Obliqueness effects in asymptotic cross-sectional analysis of composite beams

Cross-sectional stiffness properties for general composite beams, along with the interior solution for warping, are derived in such a way as to provide the extra freedom to choose a cross-sectional plane that is not perpendicular to the local beam reference line. The present development treats prismatic as well as initially twisted and curved beams. The main purpose of allowing such a choice of coordinate systems is for the convenience of the analyst; one, of course, should not expect the final 3D results to change. The 3D strain field is derived for the corresponding nonorthogonal curvilinear set of coordinates which describe the undeformed and the deformed geometry of the beam. The analysis is carried out based on the variational-asymptotic method which is used to determine the warping field. For the development of the solution, it is shown that there is a fundamental difference between the solutions for a prismatic beam and a beam with initial twist and curvature, which in turn produces a limitation of the angle of obliqueness: the effect of obliqueness is treated exactly for a prismatic beam, while for the initially twisted and curved beam the obliqueness is regarded as a small parameter. The ultimate goal of the analysis is the determination of the cross-sectional stiffness matrix, which can then be used as input for the 1D problem. Then, using the recovering relations, one can recover the strain field over the entire cross section, once the previous problem has been solved. The validity of the method is demonstrated by recovery of accurate cross-sectional stiffnesses associated with a normal cross-sectional plane from those associated with an oblique cross-sectional plane, using a simple rotation-of-axes transformation.