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and Centre Cuttings, leaving wider spaces on either side of the latter column for the record of the various measurements to the left and right of the centre stake. Transfer from the sectionlevel notes the distances and corresponding cut or fill, for each stake of that survey. Filling in the cross-section notes is designated as minus cutting.
II. Having set the level in convenient proximity to a proposed cross section, take a reading of the rod at the centre stake. Add this reading to the centre cutting, (regarding the sign of the latter), to obtain the "height of instrument.”
III. Lay off half the width of the road-bed each side of the centre, and mark the distances, temporarily, with stakes. These are the angle stakes.
IV. Proceed to take rod readings at the angle stakes, and beyond them outward, (on a line at right angles to the direction of the line of the road), at each change of inclination of the surface. Subtract each reading from the height of instrument; the remainder is the cutting, or vertical distance of the point measured, from the proposed road-bed.
V. Record each cutting, together with its horizontal distance from the nearest angle stake, in the form of a fraction expressing the ratio of the distance to the cutting. Each fraction being recorded in its proper column either "right" or "left" of the centre. Points between the centre and angle stake, are located by measurements from the centre.
VI. To find the position of the slope stake: Measure off a trial distance from the angle stake, and determine the cut as before. Multiply the cut by the ratio of height to base of the proposed slope. If the trial distance be greater than this product,
trial until the ratio of the distance to the cut expresses the ratio of slope.
60. The cutting at the angle stake is, in cases of a tolerably uniform surface, a good guide to the distance to the slope stake. Thus, when the angle cutting of an excavation is 16 feet and the ratio of slope 1 to 1, the distance out, for a level surface, would be 24 feet; but if the ground in that distance rise 2 feet, (and which in practice may be determined pretty correctly by the eye), then the horizontal distance must be increased by something more than 1 times 2 feet.
When the surface descends, the estimated distance out, for a level surface, should in like manner be diminished. In embankments the conditions are reversed; the steeper the rise, the shorter the distance out.
61. The following examples will serve to elucidate the subject still further.
The diagram represents an embankment cross section, in
which, by reason of the small depth of filling, the height of instrument is a positive quantity.
The centre cut is 4.9; reading of the rod at the centre, 8.5; the sum of these, or "height of instrument," is 3.6. The remaining rod readings are given on the line through the instrument.
62. In the example of the following diagram, the cross section is partly in excavation and partly in embankment. The ratio of slope is 2 to 1. The centre cut is 2.4. The centre reading is 7.9; height of instrument, 10.3.
The reading at H, is 6.3; at E, 6.1; at S, 5.4. The point, K, is easily found in practice, it being that point on the surfaceline where the reading of the rod exactly equals the height of instrument. The reading at F is 13.2; and at S', 15.8.
From these readings the cuttings may be found, and the notes completed as below.
63. In the following example, there is a regular rise in the surface-line of one foot in eight. The ratio of slope in the excavation is to be 1 to 1; height of instrument, 14.2.
In seeking for the position of the slope-stake S', a distance
out of 13 feet is tried; the reading of the rod at the trial point is 4.8.
How does this point compare with the true position of S"? Ans. Not far enough out.
What is the result of a trial at 16 feet out and a reading of 4.4? Ans. Too far out. What is the true cut and distance at S'? Ans.
Find the position of S.
NOTE 1.-It sometimes happens, in very hilly sections, that it is impracticable to sight to all the necessary points of a single cross section from one position of the level. In such a case, it is only necessary to work from the centre as far as the surface will permit, then establish a turning-point, precisely as in section levelling; change the position of the level so as to proceed with the work, and determine the new height of instrument, from which the readings are to be subtracted as before.
NOTE 2. The degree of accuracy desirable to be attained in setting the slope stake, varies with the kind of earth to be "staked out," so that no exact rule can be laid down.
A principle, in quite general use, permits the stake to be set when the calculated distance varies from the trial distance by less than a foot.
The limit of error should never be greater than this, but in rock and the harder kinds of earthwork, it should be made much less.
COMPUTATION OF EARTHWORK.
64. Before the work of construction of a railroad or canal commences, the calculation of the earthwork must be completed.
The cross-section levels afford the necessary data. These surveys have divided the proposed work into blocks of 100 feet, or less, in length, and which are appropriately termed prismoids. Different methods are employed for estimating their cubic contents. The most accurate, though the most laborious, is the prismoidal formula, (Leg., Mensuration, page 129),
vol. = ¦ (B + B' + 4M)
B and B' representing the areas of the end sections of the prismoid, M the area of a section midway between them, and I the entire length of the solid.
The principal difficulty in applying this formula lies in finding the dimensions of the middle section.
We will show the application of the formula by an example of road excavation.
To simplify the problem, we will suppose such a degree of regularity in the ground surface that the angle cuttings may be omitted.
The length is supposed to be 100 feet. The other dimensions are given in the diagram.
The areas of the end sections are easily found. It is only necessary, in each case, to add together the areas of the trapezoids composing the whole end figure, as represented in the diagram, and subtract therefrom the sum of the triangles which lie outside the section. The dimensions of these triangles are always expressed in the cross-section notes, by the records for