The ice has a subtle rotational motion. Assuming that the motion is horizontal, the following formula can be obtained from the equilibrium of moment at point B:
where MB is the moment with respect to point B on which the earth pressure on the sides of the ice acts, and LG is the location of the center of gravity of the ice (from point B).
From the above description, the unknown quantities are F, K and FV, all determined by the formulae (6), (7) and (9). As a result, the calculations for the ice forces can be made. Nevertheless, for dζ/dX < 0, the distribution of the subgrade reaction (K) or the location it acts on can be the unknown quantity, but it is considered to be approximately the same as for dζ/dX >0.
4. Results and discussion
Fig-4(a), (b) and (c) show the cases with a slope gradient of 1/5, and (d), (e) and (f) a slope gradient of 1/10. Each graph compares the results with the angles of inclination, θ, of 0°, 15°and 30°, at the front of the ice. The plotted symbols are the observed values, while the curves are the calculated values. Below each graph are the gouging curves corresponding to the ice forces at each angle of inclination. At θ = 0°, 15°and 30°, there was a little dispersion when comparing the ice forces and clear correlations were not confirmed, that is, the systematic differences were small. On the other hand, it was confirmed that the gouging curves corresponding to the ice forces at each angle of inclination have almost systematic relationships. As θ increases, ζ(X) increases, hence a gouging extent tends to be small. The small systematic differences in the ice forces, although the ζ(X) differed, seems to be because ζ(X) was necessarily selected to make the quantity of work of the ice a minimum. We determined the formulae for the ice forces with the help of the observed value, ζ(X). If the above descriptions prove to be true, ζ(X) can be theoretically derived by introducing further parameters of the mechanical properties of sand. The ice forces increased while fluctuating, which shows a close relationship to the form of ζ(X). Actually, it was confirmed that when the inclination of ζ(X) (at the plotted point) is great, the ice force suddenly increases a little, and that when the inclination is gentle or negative, the ice force declines a little. This can be explained theoretically. However, we could not precisely reproduce this ice force by calculation because the observed values of ζ(X) and their least squared curves were not appropriate or themselves were not precisely reproduced.
Many of the calculated values agree relatively well with the observed values at i = 1/5. However, at i = 1/10, the observed values are a little greater, and they do not agree with the calculated values very well, probably because at dζ/dX < 0, the location on which the subgrade reaction (k) acted was not clear, and because the angle of rotation of the ice was neglected. In addition, the following causes are included: 1) During the movement of the ice, the consecutive changes in the mechanical properties of sand were neglected. 2) The coefficient of dynamic friction between the ice and sand was not constant. 3) The gouging curve, ζ(X). was not precisely reproduced. For these reasons, the formulae for calculations must be improved in future. At the same time, the behavior of the gouging curve must be clarified by theory or by systematic experiments, which remains to be solved in future because this study was made to obtain a basic knowledge.
5. Conclusions
The results of this study are:
1) The depth of gouging by ice varied according to the angle of inclination, 0, of the front of the model ice,
