80 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[March, 



Fig. 2. 



Put s for tlie mean weight in lbs. of a cubic foot of the materials in 

 the (lam, h for its height, in feet, d for its width in feet, and c for the 

 deptli of high water in feet. We then have b d s equal the weight of 

 one foot in lenght. It is evident that the dam fails only when the 

 force of the water is able to turn it round the point D, and as the 



d 

 weight hds acts at the distance - from D in the figure its force is pro- 

 perly represented by bds X - =; — — . The pressure of the water 



125c- c 



is equal — — - (problem 1,) which acting at the distance ^above D 

 2 o 



125c" c 125c' 



(problem 3) has its force represented by -— — X-^: . Therefore 



2 3 6 



in case of equilibrium we have =: — — from which equation we 



find d- 



125c^ 

 367 



and d: 



V125c' 

 36i" 



From this proposition it is clear 



tliat when h^c, tlie width d of the dam is proportional to the depth of the 

 teutcr, and that the power of water to overturn a clam is as the cube of its 

 depth. The value of s will depend on the nature of the puddle used 

 in the dam, and the proportion it bears to the quantity of timber and 

 iron in the width d. In the examples to this and the following jiro- 

 blems, « is supposed to be equal to 90 lbs., which in most cases may not 

 be far from the true value, except in those cases where the water 

 penetrates under the dam when it must be reduced to about one-third, 

 or to 3U lbs. nearly. , 



F,g. 3. 



Example 1.— Find the width of a cofier-dam sufficient to resist the 

 pressure of 17 feet of water on the outside, the height of the dam 

 being 19 feet. 



Here we have 



d=^ /l^- K /i!5M.^ -^ /lilil^=V116-4=10-8feet. 

 -V 3 6s~'V 3X 19X90"" -V 5130 



/125 

 -— r=ll-Gfeet. Ifwe suppose from 



want of proper precaution the water to penetrate under the dam, s is 



s s 



reduced to about - for the height c, say -, we then get 



« d 125c' 

 (J — c) rfs-fc«?X-X-= — r^ for the equation of equilibrium from 



which we find d-X (3 m6s+3cs 

 125hc' 



3 ncs) := 12 5 c'h and 

 When ;!=3 as would be nearly the case 



d- /__J^^nc^___ 

 A/ 3n6s-|-3cs — 3!(cs' 



V125c' 

 ■Xr36— 2 1 ' ^-^ "^'"^ ^^^^ numbers in example 1, 



we get by this formulae 



d=. /Zi1!HI!L:= . /^^=^296-7=17-2 feet,shewing 

 /y 90 X (57— 34) ■V 2070 ^ ^ 



under these circumstances a necessary increase of nearly six feet in 

 width. 



Example 2. — What width of dam is sufficient to resist the pressure 

 of 17 feet depth of water, the dam to rise 4 feet above the surface, 

 when the bottom is porous gravel communicating with the water. 



In this case we have 



d— /__^^^___ / 125 X 17' _ /G14125_ 



'V 90(3 6—2 cf 'V 90 X (1^3— 34) '\/ 90 X 29~" 



V 



614125_ 

 2610 " 



:V235-3=15-3 feet. 



Problem VII. 



To find the strength of a coffer-dam (fig. 3) sufficient to resist the 

 pressure of a given depth of water so that by the intervention of stays, 

 &c. the coffer-dam could only fail by the failure of the point D. 



Put k for the distance E D, rf for the distance E F, and by using the 

 same notation as before for the other dimensions, we get by the pro- 



perties of the lever bds')(.(--\-l{)z 

 brium, and by reduction d'--\-2 kd- 



12 5c^ c 

 - Xq for the equation of equili- 



125c' 



36s 



from which we find 



V123c' 

 368 ^ 

 Example 1. — Find the width d when fc=18, c=17, and 6=21 feet, 



, /l25e' , / 125X17' „ 



'"'■^ y-Sb^ +' '-'= V 3^X21X90 +^S'-18= 



V 614125 



-g^J7^+32-4— 18=1/649— 18=25-5— 18=7-5 feet=rf. These 



were nearly the dimensions of tlie coffer-dam for building the river 

 wall at the New Houses of Parliament (see Journal, vol. 1, page 31). 

 But this coffer-dam was still held more firmly on its base by the re- 

 sistance to the piles penetrating the silth and clay substratum requiring 

 a considerable force to overcome it, over and above that which was 

 already sufficiently resisted by the upper portion of the coffer-dam. 



When d is given we find from the equation d--\-2d/c^-^. — , 



i = 



125e^ 

 Gbsd 



d 

 "2* 



Example 2. — At what distance from the imier sheet pilcing of a 

 cofier-dam 10 feet wide shall we place the brace pileing D, so that 

 when properly braced the dam shall resist the pressure of 30 feet depth 

 of water outside. The dam rising 4 feet above the surface. 



„ , 125X30'i 10 3375000 ^ ,^ _ ^ ,o . r . 



Here A= = 5=18'4-5=13'4 feet 



ex 34X90X10 2 1B360 



the distance required. Ifsi=30as would be nearly the case if the 



