Euler–Bernoulli beam theory
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity and provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case for small deflections of a beam that are subjected to lateral loads only, and is thus a special case of Timoshenko beam theory. It was first enunciated circa 1750, but was not generally applied until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.
- Galileo does not attempt any theory to account for the flexure of the beam. This theory, supplied by Hooke's law, was applied by Mariotte, Leibnitz, De Lahire, and Varignon, but they neglect compression of the fibres, and so place the neutral in the lower face of Galileo's beam. The true position of the neutral plane was assigned by James Bernoulli 1695, who in his investigation of the simplest case of bent beam, was led to the consideration of the curve called the "elastica." This "elastica" curve speedily attracted the attention of the great Euler (1744), and must be considered to have directed his attention to the elliptic integrals. Probably the extraordinary divination which led Euler to the formula connecting the sum of two elliptic integrals, thus giving the fundamental theorem of the addition equation of elliptic functions, was due to mechanical considerations concerning the "elastica" curve; a good illustration of the general principle that the pure mathematician will find the best materials for his work in the problems presented to him by natural and physical questions.
- We consider the case of a horizontal rod or beam slightly bent by vertical forces applied to it. The state of strain is no longer of the simple character appertaining to pure flexure; in particular there will be a relative shearing of adjacent cross-sections, and also a warping of the sections so that these do not remain accurately plane. We shall assume, however, that the additional strains thus introduced are on the whole negligible, and consequently that the bending moment is connected with the curvature of the axis...
- The assumption that the bending moment varies as the curvature is the basis of the 'Euler-Bernoulli' theory of flexure. This was developed in memoirs by James Bernoulli (1705) D. Bernoulli (1742), L. Euler (1744).
- [C]alculations... based on the simple theory of bending... are approximate only. While the simple (or Bernoulli-Euler) theory gives the deflections due to the bending moment with sufficient accuracy, the portion of the total deflection which is due to shearing cannot generally be estimated with equal accuracy from the distribution of shear stress... It becomes desirable, then, to check the results by those given in the more complex theory of St. Venant... if a very accurate estimate of shearing deflection is required. In a great number of practical cases, however, the deflection due to shearing is negligible in comparison with that caused by the bending moment.
- Arthur Morley, Theory of structures (1912) p. 259.
- The assumptions in the design of reinforced concrete beams are those of the ordinary beam theory, namely: the Bernoulli-Euler theory of flexure. The fundamental premise is that a plane section before bending, remains a plane section after bending, with the further assumption that Hooke's Law, i.e., the stress is proportional to the strain, is true.
Although the brilliant researches of Barre de St. Venant, have shown that plane sections do not remain plane during bending, the error becomes appreciable when the ratio of depth of beam to span exceeds one-fifth. Since for such ratios, stresses, other than those induced by bending moment, usually govern the required reinforcement and depth of beam e.g. the unit shear and adhesion, these assumptions of plane sections may be taken as valid, so long as the stresses induced by the bending moment govern the required depths and amounts of steel reinforcement. The concrete is assumed to take no tension.
- George Paaswell, Retaining Walls: Their Design and Construction (1920) p. 85.
A Treatise on the Mathematical Theory of Elasticity (1892-1893)
- The first investigation of any importance is that of the elastic line or elastica by James Bernoulli in 1705, in which the resistance of a bent rod is assumed to arise from the extension and contraction of its longitudinal filaments, and the equation of the curve assumed by the axis is formed. This equation practically involves the result that the resistance to bending is a couple proportional to the curvature of the rod when bent, a result which was assumed by Euler in his later treatment of the problems of the elastica, and of the vibrations of thin rods.
- Historical Introduction
- In Euler's work on the elastica the rod is thought of as a line of particles which resists bending. The theory of the flexure of beams of finite section was considered by Coulomb... [by investigating] the equation of equilibrium obtained by resolving horizontally the forces which act upon the part of the beam cut off by one of its normal sections, as well as of the equation of moments. He... thus... obtain[ed] the true position of the "neutral line," or axis of equilibrium, and he also made a correct calculation of the moment of the elastic forces. His theory of beams is the most exact of those [that assume] the stress in a bent beam arises wholly from the extension and contraction of its longitudinal filaments, and... Hooke's Law.
- Historical Introduction
- Daniel Bernoulli
- Jacob Bernoulli
- Leonhard Euler
- Galileo Galilei
- A History of the Theory of Elasticity and of the Strength of Materials
- Robert Hooke
- A Treatise on the Mathematical Theory of Elasticity
- YouTube videos
- Beam Theory - Parts 1 & 2 from Structure Pro
- ENG1001atMonash) Playlists: Beams, Beam Design, from Monash University.
- Beams: Simple Beam Theory, Derivation of Euler Bernoulli and Bending Stress Formulae at YouTube, by Monash University, presenter Lizi Sironic.
- Euler-Bernoulli vs Timoshenko Beam Theory from Peter Wajda.