which the integration is performed.* Note that this latter equality is independent of the loca- 

 tion of H, the point from which R^ is measured. Thus, although Equation [21] is derived 

 based on rotation about H, it may be rewritten (for convenience in calculating), with R^ 

 replaced by R, the distance from the origin to ds, measured perpendicular to the direction 

 of ds. Thus 



rf Ids =(^ Rds)G ^ =2A G — - [22] 



For the idealized structure of this report, the loops are comprised of a finite number of 

 plates, each of constant thickness t^, so that the integrals may be replaced by summations:** 



dd. 



Herei^S indicates a summation of only those plates j which comprise loop£, with signs alter- 

 ed, when necessary, to conform to the polarity of loop/; see page 56. This operation can be 

 indicated by premultiplying the terms to be summed by the appropriate column of the [Ljf 

 matrix: 



V As ^^K ^^. 



^ J L. q. =(2 L..,R. As.) G = 2AG [24] 



j t. Jl^J ^ j Jt J j' dx I dx 



Now this summation is carried out over all plates j. For convenience in calculating, we can 

 replace R. As. by y^. Zj^. - yj^.z^., where yj^., z^. locate the head end of plate j, and y^, , z^. 

 locate the tail end.f Also substituting for q. from Equation [II] and [13] yields: 



i t, 



Lj2(qpart j+^L.jK^)= [fL,j^(ytjZhj-yhj^j)] G 



dd 



X 



dx 



*The loop J^ about which we integrate to get Equation [2lJ is any one of the L loops,/ = 1,2 , L. For 



the example of Figure 17 in which L = 3, the integration would be done around loop 1 (Figure 13), then around 

 loop 2 (Figure 14), and finally around loop 3 (Figure 15); A^is then the cross-sectional area enclosed by the 

 loop around which the integration is performed, and it is that area which is indicated in Figures 13, 14, or 15, 

 depending on whether /= 1, 2, or 3. 



**The q of Equation [22] and the q. of Equation [23] and Equation [24J are the total plate shears q. of Equation 

 [ll]. In the steps from Equation [23l to Equation [25], the elements of the loop matrix Ljrtare introduced twice, 

 once to ensure that the plate shears q. are correctly computed in terms of the loop shears Kji and once to ensure 

 that the summation of (As./t.)q. overall plates j is restricted to those plates comprising the loop x. over which 

 the summation (or integration) is to be performed. 



I Twice the area of the triangle formed by the vector As; and two position vectors A and B from the origin to the 



head and tail ends of the plate, respectively, are \r. x As . | = |a x b] = |(iz. . + jy^) x (iz^. + j y^.)! or 



R . As ■ = y , ■ z. . — y. . • z,. ■ . 

 J J 'tj hj ^hj tj 



72 



