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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 }

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*(1-nu^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