TA 
 
 r 
 73- 
 
 SB 55^ 5T4 
 
 JRCHARUKJJ 
 
 DIFEEBENT FOBMS 
 
 RETAINING WALLS. 
 
 JAM2S S. T\TE, G. E. 
 
 f D. VAN 
 
 NEW YOTiK: 
 NOSTRANIJ, PUBLISHER, 
 
 23 ML'KKAV AND H", WARHHX 
 
 1873. 
 
LIBRARY 
 
 OF THE 
 
 IVERSITY OF CALIFORNIA 
 Qa*s 
 
SURCHARGED 
 
 AND 
 
 DIFFEBENT FOEMS 
 
 OP 
 
 RETAINING ALLS. 
 
 BY 
 
 JAMES S. TATE, 0. E. 
 
 NEW YOBK: 
 D. IN NOSTRAND, PUBLISHER, 
 
 23 MURRAY AND 27 WARREN STRB-KT. 
 
 1873. 
 

PREFACE. 
 
 This little Work is intended to supply 
 what has no doubt been often wanted by 
 many Engineers a certain and ready means- 
 of correctly and easily ascertaining the Pres- 
 sures of Embankments, Submerged or other- 
 wise, composed of different materials ; also 
 the Moments of Retaining Walls of differ- 
 ent forms of cross- section, to successfully 
 withstand those pressures ; so that, by know- 
 ing the exact value of each, the right dimen-^ 
 sions of the most suitable form of wall for 
 the purpose required can be at once ascer- 
 t ained. 
 
 As the method adopted does not involve 
 the use of any long or laborious calculations, 
 it is hoped it will prove useful to the Pro- 
 fession generally. 
 
RETAINING WALLS. 
 
 Retaining walls are adopted as a neces- 
 sary expedient in railway and other practice, 
 often under very peculiar circumstances, as 
 when there is not sufficient room for the 
 slope of the embankment ; it being some- 
 times perched high on a steep mountain's 
 side, and where it would have been hardly 
 possible to construct a railway at all, except 
 by securing it with a massive wall occupy- 
 ing comparatively little space. 
 
 When it is also remembered how fear- 
 fully terrible any accident would be if it ' 
 was to occur in such a dangerous situa- 
 tion if by any erroneous calculation or 
 mistaken judgment on the part of the en- 
 gineer sufficient strength had not been given 
 to the work, the wall which was to have/ 
 supported the embankment, suddenly giving 
 way, falling over into a deep ravine or chasm, 
 a large portion of the embankment going 
 
6 
 
 with it, and, it may be also, a passing train 
 there can be no doubt but that the nature 
 of the material of which the embankment 
 is to be made should be understood, and 
 the best* form and requisite dimensions for 
 the wall should be well considered and ac- 
 curately ascertained beforehand, so that it 
 may be amply strong enough. 
 
 At the same time that the wall should be 
 made perfectly secure, it is also often desir- 
 able that any unnecessary excess of strength 
 should not be given to it, and so thereby 
 avoid increasing its cost considerably, as 
 the value of work is often very much en- 
 hanced when it has to be executed in such 
 inaccessible situations as before mentioned, 
 where all the materials for building it may 
 have to be brought from a great distance. 
 
 The engineer thus may be at a loss to 
 determine of what size a retaining wall 
 should be built, so as to be safe against all 
 contingencies that can occur, and yet also 
 to be economical. 
 
 In many cases there have been failures 
 which may have arisen from not correctly 
 ascertaining beforehand how the material of 
 
which the embankment is composed will be 
 affected by the alternations of wet and dry 
 weather before it is thoroughly consolidated, 
 and the precise angle at which its slope will 
 stand in either case, thereby causing a con- 
 siderable difference in its pressure against 
 the wall. 
 
 A retaining wall also, as in the case of 
 the wing-walls of a bridge, being built at 
 the same time that the embankment is being 
 filled in behind it, has often to withstand 
 then a considerable greater pressure than it 
 will have to do afterwards when the em- 
 bankment is settled ; this also perhaps 
 when its work is green, and not prepared 
 to resist the pressure intended forit. Some- 
 times also the punning of the material be- 
 hind it has (as is often the case) not been^ 
 done effectually, and a heavy rain changes 
 the dry Dearth or clay into a wet sludge, 
 causing it to swell considerably. 
 
 It therefore being such an important 
 point in railway construction, it would no 
 doubt be very desirable if some simple form 
 of calculation were used, not only strictly 
 accurate, but easily adapted to any circum- 
 
8 
 
 stances that may occur. In the case of a 
 wall where the embankment is level with 
 its top the calculation of the pressure is 
 well known, being very simple, and is as 
 follows : 
 
 Let B D be the back of a retaining wall, 
 
 D E the natural slope of the embankment, 
 
 i 
 
 A B G E 
 
 then if we bisect the /_ B D E by the line 
 D G, B D G is the portion of the embank- 
 ment supported by the retaining wall. 
 
 Now the weight of B D G : pre'ssure of 
 its weight against the wall : : B D :.B G : : 
 H : H tang. L B D G. The weight of 
 
Pressure of weiht of 
 
 Moment of pressure of weight of 
 
 BDG = H'X ta " g -^ BPG Xjrx* 
 
 Ja 3 
 
 H 3 
 
 = _ x tang. 3 L B D G X IF, 
 b 
 
 and the double of this moment for stability 
 
 In the case of a vertical wall, as A B C D, 
 its weight = W H B, and the moment of 
 its weight 
 
 _ W HB2 
 
 3 " 
 then for equilibrium, 
 
 - 2 = X tang. 2 L B D G x W, 
 
 'W 
 
 n/ 
 and B = H tang. _ B D G 
 
10 
 
 and for stability, 
 
 W H R 2 TT 3 
 
 _J1- = L x tang.' L BDG x W, 
 & o 
 
 fi~n 
 V T" 
 
 and JB = H tang. L B D G . 
 
 V W 
 
 The figures in the columns of. Table No. 
 1, are calculated from this last formula, 
 and are 
 
 /oTrf 
 Htang. Z.BDG J, 
 
 BO if divided by the square root of the 
 weight of a cubic foot of the wall, they will 
 give the thickness of the wall. 
 
 Table No. 2 gives double the moments of 
 the pressure of the weight of different ma- 
 terials to form the embankment, calculated 
 from the formula 
 
 -* Xtang. 2 ^BDGx TF, 
 o c 
 
 and which, if made equal to either of the 
 moments of the weight of different forms of 
 retaining walls given, the dimensions of 
 that form of retaining wall required can be 
 readily ascertained. 
 
 Having now given the usual formulae and 
 
11 
 
 Tables for easy calculation deduced from 
 them, for calculating the dimensions of a 
 retaining wall with an embankment level 
 with its top, what is next required is a con- 
 venient and ready method of accurately 
 calculating the pressure of a surcharged 
 embankment. The author is not aware if 
 the method of calculation and formulae he 
 gives here are new, but the Tables for gen- 
 eral use have, he thinks, the merit of sim- 
 plicity. 
 
 When the embankment slopes away up- 
 wards above the top of the wall, the calcu- 
 lation of its pressure is a little more com- 
 plex, and no method of finding it has yet 
 been given that is simple, or that can be 
 easily used in practice. Moseley, Hann, and 
 Rankine, in their works give equations very t 
 abstruse, and apparently of no practical ap- 
 plication. Hann also takes into account 
 the pressure of the slope of the embankment 
 resting on the top of the wall, a refinement 
 of the calculation practically altogether un- 
 necessary, and which, by complicating the 
 original equation, renders mistakes more 
 likely to occur. 
 
