Classic Plate Equation: Introduction
Classical Plate Equation
The small transverse (out-of-plane) displacement w of a thin plate is governed by the Classical Plate Equation,
where p is the distributed load (force per unit area) acting in the same direction as z (and w), and D is the bending/flexual rigidity of the plate defined as follows,
in which E is the Young's modulus, is the Poisson's ratio of the plate material, and t is the thickness of the plate.
Furthermore, the differential operator is called the Laplacian differential operator ,
If the bending rigidity D is constant throughout the plate, the plate equation can be simplified to,
where is called the bihamonic differential operator.
|Origin of the Plate Equation|
The outcome of each of these segments is summarized here:
where w0 is the displacement of the middle plane in z direction.
|To relate the plate's out-of-plane displacement w0 to its pressure loading pz, we combine the results of the four plate subcategories in this order:|
The plate is assumed to be constructed by isotropic material and subjected to transverse loading. Also, the Cartesian coordinate system is used.
We'll demonstrate this hierarchy by working backwards. We first combine the 3 equilibrium equations to eliminate Qxz and Qyz,
Next, replace the moment resultants with its definition in terms of the direct stress,
Note that uniform thickness is assumed.
Use the constitutive relation to eliminate stress in favor of the strain,
and then use kinematics to replace strain in favor of the normal displacement w0,
The equation of equilibrium can then be expressed in terms of the normal displacement w0
Note that homogeneous material across the plate (x and y directions) is assumed.
As a final step, assuming homogeneous material along the thickness of the plate, the bending stiffness of the plate can be written as
We then arrive at the Classical Plate equation,
or a slimmer form
where w0 is replaced by w and pz replaced by p to be consistent with the notations in most published literatures.
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