A multivariable hierarchical finite element for static and vibration analysis of beams

Formulations of a multivariable hierarchical beam element for static and vibration analysis are presented based on the generalized variational principle with two kinds of variables. Two forms of shifted Legendre hierarchical polynomials are used as interpolating basis functions of displacement and generalized force field functions for the beam element respectively, which will simplify the computations of the relevant matrices. The multivariable hierarchical beam element formulations, in which the displacement and generalized force field functions are independently constructed, are derived by applying the generalized variational principle with two kinds of variables. Since differential operations to obtain stress fields in conventional displacement based finite element methods are not required, the present method has very high accuracy for the two kinds of independent variables simultaneously, especially for the generalized forces. Static and vibration numerical examples demonstrate the applicability of the proposed method. The proposed method can be easily extended to deal with structural analysis of shells or plates.