12 
 
 If A be the natural slope of the em- 
 bankment rising upwards above the top of 
 the wall A B G H, B E a line parallel to it 
 from the foot of the wall, B C bisecting the 
 L A B E, then A B C is the portion of the 
 embankment to be retained by the wall. 
 Now when A B is vertical, the length of 
 the slope to be retained, A 0, will be equal 
 to the height of the wall. If L E B F = the 
 angle of the slope of the embankment = 0, 
 then 
 
 and if H = height of the wall, then 
 
 and the weight of 
 
 W being the weight of a cubic foot of the 
 embankment. Pressure of the weight of 
 
 tt A P 
 
 B A C = 
 
 moment of pressure of weight of 
 
13 
 
 RAP 
 
 ^ 2 v* 90-0 H 
 
 _ ^ x tang. __? X 3 
 
 WR* / 0*" 
 - 0^ l- 4 
 
 double this moment for stability 
 
 " H* / 03- 90 
 
 = -^~ ^ -/ 1 ~ 4 X tang, - 
 
 H A 
 
 -,'D 
 
 7L 
 
 G B 
 
 Table No. 3 gives the value of 
 
 for every deg. of inclination of the slope of 
 the embankment from 15 deg. to 40 deg., 
 
14 
 
 TFH 3 
 so that by multiplying by this value, 
 
 double the moment of the pressure of the 
 embankment will be given, and Table No. 
 5 gives double the moments of different 
 kinds of material accordingly. 
 
 In the case of a vertical wall, the moment 
 
 W H B 2 
 
 of its weight = : - W being the 
 
 weight of a cubic foot of the wall. Then 
 for equilibrium, 
 
 ,., 
 t> y \y' 
 
 and for stability, 
 
 Table No. 4 is calculated from the for- 
 
 mula .81649 H VcW^so that the figures 
 in that Table, divided by the square root of 
 the weight of a cubic foot of the wall, will 
 give the thickness of the wall required for 
 stability. 
 
 Table No. 5 gives double the moments of 
 the pressure of the weight of different ma- 
 terials to form a surcharged embankment, 
 
15 
 
 with a retaining wall up to 30 ft. in height, 
 and which if made equal to either of the 
 moments of the weight of different forms 
 of retaining walls given afterwards, the di- 
 mensions required for that form of wall 
 can be at once found. 
 
 The moment of a wall of this section is 
 
 where B is the vertical portion of the wall, 
 
 H 
 
 S B 
 
 and S is the slope. If S = 1, or 3" to a ft., 
 its moment 
 
 WH ff^ . H^ 9 H 2 ^ 
 
 The moment of a battering wall of equal 
 thickness 
 
 WHB 
 = - J3 -f- S H), 
 
16 
 
 where B = thickness of wall, and S H = 
 the batter of the slope on the face. If 
 
 I W H B c^ . HA 
 
 S r- f - its moment = I B + I, 
 
 and if E F, the perpendicular from its cen- 
 tre of gravity, falls on its inside corner, its 
 moment = W H B 2 , and the wall then will 
 have the greatest amount of resisting power 
 with security, and also with a minimum 
 amount of material in it. In that case, if 
 M = moment of earth, W = weight of a 
 cubic foot of the wall ; for stability, 
 
 S = -y/^-ga i S H being = B. 
 
 To exemplify this, let H = 20 feet, 
 S = -, W F= sand of 120 Ibs. to the cubic 
 foot in a surcharged embankment, W = 
 brick of 120 Ibs. to the cubic foot in the 
 
17 
 
 wall. Then by Table No. 5, the double 
 moment of that kind of sand = 160,000. 
 T hen for the first section of wall, 
 
 
 
 -a 
 
 B = 6.9. In this case, weight of wall 
 
 = 120 ((20 X 6.9) + (1^??)J = 22,560. 
 For second section of wall, 
 
 ^^(B + 2 ^)= 160,000, 
 
 and B = 9.31, weight of wall = 9.31 X 
 20 X 120 = 22344. For second section 
 of wall, and a perpendicular from its cen- 
 tre of gravity to fall on its inside corner, 
 120 X 20 X ^= 160,000, and B = 8.16, 
 weight of wall = 8.16 X 12 ^ X 20 = 19593 
 only, showing a considerable saving of ma- 
 terial with this wall. 
 
 At the same time, though this wall has 
 the greatest amount of resisting power with 
 the smallest amount of material in it, yet 
 perhaps it may be a question if it would not 
 be advisable to make walls of great height 
 thicker from their base upwards to one- 
 
18 
 
 third of their height, which is the centre of 
 pressure. 
 
 If we now consider a wall of this form 
 of cross-section, the outside slope of which is 
 
 B 
 
 A E C F 
 
 S to 1, and the inside slope next to the em- 
 bankment S' to 1, we find that its weight is 
 
 WH 2 
 
 H B + 
 
 (S - SO, 
 
 and the moment of its weight 
 
 -f B(SH + B)), 
 
 or if we call it C E and F, where C E is 
 the difference between the slopes of the 
 front and back of the wall, D E being 
 drawn parallel to the face A B, and F is 
 
19 
 
 the batter of the back of the wall, then its 
 weight is 
 
 and the moment of its weight 
 
 "(+.+?)) . 
 
 Then, if the height of the wall be 20 ft., 
 and its weight be 120 Ibs. per cubic foot, as 
 
 before, its outside slope to 1, and its in- 
 side slope next to the embankment to 1, 
 
 then C F = 2| ft., E C = 2| ft., and its 
 moment 
 
 = 160,000, the double inoment of the em- 
 bankment. 
 
 From this equation we find B = 8.088, 
 and therefore the weight of that wall 
 
 = 120 X 20 s.088 + 2 ~\ = 22411, 
 
 (s. 
 
 and which is, what might have been ex- 
 pected from the 'form of its cross- section, 
 
20 
 
 being between that of the first form of wall 
 before mentioned, whose weight was 22560, 
 and that of the second form, whose weight 
 was 22344, less than the one and more than 
 the other. 
 
 The form of cross-section of wall, having 
 its front and back parallel, with the perpen- 
 dicular from its centre of gravity falling on 
 its inside corner, having been proved to be 
 the most economical in material, it may be 
 asked, why should not this principle be car- 
 ried further, and walls generally be built 
 thicker at the top than at the bottom, so as 
 to have their centre of gravity higher up ? 
 This, by increasing the distance of a per- 
 pendicular from it to the outside edge of 
 the wall at its foot, would much increase its 
 resisting power to the overturning force of 
 the bank. It no doubt could be done, and 
 where the wall is of great thickness it may 
 be safe to do so, but as there is a fear, how- 
 ever, of too much reducing the thickness of 
 the wall at one-third of its height, where is 
 the centre of pressure, perhaps it may be 
 advisable to make the form of equal thick- 
 ness throughout, the limit of our endeavor 
 
21 
 
 to economize material with these forms of 
 wall. 
 
 The moment of this form of wall, with its 
 vertical side against the embankment, is 
 W H B 2 
 
 ~~3 ' 
 and if it be required to support water, whose 
 
 double moment is 20.83 H 3 , we find from 
 the equation 
 
 ^1 = 20.83^,3 = !^, 
 V W 
 
 W being the weight of a cubic foot of the 
 wall. 
 
 When the sloping side of the wall is next 
 to the water, the pressure of the water on 
 it assists the resisting power of the wall. 
 
