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 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 Kirchhoff-Love platesEdit

 
Forces and moments on a flat plate.

DefinitionsEdit

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

 

MomentsEdit

The bending moments per unit length are given by

 
 

The twisting moment per unit length is given by

 

ForcesEdit

The shear forces per unit length are given by

 
 

StressesEdit

The bending stresses are given by

 
 

The shear stress is given by

 

StrainsEdit

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

 
 
 

DeflectionsEdit

The deflections are given by

 
 

DerivationEdit

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 platesEdit

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 platesEdit

This is governed by the Föpplvon Kármán plate equations

 
 

where   is the stress function.

Circular Kirchhoff-Love platesEdit

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 edgesEdit

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 platesEdit

 
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 loadEdit

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 solutionEdit

Double trigonometric series equationEdit

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 loadEdit

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 loadEdit

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 solutionEdit

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 edgesEdit

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 loadEdit

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 loadEdit

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 bendingEdit

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 endsEdit

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 platesEdit

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 equationsEdit

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 platesEdit

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 platesEdit

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 alsoEdit

ReferencesEdit

  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.