[188] H. COX, ESQ., ON THE DEFLECTION OF IMPERFECTLY ELASTIC BEAMS 



which is the required equation connecting the formulae for direct extension and compression 

 with that for the deflection of a beam of rectangular section. 



On the hypothesis of perfect elasticity /3 and & = 0, and = 7; 



Pa 3 



which coincides with Poisson's formula for the deflection of a perfectly elastic beam of rect- 

 angular section [Traite de Mecanique, No. 324). 



The Breaking Weight of Rectangular Beams. 



Cast iron is much stronger to resist rupture by compression than by extension. Conse- 

 quently a rectangular bar of that metal when deflected is always broken by the extension of 

 its convex side. 



Suppose the ultimate extension of a rod of cast iron, or the greatest extent to which it can 



be stretched before breaking, to be the n th part of its length. We have shewn that if R be 



j 

 the radius of curvature of the neutral surface and d half the depth of a deflected bar, — is the 



R 

 ratio of the extension of a filament on the convex side of the bar to its unstretched length ; 



therefore when the bar breaks, d is the w tb part of R or d = nR. 

 Also we have found that 



px 3 4 a + 7 



At the centre of the beam x = a. Let B be the breaking weight, or twice the value of P 

 the pressure on the fulcrum. Then putting d = nR, 



d j, a + 7 s 7 a /3 + 7^ 



- = 2fjid 3 ' d— '— 



n 3Ba 4 a + 7 



B = 2ad 2 r - - + - . -*- '— 



Sa \u 4 a + 7 



or the breaking weight varies as the thickness and the cube of the depth directly and as the 

 length inversely. In respect to this law the preceding formula agrees with that hitherto 

 used and is confirmed by experiment. It is observed indeed in the Report of the Iron 

 Commission that the law is not accurately true for bars differing considerably in magni- 

 tude, but the anomaly is satisfactorily accounted for (p. Ill) as arising "principally from 

 the superior hardness of smaller castings." 



Direct Compression of Cast Iron. 



From the formulae for direct tension and for deflection, by substituting numerical values 

 of the coefficients deduced from experiment, may be obtained the numerical values of the 

 coefficients in the formula for direct compression. 



