Finite element static and dynamic analysis of axially functionally graded nonuniform small-scale beams based on nonlocal strain gradient theory

In this article, a comprehensive study on mechanical behaviors of nonhomogenous nonuniform size-dependent beams is presented. Beam is modeled with axially functionally graded materials with different types of materials and functions for material variation. Also, the beam is assumed with nonuniform cross-section and scale effects are presented by having both strain gradient and nonlocal effects. Nonuniformity is modelled in several ways by having tapered, parabolic and higher-order cross-section variation in both thickness and width directions. By having these assumptions, a great majority of small-scale beams from uniform homogeneous beams to nonuniform axially functionally graded beams are modeled and formulated. For this general beam model, static deformation, stability, and free vibration response, which are the three most important mechanical behaviors of such structures, are investigated. Finite element method in conjunction with numerical integration, Gaussian quadrature method, and Wilson's Lagrangian multiplier are employed to solve different problems investigated in this study by combining different assumptions. In order to show the influence of different parameters on mechanical behavior of such structures, a parametric study is presented and the effects by varying material, cross-section, using different type of functions, etc. on bending, buckling, and free vibration response of size-dependent beams are illustrated. It is seen that by combining different cross-section and material variations through the length, mechanical behavior of nanobeams change significantly which could be a step-forward in optimize designing nanostructures.