136 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[May, 



SETTING OUT CUTTINGS AND EMBANKMENTS. 



Sir— Permit rae through your columns to submit the following simple 

 formulit, for the solutioo of a most important problem in field engineering, 

 viz., the correct determination of the half widths of cuttings and embank- 

 ments in sidelong ground. While the methods usually adopted are for the 

 roost part tedious, and all, I believe, merely approximative, the following 

 will be found to possess not only the great advantage of facility in practice, 

 but the result is obtained by a very simple calculation which may be made 

 in the field. 



I need hardly allude, more especially at the present time, to the practical 

 importance of this problem, since upon a correct staking out of the half 

 widths, not only the quantity of land required but the amount of the work 

 to be executed is mainly dependent ; an error, either in excess or the con- 

 trary, being equally objectionable— perhaps even more so in the latter 

 case, since an extra quantity of land would be required, under probably 

 less favourable circumstances, for purchase, and the additional increase to 

 the slopes, subsequent to the first formation, would be attended with an 

 increased cost in the labour. 



If, in the accompanying diagram of the cross section of a catting in 

 sidelong ground, B'C E represents the ground surface, H C the depth of 

 the cutting at the centre point, corresponding to the same point on the lon- 



gitudinal section, H F and H G the half-widths at the formation level, it is 



obvious that the simple addition of each slope (calculated to the depth at 



the centre) to the half-widths at the bottom would, in the present case, be 



inapplicable— since the widths so obtained would be too small for the 



upper side and too much for the lower, and the contrary for embankments. 



The method now proposed is simply this :— measure off from the centre 



8take the half-widths, supposing the ground to be horizontal, and between 



the centre stake and each of the points so obtained, E and E', take the 



difference of level, viz., D E and D' E' ; then the distance H C, or the 



true position B of the side stake, indicating the superior edge of the slope 



CD' „^ 



on the lower side, is equal to ^ „ . j^ g v'O 



where r represents the ratio of the slope, or of the horizontal to the vertical 

 distance. In other words, it is equal to the square of the half-width on 

 leiel ground, divided by the sum of the same half-width and the product 

 of the difference of level before observed and the ratio of the slope. 



The distance C A', or the position B' of the slake on the upper side, is 

 CD' 



equal to 



(2.) 



CD -D'E' X r 



that is, equal to the square of the half-width on level ground, divided by 

 the difference between the same half-width and the product of the differ- 

 ence of level, before determined, and the ratio of slope. 



Thus, for example, in the diagram, if the depth of the cutting H C be 

 20 feet, the half-widths H F, H G = I5 feet, and the ratio of the slopes H 

 to 1, it is evident that the half-widths C D', C D on level ground would be 

 45 feet :— Let D' E', D E (the difference of level between the above points 

 and the centre) be respectively 8 feet and 40 feet. Then, by formula No. 1, 



CD» 

 C A the half- width on the lower side 3 c D -4- D E y r ~ ^^'** ^*^'' ^^^ 



the half-width C A' (by formula No. 2), the half-width on upper side = 



CD'S 

 C'D'-D'E'xr = "'f««'- 



For embankments, the values for the upper and lower sides would be 

 exchanged, which is evident by reversing the figure : — 



Demonstration.— Dnvt A B, D E, B'A', and E' D', perpendicular to 

 the horizontal A' C D. 



Then let A C or A' C = x 

 C D or C D' = o 

 D EorD'E' =d 

 A D and A'D' =: o — x and x — a 



A Band A' B' 



Where r represents the ratio of the slopes, 

 Then by similar triangles — 



a— X x—a 

 : and 



AC : AB : 

 a* — ax 



C D : D E ; or, X : 



dx. 



and drx-(-ax = o'*.'. x= , ^^ 



d 

 = CA. 



In the same manner for the upper side — 

 A A' C : D E' : : A' B' ; D' E' or, X ; a : 



. ox-a= 

 ax= and a.T — orx = o^ . ' 



= CA'. 



(!•) 



(2.) 



o — dr 



Q. E. D. 



It is scarcely necessary to mention that the above formulae presuppose 

 that the points C, B, E, and C B' E' fall on the surface, and are respec- 

 tively reducible to one plane, which in fact, unless in very extreme cases, 

 may be assumed without leading to material error in the result — at all 

 events, the correction would be so slight that it might be made by the eye; 

 taking the distances CA, CA' rather in excess when the point B falls 

 within the surface, and the contrary when it falls without. 



G. Hawkins. 



BROAD AND NARROW GAUGE. 



Sir — I have not a single share in a railway, broad or narrow, and, in a 

 pecuniary point of view, care not one iota which way the Gauge Question 

 may be settled ; but, for the last sixteen years, I have watched, with the 

 greatest interest, the progress of railway locomotion, and at every increase 

 in the capabilities of the locomotive engine I have rejoiced, as every man 

 ought to do who purely desires the advance of art and science. From 

 recent experiments and from daily experience it appears the engines on 

 the narrow gauge have attained their limit as regards power, while the 

 broad gauge engines are susceptible of an increase of power to any extent 

 compatible with safe velocities. The engines on the Great Western Rail- 

 way have not varied in their dimensions almost from the opening of the 

 line, — having a boiler remarkably short in proportion to the other parts of 

 the engine ; but there is now building, at Swindon, an engine, the boiler 

 of which will be of a length corresponding to the long boilers on the nar- 

 row gauge lines ; the driving wheels will be 8 feet in diameter, and the 

 calculated average velocity with an ordinary train will be 70 miles an 

 hour. We are told that narrow gauge engines are not safe at velocities 

 exceeding 45 miles. What an exceedingly humiliating reflection that, 

 probably, the whole of the kingdom will be tied down for ever to a velo- 

 city of 45 miles. What is to be done? Cannot our great engineers and 

 scientific men rescue us from this unhappy fix ? — There is but one hope 

 left for the narrow gauge, and that is, the success of the atmospheric sys- 

 tem; but most provokingly, it so happens, that the opponents of the broad 

 gauge are also the enemies of the atmospheric system, and Jlr. St«phen- 

 son, instead of holding his post as driver of railway science, by the force 

 of circumstances finds himself in the unenviable position of its bnaksman. 

 It is most devoutly to be wished, for the honour of the narrow gauge, that 

 the engineers and scientific gentlemen who are most unmercifully attack- 

 ing the unfortunate atmospheric system month after month in your valu- 

 able journal, would cease their enmity, and, contrariwise, take the almost 

 prostrate system, try what they can make of it, and adopt it as a bantliDg 

 of their own. I remain. Sir, 



Your obedient servant, 

 April 12th, 1846. S. T. 



