Error estimation for geometrically non-linear analysis of framed structures

This paper is concerned with the estimation of errors that arise in the geometrically non-linear analysis of framed structures using 1D beam-column finite elements. A quartic element is used to illustrate specific issues of error estimation relating to geometrically non-linear analysis, this element exhibiting most of the sources of error that arise with other 1D elements. After a brief description of the quartic element, the sources of error that it manifests in geometrically non-linear analysis are highlighted. The first source of error is due to equilibrium defaults, including contributions from finite element discretization and from the approximation of equivalent loads, whereas the second source of error is due to compatibility defaults. Next, a stress recovery method, termed the ‘Exact equilibrium recovery’ (EER) method, is proposed for the estimation of equilibrium errors due to finite element discretization. This method, specifically developed for framed structures, is element based and is shown to be more effective than other patch-based recovery methods. An overall error estimation method is then proposed, providing separate yet combinable measures for equilibrium and compatibility errors. Finally, several examples are presented to demonstrate the effectiveness of the proposed error estimation method in the geometrically non-linear analysis of framed structures, illustrating also the relative importance of the various sources of error that are manifested in typical problems