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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. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.
Contents
- Definitions
- Moments
- Forces
- Stresses
- Strains
- Deflections
- Derivation
- Small deflection of thin rectangular plates
- Large deflection of thin rectangular plates
- Circular Kirchhoff Love plates
- Clamped edges
- Rectangular Kirchhoff Love plates
- Sinusoidal load
- Double trigonometric series equation
- Simply supported plate with general load
- Lvy solution
- Moments along edges
- Simply supported plate with uniformly distributed load
- Uniform and symmetric moment load
- Cylindrical plate bending
- Simply supported plate with axially fixed ends
- Bending of thick Mindlin plates
- Governing equations
- Simply supported rectangular plates
- Bending of Reissner Stein cantilever plates
- References
Definitions
For a thin rectangular plate of thickness,
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
The quantity
For isotropic, homogeneous, plates with Young's modulus
where
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
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
The governing equation in coordinate-free form is
In cylindrical coordinates
For symmetrically loaded circular plates,
Therefore, the governing equation is
If
where
For a circular plate, the requirement that the deflection and the slope of the deflection are finite at
Clamped edges
For a circular plate with clamped edges, we have
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 moment resultants (bending moments) are
The maximum radial stress is at
where
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 (a single Fourier component), 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
Since the plate is simply supported, the displacement
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
Double trigonometric series equation
We define a general load
where
The classical rectangular plate equation for small deflections thus becomes:
Simply-supported plate with general load
We assume a solution
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
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
Let us assume that
For a plate that is simply-supported along
Moments along edges
Let us consider the case of pure moment loading. In that case
Plugging the expression for
or
This is an ordinary differential equation which has the general solution
where
Let us choose the coordinate system such that the boundaries of the plate are at
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
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
The stresses are
Cylindrical plate bending
Cylindrical bending occurs when a rectangular plate that has dimensions
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
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
In Mindlin's theory,
The solutions to the governing equations can be found if one knows the corresponding Kirchhoff-Love solutions by using the relations
where
Simply supported rectangular plates
For simply supported plates, the Marcus moment sum vanishes, i.e.,
In that case the functions
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
and the boundary conditions at
Solution of this system of two ODEs gives
where
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