Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39.48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. Digital file copy- right by Cornell University Library 1991.A MANUAL OK EARTHWOEK, ALEX. J. S. GRAHAM, C.E. LONDON: LOCKWOOD & CO., 7 STATIONERS’-HALL COURT 1863.LONDON PRINTED BY SPOTTISWOODE AND CO. NEW-STRPET SQUARETO E. P. BEEEETON, ESQ. CIVIL ENGINEER, THE FOLLOWING ‘MANUAL ON EARTHWORK’ is RESPECTFULLY DEDICATED, AS A TRIBUTE OF EARLY FRIENDSHIP AND ESTEEM, BY HIS OBLIGED SERVANT, THE AUTHOR. 18 Duke Street, Westminster: August 17, 1863.PEEFACE. ---♦--- This Manual contains an investigation of all the prin- ciples necessary for the measurement and calculation of earthwork. A general formula is given for curving, which renders the use of tables unnecessary. It may be easily retained in the memory for field practice. Several useful methods for treating curves are also given. The manner of setting out half-widths will be found simple and accurate, especially in sidelong ground. It affords all the data necessary for calcu- lating contents, without the aid of cross sections, wherever the surface of the ground maintains a regular inclination; as may be seen by the tabular form proposed. But the principal advantages of the following system are the correct estimation of excavations in sidelong ground, as to the cubical capacity, the acreage occu- pied, and the areas of the slopes; besides some expres- sions for the volume and surfaces of the wedge-shaped ends, whether their edges be square or oblique to the6 A MANUAL ON EARTHWORK. centre line; and for the several forms of trapezoidal solids that occur in practice. The system of Tabulation proposed is less liable to error than that obtained from tabular numbers, as the figures are taken directly from the section, or half- width book. For general or Parliamentary estimates, formulae (A), (/), (q), and (s), as tabulated in examples 1, 4, 7, and 8, will be found, in most cases, sufficiently cor- rect. But, for accuracy, tabulation from cross- sectional areas, as by formulae (/), (o), and (w), and examples 3, 6, and 9, will be necessary, wherever the lateral inclination of the surface is considerable. Where the surface warps, or where the rate of incli- nation, transversely, differs within two chains’ distance, the most accurate methods to be followed are those pointed out by formulae (A), (i), (n), and examples 1, 2, and 5. Where cross sections are necessary, the central and side heights can be obtained by scale from the line that equates the contour. Alex. J. S. Graham, Resident Engineer, Forest of Dean Central Railway. Nibley: January 1863.A MANUAL ON EARTHWORK. ON CURVES. The terminal points of curves are always equidis- tant from the intersection of the tangents to those points. In marking out curves the angles fac, agc, dba, and jfa (fig. 1), are generally used. The Z. fac is equal The Z. JFA is equal to the internal and opposite angles fac and fca, which are equal. Therefore the Z. jfa is equal to twice fac, or to the central Z. agc. When the L fac cannot be had by observation, Fig. 1. to half the Z. ago at the centre. The Z. dba is equal to the internal and op- posite angles bac and bca, or the s z. fac subtended by the same arcs. Therefore the Z. dba is also equal to half the central angle agc.8 A MANUAL ON EARTHWORK. any Z fcd may be set off, and the Z fi>c taken; then /_ fcd 4- Z fdc_ ^ FACj w^ic]1 may be set 0ff at c or 2 d, or any point along the tangent selected for the commencement of the curve. If the radius of the arc abc is required, the angle fac=Z dac is obtained; as also the length of the chord ac, or the latter may be calculated by measuring ad, and taking the angle adc ; for 180°—(Zi>ac + Z adc) : ad :: z ADC : AC. This saves the trouble of mea- suring the line ac. Then as sin Z fac = sin Z age, : half the chord = ae 11 sin 90° : to the rad. : = ag. If the point b is known to be in the curve, as in the renewal of curves previously staked, it is preferable to use the angle dba in the above proportion. Or use the Z. fab and half the chord ab. In some cases the irregularity of the ground may prevent observations of the point c, from either A, b, or d; then half the external angle jfa formed by the intersection of the tangents af and cf, or half the central angle agc formed by the intersection of the radii, may be used, or the line ag or gc may be measured. Having found the radius, the angle of deflection per chain may be found as follows. Let 0=the angle of deflection from the tangent for one chain’s length of the curve, and r=the radius. Then 20 ; 1 :: 360 : 2rx 3*1416; whence —=3-1416r or —=3-1416r; whence 40 0ON CURVES. 