 Its moment is 
 
 WEB 2 
 6 ' 
 and the pressure of the water on the slope 
 
 S = 62.58 x3 =31.25SH. 
 
22 
 
 Thus, when resolved into the horizontal and 
 vertical forces, the former is 
 
 31.25 S H X sin. L a = 31.25 S H X 3 _. 31.25 H, 
 
 fci 
 
 and the latter is 
 
 31.25 S H X cos. a = 31.25 S H X 5 = 31.25 HB. 
 
 to 
 
 The moment of the former force 
 
 = 31.25 H 2 X ?- = 10.416 H 3 , 
 & \ 
 
 and which tends to overturn the wall ; and 
 the moment of the latter force 
 
 O T> 
 
 = 31 .25 H B X f- = 20.83 H B 2 , 
 
 and which tends to assist the wall. The total 
 moment of the wall for stability must there- 
 fore = 2 (moment horizontal force mo- 
 ment vertical force) 
 
 = 2 (10.416 H 3 - 20.83 H B 2 ) = 
 20.83 H(H 2 - 2B 2 ). 
 
Then 
 
 and 
 
 WHB 2 
 
 6 
 
 = 20.83 H(H 2 -2 B 2 ), 
 11 18 H 
 
 V W -|- 250 ' 
 
 If we take H = 20 feet, and W = 120 Ibs. 
 per cubic foot, then in the first case, 
 
 3=14^2 =14.4-2, 
 
 and the weight of the wall 
 120 X20X 14.42 
 
 17304 ; 
 
 and in the second case 
 11.18X 
 
 1/l2u + 250 
 and the weight of the wall 
 
 11.68, 
 
 The moment of a wall of this section is 
 
24 
 
 as before mentioned, when the water presses 
 against the vertical side, but if it is on the 
 slope, the moment is 
 
 W H /" S 2> v 
 
 ^L(B(B + S)+-). 
 
 If we have an embankment of this form of 
 
 cross-section, where the slopes are the same 
 on both sides, its moment is 
 
 If the steeper slope is on the inside of the 
 embankment, its moment is 
 
 S B S' 
 
 If the steeper slope is on the outside of the 
 embankment, its moment is 
 
25 
 
 
 H 
 
 S' B $ 
 
 If in these last five equations W = 120 
 Ibs. to the cubic foot, H = 20 feet, S = 20 
 feet, S' = 10 feet, and B = 10 feet, then 
 the moment of the first section 
 
 "0X20 
 
 4 
 
 2())2 
 
 = 920,000 ; 
 
 of the second section 
 
 
 
 O 
 
 of the third section 
 
 - 120x20(10 ( 1 
 
 = 1,800,000;' 
 of the fourth section 
 = 120 X 20 (10'(20 + ! 
 
 (-20 + 
 
26 
 
 of the fifth section 
 
 In these equations the moments of the 
 walls are to be made equal to twice the 
 difference of the moments of the horizontal 
 and vertical forces of the water, as before, 
 when the sloping side is next to the water. 
 If the wall is to be built with a curved 
 batter instead of a slope, to facilitate the 
 calculation of its moment we may assume 
 the curve to be of a parabolic form, and 
 from which, in the curves generally used 
 for that purpose, it will not sensibly differ. 
 The calculations of the moments of a few 
 forms of wall with curved batter are given, 
 to show how they have been arrived at. 
 
 To find the moment of a retaining wall 
 with curved batter generally, let ABE be 
 of the parabolic form, then the area of 
 
 . 
 o 
 
 Now the centre of gravity of A B E will be 
 found sufficiently correct for all practical 
 
27 
 
 purposes if it is taken to be in the perpen- 
 dicular line G F, which will bisect ABE. 
 
 A N 
 
 F 
 
 8 
 
 Now 
 
 H X EF- ^ 
 
 o 
 
 AEFG=AEFN-AGN= 
 ABE H v 
 
 GN =H 
 
 /KF 
 
 V 
 
 HXEF-H 
 
 = BE, 
 
 2 /E~F __ 1 _ BE 
 af/lBE" 6EF' 
 F 9 3 BE , BE 2 
 
 BE~~4 4EF^16EF 2 ' 
 
28 
 
 BE 3 -12BE 2 XEF-f 36BEXEF 2 -16EF 3 =0, 
 B E-4 E F) (B E 2 - 8 B E X E F -M E F 2 ) = 0, 
 
 BE-4EF = 0, BF = ^^ 
 4 
 
 Moment of 
 
 moment of 
 AECD 
 
 moment of 
 ABCD 
 
 If we take a triangle of equal area with 
 A G B E, and similar to a triangle ABE, 
 we shall find that its base will 
 
 = BE y/| =.8165 BE, 
 
 and therefore the distance of a perpendicu- 
 lar from its centre of gravity to 
 
 O 
 
 and therefore, BE- ,2722 B E = .7278 B E 
 from B, or nearly the same as before. Let 
 C E = 6, and other values as before, then 
 
29 
 
 160,000, B E = 6.403, 
 and weight of wall 
 
 120 Ao X 6 + i 20 X 6.A = 19523. 
 
 D A 
 
 To find the moment of A B C D when 
 ABE^AECD. Then 
 
 T) "HI 
 
 BE = 3CE, areaof ABE = HX ^===HXCE. 
 
 Moment of 
 
 = (HXCE) 
 
30 
 
 . 9 
 
 =H CE 
 
 64 
 Then 
 
 120 X 20 15 B C 2 = 160,000, 
 64 
 
 B C = 13.62, C E = 3.4, B E = 10.21, 
 and weight of w^all 
 
 = 120 (20 X 3.4 + 20 ^3^) = 16344. 
 
 When both the front and back of the 
 wall are curved and parallel. When E F 
 passes through the centre of gravity, to find 
 E A. Area of 
 
 E A B F = HXEA-f-? (B F - E A) = 
 
 o 
 
 area of 
 
 2 H,, . H r 
 _BA+-g- BF, 
 
 E F C D = H X C F + -. (E D-C F) 
 o 
 
 = ) 
 
 then, when E F bisects A B C D, 
 
31 
 
 For stability, 
 
 
 SBC SAD 
 
 OF --- 
 
 4 4 
 
 = ^? + 2(ED-EA). 
 
 r> E xi 
 
 and as 
 
 = 8EA, EA=-AD. 
 
 To find E A when the perpendicular 
 which bisects A B C D passing through its 
 centre of gravity falls on its inside corner. 
 Area of 
 
 = ?-H(AD-EA). 
 
Area of 
 
 32 
 
 (A D - E A) = H X E A -f- 5. (B C - E A), 
 
 O 
 
 C .5 
 
 To find the moment of A B C D when 
 the curves of the front and back of the wall 
 are of different radii. Area of 
 
 EF 
 
 2H, 
 
 C D = H X C F + - - (D E - C F) 
 o 
 
33 
 
 area of 
 
 area of 
 
 = HXEA-f-(BF-EA) 
 o 
 
 2H H 
 
 = __ E A + - B F, 
 
 ~ 
 o 
 
 
 - B F = (B C + 2 D A) ; 
 
 A 
 
 moment of A B C D for stability 
 
 As 
 
34 
 
 adding 2 E A to both sides, 
 
 C F + 2 DA = 4 EA + B F, 
 2D A+CF-BF 
 ~T~ 
 
 generally, and for stability, 
 BC 3BC 
 
 EA- DA 4- -i- -- -^A _*^ 
 
 T 7 44 28' 
 
 If D A is to be B C, then the moment 
 
 will be 
 
 HXBC / 3 \ 5BC 2 
 
 3 + ~~"* 
 
 Then, with values as before, 
 
 120 X 20 5 -?r^ = 160,000, B C = 10.32, 
 
 o 
 
 = - 10.32 = 7.74, 
 
 weight of wall 
 
 90 
 
 = 120 X ^ (10.32 -f 2 X 7.74) = 20640. 
 o 
 
 If D A is to be 4- B C, then the moment 
 
 o 
 
 HX . BC (B C + 2 A) 
 
 will be 
 
35 
 
 Then, with values as before, 
 
 120 X 20 - = 160,000, B C = 10.69, 
 12 
 
 D A = 4- 10.69= 7.13, 
 o 
 
 weight of wall 
 
 = 120 X ^ (10.69 + 2 X 7.13) = 19960. 
 rf 
 
 If D A is to be 1 B C, then the moment 
 2 
 
 will be 
 
 ?2iBC ( BO + BC) = H ] ^. 
 
