Irregular Sections.-When the cross-sections are irregular, the process of determining the volume of earthwork is essen m FIG. 8 tially the same as for three-level sections, ex cept that a more accurate method for computing the areas of the cross-sections is applied. After the area has been determined, the volume by the endarea method, and the prismoidal and curva ture corrections are obtained just as if the figures were three-level sections. Areas of Irregular Sections.-Using the notation of Fig. 8, the area npqgfckm is determined by the formula A = (by3+x3y2+x2y1+x1d+dxı'+yı'x2' This long expression for the area may be very easily formed as follows: Write the successive slope-stake fractions in order, in a horizontal row, beginning with the extreme left d slope stake; and for the center stake write the fraction. the beginning and end of the row, write the fraction At 0 ; for the point m, it etc.; so that the row of fractions for Fig. 8 will be as fol Thus, the fraction for the stake at n is X3 Next, multiply each denominator by the numerator that follows it and each numerator by the denominator that follows it, giving to those products connected with full lines the plus sign, and to those connected with dotted lines the minus sign. One-half of the algebraic sum of these products. will be the desired area. This is evident, since, proceeding according to the directions, the positive products are byз, x3y2, x2y1, xıd, dxı', yı'x2', and y2'b and the negative products are -Y3x2,y2x1, and -x'y' One-half of the algebraic sum of these is identical with the second member of the preceding formula. NOTE. The method just described for determining areas of irregular sections is general and may also be used for threelevel sections. Following is an illustrative example showing the application of the preceding method of determining the areas of irregular sections. The field notes are given in the accompanying table. The station numbers in column 1 run from the Roadbed 24 feet wide in cut. Slope 1.5 1. bottom of the page upwards, so that when one stands on the line of the road looking forwards, the slope-stake fractions, which give for each point the height and distance from the center, will have on the notebook the same relative position as they have on the ground. These figures for the left-hand side are always given at the extreme left of the space in column 3. The line between columns 3 and 4 may then represent the center line; the intermediate points between the left-hand slope stake and the center are given in their order in column 3. Similarly, the points on the right side are placed in column 4. The figures for the right-hand slope stake are always placed at the extreme right-hand side of that column. The preceding table shows how the computations are arranged. Take, for example, the section between Sta. 128 +40 and Sta. 129. To find the end area at Sta. 128+40, the following fractions are written: Q 22.8 20.4 18.2 13.2 12.8 0 12.0 The products of the numbers connected by full lines, 12.0 X22.8, 46.2X20.4, etc., are written in column 2, and the products of those connected by dotted lines, 22.8X31.0, 20.4 X 19.5, etc., are written in column 3. The sum of the double plus areas is 2,696.6, and the sum of the double minus areas is 1,247.1. The area of the section is, therefore, X (2,696.6-1,247.1)=724.8 sq. ft. The area at Sta. 129 is obtained in a similar manner; thus, (1,144.8-407.7) 368.6 sq. ft. The volume for a 100-ft. section as figured by the average end-area method is These figures are entered in column 4 (a) of the table of computations. If the prismoid were 100 ft. long, the volume Va would be 683+1,342=2,025 cu. yd. As the prismoid is but 60 ft. |