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Generalized vibration analysis of beams including warping effects by isogeometric methods

In this paper, the Isogeometric tools, either integrated in the Finite Element Method (FEM) or in a Boundary Element based Method (BEM) called Analog Equation Method (AEM), are employed for the vibration analysis of homogeneous beams of arbitrary cross section (thin- or thick- walled) taking into account nonuniform warping and shear deformation effects (shear lag due to both flexure and torsion). The beam is subjected to the combined action of arbitrarily distributed or concentrated axial and transverse loading, as well as to bending, twisting and warping moments. Its edges are subjected to the most general boundary conditions. By employing a distributed mass model system accounting for longitudinal, transverse, rotatory, torsional and warping inertia, ten boundary value problems with respect to the variable along the beam time-dependent 1-D kinematical components are formulated. The numerical solution or the spectrum analysis of the aforementioned problems is performed through IGA, FEM and AEM, leading to a system of second-order differential equations, which are quasi-static and solved for the free vibration case, formulating a generalized eigenvalue problem. Special cases of the generalized problem have also been studied in order to demonstrate the efficiency of AEM in reducing computational effort and improving accuracy, especially when combined to Isogeometric tools, such as NURBS and B-splines.