Contenido

La hipótesis cinemática es una especificación matemática de los desplazamientos de un sólido deformable que permite calcular las deformaciones en función de un conjunto de parámetros incógnita.

El concepto se usa especialmente en el cálculo de elementos lineales (por ejemplo, vigas) y elementos bidimensionales, donde gracias a la hipótesis cinemática se pueden obtener relaciones funcionales más simples. Así pues, gracias a la hipótesis cinemática se pueden relacionar los desplazamientos en cualquier punto del sólido deformable de un dominio tridimensional con los desplazamientos especificados sobre un conjunto unidimensional o bidimensional.

Kinematic hypothesis in linear elements

The resistance of materials proposes for linear elements or mechanical prisms, such as beams and columns, in which the displacement of any point can be calculated from specified displacements and rotations about the barycentric axis. This means that, for example, to calculate a beam, instead of specifying the displacements of any point as a function of three coordinates, we can express them as a function of a single coordinate on the barycentric axis, which leads to relatively simple systems of differential equations. There are various types of kinematic hypotheses depending on the type of stress on the beam or one-dimensional element:

Kinematic hypothesis in surface elements

For plates and sheets subjected to bending "Bending (engineering)") two hypotheses are used, which can be put in correspondence with the beam hypotheses:

The constitutive equations of the resistance of materials are those that explain the behavior of the material; the Lamé-Hooke equations of linear elasticity "Elasticity (solid mechanics)" are generally taken as constitutive equations. These equations can be specialized for linear and surface elements. For linear elements in the calculation of the sections, the stresses on any point (y,z) of the section can be written as a function of the deformations as:.

On the other hand, for surface elements predominantly subjected to bending such as plates, the specialization of Hooke's equations is:

In addition to elastic constitutive equations, in structural calculation several regulations include plastic calculation methods where constitutive equations of plasticity "Plasticity (solid mechanics)" are used.

Las ecuaciones de equivalencia expresan los esfuerzos resultantes a partir de la distribución de tensiones. Gracias a ese cambio es posible escribir ecuaciones de equilibrio que relacionen directamente las fuerzas aplicadas con los esfuerzos internos.

linear elements

In straight linear elements, the Cartesian coordinates to represent the geometry and express tensions and efforts are normally chosen with the X axis parallel to the barycentric axis of the piece, and the Y and Z axes coinciding with the main directions of inertia. In this coordinate system the relationship between normal stress (N), shear stresses (V, V), the torsional moment (M) and the bending moments (M, M) is:

Where the stresses that appear are the components of the tension tensor for a prismatic piece:

two-dimensional elements

For two-dimensional elements it is common to take a system of two coordinates (Cartesian or curvilinear) coinciding with the average surface, with the third coordinate aligned with the thickness. For a flat plate of thickness 2t and with a coordinate system in which the XY plane coincides with its median plane. The forces are composed of 4 membrane forces (or axial forces per unit area), 4 bending moments and 2 shear forces. Membrane stresses using a set of orthogonal coordinates on a Reissner-Mindlin sheet:

Where are the radii of curvature in each of the coordinate directions and z is the height above the average surface of the sheet. The shear stresses and bending moments per unit area are given by:.

The tension tensor of a general sheet for which the Reissner-Mindlin hypotheses apply is:

A particular case of the above is the flat sheets whose deformation conforms to the Love-Kirchhoff hypothesis, characterized by the vector normal to the average deformed surface coinciding with the deformed normal. This hypothesis is a very good approximation when the shear stresses are negligible and in that case the bending moments per unit area as a function of the tensions are given by:.

Where the tensions that appear are the components of the tension tensor for a Love-Kirchhoff sheet:

Las ecuaciones de equilibrio de la resistencia de materiales relacionan los esfuerzos internos con las fuerzas exteriores aplicadas. Las ecuaciones de equilibrio para elementos lineales y elementos bidimensionales son el resultado de escribir las ecuaciones de equilibrio elástico en términos de los esfuerzos en lugar de las tensiones.

Las ecuaciones de equilibrio para el campo de tensiones#Ecuaciones_de_equilibrio "Elasticidad (mecánica de sólidos)") generales de la teoría de la elasticidad lineal "Elasticidad (mecánica de sólidos)"):.

