RESISTANCE OF MATERIALS 65 



unsupported end, while maintaining the same resistance to 

 bending, as if the cross section had been uniform throughout 

 the length of the beam. The resultant profile is parabolic. The 

 beams of old-fashioned steam engines have this contour. Such 

 designs are examples of the economical use of material. 



In the structure of the stalks of vegetables and the wings of 

 birds we similarly find that the substance of such members is 

 disposed to the best advantage for sustaining the various stresses 

 to which they are subjected. 



46. Torsion. A third method of deformation takes place when 

 a tangential effort, a' "couple "of forces, act on a cylinder and 

 make the layers of matter slide one upon another. The sliding 

 does not reach to the centre, which constitutes the neutral axis. 

 For an angle of torsion 6, a layer OO'AB will be brought to 

 OO'AB' (fig. 81) presuming the base OA to be 

 fixed, and the couple applied perpendicularly to 

 the axis, at O'B. The angle of torsion is pro- 

 portional to the twisting moment M and to the 

 length L of the axis, but inversely proportional 

 to the 4th power of the radius, and to a co- 

 efficient G, called the modulus of torsion or of 

 rigidity. Then 



(3) 

 8 



In the case of a hollow cylinder, of which the 



fw 8| radii are r l and r\ 



= 



Cylinder. 



It is evident that with equal quantities of material r L and r\ 

 have such values that : 



which increases the resistance to torsion of the cylinder considered. 



It is usual, in practice, to substitute the diameter d for the 

 radius r and to write 



d* , - 32ML 



y 4 = , whence 8 = 

 ID 



2ML 



Torsion of a ~ _.ri t v _ y ' 



^ 



p 



The expression is denoted by the symbol y, known as the 



oZi 



co-efficient of Coulomb, 



e = ^; 



yfl 4 

 whence the moment : 



M dt& 



M = -- 



