decoupling the shear and torsion terms, so that this location of point s is called the shear 

 center of the beam. 



The first application of Equations [25] sets V^ = M^ = 0. By Equations [30] and [31], 

 d6 dd 



X X 



this implies that = = also. Using these relations and the fact that the terms 



'Ipart j ^''® l^nown multiples of V^ (by Equations [12] and [20]), Equations [25] become: 



j i j 



As. As. As. 



j j j 



As. As. ^As 



E-'L„L_,E^L,,L,^...Z:-L,,L„ 





^< 





< 





'>-- 



As. 



As. 



Ef— L.2 qp^^j . 



As 



JL 5[ 



[32] 



These equations are solved for K^, K2, - • • , Kj^. Then, substituting into Equation^, [11], 

 [12], and [13], we write q^ in terms ofY^ = V^, giving Qvyi'Qvy2' ^Vys' • • • ^ the 

 solution of plate shear flows per unit V . Having these, we substitute into Equation [26] 

 to get: 



'^x= [fQvyj(Yt, ^hi-yhj^tj)]v, [33] 



The third equation of Equations [30] then gives the z location of the shear center: 

 M^ -M^ + yV^ M^ 



[34] 



The second application of Equations [25] sets V = M 



0, leading to 



dx d\ 



equations identical to Equations [32] except that the terms on the right side are now all 



proportional to V^,. The solution gives Kj, K2, ■ • • , K^ in terms of V and , by Equations 



[11], [12], and [13], q. arc written in terms of V^ = ^z ' §'^^"8 the terms Q„ . of matrix 



Equations [27] and [28]. Again we substitute into Equation [26] to get: 



X = [f^vzj (ytj^hj -.yhj=^tj)] V, 



M = 



The third equation of Equation [30] now gives the y location of the shear center: 

 _ M^ - M^ + z V M^ 



[35] 



[36] 



75 



