Contenido

Una lámina es un elemento estructural bidimensional curvado. Si las placas se tratan análogamente a las vigas rectas, las láminas son el análogo bidimensional de los arcos "Arco (construcción)").

Usando coordenadas curvilíneas ortogonales sobre la superficie se pueden escribir las ecuaciones de equilibrio para los esfuerzos internos para una lámina de Reisner-Mindlin como:[3]​[4]​.

Donde:.

Dome under its own weight

As an example of the previous equations we can consider a dome in the shape of a spherical cap subjected to its own weight. Each point of the two-dimensional dome can be parameterized using the coordinates:.

With which we have the following geometric factors:

And therefore the previous equations will be reduced to:.

Axisymmetric sheet

The general case of a general sheet requires using general curvilinear coordinates to parameterize its surface, which leads to governing equations that are partial differential equations whose integration is complicated. However, many cases of interest involve sheets with axial or revolution symmetry, with loads that also respect axial symmetry. In those cases the geometry of the surface can be parameterized by a coordinate (which gives its radial "profile"), and the governing equations in that case involve derivatives with respect to a single coordinate, and are therefore a system of ordinary differential equations. Thanks to this, the real behavior can be estimated using classical methods, since it is feasible to integrate the governing equations in some cases. This contrasts with the general case for which analytical solutions to the governing equations are not known, so in the general case the behavior can only be investigated by searching for approximate numerical solutions to the governing equations.

The simplest case of axisymmetric sheet theory is Cosserat's special static theory, which describes the behavior of axisymmetric plates susceptible to bending in their middle surface, extension of the same and shear in the thickness.

Sometimes it may occur that a liquid at a certain pressure, viscosity and temperature, or a gas at a certain pressure, penetrates into a container, limited on the outside by a very thin metal sheet, which would be, for example, subjected on one side to the atmospheric pressure P and on the other side to the pressure of the liquid or gas P. In this situation the metal deforms and two phenomena can happen: that when the pressure decreases, and the metal having elastic properties, it returns to the original situation, fulfilling the Hooke's law, or that, for any reason, the metal enters the yield zone. In this case, the metal no longer returns to its original shape, that is, it deforms and if the pressure is very high, the metal could reach the breaking point.[5][6]​.

See in this image how the thin metal sheet returns to its original shape:

This phenomenon can occur in an instrument called a Bourdon tube, which could be considered a case of a flexible metal sheet subjected to two pressures, the internal pressure, which is a liquid or gas that penetrated the Bourdon tube, and the atmospheric pressure, at a certain pressure. The instrument can measure both pressure and temperature

.[7]​.

Contenido

Una lámina es un elemento estructural bidimensional curvado. Si las placas se tratan análogamente a las vigas rectas, las láminas son el análogo bidimensional de los arcos "Arco (construcción)").

Usando coordenadas curvilíneas ortogonales sobre la superficie se pueden escribir las ecuaciones de equilibrio para los esfuerzos internos para una lámina de Reisner-Mindlin como:[3]​[4]​.

Donde:.

Dome under its own weight

As an example of the previous equations we can consider a dome in the shape of a spherical cap subjected to its own weight. Each point of the two-dimensional dome can be parameterized using the coordinates:.

With which we have the following geometric factors:

And therefore the previous equations will be reduced to:.

Axisymmetric sheet

The general case of a general sheet requires using general curvilinear coordinates to parameterize its surface, which leads to governing equations that are partial differential equations whose integration is complicated. However, many cases of interest involve sheets with axial or revolution symmetry, with loads that also respect axial symmetry. In those cases the geometry of the surface can be parameterized by a coordinate (which gives its radial "profile"), and the governing equations in that case involve derivatives with respect to a single coordinate, and are therefore a system of ordinary differential equations. Thanks to this, the real behavior can be estimated using classical methods, since it is feasible to integrate the governing equations in some cases. This contrasts with the general case for which analytical solutions to the governing equations are not known, so in the general case the behavior can only be investigated by searching for approximate numerical solutions to the governing equations.

The simplest case of axisymmetric sheet theory is Cosserat's special static theory, which describes the behavior of axisymmetric plates susceptible to bending in their middle surface, extension of the same and shear in the thickness.

Sometimes it may occur that a liquid at a certain pressure, viscosity and temperature, or a gas at a certain pressure, penetrates into a container, limited on the outside by a very thin metal sheet, which would be, for example, subjected on one side to the atmospheric pressure P and on the other side to the pressure of the liquid or gas P. In this situation the metal deforms and two phenomena can happen: that when the pressure decreases, and the metal having elastic properties, it returns to the original situation, fulfilling the Hooke's law, or that, for any reason, the metal enters the yield zone. In this case, the metal no longer returns to its original shape, that is, it deforms and if the pressure is very high, the metal could reach the breaking point.[5][6]​.

See in this image how the thin metal sheet returns to its original shape:

This phenomenon can occur in an instrument called a Bourdon tube, which could be considered a case of a flexible metal sheet subjected to two pressures, the internal pressure, which is a liquid or gas that penetrated the Bourdon tube, and the atmospheric pressure, at a certain pressure. The instrument can measure both pressure and temperature

.[7]​.