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In this paper the formulation of the boundary problem of the natural vibration frequency on the basis of Hamiltons principle with consideration of Timoshenko theory has been presented. The investigated column is loaded by external compressive load with constant line of action (Eulers load). Introduction of discreet elements such as springs on both ends of the column allows one to create the general form of boundary conditions. The proper set of their stiffness corresponds to different boundary conditions. In this study an influence of slenderness and external load as well as boundary condition on the shape of characteristic curves calculated with consideration of Bernoulli- Euler and Timoshenko theories has been shown. The presented results show the direction of proper choice of the beam theory in the studies on natural vibration frequency of slender supporting systems.

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