184 



END CONDITIONS. 



Art. 112. 



of safety to s c , the working unit stress is gotten. The same factor 

 of safety is usually used for all values of . It is difficult to 



assign a rational value. 



Equations (54) and (55) give the same value of s c when 



e = I-T^E-^-, or when 



(56) 



I 



That is, equation (56) gives the value of at which equation 



(54) begins to be applicable and equation (55) ceases to be ap- 

 plicable. 



It may be noted that the stress due to direct compression is 



\ ?'i 



negligible when y 



in equation (51) is very much greater than 



unity ; this is the case when ^ is large, that is, when y max is large ; 

 but i/ max is infinite when sec 61 a or when equation (54) holds. 

 Long columns, therefore, fail by bending, and short columns by 

 direct compression. In ordinary cases, however, both kinds of 

 stress are combined, and buckling may take place for a load less 

 than P max as given by equation (53) ; that is, the elastic limit is 

 passed. The difference in these loads depends upon the eccen- 

 tricity and the initial curvature of the column. 



If the effect of the initial curvature be investigated, it will 

 be found to vary in a manner quite like the effect of eccentricity 

 as shown by the application of equation (50). This difference, 

 then, increases as the eccentricity and initial curvature increase. 



The effect of the usual variations from theoretical conditions 

 can only be found by experiments. This is discussed in Art. 113. 



112. End Conditions. The column shown in Fig. 137 has 

 a fixed end and a free end. At a fixed .end the direction of the 

 tangent to the elastic curve corresponds with that of the axis of the 



column before it is bent, that is -y-=0. When the end of a col- 



dx 



umn is perfectly free to turn so that its end condition has no 

 influence whatever on its bending, the end is said to be pivoted. 

 A round end would be formed by a hemispherical end on a flat 

 surface ; the deflection of the column shifts the point of applica- 

 tion of the load slightly. When there is a rocker bearing in place 



