W

130 mm

13 mm

The properties of the section are as follows:

J= 19 mm, v = 45 mm,IXX =4 x 10"6 m4./vv = 1.1 x 10"6 m4,/.,.,, = 1.2 x 10"6 m4.

Determine:

(a) the magnitude of the principal second moments of area together with the inclination of their axes relative to XX:

(b) the position of the neutral plane (N-N) and the magnitude of Inn',

(c) the end deflection of the centroid G in magnitude, direction and sense.

[444 x 10"8 m4, 66 x 10"8 m4, -19°51' to XX, 47°42' to XX, 121 x 10~8 m4, 8.85 mm at -42° 18' to XX.]

1.7 (B). An extruded aluminium alloy section having the cross-section shown in Fig. 1.21 will be used as a cantilever as indicated and loaded with a single concentrated load at the free end. This load F acts in the plane of the cross-section but may have any orientation within the cross-section. Given that Ixx = 101.2 x 10~8 m4 and lyy = 29.2 x 10"8 m4:

(a) determine the values of the principal second moments of area and the orientation of the principal axes;

(b) for such a case that the neutral axis is orientated at —45° to the X-axis, as shown, find the angle a of the line of action of F to the X-axis and hence determine the numerical constant K in the expression it = KFz, which expresses the magnitude of the greatest bending stress at any distance z from the free end.

[City U.] [116.1 x !0-8, 14.3 x 10"8, 22.5°, -84°, 0.71 x 105.]

1.8 (B). A beam of length 2 m has the unequal-leg angle section shown in Fig. 1.22 for which /„ =0.8 x 10"6 m , / vy = 0.4 x 10 m and the angle between X — X and the principal second moment of area axis X| — X\ is 30°. The beam is subjected to a constant bending moment (Mx) of magnitude 1000 Nm about the X — X axis as shown. Determine:

(a) the values of the principal second moments of area Ix\ and ly\ respectively;

(b) the inclination of the N.A., or line of zero stress (N — N) relative to the axis — and the value of the second moment of area of the section about N — N, that is /jv;

Fig. 1.21.
Centroid Structural Steel Angle
Fig. 1.22.

(c) the magnitude, direction and sense of the resultant maximum deflection of the centroid C.

For the beam material, Young's modulus E = 200 GN/m2. For a beam subjected to a constant bending moment M, the maximum deflection <5 is given by the formula s - ML2

[1 x 10~6,0.2 x 10"6 m4,-70°54'to XiXi, 0.2847 x 10~6 m4, 6.62 mm, 90° to N.A.]

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