Single-span beams
Tool information
What this page computes
The calculator gives the maximum bending moment and deflection for a single-span beam, using the classical formulas of strength of materials. It covers six typical configurations at once, combining three support schemes with two load types:
| Support | Point load P | Uniform load q |
|---|---|---|
| Simply supported | at midspan | over the whole span |
| Cantilever (fixed at one end) | at the free end | over the whole length |
| Fixed at both ends | at midspan | over the whole span |
Why the support scheme matters
The same load and span give very different moments depending on how the beam is supported. For a uniform load \(q\) over a span \(L\):
- simply supported: \(M_{max} = qL^2/8\) (at midspan)
- fixed at both ends: \(M_{max} = qL^2/12\) at the support and \(qL^2/24\) at midspan
Fixity moves the maximum moment to the supports and reduces it — hence the advantage of continuity. The cantilever, by contrast, concentrates all the moment at the fixed end and gives large deflections at the free end.
Deflection and the serviceability check
Besides strength, the beam must be checked for deflection (Serviceability Limit State). The deflection depends on the stiffness \(EI\): for a simply supported beam under uniform load,
\(f_{max} = \frac{5\,qL^4}{384\,EI}\)
The calculator compares the computed deflection with the admissible deflection, taken by default as \(L/300\) — a common limit for ordinary members. The \(L^4\) dependence shows why the span is the dominant parameter: doubling the span multiplies the deflection by 16.
Input data
- The span \(L\) of each beam.
- The load — the point force \(P\) or the uniform intensity \(q\).
- The section — the moment of inertia \(I\) (from a profile or entered), for the deflection.
- The admissible deflection — the limit ratio (default \(L/300\)).
Assumptions and limitations
- A straight, elastic beam of constant section; linear behaviour (no plastic hinges).
- Static loads; dynamics, fatigue and second-order effects are not treated.
- The cases are the canonical ones (a centred or full uniform load); for arbitrary schemes, multiple spans or intermediate supports, use the finite-element analysis.
- The section strength check (whether \(M_{max}\) is within resistance) is done separately — see the steel or concrete calculations.
Related calculations
- Continuous beam (finite element) — for multiple spans and intermediate supports
- Bending resistance (steel) — the section check at \(M_{max}\)
- Reinforced section (concrete) — the concrete equivalent
- Slab moments — for elements spanning in two directions