 Then, with values as before, 
 
 120 X 20 ~- = 160,000, B C = 11.54, 
 D A = i 11.54 = 5.77, 
 
 a 
 
 weight of wall 
 
 20 
 
 = 120 X ~ (11.54 -f- 2 X 5.77) = 18464. 
 o 
 
 If D A is to be B C, then the moment 
 TT x ~R P 
 
 will be 
 
36 
 
 Then, with values as before, 
 
 120 X 20 = 160,000, B C == 13.3, 
 
 = ~ 13.3 = 3.3, 
 
 weight of wall 
 
 20 
 = 120 X ~ (13.3 + 2 X 3.3) = 16000. 
 
 o 
 
 If a wall of this section is required, its 
 moment is 
 
 and if it supports water level with the top, 
 
 T) /~2 
 
 120 X 20 - ~ = 166,666, B C = 16.6 ; 
 and weight of wall 
 
37 
 
 = 120 X 20 X 
 
 = 13333. 
 
 Now as 17304 was required for the trian- 
 gular form of wall with, the same values, 
 there is shown to be a great saving of ma- 
 terial with the form of wall with curved 
 batter. 
 
 As the form of wall with a curved batter 
 of the semi- cubical parabolic section, has 
 been proved by several writers to be every- 
 where of equal strength, the calculations 
 for finding the dimensions of retaining 
 
 AN K 
 
 8 
 
 walls with a batter of that curve are also 
 given, as they may be found useful in some 
 cases. Let A G B in this figure be a curve 
 
38 
 
 of that form, with G F passing through its 
 centre of gravity. Then 
 
 2 = B E 2 : B K 3 = H 3 : : A H 2 = E F 2 : G N 3 , 
 
 /E F 2 
 
 area of A G N = 3 A ^ x G H 
 
 5 ' 
 
 ~~ If 
 ~ ~5~ "~ ~&\ 
 
 o ^ /V T?2 TJ 
 
 HXEF- ^EFXH./ =4 = ? 
 
 O r X> Jli O 
 
 3 3 /El^" B E 
 
 5 v JD lj 
 
 5 *" ^ V BE 2 
 
 EF i/: 
 
 BE 2 5 ' 
 
 BE 
 
 5EF BE 
 
 BE 2 
 
 3 
 
 BE 
 
 /E F 2 5 BE 
 
 V Bl^ = 3~ "" ?E^ " 
 
 A B ^ 3 A B ^ 
 V^EF) ( 5 -Wp) 
 
 _ 
 BE 2 
 
 27 
 
 _ _ 27 
 
 EF*' 
 
39 
 
 BE*/. BEV 27 BE 
 = 27 ' 
 
 B E = 3.759 E F = 3.759 (B E - B F), 
 3.759 B F = 3.759 B E - B E, 
 
 2. 759 BE 
 3.759 = 
 
 Moment of 
 
 A B E = B E X .734 B E = .2936 B E* X H ; 
 5 
 
 moment of 
 
 moment of 
 
 ABCD =H (^~-f CEXBE + .2930BEA 
 
 = -5 ((0 E + B E) 2 - .4f28 B E 2 ). 
 a 
 
 Let C E = 6, and other values as before, 
 then 
 
 20 
 120 ((6 + B E) 2 - .4128 B E 2 ) = 160,000, 
 
 m 
 
 B E = 6.22, and weight of wall 
 
 = 120 (20 X 6 + -| 20 X 6.22") = 20369. 
 To find the moment of A B C D when 
 
40 
 
 ABE = AECD. Then B E = - C E, 
 
 area of A B E = H X - B E = H X C E. 
 
 (.734 B E) = (H X C E) 
 
 | C E + 5^) + (H X C E) (. 734 X | C E) 
 
 = (HXCE)3CE + (HXCE) 1.835CE 
 = 4.835 HXCE 2 . Then 120X20 x 4.835 CE 2 
 
 = 160,000, CE = 3.71, BE=|- 3.71 = 9.28, 
 
 SB 
 
 and weight of wall 
 
 = 120 X 20 (3.71 + -| 9.28^ = 17808. 
 
41 
 
 When both the front and back of the 
 wall are curved and parallel. When E F 
 passes through the centre of gravity, to find 
 E A. Area of 
 
 H . 
 
 -/<+.!?* 
 
 E A 
 
 area of 
 
 OF B 
 
 then when E F bisects A B D, 
 
 ^^T?Ai^-^T5T? ^-^-rT?l^^-l?Tk 
 
 E A + -g- B F = - 6 - F -I- -y- E D, 
 
42 
 
 for stability, 
 
 SO 3 A L> _ A JLJ + 3 j 
 
 AD = 3ED-3EA, and as 
 subtracting, 
 
 : 6 E A, E A = 
 
 = 3ED + E A, 
 
 To find E A when the perpendicular 
 which bisects A B D passing through its 
 centre of gravity, falls on its inside corner. 
 Area of 
 
 area of 
 
 2H 
 5 
 
43 
 
 ~ (A D - E A) == H x E A + ?_? (B C - E A) 
 o o 
 
 3H 2H 
 
 To find the moment of ABCD when 
 the curves of the front and back of the wall 
 are of different radii. Area of 
 
 EFCD^HXCF + 
 
 area of 
 
 area of 
 
 5 
 3H^ , 2H, 
 
 = OF+ ^F 
 
 > o 
 
 EA+^BF = ^(2BC 
 o o o 
 
 moment of A B D for stability 
 
44 
 = ^g^(2BC + 3AD). 
 
 D 
 
 E A 
 
 C B 
 
 adding 3 E A to both sides, 
 
 3DA+2CF-2BF 
 EA=- g- 
 
 generally, and for stability, 
 
 BC 3BC 
 3PA + lr ^- DA BC 
 
 ~6~ ~2~ ~6~' 
 
 q 
 
 If D A is to be B 0, then the moment 
 3 H B C 
 
 (2 B C + 3 A D) 
 
 will be 
 
45 
 
 Then, with values as before, 
 
 120 X 20 51 ^ 2 = 160,000, B C = 10.23, 
 
 oU 
 
 D A = f- 10.23 = 7.67, 
 
 4 
 
 weight of wall 
 
 = 120 X ^ (2 X 10.23 + 3 X 7.67) = 208CO. 
 5 
 
 2 
 If DA is to be - B C, then the moment 
 
 will be 
 
 Then, with values as before, 
 
 q T> p2 
 
 120 X 20 = 160,000, 
 
 o 
 
 B C = 10.54, D A = 1-10.54 = 7.03, 
 o 
 
 weight of wall 
 
 90 
 
 = 120 X ^ (2 X 
 5 
 
 If D A is to be -^ B 0, then the moment 
 
 20 
 
 = 120 X - (2 X 10.54 + 3 X 7.03) = 20238. 
 o 
 
46 
 
 3HX BC 
 
 will be 
 
 20 
 
 (2 B C -f 3 A D) 
 
 Then, with values as before, 
 
 120 X 20 21 ^ 2 = 160,000, 
 40 
 
 B C = 11.27, D A = i- 11.27 = 5.63, 
 SB 
 
 weight of wall 
 
 90 
 
 = 120 X =p (2 X 11.27 + 3 X 5.63) = 18930. 
 o 
 
 If A D is to be -5- B C, then the moment 
 
 3 
 3HXB C 
 
 20 
 
 (2 B -f 3 A D) 
 
 will be 
 
 3H x B C 
 
 . 
 
