COLUMNS AND STRUTS 



101 



column are free to turn, and therefore formula (48) applies only to 

 long columns with round or pivoted ends. 



If the ends of a column are rigidly fixed against turning, the 

 elastic curve has two points of inflection, say B and D. From sym- 

 metry, the tangent to the elastic curve at the center C 

 must be parallel to the original position of the axis of 

 the column AE, and therefore the portion AB of the 

 elastic curve must be symmetrical with BC, and CD 

 with />/;. Consequently, the points of inflection, B and 

 D, occur at one fourth the length of the column from 

 either end. The critical load for a column with fixed 

 ends is, therefore, the same as for a column with free 

 ends of half the length ; whence, for fixed ends, Euler's 

 formula becomes 



(49) 



P = 



Columns with flat ends, fixed against lateral movement, are usually 

 regarded as coming umlrr formula (49), the terms "fixed ends "and 

 " flat ends " being used interchangeably. 



ne 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. 79. Therefore the critical load in this case is ap- 

 imately the same as for a column with both ends 

 free, of len-th //<"/>. that is, of k-n-tli '<|iial to f BE 

 or J / ; whence, for a column with one end fixed and the 

 other free, Eider's formula becomes 



O 77 // 



(50) P = a approximately. 



4 I 



84. Independent proof of formulas for fixed ends. 

 The results of the preceding article can be established 

 independently as follows. 



Suppose both ends of the column fixed against turn- 

 ing by a moment M at each support. Then the moment 

 at any point C, distant x from (Fig. 80), is M = M -f- Py, and 

 therefore the equation of the elastic curve is 



Fio. 80 



