# 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

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.

In rectangular Cartesian coordinates,

## 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.

The governing equation in coordinate-free form is

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 .

### 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

Let us now consider a more general load . We can break this load up into a sum of Fourier components such that

where is an amplitude. We can use the orthogonality of Fourier components,

to find the amplitudes . Thus we have, by integrating over ,

If we repeat the process by integrating over , we have

Therefore,

Now that we know , we can just superpose solutions of the form given in equation (1) to get the displacement, i.e.,

#### Uniform load

Consider the situation where a uniform load is applied on the plate, i.e., . Then

Now

We can use these relations to get a simpler expression for :

Since [ so ] when and are even, we can get an even simpler expression for when both and are odd:

Plugging this expression into equation (2) and keeping in mind that only odd terms contribute to the displacement, we have

The corresponding moments are given by

The stresses in the plate are

### Levy solution

Another approach was proposed by Levy 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.

Let us assume that

For a plate that is simply supported at and , the boundary conditions are and . The moment boundary condition is equivalent to (verify). The goal is to find such that it satisfies the boundary conditions at and and, of course, the governing equation .

#### 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.

#### 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.^{[4]}

### Governing equations

The canonical governing equation for isotropic thick plates can be expressed as^{[4]}

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^{[5]} 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

- ↑ Reddy, J. N., 2007,
**Theory and analysis of elastic plates and shells**, CRC Press, Taylor and Francis. - ↑ Timoshenko, S. and Woinowsky-Krieger, S., (1959),
**Theory of plates and shells**, McGraw-Hill New York. - ↑ Cook, R. D. et al., 2002,
**Concepts and applications of finite element analysis**, John Wiley & Sons - ↑
^{4.0}^{4.1}Lim, G. T. and Reddy, J. N., 2003,*On canonical bending*relationships for plates*, International Journal of Solids and Structures, vol. 40,*pp. 3039-3067. - ↑ E. Reissner and M. Stein. Torsion and transverse bending of cantilever plates. Technical Note 2369, National Advisory Committee for Aeronautics,Washington, 1951.