EXACT DYNAMIC STIFFNESS MATRIX FOR BEAMS OF ARBITRARILY VARYING CROSS SECTIONS

In this paper, the exact dynamic sti¤ness matrix is derived for the transverse vibration of beams whose cross-sectional area and moment of inertia vary in accordance to any two arbitrary real-number powers. This variation represents a very large class of arbitrary varying beams and thus, Þlls the void currently existing in this area of research. With this approach, most beams can be modelled by just one element, and for beams having abrupt proÞle changes or with very complex proÞles, they can be divided into separate distinct parts, with each of the part modelled by just one element, and then assembled together. The method is exact; however, the accuracy of the results depends only on the solver used to solve the exact frequency equation. To demonstrate the procedure, beams of non-linearly varying circular and elliptical cross-sections, and a combination beam consisting of a linear-tapered section, a uniform section and a non-linearly varying-section are analysed for their natural frequencies. Since there are no known solutions for these structures, comparison with Þnite element results was made and very good agreement was observed