How much does your beam sag?

What the result tells you

A beam bends under load, and two questions follow. Does it stay strong enough, that is, is the bending stress below what the material can carry? And does it stay stiff enough, that is, is the sag small enough that floors do not bounce, plaster does not crack and doors still close? This calculator answers the second question, and gives the numbers for the first: the reactions at the supports, the largest bending moment M, the bending stress σ = M ÷ S and the maximum deflection δ. It handles the two cases of the textbook, a beam resting on two supports and a cantilever fixed at one end, loaded by a uniform load or one point load, and checks the sag against a limit of L/360, L/240, L/180 or L/120.

The formulas

With span L, uniform load w (force per length), point load P, modulus of elasticity E and moment of inertia I (the formulas are the standard ones of the American Wood Council's Design Aid No. 6 and of every statics handbook):

The stress is σ = M ÷ S, where S is the section modulus (I divided by the distance to the outer fiber; b × d² ÷ 6 for a rectangle). Deflection grows with the fourth power of the span for a uniform load and the third power for a point load, and falls in proportion to E and I, which is why a deeper section (I grows with the cube of the depth) helps much more than a stronger material of the same size.

Worked examples

Floor joist. A 2×10 of No. 2 Douglas fir-larch (E = 1,600 ksi, I = 98.93 in⁴, S = 21.39 in³) spans 14 feet at 16 inches on center under 40 psf, a uniform load of 53.3 pounds per foot. The reactions are 373 pounds each, the largest moment 1.31 kip-feet, the bending stress 733 psi and the deflection 0.291 inch, or L/577. The L/360 limit is 0.467 inch, so it passes the live-load check. Steel cantilever. A 6-foot cantilever with I = 20 in⁴ and a 500-pound load on its tip deflects 0.107 inch and sees a moment of 3.00 kip-feet; the code takes the span as 12 feet, so the limit is 0.400 inch. Aluminum bar. With I = 3.5 in⁴, an 8-foot span and 300 pounds placed 2 feet from one support, the reactions are 225 and 75 pounds, the moment is 450 pound-feet and the deflection 0.110 inch, at 3.53 feet from the left support.

Limits of this check

The formulas are for a straight beam of constant section, loaded in its stiff plane and not stressed beyond the elastic range. They leave out the weight of the beam, shear deformation (important for short, deep beams, and in wood), the creep of wood and concrete under long loads, the sag that comes from supports that move, and buckling of slender compression flanges. The code limits themselves come with conditions: footnotes to IBC Table 1604.3 let steel ignore the creep part of the dead load and ask that roofs without slope be checked for ponding. And the modulus E of wood is an average value that varies a lot from piece to piece, so deflection of a lumber beam is a good estimate, not a promise.

What deflection limit should I use?

Table 1604.3 of the International Building Code sets limits as a fraction of the span: L/360 for floor members under live load, L/240 for floor members under dead plus live load, and for roof members L/360, L/240, L/180 or L/120 depending on the load case and whether the roof supports a plaster ceiling, another ceiling or none. Pick the row that fits your member in the limit field. Your own project may be stricter: a plaster ceiling, tile floor or brittle finish, or a machine that needs a rigid frame. For a cantilever the code takes the span as twice the length of the cantilever, and so does this calculator.

What values of E and I do I enter?

E is the modulus of elasticity of the material: 29,000 ksi for structural steel, about 10,000 ksi for aluminum alloy 6061 and 1,600 ksi for No. 2 Douglas fir-larch dimension lumber. I is the moment of inertia of the cross section about the axis of bending, in⁴ (cm⁴), listed in steel and aluminum shape tables, or b × d³ ÷ 12 for a rectangle. S is the section modulus, b × d² ÷ 6 for a rectangle; the calculator needs it only for the stress. Dressed 2×10 lumber, for instance, has I = 98.93 in⁴ and S = 21.39 in³ (NDS Supplement, Table 1B).

How does this differ from the joist span and deck beam calculators?

Those two read ready-made span tables from the International Residential Code (R502.3.1 and R507.5) and tell you the longest span of a given board. This one does the mechanics: you give the load, the span, the material and the section, and it returns the reactions, bending moment, bending stress and deflection of a simply supported beam or a cantilever, whatever the material. It suits steel, aluminum, a glulam you have the I of, a shop frame or a shelf.

Why is a cantilever limit based on twice the length?

Because the free end of a cantilever moves much more than the middle of a beam of the same length. Footnotes to IBC Table 1604.3 define the span for a cantilever as twice its length, so a 6-foot cantilever is checked as if it were a 12-foot span: the limit L/360 is then 0.4 inch, not 0.2 inch.

Does the result include the weight of the beam?

No. Add the weight per foot of the beam to the uniform load yourself (steel is about 490 pounds per cubic foot). The IBC deflection check also depends on which loads you include: for the D + L column of Table 1604.3, only the creep part of the dead-load deflection plus the live-load deflection is limited, and the table says steel members may take the creep part as zero. Ask your designer which combination applies.

Is a passing result enough to build?

No. Deflection is a serviceability check. A beam can pass L/360 and still be too weak in bending, shear, bearing or buckling, or fail by lateral-torsional buckling if it is not braced. Compare the bending stress with the allowable stress of your material and code, check shear and the supports, and have a licensed engineer design anything that carries people or a roof.

Sources: American Wood Council, Design Aid No. 6, “Beam Formulas with Shear and Moment Diagrams” (figures 1, 7, 8, 12, 13, 14), whose formulas the tests re-derive by numerical integration of EI·y″ = M; International Building Code 2021 and 2024, Table 1604.3 “Deflection limits” and its footnotes (span of a cantilever taken as twice its length); AWC, NDS 2018 Supplement, Table 4A (reference E of Douglas fir-larch No. 2 = 1,600,000 psi) and Table 1B (section properties of 2×4 to 2×12 lumber); AISC 360, Specification for Structural Steel Buildings (E = 29,000 ksi); aluminum 6061 about 10,000 ksi (ASM data sheet). Not an engineering design: for anything that carries people or a roof, use the stamped drawings. For code span tables of boards see the joist span calculator and the deck beam span calculator; for the section properties of shapes, the moment of inertia calculator; for steel weight, the metal weight calculator. How we check our calculators.