to Transverse Strain in Beams. 415 



The radius of curvature at the same point is determined by 

 the equation 



i_wi wi 



R~4EI 1 m.r(3-r)*' { } 



where W is the central load, and I the length of the beam. 



And the deflexion of the original beam, on the supposition 

 of stress and strain varying together, would be 



1_ 48EI ; 



1 486\ 



..... (9) 



and the elongation of the fibres at a depth — below the neutral 

 axis 



R r(3-r) 2 / 2 ' 



es at 



24S X < 



r(3-r) 2 l 2 



and the elongation at the lower side of the section 



(10) 



68^4 + 0-1) 



e i~ p r (3-r) 2 (11) 



The amount of elongation varies directly with the deflexion 



being -~ so long as the limit of elasticity is not exceeded. 



But when this is the case, as indicated by an excess of actual 



deflexion over the normal amount, an increase of elongation 



will be occasioned in a larger ratio than the increase in the 



deflexion, owing to the excess of curvature being confined to 



the portion of the beam which is overstrained, that is to say, 



to the central part, and being greatest in the centre of the 



length. The distance from the centre of the beam to which 



this overstrain will extend will be as far as the point at which 



the maximum tensile stress S is the same as the calculated 



maximum tensile stress ; that is to say, since the moment of 



stress varies as the distance from the point of support, at a 



j, S I L 



distance ~- ^ = — *. 

 ►Sj 2 2r 



* The overstrain actually continues to some extent to a greater dis- 

 tance, as the stress does not in fact maintain a strict proportion to the 

 strain until it attains its maximum, as has "been assumed for convenience 

 of calculation. The exact solution of the problem would require S to be 

 treated as a variable, which would introduce too great complexity. The 

 error occasioned by the treatment of the question which has been adopted 

 is not considerable, but causes the calculated deflexions, especially for the 

 intermediate loads, to be somewhat less than the true amounts. 



2 F2 



