Beam Deflection
Maximum deflection for the two textbook cases: cantilever and simply-supported.
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The engineering
The two most-reached-for cases in Roark's. Deflection scales with the cube of length — doubling the span means eight times the sag — and inversely with EI, the bending stiffness.
The span ratio row is the serviceability check: floor structures commonly target L/360 or better under live load.
Steel is E ≈ 200 GPa, 6061 aluminum ≈ 68.9, typical FR-4 ≈ 24, Douglas fir ≈ 13.
Where this math comes from
Galileo posed the cantilever problem in his 1638 Two New Sciences — the first serious attempt to calculate a beam's strength — and got the stress distribution wrong, an error that stood for decades. Jacob Bernoulli connected a beam's curvature to bending moment in the 1690s, and his nephew Daniel, working with Leonhard Euler around 1750, produced the Euler–Bernoulli beam equation this card solves.
It stayed academic until Claude-Louis Navier's 1826 lectures turned it into design formulas engineers could actually use — arguably the birth of structural engineering as a discipline. Raymond Roark's 1938 Formulas for Stress and Strain collected the worked cases into the desk reference that still sits behind half the answers in any stress group.
- 1638Galileo GalileiPoses the cantilever strength problem in Two New Sciences — brilliantly, and wrongly.
- 1694Jacob BernoulliLinks beam curvature to bending moment.
- 1750Leonhard Euler & Daniel BernoulliThe Euler–Bernoulli beam equation — the math under this card.
- 1826Claude-Louis NavierTurns beam theory into usable engineering design formulas.
- 1938Raymond J. RoarkFormulas for Stress and Strain — the compilation on every stress engineer's shelf.
See the full timeline of the math behind every calculator →
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