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 non-classical 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.

非古典理论由于具有捕获微观/纳米尺度结构行为的巨大潜力而​​吸引了许多研究人员的注意力。与经典理论不同，非经典理论的数值处理很复杂，涉及到高阶微分方程的求解，需要精确表示经典和非经典自由度以及相关的边界条件。在目前的工作中，在微分正交方法的框架内开发了一种梁单元，用于非经典应变梯度Euler-Bernoulli梁理论的弯曲，自由振动和稳定性分析。该元素是通过将控制方程和应力合成方程与拉格朗日插值作为测试函数组合而成的。介绍了该元素的详细数学公式及其数值实现。通过数值示例说明了在不同载荷和边界条件下所提出元件的收敛性，准确性和效率。此外，通过求解具有几何形状和载荷不连续性的示例来验证元素的通用性。最后，将结果与梯度梁的有限元解进行比较，以评估两种方法的准确性和收敛性。