a value for Av,' in terms of Ay, and AS is found thus,

Substituting this value in the equation for Ar. there is obtained

Then substituting this value for Ar in Equation 37 for the new hydraulic radius, there results an equation in which Ay, is expressed as an implicit function of AS, the other terms being known and having predetermined numerical values, viz.:

As the term involving Aye is practically insignificant on account of the smallness of its coefficient, in comparison with the other two terms to which it is added, it may be dropped, making Equation 53 reduce to

For the particular problem in hand the following numerical values are substituted in the foregoing formula.

m =

0.71

(An average for the range of values given by equations 18 to 21, inclusive.)

K = 0.0168 X (1.811 + 41.6 +

.028

r = 1.67

f'

=

4

.00281 .000284.

=

Substituting these values in Eq. 54 and solving, we find for the depth of scour, Aye, a value of 11.9 feet. As there is an old saying among those who are familiar with silt-bearing rivers to the effect that for each foot of rise there are about two feet of scour, this result appears to be correct. Such a scour, however, would involve cutting down to the gravel, but the superior resistance of this material would interfere, and hence it is probable that the scouring would extend horizontally, possibly over a large portion of the width of the bed.

Referring now to Eq. 28 and substituting therein 7 for AS and 11.9 for Aye, we find the increment of area to be 6,867 square feet, and adding to this the original area of cross-section, 1,260 square feet, gives 8,127 square feet as the total area of the new cross-section of the river proper. The new wetted perimeter is found from Eq. 36 by substituting therein. the same values of AS and Aye, making the increment about 46 feet and the total 496 feet.

The hydraulic radius is

8,127

= 16.38 feet,

Sine of slope = 0.000284, and V0.000284 = 0.0168
n = .028, as before.

Substituting these figures in Eq. 17 gives

Q

=

=

5.8 X 8127 47,137 cubic feet per second.

The next step will be to estimate the discharge taking place through the old slough. It is stated that there is considerable vegetation therein and on the adjacent slopes, so that it will be necessary to allow a higher coefficient of roughness, hence n is taken at 0.035.

From this we conclude that the minimum discharge that should be provided for should not be less than the foregoing amount. On the other hand, the critical storm method shows that provision should be made to pass about 88,000 cubic feet per second, and the "curve of unit run-offs" indicates that a flow of 82,500 cubic feet per second is the probable amount. The C. B. & Q. formula, previously quoted, gives 84,000 and the Murphy formula 92,000. Hence the further conclusion is reached that the observed high-water line is not that of extreme floods. To be safe against damage to the bridge or adjacent properties, provision should be made for a discharge of 82,500 cubic feet per second.

The probable extreme flood line will next be determined. We shall assume an additional height of four feet, making the elevation of the extreme flood line 105.0. Increment to the area of main channel = 4(488 + 4) = 1,968 square feet.

=

Substituting these values in the formula gives C 87.8; hence 87.8 X 4.47 × 0.0168

ν =

=

6.59 feet per second, and

10,095 = 66,526 cubic feet per second passing through the main channel.

For the slough we shall assume that the brush and other vegetation will prevent scour, hence we shall have the following:

Substituting these values in the formula gives C = 56.7; hence

v = 56.7 X 2.19 X .0168

Q

=

2.09 X 8376

Total

=

=

=

2.09 feet per second, and 17,506 cubic feet per second.

66,526 +17,506 = 84,032 cubic feet per second.

This volume is somewhat in excess of the assumed amount; hence it will be conservative to take 105.0 as the elevation of the extreme flood line. This gives a maximum depth of water in the channel of twentyseven feet and in the slough one of seven and a half feet, while the extreme width of flood will be about 2,250 feet, and the total area of waterway, 18,470 square feet.

If all this flood were confined to the main channel by building a levee along the low bank, a calculation similar to the preceding shows that the flood line would be raised to about elevation 108.0.

This would require a levee at least nine feet high. It is hardly probable that such a levee along the low bank would be justified, unless the land were valuable and worth protecting. It then becomes a question whether to build and maintain for railroad purposes a solid embankment across the slough and the adjoining low lands, or to leave an opening at the said slough and put in a trestle. The length of such a trestle would depend on how high the flood line might be raised without serious injury. If it be permissible to increase the extreme high water to elevation 106.0, we find by interpolation that the main channel would carry about 72,000 cubic feet per second. This leaves 10,500 cubic feet per second to be carried through the slough. A four-hundred-foot trestle will provide an opening of about 3,340 square feet, while the velocity will approximate 3.0 feet per second, which gives a discharge capacity of some 10,020 cubic feet per second, which is almost exactly right. However, the bents of the trestle will obstruct the flow somewhat, hence it is not desirable to limit the opening to a bare sufficiency. This layout gives a total waterway for the entire crossing of about 13,930 square feet with a flood line. at elevation 106.0. For an unobstructed flow, the area as previously noted would be 18,470 square feet with the flood line at elevation 105.0. With the flood confined to the main channel, the area becomes about 11,600 square feet and the flood line rises to elevation 108.0. It is thus seen that the most efficient discharge section is the restricted area. this connection it is to be noted that Dun's Drainage Table in the "Missouri Column" gives, by interpolation, for a drainage area of 3,300 square miles, a required area of waterway of 11,123 square feet, which is some

In

what less than the area of the opening provided by the main channel and the four-hundred-foot trestle at the slough, showing that the proposed layout is satisfactory.

This shows a substantial agreement between values derived from Dun's Table, the unit run-off curve, the critical storm method, the method of determining waterways from velocities estimated by Kutter's Formula, and the C. B. & Q. Formula, while the results derived from the Murphy Formula are not out of range. Fanning's Formula calls for over one hundred (100) per cent excess area for waterway, while Kuichling's Formula indicates about thirty (30) per cent excess, as compared with the figures obtained by means of Fig. 496.

In order to test further the C. B. & Q., the Murphy, and the Kuichling formulæ so as to see how they agree with the diagram method for small areas, it will be well to assume, as at the beginning of this chapter, a drainage area of 250 square miles.

Q = 38 X 250 = 9,500 cubic feet per second.

As the run-offs to fit the C. B. & Q., the Murphy, and the Kuichling formulæ are respectively 87, 97, and 125, and as the former figure exceeds all but 36 of the 447 records in the table of the A. R. E. A., and the latter all but 15 of them, it is evident that none of the formulæ can be considered satisfactory. Moreover, the 36 exceptional areas of the table exceeding a run-off of 87 have an average drainage area of only 138 square miles.

In view of the foregoing, the author feels justified in advising his readers to place no reliance whatsoever on any of the formulæ for area and discharge of streams, but to adopt instead as a standard Dun's Tables and the Run-Off Diagram presented in this chapter-bearing in mind, however, that when the anticipated area and discharge are unusually high, every practicable investigation should be made so as to determine their probable maximum values, following the methods herein explained.