Microsoft Word - WCE2012_Finite differences method torsion_Lupoae_MTA revised

Torsion of cylindrical shafts has been a topic in the classical theory of elasticity for a long time (Timoshenko and Goodier, 1970). The stiffness of a cylindrical shaft under torsional loading is often of interest in the study of torsion problems. Therefore, the uniform (St. Venant) and non-uniform torsion problem of structural components has long been the subject of theoretical and practical study in the field of solid mechanics (Chen et al., 2001). Structural elements with very different cross sectional shapes are widely used in various engineering structures. The exact solutions for torsion have been found for some simple cross-sectional shapes such as circles, ellipses, and triangles. However, in the theory of elasticity, it is difficult to obtain analytical solutions for complicated cross-sections. To solve general cross-sectional problems, numerical methods are usually necessary. For more complicated shapes, numerical methods are usually employed. Examples include the finite difference method (Ely and Zienkiewicz, 1960), the finite element method (Herrmann, 1965; Karayannis, 1995; Li at al., 2000), and the boundary element method (Jawson and Ponter, 1963; Friedman and Kosmatka, 2000; Sapountzakis, 2001; Sapountzakis and Mokos, 2001, 2003).The first two of these methods require the whole cross-section to be discretised into elements or grids. For a complicated section, the finite difference method