In calculating the correction for curvature in three-level section work, it is sufficiently accurate to use in the general formula for curvature correction the values ei, e2 and A1, A2 for the full sections qcpn= At instead of the actual area gcpmt. The values of e1 and e2 are then too small, and the resulting error nearly neutralizes the one due to the inclusion in the area of the triangle tmn. The eccentricity of the area qcpn is e1 = (wrw,), and, using the same notation as before, the curvature correction becomes

The form in which the computation of volume should be arranged when the cross-sections are three-level sections is shown in the table on page 205. The figures in the first four columns are written while the survey is being made; those in columns 5, 6, and 7 are used for computing the averageend area volume Va; those in columns 8, 9, and 10 are employed in computing the prismoidal correction; and the figures in the last two columns are used for computing the correction for curvature.

The values of Va for the prismoids included between the successive cross-sections are found as follows: Since the results always are expressed in cubic yards, the preceding formula for Va becomes, for the volume between two full stations (100),

If the slope s= 1:1 and the width of the roadbed b=22 ft., then a for all stations is

The sums of the constant depth a and the variable depths d in the second column are written in the fifth column. Thus, at Sta. 22, a+d=7.3+6.2=13.5 ft.; at Sta. 23, a+d=7.3 +9.4-16.7 ft. The total width at each station is written in the sixth column. Since, in Fig. 7, w=w1+w,, and since the measured distances wy and w, are the denominators of the fractions in columns 3 and 4 respectively, it is only necessary

to add the two denominators at each station to obtain the numbers in column 6. Thus, at Sta. 22, w=16.1+30.2 = 46.3; at Sta. 23, w= = 18.2+31.449.6 ft.

To compute the value of Va between Sta. 22 and Sta. 23, the proper values must be substituted in the formula for Va This gives

The number 579 is written in column 7 (a) opposite Sta. 22, and 767 in the same column opposite Sta. 23. The result, 1,049 cu. yd., is written opposite Sta. 23, in column 7 (b).

In a similar manner, for the volume of the prismoid between Sta. 23 and Sta. 24,

The first term of this expression has already been computed, and its value, 767 cu. yd. has been written in column 7 (a) opposite Sta. 23. The last term is the constant volume 297 cu. yd. It is therefore necessary to compute the second term only. Its value is found to be 1,132 cu. yd., and this is written in column 7 (a) opposite Sta. 24. Then, Va=767+1,132 -2971,602 cu. yd., and this result is written in column 7 (b).

It is thus seen that, at each station, it is necessary to compute but one term of the formula for Va; this term is the value 100 of (a+d)w for that station. The value of this term for 4×27 each station is written in column 7 (a). If the stations are 100 ft. apart, any number in column 7 (b) is obtained by adding the number opposite and the one preceding it in column 7 (a) and subtracting 297 cu. yd. from the resulting sum. The result so obtained is the value of Va for a prismoid 100 ft. long. But if the two stations are less than 100 ft. apart, the result must be multiplied by the ratio of their distance to 100 ft. to obtain the volume of the prismoid. This volume is then written in column 7 (b). For example, for the prismoid

-23

is

between Sta. 24 and Sta. 24+35, there should be obtained, provided the prismoid is 100 ft. long, 1,132+684-297 = 1,519 cu. yd. As the length is but 35 ft., the actual value of Va X1,519 532 cu. yd., which is written in column 7 (b). It is usually more convenient to compute all the numbers in each column before passing on to the next column. When column 7 (b) has been filled up, the number in this column opposite each station is the approximate number of cubic yards, computed by average end areas, contained between that station and the preceding station. Thus, 1,048 is the approximate number of cubic yards between Sta. 23 and Sta. 22; 531 is the approximate number between Sta. 24+35 and Sta. 24; etc. The total approximate number of cubic yards, between Sta. 22 and Sta. 25, as computed by average end areas, is, therefore, 1,049+1,602+532+426=3,609 cu. yd. The prismoidal correction must now be computed.

Since the result is to be expressed in cubic yards, the preceding formula for C becomes

=

The successive values of w-w' in column 8 are obtained by subtracting each number in column 6 from the number just below it in this column. Thus, for the prismoid between Sta. 22 and Sta. 23, w=46.3, w' 49.6; and w-w-3.3 ft. Similarly, the values of d'-d in column 9 are obtained by subtracting each number in column 2 from the number just above it in this column. Thus, for the first prismoid, d= 6.2, d'9.4, and d'-d=+3.2 ft.

The numbers in column 10 are the computed values of the prismoidal correction C. Thus, for the first prismoid, since 1 = 100

and similarly for the remaining prismoids.

The volume of the first prismoid, as obtained by the

prismoidal formula, is, therefore, 1,049-3=1,046 cu. yd.; that of the second, 1,602-11-1,591 cu. yd., etc.

Now assume that the portion of the track just calculated is on a 7° curve to the right. Applying the formula for Cc for stations of 100 ft. and in cubic yards,

At Sta. 22, w;= 16.1, w=30.2, and, hence, wi-w,= 16.1 -30.214.1. At Sta. 23, w¿'= 18.2, w,' 31.4, and, hence,

wy'-wy' 18.2—31.4=-13.2.

100At

The values of

and

2X27

1004

are those already tabulated in column 7 (a); thus,

2X27 100At

=579 and

2X27

1004't
2X27

=767. Substituting

all of these

values, and the value of R=819 for a 7° curve,

3X819

-X(579X-14.1+767×-13.2)=-7 cu. yd.

Since wy and we are smaller, respectively, than w, and w,', the centers of gravity of the sections lie on the right of the center line of the roadbed; and, as the curve turns to the right, the centers of gravity lie inside of the center line, and the correction is to be subtracted. The volume for this section computed by the prismoidal formula is 1,049-3=1,046 cu. yd., and, corrected for curvature, the final result is 1,046-7 = 1,039 cu. yd. The curvature corrections for other sections are figured in a similar manner, except for sections less than 100 ft. long, when the result must be multiplied by the ratio of the length of the section to 100 ft. To find, for instance, the curvature correction for the section between Sta. 24 and Sta. 24+35, determine, as before, the correction just as if the station were 100 ft. long and multiply the result by . Thus, 1 3X819

Cc=

(1,132X −15.6+684 × −8.2) × 1-3 cu. yd.

As in the previous case, the actual volume is less than the one computed for a straight track; therefore, the actual volume V-532-6-3-523 cu. yd.