Structural · Mechanics of materials
Beam Stress & Deflection Calculator
Model supports and loads on a span, then watch reactions, shear, moment, deflection, and bending stress update live on the diagrams below.
Free: 2 supports, 2 point loads. Pro adds fixed ends, UDLs, moments, save & PDF.
Cross-section
Profile library or direct I — drives stiffness and bending stress.
Dimensions
Rectangle
Teal highlight follows the active dimension field above
I = 6.67e+7 mm⁴ · A = 20,000 mm²
Material
Young's modulus for deflection and optional stress check.
Results
Max deflection
δ_max from EI·v″ = M
AISI 1045 Steel · SF 19.73
Support reactions
- Pinned @ 0 m5 kN
- Roller @ 5 m5 kN
Reactions, shear, moment, and deflection update live as you edit supports and loads.
Euler–Bernoulli bending, sign conventions, and what the solver assumes.
The tool assembles a direct-stiffness finite element model along the span, so it handles cantilevers, simply-supported and overhanging beams, continuous multi-span layouts, and many statically indeterminate cases. Results include support reactions, shear V, bending moment M, deflection δ, and bending stress σ = M·y/I when the extreme fibre distance is known.
- LoadsDownward forces and clockwise applied moments are positive
- Shear VNet upward force on the segment left of a cut
- Moment MSagging (tension on bottom fibre) is positive
- Deflection δDownward displacement is positive
Supports
Pinned and roller supports restrain vertical motion; fixed supports also restrain rotation and develop a reaction moment.
Section
Standard profiles compute I and y automatically. Enter I directly when you already have handbook values.
Bending stress
Peak σ is compared with catalog yield strength for a quick safety factor — not a substitute for code-compliant design checks.
- Euler–BernoulliEI d⁴v/dx⁴ = w(x)
- Bending stressσ = M·y / I
- Safety factorSF = σ_y / σ_max
- DeflectionFrom curvature κ = M/(EI)
Worked example: Simply-supported 5 m beam with a central 10 kN point load
- Reactions: each support carries P/2 = 5 kN
- Max bending moment at mid-span M = PL/4 = 10 × 5 / 4 = 12.5 kN·m
- Max deflection δ = PL³/(48EI)
M_max = 12.5 kN·m at mid-span, with peak shear of 5 kN at the supports
What beam types can this calculator analyse?
It uses the finite element (direct stiffness) method, so it handles cantilevers, simply-supported, overhanging, propped and continuous multi-span beams, including statically indeterminate cases. Add any combination of pinned, roller and fixed supports.
Do I have to design the cross-section?
No. You can pick a standard profile (rectangle, circle, I-beam, hollow rectangle or tube) to compute the second moment of area automatically, or simply type in the second moment of area I from a datasheet and skip the geometry.
What sign conventions are used?
Downward loads and clockwise applied moments are positive, the bending moment is sagging-positive, positive shear is the net upward force on the segment to the left of a cut, and deflection is reported as positive downward.
How is the bending stress calculated?
Bending stress is σ = M·y/I, evaluated along the span using the local bending moment M and the distance y from the neutral axis to the extreme fibre. The peak stress is compared with the material's yield strength to give a safety factor.
What is free and what needs Pro?
Single-span beams (two supports) with point loads and uniform distributed loads, plus on-screen shear, moment and deflection diagrams, are free. Multi-span beams, applied moments, linearly varying loads, bending-stress utilisation, saving and branded PDF export are Pro features.