Free vibration analysis of a single edge cracked multi-layered symmetric sandwich stepped Timoshenko beams, made of functionally graded materials, is studied using finite element method and linear elastic fracture mechanic theory. The cantilever functionally graded beam consists of 50 layers, assumed that the second stage of the beam (step part) is created by machining. Thus, providing the material continuity between the two beam stages. It is assumed that material properties vary continuously, along the thickness direction according to the exponential and power laws. A developed MATLAB code is used to find the natural frequencies of three types of the stepped beam, concluding a good agreement with the known data from the literature, supported also by ANSYS software in data verification. In the study, the effects of the crack location, crack depth, power law gradient index, different material distributions, different stepped length, different cross-sectional geometries on natural frequencies and mode shapes are analysed in detail.

中文翻译：

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单边裂纹对称功能梯度阶梯梁的自由振动分析

使用有限元方法和线弹性断裂力学理论研究了由功能梯度材料制成的单边裂纹多层对称夹层阶梯铁木辛哥梁的自由振动分析。悬臂功能梯度梁由 50 层组成，假设梁的第二阶段（台阶部分）是通过机械加工创建的。因此，提供了两个光束级之间的材料连续性。假设材料特性根据指数和幂定律沿厚度方向连续变化。用开发的MATLAB代码求出三种阶梯梁的固有频率，与文献中的已知数据吻合良好，数据验证也得到ANSYS软件的支持。在研究中，裂纹位置、裂纹深度、