9 0= 0 = 90 , 90 ana r— 3*1416r 28*64789 or and r— 3*14160 28*64789 multiplying the numerators by 60, these give a 1718*8734 , 1718*8734 0=---------- and r=------------, r 0 answering to minutes and decimals. As derivable from these it may be stated, that if _ . 2 r0' 6'= any angle at the centre, 2’8.64789 = 57-29578 = r the length of the arc; and if 0"=any angle at the cir- 7*0^ cumference - = the length of the arc. zo“oi± / If the length of the arc abc is known, the angle of deflection is equal to the angle fac divided by it. The length abc may not be had always accurately enough from the survey. However, if it be obtained from the survey, and if the curve, set out by the angle = —------, does not terminate in c, it will come so arc abc close to it as to give the actual length of the arc abc by measurement; and then the curve can be staked out with the exact angular deflection without any farther calculation. As the angle fac is always equal to the angle dba, if the line ab is set off at will, and the instrument moved along it, at that angle, until the side db, when10 A MANUAL ON EARTHWORK. produced, passes through the point c, b will be a point in the curve, and sin £_ dab \ • • sin 90° : rad. AG. This appears to be the easiest method of finding the radius in long curves, and is sufficiently accurate when ab is not taken less than 10 chains. In this instance, the point to commence the curve is b. To do this, an angle kba= the angle dab must be set off at b from the line ba. Then the instru- ment pointing in the line kl will be a tangent to the curve; from which the curve may be marked off back- wards to A and forwards to c. In this manner a tangent may be set off to any point in a curve; as it is only necessary to sight the chord of an arc backwards, and set off from it as many times the angle of deflection as there are chains in the arc. This is always necessary when the curvature changes from one radius to another without an inter- vening tangent; as, for instance, the angle of deflec- tion per chain might be 1° from b to A, and 1° 30' from b to c and the common tangent kl. The junc- tion of curves and tangents is seldom at the end of a chain ; therefore the deflection for the additional links is found by multiplying the angle for one chain by them. Thus, a curve of 1° deflection per chain gives *30 degrees, or 18 minutes for 30 links. If in curving along abc it is required to find the point c, such that a tangent near that point shall be also a tangent to the curve his, it will be seen that the production of the chord ch, cutting at or near i, shows the tangent point m to be somewhere betweenON CURVES. 11 C and h. Produce the tangent cn, and from the point of intersection o, between it and the tangent po of the curve ris set off qo = oc. Measure the arc CQ, find the deflection the curve gives for that length, and if it agree with the angle ocq, taken by observation, the curve terminates in Q. If it disagree, set off a tangent from Q, and if it cut inside the other curve at r and s, bisect rs, and set a pole at right angles to it at i. Take the angle between qs and iq, continue the curve to M, a farther length due to half the angle sqi, and the tangent from thence will also be very nearly a tangent to the curve ris. However it will still cut a little inside ris, and it may be necessary to move the point M a few links farther on. If a table of natural sines is not at hand, and the point b is fixed, we have, by putting ab=a and db, a perpendicular dropped from b upon AF,=y, a2 r—— for the radius of a curve, and 2y 57-295T8w 3437-747y „ . . . , , . , ---------e == ------—ll = 0 m minutes and decimals a2 a2 of a minute. Putting AD=a' and DB=y, we have r~a - for the radius of a curve, and 2y' CT-29578,/ = M3774V = , in minates „a decimak of a minute. In deep rock cutting, or where the slopes are steep, and in rough ground, the operation of curving12 A MANUAL ON EARTHWORK. is sometimes entirely confined within the formation width, and no straight lines can be had. In this case a random curve abh (fig. 2) may be run out with an assumed angle of deflection to any point b, from which the opposite end c of the re- quired curve ac may be observed, and the L fbc and the line bc obtained. =r=BD, and ef being tangent at b, Thus: G n 28*64789 0 90°— L abe= l abd, 180°—( L abe -f L fbc)= zL abc, and 2 Z. abd = L adb. Putting a=AB, b — bc, and c=AC, we have . 28*6478.. • . . 28-6478 sin Z adb sin Z abd : --- :: sin L adb : a =--;-------- 0 6 sin L ABD also putting Zbac=a, and Z.acb=b, and Zabc = C, we have 180°— Z abc= Z (a + b), and b , a + b , a—b ---- y tan —L- = tan —— a + b 2 2 The angle answering to this tangent added to or sub- A -f- B tracted from —-—, according as b is greater or less than a, gives the angle A, which added to the Z eab gives Z eac, the angle of the curve. To find r\ we have sin Albi: sin C : c, and sin Z eac = L agc : C- ! I sin 90° : aG=r'; thenON CURVES. 