Bending of plates

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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.

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[edit]

Forces and moments on a flat plate.

Definitions[edit]

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[edit]

The bending moments per unit length are given by

The twisting moment per unit length is given by

Forces[edit]

The shear forces per unit length are given by

Stresses[edit]

The bending stresses are given by

The shear stress is given by

Strains[edit]

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[edit]

The deflections are given by

Derivation[edit]

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[edit]

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[edit]

This is governed by the Föpplvon Kármán plate equations

where is the stress function.

Circular Kirchhoff-Love plates[edit]

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[edit]

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[edit]

Bending of a rectangular plate under the action of a distributed force per unit area.

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[edit]

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[edit]

Double trigonometric series equation[edit]

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[edit]

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[edit]

Displacement ()
Stress ()
Stress ()
Displacement and stresses along for a rectangular plate with mm, mm, mm, GPa, and under a load kPa. The red line represents the bottom of the plate, the green line the middle, and the blue line the top of the plate.

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[edit]

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[edit]

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[edit]

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[edit]

For the special case where the loading is symmetric and the moment is uniform, we have at ,

Displacement ()
Bending stress ()
Transverse shear stress ()
Displacement and stresses for a rectangular plate under uniform bending moment along the edges and . The bending stress is along the bottom surface of the plate. The transverse shear stress is along the mid-surface of the plate.

The resulting displacement is

where

The bending moments and shear forces corresponding to the displacement are

The stresses are

Cylindrical plate bending[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

References[edit]

  1. ^ Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis.
  2. ^ Timoshenko, S. and Woinowsky-Krieger, S., (1959), Theory of plates and shells, McGraw-Hill New York.
  3. ^ Cook, R. D. et al., 2002, Concepts and applications of finite element analysis, John Wiley & Sons
  4. ^ Lévy, M., 1899, Comptes rendues, vol. 129, pp. 535-539
  5. ^ 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.
  6. ^ E. Reissner and M. Stein. Torsion and transverse bending of cantilever plates. Technical Note 2369, National Advisory Committee for Aeronautics,Washington, 1951.