A novel weighted local averaging for the Galerkin method with application to elastic buckling of Euler column

Abstract

Abstract This paper presents a dual approach to the conventional averaging (CA) in order to construct a weighted averaging. A one-parameter weighting function for the novel weighted local averaging (WLA) is presented and analyzed in detail. The advantage of the proposed WLA is demonstrated by its application to the Galerkin method, which leads to a so-called the Galerkin method with weighted local averaging (GWLA). Application of the GWLA to the problem of elastic buckling of columns shows that the new idea can improve significantly the accuracy of the first order approximate solution of the Galerkin method. In order to solve the problem of selecting a specific weighting function among the classes of one-parameter weighting functions, the global-local approach is implemented. Further approximations resulting in the simplified GWLA (SGWLA) have been made to reduce the computation cost while still maintaining the accuracy of the solutions obtained by the GWLA. In addition, the effectiveness of the WLA is demonstrated by its combination with the least squares method to transform column with variable cross-section into equivalent column with constant cross-section. Numerical calculations show that the approximate critical buckling loads obtained by the newly developed GWLA and SGWLA outperform those obtained by the Galerkin method with conventional averaging (GCA). These new numerical algorithms could provide a novel and potential effective alternative tool for engineering calculation in designing structures with varying cross-sections.