182 



COLUMNS CONCENTRICALLY LOADED. 



Art. 111. 



111. Columns Concentrically Loaded. It is practically 

 impossible to apply loads to either tensile or compressive members 

 without a slight eccentricity ; this causes the member to bend. In 

 a tensile member the load has a tendency to reduce this bending, 

 but in a c< impressive member the tendency is just the opposite, 

 and this is of great importance as is shown by experiments, and 

 as will also appear from a discussion of the equations in the pre- 

 ceding article. 



An ideal column is one having a perfectly straight axis, and 

 the material of which is homogenious and in the same condition 

 throughout. If such a column be loaded concentrically it will 

 not bend. The practical column will bend under any load until 

 the load and the stresses are in equilibrium or until the column 

 fails. 



If the load is not great enough to cause failure, the column 

 is in stable equilibrium, for if its deflection be changed by apply- 

 ing a small transverse force, and this force is then removed, it 

 will return to its original deflected position. If the load be ap- 

 plied gradually, a point will be reached at which failure takes 

 place suddenly. The load under which stable equilibrium just 

 ceases must, therefore, be the ultimate load which the column 

 will carry. This critical load is easily found from equations (49) 

 and (50), for it is the load wliich will make i/ max =oc, which is 

 true when 9ly 2 Tr. As shown by the example in Art. 110, this 

 is quite possible in an ordinary column. From equation (49), 

 which applies to a column fixed at one end and free at the other, 



p 



-* mnT 



AE 



from which 



(53) 



This then is the load that will produce failure when the 

 eccentricity is so small that there is no danger that the elastic 

 limit will be exceeded for a less load, as was illustrated by the 

 example in Art. 110. It should be remembered that initial stresses 

 and bends, and non-homogenity of the material in a column, hav<> 

 an influence quite similar to eccentric loading; when these all 

 happen to have a tendency to produce bending in one direction. 

 they form the most unfavorable case, which must always be as- 

 sumed. It is assumed, therefore, that care has been taken to 

 have the column straight and the eccentricity very small. Equa- 



