Shape functions based on Legendre Polynomials

The plate solution uses an energy method in which the displacement distributions are assumed functions. The assumed functions use a summation series with unknown coefficients. For example

w(x, y) = Underoverscript[∑, m = 0, arg3] Underoverscript[∑, n = 0, arg3] c_mn ψ_m(x) ψ_n(y)

where c_mn is an unknown coefficient to be determined through minimization of potential energy. ψ_n[x] is a function that satisfies the required displacement boundary conditions.

The code a assumed functions that are constructed from Legendre polynomials. Following  Szabo and Babuska ("Finite Element Analysis", pg 38, Wiley, 1991), we take the first two terms of the series as

ψ_0(x) = (1 - ζ)/2 ; ψ_1(x) = (1 + ζ)/2 ;

Where in the local coordinate system, ζ has the range -1/2 to +1/2. The remaining terms of the series have the form

ψ_j(ζ) = (P_j(ζ) - P_ (j - 2)(ζ))/(2 (2 j - 1))^(1/2)

where P_j is a Legendre polynomial. The ψ functions have the characteristic of passing through zero at the endpoints. ψ_2 though ψ_4are shown in the following plot.

[Graphics:HTMLFiles/index_9.gif]

Because the higher order functions have a value of zero at the boundary, specified boundary conditions are governed by the the first two interpolation terms ψ_0and ψ_1.

Created by Mathematica  (July 9, 2004)