CHAP. VI.] MOMENT OF INERTIA. 81 



The radius of gyration squared is, therefore, 



and hence the radius of gyration is /3 = 0.44721 h. Laying 

 this off from 0, we obtain 1 and 2, and can therefore now draw 

 the central ellipse. 



63. Compound or Irregular Cross-Section. Every cross- 

 section may be divided up into trapezoids, triangles, parallelo- 

 grams and parabolic segments, and the above cases will aid us, 

 therefore, in the application of the graphic method to compound 

 or irregular cross-sections. The engineer is often called upon 

 to determine the moment of inertia of such sections as the T, 

 double T, or different combinations of these in proportioning 

 the different pieces of bridges, such as chords, struts, floor-beams, 

 etc., as also in many other constructions. The calculation for 

 such cross-sections is sometimes very laborious. As an example 

 of the application of the graphical method best illustrating the 

 above principles, we take the cross-section shown in Fig. 40, 

 PI. 12. 



First we divide the cross-section into a series of trapezoids. 

 The first segment, bounded by a curve, we may consider a para- 

 bolic area. These trapezoids we reduce to equivalent rectangles 

 of common base a [Art. 32], and take the corresponding heights 

 as forces. These forces we lay off in the force polygon and 

 choose a pole C at distance H from force line, drawing C 0, C 1, 

 C 2, etc. Parallel to these lines we have the first equilibrium 

 polygon I II HI .... VIII, the intersection of the two outer 

 sides of which gives the point of application of the resultant. 

 The intersection S of the resultant with the axis of symmetry 

 gives the centre of gravity of the cross-section [Art. 30]. The 

 segments o 1', 1'2', 2'3', etc., cut off from o S, give the statical 

 moments of the forces with reference to o S to the basis H. 

 We now choose another pole C' at distance H', and form another 

 force polygon, considering these moments as forces, and applied 

 at the centres of action of the moments of the separate areas 

 into which the whole cross-section has been divided. These 

 centres of action can be determined by forming the central 

 curve for each area according to Art. 62, and then applying the 



