Modeling of transverse vibration of non‐uniform beams with elastic boundary conditions and thermal effect using two‐phase local/nonlocal integral model

In this paper, a theoretical model for predicting the size‐dependent vibration response of non‐uniform Bernoulli–Euler beams under thermal effects is developed based on the two‐phase local/nonlocal elasticity model. It is assumed that the ends of the beam are constrained by elastic springs with translational and rotational stiffnesses to model general boundary conditions. In contrast to most of the references in this area, the bending and thermal deformations of the beam are simultaneously described as local/nonlocal in both phases in this study, that is, the size‐dependent properties of the thermal loads are comprehensively taken into account. Subsequently, the problem is solved numerically with the aid of the Generalized Differential Quadrature Method (GDQM). Through the validation study, a series of parametric studies are given to examine the effect of beam thickness and width variation slope, nonlocal relevant parameters, and elastic support conditions on the vibration of non‐uniform beams. The results show that the stiffness of the restraint spring significantly affects the vibration frequency of the beam. In addition, neglecting the effect of nonlocal thermal loads leads to an underestimation of the practical vibration frequency values of the beams.