Eigenfrequency and Euler's Critical Load Evaluation of Transversely Cracked Beams with a Linear Variation of Widths

Abstract

For a truthful evaluation of the mechanical response of structures reliable and adequate computational models are essential. Consequently, various researches have been devoted to the mathematical representation of cracked structures. This paper studies the performance of the simplified crack model in estimations of fundamental eigenfrequency as well as elastic Euler's critical load for transversely cracked beams of rectangular cross-sections with linearly-varying widths. To obtain these solutions for different beams with diverse boundary conditions Rayleigh’s energy method which requires an assumed transverse displacement function can be applied. After the appropriate displacement function is being selected, kinetic and strain energy, as well as the work done by an external axial compressive force P are evaluated. From these values, the estimations of the fundamental eigenfrequency, as well as the critical load, are assessed. To obtain these preliminary estimates, static deflection functions were applied initially. These functions represent a wide group of suitable functions since they automatically satisfy the required kinematic boundary conditions. Afterwards, alternative functions constructed from a dedicated polynomial solution were applied. Since this mathematical form offers straightforward integration, the genuinely applied displacement functions were further upgraded, separately for eigenfrequency as well as for critical load estimation. All obtained simplified model’s solutions were afterwards compared to the results from equivalent and more detailed 3D finite models of the examined structures. The comparisons of the results demonstrated very fine agreements with the results from 3D FE models for all performed analyses. The considered simplified model thus clearly yields a suitable alternative in modelling of cracked beams with a linear variation of width in those situations, where cracks have to be considered within the analysis.