VIBRATION ANALYSIS OF BEAM WITH COMPLEX CROSS-SECTION BASED ON ASYMPTOTIC ANALYSIS METHOD

The beam structures play a very important role in the engineering industry. The free vibration problem of the beam structures reflects their basic dynamic characteristics, which is also a vital problem in the dynamics analysis. The classical theories of the beam structures are often based on different assumptions. The present paper uses the asymptotic analysis method to strictly derive a one-dimensional equivalent beam model for the vibration analysis of the three-dimensional beam structures with the ratio of the section size to its length as the characteristic parameter. The parameter which is based on the geometric characteristic of the beam structures, helps to determine the true order of the magnitude relationship between physical quantities of the beam structures, such as the stress field and the displacement field. The asymptotic analysis results show that the pure bending deformation is only the first order term in the vibration analysis process of the beam structures. For the slender beams with complex cross-sections, the one-dimensional equivalent beam model with similar complexity to the Euler beam model can still be used for the vibration analysis. Based on the results of the precise three-dimensional finite element method, the accuracy and effectiveness of the proposed one-dimensional equivalent beam model are verified when it is applied in the vibration analysis. The proposed one-dimensional equivalent beam model can be conveniently implemented in the commercial finite element software through the user-defined section and other embedded modules. Comparing the natural frequency calculated by various beam elements in the finite element software, it is found that with the similar computational complexity, the proposed equivalent beam model is significantly superior. The results of the slender beam structures calculated by the proposed method are more accurate and stable than that calculated by selecting the complex cross-sections and using the beam elements directly in the commercial finite element software in the vibration analysis.