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8.43 (C). A hollow steel shaft is subjected to combined torque and internal pressure of unknown magnitudes. In order to assess the strength of the shaft under service conditions a rectangular strain gauge rosette is mounted on the outside surface of the shaft, the centre gauge being aligned with the shaft axis. The strain gauge readings recorded from this gauge are shown in Fig. 8.47.

Fig. 8.47.

If E for the steel = 207 GN/m2 and v = 0.3, determine:

(a) the principal strains and their directions;

(b) the principal stresses.

Draw complete Mohr's circle representations of the stress and strain systems present and hence determine the maximum shear stresses and maximum shear strain.

[636 x 10"6 at 16.8° to A, -204 x 10"6 at 106.8° to A, -360 x 10~6 perp. to plane; 159, -90,0 MN/m2;

8.44 (C). At a certain point in a material a resultant stress of 40 MN/m2 acts in a direction making angles of 45°, 70° and 60° with the coordinate axes X, Y and Z. Determine the values of the normal and shear stresses on an oblique plane through the point given that the normal to the plane makes angles of 80°, 54° and 38° with the same coordinate axes.

If crfv = 25 MN/m2, axz = 18 MN/m2 and ayz = —10 MN/m2, determine the values of aIX, ayy and az? which act at the point. [28.75, 27.7 MN/m2; -3.5, 29.4, 28.9 MN/m2.]

8.45 (C). The plane stress distribution in a flat plate of unit thickness is given by

<*xx = x3y - 2y3x ayy = y3* - 2pxy + qx y4 3 2 2 2 axy = ~2 ~ 2X y + Px +s

If body forces are neglected, show that equilibrium exists. The dimensions of the plate are given in Fig. 8.48 and the following boundary conditions apply:

b at >• = ± - axy = 0 b and at y = — - ayy = 0

Fig. 8.48.

Determine:

8.46 (C). Derive the differential equation in cylindrical coordinates for radial equilibrium without body force of an element of a cylinder subjected to stresses o>, og.

A steel tube has an internal diameter of 25 mm and an external diameter of 50 mm. Another tube, of the same steel, is to be shrunk over the outside of the first so that the shrinkage stresses just produce a condition of yield at the inner surfaces of each tube. Determine the necessary difference in diameters of the mating surfaces before shrinking and the required external diameter of the outer tube. Assume that yielding occurs according to the maximum shear stress criterion and that no axial stresses are set up due to shrinking. The yield stress in simple tension or compression = 420 MN/m2 and E = 208 GN/m2. [C.E.I.] [0.126 mm, 100 mm.]

8.47 (C). For a particular plane strain problem the strain displacement equations in cylindrical coordinates are:

Show that the appropriate compatibility equation in terms of stresses is dar dog vr—--(1 - v)r—- + (7r - og = 0

dr dr where v is Poisson's ratio.

State the nature of a problem that the above equations can represent. [C.E.I.]

8.48 (C). A bar length L, depth d, thickness t is simply supported and loaded at each end by a couple C as shown in Fig. 8.49. Show that the stress function 0 = Ay3 adequately represents this problem. Determine the value of the coefficient A in terms of the given symbols. [A = 2C/td3]

8.49 (C). A cantilever of unit width and depth 2d is loaded with forces at its free end as shown in Fig. 8.50. The stress function which satisfies the loading is found to be of the form:

where the coordinates are as shown.

0 0

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