Basic Fundamental Approach of 2nd Order Geometric Nonlinear Analysis: Concepts and Computational Implementation

Resumo

One of the main objectives of structural engineering has been to make the structures slenderer and more
economical, reducing their weight and the consumption of materials without, however, compromising their
stability. The increase in the slenderness of the structural elements makes them more susceptible to large lateral
displacements before their rupture occurs. The stability analysis of slender structural systems currently involves
the application of the Finite Element Method (FEM). As a consequence, a system of non-linear algebraic equations
is generated and its solution is obtained, in general, through incremental-iterative procedures. This article initially
presents structural single-degree-of-freedom systems (SDF, scalar variables), subjected to geometric nonlinear
behavior, showing their analytical and numerical solutions, using the Principle of Stationary Total Potential
Energy. Four SDF systems, with different behaviors, are presented: stable or unstable post-critical behavior and
with and without bifurcation. Geometric imperfections are incorporated. The work ends with the presentation of
two-degree-of-freedom systems, where the variables become matrix and vector, expanding the concepts of SDF
systems for the multi-degree-of-freedom (MDF) ones.