Figure 3 — Coordinate Systems 



U , U . and U are displacements: 6„, , and are rotations. The forces V^, V„, and V 

 X y z ^ ' X y z x y ^ 



and moments M , M , and M act upon the section shown 



are effective, and that the sign of the summation is negative since the positive sense of 

 Plate 2 is away from the root. Hence qp^.^cuiar = ^part = (") ^ ^out = "'iout 3 "lout 4 

 -q ^.* . means Q ,. , , or a particular solution of the shear flows out of the 



Tout 5 ^part ^particular' '^ 



nodes, as discussed in .Appendix A. 2. However, since Qpa^t ^® based on a tree which omits 

 several plates or paths of flow (2 or 3 in the example) it is not completely general. Additional 

 shear patterns (one for each plate omitted in the tree) are superposed. These are the Qjoq 

 terms. The amount of shear flow in each loop to be added to the particular solution is unknown 

 a priori, and is indicated by the coefficients Kj, K2, or K^, Kj, K3 (2 if symmetric and 3 if 

 antisymmetric). For the method of solving for the K's, the matrix operations on Sheets 2 and 

 3 of Table 4 illustrate this for the sample problem and it is further discussed below. In 

 general, the method of solution is outlined in section Shear and Torsion in Appendix A. 2. 



The solution of simultaneous equations for K^, K,, etc., is by matrix inversion and 

 multiplication in the sample calculation, as indicated on Sheets 2 and 3 of Table 4. In the 

 sample calculation, as indicated on Sheets 2 and 3 of Table 4- In the digital computer 



♦Columns (2) andfa) include the sign associated with each value of <i^^^^ j and, therefore, represent LT-- J 

 [q , ■]■ Hence a separate computation for [T-] is unnecessary. This is the reason Column (?) is multiplied by 

 a factor of 1. It is possible, of course, to treat q . (without regard to sign) and T-^^ separately as on page 65 



(see Equation ImJ). This is less convenient for calculation. 



12 



