138.30 =1 ® -(5)2 



- 97.80 = £® •© v® 

 132.88 = I ® • ®2 



and the numerical value of the elements in the right-hand matrix of Equation [32] are 



132.55 = S® •© • ® 



- 94.68 = S ® •© •© 

 obtained by matrix inversion and multiplication, as indicated 







r- 0.9479"! 



■i }■ - The solution 



\+0.0148j 



in Sheet 2 of Table 4. 



In accordance with the statement following Equation [32], we substitute the Kj's in 

 Equations [11], [12], and [13] of Appendix A. 2. This requires the use of Equations [14] and 

 [15]. For the present problem, the procedure is then as follows: 



Multiply Column @ by 1.0 (see footnote on page 12), Column ® by (Kj) = (-0.9479) 

 and Column ® by (K2) - 0.0148, and add to find Column ® , which is the shear flow distri- 

 bution due to a unit y-shear.* The YY Flexibility is calculated at the right of Sheet 2 (see 

 Equations [46J— [48] of Appendix A. 2 and pages 84 and 85.) The z-shear center is obviously 

 zero. In general, it is calculated by Equation [36] of Appendix A. 2. 



The z-shear calculation is almost the same, but now there is a third loop for anti- 

 symmetric forces, which is from nodes 1 to 6 (along centerline) and return via tree (see 

 Column (26) on Sheet 2 and footnote on page 50). 



Figure 6 and Sheet 2, Table 4 show how the sample hull cross section, which is 

 symmetric and has five compartments, is treated with two loops for symmetric loading (y-=hear), 

 and with three loops for antisymmetric loading (z-shear). It is evident that the third loop 

 cannot carry shear symmetrically. Column (26) defines the third loop. The z-shear solution 

 now involves the solution of three simultaneous equations for Kj, K,, and K, as follows 

 (See Appendix A. 2, Equation [32]). 



138.30 - 97.80 67.12 



-97.80 132.88 -52.80 



67.12 - 52.80 99.64 



I@(g)2 I(g)(g)(25) I(27)(24)(26) 



S@(24)(g) S(27)(g)2 1 @ @ @ 



1 @ @ @ S(27)(25)(26) 2(g)(26)2_ 



*The operation performed here is (1) = ® + ©Kj I-© Kj, which is equivalent to Iqj! = ^%an jS + [Ljj] IKjI 

 (see Appendix A. 2). Here, Column ® is iq g^f :! by the indicated operation Upa^t j^ ^ ['^ji^ ^''out i^ ^"'^ 

 Columns ® and ® constitute [L-,]. Since this is done with V^ = M^ = and V = 1, the values assumed by 

 Iq-i are those of Q„ . 



39 



