18+8.1 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL 



19» 



NOTES ON ENGINEERING— No. IX. 



By HoMEBSHAM Cox, B.A. 



Synoptic Trihfcs for calculation of Earthworks in Level and Sidelong 

 Ground on Railwayii. 



No eartliMo'rk tables liave hitherto been published for the ex- 

 press purpose of facilitating calculations for SIDELONG 

 GROUND. The present is an effort to supply tliis want, which is 

 much felt on account of the number and complexity of tlie opera- 

 tions usually required for adapting the published tables to the cases 

 referred to. According to the existing methods, it is necessary 

 when the ground slopes laterally, to calculate the areas of the 

 sections, and extract the square root pre^•iously to reference to the 

 tables. By tlie method here proposed, these antecedent calcula- 

 tions are wholly avoided : the numbers are taken from the table 

 without any pre\ious computation, and require only to be multi- 

 plied by the natural and artificial slopes. 



The tables are also easily applicable in calculation for LEVEL 

 GROUND ; and it is believed tliat for both purposes the methods 

 will be found very simple and expeditious. The tabular numbers 

 are calculated for every half foot. When greater exactness is ne- 

 cessary, the calculator is referred to the admirable tables of Mr. 

 Bashforth. The manuscript computations for those tables have 

 been kindly placed at the disposal of the jjresent writer, and have 

 enabled him to check the accuracy of a great part of his own re- 

 sults. 



It is not intended to demonstrate at length the formula of com- 

 putation, as they depend on well-known theorems : the following 

 brief account of them is sufficient for the present purpose. 



Let A C D B be a section in a raihray cutting in level ground, 

 C D being the formation level, and A C, B D, the artificial slopes. 

 As A B is horizontal, the points A, E, and B, ai-e aU at the same 



vertical height above the formation level : call tliis height a, 

 measured in feet. 



Let a similar section be taken at a distance along the railway of 

 66 feet, or one chain from the above, and let the vertical height 

 there be 6 feet. The solid content in cubic yards of the solid 

 terminating in E D B, is %f (a"- + a h -f 6-J r. 



The solid content is y a r w ; r being the " slope" of the 

 embankments, the measurement ol which will be explained more 

 fuUy presently ; w the width in feet of tlie formation level. 



Next, let A C D B be a section in sidelong ground — that is, 

 gi-ound inclined laterally or transvei'sely to the railway. Here 



E^-' 



Fig. I!. 



there are two sets of heights to be considered — tliose on the right 

 hand, and those on the left hand of the railway: and there are 

 two " slopes " — that of the natural ground depending on tlie in- 

 clination of A B ; and that of the artificial embankment, depend- 

 ing on the inclination of either A C or B D. Call the natural 

 slope R, and the artificial slope (as before) r. If a and A be the 



heiglits (in feet) of the points A and B respectively, above tlie 

 formation level C D. the area of the triangle AFC = J (R -)- j-)a^- 

 and of tlie triangle B F D = i (R - r) A'. ' 



Similarly, if another section be taken at a distance 66 feet along 

 the railway, and 6, B, be the coi responding heights, the areas of 

 the two triangles similar and similarly situated to A FC, B F D, 

 respectively, are i (R + r) l>- ; and i (R-»-) B". 



The solid content (in cubic yards) of the solid terminating- in 

 AFC=:: (R -!-?•) rrlA=-(- A JB-V-B). 



The larger of the accompanying tables gives values, or 

 ^i («- -I- all + ft-), for every half foot of the two heights, up to 

 jO and 60 feet, respecti\ ely. The smaller table gives values of 

 If" >< ''.V various widths of the formation level. 



.MKTHOOS OP USINli THE TABLES. 



For Level-lyiaii Ground. — JIultiply the tabular number in the 

 larger table corresponding to the heights of two successive sec- 

 tions a chain apart, by twice tlie slope, and add the number from 

 the smaller tatile, corresponding to each heiglit separately. The 

 result is the number of cubic yards required. For instance, let 

 the heights be 295 feet and io feet ; the base, 30 feet; and the 

 slope 2 (to one). In tlie larger table, the number corresponding td 

 {295, 45} is 1710. This, multiplied by twice the slope = 6840. 

 Add, from the column for base 30 in the second table, tlie number 

 for 29i (which is 1082) ; and also the number for 45 (which is 

 1650): and the total (9572) is the quantity of cubic yards re- 

 quired. 



The following is an example of the quantities corresponding to 

 four sections, a chain apart, the corresponding heights being 16, 

 20^, 30, and 44J, respectively : the base, 33 feet : the slope, 2^ (to 

 one — consequently, all the first tabular numbers are to be multi- 

 plied by twice 2^ (or 5). 



HeightE. 



rl6 

 L20§ 



r20j 

 L30 



r30 

 L444 



1st Tjb. Nos. 



Multiplied by 

 twice Slope. 



398 



1990 



788 3940 



.1717 



8585 



2nd Tab. Nos. 

 to Base '.^. 



645 



.... gg^ ... 



827 

 1210 



1210 

 1794 



Sum. 



3462 



5977 



.11589 



Answer 21028 



When the sections are at greater or less distances than one chain 

 apart, quantities between each two sections must be multiplied by 

 the corresponding distances. For instance, suppose in the above 

 example the sections had been IJ, 2, and 3 chains apart: re- 

 peating the xuin» in the last column of the above scheme, we 

 have 



51914 A)u. required. 



For Sidelong Ground. — Here the larger table alone is used. 

 There are two sets of heiglits, those on the right-hand side of the 

 railway, B/; and those on the left-hand side, Ae (fig. 2). These 

 sets are to he kept quite distinct. i\f ultiply the tabular numbers cor- 

 responding to the greater heights by the difference between the 

 natural and artificial slopes, and the tabular numbers correspond- 

 ing to the less heights by the ■■mm of artificial and natural slopes : 

 the difference between these products is the result required. 



For instance, let the natural slope be 6^ (to one), and the artifi- 

 cial slope IJ (to one) : the sum of the slopes is 8, the difference 5. 

 Also, let the lieights be : 1st section, 20i, 10 ; 2nd section, 52, 30. 

 The two major heights, 20^, 52, are taken together ; and the two 

 minor heights, 10 and 30, are taken together. The number in the 

 table for the first pair is 1707 ; which, multiplied by 5, gives 8535. 

 The height for the second pair is 529 ; which, multiplied by 8, 

 gives 4232. The difference of the two products, or 4303, is the 

 answer required. 



Where the slopes remain unchanged for several successive sec- 

 tions, the sums of each set of the tabular numbers may be multi- 

 plied by the slopes, instead of multiplying each tabular number 

 separately. For instance, let the natural slope be 3 (to one) ; the 

 artificial slope, 1 (to one). Also, let the heights be, first section, 



