MOMENT OF RESISTANCE.—CENTRAL ELLIPSE AND KEEN OF A SECTION. 103

tively, parallel and perpendicular to XX intersect (§ 69) in M the centre of reducedmoments about X X. Thus M Q is the direction of the diameter of the ellipse ofinertia conjugate to X X.

To find the length of this diameter it is necessary (§ 72) to obtain the radius ofgyration (k) of the system about X X. By § 70, k = ^/e f . OH; where e and / arethe intersections of the first and last sides of the funicular polygon I x . .. Yj with X X,and OH is the polar distance in the original polygon of forces. A line parallel toand at a perpendicular distance h from XX therefore (§ 71) cuts Y Y; i.e. M Q pro-duced in the extremity of the required diameter of the ellipse of inertia.

In order to find the length of the other diameter it is necessary to obtain A theradius of gyration about YY. The construction, which is precisely similar to thatdescribed above, is carried out by means of the two funicular polygons I 2 ... V 2 andI 2 '... V 2 ', and the last sides of the latter cut Y Y in e lt f 1 . Then k l = v /e,/ 1 . O.H,where 0 2 H 2 the polar distance of the new polygon of forces has been taken equal toOH. A line parallel to and at a perpendicular distance A from YY cuts XX in theextremity of the other diameter, and the ellipse of inertia can now be drawn (§ 71).

The centre C of the system is determined by means of the two funicular polygonsI...Y and I'... Y' (§ 67). Joining M the centre of reduced moments to 0, thedirection of the diameter of the central ellipse conjugate to a line through 0 parallel toX X is obtained. The radius of gyration about the axis through M and C, obtainedprecisely as above, determines the length of one semi-diameter of the central ellipse.The radius of gyration K about the conjugate diameter through C parallel to X X canbe deduced from the already obtained radius of gyration (k) about X X, since F = K 2+ <P, where d is the perpendicular distance from C to X X.

To avoid confusion of lines the construction for the central ellipse is not carriedout in Fig. 94, PI. IX. This figure appears complicated, but its construction will befound extremely simple, and it affords an excellent exercise in dealing with funicularpolygons.

CHAPTER X.

MOMENT OF EESISTANCE.—CENTRAL ELLIPSE AND KEEN OF A SECTION.

75. Bending Stress. —A beam defined as in § 41 is subject to simple bendingstress only, when the loading can be reduced to a couple acting in the plane ofsymmetry normal to the axis. If all, or any of the external loads act out of this plane,a stress due to twisting arises. If they are not normal to the axis, the beam has to