276 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[Sept 



Setting out Gradients. 



A fixed point to which the gradients are referred is always determined 

 at one end, at least, of a proposed line of railway, and from this point the 

 levels must be carried forward, following the centre line, w hich, it is as- 

 sumed, has already been marked by pegs or stumps at distances which 

 ought not to exceed four or five chains. Two levelling staves, a chain of 

 lifty feet in length, and a perfect spirit level in complete and accurate ad- 

 justment, are the iustruments with which the engineer should be equipped, 

 and which he will require to use with skill and expedition. 



The gradients should be laid down in the field, — 



1st, — By measuring the length of each from end to end, and fixing ordi- 

 nal y pegs to mark the beginning and termination. 



2nd, — P.y carrying a series of levels over the whole length of the gra- 

 dient without using the chain, so as to determine the difTerence of level of 

 the ground surface at each end of the gradient. 



jrd, By planting permanent bench marks within a convenient distance 



of the line, but secure from being buried or destroyed during the progress 

 of the works. 



These bench marks should be formed of posts of squared timber, 

 inches x 6 inches, secured round the head with an iron hoop, and either 

 driven into the ground as a pile, or, being squared at the foot, set upon a 

 slate, tile, or flat stone, in a hole dug to the depth of three feet, or there- 

 abouts; and they should stand 2' 6" out of the ground. Upon this post a 

 saw kerf, nick, or other mark, to be described iu the level book, must be 

 made as a reference to the height of embankment or depth of cutting at 

 the end of the gradient, already marked by a peg on the centre line. 



It is a matter of most essential convenience during the whole progress 

 of tlie construction of a railway, that these bench marks should be readily 

 accessible and properly protected from injury or derangement , and a little 

 expense incurred to attain this end will save ultimate cost, in the shape of 

 tim--, to all parties interested, whether contractor or engineer. 



Fig.:, 

 fl^^-^^ .. 1^ 



1 i 



! i 



The only calculation requisite to complete this preliminary operation is 

 readily understood from the figure, in which a 6 is the ground surface, 

 c d the gradient line, and g b the difference of level at the surface between 

 the limits of the gradient cd. Then, as we know the position of the point 

 f, with reference to a, at the beginning of the gradient, that is, in the 

 present figure, the depth of cutting ac — gh, then gb — gh-\-hd = bd 

 the height of embankment at h. 



Now, to render this process perfectly general and free from any hesita- 

 tion in applying its result, the following conventionality must be borne in 

 constant remembrance : — 



Calling, B the sum of all the back sights. 

 F the sum of all the fore sights, 

 G the rate of inclination of the gradient or the height dh 



through which it rises or falls in its whole length. 

 (". the depth of cutting or height of embankment at one end, 

 C the same depth or height at the other end. 

 Then the general formula is B — F -f- C — G = C. 

 Iu which we must use the positive or negative sign to C and G, as the 

 circumstances may be, thus — 



When C is in cutting it is always taken with the positive sign ; when in 

 embimkment, with the negative sign. 



VVheu G is a falling gradient, it is to be prefixed by the negative sign ; 

 when a rising gradient, by the positive sign. 



Then the sign which will be the result attached to C will indicate 

 whether the end of the gradient is in cutting or embankment. 



Suppose a case, in which B = 200; F = 208; C = -f G; andG = -f 3; 

 then, 200 — 208 -|- 6 — 3= — 5, where we have a cutting of G feet at 

 the beginning of the gradient, and an embankment of 5 feet at the other. 



Suppose we nse the same figures, but work the reverse way ; then 

 20S — 200 — 5 -f 3 = -f 6. 



