Bending analysis of laminated beams using isogeometric variational asymptotic method

Variational asymptotic method (VAM) is a robust mathematical method employed to arrive at a reduced order governing equations in solid mechanics. VAM for beam analysis is accomplished in two stages: first stage involves performing a two-dimensional cross-sectional analysis, and the last stage is one-dimensional beam analysis. Cross-sectional analysis stage involves determining the stiffness coefficients for the beam cross section. VAM can be implemented either by developing analytical solutions or by adopting numerical methods. For beams with arbitrary cross-sectional shapes and material heterogeneity, cross-sectional analysis in VAM is generally carried out by adopting a finite element type procedure. In this study, a numerical procedure for the two-dimensional analysis and the one-dimensional beam analysis based on isogeometric framework is presented. NURBS functions are employed to model the geometry of a cross section and the warping displacement field. Warping represents the deformation of cross section of the beam. Stiffness coefficients obtained by cross-sectional analysis are input to the one-dimensional analysis. A NURBS-based one-dimensional method for bending analysis of laminated composite beams is proposed which can handle axial forces, twisting moments, in-plane and out-of plane bending moments. The one-dimensional analysis features four unknown variables, namely axial displacement, twist and two transverse displacements. The one-dimensional beam geometry and the unknown variables are modelled using NURBS functions. The one-dimensional formulation requires C1 continuity which can be easily satisfied by NURBS functions. Numerical examples are solved to ascertain accuracy of the proposed methods. Results obtained were compared with exact solution and was found to be accurate. The one-dimensional analysis based on NURBS functions is non-intrinsic. Application of boundary conditions is far simpler to implement in comparison with the intrinsic one-dimensional analysis methods existing in traditional one-dimensional analysis of VAM literature.