Flexure of long Pillars under their own Weight. 

 d z y 



429 



Integrating the equation -tH — 



dh 



t m series, we get 



,2^.3 



where 



TT _ r mxZ mx> m ° x * 



X \ 27371 + 2.3.5.6.7""2.a.5.6.8.y.l0 + *- 



V is another series, having x 2 as a factor, and A and B are 

 arbitrary constants. 



Calling the first derived function, with respect to x, of U, 

 U 7 and so on, the condition of a pillar free at top, and fixed 

 initially vertically to a rigid base is expressed by 



and 



% =AU' + BV =0 when x=l, i. e. at foot, 

 dx 



^jL = AU" + BV" = when x=0, i. e. at top, 



since there is no bending-moment at top. 



As V contains x 2 as a factor, the second of these gives B = 0, 

 and the first then requires U^O when x=l. It will be 

 found, on inspecting the curves plotted in fig. 2, that a value 



Fig. 2. 



TTr ^U 



dy 



— mx -£ 



ax 



+ 1-0 























+0*9 



+0-8 



+07 

 +0-6 



+ 0'5 



















































































+ 0*4 























+ 0'3 

 + 0-2 



































1 









+0'1 











































m= 







' HE 



B£ 









SHE 





r^iw* 



-01 























-0*2 























— 0*3 

 -0'4 











































— 0'5 



— 0*6 























-07 



—0*8 











































-0-9 

 -1*0 











































P/hV. Me<7. S. 5. Vol. 33. No. 204. J% 1892. 2 G 



