y = 



d\ 











N- 



M. 



where the definitions of N^^ , N^2 ' ^tc, are obvious. Such a choice of j^and T uncouples 

 shear deformation from torsion deformation and is said to locate point s at the "shear center" 

 of the beam. The N matrix may be written:* 



' 1 1 



KA^^G KA G 



yy yz 



1 1 



KA G KA G 



yz zz 









 



1 



which yields Equations [2a, b, c]. The uncoupled form of this expression is validated by the 

 above development. The symbols are arbitrary but are chosen to be written in conventional 

 form. This expression (matrix) itself is a definition of the symbols A . ■ , or A , A , A 



' \ / .' ij ' y y ' yz' zz ' 



and J . 



e 



In the accompanying program, each of the above coefficients / \ appear as a 



l^KA^^cy 



single number. However, here they are written as products of several terms for comparison 



1 \ /I 



with the conventional shear and torsion flexibility coefficients 



KAG 



and 



GJ„ 



Equations [la, b, c] and [2a, b, c] are the six elastic equations for beam theory. 

 For ship problems. Equation [la] is usually not used. For motions symmetric with respect 

 to the x-y plane, use Equations [Ic] and [2a]. By rotating the y-z coordinates, it would be 

 possible to completely uncouple the equations (i.e., choose principal axes so that I =0), 

 but this has not been done here.** For symmetric sections typical of ship hulls, the axes 

 chosen are principal axes. 



*The N matrix may be written as shown because the left-hand side of the matrix equation above is related to 

 the shear and moment terms on the right side respectively, by constants which are called the shear and torsional 

 flexibilities having the form 1/KAG and 1/GJg , respectively. 



**The terms A., are defined by the above matrix. The program of this report is applicable to sections of any 

 structures which are prismatic and may be treated as beams. These structures may be symmetric or unsymmetric. 

 The sample problem chosen is symmetric (as are most ship hulls) and is, therefore, a special case of the general 

 theory presented. The following terms exist in general, but are zero in the special case (symmetric with respect 

 to the y-axis); I ,"z, 1/A , z. It is true that Figure 1 appears to have a symmetric outline, but it need not have. 



yz- 



yz' 



51 



