1840.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



79 



Example 1. — Given the dejith of water on the outside of a dam 

 equal 20 feet, that inside equal 6 feet, and tile giitli GO feet, what is 

 tlie cirecti\'e pressure against the dam ? 



We have c+(i:=2G, c — d-=.\i, and ^=G0 therefore 



l-25x(e+^)X(c-i)Xg^l25x2Gxl4xGO^^^^^^^^^ 14x30 



=45,500x30=T,3G5,00011)s. for tlie pfFective pressure. 

 When the inside and outsiiU' girths dilter, by putting g fur tlie out- 



side girth, and ^' for that inside, we get in this case 



125(c-g— i'g') 



for the effective pressure. 



Example 2. — Given the height of water on tlie sill to the upper 

 gates of a lock above, 10 feet and girth 21 feet; below 4 feet and 

 girth 25 feet — what is the effective pressure on the gates? 



„, . , ]25fc=g— rf=g') 125(100x24—10x25) 

 The pressure is equal • ^ — -S~'z= i 



=rl25(100x24— 8x25)= 125x1000= 125,0001bs. the pressure re- 

 quired. 



Example 3. — Find the effective pressure against a coffer dam, the 

 exterior depth and girth respectively being 2/ and 120 feet ; and the 

 interior depth 5 feet, and girth 100 feet. 



„ , ,, , , 125(27x27x120—5x5x100) 

 Here by the formula; ^ = 



' 2 



125(729x00— 25x50)=:125x424,90=5,311,250 lbs. the pressure re- 

 quired. 



Problem III. 



To Jind the centre of pressure in a given depth of water : or iliat point 

 where the force of the whole pressure is equal to the sum of the forces 

 arising from the pressures at different depths from the surface. 



The whole pressure (problem 1) is represented by a right angled 

 triangle having its base and perpendicular each equal to the depth of 

 water, and as the pressure at each point along tlie depth is propor- 

 tional to the depth of sucli jioint from the surface, or which is the 

 same thing to a line parallel to the base at that point meeting the 

 hypothenuse ; the centre of pressure is evidently on the same liori- 

 zontal line with the centre of gravity of the triangle. But the latter 

 is at one third of the perpendicular from the base, uierefore the centre 

 of pressure is at one-third of the depth of water from the bottom, 

 or ic. 



Examples. The centre of pressure in 15 feet of water is 5 feet 

 above the bottom : in IS feet of water at 6 feet above the bottom : and 

 in 30 feet of water at 10 feet above the bottom. 



Problem IV. 



To find the centre of pressure mhen given depths of water are inside and 

 outside a coffer-dam. 



By putting as before c for the depth outside, and dior that inside, 

 we find the outside pressure acting at the distance ^c from the bottom 



125c- 

 equal — — — (problems 1 and 3), and the inside pressure acting at the 



u 



d 125(/^. 



distance 5 equal -^ — The centre of pressure is now therefore in 

 tj 2 



the fulcrum of a lever, whose length is — ^— i which lever is acted on 



i— - — '■ — — — the distance 



125c' 125^^ 



at its ends by the two pressures -=- and — —. To find this point 



125c^ , 125d 

 we have —^ + — ^ 



c—d . . 125<;' 

 3 • ■ 2 



3(c■■^-<^■ 



of the fulcrum from a point corresponding to ^c, tlierefore =— . 



o, . I ,0- - — o, o I J..-. — — o. .. I ..., • The distance of the point 

 3(o-fa-) 3(c=+rt-) 3(c +rf') "^ 



required from the bottom of the water frsm which w'e deduce tlie fol- 

 lowing rule : — 



Divide the sum of the cubes of the inside and outside depths by 

 three times the sum of their squares, the quotient will be the distance of 

 the centre of pressure from the bottom of the mater. 



£»»)»?/?,— Take i=^2Q and d~V) we then have 



0000 



: 6 feet for the distance of the centre 



20'-f iO' _ S000+1000_ 

 3(20^+10')" 3x500 ~1500' 

 of pressure from the bottom. 



Prodlem V. 



To find the centre of pressure in a depth of water lying between the 

 depths c and d below the surface. 



Let c be the greater depth, and put .r for the distance of the centre 

 of pressure in the depth c — rf, from the centre of pressure in the depth 

 c ; we then have from the properties of the lever 



.rX 



V25(c'—d^)_2(c- 



-d) I25d' 

 — X— r-, 



from which equation by an easy reduction we find, 

 2d- 



■2d- 



there- 



fore — 



3 3(c+f/) 



of the depth c, and — + 



3(c+d) 

 is the distance of the point required from the bottom 

 2d' 



3{c+d) 



its distance from the surface of the 



water. 



Example 1. — In 15 feet depth of water what is the distance of the 

 centre of pressure of the lowest 5 feet from the bottom ? 



Here 



:5 and 



2d- 



2x10- 



3(c+rf) 3X( 15+10)" 



200 8 , , , 



z-—--^- leet, therefore 



-=5 — -=- feet the distance required. 

 00 



3 3(c+rf) 



Example 2. — Two stays support a coffer-dam at depths of 20 and 

 10 feet below the surface of (he water, and it being found necessary to 

 place another between these, at what distance shall we place it from 

 the lower stay, so that it may afford the greatest assistance possible ? 



It is easy to see that the third stay must be applied opposite the 

 centre of pressure. To find this point we have c:=20 and rf=;10, 



e ^ 2d- _20 2xlCr- _20_200_20 

 therefore g a^c+f/) - 3 3(20+10) ~ 3 90 ~ 3 ' 



40 44 ^ 

 =:-„-=.- feet, the distance required. 



20 



■ 9" 



60—20 

 9 



A proper knowledge of the position of the centre of pressure will 

 enable us to place our stays with advantage and economy, particularl)' 

 in those cases where a coffer-dam is surrounded with water. If the 

 top and bottom of such a coffer-dam (fig. 1) are Ijept from approaching 



Fig. 1. 



each other, the next best point to secure is evidently at the centre of 

 pressure of the whole depth of water, or using the same notation as 

 before at ^c from the bottom. If more stays are necessary, the most 



7c 5 c 

 important points to be secured are those at the distance — and —- 



from the bottom, or in other words at the points corresponding to the 

 centres of pressure in the lower and upper portions of the depth 



c , 2c 



Problem VI. 



To find the dimensions of a coffer-dam fig. 2 suflScient to resist the 

 pressure of a given depth of water when the section is rectangular. 