Si en ellas se trata de substituir las tensiones por los esfuerzos internos, se llega entonces a las ecuaciones de equilibrio de la resistencia de materiales. El procedimiento, que se detalla a continuación, es ligeramente diferente para elementos unidimensionales y bidimensionales.

Balance equations in straight linear elements

In a straight horizontal beam, aligned with the X axis, and in which the loads are vertical and located on the

Balance equations in two-dimensional plane elements

The equilibrium equations for two-dimensional elements (plates) in bending, analogous to the equations in the previous section for linear elements (beams), relate the moments per unit width (m, m, m), with the shear stresses per unit width (v, m) and the vertical surface load (q):.

El diseño mecánico de piezas requiere:.

En general, el cálculo de tensiones puede abordarse con toda generalidad desde la teoría de la elasticidad, sin embargo cuando la geometría de los elementos es suficientemente simple (como sucede en el caso de elementos lineales o bidimensionales) las tensiones y desplazamientos pueden ser calculados de manera mucho más simple mediante los métodos de la resistencia de materiales, que directamente a partir del planteamiento general del problema elástico.

Linear or one-dimensional elements

The stress calculation can be obtained from the combination of the Navier formula for bending, the Collignon-Jourawski formula and the stress calculation formulas for torsion.

The calculation of displacements in linear elements can be carried out using direct methods such as the elastic curve equation, Mohr's theorems or the matrix method or from energy methods such as Castigliano's theorems or even by computational methods.

Surface or two-dimensional elements

The Love-Kirchhoff plate theory is the two-dimensional analogue of the Euler-Bernouilli beam theory. On the other hand, the calculation of sheets is the two-dimensional analogue of the calculation of arches.

The two-dimensional analogue for a plate of the elastic curve equation is the Lagrangian equation for the midplane deflection of the plate. For the calculation of plates, the use of variational methods is also common.

Another important problem in many applications of strength of materials is the study of stiffness. More specifically, certain applications require ensuring that under the acting forces some resistant elements never exceed displacements above a certain preset value. The calculation of deformations from the forces can be determined by several semi-direct methods such as the use of Castigliano's theorem, the Navier-Bresse vector formulas, the use of the elastic curve equation, the matrix method of stiffness and other numerical methods for the most complex cases.

Contenido

La hipótesis cinemática es una especificación matemática de los desplazamientos de un sólido deformable que permite calcular las deformaciones en función de un conjunto de parámetros incógnita.

El concepto se usa especialmente en el cálculo de elementos lineales (por ejemplo, vigas) y elementos bidimensionales, donde gracias a la hipótesis cinemática se pueden obtener relaciones funcionales más simples. Así pues, gracias a la hipótesis cinemática se pueden relacionar los desplazamientos en cualquier punto del sólido deformable de un dominio tridimensional con los desplazamientos especificados sobre un conjunto unidimensional o bidimensional.

Kinematic hypothesis in linear elements

The resistance of materials proposes for linear elements or mechanical prisms, such as beams and columns, in which the displacement of any point can be calculated from specified displacements and rotations about the barycentric axis. This means that, for example, to calculate a beam, instead of specifying the displacements of any point as a function of three coordinates, we can express them as a function of a single coordinate on the barycentric axis, which leads to relatively simple systems of differential equations. There are various types of kinematic hypotheses depending on the type of stress on the beam or one-dimensional element:

Kinematic hypothesis in surface elements

For plates and sheets subjected to bending "Bending (engineering)") two hypotheses are used, which can be put in correspondence with the beam hypotheses:

The constitutive equations of the resistance of materials are those that explain the behavior of the material; the Lamé-Hooke equations of linear elasticity "Elasticity (solid mechanics)" are generally taken as constitutive equations. These equations can be specialized for linear and surface elements. For linear elements in the calculation of the sections, the stresses on any point (y,z) of the section can be written as a function of the deformations as:.

On the other hand, for surface elements predominantly subjected to bending such as plates, the specialization of Hooke's equations is:

In addition to elastic constitutive equations, in structural calculation several regulations include plastic calculation methods where constitutive equations of plasticity "Plasticity (solid mechanics)" are used.

