Shear Force and Bending Moment Diagrams
Sign conventions, shear force and bending moment diagrams for cantilevers and simply supported beams under point loads, UDL and moments, the load-shear-moment relations and points of contraflexure.
A beam carries loads across its length. At each section it resists them with a shear force V and a bending moment M. Their diagrams (SFD and BMD) show where the beam is most stressed.
Sign convention
- Shear force at a section: the sum of vertical forces to the left of it (upward positive).
- Bending moment: the sum of moments of the forces to the left about the section; sagging (concave up) is positive, hogging negative.
Relations between load, shear and moment
- dV/dx = −w (w = downward load per length)
- dM/dx = V
So:
- No load between two points → V constant, M linear.
- UDL → V linear, M parabolic.
- M is maximum where V = 0 (or changes sign).
- A point load causes a jump in V; a concentrated moment causes a jump in M.
Standard cases
| Beam and load | Max shear | Max moment |
|---|---|---|
| Cantilever, end load P | P | PL (at the fixed end) |
| Cantilever, UDL w | wL | wL²/2 (at the fixed end) |
| Simply supported, central load P | P/2 | PL/4 (at the centre) |
| Simply supported, UDL w | wL/2 | wL²/8 (at the centre) |
| Simply supported, load P at a from A (b from B) | Pb/L or Pa/L | Pab/L (under the load) |
Point of contraflexure
A point where the bending moment changes sign (passes through zero) is a point of contraflexure. It occurs in overhanging beams and in fixed or continuous beams, never in a simply supported beam with only downward loads.
Worked example
A simply supported beam of span 8 m carries a UDL of 5 kN/m over its whole length and a point load of 20 kN at mid-span.
- Reactions: each = (5 × 8 + 20)/2 = 30 kN.
- Max moment at mid-span = wL²/8 + PL/4 = 5 × 64/8 + 20 × 8/4 = 40 + 40 = 80 kN·m.
Practice
A cantilever 3 m long carries a UDL of 4 kN/m. What is the maximum bending moment in kN·m?
Show answer and explanation
Answer: 18M = wL²/2 = 4 × 9/2 = 18 kN·m at the fixed end.
A simply supported beam of span 10 m carries 40 kN at 4 m from the left support. What is the maximum bending moment in kN·m?
Show answer and explanation
Answer: 96M = Pab/L = 40 × 4 × 6/10 = 96 kN·m, under the load.
Over a length of beam with a uniformly distributed load, the bending moment diagram is:
Show answer and explanation
Answer: CdV/dx = −w makes V linear, and dM/dx = V makes M a second-degree curve.
Where does the maximum bending moment occur in a beam?
Show answer and explanation
Answer: AdM/dx = V, so M has its extreme value where V = 0 (or jumps through zero).