The elasticity and resistance of the materials of engineering . In Fig. 12 ^ = moment of inertia aboutFB\s\ /-Y^; . (37) and about CD : / = 12 (38) / /i^ b^{Radius of gyrationf = r^ =z — = — or — jrL 12 12 If the rectangular section is square, b = h. Hollow Rectangular Sections, The area of the section shown in Fig. 13 is: ^4 r= bh — moment of inertiaabout FB is: bk3 _ ^7/3 ^=—Y-^—; . (39) ^ c 1 1 1 h1 -hB 11 1 1 \ and that about CD is : ^ .__^- Fig. 13. Art. 49-] CIRCULAR SECTIONS. 423 / = hb^ - hb^ 12 (40) / 3 K-f J (Radius of gyratioiif = r^ = All the equations of this case (e
The elasticity and resistance of the materials of engineering . In Fig. 12 ^ = moment of inertia aboutFB\s\ /-Y^; . (37) and about CD : / = 12 (38) / /i^ b^{Radius of gyrationf = r^ =z — = — or — jrL 12 12 If the rectangular section is square, b = h. Hollow Rectangular Sections, The area of the section shown in Fig. 13 is: ^4 r= bh — moment of inertiaabout FB is: bk3 _ ^7/3 ^=—Y-^—; . (39) ^ c 1 1 1 h1 -hB 11 1 1 \ and that about CD is : ^ .__^- Fig. 13. Art. 49-] CIRCULAR SECTIONS. 423 / = hb^ - hb^ 12 (40) / 3 K-f J (Radius of gyratioiif = r^ = All the equations of this case (except Eq. (40)), just as theystand, apply directly to the rect-angular cellular section of Fig. 14,considered in reference to the axisFB. If there were n cells insteadof 3, the space between any adja-cent two would have the width n 3 Solid and Hollow Circular Sections. First consider a solid cylindrical column whose cross section has the radius r^^ as shown in Fig. moment of inertia about any di-ameter is : 7rr/. / = {Radius of gyrationf = (41) Ttr^*47rr, 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 gravit
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