TESTING OF STRUTS. 285 



If the pillar is fixed at the two ends the formula 

 becomes (Fig. 148) 



p _ 4 



and, if fixed at one end and rounded at the other (Fig. 149) 



When the pillar assumes more nearly the proportions 

 usually met with in practice, it fails by combined 

 crashing and bending, and neither of these extreme 

 formulas holds. 



In this case consider a pillar, Fig. 147, of length /, 

 pivoted or free to revolve at the ends, and subjected to 

 a compression load, P. The load will induce a compres- 

 sive stress in the material, f c , such that 



A being the area of the cross-section of the pillar. 



The pillar will also be bent under the load, this bend- 

 ing being immediately caused by the bending moment 

 represented by the product of the load on the column P, 

 and the deflection, o, which it maintains. It is conceiv- 

 able that if the load could be applied truly along the axis 

 of the pillar, and there existed no inequalities in the 

 material, no flexure would be produced, and failure would 

 take place by crushing. This state of things is not 

 possible of attainment, the pillar being in a state of 

 unstable equilibrium ; the load is never quite axial, the 

 axis is not perfectly straight, and there are always small 

 inequalities in the material. So that there is a bending 

 moment on the pillar from the time the load first comes 

 on. As the load increases, the deflection increases with 

 it, and, therefore, the moment also, both by reason 

 of the increase of load and of the increase of the deflec- 

 tion, so that the bending moment increases more rapidly 

 than in the simple ratio of the increase of load. 



This continues until the maximum stress in the material 

 exceeds the elastic limit stress. Various formulas have 

 been constructed to express the relation between the 

 dimensions of pillars and the loads required to produce 



