COLUMNS AND STRUTS 99 



Assume also that the ends of the column are free to turn about their 

 centers of gravity, as would be the case, for example, in a column 

 with round or pivoted ends. 



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 col- 

 umn to retain this lateral deflection when the lateral force is removed. 

 Let OX and OF be the axes of Xand Y respectively (Fig. 78). Then 

 if y denotes the deflection of a point C at a distance x from 0, the 

 moment at C is M = Py. Therefore the differential equation of the 

 elastic curve assumed by the center line of the column is 



which may be written 



dy * 

 To integrate this differential equation, multiply by 2 -^ Then 



Vydy 2P dy = 

 ^ Ei y dx 



and integrating each term, 



dy\> Py> 



where C^ is a constant of integration. This equation can now be written 



Integrating again, 



ima \EI X + CV 



p 



where (7 a is also a constant of integration ; whence 



\EIC, I \T 



* See Elements of the Differential and Integral Calculus, pp. 438, 444, by W. A. Gran- 

 Tille, Ph.D., with the editorial cooperation of Percey F. Smith, Ph.D. Ginn & Company, 

 1904. 



