403 On the Pressure of the Earth ngalnst Revetements 



This force is opposed by the resistance of the wall, which 

 niav be supposed to act in a horizontal direction ; and when 



_._ . , R. (cos a + fiin. n) ,„. 



reduced to the direction DL it becomes = 7ZA\m ' ' 



Also, the pressure against the back of the wall is equal to R ; 

 consequently the friction u — R/; and vvlien this force is re- 



1- • T-vr" • • R.(/sin. a— /■- C03.a) ,^.. 



duced to the direction DL, it is = Tadilll""^ * ^ ' 



And, when the forces (A), (B), and (C) are in equilibrio, we 

 have 



R.(cos. a + 2/ sin. a — /- cos. a) =W.(.sin. a —/cos. a). 



But, making the radius =1, this equation may be put under 

 a form better suited to the present purpose ; that is 



l+2ytan. o— y- ^ ' 



Or, because W=i h^ S x ; 



' " tan. ^ 



_ j/t-Sftaa.q-/) ,j,, 



tan.n + '4/ tan.* a — y-'tan.a" ^ 



The second part of this expression becomes a maxmum when 



tan.a=/+y(ii^> 

 And if the angle which the plane of repose makes with tl»e 

 horizon be denoted by c; then/= ^j consequently 



sin. t + /J k -i-v 



tan. a = ^^-^. (F) 



COS. c ^ ' 



If the friction against the back of the wall had been neglected, 

 the expression for the tan. a would have become eijuivalent to 

 the simple and elegant one obtained by M. de Prony. 



The value of the tan. a (F) being introduced in the equation 



(E), it becomes 



j^ _ .. ~ r: — . ((jr) 



— f in.3 c <ij 2 + sin. '■ c V 2 

 siu.c a/ i + 1 + + -; 



COS. * C - COS. c 



But the numeral value of the denominator will be constant 

 for the. same kind of earth 5 and let this value be = / ; 



thenR = ^. (H) 



And as the wall may either slide on its base, or turn on the 

 point C as a centre of motion; it may be shown, that in the 

 latter case, tlie leverage Dp = 4-/i. 



The preceding inquiry extends only to retaining walls, coun- 

 terscarp rcvetement.';, terrepleins without parapets, (Sec. But a 



simple 



