LAUSR

In the current study, bending, buckling and vibration behaviours of functionally graded in depth direction non-uniform nanobeams are investigated in the framework of nonlocal strain gradient theory. Material variation is assumed through the thickness and modelled using exponential, sigmoid and power-law functions. Moreover, cross-sectional variation is also considered using exponential and power-law functions. Using these general non-uniformity and non-homogeneity in conjunction with nonlocal strain gradient theory, a general beam is modelled. For this general model, equations of motion are derived using Hamilton’s principle and solved by using a novel technique in combining finite element method with Lagrangian interpolation technique, Gaussian quadrature method and Wilson’s Lagrangian multiplier method. Mechanical behaviour of such structures is fully explained with parametric study for all three cases of bending, buckling and vibration response. It is shown that combination of material variation, non-uniformity and scale effects has a significant effect in changing the mechanical behaviour of such structures in both static and dynamic cases.Graphical abstract.