 C2 B o -f 
 
 ~ao 
 Then, with values as before, 
 
 n T> r]2 
 
 120x20 " =160,000, 
 
 B C = 12.17, D A = 12.17 = 4.06, 
 3 
 
 weight of wall 
 
 90 
 
 120 X TT ( a X 12 - 17 + 3 X 4 - 06 > = 17526 ' 
 D 
 
47 
 
 If D A is to be -j- B C, then the moment 
 
 3 H x B C ^ ~ _ 
 
 (2 B C -f 3 A D) 
 
 ( 2 B C + |Bc),H2Si51. 
 
 will be 
 3HxBC 
 
 Then, with values as before, 
 
 OO T> p2 
 
 120 X 20 = 160,000, 
 
 do 
 
 B C = 12.71, D A = -1 12.71 = 3.18, 
 weight of wall 
 
 20 
 ;= 120 X ^ (2 X 12.71 + 3 X 3.18) = 16780. 
 
 If a wall of this section is required, its 
 
48 
 
 moment is .2936 B C 2 X H > an(i if> & sup- 
 ports water level with the top, 
 
 120 X 20 x .2936 B C 2 = 166,666, B C = 15.38, 
 and weight of wall 
 
 = 120 + 20X 2X ^ 38 = 14764 
 5 
 
 Having now given methods for finding 
 the correct dimensions of the different forms 
 of wall that are generally used in practice, 
 the author does not wish to express any 
 opinion on the merits of any particular 
 form of wall, leaving it to the superior 
 judgment of more experienced engineers to 
 determine the section of wall they may con- 
 sider most suitable in each case. 
 
50 
 
 TABLE 1. Thickness of Vertical Retaining Walls, 
 to sustain the Pressure of Earth, Sand, etc., level 
 with its top. The Moment of the Wall is equal 
 to twice that of the Earth, etc., to insure perma- 
 nent stability. 
 
 
 Sand. 
 
 Shingle. 
 
 Dry earth. 
 
 *c 
 
 = 30. 
 
 A = 400. 
 
 A = 43. 
 
 11 
 
 
 
 
 
 
 
 
 
 <u 
 W 
 
 94 Ibs. 
 
 120 Ibs. 
 
 119 Ibs. 
 
 106 Ibs. 
 
 94 ibs. 
 
 6 
 
 27.42 
 
 30.98 
 
 24.92 
 
 23.62 
 
 20.65 
 
 7 
 
 31.99 
 
 36.15 
 
 29.07 
 
 27.44 
 
 24.09 
 
 8 
 
 36.56 
 
 41.31 
 
 33.23 
 
 31.36 
 
 27.53 
 
 9 
 
 41.13 
 
 46.47 
 
 37.38 
 
 35.28 
 
 30.98 
 
 10 
 
 45.70 
 
 51.64 
 
 41.53 
 
 39.20 
 
 34.42 
 
 11 
 
 50.27 
 
 56.80 
 
 45.69 
 
 43.12 
 
 37. SB 
 
 12 
 
 54.84 
 
 61.97 
 
 49.84 
 
 47.04 
 
 41.30 
 
 13 
 
 59.42 
 
 67.13 
 
 53.99 
 
 50.96 
 
 44.74 
 
 14 
 
 63.99 
 
 72.29 
 
 58.15 
 
 64.88 
 
 48.19 
 
 15 
 
 68.56 
 
 77.46 
 
 62.30 
 
 58.80 
 
 51.63 
 
 16 
 
 73.13 
 
 82.62 
 
 66.45 
 
 62.72 
 
 55.07 
 
 17 
 
 77.70 
 
 87.79 
 
 70.61 
 
 66.64 
 
 58.51 
 
 18 
 
 82.27 
 
 92.95 
 
 74.76 
 
 70.56 
 
 61.95 
 
 19 
 
 86.84 
 
 98.11 
 
 78.91 
 
 74.48 
 
 65.40 
 
 20 
 
 91.41 
 
 103.28 
 
 ' 83.07 
 
 78.40 
 
 68.84 
 
 21 
 
 95.98 
 
 108.44 
 
 87.22 
 
 82.32 
 
 72.28 
 
 22 
 
 100.55 
 
 113.61 
 
 91.38 
 
 86.24 
 
 75.72 
 
 23 
 
 105 . 12 
 
 118.77 
 
 95.53 
 
 90.16 
 
 79.17 
 
 24 
 
 109.69 
 
 123.94 
 
 99.68 
 
 94.08 
 
 82.61 
 
 25 
 
 114.26 
 
 129.10 
 
 103.84 
 
 98.00 
 
 86.05 
 
 26 
 
 118.83 
 
 134.26 
 
 107.99 
 
 101.92 
 
 89.49 
 
 27 
 
 123.40 
 
 139.43 
 
 112.14 
 
 105.84 
 
 92.93 
 
 28 
 
 127.97 
 
 144.59 
 
 116.30 
 
 109.76 
 
 96.38 
 
 29 
 
 132.54 
 
 149.76 
 
 120.45 
 
 113.68 
 
 99.82 
 
 30 
 
 137.11 
 
 154.92 
 
 124.60 
 
 117.60 
 
 103.26 
 
51 
 
 TABLE 1. Continued. 
 
 Do., moist or 
 natural. 
 
 Do. , dense and 
 compact. 
 
 Clay, 
 o 
 
 Clay. 
 
 A = 54. 
 
 L = 55o. 
 
 
 L- 
 
 106 Ibs. 
 
 125 Ibs. 
 
 125 Ibs. 
 
 125 Ibs. 
 
 16.39 
 
 17.27 
 
 41.27 
 
 22.69 
 
 19.12 
 
 20.15 
 
 48.15 
 
 26.47 
 
 21.85 
 
 23.03 
 
 55.03 
 
 30.25 
 
 24.58 
 
 25.90 
 
 61.91 
 
 34.03 
 
 27.31 
 
 28.78 
 
 68.79 
 
 37.81 
 
 30.04 
 
 31.66 
 
 75.67 
 
 41.59 
 
 32.78 
 
 34.54 
 
 82.55 
 
 45.38 
 
 35.51 
 
 37.42 
 
 89.43 
 
 49.16 
 
 38.24 
 
 40.29 
 
 96 31 
 
 52.94 
 
 40.97 
 
 43.17 
 
 103.18 
 
 56.72 
 
 43.70 
 
 46.05 
 
 110.06 
 
 60.50 
 
 46.43 
 
 48.93 
 
 116.94 
 
 64.28 
 
 49.16 
 
 51.81 
 
 123.82 
 
 68.06 
 
 51.89 
 
 54.69 
 
 130.70 
 
 71.84 
 
 54.63 
 
 57.56 
 
 137.58 
 
 75.62 
 
 57.36 
 
 60.44 
 
 144.46 
 
 79.40 
 
 60.09 
 
 63.32 
 
 151.34 
 
 83.18 
 
 62.82 
 
 66.19 
 
 158.22 
 
 86.96 
 
 65.56 
 
 -69.07 
 
 165.10 
 
 90.74 
 
 68.29 
 
 71.95 
 
 171.97 
 
 94.52 
 
 71.02 
 
 74.83 
 
 178.85 
 
 98.31 
 
 73.75 
 
 77.71 
 
 185.73 
 
 102.09 
 
 76.48 
 
 80.58 
 
 192.61 
 
 105.87 
 
 79.21 
 
 33.46 
 
 199.49 
 
 109.65 
 
 81.94 
 
 86.34 
 
 206.37 
 
 113.43 
 
52 
 
 TABLE 2. Double Moments of the Pressure of the 
 Weight of Embankments of Earth, Sand, etc., 
 level with the top of Wall. 
 
