94 



RESISTANCE OF MATERIALS 



If one end of the column is fixed and the other end is free to 

 turn, the elastic curve is approximately represented by the line 

 BCDE in Fig. 81. Therefore the critical load in this case is ap- 

 proximately the same as for a column with both ends free, of length 

 BCD, that is, of length equal to | BE or 1 1 ; whence, for a column 

 with one end fixed and the other free, Euler's formula becomes 



(120) 



p = 



approximately. 



If the lower end is fixed in direction but the upper end is entirely 

 free (that is, if there is no horizontal reaction to prevent it from 

 bending out sideways), it may be regarded as half of a column 

 with round or pin 

 ends and of length 

 2 1. Consequently, 

 in this case Euler's 



Round ends; 

 Position fixed 

 but not direc- 

 tion. 



direction; One end 



end 



The general expres- 

 sion for Euler's formula 

 is then 



where the constant k is determined by the way in which the ends 

 of the column are supported. The values of k corresponding to 

 various end conditions are given in Fig. 82. 



54. Modification of Euler's formula. It has been found by ex- 

 periment that Euler's formula applies correctly only to very long 

 columns, and that for short columns or those of medium length it 

 gives a value of P considerably too large. 



Very short columns or blocks fail solely by crushing, the tend- 

 ency to buckle in such cases being practically zero. Therefore, if 

 p denotes the crushing strength of the material and A the area 



