Plates - The differential equation

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The goal here is to establish an equation from the basic assumptions and principles for a thin plate that were outlined on the page prior to this. We seek a connection between lateral deformation and loads on the form:

w(x,y)=F(p(x,y))

As for the shell problem, we base the first step on Hooke's law:


We can express the strains as


because Kirchoff's hypotesis states that:

  • A straight line who prior to defomation was normal to the middle plane remains straight, also after deformation.
  • All points will keep their initial distance from the neutral axis after deformation.
Combining (1) og (2) gives us


We know that e.g. the moment in x-direction the x-stress times the distance from the middle integrated over the entire thickness (z-direction), which gives us the moments as a pure function of the displacement, w:


It is practical to define the plate stifness, D:


and we can then write the moments as:


where
w,xx is the curving/camber in x-direction
w,yy is the curving/camber in y-direction
w,xy is the torsion/twist of the plate

Utlizing (6) we can now determine the shear forces as funtions of w:


And by introducing the differential operator:


we get


and for the y-shear we get the equivalent:


Equilibrium of the plate is satisfied by:


and by inserting (9) and (10) we get:


Again, we utilize the differential operator:


The differential equation can now be written as:


This is the plates differential equation, a linear, 4.order, inhomogeneous, partial differential equation which contains all the requirements of equilibrium and kinematic compatibility.