13 28 64789 _ ^ true angje 0f Reflection per chain. This plan may be also used in open ground in some cases. Where tables are not at hand, the angles of a triangle may be had as follows:— Let bac (fig. 3) be the given triangle, and the Z bac the given angle. Fig. 3. Let bc=c, AB=a, ac—b, ad=6?, DC=e. Then c2= (a + d)2 4- e2= a2 + 2ad + d2 + e2. But b2=d2 + e2, .*. c2=a2+52-f 2ad, and -—^ — -=d. 2a Also as b2—d2=e2, and the arc DE=e2 — ~ + 1 — y &c. we have 6-2832 (a+d) : 360° :; e - 1* + |5 - f &c. : 0' Out the number of degrees in the arc de or the angle abc. The angle acb = 180°—( Z. bac + L abc). Hence 6- =CT'295” X (e - f + * + Sc.) end a+d V 3 5 7 y — DE the length of the arc. 57-29578 Vf cl2 4* b2 c2 \ 2 1 — f--------------) (See Law's Tables, page 160, formula 16.)14 A MANUAL ON EARTHWORK. Fig. 4. In laying out curves by the production of chords of equal length, as in fig. 4, we have — y — — where y = the offset r ec. a = the chord be, or ab, produced from b to e; and as the angles gba and fbc made by chords of equal arcs, ab and be, with the tangent gf, are equal, and gba = the vertical L ebf, the angle fbc — /. ebf Therefore ef = fc = — = y, or the offset from the tangent is half that 2 2 chord. from the produced offset db = ^ or 2 2v Consequently the first In setting out on this principle, the chain must be laid along ab, and the point b found by moving it until d comes in line with ga. If the length be measured along ad there will be an error. Likewise, as the angle bee cannot be set out without a curving instrument made to suit the vari- ation of that angle, the line be produced from a, through b, must be measured; a pin left at e, and the end of the chain moved from e to c until ec—y, the calculated offset. As the angle fbcor half the central angle 2 2 2.fbc= /_ebc=. I, bhe, and the triangles ecb and chb are similar; bh \ be \ \ be \ ce, or r \ 1 :: 1 \ y, or —= y or y in inches, if r be in chains, as there r r are 792 inches in a chain.ON CURVES. 15 Suppose be = 1 chain, hi = 2 chains. Then r \ 1 :: 1 ; y—ec, and r \ 2\ \ 2 \ y'= ij. Hence y=4y, or the offset increases as the square of the chords. But the offsets from the tangent fc and hi increase more rapidly. For be \bf\\ bn \ bk, but be >cn, bf > fk, or bk <2bf, the square of bk is less than four times the square of bf, and as the offset hi is four times fc, it is evident that the offsets from the tangent increase more than as the squares of the lengths measured on the tangents. The principle of ordinates is advantageously applied in setting out small curves from 100 to 500 feet radius on level ground. In fig. 5, the radius may be had by the methods previously stated, or the lines ad, db, and ba measured, then bd : ba :: ad : ae = r, the radius; or when the curve is less than 90°, let a perpendicular hf drop from h upon cb, the tangent to the other ex- tremity of the curve cgh; then putting CF=af, fh=y't aJ2 i 2 —= r, as previously stated. Then, from the equation of the circle, taking e as the origin, and ea=x, eb=-xf, ec=x", ek &c., we have, for the ordinates from the diameter ej,16 A MANUAL ON EARTHWORK. W2 — a///2=K4, 8cc., and it is evident that these values deducted from the radius, or r — vV2—x2, r — vV2—a?'2, r — W2—a;"2, r—Vr2 — a?///2, give the offsets from the tangent gi at 1, 2, 3, &c., or that the value of kc deducted from those ordinates will give 1 1', 2 2', 3 3', &c., the points 1', 2', 3', cor- responding with a, b, c, being taken at any distances apart that are suitable to the surface of the ground. The versed sine DG=r—KC=r— vV2—a?"'2. ON HALF-WIDTHS. As sidelong ground has, in most cases, a regular inclination, the following system for finding the half- widths and the values of h and h' (fig. 6) is humbly suggested for saving the trouble of making and plotting cross sections. Fig. 6. ju^r N. I ! Jr' X i i , (C it aHALF-WIDTHS 17 Let kk! represent the natural surface of the ground, w H the central height of cutting, lc=mf=:na=-^ half the formation width of a cutting, and d=cf the difference of level between l and c; also /c?=t,h and zk=rhf. Then, drawing bd parallel to ck, we have ab \ bd\\ac \ ck. But bd\ ck\\m\ zk .\ab l ruy.ac l zk and ^^=zk. But ~ : d:\rn 2 J w w j Substituting this value in the preceding formula, we have rH yac wry ac rh' or zk= . H- 2rdn. w—2rd | and putting x for ac9 we have wrx w — 2rd =zk. . . , H xzk zk Also ru : h\\zk : za=--=— or rH r za— wx w~2rd In like manner h may be found, and the half-widths w w will be g + I'M and ^ + rh respectively. On the lower side of the cutting, if a\ b\ c\ &c., B18 A MANUAL ON EARTHWORK. jo «» is + ^ 5 CM a N 28*42 o 9 cb CO 49*47 ■e »o> o 00 11 f? 04 9 9 1 00 "if bi S 1—< fc. o* ft! o o o ii 5 9 9 9 ?