secondary torsional moment deformation effect by bem

in this paper, a boundary element method is developed for the nonuniform torsion of simply or multiply connected bars of doubly symmetrical arbitrary constant cross section, taking into account secondary torsional moment deformation effect. the bar is subjected to arbitrarily distributed or concentrated twisting and warping moments, while its edges are restrained by the most general torsional boundary conditions. to account for secondary shear deformations, the concept of shear deformation coefficient is used leading to a secondary torsion constant. four boundary value problems with respect to the variable along the bar primary and secondary angles of twist and to the primary and secondary warping functions are formulated and solved employing a pure bem approach, that is only boundary discretization is used. the warping and the primary torsion constants are evaluated employing the aforementioned primary and secondary warping functions using only boundary integration, while the secondary torsion constant is computed employing an effective automatic domain integration. numerical examples with great practical interest are worked out to illustrate the efficiency and the range of applications of the developed method. the influence of the secondary torsional moment deformation effect of closed shaped cross sections is verified, while the accuracy of the proposed numerical procedure for the calculation of the secondary torsion constant compared with a fem one is noteworthy.