Classic Plate Equation: Kinematics

Kinematics

Kinematics describes how the plate's displacements and strains relate:

where u, v, and w are the displacement in x, y, and z direction, respectively.

Kinematics for Classical Plates

The above equations are too general to be useful. A few assumptions on how a plate's cross section rotates and twists need to be made in order to simplify the problem. For the classical plate, the assumptions were given by Kirchoff and dictate how the 'normals' behave (normals are lines perpendicular to the plate's middle plane and are thus embedded in the plate's cross sections).

Kirchhoff Assumptions
 1. Normals remain straight (they do not bend)
 2. Normals remain unstretched (they keep the same length)
 3. Normals remain normal (they always make a right angle to the middle plane)

Based on these assumptions, the displacement field can be expressed in terms of the distances by which the plate's middle plane moves from its resting (unloaded) position, u0, v0, and w0 and the rotations of the plate's middle plane, , , and .

With the normals straight and unstretched, we can safely assume that the shear strain in the z direction is negligible:

Using the assumption that the normals remain normal to the midplane, we can make the x and y dependance in u(x,y,z), v(x,y,z) explicit via a simple geometric expression,

The kinematics equations therefore becomes

where the strains of the middle plane are

and the curvatures (changes of slope) of the middle plane are

Note that if there are no in-plane resultants, all strains at the middle plane are zero. Thus

 

 

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