Finite element mesh orientation has been studied in terms of the effects it may have on solution accuracy in both solid and fluid mechanics. This work studies the effect of triangular finite element mesh orientation with regards to the torsion of non-circular machine elements and structural elements in terms of maximum shearing stress computation accuracy. The membrane deflection (a Poisson problem) analogy is used to guide the automatic mesh orientation by quadrilateral diagonal swapping, starting from an initially arbitrary mesh. Two sets of results for two specific orientations are reported: mesh orientation approximately perpendicular to the membrane contour lines and mesh orientation in a direction approximately parallel to the membrane contour lines. The torsion of a square cross section member was selected as a benchmark for the analysis since it possesses a known analytic series solution that readily allows the performing of accuracy estimates. This study demonstrates that mesh orientation may have important consequences on the accuracy of the solution for the type of stress estimates treated in this work. In a more general sense, it is concluded that mesh orientation may have important consequences on the accuracy of the solution in second order Poisson type problems as well. © 2005 Elsevier B.V. All rights resersved.