The third application of Equations [25] sets V^ = V^ = V^ = Vy = 0. By Equations 

 [12] and [20], this means that Iq^^^^ .! = Iq^^^ J = also. Equations [25] now become: 



(L by L coefficient 

 matrix is identical 

 to the coefficient 

 matrix of Equation 

 [32].) 



^Lji(ytjZhj -yhjZtj) 



^Lj2 (ytj^hj-yhj^tj) 



^LjL(ytjZhj -yhj^tj) 



. G [37] 



dx 



Equations [37] are solved, giving K^, K2, • • . , Kj^ in terms of G — — . Again, substituting 

 into Equations [13] and [11] gives Iqi^^p j^ = ^'^j^ '" ^®™^ °^ ^ dd^/d\. Express the latter 



Now substituting into Equation [26], using M^ = M^^ from Equation [30] gives: 



X . 



Solving Equation [39] for G and substituting into Equation [38] gives: 



dx 



iqii = 



^Q^j (yti^hj -yhjZtj) 



/hereupon, by definition of IQ^-i , we have 



'tj* ^Q^j (ytj^hj -yhjZtj) 



[38] 



[39] 



[40] 



[41] 



Thus, by three applications of Equations [25] we have determined y and z (the shear 

 center location) and the entire matrix [Q] in 



76 



