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Two novel version of weak form quadrature elements are proposed based onLagrange and Hermite interpolations, respectively, for a sec- ond straingradient Euler-Bernoulli beam theory. The second strain gradient theory isgoverned by eighth order partial differential equa- tion with displacement,slope, curvature and triple derivative of dis- placement as degrees of freedom.A simple and efficient differential quadrature frame work is proposed herein toimplement these classi- cal and non-classical degrees of freedom. A novelprocedure to com- pute the modified weighting coefficient matrices for the beamelement is presented. The proposed elements have displacement as the onlydegree of freedom in the element domain and displacement, slope, cur- vatureand triple derivative of displacement at the boundaries. TheGauss-Lobatto-Legender quadrature points are assumed as element nodes and alsoused for numerical integration of the element matrices. Numerical examples arepresented to demonstrate the efficiency and accuracy of the proposed beamelement.

Novel weak form quadrature elements for second strain gradient Euler-Bernoulli beam theory