s-c 1 



'll - c-(l/C) p-Tn^ ' 12 - ^21 - c-(l/C) - 2 



H s + g 1 2 _ c+(l/C) 1 



^22 - - c-(l/C) Elq ; ^11^22 " ^12 " " c-(l/C) (El)2q^ 



If also |c| >> l/C, the still simpler approximations become available: 



1 •• 1 



a-,-, = (tan q«. -l) t ; a = a = -tan qi^ 



Elq-^ l"^ -^ Elq 



^22 = ^^^ 'I' ■'^^ ik ' ^1^22-\2 = - -j;^ 



Several interesting oases are now discussed in the order of 

 incresLsing qH , assuming that C = 0. 



(a) Uniform rigid beam, q^ *^< 1: Smallness of q may arise from 

 smallness of either )l or u, the latter occurring in q. Then use may be 

 made of the series (note that when C = , q^ = qg = q) : 



s = sin ql = q)l - g-(qi^)^ . . . ; c = cos qJl = 1 - -(q^) + 2^ ^^^^ ' ' ' 



S = sinh qZ = q£ + 'Uql)'^ . . . ;C = cosh q«, = 1 + ^q«.) + ^ U^^ • • 



Keeping only the lowest power of q^: 



1 + cC = 2; 1 - cC = I" (qX.) ; sS = (q«.) 



•• 2 , ,3 

 sC + cS = 2q!l; sC - cS = T-(q2') 



2 

 In this case, it may be enlightening to return also to yw by sub- 



1 p 



stituting q = . Then, provided 1^ << 1: 



EI 



11 



