306 Proceedings of the Royal Society of Edinburgh. [Sess. 
the loaded face 0 — a, and there is a total load P between x = 0 and x = a, 
the face 0 — — a being free, then on the section x = a 
For uniform loading . . xy= - P y 2 /r 2 a(tain 2a - 2a) , 
For load varying as x . . xy=- 2P?//a 2 tan 2 2a , 
and here again concentration of the load to the section tends to produce 
uniformity of shear, although at y = 0 the shear must vanish in every case. 
The above examples serve to show that the manner of loading has an all- 
important bearing on the question of shear, whatever modifying influence 
the foundations may produce near the base in the case of an actual dam, 
and the same remark applies to normal pressure. 
(19) The question still remains what are the stresses at the base of the 
dam, and to this the writer can give no real answer ; and this raises the 
further question as to whether the movements at the base, as found in this 
paper, are inconsistent with what may obtain in the case of an actual dam 
resting on a good sound rock foundation. According to the elastic theory 
the stresses will be accurate only if the movements at the base are con- 
sistent with those found by equations (39) and (40). 
Suppose, however, the complete solution of the dam were obtained in 
the general case ; the question of stresses due to shrinkage of the material 
in actual dams while setting and drying, and the stresses due to changes of 
temperature, still remain untouched, and it seems clear that the stresses set 
up by either of these causes must attain their maximum value at the base, 
where the abrupt change of section takes place. It is thought the stresses 
produced by either of the above causes may far outweigh those due to the 
super-rigidity or the lack of rigidity in a good sound rock foundation, and 
this contention seems to be accentuated when it is observed that the 
movements at the extreme points of the base are less than those which 
would be produced by a change of temperature of 2° F. in a wedge of 
concrete. In view of the elastic movements being small in comparison with 
the probable temperature movements, it is held that the stresses as found 
will in all probability be reasonably correct over a considerable period of 
the year, depending on the depth of the line of contact of the foundation 
below the surface of the ground and the range of temperature at that 
depth. 
( Issued separately September 5, 1913 .) 
