Shear Force & Bending Moment Diagrams
Cantilever, simply supported and overhanging beams under point loads, UDL and moments — how to draw and sanity-check every diagram.
Key formulas
M/I = σ/y = E/R
Euler–Bernoulli bending: The flexure formula — convert M to stress at any fibre distance y from the neutral axis.
SS+central load: Mmax = WL/4; SS+UDL: Mmax = wL²/8; Cantilever+end load: Mmax = WL
Standard maxima: Memorise — interviewers expect these instantly, including shear values V = W/2 and wL/2 at supports.
Z = I / y_max; σmax = M / Z
Section modulus: Design form of the flexure formula; rectangular section Z = bd²/6.
Cheat sheet
- ▸Sign convention (common): sagging bending moment positive, upward force on the left of a section gives positive shear. State your convention on the whiteboard.
- ▸Slope of the shear diagram = load intensity; slope of the BM diagram = shear. dM/dx = V and dV/dx = −w — use this to sketch fast and to verify.
- ▸Point loads create vertical jumps in SFD and kinks in BMD; a UDL gives a linear SFD and a parabolic BMD; applied couples create jumps in BMD only.
- ▸Maximum bending moment occurs where shear force crosses zero (or at a support for cantilevers). Max BM for SS beam + central point load = WL/4; + UDL = wL²/8.
- ▸Bending stress σ = M·y/I; the neutral axis carries zero bending stress. Section with more area away from NA (I-beam) resists bending better.
- ▸For a cantilever with UDL, max shear and moment are both at the fixed end: V = wL, M = wL²/2 (hogging).
Beam cases at a glance
| Case | Max shear | Max BM | BM shape |
|---|---|---|---|
| Cantilever, end load W | W (fixed end) | WL | Linear |
| Cantilever, UDL w | wL | wL²/2 | Parabolic |
| SS beam, central W | W/2 at supports | WL/4 at centre | Triangular |
| SS beam, UDL w | wL/2 at supports | wL²/8 at centre | Parabolic |
Visual
SS beam with UDL w over span L:
SFD: +wL/2 ______ slope −w ______ −wL/2 (linear, crossing zero at midspan)
BMD: 0 __ __ 0
\__ __/ parabola, peak wL²/8 at centre
\_________/Top 4 campus interview questions
MCQ mastery quiz
1. For a simply supported beam of span L with UDL w, the maximum bending moment is…
2. The relation between bending moment and shear force is…
3. A point load on a beam produces, in the SFD and BMD respectively…
4. In a cantilever carrying a UDL, the maximum bending moment is at…