Plate Theory wiki

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Plate theory 1

Plate theory

Vibration mode of a clamped square plate

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.[1]

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.

Of the numerous plate theories that have beendeveloped since the late 19th century, two are widely

accepted and used in engineering. These are

the KirchhoffLove theory of plates (classical plate

theory)

The MindlinReissner theory of plates (first-order shear plate theory)

KirchhoffLove theory for thin plates

Note: the Einstein summation convention of summing on repeated indices is used below.

Deformation of a thin plate highlighting the displacement, the

mid-surface (red) and the normal to the mid-surface (blue)

The Kirchhoff

Love theory is an extension of

EulerBernoulli beam theory to thin plates. The theory

was developed in 1888 by Love[2]

using assumptions

proposed by Kirchhoff. It is assumed that there a

mid-surface plane can be used to represent the

three-dimensional plate in two dimensional form.

The following kinematic assumptions that are made in

this theory:[3]

straight lines normal to the mid-surface remain

straight after deformation straight lines normal to the mid-surface remain

normal to the mid-surface after deformation

the thickness of the plate does not change during a

deformation.

Displacement field

The Kirchhoff hypothesis implies that the displacement

field has the form
http://en.wikipedia.org/w/index.php?title=Displacement_%28vector%29http://en.wikipedia.org/w/index.php?title=Beam_theoryhttp://en.wikipedia.org/w/index.php?title=Augustus_Edward_Hough_Lovehttp://en.wikipedia.org/w/index.php?title=Gustav_Kirchhoffhttp://en.wikipedia.org/w/index.php?title=File%3APlaque_mince_deplacement_element_matiere.svghttp://en.wikipedia.org/w/index.php?title=Einstein_summation_conventionhttp://en.wikipedia.org/w/index.php?title=Eric_Reissnerhttp://en.wikipedia.org/w/index.php?title=Raymond_Mindlinhttp://en.wikipedia.org/w/index.php?title=Augustus_Edward_Hough_Lovehttp://en.wikipedia.org/w/index.php?title=Gustav_Kirchhoffhttp://en.wikipedia.org/w/index.php?title=Stress_%28mechanics%29http://en.wikipedia.org/w/index.php?title=Deformation_%28mechanics%29http://en.wikipedia.org/w/index.php?title=Continuum_mechanicshttp://en.wikipedia.org/w/index.php?title=List_of_structural_elementshttp://en.wikipedia.org/w/index.php?title=Bendinghttp://en.wikipedia.org/w/index.php?title=Continuum_mechanicshttp://en.wikipedia.org/w/index.php?title=File%3AESPIvibration.jpg


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Plate theory 2

where and are the Cartesian coordinates on the mid-surface of the undeformed plate, is the coordinate for

the thickness direction, are the in-plane displacements of the mid-surface, and is the displacement of the

mid-surface in the direction.

If are the angles of rotation of the normal to the mid-surface, then in the KirchhoffLove theory

Displacement of the mid-surface (left) and of a normal (right)

Strain-displacement relations

For the situation where the strains in

the plate are infinitesimal and the

rotations of the mid-surface normals

are less than 10 the

strains-displacement relations are

Therefore the only non-zero strains are in the in-plane directions.

If the rotations of the normals to the mid-surface are in the range of 10 to 15 , the strain-displacement relations

can be approximated using the von Krmn strains. Then the kinematic assumptions of Kirchhoff-Love theory lead

to the following strain-displacement relations

This theory is nonlinear because of the quadratic terms in the strain-displacement relations.

Equilibrium equations

The equilibrium equations for the plate can be derived from the principle of virtual work. For the situation where the

strains and rotations of the plate are small, the equilibrium equations for an unloaded plate are given by

where the stress resultants and stress moment resultants are defined as

and the thickness of the plate is . The quantities are the stresses.

