186 



END CONDITIONS. 



Art. 112. 



In practical column formulas, such assumptions of end con- 

 ditions are usually made as will be on the side of safety. In 

 theoretical analyses, only the pivoted and fixed end conditions 

 are investigated. Fig. 138 shows that the fundamental cases are 

 really combinations of the simple case of a column fixed at one 

 end and free at the other, provided the ends are not allowed to 

 move sidewise, so as to change their relative lateral positions, 

 for this would, of course, change the curvature. 



Figures 138 (a), (6), (c), and (d) show, respectively, col- 

 umns with one fixed end and one free end, with two pivoted ends, 

 with one pivoted and one fixed end and with two fixed ends. The 

 points F are all fixed points because at these the tangent to the 

 elastic curve is vertical; the points P are all pivoted because at 

 these the bending moment is zero. It is evident that the points F 

 and P divide the column (d) into four equal parts and it follows, 

 that for the same strength, and for the different end conditions 

 shown in Fig. 138, a column's length should vary as 1 :2 :3 :4. 



If it is desired to have the preceding equations apply to 

 columns of cases (&), (c) and (d), and to have them in terms of 

 the total length of the column, it is only necessary to substitute 

 for I its value in terms of L, namely Z=%L 2 > ^ == %-^.-5 ar> d 1= 

 14^/4 respectively. For the four cases of Fig. 138, equation (54) 

 becomes; 



(54) 



=' I r 2 t T1 (Euler's Formula). (57) 



(58) 

 (59) 



P r 2 



i^- 2 .25** E ^ 

 A U 2 



p 



* max 



For the samv lengths the strengths are to each other as 1 :4 :9 :16. 



It is to be noted that the top of column (c), Fig. 138, is not 

 on the tangent at its lower end, but is located like the point of 

 contra-flexure in column (d). When the ends are both on the 

 tangent line, the factor 2.25 in equation (58) becomes 2.05. 



The greatest stress occurs at the points F in Fig. 138, and is 

 the same in all four cases, for the same load, on each column, the 

 radius of gyration being the same. In case (r7), the points of 

 contra-flexure P are usually nearer the ends on account of lack of 