A little consideration will make this familiar, and the remembrance of 

 the proper signs will be facilitated by connecting them with the fact, that 

 when the ground fulls, the difference of level is negative; or, in other 

 words, the sum of the fore sights is greater than the sum of the bark 

 sights. In the same way, when a gradient line falls, the expression G is 

 negative. On the other hand, where the ground rises, the difl'erence of 

 level is positive, for the sum of the back sights will exceed the sum of the 

 foresights; and in like manner, when a gradient line rises, the expression 

 G is positive. Also a negative sign before T! shows that the ground sur- 

 face is below the gradient line ; and when the sign is positive, that the 

 ground surface is above the gradient line ; the first indicating an embank- 

 ment, the last a cutting. J 



The inclination of the gradient must, on do account be taken from the I 

 ratio of inclination usually given on the section, but the dillerence of the 

 heights above datum at the ends of the gradient must alone be employed 

 in all cases. 



The ground between the ends of each gradient must be levelled over at 

 least twice, and should any difference appear, it must be gone over until all 

 uncertainty as to the truth of the result vanishes ; and then — and no^ until 

 then — the height of embankment or depth of cutting, above the nick, or 

 saw-kerf, in the bench mark, should be properly painted in figures on the 

 post, say in red paint for cuttings, and in black or white for embankments. 



When the length of the gradient is, in the first instance, chained out, pegs 

 should be driven into the ground occasionally where the rise or fall amounts 

 to a whole number; thus suppose a gradient of 1 in 3U0, a peg at 300 

 would show that the rise was 1 foot ; at 900 feet that it was 3 feet ; and 

 so on. The use of these pegs is to serve as a check upon the levels taken 

 for the widths, where an accumulation of small fractional parts of a fool, 

 individually too small for reading on the staff, and therefore neglected in 

 each chain's length, might amount to a serious multiplied error if carried 

 on ; and for distinction these pegs should be painted red. 



Marking off flie Widths. 



The first step, at this stage of the work, is to divide the entire length of 

 each gradient into spaces of one chain, or 50 feet each ; for these space.-^, 

 levels to determine the height of embankment or depth of cutting should 

 be simultaneously carried on with the chaining and measurement of the 

 widths. It is not uncommon that the surface of the ground is covered 

 with soil or grans turf, which is directed in the specification of earthwork 

 to be removed from the site of each embankment, and set aside and re- 

 served from each cutting before the embankments are begun, and to be 

 used subsequently in soiling the slopes, in forming mounds for the quick 

 fences, or to be disposed of to farmers or others. In calculating, therefore, 

 the quantities of earthwork in each chain's length (for it is presumed the 

 contract quantities will be computed from the level books, and not accord- 

 ing to the lazy method of scaling heights from a section), we must bear in 

 mind that the removal of this soil increases the quantity in every embank- 

 ment, and decreases the disposable number of cubic yards in every cutting ; 

 and the thickness of soil directed to be removed must be added to the 

 heights of embankment shown, as taken from the surface, in the level 

 books ; and must in the same way be subtracted from the depth of cut- 

 tings. For the widths, however, tlie existing surface must be worked 

 from. 



We will now suppose then, that commencing at the beginning of a gra- 

 dient, a length of 50 feet has been measured along the centre line, the 

 spirit level set up, and a back sight to commencement, as well as a fore- 

 sight to the end, of the chain's length observed and entered on a level- 

 book, the form of which will be presently given. 



The height of embankment, or depth of cutting, must be then calculated, 

 as already explained, for the value of C, and also entered on the level- 

 books. Half the width of the roadway must be measured off horizontally 

 on each side of the centre line, and at right angles to it, and also a space 

 for each slope corresponding to the value of C, multiplied by the ratio of 

 the slopes. A reading from the level staff, taken at each of these last 

 points, and entered on the level-book, completes the first part of the field 

 work. 



The values of G, or inclination of the gradient, for the several distances 

 of 50 feet, should be very accurately calculated, and entered in the proper 

 column of the level-book in the office ; — and it is not sufficient that this 

 calculation be made for one distance and repeated; but the sum of the 

 inclinations for any number of separate lengths should correspond with the 

 inclinatioa for the whole length ; so as to correct any multiplication of 