■* _L GO 6 Tf* l ’"■' ,_l Sts. *3 tO CO So > o o 00 05 r“‘ + 05 bi 13 04 o 05 II > >S 9 GO « fe 1 00 t-H »b S O T3 V co o 00 * + 9 9 00 (N 00 15 ^ 1 OJ CO co w i 1 j= ) y= h! (w + 2rh) w + r (h? h)’ h(w-\- 2 rh!} w + r(h' + K)’ and TRAPEZOIDAL FORMULA. Let the adjoining fig. 8 represent a trapezoidal solid contained between two parallel cross sections of Fig. 8. k any cutting. Suppose it to be cut by a plane aefg parallel to pnmz. Also by two vertical planes azpd and bmnc ; and by another vertical plane afc along the diagonal of. Then if pd=y, nc=x> az=y\ and mb=x\ l =pz, and w—zm (x’-y')+(x-y). _ gbrfa 2 3PRELIMINARY FORMULAS. 21 (.y-yO+Ctt-y). ™ = cfdea 2 3 y’wiu = azmgfnep J,W ’ * T ^+y+2x)-4y+^L 2 =(*'+#+2* - 4#'+6/)_ = (a) = jV+^+2 (*+y)| = azmbcndp the content of the figure, when the surface slopes both ways, from cb to ad and from dc to ba9 exclusive of the truncated pyramids at the sides formed by the slopes. To find the truncated pyramids tazjdp and bvmcnk, suppose a plane passing through tpd. Then if r express the ratio of the slopes, * J= pyramid^#. L ^ = rh" (y +y') i = pyramid pdazt. 2 3 o Hence, {rhy + rh'' (y+y)}i- = r {hy+h" (y+y1)} |-= the truncated pyramid jdptaz. Similarly r A'# + A'7 (a? -f a/) J -i = kcnvbm► Combining these with A we have (b) |V+#+2 (*+#') j + r +22 A MANUAL ON EARTHWORK. =the content of the whole figure jkvtzmnp, when the surface slopes both ways from kv to jt, and from jk to vt. Putting h' and h for the central heights at the ends, when the surface is level on the cross section, and slopes from jk to tv only, K'"’ ft y\ each = h} consequently (b) becomes (w |h + h'+2(h'+h) J -f r h'2+ h (h'+ h) + h (h'+h) | Sw (h'+ h) + r (2h/2+ 2h'h + 2h2)| (c) 3w(H'+H) + 2r (h'2 + h h+h2) j- the content of the trapezoidal solid. L ~6 (b) may be also written— («) |*'+y+2(*+y')} +rl ■17=r+-jznri) L 6 or, (/3) 1*'+^+2 (#+/)}■ • | Ay + h"y' + */hy x A"/ + h'x + h"'x' + V A'* * h'"x' jPRELIMINARY FORMULAE. 23 or, (?) (*> {*'+^+2 («+y)| fh + s o ' jo\ h' + h',f , 2 I > /2N1 \ L ^ yy TTV '( +* VjT In the latter and - are the ratios for y+y x + otf correcting the truncated pyramids when the surface is warped, x, a/, y, and y', can be had by (a) and (b) or observation on the ground. On examining (a) we find thatX' ^ ^. w is the area due to the average height between m and jo, and x.^y~. w, that to the average height between z and n. 2 Therefore, this formula expresses twice the middle area on mp, and four times the middle area on zn, when x + y' > x' -f- y, and vice versa when x' + y > x+yf* Consequently this formula is to be written l w IT |*+/+2 (V+y) J when x' +y>x+yf. When the surface is much warped, twice the middle area on mp will not be equal to the sum of the areas on zm and pn, nor will four times the area on zn be equal to four times the middle area on ww; conse- quently, where tabulation by cross sections is adopted, as in formula (d), the sections are to be taken nearer than one chain apart when the surface of the ground is much warped, or the rate of inclination on the cross sections differs within a length of two chains.24 A MANUAL ON EARTHWORK. (c) may be written— 3w (h7 + h) 4- r (2h'2 + 2h'h + 2h2) j- -^-= {h' + h} + r|n'2 + (h2 + 2h'h + h2)+h2 = {h' + h} +r | h'2 + (h'+ h)2-f h2 | ^ ~ = the content. Here we have 2w (h' + h) + r (h' + h)2= four times the middle area, w (h' + h) + r (h'2 + h2)= the sum of the end areas; jj' _i_ jj for —-—= the middle height, and h' + h , h' + h h'+h h'+h . (h' + h)2 2 2 2 2 4 = the middle area, and four times the middle area = 2w (h' + h)+r (h' + h)2. Hence the practical rule for finding the volume of a trapezoidal solid of this description. ‘ To the sum of the areas of each end add four times the middle area, and multiply by one-sixth of the length.* In adopting (c) as the principle for tabulating by cross sections, it is well to remark again that wherever the cross-sectional inclination differs much within two chains’ length, it is necessary for extreme accuracy to take the observations at half chains, or nearer, as may be seen by considering the remarks upon (a). Other- wise it will be necessary to tabulate by (m), (jp), and (w), as in Examples 1, 2, and 5.PRELIMINARY FORMULAS. 25 According to this principle, dividing the cutting, fig. 9, into an even number of parts, one chain in length each, = l, and taking the solids 2 chains long, =2l, we shall have |2(c + e, &c.) + 4(b+d + f, &c.)+a + g J-?l= *(d)|2(c+e,&c.)+4(b+d+f,&c.)+a+g J | = the volume of the cutting, exclusive of the fend wedges ay' and Gz. A and G represent the terminal cross sections a1 and g7 ; c, e, &c., those at the odd points c3, e5, &c. ; b, d, f, &c., those at the even points b2, d4, f6, &c. * See Hughes’ Tables, published by Effingham & Wilson. 