6 = wall friction angle, degrees 



These symbols are further defined in Figure 7-122. Equation (7-143) may be 

 reduced to that given by Rankine for the special Rankine conditions where 6 

 is considered equal to i and 6 equal to 90 degrees (vertical wall 

 face). When, additionally, the backfill surface is level (i = degrees), the 

 reduced equation is 



(7-144) 



Figure 7-123 shows that P from equation (7-144) is applied horizontally. 



Unit weights and internal friction angles for various soils are given in 

 Table 7-15. 



The resultant force for equation (7-143) is inclined from a line 

 perpendicular to the back of the wall by the angle of wall friction 6 (see 

 Fig. 7-122). Values for 6 can be obtained from Table 7-16, but should not 

 exceed the internal friction angle of the backfill material tj) and, for 

 conservatism, should not exceed (3/4) <^ (Office, Chief of Engineers, 1961). 



2. Passive Forces. 



If the wall resists forces that tend to compress the soil or fill behind 

 it, the earth must have enough internal resistance to transmit these forces. 

 Failure to do this will result in rupture; i.e., a part of the earth will move 

 sideways and upward away from the wall. This resistance of the earth against 

 outside forces is called passive earth force. 



The general equation for the passive force P is 



P_ = 



wh 



2 r 



(7-145) 



It should be noted that P is applied below the normal to the structure 



slope by an angle -6 , whereas the active force is applied above the normal 

 line by an angle +6 (see Fig. 7-122). 



For the Rankine conditions given in Section 1 above, equation (7-145) 

 reduces to 



-p - 1- ""' «• * I 



(7-146) 



Equation (7-146) is satisfactory for use with a sheet-pile structure, assuming 

 a substantially horizontal backfill. 



7-257 



