396 Civil and Mechanical E~gineering.

FORMUL/E~ RULES AND EXAMPLES OF COMPUTATION FOR SOME OF THE MOST USEFUL CASES OF EARTHWORK UNDER

WARPED AND PLANE SURFACES, Br JoHn- W~.R~R, A. M.

Object loro2osed.--I shall here present a number of formulm which, it is believed, will be found useful in practice and interesting for their form and mode of derivation.

Definition of the Ground Surface.'--Straight Work. ~ Intending, on a future occasion, to treat more at length of the ground surface, I shall here only define it as assumed for the present investigation, and omit extended remarks on the practical sufficiency or utility of the formulm. The ground surface is assumed to consist of one or more hyperbolic paraboloidal surfaces generated by the motion of straight lines always parallel to the vertical end planes of the work,

and moving lengthwise of it upon straight directrices. The external straight bounding lines of the sur. face, which connect the outside cot. hers of the work, are always sup. posed to be directrices, and the straight surface line connecting the extremities of the centre heights at

. . . . . . . . opposite ends of the work is always taken for a directrix when this line is supposed to exist. Other straight directrices may be imag- ined to extend from points on the surface line of one of the end cross- sections to points upon the surface line of the other ; but in all cases

J no two directrices must cut the same longitudinal vertical plane within the limits of the work. The two end cross-sections and the mid cross- section of a piece of so.called three. level ground may be represented, in their general features~ by the an. hexed diagrams, which refer to ex-

L . . . . . . . . . . . . . . . . . . . .

cavation but will serve to explain

.~ The discussion of curved work is here omitted.

~b~'~ulx for Ea~lt~wor#. 397

embankment by imagining the diagrams inverted. The)- will "be understood without minute explanation. The surface lines F •, F c, on any cross-section of three-level ground are supposed to be straight ; these are the generating lines of the ground surface, which move upon straight directrices passing through ]3 ~, F F, C c. The line B F C may either be broken in F or be a continuous straight line, according to the relative dimensions of the cross-section under con. sideration. I f ~ ~' c is a broken line on any one cross-section, the ground surface will be divided into two portions, one on the right the other on the le f t 'o f the centre; either or both of which portions may be warped or plane. In like manner each of these portions of the surface may be again subdivided, and so on, as long as the inequalities of the surface lines of the end cross-sections re. quire this mode of subdivision. I f the lines F n, F c, form a con- tinuous straight line on each of the end cross-sections, the ground surface if not plane may, according to the judgment of the compu. ter, be considered either as a single warped surface having every- where the straight surface line ~ c, or the surface line B c may be considered as broken at F on all cross.sections except those of the ends. When the surface is a plane, both these hypotheses merge into one.

Of the other .Bounding Surfaces.~These are generally plane in practice. The present paper treats the roadbed and the end sur- faces as necessarily plane ; the side slopes are not always considered necessarily plane, nor their inclinations necessarily equal.

Given Dimensions include the length of the work, the centre and side heights, distances, width of roadbed, and angles o f side slope ; but all these are not necessary in every case.

5rotation.--It will perhaps be more convenient to exhibit this col- lectively than scattered through the paper, in order that the reader may refer to it as occasion requires.

:L ~ length of the work perpendicular to the cross.sections. B ~--- width of roadbed. h ~--- centre height @ F. H ~ centre height A F, from the intersection of the side slopes,

on the mid cross-section. t t t ~ centre height A F on the first end cross-section: I t " ~ centre height A F on the second end cross-section.

It, It% H " are called augmented centre heights.

Dlh --__-

S?l h

DtPII

SIII h D i l l h

Srl H

Dlr H

D/t /H _~_

398 C[w'l ~_~d Mecha~iccd £'~yineerinj.

G -----twice G A , or the double augment of the centre height ; G A is called the augment of the centre height.

,~ = whole width or total base 5[ N on the mid cross-section. y = total base M N on the first end cross-section. y t - - total base M N and the second end cross-section.

- - angle of side slope ]3 D H or B A M. ' angle of side slope C E Q or C A N. (l -----

½ d = ~ G p, the distance of a point on the roadbed from the centre, Su ----- sum of the end centre heights O F . Dh = difference of the same. S~ ~ sum of the augmented end centre heights A F, or augmented

sum of centre heights = 1=[ r + H ' . DII = difference of the same = H ~ - H ' . Sb - - sum of total bases t I Q or M N at the ends = y + Yr. Db - - difference of the same = y - - y r . SPb = s u m o f the half end widths O D o r G E of r o a d b e d - - ]3

when the width is uniform. Drb = difference of the s a m e - 0 when the width is uniform. Sty, - - sum of all the four side heights ]3 g r C Q ; B H -t- C Q may

be called the total side height of a cross-section. difference of total side heights at the ends. sum of the end side heights ]3 tI . ditt~rence of the same. sum of the end side heights C Q. ditI~rence of the same. sum of the augmented end side heights B N. difference of the same. sum of augmented end side heights C N. difference of the same.

