Beam Deflection Calculator

Background.

Take out a wall between a kitchen and a dining room and the joists above it need somewhere new to land: a header or flush beam across the opening. Whether that beam is strong enough is one question. Whether it stays flat enough is another, and it is the one that shows up first — as a crack along the drywall tape, a door that starts to bind, grout lines opening in a tile floor, or a floor that bounces when someone crosses the room.

Codes handle the second question with deflection limits written as a fraction of the span. L/360 means the member may sag no more than its length divided by 360: 0.4 in on a 12 ft beam, 0.6667 in on a 20 ft one. Under IRC 2021 Table R301.7, floors and plaster ceilings get L/360, gypsum-board ceilings and most other members L/240, rafters steeper than 3:12 with nothing attached below L/180, and a lintel carrying brick veneer the strictest value of all, L/600. IBC 2021 Table 1604.3 does the same job for commercial buildings.

Four arrangements cover most residential members: a beam resting on a support at each end with either a uniform load or one load at midspan, and a cantilever with either a uniform load or a load at its tip. For any of them you can ask how far a section you already have will sag, or turn the question round and ask what moment of inertia would land exactly on the limit — the number you take to an LVL table or a steel shape list.

Two exponents explain nearly every result. Span enters to the fourth power, so stretching a beam from 12 to 14 ft under the same load per foot makes it sag 85% more. Depth enters cubed, through the moment of inertia, so adding a second ply of the same lumber halves the sag while going from a pair of 2x8s to a pair of 2x12s cuts it by a factor of 3.7. When a floor feels springy, a deeper member or a shorter span almost always beats more of the same.

What is beam deflection calculator?

Deflection is the distance a loaded beam moves out of line, measured at right angles to its original axis at the point where the movement is largest: midspan for a symmetrically loaded beam on two supports, the free end for a cantilever. It is governed by flexural rigidity, the product E·I. E, the modulus of elasticity, belongs to the material — about 1,600,000 psi for Douglas Fir-Larch No.2, 29,000,000 psi for structural steel. I, the moment of inertia, belongs to the shape; for a solid rectangle it is breadth × depth³ ÷ 12. A deflection limit is a serviceability rule, not a safety one: wood and steel bend visibly long before they break, and the limit exists so that finishes, doors, windows and the people walking on the floor are not troubled by the movement. Because the limit is span ÷ a denominator, it grows with the member. For cantilevers both IRC Table R301.7 footnote b and IBC Table 1604.3 footnote i take L as twice the projection.

How to use this calculator.

1. 01Choose the support and load arrangement. A header, joist or girder bearing on a wall or post at each end is simply supported; a balcony joist or bay framing that projects past its support with nothing under the far end is a cantilever. Pick the uniform case for floor, roof or wall loads spread along the member and the point case when a post or girder end lands on it.
2. 02Decide what you want back: the deflection of a section you already have, or the moment of inertia that would meet your chosen limit exactly.
3. 03Enter the span as the clear distance between the faces of the supports. For a cantilever enter only the projection beyond the support — the doubling the codes require for the limit is applied for you.
4. 04Enter the load. For a uniform case, multiply the load per square foot by the width of floor the beam carries: a header picking up joists that span 12 ft on one side carries half of them, 6 ft, so 40 psf of live load becomes 240 plf. Use live load alone for an IRC-style L/360 check and dead plus live for an IBC D+L check.
5. 05Enter E and the section. The field hint lists NDS values for common framing lumber. For sawn or built-up lumber give dressed sizes — a 2x12 is 1.5 by 11.25 in, so two plies are 3.0 by 11.25 — or switch to entering I directly for steel shapes, I-joists and LVL.
6. 06Select the limit that matches what the member supports, read the verdict, and carry the maximum moment and shear into the bending, shear and bearing checks that deflection does not cover.

The formula.

Every one of the four cases comes from the same relation between bending moment and curvature, M = E·I·d²y/dx², integrated twice with the support conditions as boundary values. The results all take the form δ = K ÷ (E·I), where K depends only on the load, the span and how the member is held. With the span L converted to inches and a uniform load w converted from pounds per foot to pounds per inch (÷ 12): a simply supported beam under a uniform load has K = 5wL⁴/384; a simply supported beam with a point load P at midspan has K = PL³/48; a cantilever under a uniform load has K = wL⁴/8; a cantilever with P at its tip has K = PL³/3. Because the same K appears in every case, the required moment of inertia is simply K ÷ (E × allowable deflection), and a section with exactly that I deflects exactly to the limit. The allowable deflection is L ÷ denominator, with L doubled for cantilevers. Moment and shear come from statics and do not depend on the section at all: wL²/8 and wL/2, PL/4 and P/2, wL²/2 and wL, PL and P. The comparisons are instructive. The same total load gathered at midspan deflects a beam 1.6 times as much as when it is spread evenly. A cantilever under a uniform load sags 9.6 times as far as a simply supported beam of the same length and load, and a tip load on a cantilever produces 16 times the deflection of the same load at the middle of a simple span. The model assumes small, linear-elastic deflection of a straight member of constant section, braced against sideways buckling, and it ignores shear deformation. That last omission matters for short, deep members and for I-joists with thin webs, where manufacturers add a shear term; it also leaves out the long-term creep of wood under sustained load.

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.