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



The Curvature of Deflected Beams. 



We shall now proceed to connect the formulae for direct extension and compression with 

 that for the deflection of an elastic beam, of which the transverse section is a rectangle with 

 vertical sides, by transverse pressure at its centre. 



If for a rod of one unit of length and one unit of sectional area 



w = 



For a rod of length I, the extension by the same weight will be I times as great : therefore 

 if e be the extension of such a rod, we must have in order that w may remain unaltered, 



w = 



Also, if the sectional area of the rod be a units of area, the weight to produce a given 

 extension in such a rod will be a times as great as before, or, 



aa 



w = 



{A). 



Which is the general formula for the extension of a rod I feet long, and having a sectional 

 area a. Similarly, let the formula for the compression d of a similar rod be 



w = 



(*)> 



where y and S are empirical contents. 



To apply these formulae to determine the extension or compression 

 of a deflected beam, let A, B be the sections of parts of the upper and 

 lower surfaces of the beam, made by a vertical plane passing through 

 the points of support. Let C also be a part of the intersection of that 

 plane with the neutral surface, and 9 the small angle made by the inter- 

 section of contiguous normals of three lines A, C, B which are assumed 

 to have the same centre of curvature O. 



Then if R be the radius of curvature of an element of the curve 

 C, and r t of an element of a parallel curve below C, B9 will be the 

 length of the former, r t 9 of the latter. If the beam be subject to c~ 

 no strains of torsion, it may be assumed that the neutral surface is a 

 cylindrical surface, and that the material above and below it is bent in B 

 cylindrical laminae parallel to that surface. So that if dr t be the depth of the filament measured 

 along its radius of curvature, «. the transverse thickness of the beam at the part where it 

 is situated, /udr will be the area of the section subject to the extension .(f - R)Q. 



