Then 

 and 



y= _2xll-54 ? ^ 1 .3 Ezrir 



Knl 



7" 



Buckling of Deep Beams. 209 



. . (120) 

 41-30- |3^£^™C 



— {-mfVI}- • ™ 



Consequently, W being the total load, 

 WZ 2 =m 3 Z 8 v/4ErcCK 



I 2 v/41-3 2 V Knj 



= 12-85 VlSnCK- 23-1 |EG (122) 



Thus the correction to WP due to the distribution of the 

 load along a line at height q above the centre line of the 



beam instead of along the centre line itself is 23TyEC. 



If the load were below the centre by an amount q the term 

 involving q would be added instead of subtracted, the beam 

 being in that case more stable than with the same load on 

 the central line. 



The problem of the stability of a beam fixed at one end 

 -and free at the other was worked out in the first paper for 

 the following two cases : firstly, with a load P at the free 

 end and no other load ; secondly, with a uniform load per 

 foot and no load on the end. Now we will try to combine 

 these two loads. 



Case 12. — Beam fixed at one end and free at the other, 

 and carrying a load P at the free end and a small uniformly 

 distributed load in addition. To find the condition for 

 instability. 



It is assumed that the load on the end is much greater 

 than the uniformly distributed load. 



Let w be the uniform load per unit length. Then the 

 bending moment G at distance x from the free end is 



G = P t ? + ^ 2 (123) 



.1 IV 2 .)' 2 



Therefore, neglecting - ___ compared with unit}*, 



G 2 ^PV + Pw 3 (124) 



Phil. Mao. S. 6. Vol. 39. No. 230. Feb. 1920. P 



