Mr Hodgkinson on the Strength of' Materials. 171 



Art. XXXVIII.— ANALYSIS OF SCIENTIFIC BOOKS AND 



MEMOIRS. 



Observations on Mr Barlow's Theory of the Strength of Materials, and 

 his Conclusions respecting the situation of the Centres of Tension and 

 Compression in a Bent Body. By Eaton Hodgkinson, Esq. Com- 

 municated by the Author. 



The common theories of the lateral strength of materials have, as is well 

 known, been formed on the supposition that bodies are incompressible, 

 and, consequently, the deflection of a bent beam is assumed to arise whol- 

 ly from the extension of its fibres, the fulcrum being on the edge of the 

 beam. 



But theories derived from this supposition are much at variance with 

 experience ; and this circumstance has induced philosophers to seek for 

 others. Coulomb assumed, that a body, when bent, was contracted on 

 one side, and lengthened on the other ; and, consequently, that there was 

 some line, between both, where contraction ended, and extension began. 

 With this supposition, and that of the equality of the forces (pressures) 

 on each side of this line, which has since been denominated the " neutral 

 line," Coulomb attempted to sketch out a theory of the strength of bodies, 

 but which, though elegant, was so abstracted and concise, that it seems to 

 have escaped the notice of writers, till Dr Robison, in his excellent Essay 

 on the Strength of Materials, alluded to it, and, adopting the above sup- 

 positions of Coulomb, varied the mode of considering the subject a little, 

 but made no attempt toward its practical application, and left it, not un- 

 mixed with errors, in other respects, nearly as he found it. 



The next writer who, after a long lapse of time, seems to have paid at- 

 tention to the embryo suggestions of Coulomb and Robison, is Mr Barlow. 

 His object, however, was not to supply their defects, and furnish, from his 

 experiments, the requisites for adapting them to practice, but in part 

 unfortunately to reject them ; proposing in their stead a new theory, dif- 

 fering from Coulomb's only in this, — that the sums of the forces on each 

 side the neutral line, taken collectively, instead of being equal one to the 

 other, are inversely as their distances from it. And this supposition, as 

 will be shown hereafter, is the source of the errors mentioned above. 



The paragraph in which they occur is as follows. " The mechanical ope- 

 ration of fracture may be considered under the form shown in" the annex- 

 ed figure, PlateL Fig. 9, " where n is the neutral axis, t the centre of ten- 

 sion, and c the centre of compression ; w a weight equal to the tension of 

 all the fibres in An, and w a weight equal to all the resistances to com- 

 pression in nC, which weights and distances, or levers nt, nc, must be such, 

 that w X nt = w' X nc ; for it is this equality which determines the po- 

 sition of the point n. " And the sum of these, when the weight W is just 

 sufficient to produce the fracture, is equal to X W »N ; that is, when the 

 three forces are in cquilibrio, we must have w X nt + w' X nc = W X »N 

 = 2w> X nt.* 



* See Mr Barlow's Essay on the Strength and SUrss of Thnbci; Art. 120. 



