Kirchhoff–Love plate theory

KirchhoffLove plate theory

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

Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue)

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 mid-surface of the plate, and are the Cartesian coordinates on the mid-surface of the undeformed 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 mid-surface displacement and an out-of-plane displacement in the direction. We can write the in-plane displacement of the mid-surface 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 mid-surface, then in the Kirchhoff-Love theory

Note that we can think of the expression for the mid-surface.

as the first order Taylor series expansion of the displacement around

KirchhoffLove plate theory

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Quasistatic Kirchhoff-Love plates The original theory developed by Love was valid for infinitesimal strains and rotations. The theory was extended by von Kármán to situations where moderate rotations could be expected.

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 strain-displacement relations are Wikipedia:Please clarify

Using the kinematic assumptions we have

Therefore the only non-zero strains are in the in-plane 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 these equations are

where the thickness of the plate is

where

are the stresses.

. In index notation,

KirchhoffLove plate theory

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Bending moments and normal stresses

Torques and shear stresses

Derivation of equilibrium equations for small rotations For 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 equilibrium equations for the plate are then given by

. The

KirchhoffLove plate theory

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If the plate is loaded by an external distributed load positive

that is normal to the mid-surface and directed in the

direction, the external virtual work due to the load is

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 virtual work. 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 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. The remaining stress-strain 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

KirchhoffLove plate theory

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

Alternatively, these shear forces can be expressed as

where

Small strains and moderate rotations 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 as

Then the kinematic assumptions of Kirchhoff-Love theory lead to the classical plate theory with von Kármán strains

This theory is nonlinear because of the quadratic terms in the strain-displacement relations. If the strain-displacement relations take the von Karman form, the equilibrium equations can be expressed as

Isotropic quasistatic Kirchhoff-Love plates For an isotropic and homogeneous plate, the stress-strain relations are

The moments corresponding to these stresses are

In expanded form,

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KirchhoffLove plate theory

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where

for plates of thickness

stress-strain relations for the plates, we can show that 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

Here we have assumed that the in-plane 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 bending For an isotropic, homogeneous plate under pure bending the governing equations are

and the stress-strain relations are

Then,

and

. Using the

KirchhoffLove plate theory

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Differentiation gives

and

Plugging into the governing equations leads to

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 previous section we get

[3]

In rectangular Cartesian coordinates, the governing equation is

and in cylindrical coordinates it takes the form

.

KirchhoffLove plate theory

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

where

is a distributed transverse load (per unit area). Substitution of the expressions for the derivatives of

into the governing equation

gives

Noting that the bending stiffness is the quantity

we can write the governing equation in the form

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 and represents the special situation where . In that case

and

and the governing equations become[3]

KirchhoffLove plate theory

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Dynamics of Kirchhoff-Love 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 Kirchhoff-Love plate are

where, for a plate with density

,

and

Derivation of equations governing the dynamics of Kirchhoff-Love plates The total kinetic energy of the plate is given by

Therefore the variation in kinetic energy is

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

Then

For a Kirchhof-Love plate

Hence,

Define, for constant

Then

Integrating by parts,

through the thickness of the plate,

KirchhoffLove plate theory

The variations

and

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are zero at

and

. Hence, after switching the sequence of integration,

we have

Integration by parts over the mid-surface 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 mid-surface gives

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 a circular plate.

mode k = 0, p = 1

mode k = 0, p = 2

mode k = 1, p = 2

KirchhoffLove plate theory

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Isotropic plates The governing equations simplify considerably for isotropic and homogeneous plates for which the in-plane deformations can be neglected. In that case we are 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 Kirchhoff-Love plates For an isotropic and homogeneous plate, the stress-strain relations are

where

are the in-plane strains. The strain-displacement relations for Kirchhoff-Love plates are

Therefore, the resultant moments corresponding to these stresses are

The governing equation for an isotropic and homogeneous plate of uniform thickness displacements is

Differentiation of the expressions for the moment resultants gives us

Plugging into the governing equations leads to

in the absence of in-plane

KirchhoffLove plate theory

Since the order of differentiation is irrelevant we have

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. 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 plate and we are left with

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

References [1] 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 Woinowsky-Krieger, S., (1959), Theory of plates and shells, McGraw-Hill New York.

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