The purpose of studies on the critical tractive force which have been carried out by many researchers since du Boys is mostly to create empirical formula by finding experimentally the relation between the diameter of sand grain and the critical velocity or the critical shearing stress on channel bottom. On the other hand, the theories on the mechanism of initiation of sand movement have been studied by only a few researchers, such as C. M. White (1940) and M. Kurihara (1948), in recent years.
C. M. White expressed the effect of turbulence by a factor named as the turbulence factor, and M. Kurihara explained by a skilful application of turbulence theory that the turbulence factor is a function of Reynolds Number with respect to roughness. Since Kurihara's theory, however, is difficult to understand, another theoretical analysis is attempted by the author in this paper, based on the idea and treatment different from Kurihara's theory.
The basic idea of the author's theory is to construct the equilibrium condition using the forces which are acting on a spherical sand grain, such as the gravity force, the fluid resistance and the resistance resulting from pressure gradient, and to evaluate these resistance considering the velocity fluctuation. In this process, the concepts of the mixing length of turbulence and the minimum scale of eddies in turbulence theory are utilized.
A dimensionless function of critical tractive force like as the function by Shields is derived by the analysis based on the idea described above, and an experimental constant named as the sheltering coefficient is introduced to this function so as to agree with the experimental results obtained with a small tunnel having a uniform and square cross-section.
On the basis of the experimental results and theoretical relationships, a new formula of critical tractive force is created and moreover compared with the empirical formulas proposed by many authorities.

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The critical tractive force for gravels on a granular bed in turbulent stream is an interesting and fundamental problem connected with the critical tractive force for mixed sand and gravel, the sieving action of flowing water and the effect of rubble on river bed protection from scouring. In 1953, experiments on the incipient motion of discrete particles resting on a stream bed of fixed granular roughness were carried out by A. T. Ippen and R. P. Verma using four samples of plastic and glass spheres, and an empirical entrainment function was produced. But nothing about the mechanism and its analysis of incipient motion of discrete particles was touched.
In this paper, theoretical cnosiderations on the incipient motion of spheres on a fixed granular bed are made by means of the same treatment as in the previous paper and other modified treatment. From these considerations, it is shown that the dimensionless critical tractive forces are expressed as a function of the roughness Reynolds Number with respect to granular bed and the ratio of diameters of sphere and bed material, and that this function results in good agreement with the experimental data. Moreover, based upon the results obtainedd by the theoretical considerations described above, the critical tractive force for mixed sand and gravel is discussed.

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