92 



RESISTANCE OF MATERIALS 



Now suppose that the column is bent sideways by a lateral force, 

 and let P be the axial load which is just sufficient to cause the 

 column to retain this lateral deflection when the lateral force is 

 removed. Let OX and OF be the axes of X and Y respectively 

 (Fig. 79). Then it can be shown that the elastic curve OCX is 

 a sine curve. For simplicity, however, it will be assumed to be a 

 parabola. Since the deflection at any point C is the lever arm 

 of the load P, the moment at C is Py. The moment at any point 

 is therefore P times as great as the deflection at that point, and con- 

 sequently the moment diagram will also be a parabola (Fig. 80). 







-*-- ?/ 



P 



FIG. 79 



Moment 

 Diagram 



O 



4 



f 



X \ 



FIG. 80 



Now let d denote the maximum deflection, which in this case is 

 at the center. Then the maximum ordinate to the moment diagram 

 is Pd. Therefore, from article 17, the area of one half the diagram 



2 I Pdl 



is A = - (Pd) - = , and the distance of its centroid from one 

 o A o 



end is x =--= Hence, from the general deflection formula, 



o L lo 



the deflection at the center will be 



(116) 



, 1 1 /Pld\5l 



rf A I \ 



El EI\ 3 /16 48^7 



Canceling the common factor d and solving for P, the result is 



9.6JEJJ 



(117) 



P _ 48 EI 

 ~~ 



