## 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 thin-walled closed section beam of constant wall thickness t has the cross-section 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 thin-walled beam is circular in cross-section 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 10-3rad, 500 < z < 1000mm

450 Nm

P.9.24 A uniform closed section beam, of the thin-walled 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 cross-section 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 cross-section 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 Bredt-Batho 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.

0 0