |
| Simply supported, equal end couples M (pure bending) | ML/2EI | ML²/8EI | Slope at supports, deflection at centre |
Standard results (EI constant). In a cantilever the slope and deflection are zero at the fixed end and maximum at the free end.
Why the Other Options Are Wrong
- A. PL/(4EI): PL/4EI has the wrong dimensions for a slope from point loads (should be PL²/EI) and in any case the fixed-end slope is zero.
- B. 3PL/(4EI): 3PL/4EI is not the slope at the support.
- C. PL/(2EI): PL/2EI is not the slope at the support.
Key Point
Fixed support: θ = 0, y = 0.
Key Concepts for Students
- Free-end slope here = P(L/2)²/2EI + PL²/2EI = 5PL²/8EI.
- Read which point the question asks about before using formulas.
- Simple supports have y = 0 but non-zero slope.
Practice more: this is question 8 in Slope and Deflection MCQs – Part 1. Try the whole set and check every answer instantly.