Keywords: Timoshenko beam, semi-infinite elastic foundation, free vibration, ordinary differential equation solver, foundation stiffness, mode shape.

Many practical engineering problems can be modelled as a Timoshenko beam supported on an elastic subgrade, such as the analysis of a suspension bridge, the structure beneath a highway, etc. Thus, the mechanical problems pertinent to Timoshenko beams have been attracting the attention of many scientists and engineers. For example Han, et al. [1] presented the full development and analysis of four traditional models for transversely vibrating uniform Timoshenko beams. Morfidis [2] fomulated the respective equations of natural vibration of Timoshenko beams on a three-parameter elastic foundation within the framework of second order theory.

However, the previous studies exhibit two deficiencies. The first is how to quantify the scope of an elastic foundation or an elastic subgrade or soil. The second is that the exact solutions can only work for some simpler engineering problems and will be inapplicable for more difficult practical problems. These two deficiencies will thus limit their applications to practical engineering.

The motivation of this is to show a novel analytical method for the dynamic problem of a Timoshenko beam resting on a semi-infinite elastic subgrade using an ordinary differential equation (ODE) solver, by which an arbitrary desired precision for the solution can be obtained [3].

The deflection of the neutral surface and the cross-sectional rotation around the neutral axis of a Timoshenko beam are selected as two basic unknown functions. By using the Hamilton principle, the dynamic problem of the Timoshenko beam resting on an elastic subgrade is then formulated as ordinary differential equations composed of the two unknown functions. The dynamic property of the beam and the influence of the subgrade rigidities on the natural frequencies can be readily analysed. The numerical results demonstrate that the proposed method is rational and powerful for the dynamic analysis of various Timoshenko beams.

1

S.M. Han, H. Benaroya, T. Wei, "Dynamics of transversely vibration beams using four engineering theories", Journal of sound and vibration, 225(5), 935-988, 1999. doi:10.1006/jsvi.1999.2257

2

K. Morfidis, "Vibration of Timoshenko beams on three-parameter elastic foundation", Computers and Structures, 88, 294-308, 2010. doi:10.1016/j.compstruc.2009.11.001

3

Y. Si, "ODE conversion techniques and their applications in computational mechanics", Acta Mechanica Sinica, 7, 283-288, 1991. doi:10.1007/BF02487597

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