Closed and Open Tubes. 539 



A being the area enclosed by the central line of the section. 

 For the split circle we get 



10=— 4irrf* (64) 



Equation (1)3) could be applied to any split tube of 

 variable or constant thickness. It' it is applied to a section 

 whose open ends do not come close together, then A is the 

 area enclosed between the curve of the central line of the 

 section and tin 4 radii drawn from the axis of twist to the two 

 ends of that line. For example, if it were applied to a 

 section with a straight central line which is intersected by 

 the axis of twist the area A would be zero, and therefore w 

 would be zero. 



The difference in the behaviour of a split and a complete 

 tube is due to the freedom of the open ends of the section of 

 the split tube to move axially relatively to each other. If this 

 freedom is constrained in any way the split tube does not 

 differ from the complete tube. If, for instance, a short split 

 tube is gripped at the ends so that no relative motion of the 

 particles in the end planes is possible, then the tube behaves 

 almost exactly like a closed tube. The rules for the split 

 tube apply to a tube twisted in such a way that the end 

 sections have perfect axial freedom, or to sections in a long- 

 tube at great distances from the ends. In actual practice a 

 split tube welded into brackets at its ends would behave very 

 nearly like a complete tube because the axial displacement 

 represented by (63) would be so great that the effect of 

 preventing all this displacement at the ends would be 

 considerable even at the middle of the rod. 



The rules that we have given for split tubes will apply 

 with fair accuracy to any thin portion of a section of a 

 twisted rod where the shear lines must all go forward and 

 return across the same normal. For example, the I-section 

 shown in fig. 8 may be treated as if the two end-pieces and 



Fig. 8. 



the backbone were put end to end to form a continuous 

 strip with open ends. That is, if the two ends be supposed 

 to be cut off along the dotted lines, then the torque in anv 



2 N 2 



