Last reviewed 16 Sept 2026 · 9 min read
Beams, supports and loads
A beam is a member loaded mainly transverse to its axis, which resists the loads by bending. Before drawing diagrams, identify the support conditions and the loads.
| Support | Reactions it provides | Unknowns |
|---|---|---|
| Roller / simple support | Force perpendicular to the surface | 1 |
| Hinge (pin) | Horizontal and vertical forces | 2 |
| Fixed | Horizontal force, vertical force and moment | 3 |
| Beam | Description | Statically determinate? |
|---|---|---|
| Cantilever | One end fixed, other free | Yes |
| Simply supported | Hinge at one end, roller at the other | Yes |
| Overhanging | Simply supported with one or both ends extending beyond supports | Yes |
| Propped cantilever | Fixed at one end, simply supported at the other | No (1 degree) |
| Fixed (built-in) | Both ends fixed | No |
| Continuous | More than two supports | No |
Loads: point (concentrated) load, uniformly distributed load (UDL, kN/m), uniformly varying load (UVL, triangular or trapezoidal), and applied couple (moment).
Shear force and bending moment
At any section of a beam:
- Shear force (SF) = algebraic sum of all vertical forces on one side of the section.
- Bending moment (BM) = algebraic sum of moments about the section of all forces on one side.
Either side may be used; both give the same answer if the reactions are correct.
- Shear force is positive when the resultant force on the left of the section acts upward (equivalently, downward on the right) — the left part tends to slide up.
- Bending moment is positive (sagging) when the beam bends concave upward — the moment of forces on the left is clockwise (or on the right, anticlockwise). Negative BM is hogging.
A shear force diagram (SFD) and bending moment diagram (BMD) plot these values along the beam.
Relations between load, shear force and bending moment
Take a small element carrying load intensity (downward). Vertical equilibrium and moment equilibrium give:
Consequences used constantly in exams:
- The slope of the SFD at a point equals (minus) the load intensity there.
- The slope of the BMD equals the shear force.
- Bending moment is maximum or minimum where the shear force is zero (or changes sign).
- The change in BM between two sections equals the area of the SFD between them.
- The change in SF between two sections equals the total load between them.
Shapes of the diagrams
| Loading on the segment | SFD | BMD |
|---|---|---|
| No load | Horizontal line (constant) | Inclined straight line |
| Point load | Sudden vertical jump equal to the load | Change of slope (kink) |
| UDL | Inclined straight line | Parabola (2nd degree) |
| Uniformly varying load | Parabola (2nd degree) | Cubic curve (3rd degree) |
| Applied couple | No change | Sudden jump equal to the couple |
The BMD curve is always one degree higher than the SFD curve, which is one degree higher than the load curve.
Standard results
Cantilevers (length , fixed at B, free at A)
| Loading | Max SF (at fixed end) | Max BM (at fixed end) |
|---|---|---|
| Point load at free end | (hogging) | |
| UDL over whole span | ||
| UVL, zero at free end, at fixed end | ||
| UVL, at free end, zero at fixed end | ||
| Couple at free end | 0 | (constant along span) |
Simply supported beams (span )
| Loading | Reactions | Max BM |
|---|---|---|
| Central point load | each | at centre |
| Point load at from A ( from B) | , | under the load |
| UDL over whole span | each | at centre |
| UVL, zero at A to at B | , | at from A |
| Symmetric triangle, peak at centre | each | at centre |
| Couple at a section | Jump of at the couple |
- UDL: cantilever ; simply supported ; fixed beam at supports and at mid-span.
- Central point load: simply supported ; fixed beam at supports and centre.
- For the same total load , a simply supported beam with UDL has max BM — half that of the central point load.