Art. 112. 



END CONDITIONS. 



187 



perfect fixity, and the maximum stress is at the middle of the 

 column. If then the location of the points of contra-flexure are 

 known, the part of the column between them may be treated as a 

 column with true pivoted ends. 



Columns do not necessarily bend into the simple curves 

 shown in Fig. 138. In case (6), for example, double and triple 

 flexure occasionally occur, and the column carries a much greater 

 load. Experiments show that these conditions are unstable. 

 With an increase of load the column goes over into the weaker 

 condition of single flexure. 1 Multiple flexure is due to variations 

 from the ideal conditions tending to produce bending in opposite 

 directions in different parts of the column. Its possibility fol- 

 lows from equation (49) because, sec 01= oc when Ol=y 2 Tr, 61= 



fa 61= fa etc. 



In Fig. 138 (6), on account of the symmetry, the tangent to 

 the elastic line is vertical at the middle, and each half of the 

 column is in the condition of a column fixed at one end and free 

 at the other case (a). This condition will not be changed if the 

 column be cut in two, provided there is no tension on the convex 

 side. It follows that a column may have a fixed end without be- 

 ing fastened, or that a flat end may be a fixed end. An elastic 

 base, however, will deform until the stresses developed are equal 

 to the pressure from the column. Since this pressure is greater 

 on one side than the other, the column, if free at one end, will 

 lean; the elastic curve will be tangent to the inclined position, 

 and the stresses will be but slightly different from those in the 

 case of a true fixed end. In a column with two fixed ends, for 

 example, any turning of the ends will move the points of contra- 

 fiexure toward the ends the column will approach the condition 

 of a column with pivoted end. If the ends abutted against per- 

 fectty inelastic surfaces, they would be in the condition of perfect 

 fixity so long as there was no tension on the convex side, near the 

 ends: the intensity of the direct compression would have to be 

 greater than the intensity of the tension due to bending, 



P ^ 

 A > I 



'See A report on Experiments on the Strength of Wrought Iron 

 struts, by James Christie, Mem. Am. Soc. C. E., Trans. Am. Soc. C. E., 

 Vol. XIII, page 88. 



