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:.
• - La hipótesis de Navier-Euler-Bernoulli. En ella las secciones transversales al eje baricéntrico se consideran en primera aproximación indeformables y se mantienen perpendiculares al mismo (que se curva) tras la deformación.
• - La hipótesis de Timoshenko. En esta hipótesis se admite que las secciones transversales perpendiculares al eje baricéntrico pasen a formar un ángulo con ese eje baricéntrico por efecto del esfuerzo cortante.
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 () can be considered with the distance along the axis of the beam and () 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 is named ). For a beam with a straight section, the stress in the case of deviated compound bending, the stress is given by :.
Dynamic bending test
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:.
• - La hipótesis de Navier-Euler-Bernoulli. En ella las secciones transversales al eje baricéntrico se consideran en primera aproximación indeformables y se mantienen perpendiculares al mismo (que se curva) tras la deformación.
• - La hipótesis de Timoshenko. En esta hipótesis se admite que las secciones transversales perpendiculares al eje baricéntrico pasen a formar un ángulo con ese eje baricéntrico por efecto del esfuerzo cortante.
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.
s, y, z
s
y, z
s
x
Navier's formula
Where:.
If the direction of the coordinate axes (y, z) are taken to coincide with the principal directions of inertia, then the products of inertia cancel out and the previous equation is greatly simplified. Furthermore, if the case of simple non-deviated bending is considered, the stresses along the axis are simply:
On the other hand, in this same case of simple undeviated bending, the displacement field, in Bernoulli's hypothesis, is given by the elastic curve equation:
Where:.
Timoshenko theory
The fundamental difference between the Euler-Bernouilli theory and the Timoshenko theory is that in the former the relative rotation of the section is approximated by the derivative of the vertical displacement, this constitutes a valid approximation only for long pieces in relation to the dimensions of the cross section, and then it happens that the deformations due to the shear stress are negligible compared to the deformations caused by the bending moment. In Timoshenko's theory, where deformations due to shear are not neglected and therefore is also valid for short beams, the equation of the elastic curve is given by the most complex system of equations:
Deriving the first of the two previous equations and substituting the second into it we arrive at the equation of the elastic curve including the effect of shear stress:
Bending in plates and sheets
Una placa es un elemento estructural que puede presentar flexión en dos direcciones perpendiculares. Existen dos hipótesis cinemáticas comunes para representar la flexión de placas y láminas:.
• - La hipótesis de Love-Kirchhoff.
• - La hipótesis de Reissner-Mindlin.
Siendo la primera el análogo para placas de la hipótesis de Navier-Bernouilli y el segundo el análogo de la hipótesis de Timoshenko.
Love-Kirchhoff theory
The Love-Kirchhoff plate theory is derived from the Love-Kirchhoff kinematic hypothesis for plates and is analogous to the Navier-Bernouilli hypothesis for beams and therefore has similar limitations, and is appropriate only when the thickness of the plate is sufficiently small in relation to its length and width.
For a plate of constant thickness h we will use a Cartesian coordinate system with (x, y) according to the plane that contains the plate, and the z axis will be taken according to the direction perpendicular to the plate (taking z = 0 in the median plane). With these coordinate axes the tensions according to the two perpendicular directions of the plate are:
Where:.
To find the arrow that appears in the previous equation, it is necessary to solve a partial differential equation that is the two-dimensional analogue of the elastic curve equation:
The factor:
is called flexural stiffness of plates where:.
Reissner-Mindlin theory
The Reissner-Mindlin theory is the analogue for plates of the Timoshenko theory for beams. Thus in this theory, unlike the more approximate Love-Kirchhoff theory, the vector normal to the midplane of the plate once the plate is deformed does not have to coincide with the vector normal to the deformed mean surface.
• - Monleón Cremades, S., Analysis of beams, arches, plates and sheets, Ed. UPV, 1999, ISBN 84-7721-769-6.
• - Bending moment.
