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Bending of plates
Encyclopedia
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 force
s and moment
s. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory
. The stress
es 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.
for plates the governing equations are
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 thickness. The quantity
has units of moment
per unit thickness.
For isotropic, homogeneous, plates with Young's modulus
and Poisson's ratio
these equations reduce to
where
is the deflection of the mid-surface of the plate.
In rectangular Cartesian coordinates,
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
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
.
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

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
. 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.,
. 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
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 theoryKirchhoff–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

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 quantity

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...
per unit thickness. The quantity

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...
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...

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 ....


where

In rectangular Cartesian coordinates,

Circular Kirchhoff-Love plates
The bending of circular plates can be examined by solving the governing equation withappropriate 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,


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 plate (radius

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


have
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

the plate is zero, the bending moment






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


Navier solution
Let us now consider a more general load
a sum of Fourier components such that

where


to find the amplitudes



If we repeat the process by integrating over

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Therefore,
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Now that we know
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equation (1) to get the displacement, i.e.,
Uniform load
Consider the situation where a uniform load is applied on the plate, i.e.,
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Now
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We can use these relations to get a simpler expression for
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Since
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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
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The stresses in the plate are
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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 atand
, the boundary conditions
areand
. The moment boundary condition is equivalent to
(verify). The goal is to find
such that
it satisfies the boundary conditions atand
and, of course, the
governing equation.
Moments along edges
Let us consider the case of pure moment loading. In that caseand
has to satisfy
. Since we are working in rectangular
Cartesian coordinates, the governing equation can be expanded as
Plugging the expression forin the governing equation gives us
or
This is an ordinary differential equation which has the general solution
whereare 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
atand
(same as before) and at
(and not
and
). Then the moment boundary conditions at the
boundaries are
whereare 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 displacementare
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
whereis 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 quantitiesand
are the rotations of the mid-surface normal
about theand
-axes, respectively. The canonical parameters for this theory
areand
. 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
whereis 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 loadat
.
and the boundary conditions atare
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
- BendingBendingIn engineering mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically...
- Infinitesimal strain theory
- Kirchhoff–Love plate theoryKirchhoff–Love plate theoryThe 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...
- Linear elasticityLinear elasticityLinear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
- Mindlin–Reissner plate theoryMindlin–Reissner plate theoryThe Mindlin-Reissner theory of plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1951 by Raymond Mindlin . A similar, but not identical, theory had been proposed earlier by Eric Reissner in...
- Plate theoryPlate theoryIn 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...
- Stress (mechanics)
- Structural acousticsStructural acousticsStructural acoustics is the study of the mechanical waves in structures and how they interact with and radiate into adjacent fluids. The field of structural acoustics is often referred to as vibroacoustics in Europe and Asia. People that work in the field of structural acoustics are known as...
- Vibration of platesVibration of platesThe vibration of plates is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two...
- Bending