185] GENERAL FLEXURE. 495 



The stresses to be applied are the force 



140) 

 and the couple 



141) M = 



Thus we still have the theorem of the bending moment. The meaning 



of equation 141) is seen by considering the stresses across a section 



of the beam at any distance from 



the origin, and noticing that if the 



beam were cut at this section in 



order to hold it in equilibrium we 



should have to apply to the two 



cut ends couples as shown in Fig. 158, 



together with equal and opposite 



forces X, the latter being independent 



of the position of the cut, the 



former proportional to the distance 



from the free end of the beam, 



which is also proportional to the 



curvature of the central line. 



We shall close this subject by the determination of the function 

 V l for the case of an elliptic cross - section. If the equation of the 

 ellipse is 



we have 



Fig. 158. 



cos (nx) : cos (ny) = '* 



and the equation 130) becomes 



^ + a?y ^ = [f x* + (l - 1) f] Vx + (2 + ij) a*xy\ 



Vx 



As in the case of torsion, a solution is given by a circular harmonic 

 of the third degree, 



{C + 3D O 2 - 2/ 2 )} - 



if we have 



Dividing through by x, transposing and putting the coefficients of 

 # 2 , / 2 and 1 respectively proportional to > > 1, we have 



