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a broken country, we shall find that hills and valleys, rivers, canals, and roads will compel us to consider these as guiding circumstances, or we must encounter innumerable tunnels and viaducts, and disproportionate cuttings and embankments, expensive bridges, and other crossings; but a well-regulated series of inclined planes graduated in a measure proportionately to the inclination of hill and valley, skirting along the one and skimming over the other, and reaching from height to height, enables us, by moderate works, and therefore at a moderate expense, to attain the desired end; and generally the reaching from valley to hill
, and hill to valley, the crossing of rivers and roads, the equalization of excavations and embankments, will be the existing circumstances in compliance with which certain heights are to be reached, and by which the inclined planes, and therefore the gradients, will be determined. The working section will show a vertical plane passing along these various points, the relative heights of which have been ascertained by levelling, and their distances from each other being known, it remains to calculate the gradients. Let us suppose that at a distance 3 miles 8 chains, there is an incline which is to reach from 161.6+ above datum to 14:50 above a road, which is 6.50 links further on; the level of the road is 151:13 above datum, and 151.43 + 14:50=165-93 ; but 165.93 = 161.61 + 4.29, or the difference of level between a point we have reached on the section and one to be attained. But 650 links are equal to 429 feet, and this divided by 4:29, the difference of level will give 1 in 100 for the gradient ; for 4:29 : 429 :: 1:100, or AB : BC :: DE : EC, Fig. 137; therefore, in all cases, to find the ratio of inclination, or gradient, divide the length in feet by the rise or fall, and the quotient will be the answer; but having the gradient, or ratio of inclination, it is also necessary to know the rise or fall per chain,
, in order to calculate the depth of cutting or embankment, or rather the height of formation above datum, as this height, deducted from that of the surface above datum, will give the depth of cutting, whilst the deduction of the height of surface from the height of formation will give the depth of embankment, as at 2 miles 10 chains 171.70–162-96 = 874, or depth of cutting; and at 2 miles 12 chains 16+.28 – 150·83=13:45, or depth of embankment. To find the rise or fall per chain divide the difference of level by the number of chains in the incline, as 4.29: 6:50.
6.50 ) 4.290 ( .66
SETTING OUT GRADIENTS.
Therefore, to find the gradient, or ratio of inclination, divide the length of the incline in feet by the difference of level. And to find the rise or fall per chain, divide the difference of level by the number of chains, as shown above.
The ratio of inclination of a plane, or the gradient, being given to find the inclination per mile, divide the number of feet in a mile by the ratio, as
100) 5280 ( 52-8 per mile.
800 The gradient being given, to find the inclination in a chain, divide 66 (the number of feet in a chain) by the ratio as for 1 in 100, 66:100
100) 660 ( 66
600 Also, the gradient being given, to know the rise or fall per yard, divid 3 (the number of feet in a yard) by the ratio, as in 1 in 100, what is the inclination per yard ?
100) 300 (03
000 And at any time, for any length, divide the length in feet by the ratio, as in 1 in 78, how much in 12 ?
78 ) 12:0 ( 15 (sufficiently near for practice)
390 In calculating the formation heights for the contract section, from which, as before observed, the depths of cuttings and embankments are afterwards ascertained, the greatest care is required in doing this correctly, for if, in this first step, an error be committed, it will be carried on into the depth of cutting or embankment, and the contractor working from such contract section may cut to a wrong depth, for which, in justice, he is in no way to blame; to men of habits at all careful, this may almost appear impossible, but it is nevertheless, however gross the blunder, one of common occurrence. Suppose that at 2 miles 33 chains, the formation height is marked 182.48 instead of 181.48, which it should be ; now the heights for the springers
would be set out by reference to bench marks, when the arch might be turned to a height of 21:41, instead of 20:41, making the crown of the arch 1 foot too high, for 181 18–161.07 = 20-41; but 182:48–161.07 = 21:41. It is hoped that the importance of a careful calculation of the gradients at every chain stake is now distinctly understood; the ratio of inclination being ascertained, and the rise or fall per chain, every stump height will require to be calculated separately, and checked by tens, twenties, &c., and in as many places as possible—the reader may rest assured, that for whatever trouble 20 or 30 miles of such calculations may give, he will be most amply repaid by ease of mind on this important subject during the working of the contract. From formation height 161:64, before referred to, we have an incline of :66 per chain for 7 chains; and •66 x 7 + 161:64 = 166.26; but 161:64 +:66 = 162:30+.66 = 162:96 + 66=163:62, and so on until we have 166:26 at the 7th chain, if no error be made; on an incline of a greater length we check these results by multiplying the rise or fall per chain by 10, 20, 30, &c., which is very quickly done, and examining the 10th, 20th, or 30th chain stakes taken at different places. This being gone through, it only remains to get the depth of cutting by deducting formation height from the surface height, or the reverse, when we get the height of embankment.
