MOMENTS AND CENTROIDS 59 
angle POA and ¢ the angle POC, Let P and Q denote the moments of 
inertia about the axes OP and OQ respectively. 
By Art. 72, A—-B=(P—Q) cos 20, and GO-D=(P—Q) cos 29. 
Therefore OLD If 0+ $=45%, then cos 24=sin 20, 
_ Having found 6, P~Q Sey and P+Q=A+B. Hence P and 
Q can be found. 
If OA, OB, OC, OD, OP, and OQ be made equal to A, B, C,; D, P, 
and Q respectively, the inertia curve for the section may be drawn. If a 
is the area of the section, and OP’ be made equal to J and OQ’ be 
made equal to ar, 6° thenOP’ and OQ’ will be the semi-principal axes 
of the momental ellipse of the section. 
In the example illustrated in Fig. 65, P = 2°17 and Q=0'42, in inch 
units. The student should work out this example, and draw the com- 
plete inertia curve and the momental ellipse. 
75. Bending Moment and Shearing Force Diagrams for Beams.— 
When a horizontal beam is acted on by vertical forces or loads, these 
= % SUTPEED, 
== ------~"Y SHEARING FORCE DIAGRAM. Z 
hile Aptcatbiiittiiile 
> 
Sse 
BENDING MOMENT SSS 1! 
Fig. 66. Fia. 67. 
forces tend to bend the beam, and the bending action at any transverse 
section is measured by the algebraical sum of the moments of the forces 
on one side of the section about a horizontal axis in that section. For 
