ON THE MECHANICAL PROPERTIES OF STEEL. 165 



Jlexibilitij of the bar, or as the modulus of flexure, since it measures the 

 amount of deflection produced by a iiiiit of pressure for a unit of section. 

 Substituting this vahie in equation (2), we get 



===^7 P) 



which gives the mean value of the modulus of elasticity, where Dj is deter- 

 mined from equation (6). 



The work U of deflection is expressed by the formula 



^ H>""4=h'' («) 



where S is the deflection in inches corresponding to the pressure (w) in lbs. 

 If w and d be taken at, or near to the elastic limit, then this formula gives 

 the work, or resistance analogous to impact, which the bar may undergo, 

 without suffering any injury in its material. This formula, reduced to unity 

 of section, becomes 



"=24K (^) 



If C be a constant, determined by experiment for the weight (W) straining 

 the bar up to the limit of elasticity, so that the bar may be able to sustain 

 the load without injury, then 



^=CKcZ, (10) 



where C=^S, or ^ of the corresponding resistance of the material per square 

 inch at the upper and lower edges of the section, 



\V7 



•••«=fs (") 



"When the section of the bar is a square, 



«=S' (1^) 



which gives the value of C, the modulus of strength, or the unit of worl-'uig 

 strength, W" being the load, determined by experiment, which strains the bar 

 up to its elastic limit : this value of C gives the comparative permanent or 

 working strength of the bar. 



Up to the elastic limit the deflections are jyroportional to their corresponding 

 strains, but beyond this point the deflections increase in a much higher ratio. 

 Hence the deflection corresponding to the elastic limit is the greatest deflec- 

 tion which is found to follow the elastic law jiist explained. 



Tensile Strain, 6fc. — The work u expended in the elongation of a uniform 

 bar, 1 foot in length and 1 inch in section, is expressed by 



H-l-i^''-' (15) 



P 7 



where P, = —=the strain in lbs. reduced to unity of section, and Z =_ 

 ' K ^ L 



=the corresponding elongation reduced to unity of length. 



This value of u, determined for the diff'erent bars subjected to experiment, 

 gives a comparative measure of their powers of resistance to a strain analogous 

 to that of impact. 



By taking P^ to represent the crushing pressure per imity of section, and 

 ?, the corresponding compression per unity of length, the foregoing formula 

 wiU express the work expended in crushing the bar. 



