Stax
Tools

Beam Calculator

Calculate reactions, shear force, bending moment, and deflection for beams.

Reaction at A (RA)
25.00 kN
Reaction at B (RB)
25.00 kN
Max shear force
25.00 kN
Max bending moment
31.25 kN·m @ 2.50m
Bending Moment Diagram (sagging positive)
AB
Simply supported beam · Deflection δ_max = 5wL⁴ / (384EI)
Relative δ (EI=1): 81.3802 kN·m³

Simply supported beam analysis

Calculate support reactions, shear force, maximum bending moment, and midspan deflection for a simply supported beam under three common loading conditions: uniform distributed load (UDL), a single point load, or an applied moment. The bending moment diagram is plotted instantly so you can visualise the stress distribution along the beam.

Three beam loading types explained

UDL (Uniform Distributed Load)— load spread evenly over the full span, such as a floor slab’s own weight or a snow load. Maximum bending moment occurs at midspan: M = wL²/8.

Point load — a concentrated force at a specific position along the span, such as a column or a heavy piece of equipment. When placed at midspan, maximum moment is M = PL/4.

Applied moment — a pure rotational moment applied at a point, less common in practice but used in frame analysis and connection design. The beam experiences linear bending moment variation between supports.

Reading the bending moment diagram

The bending moment diagram (BMD) shows how the internal bending moment varies along the beam length. The peak of the diagram indicates where the beam is under greatest stress and where cross-section sizing is most critical. For a UDL, the BMD is parabolic with the peak at midspan. For a midspan point load, it is triangular. The sign convention used here is sagging positive — a beam bending concave-up produces a positive moment.

How to use the results for structural design

The maximum bending moment (M) is used with the section modulus (Z) to check bending stress: σ = M/Z. Compare this against your material’s allowable bending stress. The deflection result should be checked against the span/250 or span/360 limit specified in your applicable design standard. This calculator is a quick-check tool suitable for preliminary design and study — always verify critical structural members against the relevant code using full design software.

Common use cases

Civil engineering students use the calculator to verify hand-calculated bending moment and reaction values for coursework problems, cross-checking their working before submission. Structural engineers use it for rapid preliminary sizing during the concept design phase — entering span and estimated load to get a ballpark bending moment before committing to detailed analysis software. Builders and contractors use it to do a quick sanity check on beam spans for simple residential floor joists or lintels before consulting a structural engineer for sign-off. Engineering educators use the bending moment diagram output to visually demonstrate how load position affects the distribution of internal forces along a beam.

Frequently asked questions

What is a simply supported beam?
A simply supported beam is supported at both ends with pinned supports. It can rotate freely at the supports but cannot move vertically. This is the most common beam configuration in structural analysis.
What is the maximum bending moment for a UDL beam?
For a simply supported beam with uniform distributed load w over span L, the maximum bending moment is M = wL²/8, occurring at the midspan.
What is EI in deflection formulas?
EI is the flexural rigidity of the beam — E is Young's modulus (material stiffness) and I is the second moment of area (section stiffness). Higher EI means less deflection.
What is the difference between a point load and a UDL?
A point load (also called a concentrated load) acts at a single location along the beam — for example, a column sitting on a beam. A UDL (Uniformly Distributed Load) is spread evenly across the full span or a portion of it — like the self-weight of a floor slab. A UDL of w kN/m over span L is equivalent to a single point load of wL at the midspan for reaction purposes, but the bending moment distribution differs.
What units should I use for inputs?
Use consistent units throughout. If your span is in metres and load in kN (or kN/m for UDL), the bending moment output is in kN·m and deflection in metres. If you prefer mm and N, use N and mm throughout. The calculator accepts any consistent unit system.
Why is deflection important in beam design?
Excessive deflection causes cracking in supported finishes, misalignment of doors and windows, and a perception of an unsafe structure. Design codes typically limit deflection to span/250 or span/360 under live loads. The calculator outputs maximum deflection so you can check your beam against these serviceability limits alongside strength checks.

Related tools