It is not hard to see that with the advancing movement of the load, and consequently of the 

 basin of deflection, ship-like waves (in eschelon, cross waves, etc. , ) will be created in the water 

 near it, which are also transmitted through the ice. 



The deflection of ice under the weight of a load is ordinarily compared to the problem of the 

 deflection of an elastic plate on an elastic foundation. But this problem presupposes uniformity of 

 the plate and ice is extremely heterogeneous, both vertically and horizontally. The vertical 

 distribution of temperature in natural ice is extremely peculiar and changeable, and all the 

 mechanical properties of ice are functions of the temperature. The crux of the matter is that when 

 solving the problem of an elastic plate on an elastic foundation it is supposed that the load does not 

 exceed the elastic limit. But the elastic limit of ice, as we have seen, is so small that ice should 

 be considered a plastic, frangible body. Furthermore, in practice in constructing and using ice 

 crossings and ice airdromes, for instance, the type of ice deflection is important not near the 

 limit of elasticity but rather near the limit of plasticity. Finally, when loads are moved on ice, the 

 ice deflects in conformity with the "water" wave which forms under it. All these facts taken to- 

 gether create unusual difficulties for a mathematical analysis and make it necessary to have re- 

 course to formulas which even though approximate, still satisfy the practical requirements. 



Examining the graphs from this point of view, we see that the curves of deflection are very 

 similar to logarithmic curves. Such a view of the curves is also supported by certain theoretical 

 conceptions. 



Actually, let us assume that the load is placed on an ice field, whose horizontal dimensions 

 can be considered infinite (figure 66). The ice will deflect somewhat under the weight of this load 

 whereupon the maximum deflection will be at the point of application of the load and to all sides of 

 this point the deflection will decrease according to a certain law. In order to establish this law, 

 I make the following recommendations : 



1. After equilibrium is established the curve of deflection of the ice due to a load has no 

 points of inflection. This assumption is completely natural, keeping in mind the properties of ice. 



2. The decrease in amount of deflection from the point of application of the load in a direction 

 toward the periphery, is proportional to the deflection at the load site and the increase in distance 

 from the load. 



The volume of the deflection, equal to the volume of the water displaced, stabilizes the weight 

 of the load. This assumption clearly excludes certain properties of ice, but simplifies the calcula- 

 tions considerably. 



In agreement with the second assumption we obtain 



(lz= — kzdx, (1) 



where s is the arrow vector of deflection, x is the horizontal distance, and k is the coefficient of 

 proportionality which is called the coefficient of deflection. 



After integrating, we obtained 



z = z,e-kx. (2) 



where s is the length of the vector of displacement at the origin of the coordinate (at the point of 

 application of the load). 



196 



