New bar finite element for modeling massive columns with linearly variable rectangular section

In this article, the stiffness matrix of a column with linearly variable rectangular cross section along the longitudinal axis is obtained. As it is well known in structural analysis, the stiffness matrix can be obtained by inversion of the flexibility matrix. So, the terms of this matrix for variable section beam are obtained via the energetic method of the principle of virtual works. Parameters called α_i,α_f,β_f and ε are obtained and they are valid only for sections with variable dimensions. The stiffness matrix is then obtained in function of these parameters, from de inversion of the flexibility matrix.Finally, the modal analysis of the abutment is carried out for the case of the support of a bridge 100 meters high, whose numerical validation of the new bar finite element is performed by means of an exact solution by the Continuous Medium Technique (CMT) and by modeled finite elements in Ansys academic version.