106 STRENGTH OF MATERIALS 



furnish reliable results when the dimensions of the column lie within 

 these limits. 



As a factor of safety to be used in applying the formula, Rankine 

 recommended 10 for timber, 4 for iron under dead load, and 5 for 

 iron under moving load. 



Problem 101. A solid, round, cast-iron column with flat ends is 16 ft. long and 

 6 in. in diametei. What load may be expected to cause rupture ? 



Problem 102. A square wooden post 12 ft. long is required to support a load 

 of 15 tons. With a factor of safety of 10, what must be the size of the post ? 



Problem 103. Two medium steel Cambria I-beams, No. B 26, weighing 26.26 

 lb./ft., are joined by lattice work to form a column 26 ft. long. How far apart 

 must the beams be placed, center to center, in order that the column shall be of 

 equal strength to resist buckling in either axial plane ? 



Problem 104. Four medium steel Cambria angles, No. A 101, 3 in. by 6 in. in 

 size, have their 3-in. legs riveted to a f-in. plate so as to form an I -shaped built 

 column. How wide must the plate be in order that the column shall be of equal 

 strength to resist buckling in either axial plane ? 



88. Johnson's parabolic formula. From the manner in which 

 equation (51) was obtained and afterward modified by the intro- 

 duction of the empirical constants / and g, it is clear that Rankine's 

 formula satisfies the requirements for very long or very short col- 

 umns, while for those of intermediate length it gives the average 

 values of experimental results. A simple formula which fulfills 

 these same requirements has been given by Professor J. R Johnson, 

 and is called Johnson's parabolic formula. 



If equation (52) is written 



and then y is written for p, and x f or - > Rankine's formula becomes 



For this cubic equation Johnson substituted the parabola 



y = B - ez, 



in which x and y have the same meaning as above, and 8 and e are 

 empirical constants. The constants 8 and are then so chosen that 



