This correction is to be added or subtracted accordingly as the curve is convex or concave toward the higher ground.

=

Example.-Given c 28; h1 = 40; h2 = 16; d= 74 ; d' = 38 ; b=28; R = 1400; or A+a= 2090; C° 4°.09; d~d' 36. 36 table 18 0.00776, =

and 2090 × 4.09 × 0.00776 = 66.3 cyds.

=

=

If the distances to the two adjacent stations are 50 and 40 feet respectively, the correction required is

50+40
200

x66.3 66.3×0.45

= 29.8 cyds.

To find the correction for curvature in side-hill work when the transverse surface slope is regular.

Given area; degree of curve; side distance; road-bed width; and width of excavation at road-bed (A; C°; d; b; w).

RULE 6 (FORMULA 24).

Enter table 18 with d+b-w and take out the corresponding factor multiply this by the product of A by C°, and the result is Q the correction in cubic yards, to be applied in all respects as in Rule 5.

Example.-Given w=17; b = 30; d = 51; h1 = 24; R = 1600; or A 204; C° 3°.58; d+b-w = 64.

=

and 204×3.58x0.01379 10.1 cyds.

If both intervals are 50 feet, the correction required is

X 10.1 = 10.1 × 0.5 : = 5 cyds.

50+50

200

For correction for curvature when the transverse surface slope is broken, or for double-width thorough-cut, see page 24.

Rules 5 and 6 apply to excavation only. For embankment the correction is to be added or subtracted accordingly as the curve is concave or convex toward the higher ground.

Example 1, as above, is of the railroad cut given in Morris's "Earthworks,"* pp. 47-54, with contents computed by Rules 1, 2, and 4, and the auxiliary tables of the present work. As here used, the areas are supposed to belong to sections which, when carried to the intersection of the side slopes in thorough-cut, are rendered sensibly similar, and the examples as here given are intended

*Easy Rules for the Measurement of Earthworks by means of the Prismoidal Formula. By Ellwood Morris, C.E." Philadelphia: 1872.

to show only the comparative facility of arriving at the prismoidal contents by Mr. Morris's methods and those of the preceding rules when the above condition of similarity is fulfilled, and not to endorse the application of the method of "Roots and Squares" (or of the rules of this work) in cases where the hypothetical middle area materially differs from the actual one.*

*

Except by trial with the actual middle section and the prismoidal formula, it seems almost impossible in cases of dissimilar end sections to know when the application of the method of Roots and Squares, or of the preceding rules, begins to fail of practical correctness, but it may safely be assumed that if the ground is properly and sufficiently cross-sectioned, the results obtained by them will be practically the prismoidal contents.

The above tabulated example shows all the steps necessary in finding the prismoidal contents in cubic yards when the areas are given. Columns (1), (2), and (3) being written out, (4) is derived directly from (3) by averaging; (5) from (3) by adding area of grade triangle in thorough-cut; (6) from (5) by table 3; (7) from (6) by subtraction; (8) from (4) by table 4; (9) from (7) by table 5; and (10) from (8) and (9) by subtraction.

Column (4) gives the average end areas throughout the cut, including the terminal pyramids, and the only break in the routine of adding the area of the grade triangle in column (5) is at the point where the cutting runs out on the lower side. At such points two areas have to be used, the one of earthwork plus the grade triangle, for computation of thorough-cut by Rule 1, and the other of carthwork alone, for the calculation of the pyramid or side-hill work into which the thorough-cut changes, and of which the computation of contents falls under Rule 2.

Column (8) gives the contents between each two stations roughed out by the common method of " average areas," column (9) the corresponding error, and column (10) the prismoidal contents, all in cubic yards.

It is not strictly necessary to write out all of the columns given above, but errors are so much more readily detected when all of the steps are shown, that ordinarily time and labor will be saved by adopting some system of tabulating similar to the above, both as regards the number of columns and the arrangement by which the figures referring to each two stations may be recorded on a line between them.

* See article on the application of the prismoidal formula, page 16.

The prismoidal contents in cubic yards between stations 1 and 17 are given by Mr. Morris as 15,721, and by the above computation as 15,723, whilst the contents of the whole cut given by him as 16,664 appear above as 16,247. The discrepancy is in the truncated portions of the cut outside of stations 1 and 17, which by some oversight he gives as 943, instead of 524 cubic yards.

The preceding example will now be computed by equivalent level heights and Rule 4. The data of level heights are supposed to be obtained from Trautwine's diagrams, as when such accuracy is required as renders the calculation of areas necessary, Rule 1, 2, or 3 should be used for the computation of contents.

With equivalent level heights given, the above tabulated example shows all the steps required in finding the approximate prismoidal contents in cubic yards. Columns (1), (2), and (3) being written out, (4) is derived directly from (3) by averaging, and (5) from (3) by subtracting. The table of level cuttings for a base of 20 feet and slopes 1 to 1, from which column (6) should be taken, is not published in this volume, but its place may readily be supplied by adding 1. to each of the heights of column (3), and taking 70 from each of the corresponding quantities in table 12. Such remainders are the amounts in column (6). Column (7) is derived from (5) by table 14, and (8) from (6) and (7) by addition.

In ordinary ground sloping transversely, the area of earthwork of the terminal pyramid at the point where the centre height is nothing, is about one-fourth of the area of the section where the pyramid begins; and practically, as only small quantities are concerned, the equivalent level height corresponding may be taken as one-fourth of that corresponding to the area of the base of the pyramid.

The calculation of contents by equivalent level heights and tables is well suited for preliminary or approximate estimates, especially if, as in the present case, when the sum of the tenths of the end heights is uneven, the average is always taken as the tenth next greater than the actual half-sum.

The variation between the contents of the thorough-cut from 1 to 17, as given in Examples 1 and 2, is due to the fact that the equivalent level heights are carried out to tenths only. In the present case, at a height of 20 feet the increment is over two cubic yards for each 0.01 of a foot, and in embankment at the same height it is still greater. As in practice neither equivalent level heights nor those of the tables of level cuttings are carried out to hundredths, one cause of the greater accuracy of the previous method by Rules 1 and 2 is evident. It may be replied that errors as important arc involved in the field work, the cross section stakes being set only approximately; but that an element of error should voluntarily be introduced into the calculations because another such already exists in the data, is a position that will not be contended for seriously.

Example 3.-In a cutting with road-bed width 16 feet, and opposite side slopes and to 1, the given areas of two consecutive cross sections with similar transverse surface lines and at a distance apart of 100 feet, are 100 and 1000 square feet respectively: required the prismoidal contents. Here the area of the grade triangle (table 2)