giving the following L equations {£ = 1, 2, ■ ■ . , L): 

 y>As. / ^ AS; 



> t_. ^ii^P-tj- i,77"L./L.i; K,, ^^ ^ 



^As. 



Lj£^2) ^2 + . - 



As. 

 .+ 1^ — L.„L 



jj2 JL 



^L^/(ytj2hj-yhjZtj) 



[25] 



ax 



In these equations, q^^^^. ■ is determined by V and V^ according to Equations [12] and [20].* 



Another equation relates M^ , the twisting moment sustained by the entire section, to 

 the plate shear flows (see Figure 9): 



Mx =f qjRjAs-Xq. (.VtjZhj -yhj^j) [26] 



Here the summation is over all the plates. 



To find q. in terms of V^ , V^ , M^ (see Equation [30]), we solve Equations [25] for 

 Kj, K2 , . . . , Kj^ and substitute into Equations [13], [12], and [11].** This is done three 

 times: 



once for V^ = M =0, giving as solutions q. = Q,, .V ; 



z X CO Tj Vyj y 



once for V,, = M^ = 0, giving as solutions q. = 0„ .V ; 



y X ' ^ ^ 1j ^Vzj z ' 



and once for V = V, = 0, giving as solutions q. = Q„.li . 



Since V = V and V^ = V^ (see below), we then have: 



/ \ 

 V 



[Q] 



V 



[27] 



where the elements of Q are defined by the above solutions: 



*The physical significance of Equation [25] is that it represents one expression of strain compatibility for 

 each independent loop comprising the cross section of the structure. Basically, in a structure with a loop, strain 

 compatibility ensures that when you go around the loop once you return to the starting point. Contrariwise, for 

 a structure without a loop (for example, a deep channel or U-shaped section), there is no requirement that the 

 adjacent, but unconnected, edges be aligned when torsion is carried. 



**Equations [is], [l2], and [llj may be combined to give; 



!^.! + [T^.]!%uti! + hJ!%! 



Solutions of Equation [25J (under the conditions listed) give Sk^I in terms of V , V^, and M^.. Also Equation 



[20] gives )q ^ .i in terms of "v and V^. 



Thus, substituting foriq .i and IKjI in the above equation will give the plate shears Iq.i in terms of 

 V , V... and M„. 



73 



