. The action of materials under stress; . /l + ^Vl = 3^VI- = l^- 90 STRUCTURAL MECHANICS. The polar moment of inertia, about the axis x, is equalto the sum of the two moments about jj/ and rj. As, in gen-eral, y and z may have any directions at right angles to oneanother, the sum of ly and I^ must always be a constant for agiven area Two sections which have the same value for I^ do not havethe same resisting moment unless j/j is also the same in bothcases. 98. Resisting Moment about an Oblique Axis.—Whenthe plane of the external forces passes through the axis of thebeam but is not parallel to
. The action of materials under stress; . /l + ^Vl = 3^VI- = l^- 90 STRUCTURAL MECHANICS. The polar moment of inertia, about the axis x, is equalto the sum of the two moments about jj/ and rj. As, in gen-eral, y and z may have any directions at right angles to oneanother, the sum of ly and I^ must always be a constant for agiven area Two sections which have the same value for I^ do not havethe same resisting moment unless j/j is also the same in bothcases. 98. Resisting Moment about an Oblique Axis.—Whenthe plane of the external forces passes through the axis of thebeam but is not parallel to either h or b, the maximum valuesof/or the value of M max. can be found as follows:—Fig. section is A B E F; its centre of gravity is G; theplane of the applied forces and of flexure is N N; Y Y and Z Z are the usual rectangular axes, andthe angle of axis N N with Y Y is ^.Let )\ and j/g denote the dis-tances of axis Z Z from the edgesA B and E F respectively and z^?^ and Z2 the similar distances of axisY Y from B C and A bending moment of If M•^ the external torces at this section, the component in the plane of j/ will be M cos ^, and thatin the plane of z will be M sin 0. The unit-stress on thelayer A B from the former will be / = M cos Oj/^ -^ I^ andon E F, = M cos Oj/^ -^ I^ . The stress on B C from the lat-ter component moment will be / = M sin Oz^ -h ly , and onA F, = M sin 0^2 ^ ly . The points A, B, and F have stresses equal to the alge-braic sum / ± f or /= M ( (jFj or JF2) cos 0 (sj or z^ sin 0 ± L )? It is plain that the corners or points B and F have themaximum unit stresses in the above figures, as the sign of thesecond term in the above formula for these points will be -}-. /at B = M {-^-^ -|- -!-= I, compression, if M is -f. MOMENTS OF INERTIA. 9I ^ ,^ /^Vn COS 6 , Zo sin 0\ . .. ,^ . /at F = M 1-^^^- h ^ ), tension, if M is +. /at A = M (?i^- JA^j. If _jj z= j^2 or Sj = 02, the expression is simpler. Example,—8 in. steel I beam, purlin on a
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Keywords: ., bookcentury1800, bookdecade1890, booksubjectstrengt, bookyear1897