1846.26 A MANUAL ON EARTHWORK. WEDGES AND TRAPEZOIDAL SOLIDS. When the back is a regular figure and the edge parallel to it (fig. 10), we have FlG* 10*___ ^5 = content of wedge on h =content of the pyramids on 1 and 2. WLU , 7*LH Hence Fig. 11. V& \ \\ \ Vi\ IVLli , 7*LH_ /a i o \ L / \ \ ^ -H- + =H(3w + 2m) - (e) w' Z, o o gives the volume of the whole figure; where l = the length, H=the height of the back, r=the ratio of the slopes, and w?=the formation width. In the case of side- long ground forming oblique wedges (fig. 11) we have H = ^and h : l :: x: —=i/(e) H w Likewise h l l :: y \ — = l" (d). Then (x-\-lir if rh'2\ l' rh content of pyramid on 1. ,___________ n V. 2 2 ) content of pyramid on 2. rh!x 2“’ (y+h. rh _ ^ l"— rJ,y 3 2 L 3^ i/. 3PRELIMINARY FORMULAS. 27 By substituting for l' and l" their equivalents in (c) and (d), the sum of these =M (A'x2+V) (eY For that portion of the wedge on h we have . V)X j x— y \w\\x\ ----and x y *-y '.w'.'.y'. ^- = yg, x—y / L fX WX \ __ / L ‘ A 2 * 3(x-y)J V y wy _l' x2—!." y2 tv __ 2 3 (x— y)j = the content. x-y By substituting for l' and l" their equivalents in (c) and id) this becomes La;3— Ly3 w? l(#3—#3) w wh , 9 . 9 . -7----• 77 = A —( • 7t = 7- («2+^2 + ^) (/) H(«—y) 6 H(ic—y) 6 6h Combining (e) and (f) we have ^ (A'tf2 + %2) +(^2 + «/2 + = i | r (A'a;2 + hy2) + w (a:2 +^2 + xy) J = 0) the whole volume of the trapezoidal wedge, (fig. 11) By substituting for x and y their values in (a) and (b) we have, in terms of the side heights h and A',When h=o it becomesPRELIMINARY FORMULAE. 29 When the terminal wedge is a pyramid (fig. 12) we have — . 7^ = the content. This, by substituting for 2 o x and l' their values in (a) and (c) w J.xhl __ Lw h!2(w + 2rJi) 2 * 3h 6h w + r (A' + A)’ But as it takes the form of — • ——t7=(g)> the volume of the 6h w + rh! v pyramidal wedge (fig. 12). The three preceding forms of wedges are those that occur most frequently in the formation of earth- works, and have therefore been transformed to ex- pressions containing h and h! instead of x and y, as the latter may not be had, after a cutting or embank- ment is finished, without separate calculations. Fig. 12. From (a) when y'=o (h) w(x'+y + 2x)~= o content of fig. 13. When x* and yf=o (i) w(2*+y)i = content of fig. 14. Fig. 13. V30 A MANUAL ON EARTHWORK. Fig. 15. When d and y* = o, and x -f y = H, / \ HL W content of fig. 15. In (h) when there are side slopes, we have from (b); vo (x + y + 2x) + r (hy + h'x + h"y) + h(x + x) j- F (k). J In (i) | w(2x+y) + r (h'x+hy) j L 6" In (j) H(3w-f-2rH) which is the same as (e). 6 For a truncated pyramid (fig. 16) we have from the! second term of (b) When the ground is level on the cross section h—y and hff=y', .*. we havePRELIMINARY FORMULAS. 31 (p) r(y* 2-\-yyf+y'2) -5- = the content of fig. 16. Fig. 17. For a wedge like fig. 17 we have, by the first for- mula for (a), wl 2" H _ =WH 3 L ¥ (Q)- When the forma- tion-width is greater at one end than the other, or w > wf (fig. 18), we have from (q) w—w' L y' __ 2 ~ ' “2 ' 3"~ y' (w—w') 2 | = azep, and . h = bmjh 2 6 J Fig. 18. Also h+y W—w' -!._(h+p) ■ (w-wr) L 2 ' 2 ' 3 2 ' (i 7 J} 232 A MANUAL ON EARTHWORK. and ih'±*l^(w-wr'> . h = bfncd. Hence 2 6 (y^w—w’) , x'(w—to'), (h+y) . (tc—w'),(h'+x) . (w-w')\ l _ V 2 2 ^ 2 T 2 / ii O) (/ + *,+y + * + A + A')^.'. £ = the sum of the contents of the side wedges, in which (S) (y'+y+h) ■ -jr — azejdp, and (t) («/ -\-x-\-h!) . Il — bmfnck. 2 6 As from (a) we get the content of azmbcnpd, if we add (a) and (r) together we have (u) ^w'{x'+y+2(x + y')} the cubical volume of the whole figure azmbhfej, or 3/')-}-A + A' ~ when «/=0, as in this case x'=yr and x —y. In (u) when w—wf9 the second term vanishes, and we have (a). In (s) and (t) when the wedges are not similar, ep and nf are to be used instead of .PRELIMINARY FORMULAS. 33 Fig. 19. In fig. 19, where the sur- \ j \ tt face is level on the cross sec- \ • tion, and slopes only in one direction, from w to w'y , we \ \ \ have in (u) h3 y, x, h\ each = h' y' and a/ each = h, | Stv' (h'+h) + (w—w1) . (2b! + h) | = (v)=|w'(h' + 2h) + w(h + 2h') J = the cubic volume, or w (h + 2h') — when wf — o. From (u) when yf == o, fig. 20, |w'(x' + y + 2x)+ w- (a/ + y + ff + A + A') J -^-= (w) Fig. 20. Fig. 21.34 A MANUAL ON EARTHWORK. the cubic content, or ~ (x/-f 2x -f h + A') ^ when w'—o; and when a/ and yf = o, fig. 21, {«/(y+2*) + ^p- (y+a; + A+A')J.| = (x) the volume, or w 2 Fig. 22. (y+x + h + h') i when w'=o. 6 and when a?' and y=o, and # and y=H, fig. 22, H(w' + 2M>)i-= (T) o the cubic content, or 2mv i when w’= o. 6 In a truncated pyramid of the form shown by fig 23, we have from (m) , r yHh + h>)-y’i{h» + h"')\ l_ w l y-y J 6 the content; or from (l) (r{y(A + A') + (y+y) . (A" + A"')})i or from (o) And when the surface is level on the cross sectionsPRELIMINARY FORMULAE. 