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Deformation of a thin plate highlightingthe displacement, the mid­surface (red) andthe normal to the mid­surface (blue)

Kirchhoff–Love plate theoryFrom Wikipedia, the free encyclopedia

The Kirchhoff–Love theory of plates is a two­dimensional mathematical model that is used to determine thestresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler­Bernoulli beam theory and was developed in 1888 by Love[1] using assumptions proposed by Kirchhoff. Thetheory assumes that a mid­surface plane can be used to represent a three­dimensional plate in two­dimensionalform.

The following kinematic assumptions that are made in this theory:[2]

straight lines normal to the mid­surface remain straight after deformationstraight lines normal to the mid­surface remain normal to the mid­surface after deformationthe thickness of the plate does not change during a deformation.

Contents

1 Assumed displacement field2 Quasistatic Kirchhoff­Love plates

2.1 Strain­displacement relations2.2 Equilibrium equations2.3 Boundary conditions2.4 Constitutive relations2.5 Small strains and moderate rotations

3 Isotropic quasistatic Kirchhoff­Love plates3.1 Pure bending3.2 Bending under transverse load3.3 Cylindrical bending

4 Dynamics of Kirchhoff­Love plates4.1 Governing equations4.2 Isotropic plates

5 References6 See also

Assumed displacement field

Let the position vector of a point in the undeformed plate be . Then

The vectors form a Cartesian basis with origin on the mid­surface of the plate, and are the Cartesian coordinates on the mid­surface of theundeformed plate, and is the coordinate for the thickness direction.

Let the displacement of a point in the plate be . Then

This displacement can be decomposed into a vector sum of the mid­surface displacement and an out­of­plane displacement in the direction. We canwrite the in­plane displacement of the mid­surface as

Note that the index takes the values 1 and 2 but not 3.

Then the Kirchhoff hypothesis implies that

If are the angles of rotation of the normal to the mid­surface, then in the Kirchhoff­Love theory

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Displacement of the mid­surface (left) and of a normal (right)

Note that we can think of the expression for as the first order Taylor series expansion of the displacement around the mid­surface.

Quasistatic Kirchhoff­Love plates

The original theory developed by Love was valid for infinitesimal strains and rotations. The theory was extended by von Kármán to situations wheremoderate rotations could be expected.

Strain­displacement relations

For the situation where the strains in the plate are infinitesimal and the rotations of the mid­surface normals are less than 10° the strain­displacement relationsare

Using the kinematic assumptions we have

Therefore the only non­zero strains are in the in­plane directions.

Equilibrium equations

The equilibrium equations for the plate can be derived from the principle of virtual work. For a thin plate under a quasistatic transverse load theseequations are

where the thickness of the plate is . In index notation,

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where are the stresses.

Bending moments and normal stresses Torques and shear stresses

Derivation of equilibrium equations for small rotationsFor the situation where the strains and rotations of the plate are small the virtual internal energy is given by

where the thickness of the plate is and the stress resultants and stress moment resultants are defined as

Integration by parts leads to

The symmetry of the stress tensor implies that . Hence,

Another integration by parts gives

For the case where there are no prescribed external forces, the principle of virtual work implies that . The equilibriumequations for the plate are then given by

If the plate is loaded by an external distributed load that is normal to the mid­surface and directed in the positive direction,the external virtual work due to the load is

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The principle of virtual work then leads to the equilibrium equations

Boundary conditions

The boundary conditions that are needed to solve the equilibrium equations of plate theory can be obtained from the boundary terms in the principle of virtualwork. In the absence of external forces on the boundary, the boundary conditions are

Note that the quantity is an effective shear force.

Constitutive relations

The stress­strain 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 balanceand are neglected. The remaining stress­strain relations, in matrix form, can be written as

Then,

and

The extensional stiffnesses are the quantities

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

The Kirchhoff­Love constitutive assumptions lead to zero shear forces. As a result, the equilibrium equations for the plate have to be used to determine theshear forces in thin Kirchhoff­Love plates. For isotropic plates, these equations lead to

Alternatively, these shear forces can be expressed as

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where

Small strains and moderate rotations

If the rotations of the normals to the mid­surface are in the range of 10 to 15 , the strain­displacement relations can be approximated as

Then the kinematic assumptions of Kirchhoff­Love theory lead to the classical plate theory with von Kármán strains

This theory is nonlinear because of the quadratic terms in the strain­displacement relations.

