Ig.t8. 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



197 



a force (tf impact,— for the pair of buffers behind will be still 

 acting for some little time after the first pair have ceased to act. 



Again, let e'V be the velocity destroyed before the second pair 

 of buffers cease to act ; then, as before, the second carriage will 

 be suddenly brought to rest by an impact on the first carriage ca- 

 pable of destroying the velocity (l-e').V in the second carriage. 

 Next, the third carriage will be suddenly stopped; this, likewise, 

 will communicate a shock to the first, though less than it commu- 

 nicates to the second; and so on. These shocks and motions will 

 be somewliat \aried by the resiliency of the buffer-springs, and the 

 impulsive friction of tlie rails against the wheels ; — this latter dis- 

 turbing force we have altogether omitted, as being comparatively 

 insignificant. On the whole, then, it ajipears that buffers very 

 much diminisli the intensity of a shock, altliough they are in- 

 capable of utterly absorbing it, supposing it to be of great intensity. 



I. H. R. 



ON STONE WALLS AND EMBANKMENTS FOR 

 RESERVOIRS OF WATER-WORKS. 



By R. G. Clakk. 



The intention of this paper is to treat of the pressure of water 

 against walls and embankments of reservoirs for water-works and 

 canals, and to lay down some easy formulfe to find the necessary 

 dimensions so as to effectually resist the pressure of the water ; the 

 demonstration of these formula> being effected by the simplest 

 methods of investigation. We shall first exhibit some investiga- 

 tions for stone walls. The level of the surface of the water in all 

 cases to be supposed on a le\el, or co-incident with the top of tlie 

 wall or embankment, so as to favour the stability of the structure 

 in case of floods or violent agitation of the water by storm winds, 

 — altliough the water might be when in a quiescent state but two 

 feet from the top. The following description of walls are required 

 to inclose a few acres of water, when there is no suitable kind of 

 earth to be obtained in the locality for forming an embankment. 

 The walls are to be constructed in solid masonry, of a uniform 

 connection in all its parts. 



I. Given the height of the wall, the depth of the water being the 

 same, and the batters on eacli side of the wall equal ; to determine 

 the thickness of the wall at the bottom : — 



Let A B C D be a vertical section of the wall ; D W the level 

 of the water; let .i- denote the required thickness, A B, of the wall; 

 the batter, B F or E A, by b ; the height of « all by « ; the specific 

 gravity of water by unity ; and that of the material by .v. After 

 some reduction, we have for the equation of equilibrium. 



4 «6.r-^&-n — ^n'-{- .9Mh( := («) ; 



(see " Moseley's Hydro- 

 statics," Art. 51), wliere 

 M = area, and ni hori- 

 zontal distance of cen- 

 tre of gravity G from A ; 

 M = i a (2 ,i- — 2 ft) = 

 n (.r — 6) ; and ?n =i 2 .*' ; 



Therefore, 

 s M ))j = 2 a X i'V—b) s = 

 moment about A ; which 

 substitute in the above 

 general equation, 



"1+1: (1). 



3* 



E.f. Given the height of the wall = 24 feet ; batter each side, 

 Ifeet; and the specific gravity of the material, 9 : to find the 

 tliiekness at bottom and top. 



Here a = 24 ; ft ^ 4 ; and s = 2. 



i^ubstitute these values in (1), we have ,1- - 2 j- = 98'6 ; 



. • . X = 11 nearly = A B ; and C D = 3. 



II. Let the vertical section of the wall be rectangular, or the 

 sides vertical ; to find the thickness at bottom : — 



Now area A B CD X J distance of centre of gravity from A = ^ »^ 



we have .c- + (4- ft) 



^ + Q-0 



(3). 



a? J^= i a^ ; 



— ; hence, j 



3s 



(9). 



£•.1'. Given the height of the wall = 24 feet, and the specific 



gravity of the material, 2 : to determine A B. 



■1/ / 24^ \ 

 ,r =: {- — - ) = 9'8, the thickness required. 



III. Let the side of the wall next the water be battered, and 

 the side behind vertical ; to determine A B, the thickness at the 

 base. 



Let A B C D be the vertical section, and let fall the perpendicu- 

 lar C E. 



The momentum of the triangle 

 C E B about A from its horizontal 

 distance of centre of gravity g, 

 s . ^ a b X (i; b + X — b). 

 The momentum of rectangle 

 D C A E about A from its centre of 

 gravity, 



s'a (x— b) X i{x — b). 

 Adding these two together, and 

 substitute for s M m in equation (o), 

 as in first case, we have, after 

 transposing, 



t±JL - i If 

 • 3.y 3 



Et. Gi\en the height of waU, 24 feet ; batter, 4 feet ; and 

 specific gravity of stone, 2 : required the thickness of the top and 

 bottom. , . ^. ,„. 



By substitution of the above values m equation (3), 



wehave, .r^ — 2.r= 92-r. 

 Solving this quadractic, we have ,1 = A B = 9-2 ; top, 5-2. 



IV. When the wall is battered behind, and the side facing the 



water perpendicular. „ , ^ . , , 



The two first terms of («) vanish when 

 B D is vertical ; .-. sUn — ia\ 



The moment of triangle A EC about A, 

 by its horizontal distance of its centre of 

 gi-avity, 



« . ^ a ft X I ft ; 

 and also of rectangle C E B D about A, 



(X — ft \ 



. • . substituting the sum in equation (1), 



we have x- =---)-- ft . 

 3s 3 



Ex. Given the dimensions of the w all as in last example ; to 



determine the thickness at bottom : — 



,r2 = — + — = 96 -i- 5-3 =: 101-3 ; 

 6^3 



. ■ . ,1 = 10- = A B ; and C D = 6- 



For additional strength to the above walls, it would l)e well to 

 insert at the centre of them one tier of bond, about two-thirds the 

 height from tlie top, which will be at tlie centre of pressure. 



V. We shall now give a case where earth shall he required in the 

 construction of an emliankment, of the form of a trapezoid, 

 having a vertical clay puddle-wall in the middle, and the slope 

 facing the water being paved with suitable material, with a puddle 

 under. In case of any contraction of the clay, there would be » 

 separation of the clay from the earth ; therefore, the triangle 

 DEB should be of sufficient strength alone to resist the fluid 

 pressure, either against sliding or revolving on D. The water i» 

 supposed to be co-incident with the top of the embankment. 



Fig. 4. 



Let D B =: a ; B E = «. Draw P M parallel to F £ ; and the 

 horizontal pressure of water = momentum B P M X horizontal 

 distance of centre of gravity from D ; 



.-. ^^=|y7j,^rf^; .-.f^lx^; .•.DE=1/(J). 



