Q

This has been added to Fig. 1.15 and indicates that the points A and B are on either side of the N.A. and equidistant from it. Stresses at A and B are therefore of equal magnitude but opposite sign. Now a = Px + Qy stress at A = 5725 x 106 x 9 x 10~3 + 1533 x 106 x 120 x 10"3

Similarly,

stress at B = 235 MN/m2 (compressive)

(b) The principal second moments of area may be found from Mohr's circle as shown in Fig. 1.16 or from eqns. (1.6) and (1.7), i-e. /„,/,. = Ulxx+Iyy)±Ulxx-Iyy)sec7B

Iyy-Ixx (4.4 -48.3)10-6 = 0.451 29 — 24° 18', 9 = 12°9' /«./„ = 5[(48.3 + 4.4) ± (48.3 — 4.4) 1.0972] 10" = ^ [52.7 ± 48.17]10~6 /„ = 50.43 x 10"6 m4 Iv = 2.27 x 10~6 m4 The required stresses can now be obtained from eqn. (1.18).

Fig. 1.16.

and M„ = 10000cos47°51' x 2 = 13422 Nm and, for A, u — x cos 9 y sin 9 = (9 x 0.9776) + (120 x 0.2105) = 34.05 mm v = ycos9 —xsin9 = (120 x 0.9776) - (9 x 0.2105) = 115.4 mm

_ 14828 x 115.4 x 10~3 13422 x 34.05 x 10~3 ° ~ 50.43 x 10"6 + 2.27 x 10"6

(c) The deflection at the free end of a cantilever is given by

0 0