35 as h and h'=y, and hn and h!n~y\ we have (fig. 24) from (z) Fig. 23. Fig. 24. (2r . ^ 5 = the content; or \ h' —h ) 6 (At) 2r (h2 + h'h + h'2) ~ the same as twice (p). 6 (q), when there are side slopes, becomes H{«,+r(A'+A)}5(B2). Fig. 25 represents fig. 18 with side-slopes. Its cubic capacity may be . found by adding the lg* 25* side figures jdpeatz and kcnfbvm to the central portion azmbcnpd, as given by formula (a). Let je and kf be produced until they meet dp and cn pro- duced in g and n', and join gz and n'm. 36 A MANUAL ON EARTHWORK. mi W----w' W — tv' . , ' Then as r x pg = ep = —-—pg=z — = si=ul. & 2r Add w~~w. . to is and we have jL which call h. Add 2r it to ku and we have kl, which call h'. Also calljt dg9 y, and cn'9 x. Now from the second term of (b) r < hy + h'x + A" (Y+y')+hf/r(x.+x') the content of the truncated pyramids jdgatz and! hcn'bvm. Also ; rw—w w—w os- V 2 2r the pyramid epgz9 and 2 fw—wf w - wr\ l _(w—w')2 os \ 2 2r ) 6 2r o i the content of both pyramids, epgz and fnn'm. Therefore (c i) | hy + h'x -f h" (y + y') + h/" (x+o') j* ___(w — w’ )2\ L _ - 2r the content of jdpeatz and kcnfbvm. On combining this with (a) we have 0>i) (w1 jV+y+2 (x+y1) j- -|r|HY+h'x+h" (y+yl')+h(x + *0 J __ (w — w')2\ L 2r J 6PRELIMINARY FORMULAE. 37 the whole cubical volume jkfetvmz. When wf=zo this becomes, as x-y and x/=yl, ( {*(h + h') + (y+/) . (A" + A'")} When there are side slopes and the surface is level on the cross section (see fig. 19), we have from (Dj) {3«%+y)+2KH2+n/ +y2)J i =(Ei) and -f 2r (n2 + Hy +y'2) — \ when wr—o. I 2r J 6 When y'=o (see fig. 20) we have from (dx) w' (a?' -f y + 2x) -f r {hy -f h'x + ^"'(x + x')} = {hy + h'x + h!u (x-f x')— when w'-=o. When x' and y'=o we have likewise (see fig. 21) |w'(2x+y)+r (hy + h'x) — |i = (g,) = \ r (hy-|-h'x) —— \ i when w/=o; and ,, t 2rJ 6 jjwhen a/ and y1 h'=x=y=h, and a:=y (see fig. |j22), we have (3w'y+2r h2^ il == (h^) where y=. H, in \ 2r J 6 ifig. 22.38 A MANUAL ON EARTHWORK. APPLICATION OF THE PRECEDING FORMULAS. As sometimes cuttings and embankments are formed with retaining walls, and without slopes, of the form shown by azmbcnpd (fig. 8); the formula (a) gives in such cases l! for the volume of a cutting or embankment in cubic yards; where s = the sum of the intermediate heights on the lower and upper side; at and yr, the terminal heights at the beginning; x and y those at the end| For if we suppose a similar figure to extend from dpnc (fig. 8) on the one side, and another from azml on the other, we shall find that xf will be taken once for fig. 8, and twice for the adjacent figure; also y twice for fig. 8, and once for the adjacent figure Therefore at and yf will be taken three times each, sc likewise will x and y, for any number of intermediate sections. At the first section x1 will be taken once, y twice. At the last section x will be taken twice, anc y once, we have therefore ; w^*4074 {at + y + 2(x+y') -b3s}J = (h) = *2037 w {2 (xf+y) + 4 (x +y') + 6 s} for the content of such a cutting, exclusive of the enc wedges. w and the constant -4074 = ---. 6x27APPLICATION OF THE FORMULA. 39 When the ground is much warped, and there are side slopes, the quantities can be had most accurately by combining the latter equation (h) with the second term of (b). We shall then have O') -2037m>|6s +2 (V + y)+4(a:+y) | ■f *4074 r (s -j- sf+s/f + s'")+y' -f- z' for the content of a cutting; where d and y\ x and y are as in fig. 8, supposing several sections to intervene between them; s = the sums of x and y at all the intermediate sections, s= the sums of hn (y' +y)> the sums of h,n (pc' -fa?) at all but the last section, the sums of hy, and 5"'= the sums h' x at all but the first section, and y' + zthe contents of the end wedges, Ay* + gz' (fig. 9) to be found by formulae (e), (f), (g), &c. In order to apply formula (d)* we have to find a general expression for the cross-sectional areas ai, b2, &c., as shown by abcd (fig. 26). Fig. 26. * This formula (d), which is the application of the Prismoidal, hclds good where the ground is not much warped, and the cross sections are taken closely; also in all cases where the heights are taken from the longitudinal section.40 A MANUAL ON EARTHWORK. (1) dKpA±bc>-d6xa/= area abcd. v J 2 Put T>k=h\ ag—h, BC—w, and r— the ratio of the slopes; then njy=rh\ also mb = rh\ and mn — w; r>6=2rA/-f w, and nb + Bc=z2rh'+ 2w. Also vbx Af=(2rh' + w) x (h'—h). Hence from (1) h' (2rh'+ 2w) - (2rh' + w) . (h'-h)_ 2 h'w + 2rhh'+hw w f,, , , ,,, -------2--------=-(A'+/i) + rM' (2) gives the area abcd. The multiplier in the general formula (d) becomes for cubic yards, or -8148 when l = 66 feet, and 1-2345 when l=100 feet. Consequently, putting yr and z for the content of the end wedges we have (j) *8148 2(c + e, &c.)