The New Approach to Analysis of Thin Isotropic Symmetrical Plates

A new approach to solve plate constructions using combined analytical and numerical methods has been developed in this paper. It is based on an exact solution of an equilibrium equation. The proposed mathematical model is implemented as a computer program in which known analytical formulae are rewritten as wrapper functions of two arguments. Partial derivatives are calculated using automatic differentiation. A solution of a system of linear equations is substituted to these functions and evaluated using the Einstein summation convention. The calculated results are presented and compared to other analytical and numerical ones. The boundary conditions are satisfied with high accuracy. The effectiveness of the present method is illustrated by examples of rectangular plates. The model can be extended with the ability to solve plates of any shape.

Pure torsion problem in tensor notation

The paper examines the application of the tensor calculus to the classic problem of the pure torsion of prismatic rods. The introduction contains a short description of the reference frames, base vectors, contravariant and covariant vector coordinates when applying the Einstein summation convention. Torsion formulas were derived according to Coulomb’s and Saint-Venant’s theories, while, as a link between the theories, so-called Navier’s error was discussed. Groups of the elasticity theory equations were used.

Notation

This chapter explains the notation for the basic construction of the correction. It employs the Einstein summation convention, according to which there is an implied summation when a pair of indices is repeated, and the conventions of abstract index notation, so that upper indices and lower indices distinguish contravariant and covariant tensors. It also presents the notation concerning multi-indices which will later prove helpful for expressing higher order derivatives of a composition. In this notation, a K-tuple of multi-indices is said to form an ordered K-partition of a multi-index if there is a partition whereby the subsets are pairwise disjoint and are ordered by their largest elements.