For the sake of simplicity the exact elastic theory solution may be 

 replaced by an approximate relationship which is plotted in Figure 16 and 

 expressed analytically as follows: 



^3 



4 



z 



Yd 



< z < 2D 



(3) 



= 



2D < z < 



where D is the plate diameter. 



Substituting Equation 2 into Equation 3 and integrating according 

 to Equation 1 yields the solution: 



SK 



p '[>" 



H/D 



;h/d) + (1/4) 



2 + 



(H/D) + 2 



(H/D) + (1/4) 



(4) 



This relation is plotted in Figure 17. It should be noted that this is not an 

 exact theoretical solution because of the approximations involved in 

 Equation 3 and the violation of the Boussinesq homogeneity assumption. 

 The trends displayed by this solution and summarized below should be 

 correct, however. 



1. The settlement, S, varies linearly 

 with the surface pressure, p, and the 

 inverse of rate of stiffness increase, 

 1/K. 



2. S is a nonlinear function of the 

 quantity H/D. If H, which is an 

 indicator of the stiffness at the 

 surface, is equal to zero, then S 

 is independent of D. As H increases, 

 S becomes more and more dependent 

 on D. 



These results also seem to be 



qualitatively correct. However, since 



it is not possible to determine directly 



elastic parameters for the soils under 



Figure 17. Plot of Equation 4 relation. consideration using available data, an 



8.0 

 6.0 









/ 



/ 









' 







/ 



/ 







2.0 



/ 



/ 









'^ 











2.0 3.0 



H/D 



21 



