Beam Deflection Calculator — Simply Supported, Cantilever, Fixed & Propped

Every beam sags under load; the question is whether the sag breaks a code limit. This structural beam calculator is three tools in one: a beam load calculator that solves reactions, shear, and moment; a beam span calculator that reports deflection against L/360–L/180 limits for any span; and a beam stress calculator that reports bending stress alongside — so serviceability and strength are checked together. Pick a support, pick a load, enter span and stiffness, and read peak deflection with its location, live diagrams, and a green-or-red code verdict in SI or Imperial units.

Beam deflection formula: the six cases that matter

The beam deflection formula changes with supports and loading, but six closed forms cover nearly every field check. Simply supported center point load: δ = PL³/48EI. Simply supported full uniform load: δ = 5wL⁴/384EI. Cantilever tip point load: δ = PL³/3EI. Cantilever full uniform load: δ = wL⁴/8EI. Fixed-fixed center point: δ = PL³/192EI. Fixed-fixed uniform load: δ = wL⁴/384EI. Here P is point load, w is load per length, L is span, E is Young's modulus, and I is the moment of inertia — stiffness is the product EI. Partial uniform loads, off-center points, and end moments use the same theory with position terms, and combined point-plus-uniform loading adds by superposition for small elastic deflections. For all ten cases with derivations, see the beam deflection formula reference.

Simply supported beam: joists, lintels, and everyday spans

The simply supported beam — a joist resting on two walls, a lintel over an opening — is the default assumption for floors and roofs, and the case every other support is compared against. Two supports share the curvature, which is why the center-point formula carries that forgiving 48 in the denominator. Enter span, E, I, and load to get midspan-dominated deflection plus the L/δ span ratio that codes actually limit. For the full treatment with floor-joist examples, open the simply supported beam guide.

Cantilever beam calculator: balconies, eaves, and overhangs

This cantilever beam calculator mode covers balconies, eaves, canopies, and overhangs — one fixed end, one free tip, and no forgiveness. With identical P, L, E, and I, a cantilever tip deflects 16× more than a simply supported center (PL³/3EI vs PL³/48EI), because one support must develop all curvature. Codes answer with the L/180 cantilever limit and the ×2 length rule, but the engineering answer is shorter arms and deeper sections: deflection scales with L³ for point loads and L⁴ for uniform loads, so small span cuts pay enormously. The calculator also reports the fixed-end moment your wall or column connection must develop. Details live in the cantilever beam guide.

Fixed, propped, and the stiffness between

Fixed-fixed ends — welded frames, cast-in-place concrete — are 4× stiffer than simply supported for point loads and 5× for uniform loads, when connections can truly develop moment. Real connections rarely achieve full fixity, so honest answers sit between pinned and fixed: the propped option (one fixed end, one pinned) models that middle ground. If a “simply supported” check fails by a small margin, do not quietly assume fixity to make it pass — verify what the connection can actually develop.

Steel beam calculator: W-shapes, E, and I-beam deflection

As a steel beam calculator, this page takes AISC sections directly: enter the tabulated inertia — for example, W12×26 with I = 204 in⁴ — select steel at E = 200 GPa (29,000 ksi), and apply span and loads for an I-beam deflection answer in seconds. The 20 ft, 1,000 plf example above passes L/360 with room to spare using exactly that section. No W-book handy? The built-in section helper computes I for solid and hollow rectangles and circles (rectangle I = b·h³/12), and presets cover aluminum at 69 GPa, SPF wood at 9.65 GPa, and concrete near 25 GPa. I-beam deflection, wood-joist sag, and slab checks all run through the same engine with units kept consistent. Whether you type I-beam or I beam deflection, the math is identical: stiffness EI working against span L.

IBC deflection limits: L/360, L/240, L/480, L/180

IBC Table 1604.3 sets the lines this page enforces: L/360 for floor live load, L/240 for total dead-plus-live load, L/480 where tile, plaster, or brittle finishes crack first, and L/180 for cantilevers. A 20 ft floor beam allows 0.667 in under live load and 1.0 in total; the limits table further down this page converts every ratio to inches for that span. The calculator reports utilization as a percentage of the selected limit, so 107% means find a stiffer section, not a friendlier code. Compare every ratio with metric examples in the allowable deflection limits guide.

Beam stress calculator: deflection is not strength

Passing deflection never proves a beam is strong enough — sag is serviceability, collapse is strength. Alongside δmax, this page reports maximum moment, maximum shear, and bending stress σ = M·c/I, so the beam stress calculator half of the check runs on the same inputs: compare σ against your material allowable, shear against capacity, and add buckling checks for thin or laterally unsupported sections. Long-span floor beams often fail deflection before stress governs, which surprises first-time users and is precisely why both numbers appear together.

Related structural resources

For derivations, use the beam deflection formula reference. For worked numbers, use the beam deflection how-to guide with bench, joist, balcony, and steel examples. For the full collection, browse all structural calculators.