Art. 110. COLUMNS ECCENTRICALLY LOADED. 181 



If in equation (51) e is put equal to e, that is if the bending 

 is neglected, s max will be the same as for a block (Eq. 45a) and s c 

 the same as given by equation (47) ; but according to equation 

 (52) this is true only when sec 61=1 or 01=0, and according to 

 equation (49), 61 has this value only when 1=0. It follows that 

 a column may be treated as a block when it is short, the boundary 

 line between the two depending upon / and e (equations (50) and 

 (49) ). When y max is small compared with e, 2/ max is negligible. 



To show the influence of the bending, y m ^ is given below in 

 terms of e for a column composed of two steel channels 10 in.X^O 

 Ibs. X 200 inches long. Applying equation (50) for a column 

 fixed at one end; 



for P= 50,000 Ibs. 2/ max = 50000 



29000000 X 157.4 

 0.662 l)e= 0.269e 

 for P=100,000 Ibs. ?/ max =(sec. 0.936 l)c= 0.6880 

 for JP=150,000 Ibs. y m ^=(sec. 1.146 l)e= 1.430e 

 for P200,000 Ibs. 2/ max =(see. 1.324 l)e= 3.1050 

 for P==250,000 Ibs. ?/ mas =(sec. 1.478 l)c== 9.628e 

 for P=270,000 Ibs. i/ max =(sec. 1.540 l)e=30.882e 

 for P=281,000 Ibs. 2/ liia r=(sec. 1.570 l)e=oc 



It is to be noted that the' influence of the bending increases 

 at first slowly and then very rapidly, so that for the greater loads 

 e is negligible compared with ?/ max . In fact, 2/ max = a f r a l a d of 

 only 281,000 Ibs., or an average stress of 23,900 Ibs. per sq. in., 

 no matter how small the eccentricity may be, so long as it is not 

 zero. This must be interpreted to mean that the equilibrium 

 between the load and the induced stresses is not possible except 

 for loads less than this ; under greater loads the column fails. Of 

 coarse if e is large, the elastic limit may be exceeded for a less 

 load than 281,000 Ibs., and it should be remembered that the 

 equations are not applicable beyond the elastic limit. Thus if 

 e=% in. and P=250,000 Ibs., j/ max =- 9.628 X%=4.814 in. s max = 



=63,500 Ibs. per sq. in. Since this 



exceeds the elastic limit, the deflection would be much greater 

 than that given by the formula and the column would fail. 



The important case in which the eccentricity is very small, 

 for practically concentric loads will be considered in the next 

 article, 



