The transformation matrix in the 7DOFs beam formulation

• Published in: Thin-walled structures Vol. 190; p. 110951
• Main Authors: Bernuzzi, Claudio, Simoncelli, Marco
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-09-2023
• Subjects: 7 degrees of freedom (7 DOFs); Bimoment; Mixed torsion; Non-bisymmetric cross-section; Transformation matrix; Warping effects; Transformation matrix; Non-bisymmetric cross-section; 7 degrees of freedom (7 DOFs); Warping effects; Mixed torsion; Bimoment
• DOI: 10.1016/j.tws.2023.110951
• EISSN: 1879-3223
• ISSN: 0263-8231

A non-negligible percentage of steel products for civil and industrial constructions is realised with members characterised by a mono-symmetric cross-section. The non-coincidence between the centroid and the shear centre reflects in a quite complex behaviour that is remarkably influenced by the warping effects. Usually, frame design is based on the output data of commercial Finite Element Analysis Packages (FEAPs) but only few of them offer a refined beam element formulation able to reproduce the response of non-bisymmetric cross-section members (like the 7 degrees of freedom, DOFs, beam element). As a consequence, warping effects are usually neglected in routine design, leading in several cases to assess non-correct internal stresses, global displacements and critical buckling multipliers.
This paper is focussed on mono-symmetric cross-section members and deals with the influence of the interaction between axial force, bending moments and bimoment in linear elastic range. In particular, the two strategies that FE developers usually adopt, in the 7 DOFs beam formulation, to pass from the local to the global reference system are herein shortly introduced and discussed. The effects of the associated transformation matrices have been investigated in few examples, also reproduced via closed expressions derived from the mixed torsion theory and FE shell models. Paper outcomes show that the results in terms of generalised displacements, internal forces and, consequently, stress distribution along the member cross-section are significantly different, strictly depending on the adopted transformation matrix. The importance of a correct evaluation of the bimoment distribution is stressed also by the fact that the new edition of the EN1993-1-1, expected in the next months, includes also the bimoment contribution for member checks.
•Interaction between bending moments and bimoment in linear elastic range for mono-symmetric cross-section members.•The two strategies adopted in 7DOFs beam formulation, to pass from the local to the global reference system are discussed.•Vlasov’s theorems are discussed and applied to example of practical interest for structural design.•Displacements and internal forces are significantly different, strictly depending on the adopted transformation matrix.