In the following, we shall use the Theorem of Castigliano, '' which is a corollary of 

 the energy theorem. The theorem states that if the strain energy is written in terms of the 

 applied forces, the displacements at the point of application of any force (in the direction of 

 that force) is the partial derivative of the strain energy with respect to the force. Thus, if 

 W is the strain energy in that part of the beam corresponding to a value of x, then 



U = 



(9W 



u.. = 



aw 



u„ 



(JW 



_aw 



6W 

 dU 



aw 

 aM„ 



Consider a short segment going from x to x + Ax. The foregoing expressions will 

 give the elastic deformations, to which we add the rigid-body motions,' due to deflections at 

 station X. Let the strain energy between x and x + Ax be Ax W. Then the total deflections 

 at X + Ax are given by 



U. 



+ A> 



X + Ax 



U 



X + A; 



= u 



X + Ax 



Ax 



X + Ax 



Ax 



aw 



- A\e.. 1 



aw 



Ax 



aM„ 



+ Ax 



+ Ax 



dV/_ 



av.. 



aw 



+ Ax 



+ Ax 



aw 



aw 



If Ax goes to zero, W becomes the strain energy per unit length. Then 



aw 



dU 



d^'. 



" "d^ 



aw 



de^ 



aM„ 



d7 



aw 



dU 



^v. 



d^ 



aw 



1!l 



aw,, 



dx 



aw 



dU 



av^- 



^ 



aw 



de^ 



aM, 



dx 



If the strain energy per unit length W can be expressed in terms of V^, V 

 and M , these six equations will give the desired elastic equations. 



V , M^ , M „, 



45 



