vary greatly for sections ordinarily used which have an axis of symmetry 

 perpendicular to the plane of bending. 

 Setting 



T = E/E (77) 



Equation (75) can be modified so that a general column-buckling formula 

 applicable in both the elastic and inelastic ranges can be written, i.e., 



^^E 

 Thus, to determine the critical buckling stress from Equation (78), it is 



necessary to knowT, which, in turn, requires a knowledge of the shape of 



the stress-strain. curve. We will now consider a method of determining T. 



37 

 Osgood has developed the following general column-buckling formula: 



l+j/2 . 



Different values of the parameter j give rise to parabolas of different 

 character. The case j ■ 2 represents a column formula suitable for 

 ductile materials like mild steel. Substituting j = 2 in Equation (79) leads 

 to the well-known Johnson parabola, i.e., 



S = 1 - T \?, (80) 



or 4 E 



Since the Johnson parabola is an empirical curve which "fits" well the 

 experimental data in the inelastic -buckling range for columns of mild 

 steel, it affords a means of finding T analytically without knowing the 

 actual stress-strain curve. Solving Equations (78) and (80) yields the 



59 



