  In this paper, first we present the variational formulation for a second
strain gradient Euler-Bernoulli beam theory for the first time. The governing
equation and associated classical and non-classical boundary conditions are
obtained. Later, we propose a novel and efficient differential quadrature
element based on Lagrange interpolation to solve the eight order partial
differential equation associated with the second strain gradient
Euler-Bernoulli beam theory. The second strain gradient theory has
displacement, slope, curvature and triple displacement derivative as degrees of
freedom. A generalize scheme is proposed herein to implement these
multi-degrees of freedom in a simplified and efficient way. The proposed
element is based on the strong form of governing equation and has displacement
as the only degree of freedom in the domain, whereas, at the boundaries it has
displacement, slope, curvature and triple derivative of displacement. A novel
DQ framework is presented to incorporate the classical and non-classical
boundary conditions by modifying the conventional weighting coefficients. The
accuracy and efficiency of the proposed element is demonstrated through
numerical examples on static, free vibration and stability analysis of second
strain gradient elastic beams for different boundary conditions and intrinsic
length scale values.
