By Prob. II. w+2 ar+2f=33+3 × 22+2x9=117 w+2br+2ƒ = 33 +3 × 38+2×9=165 or, by putting = given length, and taking half the sum of the widths, there will result (w+ra+b+2f)l =(33+1 × 22+38+2x9) 18 660 = 3.84545 acres. 660 CASE II. When the exact Quantity of Land for the Railway is required. RULE. Take the whole widths, at the end of every chain, from the 6th column of the Level-Book, for the several widths; add continually together the first and last widths, and twice the sum of all the intermediate widths, and divide the whole sum by 1320 for the area in acres. Ex. Required the area corresponding to the several widths in the Level-Book at the end of Prob. V. 12169.584 11 14.132 NOTE. It is very common in practice to find the areas of the quantities of land, required from the several proprietors, by actual measurement from the 2 chain maps, made for the use of the contractors, after the several widths have been laid down thereon: copies being taken, at the same time, from the maps, on tracing paper, showing the position and quantity of land required from each proprietor. SECTION III. ON RAILWAY CUTTINGS IN GENERAL AND METHODS OF FINDING THEIR CONTENTS. In preparing the preliminary estimates for a railway, the contents of the cuttings are usually found by tables for the purpose, the surface of the ground in the several cross-sections being assumed to be on a level with the centre of the line. But when power has been granted for constructing the line, the cross-sections are carefully taken at the end of every prominent variation of the surface of the ground, or, if consistent with accuracy, at the end of every one, two, or three chains in length; the several cross-sections are then plotted on a large scale (which may be done by the methods given in the preceding Problems), and their areas found by actual measurement; or reduced, where the surface of the ground is laterally sloping or curved, to horizontal sections, preparatory to finding the contents from the tables, by using the mean depths of the several sections, which method is correct; but the mean depth, used in this method, cannot be accurately found in many cases without considerable calculation, which may be avoided by adopting the methods given in Prob. II. p. 46. Some take the mean of every two succeeding sections, and others use a mean of the mean depths as the basis for a mean area, both of which methods are very inaccurate; especially where the areas of the extreme sections differ greatly. The magnitude of the errors in both cases, will be pointed out in the investigations at the end of these Problems. On existing Earthwork Tables. Tables for this purpose have been published by Sir John M'Neill, Mr. Bidder, Mr. Bashforth, Messrs. Sibley and Rutherford, and others; all of which are well adapted for finding the contents of cuttings, assuming the surface of the ground to be laterally level with respect to the direction of the cutting. But none of these tables are properly adapted to the finding of the contents from sectional areas, i. e. from the areas of working drawings, excepting Mr. Bashforth's tables; but his mathematical investigation of the rule for using them in finding the contents from working drawings, where the surface of the ground is laterally sloping or curved, is founded on a false assumption, and, therefore, his results are erroneous, and especially so where the sectional areas differ considerably. (See Preface, and the Investigations at the end of these Problems.) D THE GENERAL EARTHWORK TABLE. (At the end of the book.) This Table, with the help of the Auxiliary Earthwork Tables, Nos. 1 and 2, on the same sheet, possesses the advantage of being general for all varieties of slopes and bottom-widths in common use, as well as for decimal parts of feet in the depths. It may also, with a very trifling preliminary calculation, be made to extend to every variety of bottom-width and ratio of slope that can occur, if even the slopes of the two sides differ in the same cutting; and with the help of a table of square roots it will apply, with all attainable mathematical accuracy, to cuttings where the surface of the ground is uneven. The investigation of the method of forming the Tables and using them, will be given at the end of these Problems. The contents in the general Table, and those in Table No. 2., are calculated to the nearest unit for one chain in length, and checked by differences; the sideslopes being assumed to be extended till they intersect. The auxiliary Table, No. 1., gives the depths of the intersection of the side-slopes below the balance-line, and the corresponding number of cubic yards to be deducted from the contents for each chain in length. PROBLEM I. CASE L-To find the Contents of Cuttings by the general Earthwork Table, and the Auxiliary Table, No. I., at the end of the Book. Let A B b dc C be a cutting, A B = a b = bottom-width on the formation level, M M' and m m' the perpendicular depths at the middle of the two ends of the cutting; A C, B D, a c, b d the side-slopes, which, being prolonged two and two, will meet at the points N and n; also M M' and m m', being prolonged, will meet at the same points. The distance M' N m'n in feet, and = decimals is given in the Auxiliary Earth work Table, No. 1., for all bottom-widths m M and ratios of slopes in common use, at which distance a line must be ruled on the section, parallel to the balance-line, or at the same distance + 2 feet from the line of the rails, in which latter case the balanceline need not be drawn. From the line thus ruled, the depths of the cutting must be measured to adapt them to the General Earthwork Table; or a mark might be made on the vertical scale with Indian ink (which is easily washed off) at the same distance, which mark might then be applied to the line of the rails in measuring off the -- depths. For measuring the depths of embankments, the line must Ex. 1. Let the several depths of a rail- way cutting to the intersection of the slopes, at the end of every chain, be as in the an- nexed table, the bottom width 30 feet, and NOTE. By the Table No. 1. the depth to be added to the depths of the cutting, for bottom width 30 ft. and ratio of slopes 1 to 1, is 10 ft., therefore the line from which the depths in the annexed table are mea- sured is 10+2 12 ft. below the line of the rails. The corresponding number of cubic yards to be sub- tracted is carried to two places of decimals, or, if the nearest whole number had been taken, the quantity 3.00 33 2582 Ex. 2. Let the several depths to NOTE. When any of the distances is 2 to 1........... 3410304 396 x 278 = 110088 66)3300216 Content in cubic yards 50003 NOTE. When the distances are given in feet, the quantities from the General Table must be multiplied by their respective distances; also the quantity from Table No. 1. must be multiplied by the whole distance, and the final result divided by 66, as in the annexed example. See Demonstration at the end of these Problems. CASE II. To find the Contents of Cuttings by the Tables, when the Depths are given in Feet and Decimals of Feet. RULE. Let any two succeeding depths be denoted by a and b, and let the decimal parts of the depths be respectively denoted by a' and b'; find the quantity corresponding to a and b from the General Table, as in the former case; then, 2a+b, or its nearest whole number, and the decimal a' will show the 10 26+a, 10 number of cubic yards to be added in Table No. 2., and or its nearest whole number, and the decimal b' will show the cubic yards to be added in the same Table. Ex. 1. Let the depths to the intersection of the slopes be 61.6, and 39.4 feet, their distance 1 chain, the bottom width 36 feet, and the ratio of slopes 2 to 1; required the content of the cutting in cubic yards. |