If a cantilever beam carries a uniformly distributed load over its entire length, the shape of the bending moment diagram is:
Try it first: pick an option, then read the explanation.
Correct answer: B. quadratic parabola
Asked in SSC JE 2004
Detailed Explanation
Measure x from the free end. The load beyond the section is wx, acting at x/2.
\[ M_x = wx\cdot\frac{x}{2} = \frac{wx^2}{2} \]
Why “quadratic parabola” Is Correct
M varies with x², so the BMD is a quadratic (second-degree) parabola — option (b), reaching wl²/2 at the fixed end.
| Load on a segment | Shape of SFD | Shape of BMD |
|---|---|---|
| No load | Constant (horizontal line) | Linear (inclined straight line) |
| Point load | Sudden step equal to the load | Kink (change of slope) |
| UDL | Linear (1st degree) | Quadratic parabola (2nd degree) |
| Uniformly varying (triangular) load | Quadratic parabola | Cubic parabola (3rd degree) |
| Applied couple | No change | Sudden jump equal to the couple |
Why the Other Options Are Wrong
- A. linear: A linear BMD (triangle) comes from a point load at the free end.
- C. cubic parabola: A cubic parabola comes from a triangular load.
- D. triangle: A triangle is the BMD for an end point load.
Key Point
Cantilever + UDL: SFD triangle (linear), BMD quadratic parabola.
Key Concepts for Students
- Point load → linear BMD; UDL → 2° parabola; triangular load → 3° parabola.
- Max BM wl²/2 at the fixed end.
- The BMD is tangent to the axis at the free end (zero shear there).
Practice more: this is question 61 in Bending Moment and Shear Force MCQs – Part 5. Try the whole set and check every answer instantly.