  In this paper, we propose a novel and efficient differential quadrature
element based on Lagrange interpolation to solve a sixth order partial
differential equations encountered in non-classical beam theories. These
non-classical theories render displacement, slope and curvature as degrees of
freedom for an Euler-Bernoulli beam. A generalize scheme is presented herein to
implementation the multi-degrees degrees of freedom associated with these
non-classical theories in a simplified and efficient way. The proposed element
has displacement as the only degree of freedom in the domain, whereas, at the
boundaries it has displacement, slope and curvature. Further, we extend this
methodology and formulate two novel versions of plate element for gradient
elasticity theory. In the first version, Lagrange interpolation is assumed in
$x$ and $y$ directions and the second version is based on mixed interpolation,
with Lagrange interpolation in $x$ direction and Hermite interpolation in $y$
direction. The procedure to compute the modified weighting coefficients by
incorporating the classical and non-classical boundary conditions is explained.
The efficiency of the proposed elements is demonstrated through numerical
examples on static analysis of gradient elastic beams and plates for different
boundary conditions.
