304 Proceedings of the Royal Society of Edinburgh. [Sess. 
At Front Face. 
At Back Face. 
u (in feet). 
v (in feet). 
u (in feet). 
v (in feet). 
Case I 
„ II 
„ III. . . 
Net movement . 
0-000,0000 
+ 0-000,0946 
- 0-000,2719 
- 0-000,2425 
-0000,2170 
+ 0-000,1976 
0-000,0000 
+ 0-000,0946 
-0-000,2719 
-0-000,1763 
- 0-000,2425 
+ 0-000,2170 
- 0-000,0695 
-0000,1763 
-0-000,2619 
- 0 000,0550 
The movements at the axis of 0 * * * § due to the same forces are as follows : — 
Case I. u= O'OOO, 0000 feet, v — - 0*006,895 feet. 
„ II. u= +0-000,1577 „ v= 0-000,000 „ 
„ III. u= +0-000,8586 „ v= +0-001,294 „ 
The data used in the calculation of the above were 
pg/ 2-3 = P = 62 5 lbs. weight. 
and 
X = fx — 2E/5 = 1,000,000 lbs. weight per square inch. 
(16) The complete solution of the problem of the dam would require 
the determination of the stresses not only in the dam itself but also in the 
foundation, with the necessary condition of continuity of displacement 
throughout the whole mass, and for the simplest form of dam would 
require the satisfaction of the boundary conditions along four separate 
boundaries, and to obtain this seems practically hopeless. 
The calculation of the stresses by the great writers Messrs Graeff and 
Delocre,f Rankine,J Bouvier,§ etc., has been worked out in the case of 
practical masonry dams on the well-known theory of the line of resistance, 
or what may be called the point load and bending moment method, and it 
has usually been assumed that at least the normal stresses on any section 
(generally taken horizontal) vary uniformly. (At the same time, it should 
be stated that some at least of the above writers were well aware that 
uniform variation of stress might not have been true.) Further, it has 
generally been taken for granted, tacitly or otherwise, that what applies to 
a section above the base also applies at the base. 
The usual methods of calculation of stresses, while giving the total 
shear on a section, do not indicate how it is distributed, and this has led to 
* For actual movements in the Yyrnwy dam, see Proc. Inst. Civil Eng., vol. cxv. p. 117. 
+ Annales des pouts et chaussees , 1866, pp. 186, 212. 
+ Kankine’s Miscellaneous Scientific Papers , p. 550. 
§ Annales des pouts et chaussees , vol. x., 1875. 
