1 Example.-8=;s' = 1; b = 16; h = 20. 2 To calculate the area abcdefg b'c'd'e'f'g. The elevations and horizontal distances apart of the points a, b, c, d, e, f, g, must be determined in the usual manner before the surface is disturbed, and of b', c', d', e', f', g', after the excavation is made. Calculate the area A abcdefg B between the surface line and the assumed datum plane AB; also The area Aab' c' d'e' f'g' gB between the bottom of the pit as excavated and the same datum plane AB. The difference between the results so obtained, gives the area required. When the cross sections of the line have the surface broken transversely, if the slope stakes are supposed to be at a and g (fig. 3), and AB is the plane of the road-bed, calculate The difference between the above two results will give the area of earthwork required. For side hill work the process is similar, except that only one triangle of excess = 2 is to be deducted. This of course applies to embankment as well as excavation. None of the preceding cases require that the cross section shall be drawn before calculating its area. CONTENTS.-FRUSTUM FORMULA. Fig. 4. B If ABCD and A'B'C'D' be two consecutive cross sections with like surface lines and side slopes but unequal bottom widths, by producing the side slopes until they meet at E and E', the whole figures ABE and A'B'E' are similar as well as the triangles CDE and C'D'E'. But the solid ABCDA'B'C'D' being the difference between the frustums ABEA'B'E' and CDEC'D'E' its cubic contents are ABE + A'B'E' + √ABE × A'B'E' in which 7 represents the distance between the cross sections. If areas ABCD, A'B'C'D', CDE and C'D'E' be represented by A, A', a and a' respectively, then taking 7 as 100 feet, and representing the contents in cubic yards by C, we have : C= (A+a)+(A'+a')+√(A+a)(A'+a')—(a+a'+√aa')100 3 If CD = C'D' then a' = a, and the formula becomes : c=((A+a)+(A'+a)+√(A+a) (A'+a) 3 -a) 100 27 which is the formula for the frustum of a pyramid. X (8) 27 By formulæ (8), (9), and (10) the whole of the formulæ for cubic contents hereafter given may be conveniently tested. As the solid resulting from connecting the homologous sides of two similar and parallel sections of unequal areas is the frustum of a pyramid, formula (10) is applicable to any plane solid with such end sections. Let ABCDF be a given cross section, with a base FD = b, and s and s' the ratios of its side slopes to 1; also let IKDF be an equivalent cross section with level surface, height MN = h, and with same base and side slopes. Produce the side slopes to their intersection at E, and from E let fall the perpendicular EL on IK, intersecting the base in G. Let area ABCDF = IKDF A, and FDE =α. = In the triangle FDE, FG EG × s, and GD = EG × s', or b :=(h+ IK = (^+~+~+) (+), and area IKE=(^++) (+) =A+a; s+ 2 (11) For a second section with corresponding parts b', H', s and s', and areas A' and a' and for the area M of a cross section midway between A and A', 2 M = H+H' g+g 2 (12) The prismoidal formula for the contents C between two end areas A and A' at a distance apart = 7, with an area M midway be The two last expressions for the value of C show that the calculation of contents by averaging the end areas requires a minus correction; and by the middle area (or, what is equivalent, taking the amount corresponding to the average of the end heights from a special table) a plus correction of exactly half as much. The actual minus correction will now be found. By substituting the values of A, A' and M in the second term of (14) we have: 2H2-292-2H-2g2-H2-2HH'-H2+g2+2gg'g' 2 2 |