32 



PRACTICAL STRUCTURAL DESIGN 



the span, a semicircle may be used. For cantilever beams the 

 vertex of the parabola is at the support. For beams on two or 

 more supports the vertex of the parabola between the supports 

 is at the top. 



For one concentrated load in the middle of a beam resting freely 

 on two supports drop a perpendicular line under the load to repre- 

 sent the amount of the 

 bending moment, as 

 shown in Fig. 25. Con- 

 nect the lower end, C, 

 of the line by straight 

 lines to A and B. 

 Measuring vertically 

 from any point on the 

 center line of the beam 

 to the bounding line, 

 ACB, gives the bend- 

 ing moment at that 

 point. The moment 

 at any point may be 

 computed also if the 

 arithmetical method is preferred to the graphical method. The 

 load being concentrated at the middle of the span, 



PL 



4 



Fig. 26 Graphical Method for Two Concen- 

 trated Loads Resting on a Beam 



M = -=-- 



To compute the bending moment at any section, xx, distant 

 y from the end, use the principle of similar triangles 



L PL 



from which (for the load only) 



PL 



jT^ PLy 2 = Py 

 .L 4 * L ~ 2 ' 



2 



The shear diagram is shown below. For a concentrated load 

 the shear is constant from the load to each end and the end shear 

 is always equal to the reaction. 



In Fig. 26 two concentrated loads are shown. Call the length 



