1846.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



333 



EARTHWORK ON SIDE-LONG CIROUND. 

 By John Hbghes, C.E., A.I.C.E. 



The corrections to quantity arising from an inclination of the ground 

 surface, in cross sections, are most frequently neglected, when estimating 

 the cuttings and embankments of a railway or canal. This omission is in 

 general expressly mentioned in the engineer's specification, so as to avoid 

 all question upon the terms of the contract ; it gives rise, therefore, to no 

 positive injustice to any party, whilst a statement of the number of cubic 

 yards to be removed, as ascertained from the heights taken along the centre 

 line, is considered sufficiently near the truth to enable a contra' tor to ap- 

 preciate the means he is required to supply. This practice obtains, be- 

 cause the corrections for excess and defect are assumed as balancing each 

 other, and because the process of calculating them has been considered a 

 tedious and troublesome labour. My purpose is to show thut neither of 

 these grounds should be acted upon ; — the tirst is mathematically unsound, 

 and the strength of the second may be materially abated. 



If the ground surface is supposed to be generated by the motion of a 

 straight line, which is alvpays at right angles with the centre line, its ends 

 resting on the outer edges of the slopes, the direction of such motion being 

 coincident with the centre line, and all positions of the generatrix being 

 parallel, then the correction to the area of a cross section of any cutting or 

 embankment first computed on the assumption that this line is horizontal, 

 and that the height or depth is measured on the centre, will, when it has any 

 inclination with the horizontal, always be additive. The corrections to 

 widths due to this iticlinatiun I will designate by -|- J^ on one side, and by 

 — x' on the other ; the correction to area will then be represented by the 



X x' 

 expression — — ; r being the ratio of the slopes. 



If the ground surface is supposed to be generated by the motion of two 

 lines, situated in the same vertical plane, having their ends resting on the 

 centre line and on the outer edges of the slopes, all other circumstan- 

 «es being the same as already described, then the correction to area will be 



oa one side, as an addition, supposing the ground to rise(B-j-r C) — ; and 



x' 

 on the other side, as a deduction, supposing the ground to fall (B + r C) r— 



Consequently, the whole correction to the area of the cross section, calcu- 

 lated to the height C, in the ordinary way, is — — — (.r — x'). 



I take leave, at this point, to refer to the diagrams and characters em- 

 ployed in a paper on " Setting out Railways," in the Journal for Septem- 

 ber last (p. 277), as explanatory of these I now adopt. Moreover, let 

 -f r and — 2 represent the corrections to widths in a cross section parallel 

 to the first at a distance from it = L. 



Then, in the case first named, where the cross sections all exhibit the 

 ground surface as a right line, the solid, which is constantly to be added, 

 has for its value 



-[x(2x' + 2') + = (2:'-i-x')] ..A 



And in the second case 



L/B + rC\ ; L/B-HrC'\ 



g-V^-7— ^(:r-x') + J(2-s')+-^^— ^r-j(2-2') + i(^-x')....B. 



However complicated these formulEe may appear as algebraic expres- 

 sions, their numerical application is perfectly simple and easy. To com- 

 pare the two, let us first consider the increment to the area of one cross 



V u • ^^' , , /B4-rC\ 

 section only, which is -y, and also I — — — j (^ — x'); by assigning 



the following values : — 



B-|-rC = 31;r = 2;x=:5; x' = 3 78. 



xx' 5 X 3-78 

 Then, — = = 945 Square feet 



B-f rC „ 31 X 1-22 

 •nd, —^7- (-T — x') = 2x'i ' ~ ^^^ Square feet. 



Also, in the other cross section, 



if B -I- r C = 35, r = 2, s — 7 241, s' =: 5122, 



xx' 7-241 X 5-122 

 we have, — = =: 18-54 Square feet ; 



B + r C 35 

 and, — ^-j— {z—z') = — x 2-110 = 18-54 Square feet. 



Upon the choice between these two formula;, it is proper to remark that 



X x' 

 although the process according to is the shortest, perfect accuracy re- 



quires that the values of x and x> should be nicely calculated, even to a 



third place of decimals, to ensure exact coincidence with the result by th« 



B + rC B+rC 



formula — ^-:— (x — x') ; and more especially when — - — - is large. 



In the next place, let us compare the facilities of the two formula for 

 the increment to the solid due to the side-long inclination of the ground 

 with the values above given, the cross sections being 50 feet asunder =: L. 

 First, x' =: 3 78 5122 



X = 3-78 5- 122 



2' = 5 122 3-781) 



(2 x' + 2') = 12-682 (2 2' -I- J^') - • - 14-024 



Multiplied by 5 7-241 



63- 41 101-548 



63-41 



2)164 958 =82 5. 

 With this number enter Table No. I of my Tables* for calculating earth- 

 work, where I find, on the first line of figures, 



opposite 82 10123 



„ -5 -0062 



For 82-5 1-0185 



.-iO 100 



and multiplying this by — ~ we get . . . 25-46 cubic yards. 



2 4 



By the second formula for the increment to the solid, the process 



will be — 



x — x' 5 — 3 78 =122 this -^2 = 61 



7-241 — 5122 ' 

 \{z — z') = 1-059 this X 2 = 2-119 



(x-x)+i(2-2') 

 B + rC 31 

 r "'T — 



2-279 (2-2')-f i(x-x).. 

 B + r C" 35 



35-324 



1-0247 from Table No. 1 



-0009 



83-081 



1 0256 



and multiplying by - 



50 



. . 25 64 cubic yards. 



A slight examination will enable us to perceive that the last is the pre- 

 ferable process to be adopted in practice ; for as an algebraic expression 

 it involves less trouble, although apparently the contrary ; and avoiding 

 the extreme nicety necessary iu the values of x, x', 2, and 2' in the first 

 is, even in the hypothesis of the generatrix of the ground surface being on« 

 straight line, the most accurate of the two. 



This hypothesis will never be satisfied over the most uniform ground, 

 and in this sense also the second formula claims the superiority. Ex- 

 pressed in words, it may be stated as follows : — 



1. Take the difl'erence between the additive and subtractive corrections 

 to the widths at each cross section terminating the length of cutting or of 

 embankment under calculation. 



2. Add together half the width of the railway previously divided by the 

 ratio of the slopes to the height at the centre of each cross section; and 

 call these the multipliers. 



3. To the dilTerence of the corrections to widths in each cross section add 

 half the difl'erence of the corrections in the adjoining cross section, and 

 multiply these sums by their respective multipliers. 



4. The last two results added together and multiplied by one-sixth 

 of the distance betwien the ciuss sections gives the solid content sought. 



The rule thus expressed requires moditication according to the signs 

 attaching to x, x', 2, 2', as will be evident to every oue familiar with the 

 first principles of algebra, and needs no further remark here. 



* " Concise Tables to facilitate the Calculation of Farlhvvorlv and Land required in Iht 

 Construction of llailways. Canals, and other Public Worlis: adapted to the Practice of 

 the Engineer, Architect, and Surveyor." By John Hughes, engineer. London; Elflng- 

 tiam Wilson, Royal Exchange. 



43 