If the plate is loaded by an external distributed load that is normal to the mid-surface and directed in the

positive direction, the principle of virtual work then leads to the equilibrium equations
http://en.wikipedia.org/w/index.php?title=Principle_of_virtual_workhttp://en.wikipedia.org/w/index.php?title=Theodore_von_K%C3%A1rm%C3%A1nhttp://en.wikipedia.org/w/index.php?title=Infinitesimal_strain_theoryhttp://en.wikipedia.org/w/index.php?title=File%3APlaque_mince_deplacement_rotation_fibre_neutre_new.svghttp://en.wikipedia.org/w/index.php?title=Normal


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Plate theory 3

For moderate rotations, the strain-displacement relations take the von Karman form and the equilibrium equations

can be expressed as

Boundary conditionsThe boundary conditions that are needed to solve the equilibrium equations of plate theory can be obtained from the

boundary terms in the principle of virtual work.

For small strains and small rotations, the boundary conditions are

Note that the quantity is an effective shear force.

Stress-strain relations

The stress-strain relations for a linear elastic Kirchhoff plate are given by

Since and do not appear in the equilibrium equations it is implicitly assumed that these quantities do not

have any effect on the momentum balance and are neglected.

It is more convenient to work with the stress and moment results that enter the equilibrium equations. These are

related to the displacements by

and

The extensional stiffnesses are the quantities

The bending stiffnesses (also called flexural rigidity) are the quantities


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Plate theory 4

Isotropic and homogeneous Kirchhoff plate

For an isotropic and homogeneous plate, the stress-strain relations are

The moments corresponding to these stresses are

Pure bending

The displacements and are zero under pure bending conditions. For an isotropic, homogeneous plate under

pure bending the governing equation is

In index notation,

In direct tensor notation, the governing equation is

Transverse loading

For a transversely loaded plate without axial deformations, the governing equation has the form

where

In index notation,

and in direct notation

In cylindrical coordinates , the governing equation is
http://en.wikipedia.org/w/index.php?title=Pure_bending


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Plate theory 5

Orthotropic and homogeneous Kirchhoff plate

For an orthotropic plate

Therefore,

and

Transverse loading

The governing equation of an orthotropic Kirchhoff plate loaded transversely by a distributed load per unit area is

where

Dynamics of thin Kirchhoff plates

The dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves and

vibration modes.

Governing equations

The governing equations for the dynamics of a KirchhoffLove plate are

where, for a plate with density ,

and

The figures below show some vibrational modes of a circular plate.
http://en.wikipedia.org/w/index.php?title=Orthotropic_material


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Plate theory 6

mode k= 0,p = 1 mode k= 1,p = 2

Isotropic plates

The governing equations simplify considerably for isotropic and homogeneous plates for which the in-plane

deformations can be neglected and have the form

where is the bending stiffness of the plate. For a uniform plate of thickness ,

In direct notation

MindlinReissner theory for thick plates

Note: the Einstein summation convention of summing on repeated indices is used below.

In the theory of thick plates, or theory of Raymond Mindlin[4]

and Eric Reissner, the normal to the mid-surface

remains straight but not necessarily perpendicular to the mid-surface. If and designate the angles which the

mid-surface makes with the axis then

Then the MindlinReissner hypothesis implies that

Strain-displacement relations

Depending on the amount of rotation of the plate normals two different approximations for the strains can be derived

from the basic kinematic assumptions.

For small strains and small rotations the strain-displacement relations for MindlinReissner plates are

The shear strain, and hence the shear stress, across the thickness of the plate is not neglected in this theory. However,

the shear strain is constant across the thickness of the plate. This cannot be accurate since the shear stress is known
http://en.wikipedia.org/w/index.php?title=Eric_Reissnerhttp://en.wikipedia.org/w/index.php?title=Raymond_Mindlinhttp://en.wikipedia.org/w/index.php?title=Einstein_summation_conventionhttp://en.wikipedia.org/w/index.php?title=File%3ADrum_vibration_mode12.gifhttp://en.wikipedia.org/w/index.php?title=File%3ADrum_vibration_mode01.gif


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Plate theory 7

to be parabolic even for simple plate geometries. To account for the inaccuracy in the shear strain, a shear

correction factor ( ) is applied so that the correct amount of internal energy is predicted by the theory. Then

Equilibrium equationsThe equilibrium equations have slightly different forms depending on the amount of bending expected in the plate.

