Dynamic analysis of beam-like structures is significantly important in modeling actual cases such as tall buildings and several other related applications as well. This article studies free vibration analysis of tall buildings with nonuniform cross-section structures. A novel and simple approach is presented to solve natural frequencies of free vibration of cantilevered tall structures with variable flexural rigidity and mass densities. These systems could be replaced by a cantilever Timoshenko beam with varying cross-sections. The governing partial differential equation for vibration of a nonuniform Timoshenko beam under variable axial loads is transformed with varying coefficients to its weak form of integral equations. Natural frequencies can be determined by requiring the resulting integral equation, which has a nontrivial solution. The presented method in this study has fast convergence. Including high accuracy for the obtained numerical results as well. Numerical examples including framed tube as well as tube-in-tube structures are carried out in the study and compared with available results in the literature, and also with the results obtained from finite element analysis in order to show the accuracy of the proposed method in the study. Obtained results indicate that the presented method in this study is powerful enough for the free vibration analysis of tall buildings. KEYWORDS axial force, free vibration, nonuniform, tall building, Timoshenko beam, weak form integral equation 1 | INTRODUCTIONFree vibration analysis plays an important role in the structural design of tall buildings, especially for the first mode because the first mode shape is a dominant component in wind and earthquake-induced vibrations of tall buildings. Therefore, it is important to investigate the calculating methods of natural frequencies and mode shapes for tall buildings. Many researchers in structural engineering have devoted to obtain accurate theoretical results for the free vibration of tall buildings in the past decades. In free vibrations analysis of cantilevered tall structures, it is possible to regard them as beams with variable cross-section. [1,2] Wang [3] derived the closed-form solutions for the free flexural vibration of a beam with variably distributed stiffness, but uniform mass. Li et al. [1] and Li [4] studied the free flexural vibration of tall buildings as well as high-rise structures, which have variably distributed stiffness and mass. However, the effect of axial forces acting on the cantilevered structures with variable cross-section on natural frequencies and mode shapes was not considered. The subject of free vibration of nonuniform beams has been paid attention by many researchers. However, few of them have considered the effects of axial forces on beam vibration frequencies. Most of these studies include simplifications in calculation process such that the governing differential equation becomes manageable to solve. Li et al. [5] investigated free vibration of cantilevered tall structures under various ...

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