Bending of plates - AbsoluteAstronomy.com

Bending of plates or plate bending refers to the deflection

Deflection

Deflection or deflexion may refer to:* Deflection , the displacement of a structural element under load* Deflection , a technique of shooting ahead of a moving target so that the target and projectile will collide...

of a plate

Plate

Plate may refer to:* Plate , a broad, mainly flat vessel on which food is served* Silver or the plate, dishware and cutlery made of sterling, Britannia or Sheffield plate silver* Plating, the deposition of metallic layers...

perpendicular to the plane of the plate under the action of external force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

s and moment

Moment

- Science, engineering, and mathematics :* Moment , used in probability theory and statistics* Moment , several related concepts, including:** Angular momentum or moment of momentum, the rotational analog of momentum...

s. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory

Plate theory

In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions . The typical thickness to width ratio of a plate...

. The stress

Stress (physics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

es in the plate can be calculated from these deflections. Once the stresses are known, failure theories

Failure theory (material)

Failure theory is the science of predicting the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure or ductile failure . Depending on the conditions most materials can fail in a brittle or ductile...

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

Kirchhoff–Love plate theory

The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love using assumptions...

for plates the governing equations are

and

In expanded form,

and

where

is an applied transverse load

Load

Load may refer to:*Structural load, forces which are applied to a structure*Cargo*The load of a mutual fund *The genetic load of a population*The parasite load of an organism...

per unit area, the thickness of the plate is

, the stresses are

, and

The quantity

has units of force

Force

per unit thickness. The quantity

has units of moment

Moment

per unit thickness.

For isotropic, homogeneous, plates with Young's modulus

Young's modulus

Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

and Poisson's ratio

Poisson's ratio

Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....

these equations reduce to

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

In cylindrical coordinates

For symmetrically loaded circular plates,

, and we have

Therefore, the governing equation is

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

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

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

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.

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

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

Bending of Kirchhoff-Love plates

Circular Kirchhoff-Love plates

Clamped edges

Rectangular Kirchhoff-Love plates

Sinusoidal load

Navier solution

Uniform load

Levy solution

Moments along edges

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

See also