Figure 17 - Torsion in Two Sections of a Prismatic Structure 

 Now we can write L equations, introducing the unknown rate of twist 



d\ 



,by 



integrating around each of the loops an equation relating geometry to shear strain as shown 

 in Figure 17. This figure shows two sections of a prismatic structure undergoing torsion, the 

 section at x = (dotted) and the section at x = Ax (solid). The solid section exhibits rota- 

 tion of magnitude A^ with respect to the dotted section. At some point H, not necessarily 

 known, there is no relative translation between the two sections. As shown by the following 

 detailed development, a segment of plate (shown), ds wide by Ax long by t thick, has a shear 



. , ^^x q ^^x 



strain of R„ — a shear stress* of T- — = R„ G , where R„ is the distance from 



" Ax t " Ax " 



H to ds, measured perpendicular to the direction of ds. 



Figure 18 shows the shear strain in the plates comprising a loop. The definition of 

 shear strain in the plates is conventional; assuming the shearing forces are applied in the 

 x-direction (longitudinal) and the s-direction (circumferential), shear strain is 



du.. du 



ds 



dx 



Thus, if lines are inscribed on the undeformed plate which are parallel to the x- and s-axes, 

 respectively, they form a right angle; after the plate undergoes shear deformation, the differ- 

 ence between the angle of intersection of these two lines and 90 deg is the shear strain. 



*See pages 53 and 54. 



69 