 
 Shingle. 
 
 Dry earth. 
 
 Sand. 
 
 
 
 cTFH* 
 
 =10.4 H 
 
 13.3H3 
 
 8.62522 H : * 
 
 7. 6829 H 3 
 
 5. 92394 H^ 
 
 3 
 
 
 
 
 
 
 6 
 
 2256 
 
 2880 
 
 1863 
 
 1659 
 
 1280 
 
 7 
 
 3582 
 
 4573 
 
 2958 
 
 2635 
 
 2032 
 
 8 
 
 5347 
 
 6827 
 
 4416 
 
 3934 
 
 3033 
 
 9 
 
 7614 
 
 9720 
 
 6287 
 
 5601 
 
 4318 
 
 10 
 
 10444 
 
 13333 
 
 8625 
 
 7683 
 
 5924 
 
 11 
 
 13901 
 
 17747 
 
 11480 
 
 10226 
 
 7885 
 
 12 
 
 18048 
 
 23040 
 
 14904 
 
 13276 
 
 10236 
 
 13 
 
 22946 
 
 29293 
 
 18949 
 
 16879 
 
 13015 
 
 14 
 
 28659 
 
 36587 
 
 23667 
 
 21081 
 
 16255 
 
 15 
 
 35250 
 
 45000 
 
 29110 
 
 25929 
 
 19993 
 
 16 
 
 42780 
 
 54613 
 
 35329 
 
 31468 
 
 24264 
 
 17 
 
 51313 
 
 65507 
 
 42376 
 
 37745 
 
 29104 
 
 18 
 
 60912 
 
 77766 
 
 50302 
 
 44805 
 
 34548 
 
 19 
 
 71638 
 
 91453 
 
 59160 
 
 52696 
 
 40632 
 
 20 
 
 83555 
 
 106666 
 
 69002 
 
 61461 
 
 47391 
 
 21 
 
 96726 
 
 123480 
 
 79878 
 
 71149 
 
 54862 
 
 22 
 
 111212 
 
 141973 
 
 91841 
 
 81805 
 
 63078 
 
 23 
 
 127077 
 
 162227 
 
 104943, 
 
 93475 
 
 72077. 
 
 24 
 
 144384 
 
 184320 
 
 119235 
 
 106206 
 
 81892 
 
 25 
 
 163194 
 
 208333 
 
 134769 
 
 121042 
 
 92561 
 
 26 
 
 183571 
 
 234346 
 
 151597 
 
 135035 
 
 104119 
 
 27 
 
 205578 
 
 262440 
 
 169770 
 
 151222 
 
 116601 
 
 28 
 
 229276 
 
 292693 
 
 189341 
 
 168655 
 
 130042 
 
 29 
 
 254729 
 
 325186 
 
 210360 
 
 187378 
 
 144479 
 
 30 
 
 282000 
 
 360000 
 
 232881 
 
 207438 
 
 159946 
 
53 
 
 TABLE 2. Continued. 
 
 Do., moist 
 or natuial. 
 
 Do. , dense 
 and 
 compact. 
 
 Clay. 
 
 Water. 
 
 3. 73024 H 3 
 
 f 
 4. 14222 H 3 
 
 23.66012 H 3 
 
 7.14887 H 3 
 
 20.83 H 3 
 
 806 
 
 895 
 
 5110 
 
 1544 
 
 4500 
 
 1279 
 
 1421 
 
 8115 
 
 2452 
 
 7146 
 
 1910 
 
 2121 
 
 12114 
 
 3660 
 
 10666 
 
 2719 
 
 3020 
 
 17243 
 
 5211 
 
 15187 
 
 3730 
 
 4142 
 
 23660 
 
 7149 
 
 20833 
 
 4965 
 
 5513 
 
 31492 
 
 9515 
 
 27729 
 
 6446 
 
 7158 
 
 40885 
 
 12353 
 
 36000 
 
 8195 
 
 9100 
 
 51981 
 
 15706 
 
 45771 
 
 10236 
 
 11366 
 
 64923 
 
 19616 
 
 57166 
 
 12590 
 
 13980 
 
 79853 
 
 24127 
 
 70312 
 
 15279 
 
 16966 
 
 96912 
 
 29282 
 
 85333 
 
 18327 
 
 20251 
 
 116242 
 
 35122 
 
 102354 
 
 21755 
 
 24157 
 
 137986 
 
 41692 
 
 121500 
 
 25586 
 
 28411 
 
 162285 
 
 49034 
 
 142896 
 
 29842 
 
 33138 
 
 189281 
 
 57191 
 
 166666 
 
 34546 
 
 38361 
 
 219116 
 
 66206 
 
 192937 
 
 39720 
 
 44106 
 
 251933 
 
 76121 
 
 221833 
 
 45386 
 
 50398 
 
 287873 
 
 86980 
 
 253479 
 
 51567 
 
 57262 
 
 327077 
 
 98826 
 
 288000 
 
 58285 
 
 64722 
 
 369689 
 
 111701 
 
 325521 
 
 65563 
 
 72804 
 
 415850 
 
 125650 
 
 366166 
 
 73422 
 
 81531 
 
 465702 
 
 140711 
 
 410062 
 
 81886 
 
 90930 
 
 519387 
 
 156932 
 
 457333 
 
 90977 
 
 101025 
 
 577046 
 
 174354 
 
 508104 
 
 100716 
 
 111840 
 
 638823 
 
 193019 
 
 562499 
 
54 
 
 TABLE 3. For Surcliarged Embankments. 
 
 of slope = 0. 
 
 U,c. 
 
