Elastic Eqiiilibrium of Semi-Infinite Solid. 



55 



i, = — . > — ^^ ^ — - — - sm •2;i + -2)ti' 



_ A X- (-i)"(2^0 



■^-a{). + [x) t,n\ («+!)! 2"-^"+i \ ^z' + d' 



n 



^ (-lÄ 2n)!/ r V^", 



^2 Zj ^^|\29rn \ /,2_i_^2y 



4-a// V-^'+a^^o (w!)'2-» \ V'^H 



sin(2» + l)^A, 



sill (2??,+ \)ip 



+ 



where 



-'^^^±^i:^=^^:^(-..^) -'w.- 



4-ii(;. + ^);/„^o (w!f2--^" \ Vr + ^^ 



cA = tail"' 



_i a 



V(116) 



vi40. At the surface {z = o) they reduce simply to'^ 



for /'é <i'. and 



{uX = - 



fj a — y/à' — r- 



1 



1 



47T{À + /i) ' r 



A- A-\- a)na 



(117) 



(118) 



for r ^ a. 



As seen from the formula (117) the vertical displacement is 

 constant over tlie loaded area, and therefore this solution is 

 applicable when an absolutely rigid body of circular base is pressed 

 normally against an infinite elastic body; the problem has been 

 attacked by various writers from this point of view. 



§41. Similarly the expressions for the stresses at the surface 

 will be found to be 



1) For n = o we have to take -L instead of zero. 



2) If we wish to know the expressions for displacement only at the surface, these can 

 be olitained with less calculation. See Lamb I.e. 



