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Now there is no excavation or embankment solid such as we have supposed, that cannot be divided into prisms, prismoids, pyramids, or wedges, or some combination of them, having a common length or height, equal to the distance between the end areas or cross sections. And the height or length being common to all, it will be evident on reflection, that if a given portion of excavation or embankment be composed of any number of the solids named, the area of one end section will equal the sum of the areas of the bases or tops of those solids, the other end area the sum of their tops or bases, and the area of the mid-section will equal the sum of the areas of their middle sections; and, hence, if (as has been proved) the capacities of the separate solids are reducible to one general rule, the solidity of a whole body composed of such solids, and having the height as one common dimension, may therefore be computed by the same rule.
The general process, then, the adoption of which we suggest as a valuable succedaneum to those in common use, will be to compute by the general formula from the sections usually taken in the field, in the following manner : Draw the sections in a book, leaving between each two space enough for the middle section, which will be subsequently deduced from those drawn; on each left hand page should be placed either three sections, (including the mid-section) or some multiple of three, depending on the character of the ground, and the size of the leaf; the right hand page being left open to record the calculations upon. The scale we would recommend to be twenty feet to the inch.
To prevent misapprehension, we will here observe, that in speaking of excavation or embankment, the centre line is always supposed to be a tangent, that being the universal presumption, in practical calculations; although upon curves, owing to the convergence of the cross sections, (they being taken on the radii,) this hypothesis is not exact, and consequently occasions some error, not often, however, of much importance, though cases will sometimes arise (where the primary angle of deflection is unusually large), that ought to receive correction for curvature.
Though not absolutely indispensable, it will be found convenient in using the prismoidal method of calculation, as well as conducive both to expedition and accuracy, to observe the following rules in "taking the cuttings,' as far as the character of the surface may admit, viz:
1. On sidehill, at each section of cuttings where the work runs partly in filling, and partly in cutting, ascertain the point where grade or bottom strikes ground surface.
2. On every transverse section take a cutting at both edges of the road, or, at the distance out right and left of one half the base.
5. Always take a cross section, whenever either edge of the road or base passes from excavation to embankment, or vice versa.
4. On sidehill, if the ground admits, take the cuttings (not otherwise provided for) uniformly at ten feet apart.
5. Wherever the ground admits, so place the cross sections as to be at some decimal division of 100 feet apart, as 10, 20, &c.
Excavation and embankment solids, naturally divide themselves into three classes or cases, with modifications, and under one or another of these cases or their modifications, will fall nearly every kind of ground; though on a very intricate surface, such as a rocky hillside, cases may arise, requiring additive or deductive solids, but the Engineer will find little difficulty in managing such, without violating, or interfering with, the general process.
Case 1. Prisms.
Embankment or excavation, either on level ground, or on ground inclined transversely, and level longitudinally, at the same distance out.
Modification 1: all excavation, or all embankment. Modification 2: both filling and cutting.
Case 2. Prisms, Prismoids, and Pyramids.
Embankment or excavation, on ground inclined longitudinally in one plane, and level transversely.
Modification 1: all excavation, or all embankment. Modification 2: passage from excavation to embankment, or the reverse.
The above two cases do not often exist in practice, that following being of he most general occurrence.
Case 3. Prisms, Prismoids, Pyramids, & Wedges, or a combination of them. Excavation or embankment on ground inclined both longitudinally and transversely.
Modification 1: all excavation, or all embankment. Modification 2: cutting and filling both. Modification 3: passage from cutting and filling to either cutting or filling. Modification 4, complete passage from excavation to embankment on sidelong ground.
The General Formula admits of a modification, more convenient for use in computing excavation and embankment. It is as follows:
b+ 4 m + t
6 × h = S. in lieu of b + 4 m +1 × = S; this modification we shall employ; and now proceed to give examples in figures of each case, but it may be as well previously, to make some remarks relative to deducing the middle section between any two which have been taken in the field, and sketched in the section book.
To average for the cuttings of the middle section, commence either at centre or at grade, if there be a grade point upon the cross section, and having regard to the inclination of the ground, proceed each way, averaging the cuttings as they occur, for a corresponding cutting of the middle section, and their distances out, or rather their distances apart, for a corresponding distance apart; and if there be more cuttings in one section than in the other, the surplus cuttings (of the same kind) on each side, all average with the outer cutting on that side, and their distances apart divided by two (or averaged with 0,) give the corresponding distances of the cuttings which answer to them upon the mid-section. But the averages may be made in any other way demanded by the transverse slopes of the surface, provided all the cuttings are used, and that lines drawn to join any two cuttings averaged, do neither meet between the end sections nor cross. In the mid
section will always appear the same number of cuttings as are contained in that end section which has the most; and its correctness admits of verification thus,
1. sum of distances between the extreme cuttings of the end sections, equals the distance between the extreme cuttings of the mid-section. This proves it horizontally: to verify it vertically,
2. Where the number of cuttings of both kinds is the same in each end section, sum of all the cuttings of the end sections, equals the sum of all the cuttings of the middle section.
3. Where the number of cuttings in the end sections is different, to prove the cuttings of excavation, sum of the cuttings of the end sections, equals the sum of the cuttings of the mid-section; minus, least outer cutting left of centre multiplied by the difference in the number of cuttings in the end sections on the left; plus, least outer cutting right of centre multiplied by the difference in the number of cuttings on the right.
Though this last rule is long in words, it is short in practice, and of course only refers to the excavation or plus cuttings in proving excavation, whilst the same process applied to the minus cuttings will verify the embankment of the middle section.
The exemplifications which will be given apply to the graduation of a road, or rail road, but the principles apply equally to a canal, as the tow path and berm banks above bottom are constant quantities.
In all the following examples the slopes are considered to be the same on both sides of the centre, which is supposed to divide equally the surface of grade, or the base as it will be called. The sections numbered 1: and S: will uniformly be presumed to be those taken in the field, whilst No. 2: will represent the middle section deduced from the end sections, 1: and 3: the distance between which, will, for convenience, be assumed at 30 feet in every case. Excavations, as to figure, are merely embankments inverted, and hence, as a matter of course, the same principles apply to both.
In all the examples, the results obtained by rules No. 1 and 2, will also be set down for the sake of exhibiting how great in some cases the differences are: the base will be assumed at 30 feet, the slopes at 2 to 1, and C, C, will represent the centre line of the road.
Example of Case 1: Modification 1: fig. 6, plate 1.
Example of Case 1: Modification 2: fig. 7, plate 1.
Example of Case 3: Modification 1: fig. 10, plate II.
By Rule No. 2,