If the strain­displacement relations take the von Karman form, the equilibrium equations can be expressed as

Isotropic quasistatic Kirchhoff­Love plates

For an isotropic and homogeneous plate, the stress­strain relations are

The moments corresponding to these stresses are

In expanded form,

where for plates of thickness . Using the stress­strain relations for the plates, we can showthat the stresses and moments are related by

At the top of the plate where , the stresses are

Pure bending

For an isotropic and homogeneous plate under pure bending, the governing equations reduce to

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Here we have assumed that the in­plane displacements do not vary with and . In index notation,

and in direct notation

The bending moments are given by

Derivation of equilibrium equations for pure bendingFor an isotropic, homogeneous plate under pure bending the governing equations are

and the stress­strain relations are

Then,

and

Differentiation gives

and

Plugging into the governing equations leads to

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Since the order of differentiation is irrelevant we have , , and . Hence

In direct tensor notation, the governing equation of the plate is

where we have assumed that the displacements are constant.

Bending under transverse load

If a distributed transverse load is applied to the plate, the governing equation is . Following the procedure shown in the previoussection we get[3]

In rectangular Cartesian coordinates, the governing equation is

and in cylindrical coordinates it takes the form

Solutions of this equation for various geometries and boundary conditions can be found in the article on bending of plates.

Derivation of equilibrium equations for transverse loadingFor a transversely loaded plate without axial deformations, the governing equation hasthe form

where is a distributed transverse load (per unit area). Substitution of the expressions forthe derivatives of into the governing equation gives

Noting that the bending stiffness is the quantity

we can write the governing equation in the form

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In cylindrical coordinates ,

For symmetrically loaded circular plates, , and we have

Cylindrical bending

Under certain loading conditions a flat plate can be bent into the shape of the surface of a cylinder. This type of bending is called cylindrical bending andrepresents the special situation where . In that case

and

and the governing equations become[3]

Dynamics of Kirchhoff­Love plates

The dynamic theory of thin 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

Derivation of equations governing the dynamics of Kirchhoff­Love plates

The total kinetic energy of the plate is given by

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Therefore the variation in kinetic energy is

We use the following notation in the rest of this section.

Then

For a Kirchhof­Love plate

Hence,

Define, for constant through the thickness of the plate,

Then

Integrating by parts,

The variations and are zero at and . Hence, after switching the sequence of integration, we have

Integration by parts over the mid­surface gives

Again, since the variations are zero at the beginning and the end of the time interval under consideration, we have

For the dynamic case, the variation in the internal energy is given by

Integration by parts and invoking zero variation at the boundary of the mid­surface gives

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If there is an external distributed force acting normal to the surface of the plate, the virtual external work done is

From the principle of virtual work . Hence the governing balance equations for the plate are

Solutions of these equations for some special cases can be found in the article on vibrations of plates. The figures below show some vibrational modes of acircular plate.

mode k = 0, p = 1

mode k = 0, p = 2

mode k = 1, p = 2

Isotropic plates

The governing equations simplify considerably for isotropic and homogeneous plates for which the in­plane deformations can be neglected. In that case weare left with one equation of the following form (in rectangular Cartesian coordinates):

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

In direct notation

For free vibrations, the governing equation becomes

Derivation of dynamic governing equations for isotropic Kirchhoff­Love plates

For an isotropic and homogeneous plate, the stress­strain relations are

where are the in­plane strains. The strain­displacement relations for Kirchhoff­Love plates are

Therefore, the resultant moments corresponding to these stresses are

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The governing equation for an isotropic and homogeneous plate of uniform thickness in the absenceof in­plane displacements is

Differentiation of the expressions for the moment resultants gives us

Plugging into the governing equations leads to

Since the order of differentiation is irrelevant we have . Hence

If the flexural stiffness of the plate is defined as

we have

For small deformations, we often neglect the spatial derivatives of the transverse acceleration of the plateand we are left with

Then, in direct tensor notation, the governing equation of the plate is

References1. A. E. H. Love, On the small free vibrations and deformations of elastic shells, Philosophical trans. of the Royal Society (London), 1888, Vol. série A, N° 17 p. 491–

549.2. Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis.3. Timoshenko, S. and Woinowsky­Krieger, S., (1959), Theory of plates and shells, McGraw­Hill New York.

See also

BendingBending of platesInfinitesimal strain theory

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Linear elasticityPlate theoryStress (mechanics)Stress resultantsVibration of plates

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