Bending of plates


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

Bending of Kirchhoff-Love plates

Definitions

For a thin rectangular plate of thickness, Young's modulus, and Poisson's ratio, we can define parameters in terms of the plate deflection,.
The flexural rigidity is given by

Moments

The bending moments per unit length are given by
The twisting moment per unit length is given by

Forces

The shear forces per unit length are given by

Stresses

The bending stresses are given by
The shear stress is given by

Strains

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

Deflections

The deflections are given by

Derivation

In the Kirchhoff–Love plate theory for plates the governing equations are
and
In expanded form,
and
where is an applied transverse load per unit area, the thickness of the plate is, the stresses are, and
The quantity has units of force per unit length. The quantity has units of moment per unit length.
For isotropic, homogeneous, plates with Young's modulus and Poisson's ratio these equations reduce to
where is the deflection of the mid-surface of the plate.

Small deflection of thin rectangular plates

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.

Large deflection of thin rectangular plates

This is governed by the Föppl–von Kármán plate equations
where is the stress function.

Circular Kirchhoff-Love plates

The bending of circular plates can be examined by solving the governing equation with
appropriate boundary conditions. These solutions were first found by Poisson in 1829.
Cylindrical coordinates are convenient for such problems. Here is the distance of a point from the midplane of the plate.
The governing equation in coordinate-free form is
In cylindrical coordinates,
For symmetrically loaded circular plates,, and we have
Therefore, the governing equation is
If and are constant, direct integration of the governing equation gives us


where are constants. The slope of the deflection surface is
For a circular plate, the requirement that the deflection and the slope of the deflection are finite
at implies that. However, need not equal 0, as the limit
of exists as you approach from the right.

Clamped edges

For a circular plate with clamped edges, we have and at the edge of
the plate. Using these boundary conditions we get


The in-plane displacements in the plate are
The in-plane strains in the plate are
The in-plane stresses in the plate are
For a plate of thickness, the bending stiffness is and we
have


The moment resultants are
The maximum radial stress is at and :
where. The bending moments at the boundary and the center of the plate are

Rectangular Kirchhoff-Love plates

For rectangular plates, Navier in 1820 introduced a simple method for finding the displacement and stress when a plate is simply supported. The idea was to express the applied load in terms of Fourier components, find the solution for a sinusoidal load, and then superimpose the Fourier components to get the solution for an arbitrary load.

Sinusoidal load

Let us assume that the load is of the form
Here is the amplitude, is the width of the plate in the -direction, and
is the width of the plate in the -direction.
Since the plate is simply supported, the displacement along the edges of
the plate is zero, the bending moment is zero at and, and
is zero at and.
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. We can calculate the stresses and strains in the plate once we know the displacement.
For a more general load of the form
where and are integers, we get the solution

Navier solution

Double trigonometric series equation

We define a general load of the following form
where is a Fourier coefficient given by
The classical rectangular plate equation for small deflections thus becomes:

Simply-supported plate with general load

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 with general load is given by

Simply-supported plate with uniformly-distributed load

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 with uniformly-distributed load is given by


The bending moments per unit length in the plate are given by

Lévy solution

Another approach was proposed by Lévy 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. The goal is to find such that it satisfies the boundary conditions at and and, of course, the governing equation.
Let us assume that
For a plate that is simply-supported along and, the boundary conditions are and. Note that there is no variation in displacement along these edges meaning that and, thus reducing the moment boundary condition to an equivalent expression.

Moments along edges

Consider the case of pure moment loading. In that case and
has to satisfy. Since we are working in rectangular
Cartesian coordinates, the governing equation can be expanded as
Plugging the expression for in the governing equation gives us
or
This is an ordinary differential equation which has the general solution
where are constants that can be determined from the boundary
conditions. Therefore, the displacement solution has the form


Let us choose the coordinate system such that the boundaries of the plate are
at and and at . Then the moment boundary conditions at the boundaries are
where are known functions. The solution can be found by
applying these boundary conditions. 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.

Simply-supported plate with uniformly-distributed load

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

Uniform and symmetric moment load

For the special case where the loading is symmetric and the moment is uniform, we have at,
The resulting displacement is


where
The bending moments and shear forces corresponding to the displacement are
The stresses are

Cylindrical plate bending

Cylindrical bending occurs when a rectangular plate that has dimensions, where and the thickness is small, is subjected to a uniform distributed load perpendicular to the plane of the plate. Such a plate takes the shape of the surface of a cylinder.

Simply supported plate with axially fixed ends

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.

Bending of thick Mindlin plates

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. Mindlin's theory
provides one approach for find the deformation and stresses in such plates. Solutions
to Mindlin's theory can be derived from the equivalent Kirchhoff-Love solutions using
canonical relations.

Governing equations

The canonical governing equation for isotropic thick plates can be expressed as
where is the applied transverse load, is the shear modulus,
is the bending rigidity, is the plate thickness,,
is the shear correction factor, is the Young's modulus, is the Poisson's
ratio, and
In Mindlin's theory, is the transverse displacement of the mid-surface of the plate
and the quantities and are the rotations of the mid-surface normal
about the and -axes, respectively. The canonical parameters for this theory
are and. The shear correction factor usually has the
value.
The solutions to the governing equations can be found if one knows the corresponding
Kirchhoff-Love solutions by using the relations
where is the displacement predicted for a Kirchhoff-Love plate, is a
biharmonic function such that, is a function that satisfies the
Laplace equation,, and

Simply supported rectangular plates

For simply supported plates, the Marcus moment sum vanishes, i.e.,
In that case the functions,, vanish, and the Mindlin solution is
related to the corresponding Kirchhoff solution by

Bending of Reissner-Stein cantilever plates

Reissner-Stein theory for cantilever plates leads to the following coupled ordinary differential equations for a cantilever plate with concentrated end load at.
and the boundary conditions at are
Solution of this system of two ODEs gives
where. The bending moments and shear forces corresponding to the displacement
are
The stresses are
If the applied load at the edge is constant, we recover the solutions for a beam under a
concentrated end load. If the applied load is a linear function of, then