For the situation where the strains and rotations of the plate are smallthe equilibrium equations for a

MindlinReissner plate are

The resultant shear forces in the above equations are defined as

Boundary conditions

The boundary conditions are indicated by the boundary terms in the principle of virtual work.

If the only external force is a vertical force on the top surface of the plate, the boundary conditions are

Constitutive relationsThe stress-strain relations for a linear elastic MindlinReissner plate are given by

Since does not appear in the equilibrium equations it is implicitly assumed that it do not have any effect on the

momentum balance and is neglected. This assumption is also called the plane stress assumption. The remaining

stress-strain relations for an orthotropic material, in matrix form, can be written as

Then,

and
http://en.wikipedia.org/w/index.php?title=Orthotropic


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Plate theory 8

For the shear terms

The extensional stiffnesses are the quantities

The bending stiffnesses are the quantities

Isotropic and homogeneous Mindlin-Reissner plates

For uniformly thick, homogeneous, and isotropic plates, the stress-strain relations in the plane of the plate are

where is the Young's modulus, is the Poisson's ratio, and are the in-plane strains. The

through-the-thickness shear stresses and strains are related by

where is the shear modulus.

Constitutive relations

The relations between the stress resultants and the generalized displacements for an isotropic MindlinReissner plate

are:

and

The bending rigidity is defined as the quantity

For a plate of thickness , the bending rigidity has the form
http://en.wikipedia.org/w/index.php?title=Bending_rigidityhttp://en.wikipedia.org/w/index.php?title=Shear_modulus


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Plate theory 9

Governing equations

If we ignore the in-plane extension of the plate, the governing equations are

In terms of the generalized deformations , the three governing equations are

The boundary conditions along the edges of a rectangular plate are

ReissnerStein theory for isotropic cantilever plates

In general, exact solutions for cantilever plates using plate theory are quite involved and few exact solutions can be

found in the literature. Reissner and Stein[5]

provide a simplified theory for cantilever plates that is an improvement

over older theories such as Saint-Venant plate theory.

The Reissner-Stein theory assumes a transverse displacement field of the form

The governing equations for the plate then reduce to two coupled ordinary differential equations:

where

At , since the beam is clamped, the boundary conditions are

The boundary conditions at are

where


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Plate theory 10

Derivation of ReissnerStein cantilever plate equations The strain energy of bending of a thin rectangular plate of

uniform thickness is given by

where is the transverse displacement, is the length, is the width, is the Poisson's ratio, is the Young's

modulus, and

The potential energy of transverse loads (per unit length) is

The potential energy of in-plane loads (per unit width) is

The potential energy of tip forces (per unit width), and bending moments and (per unit

width) is

A balance of energy requires that the total energy is

With the ReissenerStein assumption for the displacement, we have

and


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Plate theory 11

Taking the first variation of with respect to and setting it to zero gives us the Euler equations

and

where

Since the beam is clamped at , we have The boundary conditions at can be found by integration by

parts: where

References

[1] Timoshenko, S. and Woinowsky-Krieger, S. "Theory of plates and shells". McGrawHill New York, 1959.

[2] A. E. H. Love, On the small free vibrations and deformations of elastic shells, Philosophical trans. of the Royal Society (London), 1888, Vol.

srie A, N 17 p. 491549.

[3] Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis.

[4] R. D. Mindlin,Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics, 1951,

Vol. 18 p. 3138.

[5][5] E. Reissner and M. Stein. Torsion and transverse bending of cantilever plates. Technical Note 2369, National Advisory Committee for

Aeronautics,Washington, 1951.


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Article Sources and Contributors 12

Article Sources and ContributorsPlate theorySource: http://en.wikipedia.org/w/index.php?oldid=534223791 Contributors: Bbanerje, Buenasdiaz, Dhollm, Gpayette, Hmains, Kahoolie, Mcapdevila, Mjohnrussell, Myasuda,

Nicoguaro, OnePt618, Pdcook, R'n'B, 6 anonymous edits

Image Sources, Licenses and ContributorsImage:ESPIvibration.jpgSource: http://en.wikipedia.org/w/index.php?title=File:ESPIvibration.jpg License: Public Domain Contributors: Epzcaw

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