 ;, 
 
 Tine 1 nf 
 
 2 
 
 au e* u 't 
 
 4 tol 
 
 = 14 12' 
 15 
 16 
 
 37 
 37 
 37 
 
 54/ 
 30 
 
 
 .77847 
 . 76732 
 .75355 
 
 
 17 
 
 36 
 
 30 
 
 . 73996 
 
 
 18 
 
 36 
 
 
 
 .72654 
 
 3 tol 
 
 = 18 25 
 
 35 
 
 47^ 
 
 .72100 
 
 
 19 
 
 35 
 
 30 
 
 .71329 
 
 
 20 
 
 35 
 
 
 
 . 70020 
 
 
 21 
 
 34 
 
 30 
 
 .68728 
 
 
 22 
 
 34 
 
 
 
 .67450 
 
 
 23 
 
 33 
 
 30 
 
 .66188 
 
 
 24 
 
 33 
 
 
 
 .64940 
 
 
 25 
 
 32 
 
 30 
 
 .63707 
 
 
 26 
 
 32 
 
 
 
 .62486 
 
 2 tol 
 
 = 26 35 
 
 31 
 
 42^ 
 
 .61781 
 
 
 27 
 
 31 
 
 30 
 
 .61280 
 
 
 28 
 
 31 
 
 
 
 .60086 
 
 
 29 
 
 30 
 
 30 
 
 .58904 
 
 IK tol 
 
 = 29 44 
 
 30 
 
 8 
 
 .58045 
 
 
 30 
 
 30 
 
 
 
 .57735 
 
 
 31 
 
 29 
 
 30 
 
 .56577 
 
 
 32 
 
 29 
 
 
 
 .55430 
 
 
 33 
 
 28 
 
 30 
 
 .54295 
 
 \y to i 
 
 = 33 42 
 
 28 
 
 9 
 
 .53507 
 
 
 34 
 
 28 
 
 
 
 .53170 
 
 
 35 
 
 27 
 
 30 
 
 .52056 
 
 
 36 
 
 27 
 
 
 
 .50952 
 
 
 37 
 
 26 
 
 30 
 
 .49858 
 
 
 38 
 
 26 
 
 
 
 .48773 
 
 l^tol 
 
 = 38 40 
 
 25 
 
 40 
 
 .48055 
 
 
 39 
 
 25 
 
 30 
 
 .47697 
 
 
 40 
 
 25 
 
 
 
 .46630 
 
55 
 
 TABLE 3. Continued. 
 
 o s n. (90 + 0) 
 
 
 
 Taog. of 9 - * 
 
 u ' 
 
 o2 
 
 . 2 
 
 8i , (^) 
 
 * - "4 
 
 ( Vl -T) 
 
 1.5782 
 1.5867 
 
 0.96944 
 0.96592 
 
 c 
 0.75469 
 0.74118 
 
 1.5972 
 
 0.96126 
 
 0.72436 
 
 1.6077 
 
 0.95630 
 
 0.70762 
 
 1.6180 
 
 0.95105 
 
 0.69098 
 
 1 . 6223 
 
 0.94878 
 
 0.68407 
 
 1.6282 
 
 0.94551 
 
 0.67443 
 
 1.6383 
 
 0.93969 
 
 0.65798 
 
 1.6483 
 
 0.93358 
 
 0.64163 
 
 1.6581 
 
 0.92718 
 
 0.62539 
 
 1.6678 
 
 0.92050 
 
 0.60926 
 
 1.6773 
 
 0.91354 
 
 0.59326 
 
 1.6868 
 
 0.90630 
 
 0.57738 
 
 1.6961 
 
 0.89879 
 
 0.56162 
 
 1.7014 
 
 0.89428 
 
 0.55250 
 
 1.7053 
 
 0.89100 
 
 0.54601 
 
 1.7143 
 
 0.88294 
 
 0.53052 
 
 1 1.7232 
 
 0.87461 
 
 0.51519 
 
 1 . 7297 
 
 0.86834 
 
 0.50403 
 
 1.7320 
 
 0.8C602 
 
 0.50000 
 
 .7407 
 
 0.85716 
 
 0.48496 
 
 .7492 
 
 0.84804 
 
 0.47008 
 
 . 7576 
 
 0.83867 
 
 0.45536 
 
 .7634 
 
 0.83195 
 
 0.44515 
 
 .7659 
 
 0.82903 
 
 0.44080 
 
 .7740 
 
 0.81915 
 
 0.42642 
 
 1.7820 
 
 0.80901 
 
 0.41221 
 
 1.7899 
 
 0.79863 
 
 0.39818 
 
 1.7976 
 
 0.78801 
 
 0.38433 
 
 1.8026 
 
 0.78079 
 
 0.37521 
 
 1.80517 
 
 0.77714 
 
 0.37067 
 
 1.81261 
 
 0.76604 
 
 0.35721 
 
56 
 
 TABLE 4. Thickness of Vertical Retaining Walls 
 to sustain the Pressure of a Surcharged Em- 
 bankment of Earth, Sand, etc. The moment of 
 the Wall is equal to twice that of the Earth, etc., 
 to insure permanent stability. 
 
 
 
 
 g 
 
 
 
 1 
 
 Sand. 
 
 Shingle. g 
 
 Dry earth, co 
 
 p 
 
 A = 300. o 
 
 ^=40. o 
 
 A ^ 43. co 
 
 o 
 
 
 
 li 
 
 || 
 
 E 
 
 
 
 
 o 
 
 "5 
 B 
 
 94 Ibs. 
 
 120 Ibe. 
 
 119 Ibs. 
 
 106 Ibs. 
 
 94 Ibs. 
 
 6 
 
 33.58 
 
 37.94 
 
 31.94 
 
 30.14 
 
 26.78 
 
 7 
 
 39.18 
 
 44.27 
 
 37.26 
 
 35.17 
 
 31.24 
 
 8 
 
 44 78 
 
 50.59 
 
 42.58 
 
 40.19 
 
 35.71 
 
 9 
 
 50.37 
 
 56.92 
 
 47 91 
 
 45 21 
 
 40.17 
 
 10 
 
 55.97 
 
 63.24 
 
 53.23 
 
 50 24 
 
 44.64 
 
 11 
 
 61 57 
 
 69.57 
 
 58.55 
 
 55.26 
 
 49.10 
 
 12 
 
 67.17 
 
 75.89 
 
 63.88 
 
 60.29 
 
 53.56 
 
 13 
 
 72.76 
 
 82.21 
 
 .69.20 
 
 65.31 
 
 58.03 
 
 14 
 
 78.36 
 
 88.54 
 
 74.52 
 
 70.33 
 
 62.49 
 
 15 
 
 83.96 
 
 94.86 
 
 79.85 
 
 75.36 
 
 66.96 
 
 16 
 
 89.56 
 
 101.19 
 
 85.17 
 
 80.38 
 
 71.42 
 
 17 
 
 95.16 
 
 107.51 
 
 90.49 
 
 85.41 
 
 75.89 
 
 18 
 
 100.75 
 
 113.84 
 
 95.82 
 
 90.43 
 
 80 35 
 
 19 
 
 106.35 
 
 120.16 
 
 101.14 
 
 95.46 
 
 84.81 
 
 20 
 
 111.95 
 
 126.49 
 
 106.46 
 
 100.48 
 
 89.28 
 
 21 
 
 117.55 
 
 132.81 
 
 111 . 79 
 
 105 50 
 
 93.74 
 
 22 
 
 123.14 
 
 139.14 
 
 117.11 
 
 110.53 
 
 98.21 
 
 23 
 
 128.74 
 
 145.46 
 
 122.43 
 
 115.55 
 
 102.67 
 
 24 
 
 134 34 
 
 151.78 
 
 127.76 
 
 120.58 
 
 107.13 
 
 25 
 
 139.94 
 
 158.11 
 
 133.08 
 
 125.60 
 
 111.60 
 
 26 
 
 145.54 
 
 164.44 
 
 138.41 
 
 130.63 
 
 116.07 
 
 27 
 
 151.13 
 
 170.76 
 
 143.73 
 
 135.65 
 
 120.53 
 
 28 
 
 156.73 
 
 177.09 
 
 149.05 
 
 140.68 
 
 124.99 
 
 29 
 
 162.33 
 
 183.41 
 
 154 38 
 
 145.70 
 
 129.46 
 
 30 
 
 167.93 
 
 189.74 
 
 159.70 
 
 150.73 
 
 133.92 
 
57 
 
 TABLE 4. Continued. 
 
