In Figure 18, (a) shows a cross section of the hull, having a length Ax in a direction 



parallel to the longitudinal axis of the hull; (b) shows only those plates which comprise the 



first loop (i = 1) of the section (refer also to Figure 13); and (c) shows the same plates as 



(b) but, for clarity in what follows, the structure has been unfolded, or developed, into the 



form of a plane sheet. In (c) the plates are undeformed by shear stresses. The same plates 



are shown in (d) but now they are deformed by shear stresses, (e) gives the equation for 



shear strain in a plate, making use of the definition of shear strain and the fact that R^A^^ 



in the hull is the equivalent of du^ in the definition. This is evident by inspection of the 



geometry of Figure 17 if we consider that a point on element ds on the cross section at x = Ax 



is twisted through an angle IS.0 relative to its original position on element ds (same position 



as that shown in Figure 17 for the section at x = 0). This point moves a distance R^^ A d^ 



du 



s 



during the deformation. The point is also displaced a distance yAx = — — Ax = du (y = 



dx ^ 



angle of shearing strain and dx = Ax). Hence Rjj ^9^ = du^. In (f) this expression for shear 

 strain is integrated around the loop£, resulting in the equivalent of Equation [21]. Equation 

 [21], which gives a relation between the integral of shear strain around a loop and the rate of 

 twist of the hull structure, is obtained by integrating r around any loop jf as follows (letting 



^^ dd ' 



X X 



take its limiting value, ): 



Ax ^ dx ' 



q ^^^ ^^^ r n 



rf Tds = rf — ds = G — tf R„ ds = 2A„ G [21] 



^l ^i t (9x 7 " X dx 



du^ 



The term in the definition of shear strain will exist only if cross sections of the 



ds 

 hull are permitted to warp out of their plane when the hull is deformed, as illustrated in (d) 



of Figure 18. 



In Equation [21] the expression for shear strain neglects warping of the section and 

 omits the term du /ds. However, the integration of the simpler expression for shear strain 

 gives the correct result, as shown in (f), because the integral of the term du /ds around the 

 loop equals u (end point) - u (start point), which must equal zero if the loop is closed 

 because the end point is the start point. Therefore, Equation [21] is valid whether or not 

 warping is permitted. 



Equation [21] shows another simplification compared with (f) of Figure 18 in that the 

 shear modulus G is a constant outside the integral. The reason is that here the analysis is 

 based on the simplification that all shear elements are composed of the same material. The 

 last step in Equation [21] recognizes that (j, R^ ds equals twice the area of the loop around 



71 



