A dynamic saint-venant principle for an elastic semi-infinite strip

Abstract

We consider the problem of longitudinal deformation of an elastic semi-infinite strip under the action of various kinds of self-balanced harmonic loads. Using numerical analysis we show that in a region of relatively low frequencies, when only one propagating wave ezists in the wave conductor, one can formulate an analog of Saint-Venant's principle. When this is done, the given frequency range decomposes into two subranges. In the first of these the main role in the Fourier series ezpansion of the load is played by the non-self-balanced component, while in the second it is necessary to take account of the first harmonic as well. One figure. Bibliography: 5 titles.The problem of enlarging the area of applicability of the static Saint-Venunt principle [1] to dynamic problems of the theory of elasticity has been studied for quite some time [2; 3], However, in the majority of existing publications only nonstationary processes were studied, and it was in relation to this case that the dynamic Saint-Venant principle for a semi-infinite elastic strip was stated. The study of the stationary harmonic problem for a semi-infinite strip is of equal interest, especially when one considers that in this case there will a priori be homogeneous waves propagating at any frequency, transporting energy to infinity. For that reason the statement of a dynamic Saint-Venant principle for an elastic semi-infinite strip has intrinsic scientific interest.: The purpose of the present article is to perform a quantitative analysis of the behavior of an elastic semi-infinite strip under the action of various kinds of self-balanced harmonic loads.Consider the two-dimensi0nal problem of the forced vibrations of a semi-infinite strip 0 < z < c~,

Dynamic equivalence, self-equilibrated excitation and Saint-Venant’s principle for an elastic strip

“…4, is in agreement with the formulations of DSVP suggested by Gomilko et al (1995) and . Gomilko et al (1995Gomilko et al ( , p. 1153) formulated DSVP as follows: ''When a self-balanced system of forces acts on the end of semi-infinite strip, stresses arise as a result of this system only near the end. At a significant distance from the point of application of the forces the effect of such a load is practically zero".…”

“…The point of departure between the DSVP formulation in (A) and (B) and the formulation suggested by Gomilko et al (1995) lies not only in the extension of validity to higher frequencies (up to the first cut-off). Rather, it is a conceptual shift from load to power that distinguishes these formulations.…”

“…These conditions are considered as a deviation from the self-equilibrium conditions for static decay. In their subsequent work, they evaluate the ratio of the power generated by self-equilibrated loads to the power generated by non-self-equilibrated loads for low frequencies ; their interpretation of DSVP resembles that of Gomilko et al (1995). More references related to DSVP can be found in Karp (2005).…”

“…Gomilko et al (1995) compared the amplitude of waves induced by a self-equilibrated excitation to the amplitude of waves induced by non-self-equilibrated excitations. That ratio turned out to be extremely small for excitations with a low frequency, confirming a form of DSVP.…”

Saint-Venant’s Principle in Dynamics of Structures

“…Gomilko et al [94] compared the amplitude of waves induced by a self-equilibrated excitation to those induced by nonselfequilibrated excitations. The ratio obtained turned out to be extremely small for excitations with a low frequency, confirming a version of DSVP.…”