 CO 
 
 Do -> g 
 
 Do., dense S? 
 
 
 
 moist or 
 
 and g 
 
 Clay. 
 
 Clay. 
 
 natural. ^ 
 
 compact. S 
 
 L = 16. 
 
 L = ^ Q . 
 
 Z. - 54. || 
 
 L -55. ,| 
 
 
 
 
 
 w 
 
 
 
 106 Ibs. ^ 
 
 125 Ibs. 
 
 125 Ibs. 
 
 125 Ibs. 
 
 22.04 
 
 23.29 
 
 46.61 
 
 29.64 
 
 25.71 
 
 27.17 
 
 54.38 
 
 34.58 
 
 29.39 
 
 31.05 
 
 62.15 
 
 39.52 
 
 33.06 
 
 34.93 
 
 69.92 
 
 44.46 
 
 36.73 
 
 38.82 
 
 77.69 
 
 49.40 
 
 40.41 
 
 42.70 
 
 85.46 
 
 54.34 
 
 44.08 
 
 46.58 
 
 93.23 
 
 59.28 
 
 47.75 
 
 50.46 
 
 101.00 
 
 64.22 
 
 51.43 
 
 54.34 
 
 108.77 
 
 69.16 
 
 55.10 
 
 58.23 
 
 116.54 
 
 74.10 
 
 58.78 
 
 62.11 
 
 124.31 
 
 79.04 
 
 62.45 
 
 65.99 
 
 132.08 
 
 83.98 
 
 66.12 
 
 69.87 
 
 139.85 
 
 88.92 
 
 69.80 
 
 73.75 
 
 147.62 
 
 93.86 
 
 73 47 
 
 77.64 
 
 155.39 
 
 98.80 
 
 77.14 
 
 81.52 
 
 163.16 
 
 103.74 
 
 80.82 
 
 85.40 
 
 170.93 
 
 108.68 
 
 84.49 
 
 89.28 
 
 178.70 
 
 113.62 
 
 88.16 
 
 93.16 
 
 186.47 
 
 118.56 
 
 91.84 
 
 97.05 
 
 194.24 
 
 123.50 
 
 95.52 
 
 100.93 
 
 202.01 
 
 128.45 
 
 99.19 
 
 104.81 
 
 209 . 78 
 
 133.39 
 
 102 86 
 
 108.70 
 
 217.55 
 
 138.33 
 
 106.54 
 
 112.58 
 
 225.32 
 
 143.27 
 
 110.21 
 
 116.46 
 
 233.10 
 
 148.21 
 
58 
 
 TABLE 5. Double Moments of the Pressure of tfie 
 Weight of Surcharged Embankments of Earth, 
 Sand, etc. 
 
 Sand. 
 
 Shingle. 
 
 cTFHs 
 
 
 20 H3 
 
 14. 16945 H 3 
 
 12. 62153 H 3 
 
 3 
 
 
 6 
 
 3384 
 
 4320 
 
 3061 
 
 2726 
 
 7 
 
 5373 
 
 6860 
 
 4860 
 
 4329 
 
 8 
 
 8021 
 
 10240 
 
 7255 
 
 6462 
 
 9 
 
 11421 
 
 14580 
 
 10330 
 
 9201 
 
 10 
 
 15666 
 
 20000 
 
 14169 
 
 12621 
 
 11 
 
 20852 
 
 26620 
 
 18860 
 
 16799 
 
 12 
 
 27072 
 
 34560 
 
 24485 
 
 21810 
 
 13 
 
 34419 
 
 43940 
 
 31130 
 
 27729 
 
 14 
 
 42989 
 
 54880 
 
 38881 
 
 34633 
 
 15 
 
 52875 
 
 67500 
 
 47822 
 
 42598 
 
 16 
 
 64170 
 
 81920 
 
 58038 
 
 51698 
 
 17 
 
 76970 
 
 98260 
 
 69614 
 
 62010 
 
 18 
 
 91368 
 
 116640 
 
 82636 
 
 73609 
 
 19 
 
 107457 
 
 137180 
 
 97188 
 
 86571 
 
 20 
 
 125333 
 
 160000 
 
 113355 
 
 100972 
 
 21 
 
 145089 
 
 185220 
 
 131223 
 
 116888 
 
 22 
 
 166818 
 
 212960 
 
 150876 
 
 134394' 
 
 23 
 
 190616 
 
 243340 
 
 172400 
 
 153566 
 
 24 
 
 216576 
 
 276480 
 
 195878 
 
 174480 
 
 25 
 
 244791 
 
 312500 
 
 221397 
 
 197211 
 
 26 
 
 275357 
 
 351520 
 
 249042 
 
 221836 
 
 27 
 
 308367 
 
 393660 
 
 278897 
 
 248429 
 
 28 
 
 343914 
 
 439040 
 
 31 1048 
 
 277068 
 
 29 
 
 382094 
 
 487780 
 
 345579 
 
 307826 
 
 30 
 
 423000 
 
 540000 
 
 382575 
 
 340781 
 
59 
 
 
 Do., moist 
 
 Do., dense 
 
 
 Dry earth. 
 
 or 
 
 and 
 
 Clay. 
 
 
 natural. 
 
 compact. 
 
 
 9.96405 H3 
 
 6. 748066 H 3 
 
 7 53526 H3 
 
 30.183 H 3 
 
 12. 20375 H* 
 
 2152 
 
 1458 
 
 1627 
 
 6520 
 
 2636 
 
 3418 
 
 2315 
 
 2585 
 
 10353 
 
 4186 
 
 5102 
 
 3455 
 
 3858 
 
 15454 
 
 6248 
 
 7264 
 
 4919 
 
 5493 
 
 22004 
 
 8896 
 
 9964 
 
 6748 
 
 7535 
 
 30183 
 
 12204 
 
 13262 
 
 8982 
 
 10029 
 
 40174 
 
 16243 
 
 17218 
 
 11661 
 
 13021 
 
 52157 
 
 21088 
 
 21891 
 
 14825 
 
 16555 
 
 66313 
 
 26812 
 
 27341 
 
 18517 
 
 23677 
 
 82823 
 
 33487 
 
 33628 
 
 22775 
 
 25432 
 
 101869 
 
 41188 
 
 40813 
 
 27640 
 
 30864 
 
 123631 
 
 49986 
 
 48953 
 
 33153 
 
 37021 
 
 148291 
 
 59957 
 
 58110 
 
 39355 
 
 43946 
 
 176029 
 
 71172 
 
 68343 
 
 46285 
 
 51684 
 
 207027 
 
 83712 
 
 79712 
 
 53985 
 
 60282 
 
 241466 
 
 97630 
 
 92277 
 
 62494 
 
 69784 
 
 279528 
 
 113019 
 
 106097 
 
 72453 
 
 80235 
 
 321392 
 
 1:9945 * 
 
 121232 
 
 82104 
 
 91681 
 
 367241 
 
 148483 
 
 137743 
 
 93285 
 
 104167 
 
 417254 
 
 168704 
 
 155688 
 
 105438 
 
 117738 
 
 471615 
 
 190684 
 
 175128 
 
 118604 
 
 132439 
 
 530502 
 
 214493 
 
 196122 
 
 132822 
 
 148316 
 
 594098 
 
 240206 
 
 218731 
 
 148133 
 
 165414 
 
 662584 
 
 267897 
 
 243013 
 
 164578 
 
 183777 
 
 736141 
 
 297637 
 
 269029 
 
 182198 
 
 203452 
 
 814949 
 
 329501 
 
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