228 



RETAINING WALLS. 



CHAP. V. 



will make an angle with a normal to the back of the wall equal to z (different authorities assume 

 values of z from zero to <j>', the angle of friction of earth on masonry, or <j>, the angle of repose of 

 earth); while OR will make an angle with the normal to the plane of rupture AE equal to <f>. 

 Let P represent the pressure OP against the wall, W represent the weight of the prism of earth, 

 and w the weight. per cu. ft. 



I*--. 



FIG. 4. 



In the triangle OWR angle WOR = x - <j>, and angle ORW = 9 + <t>+z-x. Through E 

 draw E N, making the angle AEN = 9 + <]> + z x with A E. Then the triangle A E N is 

 similar to triangle ORW, and 



P_ 



W 



EN 



AN' 



and 



P = W 



EN 



AN 



But W equals warea triangle ABE = %w-AB- BE- sin (0 5), and 



AB-BE-EN 



P = 



(0 - 5) 



(4) 



Now P varies with the angle x, and will have a maximum value for some value of x, which 

 may be found by differentiating (4) and placing the result equal to zero. 

 Differentiating and substituting in (4) and reducing we have 



sin 2 (0 - 0) 



P = 



sin 2 6 sin (0 + 



sin (z + &) sin 



sin (0 + z) sin (0 



_=Y 

 -*)) 



which is the general formula for the pressure on a retaining wall. 



Now if z in (5) is made equal to <', the angle of repose of earth on the wall, 



sin 2 (6 0) 



P = \w-W 



sin 2 0- sin (0 + <t>') I i 



which is Cain's formula (20) in another form. 



sin (0 + 0')-sin (0 5) 

 sin (0 + 0') sin (0 



-*)V 

 -I)/ 



(5) 

 (6) 



(7) 



