248 On Beams fixed at the Ends. 



of the beam, which may be arbitrarily fixed ; E F G E is 

 a diagram of the values of m, easily drawn when the loading 

 is known. We are required to find two points, P and Q, 

 such that 



area ET + area QC = area PU 



and PR = QS. 



When found, these points P and Q are the points of inflexion, 

 and PR or QS is what we called c. That is, m — PR is the 

 real bending-moment M at every place; and, knowing the 

 bending-moment and depth d, it is easy to find I, as the con- 

 dition of uniform strength is given in (5), or 



Md 

 l-± 2/ , 



I being of course always positive. 



We have found it very easy to solve this problem by trial. 

 First find the area EC. Choose two points P', Q', whose 

 ordinates P'R' and Q'S' are equal, and such that the area P'U 7 

 seems to the eye nearly half the area EC. Measure the area 

 Ft)', subtract from ^EC, divide by about 4Q'U' or 4P'T' or 

 by the mean value of these two. This will give an approxi- 

 mation to the error in the positions of P' and Q\ With an 

 easy exercise of one^s judgment, it will probably not be 

 necessary to make a second approximation. 



V. If the depth of the beam is constant, the problem of 

 finding the diagram of bending-moment for a beam of uniform 

 strength is of course solved by finding two points, P and Q, 

 whose distance asunder is half the length EG of the beam, 

 and at which the ordinates PR and QS are equal. Subtract 

 PR or OS from every ordinate of the m diagram E F G, and 

 we have the M diagram. 



It will be seen that, with any system of loading and any 

 variation of section in a beam, it is quite easy to get the dia- 

 gram of bending-moment ; and also that, with any system 

 of loading and any variation of depth of beam, it is quite easy 

 to get diagrams of bending-moment and moment of inertia 

 for beams of uniform strength. Many other applications of 

 the method are obvious. 



