Plate theory - AbsoluteAstronomy.com

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams

Bending

In engineering mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically...

. 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

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

problem to a two-dimensional problem. The aim of plate theory is to calculate the deformation

Deformation

In materials science, deformation is a change in the shape or size of an object due to an applied force or a change in temperature...

and stresses in a plate subjected to loads.

Of the numerous plate theories that have been developed since the late 19th century, two are widely accepted and used in engineering. These are

the Kirchhoff
Gustav Kirchhoff
Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...

–Love
Augustus Edward Hough Love
Augustus Edward Hough Love FRS , often known as A. E. H. Love, was a mathematician famous for his work on the mathematical theory of elasticity...

theory of plates (classical plate theory)
The Mindlin–Reissner theory of plates (first-order shear plate theory)

Kirchhoff–Love theory for thin plates

The Kirchhoff

Gustav Kirchhoff

Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...

–Love

Augustus Edward Hough Love

Augustus Edward Hough Love FRS , often known as A. E. H. Love, was a mathematician famous for his work on the mathematical theory of elasticity...

theory is an extension of Euler–Bernoulli beam theory to thin plates. The theory was developed in 1888 by Love. 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:

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

Displacement (vector)

A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...

field has the form

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

Normal

Normal may refer to:* Normality , conformance to an average* Norm , social norms, expected patterns of behavior studied within the context of sociology* Normal distribution , the Gaussian continuous probability distribution...

to the mid-surface, then in the Kirchhoff–Love theory

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 Kármán

Theodore von Karman

Theodore von Kármán was a Hungarian-American mathematician, aerospace engineer and physicist who was active primarily in the fields of aeronautics and astronautics. He is responsible for many key advances in aerodynamics, notably his work on supersonic and hypersonic airflow characterization...

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

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

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.

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

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

Pure bending

Pure bending is a condition of stress where a bending moment is applied to a beam without the simultaneous application of axial, shear, or torsional forces....

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

Orthotropic and homogeneous Kirchhoff plate

For an orthotropic

Orthotropic material

An orthotropic material has two or three mutually orthogonal twofold axes of rotational symmetry so that its mechanical properties are, in general, different along each axis. Orthotropic materials are thus anisotropic; their properties depend on the direction in which they are measured...

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 Kirchhoff–Love plate are

where, for a plate with density

and

The figures below show some vibrational modes of a circular plate.

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

Mindlin–Reissner theory for thick plates

In the theory of thick plates, or theory of Raymond Mindlin 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 Mindlin–Reissner 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 Mindlin–Reissner 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 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 equations

The 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 Mindlin–Reissner 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 relations

The stress-strain relations for a linear elastic Mindlin–Reissner 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

Orthotropic

Orthotropic may refer to:* Orthotropic material is one that has the different materials properties or strengths in different orthogonal directions...

material, in matrix form, can be written as

Then,

and

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 Mindlin–Reissner plate are:

and

The bending rigidity is defined as the quantity

For a plate of thickness

, the bending rigidity has the form

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

Reissner–Stein 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 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