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The typical thickness to width ratio of a plate structure is less than 0.1. [citation needed] A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem. The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads.
In the Kirchhoff–Love plate theory for plates the governing equations are [1], = and , = In expanded form, + = ; + = and + + = where () is an applied transverse load per unit area, the thickness of the plate is =, the stresses are , and
Vibration mode of a clamped square plate. The vibration of plates is a special case of the more general problem of mechanical vibrations.The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two.
Let the position vector of a point in the undeformed plate be .Then = + +. The vectors form a Cartesian basis with origin on the mid-surface of the plate, and are the Cartesian coordinates on the mid-surface of the undeformed plate, and is the coordinate for the thickness direction.
The Mindlin hypothesis implies that the displacements in the plate have the form = (,) ; =, = (,)where and are the Cartesian coordinates on the mid-surface of the undeformed plate and is the coordinate for the thickness direction, , =, are the in-plane displacements of the mid-surface, is the displacement of the mid-surface in the direction, and designate the angles which the normal to the mid ...
For the Pohlhausen solution for laminar flow on a heated flat plate, [10] it is found that thermal boundary layer thickness defined as = + where = /, tracks the 99% thickness very well. [ 11 ] For laminar flow, the three different moment cases all give similar values for the thermal boundary layer thickness.
A plate is a broad, mainly flat vessel on which food can be served. [1] A plate can also be used for ceremonial or decorative purposes. Most plates are circular, but they may be any shape, or made of any water-resistant material. Generally plates are raised round the edges, either by a curving up, or a wider lip or raised portion.
The plate elastic thickness (usually referred to as effective elastic thickness of the lithosphere). The elastic properties of the plate; The applied load or force; As flexural rigidity of the plate is determined by the Young's modulus, Poisson's ratio and cube of the plate's elastic thickness, it is a governing factor in both (1) and (2).