1903.] Stress and Strain in the Cross-section of a, Beam. 



27 



ends, and subjected to bending by applying a transverse force at the 

 centre of its span. Let the origin be at one end and let 



x be taken in the direction of the length, 



y = the distance from the neutral surface, 



P = the stress at (x, y), 

 W = the load applied at the centre, 



8 = the deflection of the centre line at x } 



E = Young's modulus for the material, 



<t = Poisson's ratio, 



I, b, d = the length, breadth, and depth of the beam. 



If at any section, x > 1/2, the change produced in the breadth of 

 the beam be measured when the load is applied, and, at the same time, 

 the deflection of the centre line of the beam at that point be observed, 

 then the angle turned through by the side of the beam 



, ,o-P6 

 = tan 1 - , 

 2Ey' 



and the deflection, 8, is given by 



s Wl 2 Wo 

 16EI V 12EI' ' 



Combining these we obtain an expression from which o- can be at 

 once determined, namely, angle turned through by side of beam 



= tan 1 — — — . 



3l 2 - 4r.X 2 



The left-hand side of this equation is obtained from the mean 

 straight line through the points representing the lateral strain in 

 the beam. 



In conclusion the author must express his thanks to Messrs. F. C. 

 Prentice and C. M. Eushton for assistance rendered at various times 

 during the preparation of this paper. 



