PROCEEDINGS OF SECTION H. 659 



5.— THE RELATIVE STRENGTH OF CERTAIN BEAM 

 SECTIONS. 



By EDWARD VINCENT CLARK, B.Sc, Assoc. 31. Inst. C.E. 



In the course of a paper on " The Theory of Cast-Iron Beams " 

 iMin. Proc. Inst. C.E., vol. CXLIX.) the author pointed out that the 

 calculated modulus of resistance of a beam of square section set with 

 a diagonal vertical, if of cast iron or any other material considerably 

 stronger in compression than in tension, is increased by the removal 

 of metal from the corner under tensile stress up to a certain point ; 

 and that under the assumption of a perfectly elastic material the maxi- 

 mum modulus of resistance is reached when about one-eighth of the 

 total depth of the beam has been removed, being 11-34 per cent, greater 

 than the modulus for the full section. In the case of cast iron, whose 

 elasticity is by no means perfect, the precise amount of material to be 

 removed to procure maximum modulus will be slightly different ; but 

 by taking the stress-strain curves of cast iron to be parabolic and of 

 the form there determined {loc. cit.), and by making certain other as- 

 sumptions, it was deduced that the maximum modulus of resistance 

 would be reached when about one-tenth of the total depth had been 

 removed, and that it would exceed the modulus of resistance of the 

 iull section, calculated upon the same assumptions, by 6-19 per cent. 



Ordinarily {cceteris paribus) we assume that the strengths of two 

 beams are in the same ratio as their moduli : but that is quite unjusti- 

 lied here. The modulus of a beam's section, under the customary 

 assumption of the independence of the different longitudinal layers 

 of the beam, enables us from the bending moment to determine the 

 (apparent) stress in the outermost layer of the beam's section, all 

 material nearer the neutral axis having less stress. Immediately 

 prior to fracture taking place it is (under the common theory) only 

 the outer layer of material that is strained to breaking point ; and not 

 until this outer layer is ruptured and the full load cast upon the remain- 

 ing section, which being smaller has in general a lower modulus, does 

 the second layer become strained to its limit. Thus fracture of cast 

 iron by bending may be regarded as a process of tearing (though instan- 

 taneous to the eye) accompanied by a steady shift of the neutral axis 

 from its initial position to very near the compression side of the beam, 

 as is shown by the fact that the broken face exhibits throughout the 

 clear fracture of tension, with only at the very extreme edge any 

 sign of crushing. But in this particular case of a beam of square 

 section with diagonal vertical, the diminution of section as fracture 

 begins does not lower the modulus, but, on the contrary, raises 

 it ; and hence that load which strains the extreme corner to 

 its limit will not strain the next layer so severely after the first 

 is broken, so that an increase of load should be required lo 

 complete the fracture of the beam. In other words, on the usual 

 assumption of the independence of the different longitudinal layers 

 of the beam, one may expect that the same load will be required to 



