Art. 111. COLUMNS CONCENTRICALLY LOADED. 183 



tion (53) is, therefore, applicable under conditions approaching 

 very closely to the ideal, as it is independent of the eccentricity. 1 

 Equation (53) is based upon the maximum deflection and 

 not upon the maximum unit stress. It is much simpler than 

 equation (52) in which it would be impossible to assign a satis- 

 factory value of c for concentric loads. No doubt, when the col- 

 umn fails, the stress has reached the ultimate strength of the 

 material. It has been shown by many experiments on steel, 2 that 

 the yield point is the ultimate strength in columns, that is, the 

 stress -deformation diagram does not rise above the yield point 

 in steel. 3 Since the equations do not apply beyond the elastic 

 limit, or limit of proportionality, it will be on the side of safety 

 to call the elastic limit the ultimate strength. When equation 



p 

 (53) gives values of - greater than the elastic limit, (as it 



A 



will for small values of ) it is no longer applicable. Under this 



formula the column is treated as a block. Equation (53) may be 

 written in the form of equation (54) and must be used in con- 

 nection with equation (55). 



(54) 



(55) 



s e is the unit stress at the elastic limit. Applying a factor 



1 Prof. Wm. Cain has shown by a rigorous analysis th,at equation 

 (53) gives the load on the ideal column that is just sufficient to keep 

 the column deflected after it has been slightly bent by a transverse 

 force, .and he deduces a formula for the amount of the maximum de- 

 flection for loads greater than this critical load. This formula shows 

 that a load only a few pounds greater than the critical load will cause 

 a deflection so large that, practically the critical lo,ad is the ultimate 

 load. Slight variations from the ideal condition would produce the 

 same result. See Trans. Am. Soc. C. E., Vol. XXXIX, p t age 96. 



2See a paper by Chas. Marshall, Trans. Am. Soc. C. E., Vol. XVII, 

 page 53. 



3See also Johnson's Materials of Construction, page 363, for an 

 account of experiments by M. Consid^re showing that the strength of 

 a column is a function of the elastic limit of the material and is inde- 

 pendent of the ultimate strength in either tension or compression. 



