A micro-nano-scale Timoshenko-Ehrenfest beam model is investigated using doublet mechanics theory in the present study. The governing equations and all possible boundary conditions are obtained based on doublet mechanics model. The static bending, buckling and vibration problems of Timoshenko microbeams are examined in detail. Deflection, rotation, critical buckling loads and natural frequencies predicted by the present doublet mechanics model are obtained for simply supported micro-scale Timoshenko beams by the Navier solution method. The obtained results are compared to other classical and non-classical continuum theories. To illustrate the present doublet mechanics model, the influences of thickness to length scale parameter ratio of the considered material and slenderness ratio on static bending, buckling and vibration problems are investigated. It is shown that there are two frequency spectrums in the vibration of nanobeams similar to macro Timoshenko beams. It is interesting to note that acceptable physical frequencies (mode numbers) have an upper bound due to Van Hove singularity depending on geometrical and material properties of the beam. That fact is observed first time in the open literature by using scale dependent theories.

在本研究中，利用双峰力学理论研究了微纳米蒂莫申科-埃伦菲斯特光束模型。基于双峰力学模型获得了控制方程和所有可能的边界条件。详细研究了Timoshenko微梁的静态弯曲，屈曲和振动问题。通过Navier解法获得了简单支撑的微型Timoshenko梁的挠度，旋转，临界屈曲载荷和本双峰力学模型预测的固有频率。将获得的结果与其他经典和非经典连续论进行比较。为了说明当前的双峰力学模型，考虑了材料的厚度与长度比例参数比和细长比对静态弯曲的影响，研究了屈曲和振动问题。结果表明，纳米束的振动与宏观的季莫申科束相似，有两个频谱。有趣的是，由于Van Hove奇异性取决于光束的几何和材料特性，可接受的物理频率（模式编号）具有上限。在公开文献中，第一次使用比例尺依赖的理论观察到了这一事实。