Beam Deflection Calculator: Shear, Moment and Deflection Diagrams

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Beam Deflection Studio

Deflection, slope, shear and moment for a single span beam, worked two independent ways at once. The textbook closed form result is shown beside a numerical stiffness solution of the same beam, so you can see the two agree. Lengths in metres, loads in kN, deflections in millimetres.

How the studio works and when to trust the formula

The studio above solves your beam twice. The closed form column is the textbook expression for that exact case, the one you would look up in a table. The numerical column comes from a stiffness solution that knows nothing about those formulas: the beam is split into many small elements, each with its own stiffness matrix, and the whole system is solved for deflections and rotations.

They should agree to a fraction of a percent, and the status line reports the difference so you can see it. That is not decoration. It is the honest way to use a calculator: two independent routes to the same number, with the disagreement shown rather than hidden.

Two cases show a difference of a fraction of a percent rather than zero, and the reason is worth knowing. The familiar propped cantilever coefficient of one over 185 and the triangular load coefficient of 0.00652 are themselves rounded numbers. The exact values are 0.005416 and 0.006519. So the small gap you see is the textbook constant being approximate, not the solver being wrong, which is a useful reminder that a memorised coefficient is already a rounding of something else.

Where there is no simple closed form, the table says so instead of inventing one. A point load at an arbitrary position on a propped cantilever is a real case with an ugly expression, and the numerical answer is the sensible one to use.

Why deflection usually governs before strength

Most beams in buildings are not sized by bending strength. They are sized by how much they sag. A steel beam that is comfortably strong can still bounce underfoot, crack a plaster ceiling below, or stop a door from closing, and none of those show up in a moment capacity check.

That is why deflection is expressed as a fraction of span rather than an absolute number. Span over 360 is the common limit for floors carrying brittle finishes, span over 250 is often accepted where the finish is more forgiving, and span over 480 turns up where the beam supports something fussy. The studio checks whichever you pick and tells you the ratio you actually achieved.

Notice how strongly deflection depends on span. It goes with the fourth power of length for a uniform load, so doubling a span multiplies the sag by sixteen while the moment only goes up by four. Stretch a span and deflection overtakes strength very quickly, which is exactly why long span floors end up deeper than the bending calculation alone suggests.

What changes the answer, in order of leverage

Second moment of area is the lever worth pulling. Deflection is inversely proportional to it, and for a rectangular section it goes with the cube of depth. Adding fifty millimetres of depth does far more than widening the section by the same amount, and far more than switching to a stronger grade.

Material grade barely matters for deflection at all. All structural steels have essentially the same modulus of elasticity, so a higher grade buys you strength and no stiffness whatsoever. A beam that fails deflection will not be fixed by specifying better steel, only by a deeper or heavier section.

End restraint matters a lot. Fixing both ends of a uniformly loaded beam cuts the deflection to a fifth of the simply supported value, because the end moments push the middle back up. That is real, but only if the connections genuinely deliver the fixity, which in practice they often only partly do.

Related Calculators and Reference

The four classic cases, and the original calculator

The material below is the original treatment of the four standard cases. It is still the quickest way to get a feel for how each support arrangement behaves before you start changing numbers in the studio above.

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Max bending moment (kNm)
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Max shear (kN)
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Max deflection (mm)
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Deflection check

Beam Deflection and Moment Result

Generated by calculatgeek.com Beam Calculator

Four classic beam cases with their maximum moment and deflection formulas: simply supported and cantilever beams under uniform and point loads
The four textbook beam cases cover most quick span checks: simply supported or cantilever, uniform or point load, each with a closed-form moment and deflection.

Most beam questions on a real site are not exotic. A joist under a uniform floor load, a lintel with a point load from a post, a canopy cantilever, a scaffold beam: the same four textbook cases cover the overwhelming majority of quick checks, and each has an exact closed-form answer that predates the computer by a century. This calculator packages all four with the two questions every span check must answer: how much moment must the section resist, and how much will it sag.

The four cases

  • Simply supported with uniform load: Mmax = wL^2/8 at midspan, deflection 5wL^4/(384EI)
  • Simply supported with a center point load: Mmax = PL/4, deflection PL^3/(48EI)
  • Cantilever with uniform load: Mmax = wL^2/2 at the support, tip deflection wL^4/(8EI)
  • Cantilever with an end point load: Mmax = PL at the support, tip deflection PL^3/(3EI)

Worked example

A 6 m simply supported steel beam (E = 200 GPa, I = 8,360 cm4, roughly an IPE 300) carries 10 kN/m. The moment peaks at 45 kNm at midspan and the shear at 30 kN at the supports. The midspan deflection works out to 10.1 mm, comfortably inside the L/360 serviceability limit of 16.7 mm that floor structures commonly use. The bending moment diagram the calculator draws makes the shape of the problem visible: a parabola for distributed load, a triangle for the point load, and for cantilevers the peak migrating to the support, which is why cantilever connections are where the reinforcement congregates.

Why deflection so often governs

The formulas hide an asymmetry worth internalizing: moment grows with the square of span, but deflection grows with the fourth power. Stretch a span by 50 percent and the required moment capacity rises 2.25 times while the deflection, for the same section, rises five-fold. This is why long, lightly loaded members, roof purlins, floor joists in living spaces, glazing supports, routinely pass every strength check and still fail on stiffness, and why the practical fix is depth: I grows with the cube of section depth, so a modestly deeper member buys stiffness far faster than a stronger grade of the same shape. Standard limits run from L/180 for roofs nobody walks on, through L/360 for typical floors, to L/480 or tighter under brittle finishes. Two scope notes to keep the tool honest: these formulas assume elastic behavior, prismatic sections, and loads applied at the shear center, and cantilever cases assume a genuinely fixed support, which a bolted end plate only approximates; partial fixity shows up in the field as more deflection than the formula promised.

Where this fits alongside the other tools

Column-type members with axial load rather than bending are checked with the Euler buckling calculator, slab reinforcement quantities come from the rebar weight and spacing calculator, and section property conversions use the stress and modulus unit converter. A good open derivation reference is the beam deflection article.