中文摘要：

材料特征尺寸与其内禀尺寸相当时,材料表现出明显的尺寸效应.基于简化的应变梯度理论,通过半逆法,本文给出多层简化应变梯度Timoshenko梁的变分原理,通过最小总势能原理导出系统的边界条件并对其低阶和高阶边界条件进行讨论,随后给出简支梁系统屈曲载荷和振动频率的Rayleigh（瑞利）解.通过双层梁系统的振动分析算例得到内禀尺寸、长径比等因素对梁系统振动频率的影响.该文构造的Rayleigh解有望对其他数值方法,如有限元法、传递矩阵法等,提供一定的参考和对比.

英文摘要：

The mechanical properties of member materials exhibit notable size effects when the characteristic sizes of the members are comparable to their instinct length parameters.A variational formulation of the nanosize multi-layer Timoshenko beam problem was developed via the semi-inverse method within the context of the simplified strain gradient theory.This method was fit for determining all the possible loworder and high-order boundary condtions directly from the governing equations of the system,according to the minimum total potential energy principle.In turn,the Rayleigh solutions of buckling load and free vibration frequencies of the simply supported beam system were given.The numerical simulations indicate the prominent effects of the instinct length parameters and aspect ratios on the free vibration frequencies of the double-layer beam systems.As a possible benchmark for the later numerical studies with the transfer matrix method or the finite element method,the present Rayleigh solutions of buckling load and free vibration frequencies of the multi-layer beam systems will make good sense.