JOURNAL

or

THE FRANKLIN INSTITUTE

OF THE STATE OF PENNSYLVANIA

FOR THE

PROMOTION OF THE MECHANIC ARTS.

JULY, 1851.

CIVIL ENGINEERING.

For the Journal of the Franklin Institute.

COPYRIGHT SECURED.

A New Method of Calculating the Cubic Contents of Excavations and Embankments.-BY JOHN C. TRAUTWINE, C. E.

There is but one correct principle upon which to calculate the cubic contents of excavations and embankments; and that is, by means of what is known as the Prismoidal Formula, or Rule.

This Rule is as follows:

Add together the areas of the two parallel ends of the prismoid, and four times the area of a section half way between and parallel to them; and multiply the sum by one-sixth of the length of the prismoid, measured perpendicularly to its two parallel ends.

Since, in railroad measurements, the prismoids are generally 100 feet long, it becomes easier in practice to multiply the sum of the areas by 100, (by merely adding two cyphers,) and to divide the product by 6; which amounts to the same thing as multiplying their sum by 4th of 100 feet.

The very general application of the prismoidal formula to other solids than such as are generally understood by the term "prismoids," has been shown by Mr. Ellwood Morris, Civil Engineer, in an able paper published in the Journal of the Franklin Institute. Vol. xxv, 2d Ser. p. 381.

It embraces all parallelopipeds, pyramids, prisms, cylinders, cones, wedges, &c., whether regular or irregular, right or oblique; together with their frustra, when cut by planes parallel to their bases; in a word, any solid whatever, which has two parallel ends, connected together by plane or unwarped surfaces.

VOL. XXII.-THIRD SERIES.-No. 1.-JULY, 1861.

1

In the cylinder and cone, the sides may be considered as consisting of an infinite number of indefinitely narrow unwarped planes. In railroad cuttings, it rarely happens that the ground surface lying between two consecutive cross sections 100 feet apart, is absolutely unwarped; yet, for practical purposes, it may very frequently be assumed to be so. When much warped, the cross sections must be taken closer together than 100 feet.

There are generally two circumstances under which it is necessary to ascertain the cubic contents on a public work; viz: first, after a preliminary survey of one or more trial lines, for the purpose of determining approximately their actual or comparative costs; and, second, after the final adoption and staking out of the determined route, in order to know precisely the amount of work to be done.

The measurements for the latter are performed with considerably more care, and attention to detail, than those of the former, inasmuch as upon them depend the payments to be made to the person who executes the work. They, moreover, involve considerations which cannot be attended to during a preliminary survey, without incurring an expenditure of time and labor, more than commensurate with the importance of the result.

When the ground is level transversely of the line of survey, there is no difficulty whatever in ascertaining the contents from a table of level cuttings, previously calculated; but when the ground is inclined or irregular transversely, the calculations have hitherto been attended with considerable labor.

The following method by diagrams, devised by myself during my probation as an assistant, and now communicated for the first time, will, I trust, be found to render the operations in the last cases nearly as simple and expeditious as in those of level ground; or, at least, infinitely more so than the usual ones. It dispenses with a great deal of calculation, and is, therefore, comparatively free from liability to error arising from

that source.

The construction of the diagrams is extremely simple, notwithstanding that, at first sight, they appear somewhat complex. They are but few in number, since any particular road will generally require but three or four, which may be prepared by one person in a day. Before proceeding to explain the manner of drawing them, I will give one or two examples of their use, that the reader may see the object aimed at, and to what extent it is attained.

Example 1. Suppose that in a roadway of 28 feet wide, and with sideslopes of 1 to 1, the cutting at a certain station is 20 feet; and that the ground, instead of being level transversely, inclines at an angle of 150.

Turn to the diagram for a roadway 28 feet wide, with side-slopes of 1 to 1: place a finger on the centre line, at the height of 20 feet, and run it along up the curved line which commences at that point, until it strikes the inclined line marked 15°. It will be seen at once that the two coincide at the height of 22.8 feet: and this is the depth of the equivalent level cutting, which would have precisely the same area as the section under consideration.

All such cases may therefore be instantly, and without any calculation whatever, reduced to others of equivalent level cuttings.

This constitutes the main feature of the principle involved in the dia. grams.

Had the depth been 20-3, or other decimal of a foot, the proceeding would have been the same as with the 20 feet; and the equivalent level cutting would be found on the inclined line 15°, at the distance of 3 of a foot (estimated by eye) above the curved line 20.

