The paper presents the analysis of Timoshenko beam on two-parameter elastic foundation. The differential equation for dynamic equilibrium of beam compressed by conservative axial force has been derived and its solution has been analyzed depending on mutual relationships between its coefficients. The exact stiffness matrix of the basic beam element has been derived formally, and by the power series expansion of this matrix in relation to the arguments of the functions occuring in it the stiffness matrix of working element with elastic foundation and constant axial force has been determined. It has been demonstrated that the FEM matrices obtained with the use of so-called physical shape functions are the linear approximation of the formally exact matrix. In addition, a formula for critical force is found for the case of free supported beam on elastic foundation.

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