A numerical analysis of Saint-Venant torsion in strain-gradient bars

The article addresses the torsion of bars with arbitrary cross sections subject to warping employing the finite element method in the strain-gradient elasticity. Subsequent to extracting the strains and stresses given by the Saint-Venant torsion hypothesis for an infinitesimal deformation, the principle of virtual work is adopted to investigate the weak form of stationary equation. Then, two torsion-warping elements, the former has four nodes and the latter has nine nodes with 6 DoF per node, are constructed for the torsion analysis of bars. To investigate the behaviour of the devised elements, some examples with known cross sections are presented which can be used to examine the accuracy of the developed elements and to study the size-dependent trend in the bars.