Determining the Stresses in Beams Due to Short-Term Effect on Their Supports

Abstract

In many engineering cases, forces acting on structures are random process excitations. Some of them can be treated as stationary random process, however, most of them should be considered as non-stationary random process. The problems solved in this paper are posed as follows. It is believed that at some time, all the supports and fixed-end beam sections (or part of them) begin to move according to the well-known laws, and the duration of the support motion is short in comparison with the first period of transverse vibrations of the beam. It is required that the internal forces arising in the beam during its subsequent movement be determined. Solutions in the form of trigonometric series, under the assumption that the shaking motion of the supports is short, ineffective due to the fact that it is necessary to deal with the determination and summation of a large number of harmonics of the series. Solutions are built using the method of successive approximations. Zero approximation is the solution for an infinitely long beam with a single support. Consideration of the solutions show that although the original equation is not a wave one, its solutions are of quasi-wave character in the sense that each subsequent approximation substantially changes the solution only from a certain time moment, all the more, the higher the number of the approximation being made will be. In this regard, from our point of view, the most significant conclusion: no features (such as: supports, concentrated inclusions, etc.), which are distant from the bearing undergoing shaking by a greater distance corresponding to the magnitude n = 50, cannot affect the maximum value of internal forces in its proximity. The method described above can be successfully applied while solving problems of the same category. Here we limit ourselves to providing operational representations of solutions for some of the respective tasks; a simple comparison of them with the representations obtained earlier shows that the transition from these representations to the originals in its essence is not different from the above.