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 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


       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 "


   errlim = 0.005

   cubic       { Use Cubic Basis }






   xslab = 11.2

   yslab = 8

   h = 0.0598 {16 ga}

   L = 1.0e4

   E = 29e6

   Q = 14.7

   nu = .3

   D = E*h^3/(12*(1-nu^2))

Initial Values

   U =  0

   V =  0


   U:  del2(U) = V

   V:  del2(V) = Q/D


  Region 1

    start (0,0)

    natural(U) = 0

    natural(V) = L*U

    line to (xslab,0)

          to (xslab,yslab)

          to (0,yslab)

          to close




  contour (U) as "Displacement"

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

  surface(U) as "Displacement"