42 PRACTICAL STRUCTURAL DESIGN 



The shear diagram is evident. After drawing all the diagrams 

 the student should check the moment at different sections by means 

 of the shear diagram. Drop a vertical line through the shear 

 diagram from any point on the beam. The area of the shear dia- 

 gram between this section and the nearest support is equal to the 

 bending moment at the section. When a beam rests on two sup- 

 ports and the moment is desired at any point the beam is assumed 

 to be fixed there and to be pushed up by a force equal in amount 

 to the reaction. Thus it is a cantilever beam, and by reference to 

 Fig. 3 it will be found that for a cantilever beam the moment 

 at any section is equal to the area of the shear diagram between 

 that section and the free end. 



The principle of the lever applies in all cases and moment effects 

 are additive; therefore the effect of additional concentrated loads 

 on the beam may be readily found. This is recommended as an 

 exercise, arithmetical computations being checked by graphical 

 methods. 



In the figures an arrow point indicates the center of gravity 

 of the bearing area. The clear span is S and the length of the 

 beam is LI. The moment span is L. To simplify all computa- 



T 



tions use L = 



, 

 and instead of M = 



w X S x Li 



wL 2 

 use the formula M = - > 



o 



the average length being used in all cases, as it is close enough 

 for all practical purposes. 



In examples involving loads concentrated at some point one 

 side of the middle of a span the distance to the nearer support has 

 been termed a and the distance to the farther support 6. Then 



,, Pab 



M = -T- 



The custom in modern text books is to use the letter a for the 

 shorter length and designate the longer length by describing it 

 as the difference between L and a. Thus 



, 



