Beam Calculator with Diagrams and Section Properties

Reactions, shear force, bending moment, stress and deflection of a beam with a uniform load and a point load, with diagrams and the properties of common cross-sections.

How it works

The calculator solves the three most common beam cases (simply supported, cantilever and fixed at both ends) for a uniformly distributed load plus a point load at any position, using the standard closed-form equations and superposition. It gives the reactions, draws the shear force, bending moment and deflection diagrams, finds the maxima and checks the bending stress against the strength of the material.

σ = M c ÷ I, S = I ÷ c, I = b h³ ÷ 12 (rectangle), δmax = 5 w L⁴ ÷ (384 E I) (simply supported, uniform load)

Sign convention: loads act downwards, positive shear is upward on the left of a cut, positive moment makes the beam sag (tension on the bottom). A cantilever or fixed beam has hogging (negative) moment at its fixed end. The section properties come from the dimensions of the shape (or from your own I, c and A for another section). The theory is on Structures and Strength of Materials, with the formulas for the other loadings and the beam profiles.

Worked example

A 3 m simply supported steel beam (S275, E = 210 GPa, yield 275 MPa) of 100 mm × 200 mm rectangular section carries a uniform load of 5 kN/m and a 10 kN point load 1 m from the left end.

  • I = 0.1 × 0.2³ ÷ 12 = 66 666 667 mm⁴; S = I ÷ 0.1 = 666 667 mm³; A = 20 000 mm²; mass = 0.02 × 7 850 = 157 kg/m.
  • Reactions: R1 = 5 000 × 3 ÷ 2 + 10 000 × 2 ÷ 3 = 14 167 N; R2 = 7 500 + 10 000 × 1 ÷ 3 = 10 833 N. Their sum, 25 000 N, equals the total load 25 kN.
  • The shear is zero where R1 − w x − P = 0, at x = (14 167 − 10 000) ÷ 5 000 = 0.833 m, where the moment is a maximum: Mmax = 11 667 N·m = 11.67 kN·m at x = 1.00 m. Maximum shear = 14 167 N at the left support.
  • Stress σ = 11 667 ÷ 0.0006667 = 17.5 MPa; safety factor = 275 ÷ 17.5 = 15.7.
  • Deflection: the maximum is 0.72 mm at x = 1.44 m (span ÷ 4 164). A numerical double integration of M ÷ E I gives the same value, which confirms the closed-form equations in the script.

Things to know

The results are for a linear-elastic beam with small deflections and a uniform section. Plastic design, buckling of slender beams or thin flanges (lateral-torsional buckling), shear deformation of short deep beams and local crushing at supports are not included.

Check the shear stress as well as bending: for a rectangle τmax = 1.5 V ÷ A, for an I-beam the web carries most of V. Short, heavily loaded beams are governed by shear.

Deflection limits depend on the use: span ÷ 360 for floors with plaster, span ÷ 250 for general beams, span ÷ 180 for roofs, and much stricter for machine parts and precision frames.

Real supports are not ideal: a bolted end is neither a perfect pin nor a perfect fixed end. Bracket the answer by calculating both ideal cases.

Several point loads or a partial load can be found by superposition: add the results of separate calculations (reactions, moments and deflections add for a linear beam).

For standard rolled sections (I-beams, channels, angles) use the section properties from the supplier’s tables in the “Other” option.

Common questions

Where is the maximum bending moment?

Where the shear force is zero (or changes sign). For a simply supported beam with a central load it is at midspan; for a cantilever it is at the fixed end.

What does the section modulus tell me?

The bending strength of the section per unit of allowable stress: the maximum moment is S × the allowable stress. A deeper section has a much larger S for the same area because S rises with the depth squared.

Why does deflection depend so strongly on length?

For a uniform load deflection goes as L⁴; for a point load as L³. Doubling the span multiplies the deflection of a simply supported beam under uniform load by 16.

Why is a fixed beam stiffer than a simply supported one?

The end moments restrain the rotation at the supports. For a uniform load the maximum deflection of a fixed-fixed beam is one fifth of that of a simply supported beam.

Sources

  • Gere, J. M. and Goodno, B. J., Mechanics of Materials, Cengage (beam deflection tables); Young, W. C., Budynas, R. G. and Sadegh, A. M., Roark’s Formulas for Stress and Strain, McGraw-Hill.
  • AISC, Steel Construction Manual, Table 3-23 (beam diagrams and formulas); Eurocode 3 (EN 1993-1-1) for design.

Last reviewed: November 2, 2026