Proceedings, <&c, for 1884. 171 



This remark applies very directly to the case under notice, but 

 it seems to have been quite neglected by the chief authorities on 

 the strength of beams, with the result that what is really a 

 simple consequence of two facts to be presently mentioned, is 

 looked on as an obscure anomaly. 



The ordinary formula for the moment of resistance of a 

 rectangular beam (J fbh 2 ), rests on two assumptions — 



1. That elasticity is equal in tension and compression. 



2. That elasticity is constant up to breakage. 



Neither of these is strictly correct, though both are approximately 

 true till near breaking load. 



In cast-iron beams the formula gives strength only 40 per cent, 

 of the truth ; and in wrought-iron, only 60 per cent. 



To correct this, it is usual to increase/ to/ -j- <E>, <£ being arranged 

 to make results agree with experiment. 



This empirical formula gives results very closely in accord with 

 experiment, but throws no light whatever on the source of the 

 extra strength. 



To account for this strength, we need only examine the effect of 

 the variations in the elasticity — 



(1). Suppose elasticity constant up to breakage, and equal in 

 tension and compression, then the neutral axis will be at the 

 centre of the beam, and the stress will vary uniformly over the 

 section. In this case we shall have the formula (^fbh 2 ) rigorously 

 exact. 



(2). Suppose the modulus of elasticity decreases slowly with the 

 stress, but is equal in tension and compression, then the neutral 

 axis will be at the centre ; but the stress at any point increases in a 

 less ratio than the distance of the point from the neutral axis, so 

 that the stress-curve bulges and the beam is slightly stronger than 

 appears from the formula. 



(3). Suppose the modulus of elasticity is greater in compression 

 than in tension, and that elasticity varies with the stress, then the 

 neutral axis will rise above the centre of the beam, and the curves 

 bulge considerably, then the beam will fail by compression, but 

 will be considerably stronger than appears from the formula. This 

 is the case with cast iron. With wrought iron the modulus of 

 elasticity is less in compression, so that the neutral axis falls and 

 the beam fails in tension ; but as the difference between the two 

 elasticities is not so marked as in cast iron, the discrepancy from 

 the formula is not so great. 



26th November, 1884. 



Professor Kernot brought up the subject of the Richmond 

 boiler explosion of the 7th inst. This was a Cornish boiler, 



