Beam Deflection Calculator
Finite-element beam solver for deflection, rotation, shear and moment with multi-span supports, point and distributed loads, P-Delta and elastic foundation.
Continuous FEA Beam Mainframe
Timoshenko-Ehrenfest Theory // Multi-Span Continuity Matrix // Elastoplastic boundary
Material Profile Stratum
Boundary Matrix CONTINUITY
Complex Load Vectors
VIEWPORT A // Deflected Structural Profile
HOVER TO EXAMINE FIBER STRESSESVIEWPORT B // SHEAR FORCE & BENDING MOMENT DIAGRAMS (SFD / BMD)
VIEWPORT C // Live Sectional Stress Fiber heatmap
A clear path from inputs to results.
A beam calculator is only as good as its boundary conditions. This one assembles a Timoshenko beam stiffness matrix with optional P-Delta axial stiffening and Winkler foundation springs, so multi-span and continuous beams are solved properly rather than approximated by a single-span formula.
How to use
- Set the span length and choose a material class.
- Pick a section preset — W-shape, RHS, circular or solid rectangle — and its dimensions.
- Define supports: pinned, roller or fixed, at any position along the span.
- Add point loads and uniformly distributed loads at their positions.
- Optionally add axial thrust for P-Delta effects or a Winkler foundation modulus for soil-supported beams.
- Read the deflection, rotation, shear and moment diagrams and compare the peak deflection with the span/360 serviceability limit.
The formula
Reference: Timoshenko beam theory; Gere & Goodno, "Mechanics of Materials"; AISC 360 serviceability provisions.
Worked example
Simply supported 150 × 300 mm section, 8 m span, 10 kN/m
- Solid rectangle 150 mm wide, 300 mm deep.
- E = 200 GPa (steel).
- Span 8 m, UDL 10 kN/m.
- 1
Second moment: 150 × 300³ / 12 = 337.5 × 10⁶ mm⁴ = 3.375 × 10⁻⁴ m⁴.
- 2
EI = 200 × 10⁹ × 3.375 × 10⁻⁴ = 6.75 × 10⁷ N·m².
- 3
Deflection: 5 × 10,000 × 8⁴ / (384 × 6.75 × 10⁷) = 7.9 mm.
- 4
Serviceability limit: L/360 = 22.2 mm, so the beam passes.
- 5
Peak moment: wL²/8 = 80 kN·m, giving a bending stress of 80e3 / 2.25e-3 = 35.6 MPa.
7.9 mm mid-span deflection · 35.6 MPa bending stress · well inside L/360.
Interpreting the result
Deflection scales with the fourth power of span, which is why a 20% longer span deflects more than twice as much. It also scales inversely with the second moment, so depth is the cheapest lever: doubling depth cuts deflection by a factor of eight. The serviceability limit often governs before strength does on long spans.
Assumptions & limitations
- Linear elastic, small-deflection behaviour. Large deflections need a geometrically nonlinear solve.
- Plane sections remain plane, with shear deformation included through the Timoshenko correction.
- Loads are applied in the plane of bending and the section is symmetric about that plane.
- No lateral-torsional buckling, local buckling or connection flexibility is modelled.
- The Winkler foundation is linear and continuous, with no uplift or separation.
Frequently asked questions
When should I use Timoshenko rather than Euler-Bernoulli?
When the span-to-depth ratio is below about 10, where shear deformation becomes significant. Deep beams, short spans and composite sections all benefit.
What deflection limit should I use?
L/360 is the common floor limit for brittle finishes; L/240 for roofs without ceilings; L/480 or tighter for sensitive equipment. Check the governing code for the actual occupancy.
What is P-Delta and when does it matter?
It is the second-order effect of axial load acting on a deflected shape, which amplifies both deflection and moment. It matters for slender columns and for beams carrying significant axial compression.
Can I model a continuous beam over several supports?
Yes — add each support at its position and the solver assembles the full multi-span system, which is where closed-form formulas stop being usable.
Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.