Euler–Bernoulli beam theory

"[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."

"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 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)."

"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 , 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 , 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."