12
Ans. 197.2mm from vertical through booms 2 and 3.
Ans. 197.2mm from vertical through booms 2 and 3.
P.9.18 A thinwalled closed section beam of constant wall thickness t has the crosssection shown in Fig. P.9.18.
Ans. M>! = —w6 = —0.53 mm, w2 = — w5 = 0.05 mm, w3 = —h>4 = 0.38 mm. Linear distribution.
P.9.23 A uniform thinwalled beam is circular in crosssection and has a constant thickness of 2.5 mm. The beam is 2000 mm long, carrying end torques of450 N m and, in the same sense, a distributed torque loading of 1.0 N m/mm. The loads are reacted by equal couples R at sections 500 mm distant from each end (Fig. P.9.23).
Calculate the maximum shear stress in the beam and sketch the distribution of twist along its length. Take G = 30 000 N/mm2 and neglect axial constraint effects.
Arts. rmax = 24.2 N/mm2, 6 = 0.85 x 10"V rad, 0 ^ z < 500mm,
0 = 1.7 x 10_8(1450z — z2/2)  12.33 x 103rad, 500 < z < 1000mm
450 Nm
P.9.24 A uniform closed section beam, of the thinwalled section shown in Fig. P.9.24, is subjected to a twisting couple of 4500Nm. The beam is constrained to twist about a longitudinal axis through the centre C of the semicircular arc 12. For the curved wall 12 the thickness is 2 mm and the shear modulus is 22 000 N/mm2. For the plane walls 23, 34 and 41, the corresponding figures are 1.6 mm and 27 500 N/mm2. (Note: Gt = constant.)
Calculate the rate of twist in radians/mm. Give a sketch illustrating the distribution of warping displacement in the crosssection and quote values at points 1 and 4.
Ans. d0/<te =29.3 x 10_6rad/mm, w3 = tv4 = 0.19 mm, w2 = — W! = —0.056 mm
P.9.25 A uniform beam with the doubly symmetrical crosssection shown in Fig. P.9.25, has horizontal and vertical walls made of different materials which have shear moduli Ga and Gb respectively. If for any material the ratio mass density/shear modulus is constant find the ratio of the wall thicknesses ta and tb, so that for a given torsional stiffness and given dimensions a, b the beam has minimum weight per unit span. Assume the BredtBatho theory of torsion is valid.
If this thickness requirement is satisfied find the a/b ratio (previously regarded as fixed), which gives minimum weight for given torsional stiffness.

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