CHAP, in.] SUPPLEMENT TO OHAP. VH. 113 



But, as we have just seen, this integral is the moment of inertia I of the 

 cross-section with reference to the axis through the centre. Hence, 



M = - ^ . Since </> is a very small angle, it may be taken equal to its 



Ci X 



ix dy i. d (b d* y , . f,-,<Py 

 tangent, or equal to - ; hence -^~ and M = E I -2-. 

 dx dx da? dx* 



But v d <f> : v :: d x :r, where r is the radius of curvature ; 



v d d) d x d d> I 



hence = or -^- = . 



v r das r 



1 (P 11 T I 



Therefore, M = EI- = EI^-|= (8) 



r dtf v 



and T = U , . . (9) 



r 



Equation (8) is our fundamental equation. 



In any given case we have only to write down the expression M for the 



d 1 y 

 moment of the outer forces at any point, and equate it with E I -=-?. 



d x , 



Integrating once we shall then have for I constant, of course, E I -=-^ and, 



(L X 



integrating again, E I y in terms of x, or the equation of the deflection 

 curve itself. Making E I =-^ = 0, we can then find the point of maximum 



it X 



deflection, and inserting in the value for Ely the value of x thus found, 

 can find the maximum deflection itself. The discussion of any case reduces 

 thus to a simple routine, and every case is in many respects but a repetition 

 of the same processes. 



12. JBeam fixed at one end and loaded at the other 

 Constant cross-section. We shall always consider a moment positive 

 when it causes compression in the lower fibre ; negative when it causes ten- 

 sion in that fibre. Distances to the right of the origin are always positive, 

 to the left negative. Hence on the left of any section an upward force is 

 negative, a downward force positive ; while on the right of the section the 

 upward force is positive and the downward one negative. The reader 

 should always draw the Fig. for each case discussed, and in the beginning, 

 at least, review these conventions each time. 



Now let a beam of length I have the weight P at the free end, and let it 

 be fixed horizontally or " walled in " at the right end. Then the moment 

 at any point distant *'from the left or free end is M = + P x. 



(a) Change of shape. 



From (8) we have now 



