24 BOVEY UPON THE FLEXUEE OF COLUMNS. 



F {i-i, (f>) being an elliptic integral oîihejirst kind. Then 



P' Tt^ 



and F O-i, <!') = f = I (J>) 



El- P 

 2 



Let the actual thrust on the strut be 



P^ri'P' (G) 



w' being a coefficient > unity. 



The corresponding value of the modulus is given by 



Fin, <l>)=^-\^^^ = ^na = n-^ (7) 



By reference to Legendre's Tables it is found that a large increase in the value of /<, 



n 



i.e., sin -^ or 6^,, is necessary in order to produce even a small increase in the value of 



p 



F (//, 6) and therefore of re- ( = p,). Hence, as soon as the thrust P exceeds the least thrust 



which will bend the column, viz., P', B„ rapidly increases. 



The total max. intensity of stress in the skin of the strut at the most deflected point 



P M P P Y 



=/+¥«>4°>|/^ (8) 



Z being the distance of the skin from the neutral axis and / being equal to —• 



A. 



The last term of this equation includes the product / E, which is very large, and 



n 



also the factor sin -^■, which increases with B„, so that the ultimate strength of the material 



is rapidly approached, and, in fact, rupture usually occurs before the column has assumed 

 the position of equilibrium defined by the end slope w„. If there were no limit to the 

 flexure the column would take up its position of equilibrium only after a number of oscil- 

 lations about this position, and the max. stress in the material would necessarily be greater 

 than that given by Eq. 8. 



, . 1 (1—2 ^J? sin 2^i) d(l) 



Again, dx = as cos ^ = — - . -^rr- 



a ^/l — /(■' sin ■'^ 



Let X be the vertical distance between the strut ends. Then, 



^ 2\ 1-2/^' sin V _2J \ I d,/> 



~ o"^ £0 \/ 1 — /^^ sin V a[j E' \/ 1 — /<^ sin -(i- d'l> ~) E« s/'^ — h' «in ''<!>) 



E (/<, 0) being an elliptic integral of the second kind. Hence, the diminution in the length 

 of the strut 



