ON THE MECHANICAL ruOPEUTIES OF STEEL. 105 



which gives the mean value of the modulus of elasticitj^, where D, is deter- 

 mined from equation (6). 



The work (U) of deflection is expressed by the formula 



■n=-xwxY2=2l' (^^ 



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

 If lu and ^ be taken at, or near the elastic limit, then this formula gives the 

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

 suffering any injury in its material. This formula, reduced to unity of sec- 

 tion, 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 



^=CKd, (10) 



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

 inch at the upper and lower edges of the section, 



• '^-'iKd • • • ^^^ 



When the section of the bar is a square, 



c=S' (i2> 



which gives the value of C, the modulus of strength, or the unit of ivorhing 

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

 up to the elastic limit. This value of C gives the comparative permanent or 

 working strength of the bar. 



Up to the elastic limit the deflections are ^proportional 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 foUow the law just explained. 



For the reduction of the Experiments on Tension and Compression.- — The 

 work n expended in the elongation of a uniform bar, 1 foot in length and 

 1 inch in section, is expressed by 



^=rri:=k^^' (^^^ 



p I 



■where Pj=^=strain in lbs. reduced to unity of section, and Z^ = j-=the cor- 

 responding elongation reduced to unity of length. 



The value of u, determined for the different bars subjected to experiment, 

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

 to that of impact. 



By taking Pj to represent the crushing pressure per unity of section, and 

 Zj the corresponding compression per unit of length, the foregoing formula 

 will express the work expended in criishing the bar. 



Having given the formulae for calculating the resisting powers of the steel 

 bars to a transverse, tensile, and compressive strain, and the amount of work 

 expended in producing fracture, we now proceed to the experiments, as 

 follows. 



