t 428 ] 



XL VII. Flexure of long Pillars under their own Weight. 

 By Maurice F. Fitz Gerald*. 



THE origin is taken at the upper end of the neutral axis, 

 abscissae being reckoned, positive vertically downwards, 

 and ordinates horizontal. The flexure is supposed small, and 

 assumed to lie in a vertical plane. The symbols employed 

 are as follows : — 



H = total height of pillar ; 

 h = height below top of any point in it ; 

 S = total shear on a section normal to neutral axis ; 

 M = bending- moment ; 

 w = weight of pillar per unit of length ; 

 E and I, as usual, stand for the coefficient of flexural elas 

 ticity, and moment of inertia of cross section, respectively. 



Taking a plane, AB (fig. 1), 

 normal to the neutral axis, the 

 shear on this plane is the com- 

 ponent along it of the weight 

 of the upper part of the pillar 

 (whose top is supposed free); 

 for small bending we have there- 

 fore 



^ — wh~i nearly. 



Fig. 1. 



■■^rnkdm^^^zm^ 



By well-known theorems, -rr = S and M= — EI-j^, which 



give by substitution. 



ei S—4- 



By writing fj = x and m— ^r, this takes the form 



H 



EI 



d 3 y dy 



_4 = — ra#-r-> 

 doc 6 dx 



in which, putting -~ =u. we get 

 d 2 u 



dx' 



= — nixu, 



a differential equation which enters into other questions. 



The value of x( = w) runs from at top to 1 at foot of 



pillar ; m has, except for pieces ot fine wire a few feet in 

 length, or for very unusually tall and large columns, only a 

 small fractional value in practice. 

 * Communicated by the Physical Society : read February 26, 1892. 



