188 



END CONDITIONS. 



Art. 112. 



From equation (50) 



M f =P (e- 

 P Pe sec 01 vi 

 ~A> 



At* 



or 



\=PesecOl 



e sec 01 



< 1 



(60) 



If the ends are not fastened but simply press against flat 

 rigid surfaces, there will be the less liability of separation on the 

 convex side, the broader the ends are, for v^ and r refer to the 

 end surfaces here 



A column's support approaches rigidity when it is a large 

 mass like a bridge pier, and when the end of a column is so large 

 that the intensity of pressure on the support is comparatively 

 small. A further difficulty in the practical application of equa- 

 tion (60) lies in the fact that no c 

 satisfactory value of e can be as- 

 signed to take account of all the 

 variations from ideal conditions. 



If the end of the column is 

 forked, as shown in Fig. 139, it will 

 be fixed when 



i- Pfc= Jf 



M being the bending moment at 

 the end due to the column bending, 

 or this combined with the moment 

 due to transverse loads on the col- 

 umn, the support being assumed 



inelastic. 

 When the ends of a 



column fixed at both 



\ 



Fig. 139. 





Fig. 140. 



ends are not fastened, they will turn as soon as the 

 pressure on the convex side passes zero ; the curva- 

 ture between the points of contra-flexure increases 

 and beyond them it decreases the points of contra- 

 flexure move toward the ends, and when they ar- 

 rive at the ends, we have the case of pivoted ends 

 so far as the form of the elastic line is concerned, 

 but on account of the flat ends, the end pressures 

 will be eccentric, as shown in Fig. 140. 



It follows that the stress in a flat ended col- 

 umn will be less than in a column with pivoted 

 ends, because the turning of the ends produces an 

 eccentricity tending to reverse the curvature. Equa- 



