Beam Deflection Calculator

Deflection is the vertical displacement of a loaded beam measured from its unloaded position, and it governs serviceability rather than strength. A beam can be far from failing yet still sag enough to crack plaster, jam a door, or produce a floor that feels alive underfoot. Every deflection formula here derives from the Euler-Bernoulli relation that curvature equals M divided by EI; integrating that curvature twice along the span produces the closed-form coefficients. A simply supported beam under uniform load w gives 5wL^4/(384EI), the same beam with a central point load P gives PL^3/(48EI), and a cantilever loaded at its tip gives PL^3/(3EI), which is sixteen times larger than the simply supported point-load case. The variables carry very different leverage. Span appears to the third or fourth power, so lengthening a beam from 16 to 20 feet raises uniform-load deflection by roughly 2.4 times. Moment of inertia enters linearly but itself grows with the cube of depth, which is why a W12x26 at I = 204 in^4 outperforms a W10x22 at 118 in^4. Elastic modulus is fixed by material: 29,000 ksi for every structural steel grade, about 10,000 ksi for aluminum, and 1,400 to 1,800 ksi for common softwood lumber. Results assume elastic behavior, prismatic sections, and no composite action or end continuity, so a real continuous beam deflects less than this calculation suggests. Long-term creep in timber and concrete can double the initial elastic value.

The study of beam bending begins with Galileo Galilei, whose 1638 Two New Sciences posed the problem of a cantilever projecting from a wall. He correctly deduced that strength grows with the square of depth but assumed uniform tension across the section. Edme Mariotte in 1686 and Antoine Parent in 1713 corrected the internal force distribution by recognizing a neutral axis. The deflection curve itself came from the Bernoulli family and Leonhard Euler: Jacob Bernoulli proposed in the 1690s that curvature is proportional to bending moment, and Euler developed the elastica in the mid-eighteenth century. Claude-Louis Navier assembled these strands into the modern beam equation in his 1826 lectures at the Ecole des Ponts et Chaussees, giving engineers the EI stiffness product used here. Deflection limits expressed as a fraction of span are a later, empirical addition. Nineteenth-century builders observed that lime plaster ceilings cracked once sag reached roughly one three-hundred-sixtieth of the span, and L/360 entered American practice through early twentieth-century municipal codes and lumber association span tables. Those ratios were carried by the Uniform Building Code and now appear in International Building Code Table 1604.3, which pairs L/360 for live-load deflection of plastered members with L/240 for total load. The AISC beam diagram tables, printed in every edition of the Steel Construction Manual since 1927, remain the standard reference for these coefficients.