F is the seaward (active) earth force, and F. 5.s the landward (passive) force. 

 Note that the point of application of the forct- 7 is a distance ■(, below the 

 tie rod given by 



r^W" 2.82 feet 



(116) 



where 11 is given by equation (1^7 ) . 



3h5'. Pile Design - Minim-um Bending Moment . - By driving the pile some- 

 what deeper, a cantilever would be formed near the pile bottom, which would 

 develop passive pressures in the backfill. Tlie loading diagrara would be 

 changed as sho>m in Figure 110, 



3U6, For optimum, pile design, that is for the least length of pile for 

 which the maximum bending moment would be smallest, pressures p„ and p..-. must 

 be as large as can be develooed by the ground at these points. This may be 

 oroven for Pg by finding the expression for the iriaximum negative bending 

 moment. This maximum mom.ent would occxiv below G, Vn^ point of zero pile load. 

 Calling the moment at G, M^, and the f ' cc.r it the same point, S-, the maximum 

 moment X'jould occiir at the point of zsic r-hear (say Zp), and would equal M™ plus 

 the area of the shear diagram between G and Z^. Referring to Figure 111, the 

 loading, shear, and bending moment diagr3m.s for this section of pile, the shear 

 at any point Z below G would be S = Sp - RZ /2 x-rhere R is the rate of change 

 of -ressure between pq and p^,. At Z = Zp, this shear would be zero, therefore 



Z^ = —5— . The moment at Z^ is 



=M .23 fG 

 C 3 c R 



= the maximum negative mom.ent. 



3I47. Now M and S- are functions of the pressure area pq G F; as this 

 area increases, both Mp and S^ will increase. These would be smallest when 

 the rate of pressure change R^''is greatest (some lesser rate would appear as 

 the dashed line in Figure 111 and would cause point G to be lowered). But 

 the greatest possible rate of ores sure change between pq and p^ would make 

 Pp as large as the p- round would permit as that point, which was to be shown, 



3U8. Referring again to Figure 110, one of the earth forces xirhich cause 

 the oressure p^ to be developed is that due to the pressure triangle p,Q I J, 



168 



