Euler–Bernoulli beam theory

Euler–Bernoulli beam theory models slender beams by relating bending moment to curvature. It assumes cross-sections remain plane and perpendicular to the beam’s neutral axis during deformation.

Beam bending equation: A differential equation relating a beam’s transverse deflection to its loading and flexural rigidity. It translates the theory’s moment–curvature relation into a solvable equation for deflection.
Beam (structure): A structural element designed primarily to carry loads transverse to its length through bending and shear. The theory idealizes this structural element as slender and predominantly bending-dominated.
Timoshenko beam theory: A beam model that includes transverse shear deformation and cross-section rotation independent of the neutral axis slope. It corrects Euler–Bernoulli theory’s neglect of shear deformation, especially in deep beams.
Beam deflection: The transverse displacement of a beam from its unloaded position under applied forces. Deflection is the most direct response calculated from the beam equation.
Leonhard Euler: An eighteenth-century mathematician and physicist whose work included the elastic curve and column buckling. Euler developed foundational mathematical treatments of bending and elastic stability.
Flexural rigidity: A beam’s resistance to bending, given by Young’s modulus multiplied by its second moment of area. It sets how strongly a given bending moment curves the beam.
Young’s modulus: A material constant measuring the ratio of longitudinal stress to elastic strain. It supplies the material stiffness in the theory’s bending law.
Rayleigh beam theory: A beam vibration model that includes rotary inertia while retaining Euler–Bernoulli kinematics. It improves predictions of higher-frequency vibration without adding shear deformation.
Euler buckling: The elastic instability in which a slender column suddenly bends under axial compression. The classical critical-load result uses the same bending stiffness and beam assumptions.
Daniel Bernoulli: An eighteenth-century mathematician and physicist who studied elastic curves and vibrating beams. Bernoulli contributed early analysis of beam bending and vibration.

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Sophie GermainCompared with: Beam models simplify elasticity problems that Germain addressed for thin plates.