basic displacement functions (bdfs)

In order to obtain accurate results from displacement-based Finite Element Method (FEM), it is crucial to introduce accurate shape functions that interpolate the displacement field within an element. This paper attempts to provide such a new component by using Finite Element method using Basic Displacement Function (BDFs) for the free vibration analysis of plates with in-plane Functionally Graded Material (FGM). The first step is to introduce displacement functions and compute them using the energy method. Later, new shape functions are developed based on stiffness and force methods used to model the mechanical behavior of the element, wherein the shape functions benefit from the generality and accuracy of the stiffness and force methods. Last, the plate is analyzed using Finite Element method to derive the structural matrices from new shape functions. Several numerical examples demonstrate the accuracy and efficiency of the method, and a special material graded index named Ns is introduced.

The accuracy and efficiency of the elements proposed by finite element method (FEM) considerably depend on the interpolating functions namely shape functions used to formulate the displacement field within the element. In the present study, novel functions, namely basic displacements functions (BDFs), are introduced and exploited for structural analysis of nanobeams using finite element method based on Eringen’s nonlocal elasticity and EulerBernoulli beam theory. BDFs are obtained through solving the governing differential equation of motion of nanobeams using the power series method. Unlike the conventional methods which are almost categorized as displacement-based methods, the flexibility basis of the method ensures true satisfaction of equilibrium equations at any interior point of the element. Accordingly, shape functions and structural matrices are achieved in terms of BDFs by application of merely mechanical principles. In order to evaluate the competency and accuracy of the proposed method with different boundary conditions, several numerical examples with various boundary conditions are scrutinized. Carrying out several numerical examples, the results in stability analysis, free longitudinal vibration and free transverse vibration show a complete accordance with those in literature.

This paper aims at presenting a new efficient element for free vibration and instability analysis of Axially Functionally Graded Materials (FGMs) non-prismatic beams using Finite Element Method (FEM). Using concept of Basic Displacement Functions (BDFs), two- node element extends to three-node element for obtaining much more exact results using FEM. First, BDFs are introduced and computed using energy method such as unit-dummy load method. 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; free vibration and instability analysis of the FGM beam are studied using finite element method for all types of AFGM beams and the convergence of FEM has been studied. The results from both free vibration and instability analysis are in perfect agreement with those of previously published.