1819.J 



THE CIVIL ENGINEER AND ARCHITECrS JOURNAL. 



365 



When the face is vertical and the top horizontal, c=i3; in this 



case 



ens c' ,,. 



tan <f> = -: — ~ — 77 (5) 



sin e' + Vi 



trA- spn c' 



( ' y 



W2 tail c' + acce'/ ■ 



(6) 



The value of tan ^ here derived is equivalent to that of 



ill equation (F) of Tredgold;* hut the value of the resistance 

 differs materially from his, and is far more simple. Tredgold's 

 equation (G) for the value of the resistance acting horizontally, 

 after making the necessary changes to our notation, is 





sin c a/2 + 1 + 



1 



siu^c' v'2 + sin- c' 



V2 



+ n '~ — ; 



COS^ C' ' 2 COS c' 



This value, however, is erroneous, and should be 



„ /i=W 1 



^—r"- — "" — — - — — 



sin c' V2 + 1-1- 



sin^ c' \/2 + 3 sin- c' + sin c' \/2 



which, multiplied by sec c\ to find the resulting resistance, is equal 

 to the more simple form found above. 



AVhen e = /3, the top slopes upwards at the angle of repose : in 

 this case 



tan(p = tan/3 (7) 



_ K.-A- sin^ B ,„, 



The second of these equations gives the greatest of the maximum 



values of the resistance : if the face be vertical, tan /3 = ;, and 



' tan c 



R = — cose' (9) 



2 ^ ^ 



The horizontal portion of this resistance is 



toZi^ w7<- ,,„^ 



R= — cos- c'= ■ — sin=' e (10) 



2 2 



As this value is the same as (7") the limiting value of tlie hori- 

 zontal resistance, neglecting friction at the face, it appears that the 

 limiting value of the horizontal resistance is the same whether 

 friction at the face be taken in the calculation or neglected. 

 M'lien the top slopes downwards at tlie natural slope, 

 tan (p= tan |/ 

 t/'A= sin S tan e / ^/(tan 2c' + tnn $) 



trA= sin S tan e / ^/(tan 2f' + tnn e) y 



~ 2 cos 2c' \tan 2e' sec 8 + tan 2c' + tan Sj 



(11) 

 (12) 



The value of the resistance here given is the least of the maximum 

 values. If the face be vertical, 



tan(f = tan^e (13) 



«=— secc'l- ; I (11) 



2 V2 tan c' + secf'/ ^ ^ 



The value of the angle of fracture is of the same form as that 

 of Prony for a vertical face and horizontal top. 



The equations show that the stability imparted to a structure at 

 the face of a bank, by friction, arises principally from the direc- 

 tion of the resulting force, which makes an angle equal to the 

 complement of the angle of repose with the face, and that this 

 force is in general less than the horizontal force derived from the 

 equation of Prony, or any other in wliich face friction is ne- 

 glected; that the values of both forces, for ordinary banks, ai-e 

 equal at angles of repose in and about 45°; that the former are 

 least for angles of repose less than this, and the latter for angles 

 of repose that are greater; and that the direction of the resulting 

 force makes it in no small degree a crushing force. 



It also appears from the equations, that when the angle of 

 repose is 45°, the face vertical, and top horizontal, that the tan- 

 gent of the angle of fracture is (i) equal half the tangent of 

 the angle of repose. The Equation of Prony, which neglects fric- 

 tion at the face, for the same case, gives the tangent of the angle 

 of fracture equal to the tangent of half the angle of repose. 



resistance when a maximum, for any banli CDE, v hen suppcrted at the face CD, by a 

 retaining structure, taliing the friction at CU into account ; for, in this case, the resist- 

 ance, when in equilibrium with the pressure, must malce an angle equal to the comple- 

 ment of the angle of rtpose with the face, and hence iS=SC'. 

 * Philusc^hical Magayiue, vol. li. p. 402. 



In the following Table of Co-efficients, for finding the maximum 

 values of the resistances, — 



Column 1 contains the engineering names for the slopes corre- 

 sponding to some of the angles of repose in column 2. 



Column 2 contains the angles of repose from which the co- 

 efficients of whl are calculated. 



Column 3 contains the complements of the angles of repose in 

 column 2; or the angle which the direction of the resulting resist- 

 ance makes with the face, taking friction thereat into account. 



Column 4 contains the co-efHcients which, multiplied by u-AJ, 

 give the value of the horizontal resistances when the top is hori- 

 zontal and the face vertical; calculated from the Equation of 

 Prony, neglecting friction at the face. 



Column 5 contains the co-efRcients which, multiplied by m7jJ, 

 give the values of the horizontal resistances, rejecting friction at 

 the face, required to sustain banks with a horizontal top; tlie face 

 sloping 10° from the vertical: 6 = 80°. 



Column 6 contains the co-efKcients which, multiplied by wh;, 

 give the values of the resulting resistnnces when the top is horizon- 

 tal and the face vertical, as in column 4. 



Column 7 contains the values of the co-efficients as before, for 

 finding the resulting resistances when the top is horizontal and the 

 face slopes 10° from the vertical, as in column 5 : now, in this case 

 e = 80°. 



Column 8 contains the values of the co-efficients for finding the 

 values of the resulting resistances when the face overhangs 10° from 

 the vertical, and the top is horizontal: in this case 6 = 100°. 



Column 9 contains the resolved co-efficients of a7ij for finding 

 the portions of the resistances in column 6 at right angles to the 

 face, which in this case are horizontal. 



Column 10 contains the resolved co-efficients of vhl for finding 

 the portions of the resistances in column 7 at right angles to the 

 face. These, in this case, not differing much from the resolved 

 horizontal portions, may be compared with those in column .5. 



Column 11 contains the resolved co-efficients of u'AJ, for finding 

 the portions of the resistances in column 8 at right angles to the 

 face. 



Column 12 contains the values of the co-efficients which, multi- 

 plied by whf, give the ultimate or maj:imum maximorum \'alues of 

 tlie resulting resistances; the face being vertical and the top sloping 

 upwards, at the slope of repose. 



Column 13 contains the co-efficients for finding the horizontal 

 portions of the resistances determined from column 12. 



The length of the perpendicular from the toe of the face to the 

 top, or top produced, is represented hy h^; and the lengtli of tlie 

 face itself by h. 



whl is to bs multiplied by the co-efficients in columns 4 to II, 

 to find the resistances; and wh'^ by the co-efficients in columns 11 

 and 12. 



Table of Co efficients for finding the maximum Values of the Resistances 

 for different Angles of Repose; also the Co efficients for finding the ulti- 

 mate Values of the Resistances when the Face is vertical, and Scarp at the 

 natural Slope. 



The slopes marked thus * are approximate. 



In the preceding equations we have only considered the maxi- 

 mum retaining-forccs. The minimum overcoming-forces, and the 

 position of the corresponding fractures, are determined in a simi- 

 lar manner, and by similar etjuations. Retaining the same nota- 



