E is Young's modulus, 



G is the shear constant usually denoted by G, 



A is the area of the cross section, 



I is its area moment of inertia ahout its principal axis , and 



K is the usual shear-warping constant. 



If u is small enough, qi and K will both be much less than unity. 



Specifically, ql << 1 and 5 << 1 if 

 1 / EI \ 1/2 



to << 



(— j ; 0) << 2KAG^//yEr [59a,b] 



The second limit on co is obtained upon substituting for qi in the 

 expression given for 5 and setting the result << 1. 



When C is so small, all shear effects in the beam may be ignored and 

 the approximations derived in Section 3 for ql << 1 may be used, namely: 



- - - - - . . _ [60a,b,cJ 



11 „ 2 ' 12 21 „2 2 ' 22 „3 2 



Let w be small enough so that 



P ^k 12k2 



0) << —7— ; w << — ^ [6la,b] 



Then, using the approximations for a and a „, la k. I >> 1 and 

 R . p ^-|_ 22' ' 11 l' 



la k I >> 1. Hence: 

 I 22 2' 



.. 12k^k2 

 s=ak; s=ak; D= ^ \ \ 

 1 11 1 2 22 2 „2, ■ ■ 



After inserting the approximations into the formula: 



U ~ S -. S Q ■" a-j qSq -. K-1 Kp 



62 



