COLUMNS AND STRUCTURES 255 



With the both formulas it is necessary to select a column sec- 

 tion and obtain the values of e, r and n. There is little difference 

 in the values of r and n between columns of nearly the same size, 

 so one computation for the value of W e will generally be sufficient. 

 An experienced designer can usually select a trial size so nearly 

 right that but one approximation will be necessary. 



Having obtained W it is necessary to add to it the direct con- 

 centric load W and the eccentric load P; thus W e WP = Wt the 

 total equivalent concentric load, and the uniform compressive 



fiber stress becomes / = -p 



When something better than a very close approximation (good 

 enough for ninety-five per cent of columns) is wanted, the follow- 

 ing formula by Professor J. B. Johnson may be used. 



/ Mn 



WE 



in which /6 = unit flexural stress in pounds per square inch, 

 L = length of the piece in inches, 

 E = modulus of elasticity of the material. 



The Johnson formula is used for beams subjected to bending 

 as well as to direct compression or tension; and, also, to struts 

 and ties eccentrically loaded in addition to having a concentric 

 load to carry. The unit flexural stress must be added, algebrai- 

 cally, to the direct stress due to the concentric load, and the sum 

 of the two cannot exceed the safe fiber stress of the column as 

 determined by a column stress reduction formula. 



For columns, ties and struts, the load P acts parallel to the 

 piece and M - Pe. For beams the moment M is the bending 

 moment caused by the dead load of the beam plus whatever ad- 

 ditional transverse load there may be on it. Therefore, for beam 

 subjected to direct tension or compression in addition to cross 



W 



bending the P is really W, as used in the expression -p and is 



not an eccentric load, but is the direct concentric tension or 

 compression. 



The beams carried by wooden columns and steel columns rest 

 on brackets attached to the columns. There can, therefore, be 

 no uncertainty as to the amount of eccentricity, e. Concrete 

 columns are cast integrally with beams and slabs, so considerable 



