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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*(1nu^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 16gauge 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 boundaryvalue 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 }
Variables U(0.1) V(0.1)
Definitions 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*(1nu^2))
Initial Values U = 0 V = 0
Equations U: del2(U) = V V: del2(V) = Q/D 
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