Abstract
Structural analysis of axially functionally graded tapered Euler-Bernoulli beams is studied using finite element method. A beam element is proposed which takes advantage of the shape functions of homogeneous uniform beam elements. The effects of varying cross-sectional dimensions and mechanical properties of the functionally graded material are included in the evaluation of structural matrices. This method could be used for beam elements with any distributions of mass density and modulus of elasticity with arbitrarily varying cross-sectional area. Assuming polynomial distributions of modulus of elasticity and mass density, the competency of the element is examined in stability analysis, free longitudinal vibration and free transverse vibration of double tapered beams with different boundary conditions and the convergence rate of the element is then investigated.
Highlights
Graded (FG) materials are one of the most advanced materials whose mechanical properties vary gradually with respect to a desired spatial coordinate
There are relatively few works on axially Functionally graded (FG) beams whose mechanical properties vary along the axis of the beam where most of them concern the special case of uniform beams
It is instructive to know that Finite element method (FEM) is a type of stiffness method which is formulated on the basis of variational calculus; it is expected that the results provided by FEM show an upper bound of the exact ones
Summary
Graded (FG) materials are one of the most advanced materials whose mechanical properties vary gradually with respect to a desired spatial coordinate. A new element is proposed for analysis of tapered beams with an arbitrarily varying cross-section made of axially functionally graded materials. Several works have been carried out on axially FG beams; there is still a gap in analysis of axially FG beams with arbitrarily varying cross-sectional area and mechanical properties along the beam axis. This could be explained in detail by recalling that the majority of previous works have followed a semi-inverse procedure in which the displacement field and the mass density distribution are prescribed and afterwards the distribution of modulus of elasticity is obtained by satisfying the governing differential equations. It is instructive to bear in mind that semi-inverse method provides exact results while FEM does not essentially predict the deformations of the structural system accurately
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