The elasticity and resistance of the materials of engineering . rr, Fig. 15. 4 Next consider a hollow circular column whose interior andexterior radii are r^ and r^ respectively. The moment of iner-tia about any diameter is: / = -^-^— -^ = ^ J -; {A = area) . (42) (Radius of gyratiorif = I 7r(r/ - r,) + r. = r\ 424 MOMENTS OF INERTIA. [Art. 49. As tables of circular areas are very accessible, it may beconvenient to write: 7^ = ^^2^ . or r^ =r ^(^/ -f- ^i) Phcenix Section. Fig. 16 shows the section of a 4 segment Phoenix column. Let CD represent any axistaken through the centre of
The elasticity and resistance of the materials of engineering . rr, Fig. 15. 4 Next consider a hollow circular column whose interior andexterior radii are r^ and r^ respectively. The moment of iner-tia about any diameter is: / = -^-^— -^ = ^ J -; {A = area) . (42) (Radius of gyratiorif = I 7r(r/ - r,) + r. = r\ 424 MOMENTS OF INERTIA. [Art. 49. As tables of circular areas are very accessible, it may beconvenient to write: 7^ = ^^2^ . or r^ =r ^(^/ -f- ^i) Phcenix Section. Fig. 16 shows the section of a 4 segment Phoenix column. Let CD represent any axistaken through the centre of thecolumn. The moments of iner-tia of the rectangles bl aboutaxes through their centres ofgravity and parallel to CD willbe very small indeed comparedwith the moment of inertia ofthe whole section. The mo-ment of inertia of any oneof these rectangles, therefore,about CD^ will be taken asequal to the product of its areaby the square of the normaldistance from its centre of gravity to the axis CD, The mo-ment of inertia of the section about CD will then be:. 4 r,-\ j sin^ + ( ^2 H I ^os a ••• ^ = ^ + ^^^ V + 2. (43) The moment of inertia is thus seen to be the same aboutall axes, a result of the general principle established in the firstpart of this Article. The area of the cross section is: Art. 49.] TRUE EYE SECTION. 425 A = n{r^ - r/) + ^bl. (43^) {Radius of gyratiorif = r = -^ . The moments of inertia of six and eight segment columnsmay be found in precisely the same manner. The moments ofinertia of the rectangular sections of the flanges about axespassing through their centres of gravity, being very small indeedwhen compared with the moment of inertia of the whole sec-tion, may be neglected without sensible error. Let r — 2S True Eye Section,; r is then the b ~ t^batter, or slope, of the under sideof each flange to the top or bot-tom of the beam ; it ranges fromabout one-third to essentially noth-ing. If the area of the cross sectionis not deduced from the weight:
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