An investigation of static bending of a bi-directional strain-gradient Euler–Bernoulli nano-beams with the method of initial values

Investigated herein is the static bending of Euler–Bernoulli nano-beams made of bi-directional functionally graded material with the method of initial values in the frame of gradient elasticity. To the best of the researchers’ knowledge, in the literature, there is no study carried out into gradient elasticity theory for bending analysis of bi-directional strain-gradient Euler–Bernoulli (BDSGEB) nanostructures with arbitrary functions. Basic equations and boundary conditions are derived by using the principle of minimum potential energy. The transfer matrix for beams is given analytically. For an initial value problem, the transport matrix is unique. The diagrams of the solutions for different end conditions and various values of the parameters are given and the results are discussed.