Zagadnienie Dynamiczne w Obliczeniach Statycznych Budowli - Wierzbicki, Witold (Engineering Transactions, 1955)
In Sec. I the problem of the attachment points of braking force and tractive force of a locomotive in computation of bridge abutments and piers is considered. It is shown that if two axles are braked over the considered support, the centre of gravity of rolling stock should be assumed as the attachment point of the force. In the case when only one axle is 600 braked, the attachment should be taken at the level of the rail head. In general, the tractive force should be taken as acting at the level of the rail head. There are cases, however, in which it should be assumed to act at the level of the couplers. This is, for instance, the case when the track beyond the bridge runs down an incline. An analysis of bridge structures is given, from which attachment points of braking forces can be established in statical computations. Sec. II contains approximate calculation of free vibration. A method based on successive approximations and on the method of secondary moments is used. Vibrations of 1, 2, 3 and 4 degrees of freedom are discussed as well as transversal and longitudinal vibrations of elastic bodies. The frequencies of vibrations of first and second order as well as first and second approximations of those frequencies are calculated. Sec. III is devoted to the dynamic interpretation of the equation d(V-T) = 0, which constitutes the basis of the method of approximate determination of deformations. V denotes here the elastic energy of the element and T the work of external forces, assumed in the form of a sum of products of forces P and the corresponding displacements of their attachment point. The dynamic interpretation of the equation d(V - T) = 0 consists in that the forces P are considered to be attached in a sudden manner which provokes damped vibration of the structure. It follows therefore that this equation expresses the condition for a minimum of the function U = V - T, corresponding to the state of equilibrium of the structure in other words - to its static deformation.