416 Mr. B. C. Nichols on the Resistances 



At any point in the beam between this and the centre, and 



at a distance x from the point of support, the maximum stress 



which would exist on the supposition of perfect elasticity will 



be reduced in the proportion of 2x to I. In the equation for 



I 1} therefore, to obtain the moment of inertia of the reduced 



2rx 

 section at that point, — j- must be substituted for r, and . 



I= L^3Z-2^ ; 



21 



and at this point, if a?, y be the ordinates of the curve of 

 deflexion, the origin being the point of support, 



. FT ^_ WZ 3 



dx 2 r(3l-2rx) 2 



l_ 

 2' 



Integrating between x and — , and noting that at the centre 



dx w d JL- W w 



<te ~ 2r 3 (3Z-2rvc) 2r 2 (3-^) ; 



and when x— jr-. 



EI tan a^- V 2 }l ! (13) 



Integrating again between the same limits, y being equal 

 to — h at the centre, 



t,t, *n WZ 3 3Z-2nr WPx L WZ 3 „,. 



and when # = o- 5 y=#i> 



4r 3 (3-r)/ 2r*(3-r) 4r 2 (3-r) 



2? 



Again, at any point between #=s~ an d tbe point of support, 

 the moment of inertia is unaffected by overstrain, and 



