Science & Engineering
Beam bending stress.
Pick a beam case and a cross-section to get the maximum bending moment, peak bending stress (σ = Mc ÷ I) and deflection. Standard elastic formulas; runs in your browser.
Beam, load and section
Result
Fill the form and press Calculate.
Worked example
A steel bar 50 mm wide and 100 mm deep spans 2 m, simply supported, with a 5 kN point load at mid-span. Maximum moment M = PL ÷ 4 = 5,000 × 2 ÷ 4 = 2,500 N·m. For the rectangle I = 50 × 100³ ÷ 12 = 4,166,667 mm⁴ and c = 50 mm, so σ = Mc ÷ I = 2,500,000 N·mm × 50 ÷ 4,166,667 = 30.0 MPa (about 4,351 psi). Mid-span deflection δ = PL³ ÷ 48EI = 1.00 mm (about L/2000) with E = 200 GPa.
How it works
The flexure formula gives peak bending stress at the extreme fibre as σ = M × c ÷ I, where M is the maximum bending moment, I the second moment of area of the section and c the distance from the neutral axis to the outer edge. The four cases use the standard results: simply supported with a centre point load, M = PL/4 and δ = PL³/(48EI); simply supported with a uniform load w over the span, M = wL²/8 and δ = 5wL⁴/(384EI); cantilever with an end load, M = PL and δ = PL³/(3EI); cantilever with a uniform load, M = wL²/2 and δ = wL⁴/(8EI). A uniform load is entered as its total, so w × L equals the figure you type.
Young's modulus values are typical figures for each material (steel A36 200 GPa, aluminium 68, titanium 116, copper 110, brass 106); real alloys and tempers vary, so use your material's data sheet for anything that matters.
Not for structural design
Planning and learning aid only. It is not for structural, pressure-system or other safety-critical design, and it is not professional engineering advice. Have a qualified engineer check anything that carries people or could fail dangerously. This model is a prismatic, linear-elastic Euler-Bernoulli beam. It does not check yield or ultimate strength, shear, lateral-torsional buckling, connections, vibration or code limits.
Sources and basis: Fundamental Mechanics of Materials Equations (Missouri S&T, J. Thomas); Beams Supported at Both Ends with Continuous and Point Loads (Engineering ToolBox); Cantilever Beam Calculations: Formulas, Loads & Deflections (Engineering ToolBox); Young's modulus (Wikipedia). Checked 2026-10-08. Planning aid only; see the note above.
FAQ
What is the bending stress formula for a beam?
σ = M × c ÷ I: the maximum bending moment times the distance from the neutral axis to the outer edge, divided by the section's second moment of area. Equivalently σ = M ÷ S where S = I ÷ c is the section modulus.
Where is the maximum bending moment?
At mid-span for a simply supported beam (PL/4 for a centre point load, wL²/8 for a uniform load) and at the fixed end for a cantilever (PL for an end load, wL²/2 for a uniform load).
Does this include the beam's own weight?
No. Add the self-weight to the load yourself, as part of the uniform load.
Is a stress below the yield strength safe?
Not by itself. Real design applies safety factors and checks shear, buckling, connections, fatigue and code limits. This tool gives the peak elastic bending stress only.
Why does depth matter so much?
For a rectangle, I grows with the cube of the depth, so doubling the depth cuts the bending stress by about four times at the same load and width.