A new high-order non-uniform Timoshenko beam finite element on variable two-parameter foundations for vibration analysis

A new finite element model of a Timoshenko beam is developed to analyze the small amplitude, free vibrations of non-uniform beams on variable two-parameter foundations. An important characteristic of the model is that the cross-sectional area, the second moment of area, the Winkler foundation modulus and the shear foundation modulus are all assumed to vary in polynomial forms, implying that the beam element can deal with commonly seen non-uniform beams having different cross-sections such as rectangular, circular, tubular and even complex thin-walled sections as well as the foundation of beams which vary in a general way. Thus this new beam element model enables users to handle vibration analysis of more general beam-like structures. In this paper, by using cubic polynomial expressions for the total deflection and the bending slope of the beam, the mass and stiffness matrices of the element are derived from energy expressions. The element model can accommodate various boundary conditions to represent a Timoshenko beam accurately. Excellent agreement with other investigators' results and a rapid rate of convergence with relatively few elements are demonstrated. This study also brings out the fact that the complicated form of this new beam model is necessary because of its advantage over linear or uniform approximations of the non-uniform foundation and/or geometrical properties of beams. The same accuracy being achieved with fewer elements is the main advantage. Finally, an optimum design problem is illustrated to emphasize the practical application of this element. (C) 1996 Academic Press Limited