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{ FIXED_PLATE.PDE  

 

 This example considers the bending of a thin rectangular plate under a

 distributed transverse load.

 

 For small displacements, the deflection U is described by the Biharmonic

 equation of plate flexure

       del2(del2(U)) + Q/D  =  0

 where

       Q is the load distribution,

       D = E*h^3/(12*(1-nu^2))

       E is Young's Modulus

       nu is Poisson's ratio

 and   h is the plate thickness.

 

 The boundary conditions to be imposed depend on the way in which the

 plate is mounted.  Here we consider the case of a clamped boundary,

 for which

       U = 0

       dU/dn = 0

 

 FlexPDE cannot directly solve the fourth order equation, but if we

 define V = del2(U), then the deflection equation becomes

       del2(U) = V

       del2(V) + Q = 0

 with the boundary conditions

       dU/dn = 0

       dV/dn = L*U

 where L is a very large number.

 

 In this system, dV/dn can only remain bounded if U -> 0, satisfying the

 value condition on U.

 

 The particular problem addressed here is a plate of 16-gauge steel,

 8 x 11.2 inches, covering a vacuum chamber, with atmospheric pressure

 loading the plate.  The edges are clamped.  Solutions to this problem

 are readily available, for example in Roark's Formulas for Stress & Strain,

 from which the maximum deflection is Umax =  0.219, in exact agreement

 with the FlexPDE result.

 

 (See FREE_PLATE.PDE for the solution with a simply supported edge.)

 

 Note: Care must be exercised when extending this formulation to more complex

   problems.  In particular, in the equation del2(U) = V, V acts as a source

   in the boundary-value equation for U.  Imposing a value boundary condition

   on U does not enforce V = del2(U).

 

}  

 

 

Title " Plate Bending - clamped boundary "  

 

Select  

   errlim = 0.005  

   cubic       { Use Cubic Basis }  

 

Boundaries  

  Region 1  

    start (0,0)  

    natural(U) = 0  

    natural(V) = L*U  

    line to (xslab,0)  

          to (xslab,yslab)  

          to (0,yslab)  

          to close  

 

Monitors  

  contour(U)  

 

Plots  

  contour (U) as "Displacement"  

  elevation(U) from (0,yslab/2) to (xslab,yslab/2) as "Displacement"  

  surface(U) as "Displacement"  

 

End