1. Bending of Kirchhoff-Love plates
     Definitions    Moments    Forces    Stresses    Strains    Deflections    Derivation    Small deflection of thin rectangular plates    Large deflection of thin rectangular plates  

2. Circular Kirchhoff-Love plates
    Clamped edges 

3. Rectangular Kirchhoff-Love plates
    Sinusoidal load  Navier solution   Double trigonometric series equation    Simply-supported plate with general load    Simply-supported plate with uniformly-distributed load   Lévy 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  

4. Bending of thick Mindlin plates
    Governing equations  Simply supported rectangular plates 

5. Bending of Reissner-Stein cantilever plates

6. See also

7. References

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^[1]

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^[2]

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 (radius ). 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 (bending moments) 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 (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 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.^[3] 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 (of corner-origin) with general load is given by

Simply-supported plate with uniformly-distributed load

{{multiple image

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 ^[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. 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

Let us 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 (same as before) and at (and not and

). 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 ,

{{multiple image

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.^[5]

Governing equations

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

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^[6] 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

See also

• Bending
• Infinitesimal strain theory
• Kirchhoff–Love plate theory
• Linear elasticity
• Mindlin–Reissner plate theory
• Plate theory
• Stress (mechanics)
• Stress resultants
• Structural acoustics
• Vibration of plates

References

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