Beam Deflection Calculator

Background.

A beam that is strong enough can still be unacceptable. Bounce a floor and the drywall over it cracks; sag a header and the door under it binds; deflect a lintel and the brick above it steps. That is why building codes carry a separate serviceability limit — a maximum sag expressed as the span divided by a number — alongside the strength requirements. This page computes the sag, compares it against the limit you select, and tells you the stiffness you would need to satisfy that limit exactly.

The mathematics is elastic beam theory, and for the four arrangements almost every residential and light-commercial member falls into, the answers are closed-form. A simply supported beam under a uniform load deflects five times the load times the span to the fourth, over three hundred and eighty-four times the modulus of elasticity times the moment of inertia. Concentrate that same total load at midspan instead and the deflection drops to five-eighths of the uniform figure, because the load has moved on average closer to the supports. Turn the beam into a cantilever and the numbers get dramatically worse: a cantilever under a uniform load deflects forty-eight times as far as the same beam simply supported over the same length.

The two things worth internalising are the exponents. Span appears to the fourth power, so a beam spanning twelve feet sags sixteen times as far as the same beam spanning six. Depth appears cubed, because the moment of inertia of a rectangle is breadth times depth cubed over twelve — so swapping a 2x6 for a 2x12 cuts deflection to an eighth, while nailing on a second 2x6 alongside only halves it. Deeper always beats wider, and by a very large margin. This is the single most useful piece of structural intuition a builder can have, and it is why the answer to a bouncy floor is almost never more joists.

The limit itself is a code value, not an opinion. IRC 2021 Table R301.7 gives L/360 for floors including deck floors and for ceilings with plaster or stucco, L/240 for ceilings with gypsum board and for all other structural members, L/180 for rafters steeper than 3:12 with no ceiling attached, and L/600 for lintels supporting masonry veneer. IBC 2021 Table 1604.3 covers the same ground for commercial work, with separate columns for live load alone and for dead plus live. Both tables carry the same footnote about cantilevers — for a cantilever member, L is taken as twice the length of the cantilever — and this page implements that rule, which is why a four-foot cantilever checked at L/240 gets an allowance of 0.4 inches rather than 0.2.

One decision the page deliberately leaves to you: which load to deflect. IRC-style checks and the AWC span tables apply L/360 to the live load only, on the reasoning that the dead-load sag is already there when the finishes go on and is not what cracks them. IBC Table 1604.3 tabulates dead plus live separately. This calculator deflects whatever load you type into the field, so enter the live load alone for an IRC-style check and the full load for a D+L check. Guessing on your behalf would produce a confidently wrong answer in half the cases, which is worse than asking.

The worked example is a two-ply 2x12 Douglas Fir-Larch header, three inches of dressed breadth by 11.25 inches deep, spanning twelve feet and carrying 400 pounds per lineal foot, with a modulus of elasticity of 1,600,000 psi from NDS Table 4A. Its moment of inertia is 355.95703125 in⁴ and it deflects 0.32768 inches, which is L/439 — comfortably inside the 0.4 inch allowance that L/360 gives a twelve-foot span, at 81.92 percent of it. To land exactly on the limit the header would need 291.6 in⁴, so there is real margin here.

Finally, the scope, which matters more than the arithmetic. Deflection is a serviceability check and nothing else. A member that passes L/360 may still fail in bending, in horizontal shear, in bearing at its ends, or by lateral-torsional buckling if it is not braced. This page reports the maximum moment and shear so you can carry them into those checks, but it does not perform them. It also uses instantaneous elastic deflection: NDS 3.5.2 requires long-term creep to be accounted for in wood, which typically means adding half again or more of the dead-load deflection over the life of the building. Adopted code editions and local amendments vary between jurisdictions, and a licensed structural engineer or architect must sign off on a structural member before work proceeds.

What is beam deflection calculator?

Deflection is how far a loaded beam moves out of line, measured perpendicular to its original axis at the point where the movement is greatest. For a simply supported beam that point is at or near midspan; for a cantilever it is the free end. It is a serviceability quantity rather than a safety one: a beam bends visibly long before it breaks, and the codes limit the bending so that finishes, doors, windows and occupants' nerves survive.

The governing quantities are the modulus of elasticity E, which is a property of the material, and the moment of inertia I, which is a property of the shape. Their product EI is the flexural rigidity, and every deflection formula on this page has the form δ = K ÷ (E·I) where K depends only on the load, the span and the support conditions. That structure is why solving for the required moment of inertia is trivial once the deflection is known, and it is why the calculator can answer both questions from one computation.

The limit is written as the span divided by a number — L/360, L/240, L/180 — so it scales with span rather than being a fixed distance. A twelve-foot floor joist is allowed 0.4 inches of live-load sag at L/360; a twenty-four-foot girder is allowed 0.8. The denominators come from IRC Table R301.7 and IBC Table 1604.3 and vary with what the member supports: a brittle plaster ceiling gets a stricter limit than a rafter with nothing under it, and a lintel carrying brick veneer gets the strictest of all at L/600.