+4 (b + d+f, &c.)+a + gJ -fy'-l-2'= the cubical volume of the whole cutting from y* to z' in fig. 9. When the distances are 100 feet apart this becomes (&) 1-2345 | 2(c + e,&c.) + 4(b + d + f, &c.) + (a + g) J +y,-srzl In (2) if the sums of (h* + h)=s, and those of their products hxh'=s for the cross-sections at c, e, &c., then ws + 2rs gives twice the sum of their areas, or 2 (c + e, &c.) in (j) and (k). Likewise in the cross- sections at b, d, f, &c., if the sums of (^' + A)=s', and those of hxh'^s', 2i0S/+4r$/ gives four times the sum of their areas, or 4(b-|-d-|-f, &e.) Finally, ifAPPLICATION OF THE FORMULAE. 41 the sums of (A' + A)==s", and those of hxk'=:s", in the cross-sections at A and G, ^ s" + rs" gives the sum of the areas at a and g. Hence (j) may be written *8148 | (ws + 2rs) + (2ws, + 4rs) + (~ s" + rs") | + y + z' and (A) 1*2345 | (ws + 2rs) + (2ws' + 4rs') + (*f s" + rs") | +y + z or (l) *2037w (4s + 8s' + 2s") + *4074r (4s + 8s + 2s") + y + z and (m) *3086w (4s + 8s' + 2s") + *6172r (4s + 8s' + 2s") +y +z The values of '2037w>, *4074r, &c., are given in the Table of Ratios. When l=33, 50, 16*50 or 25 feet, one half or one fourth the tabular values of these constants is to be used. Also when tabulation from the heights on the longi- tudinal section is necessary, the values of 2A are used instead of those of (A' -f A), and those of A2 instead of h x A7, as may be seen on comparing examples 3 and 4. Should Y-shaped drains, in sidelong ground (fig. 23) be calculated in successive lengths, the tabulation of (z) as derived from (o) is the best. t If A" and A'", h!,n and h!nn represent the side heights, y' and yn the middle heights, at the terminal sections. Also s, the sums of A + A'; s', the sums of y; s, the sums of y*\ and s\ the sums of y'y at all the intermediate sections. Then42 A MANUAL ON EARTHWORK. 2s + h" + h!" + h"" + h" { (h) *4074r 2s'+y+y' 2s + h" + h"' + h"" + h'"" 0 ^ . ,2l ,/2\ the content, which added to the end pyramids gives the cubic volume of such a drain. The end pyramids equal the areas of the terminal sections multiplied by one third of their respective lengths. If the values of y are not taken by observation, they may be had f 2h!h from 5,=^ (3). An easier tabulation for v-drains may be had by cross-sectional areas, unless the surface of the ground be much warped. The area of a v-shaped drain (figs. 23 and 24) is h! -f h J _ ^rhn r__ r rh!h (4). consequently if s = h'xh at c, e, &c., s'=h'xh at b, d, f, &c., $"=h'xh at A and g (fig. 9), we shall have from (/) (o) •4074r(4* + 8*'+2«")+y'+*' for the cubical content when 66 feet lengths are used; and from (m) (p) -6l72r (4s+8s' + 2s")+y' + zf when the lengths are 100 feet. Should the surface of the ground be level (fig. 24), we have from (a2)CALCULATIONS FOR LAND. 43 2r (2s + s' + h2 + h'2) i = 6 (iq) *4074r |4s + 2(s' + h2 + h/2)| the content in cubic yards, exclusive of the end wedges y and z\ which may be found as in (n) from (4). Here s=the sums of h/2; s'= the sums of h'h; h2 and h'2, the squares of the terminal heights.* CALCULATIONS FOR LAND. The quantity of land to be occupied by a cutting or embankment may be found as follows:— Fig. 27. Let fig. 27 represent the plan of a cutting or * Supposing fig. 24 divided by several parallel sections, when h' at the end of the first length becomes h at the beginning of the second length, and h' at the end of the second length becomes h at the beginning of the third length, and so on.44 A MANUAL ON EARTHWORK. embankment where ^ is the distance between the top £4 of the slopes and the fence, w the formation width, and rh, r//, &c., the bases of the slopes. Dividing it into an even number of parts, and calling the ordinates at right angles to the centre line at each length yf, a, b, c, &c., we have (r) • |4(A + C + E + G) + 2(B + D + F)+y+2/1 = the area expressed in feet. Putting s for the sums of (k'+h) in A, c, E, &c., and s' for those of (A'-f A) in B, d, f, &c., and (w+c)=zyf = z!, we have (s) ^--£r(4s+2s') + 3(»+l) . (m> + c)| or jr (4s+2s') J when (tv 4- c) = o, as in Y-shaped drains, for the area of the land occupied ; where n = the number of even parts into which the cutting has been divided. (See Hughes’s Tables, note 2.) When these calculations are carried on continuously from cutting to embankment this formula holds good, whether the ends are square to the centre line or not. But when the land required for a single cutting (fig. 9) or embankment with oblique ends is to be computed, we have (fig. 11), by substituting for l' and Ln their values in (c) and (d), and for x and y their values in (a) and (b\CALCULATIONS FOR LAND. 45 LX 2h X rh'=z A on 1. x r A = A on 2. 