Strain gradient elasticity theory incorporates the material length scale coefficient which can capture the size effect. First part of study, it has been shown that sixth order boundary value problem of gradient elastic beam equation can be written as a system of differential equations, which can be solved by using the variational iteration method. Related boundary conditions have been derived via variational approach in conjunction with the gradient elasticity. Two numerical examples are given to illustrate the implementation of the method. The second part of study, weighted residual methods are examined to the static bending behavior of microbeams. The weighted residual methods described here include the gradient elasticity method of Galerkin, subdomain and collacation. The governing equation of microbeam in strain gradient elasticity is solved by using weighted residual methods. The results indicate that the variational iteration method and weighted residual methods are successfully applied to analyze the nanomechanical characteristics of carbon nanotubes and MEMS such as those in sensors, nanomachines and actuators in which lengths of beams are on the order of microns and sub-microns.