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Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections.
The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads. Of the numerous plate theories that have been developed since the late 19th century, two are widely accepted and used in engineering. These are the Kirchhoff–Love theory of plates (classical plate theory)
Thin rectangular plate of mass m, length of side adjacent to side containing axis of rotation is r [a] (Axis of rotation along a side of the plate) = Solid rectangular cuboid of height h, width w, and depth d, and mass m. [7]
Stress resultants are simplified representations of the stress state in structural elements such as beams, plates, or shells. [1] The geometry of typical structural elements allows the internal stress state to be simplified because of the existence of a "thickness'" direction in which the size of the element is much smaller than in other directions.
where , are the coordinates of a point on the cross section at which the stress is to be determined as shown to the right, and are the bending moments about the y and z centroid axes, and are the second moments of area (distinct from moments of inertia) about the y and z axes, and is the product of moments of area. Using this equation it is ...
A point force with components and is applied at the point (,) of the plate. For the situation where the plate is large compared to the size of the crack and the location of the force is relatively close to the crack, i.e., h ≫ a {\displaystyle h\gg a} , b ≫ a {\displaystyle b\gg a} , x ≪ b {\displaystyle x\ll b} , y ≪ h {\displaystyle y ...
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Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue) The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments.