Euler-Bernoulli beam theory
Introduction
In engineering, flexion is the type of deformation that an elongated structural element presents in a direction perpendicular to its longitudinal axis. The term "elongated" is applied when one dimension is dominant over the others. A typical case is beams, which are designed to work mainly in bending. Likewise, the concept of bending extends to surface structural elements such as plates or sheets.[1].
The most notable feature is that an object subjected to bending has a surface of points called a neutral fiber such that the distance along any curve contained in it does not vary with respect to the value before deformation. The stress that causes bending is called the bending moment.
Bending in beams and arches
Contenido
Las vigas o arcos "Arco (construcción)") son elementos estructurales pensados para trabajar predominantemente en flexión. Geométricamente son prismas mecánicos cuya rigidez depende, entre otras cosas, del momento de inercia de la sección transversal de las vigas. Existen dos hipótesis cinemáticas comunes para representar la flexión de vigas y arcos:.
Euler-Bernoulli theory
The Euler-Bernoulli theory for the calculation of beams is the one derived from the Euler-Bernoulli kinematic hypothesis, and can be used to calculate stresses and displacements on a beam or arch with a long axis length compared to the maximum depth or height of the cross section.
To write the formulas of the Euler-Bernouilli theory it is convenient to take a suitable coordinate system to describe the geometry, a beam is in fact a mechanical prism on which the coordinates (s, y, z) can be considered with s the distance along the axis of the beam and (y, z) the coordinates on the cross section. In the case of arches this coordinate system is curvilinear, although for beams with a straight axis it can be taken as Cartesian (and in that case s is named x). For a beam with a straight section, the stress in the case of deviated compound bending, the stress is given by :.