82 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[March, 



=v^ 



•2,500 — 3300 



17 



+ 100 



/9200 



+ 100 — 10 



= V511-2 + 100 — 10 = 25-3 — 10 = 15-3 fee(. 



If we t;ike into consideration the weight of the water above K G, 

 the vahics found for rf and X- in the examples are too high; but the 

 gravity of tlic materials in the dam being to the gravity of the water 



28 



as 8 to G2i, if we substitute /' + c — / x — , or /' + c — / ' x § 



(nearly), for/' in the general equations (1) and (2), we will find cor- 

 rect values for d and /. Thus in examples 1 and 3,/' = 10-]- 17 — lo 



X S = Vi =""1 '" examples 2 and 4, /' =: 17 + 30 — 17 x § =:y ; 

 using these values ofy we would find in 



Example 1. — d= 7-4 feet ; 

 Example 2. — d= 11-2 feet ; 

 Example 3.—*= 5-9 feet; 

 Example 4. — i= 11-4 feet ; 



all of which are intermediate between the former values and those 

 found in the examples of problem 8. It appears therefore in these 

 examples that fig. 4 is to be prefered to fig. 5. If we wish to have 

 equal strength in these two forms, we get by equating the general 

 equations in problems S and 9, s bd" + 2 s bdi -\- a /c''/=^ ^ 1^''/'+ 

 2 s di'f + d''8b, and thence 2 A rf * + F/:= *'-/' + 2 di'f, which 

 equation will furnish the value of any of the quantities when the others 



are given. By substituting/' 



2c-2/' _ 2c+/' 



for/', we take 



into consideration the weight of v^ater over K G ; in assisting the 

 stability of the dam Fig. 5, this substitution gives us Gbdi -\- 3 Wf 

 ^■2c li"' -\-4cd h' + (It' + 2 ri ''')/', for a general equation of equal 

 strength in both forms. 



The subject we have now been considering, is closely connected 

 with the consideration of the comparative strength of buttresses and 

 contreforts to retaining walls. If we put n for the weight of a cubic 

 foot of earthwork or filling, and s for that of masonry, and substitute 



/'X 



c—fy.1 



for/' in the equation 2 6rf/t + *'/=; 2 rfA'/' + /t'2/', 



we get 2 hdsk + k'-fs = {k'-^ + 2 dk') X »g + (s »)/' ^^^ ^ ^^^^^^1 



equation of equal stability between buttress and aontrefort, by which 

 we may with ease determine any of the dimensions by having the 

 others given, as none of the quantities rise higher than the second 



power. The quantity — ^ is the height of a prism of masonry 



equal in weight to a prism of clay whose height is c — /. This prism 

 acts with the clay or filling in moving out the wall, and also, from its 

 weight on the cnntrefort, gives the latter greater stability. This 

 double action often seiiarates the contrefort from the main wall when 

 both are not well bonded into each other. 



Having pointed out the method of taking the weight over K G, 

 Fig. 5 into account, where considered necessary, we will neglect it in 

 the examples to the following problem, though the formulae are general 



2c+/' 

 by substituting — —— for/'. 

 o 



Problem X. 



To find the dimensions of a coffer-dam. Fig. G, sufficient to resist 

 the pressure of a given depth of water. 



By observing the same notation as in the former problems, we get 

 from the principles of the lever, 



*/A-' + (k + d + '^) +shdx (A + |) + sfkxl " -^^ ^'3 

 for the equation of equilibrium, and by reduction we find 2 k k'/'-{- 



IOC «3 



2 k'f d + k'-f -\-2kdb + bd'--]-fk-:= i^ = m c' by putting 



125 

 in = -5—. From this equation we find 



,. , 2k'f' + '2.kh me—'xkvf—yf'—k-'f 

 a- -Y ^ Ad „_ ; 



Fit,', (i. 



A B = rf 

 B H = A 

 CE = c 



IH=/' 

 H D^* 

 FE = *' 



KE=/. 



/t2 J_ 2&rf-t-2/^'/' _ m e — 2/ d V — /' V — bd- 

 f " / 



and X-2 -f (2 (i + 2 /^) k' = 



ne — 2bdk—fk-—bd- 

 f 



From these equations we find, by quadratics, the following general 

 values for d, k and k'. 



d ^ fme-%k k'f^' W^flc ^ ^f + ^fc V:_ k'f + k b ^^^^ 



j,^ / liK^2df'k'~f'k"—bd- bd+yy^ M-fA-/' ^,, 

 V / "^ —7—1 "~7=~(^*- 



^,__ / mc' — 2 bdk —fie — bd- 



/' 



+ rf+7l= — 3. — k 



(3). 



When f^=f', as is generally the case in practice, we get by a 

 simple reduction. 



'=\/'^^ 



■fy-ik+k'^) , *'/+ 



+ 



Jl^= 



k'f+kb 



(4). 



/ mc^ — (2 rfA -^ A-a)/— bd - ba + k'f ] 

 V ft +6 



bd + k'f 



k'-- 



■{b—f)XSdZ±^dk) 



■d — k 



(5). 



(6). 



from which equations, by having any two of the widths a, k, and k' 

 given, the other may be found. 



Example 1. — Required the width of the main dam in Fig. 6, the 

 depth of the water to be resisted being 30 feet, and the other dimen- 

 sions as follows, viz./=/' — 17 feet ; A = 7 feet ; *' =r 10 feet ; and 

 6 = 33 feet. 



By equation (4) we have 



^=a/" 



X SO'— 17X 17= 



33 



I 10 X 17 -f 10 X 33"] 

 33 I 



10 X 



17 -^ 7 X 33 _ / 12,500 — 4913 4oT|° 401 ^ 



~33 ~ '\/ 33 33' 33 



t/229-9 + 147-6 = 12-2 = V377-5 — 12-2 = 19-4 -- 12-2 = 7-2 ft. 

 which nearly corresponds with the width of the principal dam in the 

 coffer-dam iissd by Telford^ for bHilding St. Katherine's docks, the 



