A beam theory from around 1750 waited for the Eiffel Tower
Engineers can predict how far a beam will sag and how much it can carry thanks to Euler–Bernoulli beam theory. It was first stated around 1750, but it saw no large-scale use until late nineteenth-century engineers built the Eiffel Tower and the Ferris wheel. After that it helped drive the Second Industrial Revolution.
The theory, also called engineer's or classical beam theory, simplifies the linear theory of elasticity into a chain of running totals. Loads pressing on a beam create internal shear forces and bending moments. The shear at any point equals the sum of loads applied up to that point; the bending moment equals the sum of those shear forces; and the sag, or deflection, is the fourth integral of the loads, tempered by the beam's flexural rigidity. For common arrangements, engineers simply look up ready-made deflection formulas in handbooks, and for trickier cases they solve the equation with techniques such as Macaulay's method or the conjugate beam method.
Its limits are clear. It handles only small deflections of beams loaded from the side and bending elastically. Twentieth-century refinements added shear deformation and rotatory inertia, and the classical theory is now seen as a special case of Timoshenko–Ehrenfest beam theory with those effects left out. Plate theory and other models exist, yet the simplicity of beam theory keeps it central to structural and mechanical engineering.
The history involves several famous names. Galileo Galilei is usually credited with the first attempt at a theory of beams, though recent studies argue Leonardo da Vinci made the crucial observations first. Leonardo lacked Hooke's law and calculus, while Galileo was tripped up by a wrong assumption.
Jacob Bernoulli proposed in 1705 that a bent beam's curvature at any point is proportional to its bending moment there, and worked out the deflection of a cantilever loaded at its free end, though he misjudged the axis about which the cross-section rotates. Leonhard Euler extended that work with differential calculus in his 1744 Methodus inveniendi lineas curvas, and Daniel Bernoulli derived the equation governing a vibrating beam.
Source: Euler–Bernoulli beam theory