2h L W = CD on 3., ( h'2 (w + 2rh)+h2 (w + 2r/ir) \ l ' 2h {w + r(h'+h)} T y for the area of each oblique end, and for the land oc- cupied by the remainder of the cutting — 4 (b 4“ i) -j- F, &C.) -f“ 2 (c+E, &C.)-|~ A-{-G or, putting s" for the sum of (A'+A) in A and G, (w) -|-|r(4s'+ 2s + s") + 3 («— 1) . (to+c)| = the area, exclusive of the oblique ends, and when w-\-c=o, as in v-drains, 2s + s") | In (s) and (u) T‘ = ’07958, and in (t) l = ’2424, O for perches, when the distances are 66 feet long. If 100 feet long, these become *1224 and *3673 respec- tively.46 A MANUAL ON EARTHWORK. AREAS OF SLOPES. As r (h! + h) expresses the bases of the slopes in fig. 27, sec. 0 (A'-f- h) will express the lengths of the sloped surface on each side at a, b, c, &c., 0 being the angle of the slope with the vertical. Consequently (/) becomes l ^sec. 0 ~-^ ~ O) l ( sec. 0 A'2 (w + 2rk) + A2 (w4- 2rA')\ 2h [w + r (A' 4- A)} / for the area in square feet of the sloped surface of the terminal triangles, and (w) becomes ~1 sec. 0 (4s' + 2s + s") ]• (w) for the area of the intermediate slopes from a to G. But if these calculations are carried on from cutting to embankment, we have from (s) L 3 sec. 0 (4s -f 2s') | (#) for the superficies. The values of sec. 0, or Vl + r2 are given in the Table. When l = 66 feet, ^ == 7*3333 becomes the mul- tiplier in (t?) for square yards, and == 2*4444 in (w) and (x). If l = 100 feet, these multipliers be- come 3*7037 and 11*1111 respectively.TABLE OF RATIOS. 47 TABLE OF RATIOS. w r •2037m; •4C74r •3086 to •6173 r Sec. 0 Vl + r2 15 3-0555 4-6297 16 l 4 3*2592 •1018 4-9384 •1543 1-0307 17 3-4629 5 2470 18 4 3*6666 •2037 5 5557 •3036 1-1180 19 3*8703 5*8643 20 1 4-0740 •4074 6*1730 •6172 1-4100 21 4-2777 6*4816 22 H 4-4814 •6111 6-7903 •9258 1-8027 23 4-6851 7-0989 24 2 4-8888 •8148 7-4076 1-2345 2-2360 25 5*0925 7-7162 26 H 5-2962 1-0185 8*0249 1-5431 2*6925 27 5-4999 8*3335 28 3 5-7036 1-2222 8-6422 1-8517 3-1622 29 5-9073 8*9508 30 CO 6*1110 1-4259 9 2595 2-1603 3-6405 31 6-3147 9-5681 32 4 6*5184 1-6296 9-8768 2-4690 4-1231 33 6-7221 101854 34 5 6-9258 2 0370 10-4941 3-0862 5*0990 35 7*1295 10-8027 36 6 7*3332 2-4444 11-1096 3-2035 6*0828Ex. 1.—Required the quantity of cutting in Fig. 9, supposing the formation width to be 30 feet, the values of h to be the side heights, and no slopes. A MANUAL ON EARTHWORK, « S + |W b + • + o o o £ Tt< N o > -I- s U 3 h a A3 | + a a Nfe° 2, O GO V + ?? d '-tf c + S I A x: I d .. «*>,_ OS OS jj + 3 O < O Tj< d d o 00 o O 00 '2037m>=* 6*111 x 686 = 4192-14Ex. 2.— Required the same with sides sloped at lj to 1. 49 DEx. 3.—Required the same from cross-sectional areas, by formula (l). 50 A MANUAL ON EARTHWORK, I © r-J © r-J « U C-'TS © I’o r£ g £ b K bO © - W u © bL^ © B £;-©<«- ■S T3 ^ ° ©©-•-> ~ © © O P. O .03 £ © q £ eg «t5 © '-S S 2 •b • H: £ — © C/j r£ © £ • cn HO Sc?3 © £ X © c 1 W) l-sc X 48 80 00 CM CM VO O CM + 00 CO 0 00 CM CM Tt* |5t, CO CO A G "co (c *cc 00 CM II II II X O CM CO 00 00 -* CO *0 t>- 05 -too CM N O (M + OO CM CO CM 00 CO 00 -J* 0 CM B B F CM O CM Tf 00 X Tfi »TS 0 ^ -H CM 0 ■< + 00 ^ 0 CO ^ -t Tt* ^ 00 CM CO — Tf< 0 « II II II to '*> "oo ''T 00 CM CO <0 © Ph S CO £ O © P. £ co © r£ X 00 o o CO * 05 00 CO — o 7 00 w m H3 © i 00 ic 00 O O h (M ^ 00 co I! H 00 JO •o CO CM t>. CO X COEx. 4.—Required the same, using the heights on the longitudinal section and formula (l). EXAMPLES. £ x x W aJ N * 4- T3 v'D 0^0^ CO o CO o Xji XN CO Oi CO Vs 05 CO 05 Tj( o || O) || X / N X s O O o « N 0 *o 00 „ . i io Tf« id *o o »o t ^ O . ^ • 05 00 < II II D 2 6552*48 c. yds. Note. The differences between this and the two preceding Examples exemplify the error that results by using the heights on the Longitudinal Section instead of those obtained by Cross- sections, or Half-widths.52 A MANUAL ON EARTHWORK. t _r! la* <3 5 o n3 cr • o o >• h! CO « 2 i ^ : + i + s c3 u p? •f3 00 *p r-. 7 If 7 ? ’ »*»> P2 ?1 Ci 't oo co >c

cn o o o (O -- O C to OOCOHO) O C) 't (N to 00 to O 00 5»i N <; o II II 00 > 7"1 O + o ®0 - os B D F 72 X 112 140 c E 00 OJ <0 CN *0 h l(N ^ I 00 Tf* CO I 00 O © "t o 0^ © N ffl O Note. Here, again, the result derived from the PrismoidaJ formula is less than that from the Trapezoidal.Ex. 7.—Required the same when the surface of the ground is level on the cross-sections. Here the values of h will be the same as those of y in Example 5. By formula (q). A MANUAL ON EARTHWORK, in o a > a a co co 7- co b o •p b oo 00 co m x (M p A- O CO CMEx. 9.—Required the same when the ends are oblique, and the side heights given by cross- sections are used. 56 A MANUAL ON EARTHWORK. + a O o co + <3 ^ + w iT + (m a a -t o Ci o II 00 ^ + 2 2 + . 2 * »p *p ..H ^ 5 «< o o 5-1 c3 £ * a4 CO »o Tt< CO cq CO l> o H o oo T* co 05 05 - 05 CO II II l"c3 o Tf<