122 



DYNAMICAL THEORY OF SOUND 



J 



J 



The value of 23 is found as follows. We take rectangular 

 axes Gy, Gz in the plane of a 

 section, the origin being at the 

 centre (i.e. the centre of gravity 

 of the area), and the axis of z 

 normal to the plane of flexure. 

 Assuming the axis of the bar, 

 i.e. the line through the centres 

 of the sections, to be unex- 

 tended, we see that if R denote 

 the radius of the circle into 

 which it is bent, the length 

 of a longitudinal linear element 

 whose distance from the plane 

 xz is y is altered in the ratio 

 of R + y to R, and that the 

 extension is accordingly y/R. 

 The corresponding stress per 

 unit area of the section is JEy/R, 

 where E is the appropriate Young's modulus. The total 

 longitudinal tension is therefore 



E 





Fig. 43. 



This justifies the provisional assumption that the axis (as 

 above defined) is on the present hypothesis unextended. For 

 the bending moment we have, taking moments about Gz, 



E 



Except in the special case just considered, viz. that of a bar 

 bent statically into an arc of uniform curvature, there will be 

 a shearing of cross-sections relative to one another, and also 

 a warping of the sections so that these do not remain accurately 

 plane. An exact investigation is out of the question, but 

 enough is understood of the matter to warrant the statement 

 that the additional sfam'rcs thus introduced are as a rule small 

 compared with those taken account of in the preceding calcula- 

 tion. We therefore adopt the formula (5) as sufficiently 



