98 



RESISTANCE OF MATERIALS 



For different materials and methods of end support Johnson's 

 parabolic formulas, obtained as above, are as follows : 



The limit for use in each case is the value of x ( = -) at the point 

 where Johnson's parabola becomes tangent to Euler's curve. For 

 greater values of - Euler's formula should therefore be used. 



Jj 



A graphical representation of the relation between Euler's for- 

 mula, Rankine's formula, J. B. Johnson's parabolic formula, and 

 T. H. Johnson's straight-line formula (considered in the next 

 article) is given in Fig. 83 for the case of a wrought-iron column 

 with hinged ends.t 



* In the formulas for timber t' is the least lateral dimension of the column. 

 t For a more extensive comparison of these formulas see Johnson's Framed Structures, 

 8th ed., 1905, pp. 159-171; also Trans. Amer. Soc. Civ. Eng., Vol. XV, pp. 518-536. 



