CHAP. I.] SUPPLEMENT TO CHAP. XIV. 277 



shown in the Fig., as A O B D perpendicular to the axis of the piece m n. 

 Let these sections be infinitely near ; then, the distance & a upon the axis is 

 d s. Let d v be the length of any fibre, as d e, before the change of form. 

 Then, after deformation, its length is = d * v + A d v . But if d <f> is the 

 small angle between the normals, dsv = ds + vd(f>, where t> is the distance 

 a c of any fibre from the centre of gravity of the cross-section. After 

 deformation, d s becomes d s + A d s, and d < becomes d tf> + &d<f>, and 

 d *y becomes d s v + A d s v . Hence the length of any fibre after deforma- 

 tion is <Z*T 



Subtracting this from the eq. for d v above, we have 



A <Z s v = Arf + A<Z0. 

 Therefore the ratio of the change of length to the original length of fibre 



A d y _ AaEa + t>A<?<ft 

 dty da + v d(fr 



If r is the radius of curvature, then rd(f> = ds, -^ = ; hence 



d s T 



_ _ 



da? \_ ds ds J r + t 



G 



From eq. (1) we have the strain on a fibre P = -r. 





From eq. (4), Q = E A .. Hence P = B . In the present case 



is given by (6) ; therefore 



(7) 



Since now from (1) Q = / P d a, we have from (7) 



Ads da &d<> *v d a 



Since from (2) M = / P v d a, we have again from (7) 



M_ A <Z * fvda Ad<f> /V d a 

 B ~ T ~d7J T+v + r ~d7J r~+v' 



But / is, when t> is very small compared to r, equal to 



I v* da, which is the moment of inertia of the cross-section X. Also, 



/da /*, i*vda 

 = Ida- I , 

 r + v J J r + v 



