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

Definitions

For a thin rectangular plate of thickness H, Young's modulus E, and Poisson's ratio \nu, we can define parameters in terms of the plate deflection, w. The flexural rigidity is given by

Moments

The bending moments per unit length are given by The twisting moment per unit length is given by

Forces

The shear forces per unit length are given by

Stresses

The bending stresses are given by The shear stress is given by

Strains

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

The deflections are given by

Derivation

In the Kirchhoff–Love plate theory for plates the governing equations are and In expanded form, and where q(x) is an applied transverse load per unit area, the thickness of the plate is H=2h, the stresses are \sigma_{ij}, and The quantity N has units of force per unit length. The quantity M has units of moment per unit length. For isotropic, homogeneous, plates with Young's modulus E and Poisson's ratio \nu these equations reduce to where w(x_1,x_2) is the deflection of the mid-surface of the plate.

Small deflection of thin rectangular plates

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

This is governed by the Föppl–von Kármán plate equations where F is the stress function.

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. Here z 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, w = w(r), and we have Therefore, the governing equation is If q and D are constant, direct integration of the governing equation gives us <blockquote style="border: 1px solid black; padding:10px; width:530px"> where C_i 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 r = 0 implies that C_1 = 0. However, C_3 need not equal 0, as the limit of r \ln r, exists as you approach r = 0 from the right.

Clamped edges

For a circular plate with clamped edges, we have w(a) = 0 and \phi(a) = 0 at the edge of the plate (radius a). Using these boundary conditions we get

<blockquote style="border: 1px solid black; padding:10px; width:530px"> 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 2h, the bending stiffness is and we have <blockquote style="border: 1px solid black; padding:10px; width:430px"> The moment resultants (bending moments) are The maximum radial stress is at z = h and r = a: where H := 2h. 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 q_0 is the amplitude, a is the width of the plate in the x-direction, and b is the width of the plate in the y-direction. Since the plate is simply supported, the displacement w(x,y) along the edges of the plate is zero, the bending moment M_{xx} is zero at x=0 and x=a, and M_{yy} is zero at y=0 and y=b. 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. We can calculate the stresses and strains in the plate once we know the displacement. For a more general load of the form where m and n are integers, we get the solution

<blockquote style="border: 1px solid black; padding:10px; width:530px">

Navier solution

Double trigonometric series equation

We define a general load q(x,y) of the following form where a_{mn} is a Fourier coefficient given by The classical rectangular plate equation for small deflections thus becomes:

Simply-supported plate with general load

We assume a solution w(x,y) 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 <blockquote style="border: 1px solid black; padding:10px; width:600px">

Simply-supported plate with uniformly-distributed load

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 <blockquote style="border: 1px solid black; padding:10px; width:600px"> The bending moments per unit length in the plate are given by <blockquote style="border: 1px solid black; padding:10px; width:600px">

Lévy solution

Another approach was proposed by Lévy 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 Y_m(y) such that it satisfies the boundary conditions at y = 0 and y = b and, of course, the governing equation. Let us assume that For a plate that is simply-supported along x=0 and x=a, the boundary conditions are w=0 and M_{xx}=0. 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

Consider the case of pure moment loading. In that case q = 0 and w(x,y) has to satisfy. Since we are working in rectangular Cartesian coordinates, the governing equation can be expanded as Plugging the expression for w(x,y) 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

<blockquote style="border: 1px solid black; padding:1px; width:800px"> Let us choose the coordinate system such that the boundaries of the plate are at x = 0 and x = a (same as before) and at y = \pm b/2 (and not y=0 and y=b). Then the moment boundary conditions at the y = \pm b/2 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 <blockquote style="border: 1px solid black; padding:1px; width:800px"> where Similarly, for the antisymmetrical case where we have <blockquote style="border: 1px solid black; padding:1px; width:800px"> We can superpose the symmetric and antisymmetric solutions to get more general solutions.

Simply-supported plate with uniformly-distributed load

For a uniformly-distributed load, we have The deflection of a simply-supported plate with centre with uniformly-distributed load is given by <blockquote style="border: 1px solid black; padding:10px; width:600px"> The bending moments per unit length in the plate are given by <blockquote style="border: 1px solid black; padding:10px; width:850px">

Uniform and symmetric moment load

For the special case where the loading is symmetric and the moment is uniform, we have at y=\pm b/2, The resulting displacement is <blockquote style="border: 1px solid black; padding:1px; width:600px"> where The bending moments and shear forces corresponding to the displacement w are The stresses are

Cylindrical plate bending

Cylindrical bending occurs when a rectangular plate that has dimensions, where a \ll b and the thickness h 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 x_1. 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. Raymond D. 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 q is the applied transverse load, G is the shear modulus, is the bending rigidity, h is the plate thickness, , \kappa is the shear correction factor, E is the Young's modulus, \nu is the Poisson's ratio, and In Mindlin's theory, w is the transverse displacement of the mid-surface of the plate and the quantities \varphi_1 and \varphi_2 are the rotations of the mid-surface normal about the x_2 and x_1-axes, respectively. The canonical parameters for this theory are and. The shear correction factor \kappa usually has the value 5/6. The solutions to the governing equations can be found if one knows the corresponding Kirchhoff-Love solutions by using the relations where w^K is the displacement predicted for a Kirchhoff-Love plate, \Phi is a biharmonic function such that, \Psi is a function that satisfies the Laplace equation,, and

Simply supported rectangular plates

For simply supported plates, the Marcus moment sum vanishes, i.e., Which is almost Laplace`s equation for w[ref 6]. In that case the functions \Phi, \Psi, \Omega 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 q_x(y) at x=a. and the boundary conditions at x=a 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 y, then