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.
Of the numerous plate theories that have been developed since the late 19th century, two are widely accepted and used in engineering. These are
Kirchhoff–Love theory for thin plates.
The Kirchhoff–Love 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 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:
Displacement field.
The Kirchhoff hypothesis implies that the displacement field has the form
where formula_1 and formula_2 are the Cartesian coordinates on the mid-surface of the undeformed plate, formula_3 is the coordinate for the thickness direction, formula_4 are the in-plane displacements of the mid-surface, and formula_5 is the displacement of the mid-surface in the formula_3 direction.
If formula_7 are the angles of rotation of the normal to the mid-surface, then in the Kirchhoff–Love theory
formula_8
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 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 formula_13. The quantities formula_14 are the stresses.
If the plate is loaded by an external distributed load formula_15 that is normal to the mid-surface and directed in the positive formula_3 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 formula_19 is an effective shear force.
Stress–strain relations.
The stress–strain relations for a linear elastic Kirchhoff plate are given by
Since formula_21 and formula_22 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 resultants 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 formula_29 and formula_30 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 formula_36, the governing equation is
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 formula_41 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 formula_44,
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 formula_48 is the bending stiffness of the plate. For a uniform plate of thickness formula_13,
In direct notation
Uflyand-Mindlin theory for thick plates.
In the theory of thick plates, or theory of Yakov S. Uflyand (see, for details, Elishakoff's handbook), Raymond Mindlin and Eric Reissner, the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. If formula_51 and formula_52 designate the angles which the mid-surface makes with the formula_3 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 (formula_56) 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 small the 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 formula_22 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
For the shear terms
The extensional stiffnesses are the quantities
The bending stiffnesses are the quantities
Isotropic and homogeneous Uflyand-Mindlin plates.
For uniformly thick, homogeneous, and isotropic plates, the stress–strain relations in the plane of the plate are
where formula_69 is the Young's modulus, formula_70 is the Poisson's ratio, and formula_71 are the in-plane strains. The through-the-thickness shear stresses and strains are related by
where formula_73 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 formula_78, the bending rigidity has the form
where 
formula_80
Governing equations.
If we ignore the in-plane extension of the plate, the governing equations are
In terms of the generalized deformations formula_82, the three governing equations are
The boundary conditions along the edges of a rectangular plate are
Reissner–Stein static 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
The governing equations for the plate then reduce to two coupled ordinary differential equations:
where
At formula_86, since the beam is clamped, the boundary conditions are
The boundary conditions at formula_88 are
where