• - Neutral fiber.
• - Slopes and deformations in beams.
• - Central core "Central core (resistance of materials)").
References
[1] ↑ Carlos Núñez, Antoni Roca, Jordi Jorba (2013). Comportamiento mecánico de los materiales. Volumen II. Ensayos mecánicos. Ensayos no destructivos. Edicions Universitat Barcelona. p. 50. ISBN 9788447537297. Consultado el 26 de mayo de 2024.: https://books.google.es/books?id=TWq_BAAAQBAJ&pg=PA50#v=onepage&q&f=false
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 Navier's formula:.
Where:.
If the direction of the coordinate axes (y, z) are taken to coincide with the principal directions of inertia, then the products of inertia cancel out and the previous equation is greatly simplified. Furthermore, if the case of simple non-deviated bending is considered, the stresses along the axis are simply:
On the other hand, in this same case of simple undeviated bending, the displacement field, in Bernoulli's hypothesis, is given by the elastic curve equation:
Where:.
Timoshenko theory
The fundamental difference between the Euler-Bernouilli theory and the Timoshenko theory is that in the former the relative rotation of the section is approximated by the derivative of the vertical displacement, this constitutes a valid approximation only for long pieces in relation to the dimensions of the cross section, and then it happens that the deformations due to the shear stress are negligible compared to the deformations caused by the bending moment. In Timoshenko's theory, where deformations due to shear are not neglected and therefore is also valid for short beams, the equation of the elastic curve is given by the most complex system of equations:
Deriving the first of the two previous equations and substituting the second into it we arrive at the equation of the elastic curve including the effect of shear stress:
Bending in plates and sheets
Una placa es un elemento estructural que puede presentar flexión en dos direcciones perpendiculares. Existen dos hipótesis cinemáticas comunes para representar la flexión de placas y láminas:.
• - La hipótesis de Love-Kirchhoff.
• - La hipótesis de Reissner-Mindlin.
Siendo la primera el análogo para placas de la hipótesis de Navier-Bernouilli y el segundo el análogo de la hipótesis de Timoshenko.
Love-Kirchhoff theory
The Love-Kirchhoff plate theory is derived from the Love-Kirchhoff kinematic hypothesis for plates and is analogous to the Navier-Bernouilli hypothesis for beams and therefore has similar limitations, and is appropriate only when the thickness of the plate is sufficiently small in relation to its length and width.
For a plate of constant thickness h we will use a Cartesian coordinate system with (x, y) according to the plane that contains the plate, and the z axis will be taken according to the direction perpendicular to the plate (taking z = 0 in the median plane). With these coordinate axes the tensions according to the two perpendicular directions of the plate are:
Where:.
To find the arrow that appears in the previous equation, it is necessary to solve a partial differential equation that is the two-dimensional analogue of the elastic curve equation:
The factor:
is called flexural stiffness of plates where:.
Reissner-Mindlin theory
The Reissner-Mindlin theory is the analogue for plates of the Timoshenko theory for beams. Thus in this theory, unlike the more approximate Love-Kirchhoff theory, the vector normal to the midplane of the plate once the plate is deformed does not have to coincide with the vector normal to the deformed mean surface.
• - Monleón Cremades, S., Analysis of beams, arches, plates and sheets, Ed. UPV, 1999, ISBN 84-7721-769-6.
• - Bending moment.
• - Neutral fiber.
• - Slopes and deformations in beams.
• - Central core "Central core (resistance of materials)").
References
[1] ↑ Carlos Núñez, Antoni Roca, Jordi Jorba (2013). Comportamiento mecánico de los materiales. Volumen II. Ensayos mecánicos. Ensayos no destructivos. Edicions Universitat Barcelona. p. 50. ISBN 9788447537297. Consultado el 26 de mayo de 2024.: https://books.google.es/books?id=TWq_BAAAQBAJ&pg=PA50#v=onepage&q&f=false