On the coupled, flexural−flexural−torsional vibrations of an asymmetric concrete beam

This paper presents the analytical solution of an L-shaped cross-section asymmetric beam (concrete terrace unit) undergoing 'triple coupling', that is, flexural vibration in two mutually perpendicular planes (vertical and horizontal) plus torsional vibration about an axis passing through its shear centre, using the classical approach. Essentially, the procedure involved the development of three governing, coupled, partial differential equations based on Euler-Bernoulli theory for beams with isotropic material properties, from which the 'exact' solution was extracted. The warping effect was considered in the torsional equation. A comparison between the analytical solution and corresponding numerical and experimental results obtained earlier was attempted, and similarity and accuracy were discussed. It is reasonable to state that the analytical method in calculating the natural frequencies of a system is the most reliable, compared to experimental (needs, skills and experience) and numerical (calibration, updating, validation etc.). However, even the analytical solution may not be as accurate as expected, as it depends on several factors/parameters beyond the full control of the investigator. Some useful comments and conclusions are drawn.