Differential quadrature-based solution for non-classical Euler-Bernoulli beam theory

The non-classical theories have attracted the attention of many researchers due to their high potentiality in capturing the micro/nano scale structural behaviour. Unlike classical theories, numerical treatment of nonclassical theories is complicated and involves the solution of higher order differential equation with accurate representation of classical and non-classical degrees of freedom and associated boundary conditions. In the present work, a beam element is developed with in the framework of differential quadrature method for bending, free-vibration and stability analysis of non-classical strain gradient Euler-Bernoulli beam theory. The element is formulated by combining the governing equation and stress resultant equations with Lagrange interpolations as test functions. Detailed mathematical formulation of the element and its numerical implementation is presented. The convergence, accuracy and efficiency of the proposed element is demonstrated through numerical examples for different loading and boundary conditions. Further, the generality of the element is verified through solving examples with geometry and load discontinuity. Lastly, the results are compared with the finite element solution obtained for gradient beams to asses the accuracy and convergence behaviour of the two methods.