bending of plates - wikipedia, the free encyclopedia

Bending of an edge clamped circular plate under the

action of a transverse pressure. The left half of the

plate shows the deformed shape while the right half

shows the undeformed shape. This calculation was

performed using Ansys.

Forces and moments on a flat plate.

Bending of platesFrom Wikipedia, the free encyclopedia

Bending of plates or plate bending refers to the deflection of a plateperpendicular to the plane of the plate under the action of external forces andmoments. The amount of deflection can be determined by solving thedifferential equations of an appropriate plate theory. The stresses in the plate canbe calculated from these deflections. Once the stresses are known, failuretheories can be used to determine whether a plate will fail under a given load.

Contents

1 Bending of Kirchhoff-Love plates

2 Circular Kirchhoff-Love plates

2.1 Clamped edges

3 Rectangular Kirchhoff-Love plates

3.1 Sinusoidal load

3.2 Navier solution

3.2.1 Uniform load

3.3 Levy solution

3.3.1 Moments along edges

3.3.2 Uniform and symmetric moment load

3.4 Cylindrical plate bending

3.4.1 Simply supported plate with axially fixed ends

4 Bending of thick Mindlin plates

4.1 Governing equations

4.2 Simply supported rectangular plates

5 Bending of Reissner-Stein cantilever plates

6 See also

7 References

Bending of Kirchhoff-Love plates

In the KirchhoffLove 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

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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. Thesesolutions 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.

Clamped edges

For a circular plate with clamped edges, we have and at the edge of the plate (radius ). Using these


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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 simplysupported. The idea was to express the applied load in terms of Fourier components, find the solution for a sinusoidal load (a singleFourier 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 .


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Bending of a rectangular plate under the

action of a distributed force per unit

area.

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 theplate 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,


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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


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) Stress ( )

mm, mm, GPa, and under a load kPa. The red line represents the bottom of the

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 theparameters 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


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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


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Bending stress ( ) Transverse shear stress ( )

te under uniform bending moment along the edges and . The bending stress is along the bottom surface of the plate. The transverse

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 acylinder.


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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 bendingsolutions 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-surfaceafter 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 rotationsof 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 therelations

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.,


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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 withconcentrated 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


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linear function of , then

See also

Bending

Infinitesimal strain theory

KirchhoffLove plate theory

Linear elasticity

MindlinReissner 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.1.

^ Timoshenko, S. and Woinowsky-Krieger, S., (1959), Theory of plates and shells, McGraw-Hill New York.2.

^ Cook, R. D. et al., 2002, Concepts and applications of finite element analysis, John Wiley & Sons3.

^ a b 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.

4.

^ E. Reissner and M. Stein. Torsion and transverse bending of cantilever plates. Technical Note 2369, National Advisory Committee for

Aeronautics,Washington, 1951.

5.

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