Size-Dependent Buckling and Vibrations of Piezoelectric Nanobeam with Finite Element Method

In the present paper, a finite element method is used to study the vibrations and buckling of a piezoelectric nanobeam. The beam theory used here is Bernoulli–Euler model. In order to achieve the goal, first, the governing equations of the piezoelectric nanobeam were obtained via the Hamilton principle, based on the higher-order theory such as modified couple stress. The shape functions were embedded in these equations, and the matrix form of the equations was obtained. In other words, the governing equations were discretized by the finite element method. By calculating the matrix form of equations, mass matrices, mechanical stiffness and electrical stiffness of the beam were obtained. It should be noted that when calculating stiffness matrices, first, the definition of a new element must be added to the effects of the electric field in the elements of the beam, and the effects of the gradient of strain should be added in the form of non-classical matrix to the stiffness matrix. With the aid of these critical load matrices, the buckling force and voltage and the natural frequency of the piezoelectric nanobeam have been calculated. Numerical results show that this method can involve the size effects.