SrP a ~ sum of end distances out G H or A M. Drra = difference of the same. SHIn ~ sum of end distances out G Q or A N. DPIfa = dit}'erence of the same. SlY d

~Div d

S v d

am Dm --

sum of the margina l distances out D t I at the ends. difference of the same. sum of margina l distances out E Q at the ends, difference of the same. sum of the end side slopes A B. difference of the same. sum of the end side slopes "21 C.

l'br.~ul.~ for Eart/zwork. 399

D. -- difference of the same. A = area of mid cross-section. A r = area of first end cross-section. A " =- area of second end cross-section. V --= volume. Vp ---- volume of the redundant prism on _A. D E. V, ~ volume contained between the ground surface and intersec-

tion of side slopes. V~ = volume of the work.

Dimensions of the severa~ Cross-seetlons.--The mid cross-section is an hypothetical one, having each of its linear dimensions and the auxilliary lines of its diagram, respectively, the mean between the like parts on the diagrams of the ends. The lines of the end cross. sections will be found by adding to or subtracting from the mean dimensions of the mid cross-section, respectively, the half difference of the like end dimensions. Before deriving these expressions we will remark that the differences of the end dimensions may, in the formulas, all be regarded as positive in relation to either one of the end cross-sections, but that in the application of the formulm, we must give each difference its proper sign, in order to know the sign of the product of the differences. This is equivalent to considering the differences to be of like sign and their product positive if, on the end cross-section chosen, the greater height is coupled with the greater base; otherwise the product is negative.

For the mid Cross-section :--

G F=½ S,, C Q + B H = ½ S ' h A B = ½ S m H Q _ - - I S ~ G E-= ½ S'b A C = ½ S ,

For the first end Cross.seelion :--

GF=½ (Sh -t- Dh ), C Q ~- B H---½ (S'h ÷ D'h ), A B = ½ ( S m + D m ) HQ=½ (Sb + Db ), G E -- ½ (S'b + D%), A N----½ (S, + D n )

For the second end Cross.section :--

G F----½ (Sh - - Dh ), C Q + B H ---- ½ (S'h - - D'h ), A B = ½ (Sm--D m) HQ=½ (Sb--Db), G =½ (s%-- D, , , ) AN=½

We may also here note the following values :-- 1 1

- - " ~ S " d - - S " d Sm~S ' t i i Gosee, ~ - S H----7 sec. ~-- ,. s i n . ~ c o s . 6

400 Civil and Mechanical $3~gineerin 2.

D m - - - D ' a c o s e c ~ = D ' r r .1 ~ D ' a s e c . ~ z D ' d 1

s i n . o Cos . o

S,~ ~-~ S'1'n cosec~'~- S '"r l I -~ S'"d sec. 5' S"'d I s i n . ~ c o s . o

1 1 D n ~ ] ) f i t n c o s e c o t ~ I ) f l f t t . . . . . . f i t Y__T~ff!

s i n . o r - D d S e C . o - - ~ ' a e o s . a t"

1 S m = (S"h + G) cosec o = (S"h + G) ; .

Sln. o

1 D m ~ D " h c o s e c ~ = Dr'j,

sin. o" 1

Sn ~ (S'"h + G) cosec ~' - - (S'"h + G) sin. o"

1 Dn ~ D"'h c o s e c o' ~--- D'q,

sin. o I"

These values of Sin, So, D,,, D~, do not change their form if we suppose the angles of side slope to be unequal, and hence the ver- tical line A F to cut the roadbed out of the centre, as at G'. But in the case of unequal angles of side slope, put G G ' = ½ d, and consider it positive when B A M is less than C A N, and negative in the opposite case. W e shall then find the following values, which will also hold for equal angles of side slope by putt ing d - - 0 .

D G ' = D G + G G ' = ½ ( B + d ) ; G ' E - - ~ G E - - G G ' = ½ ( B - - d ) . 1

S~ = (S% + B + d) sec o = (S% + B + d) - - - . COS. a

1 D m = D i ' d s e c . ~ = D i e d . . . . . . . . . .

COS. (~

1 Sn = (S" ~ + 13 - - d) see. ~' = (S ~ d + B - - d) - -

COS. a t"

1 Do = D" a see. o' = D" d COS.~"

(To be c o n t i n u e d . )