Bending of plates

Last updated
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. BendingCircularPlate.png
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.

Contents

Bending of Kirchhoff-Love plates

Forces and moments on a flat plate. PlateForcesMoments upd.png
Forces and moments on a flat plate.

Definitions

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

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 [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

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 Föpplvon Kármán plate equations

where 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 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

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

Bending of a rectangular plate under the action of a distributed force
q
{\displaystyle q}
per unit area. RectangularPlateBending.svg
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

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

Double trigonometric series equation

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

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

Wx rectangularPlate.svg
Displacement ()
Sxx rectangularPlate.svg
Stress ()
Syy rectangularPlate.svg
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

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

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

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

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

SurfRecBMIso w.png
Displacement ()
SurfRecBMIso sy.png
Bending stress ()
SurfRecBMIso syz.png
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

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. 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. [5]

Governing equations

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

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 [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

Related Research Articles

Bessel function Families of solutions to related differential equations

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation

Navier–Stokes equations Equations describing the motion of viscous fluid substances

In physics, the Navier–Stokes equations are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.

Bending

In applied mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation

Euler–Bernoulli beam theory

Euler–Bernoulli beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.

Elliptic coordinate system

In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

Toroidal coordinates

Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.

Elliptic cylindrical coordinates

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular -direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

Prolate spheroidal coordinates

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.

Oblate spheroidal coordinates

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the x-y plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.

In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen, and published by Poiseuille in 1840–41 and 1846. The theoretical justification of the Poiseuille law was given by George Stokes in 1845.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

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 structure is less than 0.1. A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem. The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads.

Kirchhoff–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 proposed by Kirchhoff. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form.

In statistics, the generalized Marcum Q-function of order is defined as

Vibration of plates

The 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. This suggests that a two-dimensional plate theory will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.

Mindlin–Reissner plate theory

The Uflyand-Mindlin theory of vibrating 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 1948 by Yakov Solomonovich Uflyand (1916-1991) and in 1951 by Raymond Mindlin with Mindlin making reference to Uflyand's work. Hence, this theory has to be referred to us Uflyand-Mindlin plate theory, as is done in the handbook by Elishakoff, and in papers by Andronov, Elishakoff, Hache and Challamel, Loktev, Rossikhin and Shitikova and Wojnar. In 1994, Elishakoff suggested to neglect the fourth-order time derivative in Uflyand-Mindlin equations. A similar, but not identical, theory in static setting, had been proposed earlier by Eric Reissner in 1945. Both theories are intended for thick plates in which the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. The Uflyand-Mindlin theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of one tenth the planar dimensions while the Kirchhoff–Love theory is applicable to thinner plates.

In mathematics, a Jackson q-Bessel function is one of the three q-analogs of the Bessel function introduced by Jackson. The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.

References

  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. 1 2 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.