Las ecuaciones de equivalencia expresan los esfuerzos resultantes a partir de la distribución de tensiones. Gracias a ese cambio es posible escribir ecuaciones de equilibrio que relacionen directamente las fuerzas aplicadas con los esfuerzos internos.

linear elements

In straight linear elements, the Cartesian coordinates to represent the geometry and express tensions and efforts are normally chosen with the X axis parallel to the barycentric axis of the piece, and the Y and Z axes coinciding with the main directions of inertia. In this coordinate system the relationship between normal stress (N), shear stresses (V, V), the torsional moment (M) and the bending moments (M, M) is:

Where the stresses that appear are the components of the tension tensor for a prismatic piece:

two-dimensional elements

For two-dimensional elements it is common to take a system of two coordinates (Cartesian or curvilinear) coinciding with the average surface, with the third coordinate aligned with the thickness. For a flat plate of thickness 2t and with a coordinate system in which the XY plane coincides with its median plane. The forces are composed of 4 membrane forces (or axial forces per unit area), 4 bending moments and 2 shear forces. Membrane stresses using a set of orthogonal coordinates on a Reissner-Mindlin sheet:

Where are the radii of curvature in each of the coordinate directions and z is the height above the average surface of the sheet. The shear stresses and bending moments per unit area are given by:.

The tension tensor of a general sheet for which the Reissner-Mindlin hypotheses apply is:

A particular case of the above is the flat sheets whose deformation conforms to the Love-Kirchhoff hypothesis, characterized by the vector normal to the average deformed surface coinciding with the deformed normal. This hypothesis is a very good approximation when the shear stresses are negligible and in that case the bending moments per unit area as a function of the tensions are given by:.

Where the tensions that appear are the components of the tension tensor for a Love-Kirchhoff sheet:

Las ecuaciones de equilibrio de la resistencia de materiales relacionan los esfuerzos internos con las fuerzas exteriores aplicadas. Las ecuaciones de equilibrio para elementos lineales y elementos bidimensionales son el resultado de escribir las ecuaciones de equilibrio elástico en términos de los esfuerzos en lugar de las tensiones.

Las ecuaciones de equilibrio para el campo de tensiones#Ecuaciones_de_equilibrio "Elasticidad (mecánica de sólidos)") generales de la teoría de la elasticidad lineal "Elasticidad (mecánica de sólidos)"):.

Si en ellas se trata de substituir las tensiones por los esfuerzos internos, se llega entonces a las ecuaciones de equilibrio de la resistencia de materiales. El procedimiento, que se detalla a continuación, es ligeramente diferente para elementos unidimensionales y bidimensionales.

Balance equations in straight linear elements

In a straight horizontal beam, aligned with the X axis, and in which the loads are vertical and located on the

Balance equations in two-dimensional plane elements

The equilibrium equations for two-dimensional elements (plates) in bending, analogous to the equations in the previous section for linear elements (beams), relate the moments per unit width (m, m, m), with the shear stresses per unit width (v, m) and the vertical surface load (q):.

El diseño mecánico de piezas requiere:.

En general, el cálculo de tensiones puede abordarse con toda generalidad desde la teoría de la elasticidad, sin embargo cuando la geometría de los elementos es suficientemente simple (como sucede en el caso de elementos lineales o bidimensionales) las tensiones y desplazamientos pueden ser calculados de manera mucho más simple mediante los métodos de la resistencia de materiales, que directamente a partir del planteamiento general del problema elástico.

Linear or one-dimensional elements

The stress calculation can be obtained from the combination of the Navier formula for bending, the Collignon-Jourawski formula and the stress calculation formulas for torsion.

The calculation of displacements in linear elements can be carried out using direct methods such as the elastic curve equation, Mohr's theorems or the matrix method or from energy methods such as Castigliano's theorems or even by computational methods.

Surface or two-dimensional elements

The Love-Kirchhoff plate theory is the two-dimensional analogue of the Euler-Bernouilli beam theory. On the other hand, the calculation of sheets is the two-dimensional analogue of the calculation of arches.

The two-dimensional analogue for a plate of the elastic curve equation is the Lagrangian equation for the midplane deflection of the plate. For the calculation of plates, the use of variational methods is also common.

Another important problem in many applications of strength of materials is the study of stiffness. More specifically, certain applications require ensuring that under the acting forces some resistant elements never exceed displacements above a certain preset value. The calculation of deformations from the forces can be determined by several semi-direct methods such as the use of Castigliano's theorem, the Navier-Bresse vector formulas, the use of the elastic curve equation, the matrix method of stiffness and other numerical methods for the most complex cases.