Di!erential equations and boundary conditions which describe physical phenomena are often obtained from physical principles by means of the variational calculus techniques. The necessary conditions for the existence of extremes of a functional lead to the Euler di!erential equation which involves unnecessary derivatives of higher order than the order of the derivatives included in the functional. Since this functional describes a certain type of energy, it is more natural, from a physical point of view, to look for a weak solution of the problem under consideration than to "nd its classical solution which does not exist for many common industrial problems.
A weak solution of a boundary value or eigenvalue problem may be obtained, under rather natural assumptions for the parameters of the problem, by variational methods. Let G be a domain in R with a piecewise smooth boundary "*G and the operator
Au"
" "," ")2 (!1) " " D? (a ?@ (x)D@u(x))"
" ", " ")2
(!1) " " * " " *x? *x?
where
with u"v or w.
Taking into account the boundary conditions (6b)}(6e) and the fact that since v3<, is v"0 in ยธ( ), we get (see Figure 1) * *x P(w)vn ds! P(w) *v *x n ds# * *x Q(w)vn ds! Q(w) *v *x n ds # * *x R(w)vn ds! R(w) *v *x n ds"r *v *x *w *x ds#r *v *x *w *x ds #r *v *x *w *x ds#r *v *x *w *x ds Then, we have
The double integral in equation ( 9) constitutes the bilinear form A(v, w) associated with the di!erential operator A de"ned in equation ( 7), and the curvilinear integrals constitute the boundary bilinear form a(v, w). The equality (8) now assumes the form B(v, w)" % q v dx"(q, v) ยธ(G) . LETTERS TO THE EDITOR #2B *w *w *w *x #4B *w *x *w *x *w #B *w *x #4B *w *x *w *x *x #4B *w *x *x *C x *w *x #2 *w *x *x # *w *x . where the constants B GH are expressed by the coe$cients of the sti!ness matrix. The integration in the plate volume, in the case of constant thickness h, leads to % D *w *x #2D *w *x *w *x #4D *w *x *x #D *w *x #4D *w *x *w *x *x #4D *w *x *x dx dx * C h 12 % *w *x #2 *w *x *x # *w *x dx dx . #2D *v *x *v *x #4D *v *x *v *x *x #D *v *x #4D *v *x *v *x *x #4D *v *x *x dx Pr *v *x dS#r *v *x ds#r *v *x ds#r *v *x ds!2 % qv dx.