The section being prepared so far, we should, as soon as the works of construction are determined on, insert notes from the working drawings or otherwise, of the angles of skew at which the line crosses roads, canals, &c., the spans of arches on the square and skew, the rise of the arch, the depth of arch stones, of puddle, if any; also, if the works be on an inclined plane, the rise or fall from centre to centre of piers; memoranda also, of nearly similar nature, should be made of girder bridges, culverts, drains, and other works occurring along the line. These remarks are more than necessary, because, when on the works, the drawings, when required, are often mislaid, or partially defaced or destroyed; it must be added, however reluctantly, that the tracings with which contractors and sub-contractors are supplied, are often wrongly figured, and the site of construction, amidst the moving to and fro of masons, labourers, and 'navvies,' is not the place where such errors may be most readily detected and corrected. The acting engineer always affixes his signature to the working drawing, but not always to the tracings, the correctness of which is often entrusted to a confidential office assistant, but who, from want of field practice, is not always aware of the value of correct figures; it will be found advantageous, and only consistent with duty, to compare the copy of contract works with the office original.
REMARKS ON CONTRACT SECTION.
The contractor is supplied with a copy of the contract section of the intended line of railway, or any portion of it, the performance of the works on which is entrusted to his execution, and if the levels are set out by his agents, the engineer has but to check them, a simple matter merely demanding correct levelling; but he may have to set out the bed moulds, which should be done in the following manner :take Fig. 137 for a cutting to be worked from both ends. At stump, 2 miles 27 chains, we are to have for formation 176.41 above datum, and the height of surface is 174:05; plant the level at a convenient point, and direct the levelling-staff to be held on the level peg at 174:05, and read off, say 7.00; but 176:41–174:05 = 2:36 for height of embankment, and 7.00-2:36 = 46+; now if the staff be raised until we read through the telescope 4:64, it is very evident that the foot of the staff will be held at 2:36 above the surface, or at 176.41 above datum, or formation height ; next, direct the staff to be moved along the nick or centre line, and towards the cutting, until we read off with the level 4:64 again, we shall then have found along the nick a point exactly level with the intended formation height at 176·41 ; but our gradient is 1 in 78, or .846 per chain ; the formation therefore at the point sought will be lower than 176.41 by 1 in 78, according to the distance from 176:41 ; let this distance be half a chain, the formation height will then be 176 41 - 846 = 175.99, or 42 lower; and if we add this :+2 to 4:64, we shall have 5:06; move the staff nearer until with the level this depth 5:06 be read off, when we shall have the desired point; for if when we first read 4:64 the foot was at formation height, as we hope is evident, by adding to it 42, and reading off 5-06, we have a point ·42 nearer to datum at half a chain distant, and therefore parallel with the incline; at this point found, drive in a stout stake, until on reading off the staff we have 5:06, and another at 176-41, until we bave 4:64, and we shall have two stakes at the required formation height half a chain apart, and by boning from these two heights any tolerable excavator can direct the formation of the bottom of his cutting until fresh levels are given. Boning is performed with boning rods, which exactly resemble T squares, in the following manner :let A and B, Fig. 138, be two stakes driven to a certain depth, and according to a given inclination ; if on both of these stakes boning rods, of exactly equal lengths, be held perfectly upright, it is plain that the tops of these will be parallel to the incline, and if a third rod be carried along the intended slope, the top of it will be in line with the top of the other two, if the incline be correct; if it is above there will be more to cut away, and if it is below the excavation will have been made too deep; this method is certainly but approximate, but quite sufficient to guide
the excavators for a time. As another example of setting out the bed moulds, we will take the other end of the cutting, Fig. 137. At formation 165:60 we have the surface height of 161-80, and therefore a bank of 3.80; plant the level, and then set the staff on the level peg, and read off, say 9:50; but 9.50–3-80= 5.70, and if the staff be held up until such depth be read off
, then again will the foot of it be at the formation height at this point; move the staff along the centre line until this 5-70 be again read off, and we shall have a point level with the intended height of embankment; but the gradient here being 1 in 100, the new point sought must be higher in that proportion, as we are now rising, and the distance is 52 feet; for this distance the rise will be ·52, and as we are rising, be it remembered, we must deduct ·52 from 5-70, when we shall have 4:18; by moving the staff along the nick until from the level we read off 4:18, we shall find the new point for formation, at which drive in a stake. Under no circumstances should this part of the work ever be considered as completed until carefully checked, which should be done by carrying on the levels to the nearest P.P., or permanent post.
The incline for the cutting being given, will be, however, but an indifferently approximate guide for the height of an embankment, particularly if of clay and tipped in dry summer weather, for under such circumstances, such an embankment may settle one-fifth of the height at which it is first formed ; let Fig. 139 be a sketch of a cutting and an embankment, in which suppose A B the inclined plane, and the required formation height at BC to be 22 feet; if the embankment be formed at once to this height, and the above mentioned settlement before perfect consolidation takes place, and it will not be much less, the general height will be something like A D, and will require reforming; it is therefore absolutely necessary to keep the bank high in proportion to the expected settlement, whatever may be the material of which the bank is formed, and a near approximation of the depression may be very easily ascertained a few weeks after the tipping has commenced, due allowance being made for extremes of weather. At 1 chain from the cutting we have embankment height of 8 feet; as soon as the tip is formed beyond this point, ascertain the height by levelling; a few weeks after, the depression from the weight of the earth and that of the loaded waggons may be ascertained by levelling again, when an approximate ratio of settlement will be determined, by which the formation of the embankment may be directed.
Viaducts and bridges are works which require the greatest accuracy as regards levels, and an error of this description is unpardonable, as attention and care are all that are required to