COLUMNS AND STRUTS 107 



the vertex of this parabola is at the elastic limit of the material 

 on the axis of loads (or Y-axis), and the parabola is also tangent 

 to Euler's curve. In this way the formula is made to satisfy the 

 theoretical requirements for very long or very short columns, and 

 for those of intermediate length it is found to agree closely with 

 experiment. 



For different materials and methods of end support Johnson's 

 parabolic formulas, obtained as above, are as follows : 



KIND OF COLUMN 



FORMULA 



LIMIT FOB USE 



Mild steel 



Pin ends 



Flat ends 



Wrought iron 

 Pin ends 



Flat ends 

 Cast iron 



Round ends 



Flat ends 

 Timber (flat ends) 

 White pine 



Short-leaf yellow pine 

 Long-leaf yellow pine 



Whir.- .,;ik 



- = 42,000 -.97 1- 

 = 42,000 - .62 (L 



= 84,000-. 48 



- 2,600- .6 



P /Z\ 2 



-=4,000-.8(-) 



f = 3,600- .8@' 



^190 



^170 

 ^210 



^ 70 

 ^120 



* 60 



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

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

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



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



