program, it is accomplished by the "Gram-Schmidt Reduction" indicated in Figure 4.* This 

 is a standard technique in matrix algebra. Equations [25] and [32] of Appendix A. 2 show the 

 matrix derivation. 



For reasons discussed below, Column (?) is formed by summing entries from (is) , 

 Sheet 1 (top) as indicated in (g) , (|) , of Sheet 2. For example, for Plate 2: 



Column ® = - (nodes 3 + 4 + 5) = - (-0.6572 + 2.6709 + 3.5423) = - 5.5560. Similarly, 

 Column (^ is generated by the same combinations of nodes; however, in this case the shears 

 are taken from the entries in Column (16) , Sheet 1 (top). 



That Columns (15) , and (16) are proper expressions for q^^^ from each node for the 

 section sustaining y-shear and z-shear, respectively, is seen from the equations for Xq. (out) 

 in Appendix A.l, remembering that I ^= 0,"z = in the example. Columns (^ and (§) are 

 loops (Column (s) is associated with Plate 5, @ with plate 9 by random selection). They 

 could as easily have been reversed. The selection of entries in (5) and © is based on the 

 following statement in Appendix A.l, "For each plate which is not on the tree, there exists 

 a closed loop through that plate and others in the tree." Reference to Figure 1 shows that 

 the loop, including Plate 5 in the positive sense (but excluding Plate 9), includes Plates 2, 

 3, 4, 7, and 8 (all in the positive sense). Also, the loop involving Plate 9 in the positive 

 sense (but excluding Plate 5) includes Plates 2, 3, and 8 (all in the negative sense). The 

 entries in (E) and (o) reflect these statements.** See also Equations [13]-[15] for the 

 !q, ! and discussion of the L-, matrix in Appendix A. 2. Column (j) of Sheet 2, Table 4, 



"loop J ^^^ Jl ^'^ ^^ 



is plate Column (l4) of Sheet 1, Table 4. Next, solve for the factors Kj and K2, which are 

 the amount of shear flow in the loops. This is indicated as a matrix operation to the right on 

 the calculation sheet (not the same method used in the program, but equivalent). * The follow- 

 ing hand calculations and those used in the program are based on the set of Equations [32] of 

 Appendix A. 2. While these equations are for a general cross section, the equations of the 

 sample problem are for cross sections with only two or three loops. 



The y-shear calculation for the symmetric hull cross section involves the solution of 

 two simultaneous equations for Kj and K2 (see Appendix A. 2, Equation [32]). The numerical 

 values for the elements in the matrix to the left of the K. matrix in Equation [32] are obtained 



as follows (see Table 4, Sheet 2):tt (Text continued on page 39) 



*The digital computer program and Flow Charts (see Figures 4 and 5) are discussed in Appendix B. 

 **For z-shear the entries in Columns ^24) andr25J are identical to those in(5)and(6), respectively. Since the 

 force is antisymmetric (see Figure 6) a third loop consisting of Plates 6, 7, 8, and 1 must be considered. The 

 entry in (26j reflects the shear flow in this loop. 



^The solution of the matrix equation is performed differently in the sample problem and in the computer program 

 because the sample calculation inverts a 2-by-2 or a 3-by-3 matrix by hand with a desk calculator, and the com- 

 puter program permits inversion of an n-by-n matrix (where n may be any integer up to 30, the maximum number of 

 loops) by high-speed digital computers (Gram-Schmidt Reduction). The optimum method is naturally different in 

 the two instances. 



' 'The rationale underlying the y- and z-shear calculations are similar. For the latter, a detailed calculation is 

 given on page 39. 



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