RECIPROCAL FIGURES, FRAMES, 



The simplest form of <j>(y) which will satisfy these conditions is 



Hence we find the following expression for the function of stress by integrating 

 (29) with respect to x, 



Y ...................... (30), 



where a is a constant introduced in integration, and depends on the manner 

 in which the beam is supported. From this we obtain the values of the 

 vertical, horizontal, and shearing stresses, 



The values of Q and of U, the vertical and the shearing stresses, as 

 given by these equations, are perfectly definite in terms of h and k, the load 

 and the weight of the beam per unit of length. The value of P, the hori- 

 zontal stress, however, contains an arbitrary function Y, which we propose to 

 find from the condition that the beam was originally unstrained. We therefore 

 determine a and /3, the horizontal and vertical displacement of any point (x, y), 

 by the method indicated by equations (13), (14), (15) 



' (35), 



where X' is a function of x only, and Y' of y only. Deducing from these 

 displacements the shearing strain, and comparing it with the value of the shearing 

 stress, U, we find the equation 



(36). 



Hence = n (by - y.) .................................... (37) , 



........... (38). 



