farther from the tree root than panel j. The sign of this summation is plus (+) if the positive 

 sense of panel j is toward the root, and minus (-) if the positive sense is away from the root. 

 The above relations and definitions may be summarized in matrix form by the following 

 equations. In these equations the vector [q .} represents shears in the plates due to 

 ^^out i' ^^ ''^® nodes, which is, in turn due to V^ and V^. Also the vector Iqi^^ i represents 

 plate shears due to all the loop shears {K^S. In Figures 13, 14, and 15 the light arrows 

 associated with the plate numbers represent plate shears. The heavy arrows on the loops 

 (heavy lines) represent loop shears. 



Ij 



part j 

 "loop j 



I "loop j I 



[iij 



[12] 

 [13] 



The formation of the [T.-land [L Imatrices, which describe the tree and the loops, 

 respectively, is illustrated for a simple example in Figures 11 to 15. Figure 11 shows a 

 section composed of nodes numbered 1 to 7 and plates numbered 1 to 9. An arrow indicates 

 the polarity of each plate. Figure 12 shows how a tree has been selected by omitting plates 

 6, 8, and 9. Node 1 is chosen as the root of the tree. The following equation gives the 

 matrix [T. J, which describes the tree (also see page 11): 



r- '^part P 





'^out 



1 -^ 









-1 



-1 -1 - 



-1 



-1 



-1 



^out 1 



^part 2 





^out 



2 













-1 -1 - 



-1 



~1 



-1 



^out 2 



^part 3 





^out 



3 













-1 - 



-1 



-1 



-1 



^out 3 



^part 4 





qout 



4 













- 



-1 



-1 







lout 4 



^part 5 



= T.. 



^out 



5 



- 



















-1 







lout 5 



■^part 6 





^out 



6 





























■lout 6 



qpart 7 





^^out 



7 J 

























+ 1 



'Jout 7 



^part 8 

































V- Ipart 9> 



(q t 1 



I ^part j 



\ 



{^out 



'1 































y [14] 



Figures 13, 14, and 15 illustrate loops 1, 2, and 3, formed by adding to the tree plates 

 6, 8, and 9, respectively. In each case, a polarity of the loop has been chosen and indicated. 

 The following equation shows how the matrix [L. J is formed to relate the plate shear flows 

 iq^^^ . ! to the (as yet unknown) loop shear flows IK. I (see page 13 also):* 



*In Equation [isj note that the loop including Plate 6 in the positive sense includes plates 1 through 6. Hence 

 Lj., Ljj L,., L.., Lcj, Lcj are denoted by +1, whereas L_,, L„,, and Lg, are designated zero. Similarly 

 for elements in Columns 2 and 3. 



65 



