FOUNDATIONS AND EETAINING WALLS 207 



3. The resultant pressure is inclined at an angle f to a normal to 

 the back of the wall. 



From the result of the theory based on these assumptions, the 

 values of the resultant earth pressure given by Coulomb, Weyrauch, 

 Rankine, and others will then be deduced as special cases by giving 

 different values to 



In Fig. 142 let AB represent the back of the wall, BD the surface 

 of the ground, AD the natural slope, and A C any line included between 

 AB and AD. Also let P' denote the resultant pressure due to the 

 wedge BAC, P l the weight of this wedge, OR its reaction against 

 the plane AC, ^ the angle between P' and a normal to the back of 

 the wall, co the angle of repose of the earth, a the angle between the 

 back of the wall and the horizontal, /3 the angle between the surface 

 of the ground and the horizontal, and x the angle between AC and 

 the horizontal. 



Then in the triangle TOS, by the law of sines, 



si 

 *s 



or, since TOR=x-coa.ud mS=180 -a-f, wehave OST=a+%-x+co, 

 and, consequently, ^ ^ sin(g-) 



1 sin (a + f + <*> x) 



To find an expression for P v let w denote the weight of a unit 

 volume of the material, say the weight of one cubic foot. Then for a 

 section of unit length in the direction of the wall 



P l = w(area ABC) = AB-ACsm BAC ; 



or, if h denotes the height of the wall, AB = > BAG = a x, 

 . , sin a 



i A ^ AT* sin ( a P) i. 

 and A C = AB - ; whence 



sin (x p) 



_ wh 2 sin (a 0) sin (a x) 

 1 " 2 sin 2 a sin (x 0) 



and, consequently, 



2 sin 2 a sin(# /3)sin(a + + co x) 



, _ wh* sin (a 0) sin (a x) sin (a? co) 