This calculator handles prismatic, laterally braced, elastic members under a single load arrangement at a time. It does not handle continuous beams over three or more supports, partial-span or triangular loads, combined uniform-plus-point loading, composite action with the sheathing, shear deflection in short deep members or I-joists, or long-term creep. For those you need a frame analysis or the joist manufacturer's software.

How to use this calculator.

1. 01Pick the support and load arrangement. If the beam rests on a wall or post at each end it is simply supported; if it projects past its support with nothing under the far end it is a cantilever.
2. 02Choose whether you want the deflection of a section you already have, or the moment of inertia you would need to meet the limit.
3. 03Enter the span face to face of supports. For a cantilever, enter only the projection beyond the support.
4. 04Enter the load. Use the uniform field in pounds per lineal foot for the two uniform cases and the concentrated field in pounds for the two point cases. Enter live load alone for an IRC-style L/360 check, or dead plus live for an IBC D+L check.
5. 05Enter the modulus of elasticity. The hint carries the NDS Table 4A and 4B values for the four common framing species; steel is 29,000,000 psi.
6. 06Define the section. Enter dressed breadth and depth for sawn or built-up lumber — a two-ply 2x12 is 3.0 by 11.25 inches — or switch to entering the moment of inertia directly for steel shapes and I-joists.
7. 07Select the deflection limit that matches what the member supports, using IRC Table R301.7 or IBC Table 1604.3 for your jurisdiction's adopted edition.
8. 08Read the verdict beside the result, then carry the maximum moment and shear into a bending, shear and bearing check — passing deflection alone never proves a beam is adequate.

The formula.

All four cases share the form δ = K ÷ (E·I), with the span converted to inches and the uniform load converted from pounds per foot to pounds per inch. For a simply supported beam under a uniform load, K = 5wL⁴/384; for a point load at midspan, K = PL³/48; for a cantilever under a uniform load, K = wL⁴/8; and for a cantilever with a point load at the free end, K = PL³/3. Because the required moment of inertia is just K ÷ (E · δ_allowable), the same K serves both directions and the two modes are exact inverses of one another — a test asserts that a section with exactly the required I deflects exactly to the limit. Working the example: the header spans 12 ft, so L = 144 in; the load is 400 plf, so w = 400/12 = 33.333… lb/in; the section is 3.0 in by 11.25 in, so I = 3.0 × 11.25³ ÷ 12 = 355.95703125 in⁴. The numerator is 5 × 33.333… × 144⁴ = 71,663,616,000 and the denominator is 384 × 1,600,000 × 355.95703125 = 218,700,000,000, giving 0.32768 in exactly. The span-to-deflection ratio is 144 ÷ 0.32768 = 439.453125, so the header deflects L/439. The allowable deflection at L/360 is 144 ÷ 360 = 0.4 in, so the utilisation is 0.32768 ÷ 0.4 = 81.92 percent and the verdict is a pass. The moment of inertia that would land exactly on the limit is 186,624,000 ÷ (1,600,000 × 0.4) = 291.6 in⁴. Statics gives the moment and shear independently of the section: M = wL²/8 = 400 × 12² ÷ 8 = 7,200 ft-lb and V = wL/2 = 400 × 12 ÷ 2 = 2,400 lb. Rounding happens only at the return boundary — every intermediate step, including the fourth powers, is carried at twenty significant digits — and the pass/fail verdict classifies the unrounded ratio, so a member that displays 0.4000 in on both sides of the limit still flips its verdict correctly. For the two cantilever cases the allowable deflection is computed from twice the projection, following IRC 2021 Table R301.7 footnote b and IBC 2021 Table 1604.3 footnote i.

A worked example.

Frequently asked questions.

References& sources.

1. [1]International Code Council — IRC 2021 Table R301.7, Allowable Deflection of Structural Members, with footnote b: "For cantilever members, L shall be taken as twice the length of the cantilever." Rows used: floors including deck floors L/360, ceilings with gypsum board L/240, rafters steeper than 3:12 with no ceiling attached L/180, all other structural members L/240, lintels supporting masonry veneer L/600.
2. [2]ICC Building Code Action Committee — code change proposal reproducing Table R301.7 in full with all footnotes, used to confirm every row and the cantilever footnote verbatim.
3. [3]International Code Council — IBC 2021 Section 1604.3 and Table 1604.3, Deflection Limits, footnote i: "l = Length of the member between supports. For cantilever members, l shall be taken as twice the length of the cantilever."
4. [4]American Wood Council — NDS 2018 Supplement, Table 4A and Table 4B, Reference Design Values for Visually Graded Dimension Lumber, giving the modulus of elasticity values quoted in the field hint and FAQ.
5. [5]American Wood Council — Span Tables for Joists and Rafters, 2021 edition, section 3, confirming the PS 20 dressed dry sizes used for the depth hint (2x6 = 1.5 x 5.5, 2x8 = 1.5 x 7.25, 2x10 = 1.5 x 9.25, 2x12 = 1.5 x 11.25) and the L/360 live-load deflection basis.
6. [6]American Institute of Steel Construction — Steel Construction Manual, Table 3-23, "Shears, Moments and Deflections", the standard tabulation of the four closed-form cases implemented here. Print and member-gated; the four expressions were independently re-derived from the moment-curvature relation and that derivation is asserted in the test suite.