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 formula_1, Young's modulus formula_2, and Poisson's ratio formula_3, we can define parameters in terms of the plate deflection, formula_4.
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 formula_26 is an applied transverse load per unit area, the thickness of the plate is formula_27, the stresses are formula_28, and
The quantity formula_30 has units of force per unit length. The quantity formula_31 has units of moment per unit length.
For isotropic, homogeneous, plates with Young's modulus formula_2 and Poisson's ratio formula_3 these equations reduce to
where formula_35 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 formula_39 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 formula_40 is the distance of a point from the midplane of the plate.
The governing equation in coordinate-free form is
In cylindrical coordinates formula_42, 
For symmetrically loaded circular plates, formula_44, and we have
Therefore, the governing equation is
If formula_47 and formula_48 are constant, direct integration of the governing equation gives us
where formula_50 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 formula_52 implies that formula_53. However, formula_54 need not equal 0, as the limit 
of formula_55 exists as you approach formula_52 from the right.
Clamped edges.
For a circular plate with clamped edges, we have formula_57 and formula_58 at the edge of
the plate (radius formula_59). 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 formula_64, the bending stiffness is formula_65 and we
have
The moment resultants (bending moments) are 
The maximum radial stress is at formula_68 and formula_69:
where formula_71. 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 (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 formula_74 is the amplitude, formula_59 is the width of the plate in the formula_76-direction, and
formula_77 is the width of the plate in the formula_78-direction.
Since the plate is simply supported, the displacement formula_79 along the edges of
the plate is zero, the bending moment formula_80 is zero at formula_81 and formula_82, and 
formula_83 is zero at formula_84 and formula_85.
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 formula_89 and formula_90 are integers, we get the solution
Navier solution.
Double trigonometric series equation.
We define a general load formula_92 of the following form
where formula_94 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 formula_79 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
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 (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 formula_113 such that it satisfies the boundary conditions at formula_114 and formula_115 and, of course, the governing equation formula_116.
Let us assume that
For a plate that is simply-supported along formula_81 and formula_82, the boundary conditions are formula_120 and formula_121. Note that there is no variation in displacement along these edges meaning that formula_122 and formula_123, thus reducing the moment boundary condition to an equivalent expression formula_124.
Moments along edges.
Consider the case of pure moment loading. In that case formula_125 and
formula_79 has to satisfy formula_127. Since we are working in rectangular
Cartesian coordinates, the governing equation can be expanded as
Plugging the expression for formula_79 in the governing equation gives us
or
This is an ordinary differential equation which has the general solution
where formula_133 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 formula_135 and formula_136 (same as before) and at formula_137 (and not formula_84 and
formula_85). Then the moment boundary conditions at the formula_137 boundaries are
where formula_142 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 formula_150 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 formula_154,
The resulting displacement is
where 
The bending moments and shear forces corresponding to the displacement formula_4 are
The stresses are
Cylindrical plate bending.
Cylindrical bending occurs when a rectangular plate that has dimensions formula_161, where formula_162 and the thickness formula_163 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 formula_164. 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 formula_47 is the applied transverse load, formula_167 is the shear modulus, formula_168
is the bending rigidity, formula_163 is the plate thickness, formula_170, 
formula_171 is the shear correction factor, formula_2 is the Young's modulus, formula_3 is the Poisson's 
ratio, and
In Mindlin's theory, formula_4 is the transverse displacement of the mid-surface of the plate
and the quantities formula_176 and formula_177 are the rotations of the mid-surface normal
about the formula_178 and formula_164-axes, respectively. The canonical parameters for this theory
are formula_180 and formula_181. The shear correction factor formula_171 usually has the
value formula_183.
The solutions to the governing equations can be found if one knows the corresponding 
Kirchhoff-Love solutions by using the relations
where formula_185 is the displacement predicted for a Kirchhoff-Love plate, formula_186 is a 
biharmonic function such that formula_187, formula_188 is a function that satisfies the 
Laplace equation, formula_189, and
Simply supported rectangular plates.
For simply supported plates, the "Marcus moment" sum vanishes, i.e.,
In that case the functions formula_186, formula_188, formula_194 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 formula_196 at formula_82.
and the boundary conditions at formula_82 are
Solution of this system of two ODEs gives
where formula_202. The bending moments and shear forces corresponding to the displacement 
formula_203 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 formula_78, then