Herleitung verbesserter hierarchischer Plattenmodelle und deren Integralgleichungsformulierung [online]

Abstract

The objective of this dissertation is the derivation of a
hierarchical formulation for the differential and integral
equations of a twelfth-order plate bending model for homogeneous
transversely isotropic materials, named after Poniatovskii and
Reissner. The hierachical characteristic means that the direct
reduction from the twelfth-order equation system automatically
leads to corresponding systems of lower-order. By the first
simplification we obtain the sixth-order equation system of
Reissner (and Mindlin) and, by taking the limit of the thickness
to zero or the shear stiffness to infinity, the fourth-order
Kirchhoff bending equation. The simplification from the
twelfth-order to the sixth-order system is performed by the
selection of particular values for the two coupling variables.
The differential equations of the twelfth-order model are
obtained by application of the Hellinger-Reissner mixed
variational principle. For the dimensional reduction we propose
as starting point an expansion of the displacements along the
plate thickness in the form of Legendre polynomials. In the
sequel we determine the corresponding stress field in order to
... mehrevaluate the expression for the complementary energy density to
be used in the variational principle.
The weighted residual method leads to the corresponding system
of hierarchical integral equations allowing for the simultaneous
BEM formulation of both, the sixth-order and the twelfth-order
plate models. It is shown that the system of differential
equations gives a better approximation of the three-dimensional
elasticity equations than displacement-based models. The
hierarchical reduction of the twelfth-order integral equations
yields the known sixth-order equations of van der Weeen, now for
transversely isotropic materials.