Example 2. Using the same diagram; let the depth of cutting be 2 feet, and the transverse slope of the ground 20°. Here, placing a finger on the centre-line, at the height of 2 feet, and running it along the curved line commencing at that point, it will be found that before reaching the inclined line of 20°, it encounters the dotted curved line drawn near the bottom of the diagram. When this occurs, we know that the ground-slope cuts the roadway, forming a cross section, partly in excavation, and partly in embankment, as in fig. 7.

This is a most useful check; for in such cases, the contents cannot be obtained by means of the diagram; but recourse must be had to a figure of the section, drawn for the purpose; as must also be the case when the ground is irregular transversely. A very simple method of proceeding, in all such cases, will be given further on.

I trust that what has been already shown, will satisfy the reader that my method partakes of utility, as well as of novelty,

On the page opposite each diagram, is a table of cubic yards for level cuttings, and for lengths of 100 feet. By means of this, the cubic contents may at once be taken out, when the equivalent level cuttings at both ends of a station are equal: but if they are not so, the prismoidal rule must be employed, thus:

Suppose the equivalent level cutting at one end to be 20 feet, and at the other 25 feet. Then, that at a point half way between them would be 22 feet. Therefore, the cubic content will be equal to one-sixth of the sum of those corresponding to each of the two end depths, and four times that of the centre depth; that is,

Cubic content by table 1x, for 20 feet depth

"25

"22

or 4 times 5146

Cubic yards contained in the station,

=

=

4296 cubic yards, 6065 (6

= 20584

6)30945 = 5157-5

It will be perceived that, instead of the areas corresponding to the differ. ent depths of cu ting, or heights of filling, our tables give the cubic yards corresponding to those areas, for lengths of 100 feet. For the purposes of calculating cubic contents, these solidities may evidently be used instead of the areas; but for such cases as require the areas themselves, a table of such is added. Its use will be shown further on.

I will here observe, that for mere comparative estimates of trial lines, the labor may be much reduced by taking from the tables, the cubic content corresponding to the average of the equivalent level cuttings at the two ends. This mode is, of course, by no means mathematically correct, and should never be resorted to for final estimates; but it will frequently

be sufficiently approximate (always a little deficient) for such trapezoids as occur in ordinary cuttings and fillings, where the depths at the two ends differ but a few feet.

For instance, in the foregoing example, the correct contents of the sta tion 20 feet deep at one end, and 25 feet at the other, were found to be 5157.5 cubic yards; while by this approximating mode, the contents of an average equivalent level depth of 22 feet, would be 5146 cubic yards; or but 11 yards less than the truth.

In this manner, an entire cut or fill may be approximately estimated at one operation, by merely adding together all the average midway equivalent level cuttings or fillings, and dividing their sum by their number for an average one.

Thus, if the average midway equivalent level depths of 6 consecutive stations of cuttings, were respectively 2, 5, 6, 4, 3, 7 feet, an average 2+5+6+4+3+7 depth might be taken of

6

4.5 feet; and the cubic

contents taken from the proper table, for a depth of 4.5 feet, and multiplied by 6, (the number of stations,) would give the contents of the whole 600 feet, but slightly too little.

I will now proceed to describe the mode of preparing the diagrams, for any width of roadway, and for any side-slope whatever.

The determination of the principle on which they are formed, is left as a

problem for such of the younger members of the profession as may choose to exercise their ingenuity upon it.

Draw a vertical line a b, fig. 1, of any given length at pleasure. (One foot decimally divided; or 12 inches, divided into ths of an inch, or 10 inches divided into oths of an inch, will generally be found convenient.) Call the length of this line unity, or 1. It represents the usual centre-line of levels, or of cuttings and fillings.

b

From the upper end of this line draw C, at right angles to it; and from b towards c, lay off and number the distances b5°, b10°, b15°, &c., contained in the following table; using as a scale, the length a b, as 1 or unity, divided into tenths and hundredths.

For example, if the side-slopes he, gf, of the excavation or embankment, are to 1, lay off (without any regard to the width of roadway,) the distances on the upper column of the table; if 1 to 1, those on the 2nd column, &c. This done, the scale of a b, as unity, will be of no further use. Distances on b c, intermediate of those in the table, may be inserted. with sufficient accuracy by eye.

Those beyond 20° will rarely be required, because, when the transverse ground-slope is great, retaining walls are resorted to as a matter of

economy.