Since the accuracy of results obtained through displacement-based finite element method (FEM) considerably depends on the accuracy of shape functions used to interpolate the displacement field within an element, this paper aims at presenting a new efficient element for static and free vibration analysis of higher-order shear deformable beams using FEM with introducing basic displacement functions (BDFs). First, BDFs are introduced and computed. Afterward, new efficient shape functions are developed in terms of BDFs during the procedure based on the mechanical behavior of the element in which presented shape functions benefit generality and accuracy from stiffness and force method, respectively. Finally, deriving structural matrices of the beam with respect to new shape functions, static and free vibration behavior of the higher-order shear deformable beam is studied using FEM. The accuracy and economy of the method are demonstrated through several numerical examples.Keywords Higher-order beam theory (HOBT) · Finite element method (FEM) · Basic displacement functions (BDFs) · Free vibration B Rahmat Kazemi Firouzjaei

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