Plate theory

Plates are defined as plane structural elements with a small thickness compared to the planar dimensions.

[citation needed] 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.

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

The theory was developed in 1888 by Love[2] 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 are made in this theory:[3] The Kirchhoff hypothesis implies that the displacement field has the form where

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.

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

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 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.

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 For an isotropic and homogeneous plate, the stress–strain relations are The moments corresponding to these stresses are The displacements

For an isotropic, homogeneous plate under pure bending the governing equation is In index notation, In direct tensor notation, the governing equation is

, the governing equation is For an orthotropic plate Therefore, and The governing equation of an orthotropic Kirchhoff plate loaded transversely by a distributed load

The governing equations simplify considerably for isotropic and homogeneous plates for which the in-plane deformations can be neglected and have the form where

, In direct notation In the theory of thick plates, or theory of Yakov S. Uflyand[4] (see, for details, Elishakoff's handbook[5]), Raymond Mindlin[6] and Eric Reissner, the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface.

axis then Then the Mindlin–Reissner hypothesis implies that 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.

This cannot be accurate since the shear stress is known to be parabolic even for simple plate geometries.

) is applied so that the correct amount of internal energy is predicted by the theory.

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

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.

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

If we ignore the in-plane extension of the plate, the governing equations are In terms of the generalized deformations

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

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