192 



the material experimented upon. If the length of pillars never ex- 

 ceeded four or five diameters, all we need do to arrive at the strength of 

 any given pillar would be to multiply its transverse area in square 

 inches by the tabulated crushing strength of that particular material. 

 It rarely happens, however, that pillars are so short in proportion to 

 their length ; and hence we must seek some other rule for calculating 

 their strength, when they fail, not by actual crushing, but by flexure. 



If we could insure the line of thrust always coinciding with the axis 

 of the pillar, then the amount of material required to resist crushing 

 merely would suffice, whatever might be the ratio of length to diameter. 

 Eut practically it is impossible to command this, and a slight deviation 

 in the direction of the thrust produces a corresponding tendency in the 

 pillar to bend, "With tension-rods, on the contrary, the greater the 

 strain, the more closely will the rod assume a straight line, and, in de- 

 signing their cross section, it is only necessary to allow so much material 

 as will resist the tensile strain. This tendency to bend renders it neces- 

 sary to construct long pillars, not merely with sufficient 

 material to resist crushing, supposing them to fail from 

 that alone, but also with such additional material or 

 bracing as may effisctually preserve them from yield- 

 ing by flexure. It is evidently, therefore, of consider- 

 able importance that we should ascertain the laws 

 determining the flexure of long pillars, which may be 

 done as follows : — 



Let the figure represent a pillar, very long in 

 proportion to its breadth, and just on the point of 

 breaking from flexure. 

 Let 1V= the deflecting weight ; 



b = the breadth of pillar ; 



d = its depth ; 



I = its length ; 



h = the central deflection ; 



H = the radius of curvature ; 



C = the resultant of all the longitudinal forces of 

 compression on the concave side at the centre 

 of the pillar ; 



T = the resultant of all the longitudinal forces of 

 tension on the convex side ; 



S = the distance between the centres of tension 

 and compression. 



The longitudinal forces acting at the centre of the pillar are three, 

 viz. the weight JF acting in the chord line of the curve, the resultant 

 C acting at the centre of compression in the concave half, and the resul- 

 tant T acting at the centre of tension in the convex half. Taking mo- 

 ments round either centre of strain, we have approximately 



^=? = ?. I. 



h 



h 



h being assumed equal to the distance between the chord-line and either 



