430 Flexure of long Pillars under their own Weight. 



of m = 7*85 nearly is that required, For dimensions in feet, 

 and for steel in which E = about 12,000 tons per square inch, 

 this gives, on putting in the numerical values, and putting 

 L = ratio of length to diameter, 



8*1 xlO 6 

 H (in feet) = ^ — for steel tubes, 



H being here independent of the thickness, supposed small ; 



and 



4 x 10 6 

 H = — r^ — for round steel rods, 



as the limiting height of pillar which can stand without bend- 

 ing under its own weight. Thus for L— 100, the maximum 

 height is about 800 feet, giving a tube 8 feet diameter. 

 For wires, L may be much greater; for instance, the limit at 

 which bending due to its own weight, of wire originally 

 straight and vertical, size No. 28 B.W.Gr., must occur is 

 about 1*8 feet. 



All columns, in practice, naturally fall far within the limits 

 here given. In connexion, however, with the inherent flexi- 

 bility of very large masses under their own weight, even when 

 direct crushing is prevented (say by external fluid pressure), 

 it may be remarked that for L = 4, H = 47 miles, approxi- 

 mately ; so that a solid steel column 12 miles diameter would 

 bend, even if prevented from bulging, if it were 50 miles high. 



The only case of interest, besides that of a column fixed at 

 its base and free at the top, above treated, seems to be that of 

 a heavy upright column, held at top and bottom by external 

 bending-moments so that the neutral axis is vertical at both 

 ends, but otherwise free. 



In this case, denoting by suffixes the values at each end, 

 we have 



AU ' + BV ' = 0, AU/ + BV/ = 0, 



AU " + BV " = Mi IP, AW + BV/' = M 2 H*. 



V(/ and U // are both zero identically; U '=l, and V / ' = 2, 

 which give A=0 ; 2B = M 1 H 2 ; and, on substitution in the 

 second and last of the above equations, we get 



BV/ = and M 1 V 1 " = 2M 2 , 



where, in V/', the value of m which makes V/ = is to be 

 inserted. The result shows that there is, in this case, a 

 definite ratio between the external bending-moments. 



Precisely similar results, as to producing bending, would 

 take place in a bar accelerated by a force applied at its back 



