A novel formulation for the weak quadrature element method for solving vibration of strain gradient graded nonlinear nanobeams

Abstract A novel formulation of the weak form quadrature element method, referred to as locally adaptive is proposed develop elements for nonlinear graded strain gradient Timoshenko and Euler–Bernoulli nanobeams. The equations motion are obtained based on Hamilton principle while accounting position physical neutral axis. use Gauss points ensure full integration variational statement. develops matrices differential method which employs Lagrange-based polynomials. These can be modified accommodate any number extra derivative degrees freedom including third-order beams higher-order without requiring an entirely new formulation. performance evaluated free vibration response linear Both frequencies a large configurations boundary conditions. It shown that results in good accuracy improved convergence speed compared other methods available literature.