Bending of plates

The amount of deflection can be determined by solving the differential equations of an appropriate plate theory.

Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.

The flexural rigidity is given by The bending moments per unit length are given by The twisting moment per unit length is given by The shear forces per unit length are given by The bending stresses are given by The shear stress is given by The bending strains for small-deflection theory are given by The shear strain for small-deflection theory is given by For large-deflection plate theory, we consider the inclusion of membrane strains The deflections are given by 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

This is governed by the Germain-Lagrange plate equation This equation was first derived by Lagrange in December 1811 in correcting the work of Germain who provided the basis of the theory.

The bending of circular plates can be examined by solving the governing equation with appropriate boundary conditions.

The governing equation in coordinate-free form is In cylindrical coordinates

are constant, direct integration of the governing equation gives us where

The idea was to express the applied load in terms of Fourier components, find the solution for a sinusoidal load (a single Fourier component), and then superimpose the Fourier components to get the solution for an arbitrary load.

If we apply these boundary conditions and solve the plate equation, we get the solution Where D is the flexural rigidity Analogous to flexural stiffness EI.

are integers, we get the solution We define a general load

is a Fourier coefficient given by The classical rectangular plate equation for small deflections thus becomes: We assume a solution

of the following form The partial differentials of this function are given by Substituting these expressions in the plate equation, we have Equating the two expressions, we have which can be rearranged to give The deflection of a simply-supported plate (of corner-origin) with general load is given by For a uniformly-distributed load, we have The corresponding Fourier coefficient is thus given by Evaluating the double integral, we have or alternatively in a piecewise format, we have The deflection of a simply-supported plate (of corner-origin) with uniformly-distributed load is given by The bending moments per unit length in the plate are given by Another approach was proposed by Lévy[4] in 1899.

In this case we start with an assumed form of the displacement and try to fit the parameters so that the governing equation and the boundary conditions are satisfied.

Note that there is no variation in displacement along these edges meaning that

, thus reducing the moment boundary condition to an equivalent expression

Since we are working in rectangular Cartesian coordinates, the governing equation can be expanded as Plugging the expression for

Therefore, the displacement solution has the form Let us choose the coordinate system such that the boundaries of the plate are at

We can show that for the symmetrical case where and we have where Similarly, for the antisymmetrical case where we have We can superpose the symmetric and antisymmetric solutions to get more general solutions.

For a uniformly-distributed load, we have The deflection of a simply-supported plate with centre

with uniformly-distributed load is given by The bending moments per unit length in the plate are given by For the special case where the loading is symmetric and the moment is uniform, we have at

are The stresses are Cylindrical bending occurs when a rectangular plate that has dimensions

is small, is subjected to a uniform distributed load perpendicular to the plane of the plate.

For a simply supported plate under cylindrical bending with edges that are free to rotate but have a fixed

Cylindrical bending solutions can be found using the Navier and Levy techniques.

For thick plates, we have to consider the effect of through-the-thickness shears on the orientation of the normal to the mid-surface after deformation.

Raymond D. Mindlin's theory provides one approach for find the deformation and stresses in such plates.

[5] The canonical governing equation for isotropic thick plates can be expressed as[5] where

, and For simply supported plates, the Marcus moment sum vanishes, i.e., Which is almost Laplace`s equation for w[ref 6].

vanish, and the Mindlin solution is related to the corresponding Kirchhoff solution by Reissner-Stein theory for cantilever plates[6] leads to the following coupled ordinary differential equations for a cantilever plate with concentrated end load