Lateral vibrations as related to structural stability

Abstract

The apparently different physical problems of lateral vibration and elastic stability are limiting cases of a single phenomenon, the most general expression being the mode of vibration with end thrust. The theory of straight beams and flat plates is discussed in detail, and it is shown that the square of the frequency of lateral vibration is approximately linearly related to the end load. The linear relationship is exact if the mode of free vibrations is identical to the buckling mode. In all cases, the load corresponding to zero frequency is the critical buckling load. The analysis is valid only if the boundary conditions do not change with load. Experimental tests were conducted on elastically restrained columns in the form of rigid rectangular frames. It is found that the relationship between the square of the frequency and the load is practically linear, and that the extrapolated load corresponding to zero frequency coincides with the buckling load. Determining the critical load by frequency measurements seems to have the advantage of predicting that load corresponding to the actual boundary conditions which prevail, whereas a theoretical calculation may unjustifiably assume certain conditions which are not exactly realized. In the case of flat plates, tests showed that the linear relationship is not achieved in practice. It is shown that this is probably due to the fact that the linear plate equations are not valid due to initial curvatures in the plate. Rigid-joint trusses were also tested. Due to the change of end restraint with load, in some cases the relationship between the square of the frequency and the load deviates considerably from linearity. The amount of deviation appears to depend on the section properties of the members of the truss.