[ 476 ] 



LIII. On the Deformation of Twisted Strips. 

 By G. H. Bryan*. 



IN the March number of the Philosophical Magazine (p. 244) 

 Professor Perry has described the behaviour of a twisted 

 strip of metal when its ends are pulled apart, and has investi- 

 gated formulae to determine the untwisting produced. It 

 seemed, however, that Prof. Perry's theory hardly afforded a 

 satisfactory explanation of the phenomena, and the idea 

 occurred to me that a better clue to the correct solution 

 might be obtained by considering the problem as a particular 

 case of the deformation of a thin plate. I have pursued this 

 method in the present paper, and the results obtained will, I 

 think, be found to agree very well with those found by expe- 

 riment and described in Prof. Perry's paper. 



It will be convenient to denote the breadth of the strip by 

 2a and its thickness by 2h, instead of using b and t respect- 

 ively. We suppose h to be small in comparison with a, so 

 that the strip can be treated as a thin plate. Let the strip, 

 originally supposed plane, be twisted about its middle line 

 into a helicoid, the twist per unit length being <£. If we 

 consider a fibre of the material whose distance from the axis 

 is x, its elongation per unit length will be 



v/(l + .?; 2 c/> 2 )-l. 



In the helicoid the lines of principal curvature will every- 

 where cut the generating lines at an angle of 45°. If % is the 

 inclination to the axis of the tangent plane at the point a, 

 we have 



tan ;>£=#<£, 



. dx_ <t> 

 " dx 1 + x 2 $ 2 ' 



whence we readily find for the principal curvatures, 



:l l _ j> 



pi" Pz "~ 1 + x 2 $ 2 



We shall now make the assumption that the twist is suffi- 

 ciently small to permit of our neglecting a 2 <f) 2 , and therefore 

 also x 2 <j> 2 , in comparison with unity. Now aij) is the whole 

 angle of twist in a length of the strip equal to half its breadth, 

 and this will certainly be small in all cases of practical interest; 

 moreover the results which we shall obtain will fully justify 

 our assumption. To this order of approximation the elongation 



* Communicated by the Author. 



