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= -4.54 x 10~4[2(—2.5903) - 1] + 0.0375C, + 13.33C2 - 2.8 x 10"3 = 0.0375C, + 13.33C2

Substituting in (5),

1 0.0375

= -7.048 x 10"2 Now taking y = 0 at r = 0.075, from eqn. (1)

0 = -3000 x (0-075)2 Q75 _ _ 7-048 xlO-2 2 8tt x 32900 4

- 11.8 x 10~6loge 0.075+ C3 = -3.4 x 10~5(—3.5903) - 99.1 x 10"6 + 30.6 x 10~6 + C3 = 10~6(122 - 99.1 + 30.6) + C3 C3 = -53.5 x 10~6

Therefore deflection at r — 0.03 is given by eqn. (1),

8n x 32900 Be 4

- 11.8 x 10"6 loge 0.03 - 53.5 x 10"6 = 10'6[+24.5 - 15.9 + 41.4 - 53.5] = -3.5 x 10-6 m

Problems

In the following examples assume that d Tr

with conventional notations.

Unless otherwise stated, E = 207 GN/m2 and v = 0.3.

7.1 (B/C). A circular flat plate of 120 mm diameter and 6.35 mm thickness is clamped at the edges and subjected to a uniform lateral pressure of 345 kN/m2. Evaluate (a) the central deflection, (b) the position and magnitude of the maximum radial stress. [1.45 x 10-5 m, 23.1 MN/m2; r = 60 mm.]

7.2 (B/C). The plate of Problem 7.1 is subjected to the same load but is simply supported round the edges. Calculate the central deflection. [58 x 10~6 m.]

73 (B/C). An aluminium plate diaphragm is 500 mm diameter and 6 mm thick. It is clamped around its periphery and subjected to a uniform pressure q of 70 kN/m2, Calculate the values of maximum bending stress and deflection.

Take Q = qR/2, E = 70 GN/m2 and v = 0.3. [91, 59.1 MN/m2; 3.1 mm.]

7.4 (B/C). A circular disc of uniform thickness 1.5 mm and diameter 150 mm is clamped around the periphery and built into a piston, diameter 50 mm, at the centre. The piston may be assumed rigid and carries a central load of 450 N. Determine the maximum deflection. [0.21 mm.]

7.5 (C). A circular steel plate 5 mm thick, outside diameter 120 mm, inside diameter 30 mm, is clamped at its outer edge and loaded by a ring of edge moments Mr = 8 kN/m of circumference at its inner edge. Calculate the deflection at the inside edge. [4.68 mm.]

7.6 (C). A solid circular steel plate 5 mm thick, 120 mm outside diameter, is clamped at its outer edge and loaded by a ring of loads at r = 20 mm. The total load on the plate is 10 kN. Calculate the central deflection of the plate. [0.195 mm.]

7.7 (C). A pressure vessel is fitted with a circular manhole 600 mm diameter, the cover of which is 25 mm thick. If the edges are clamped, determine the maximum allowable pressure, given that the maximum principal strain in the cover plate must not exceed that produced by a simple direct stress of 140 MN/m2. [1.19 MN/m2.]

7.8 (B/C). The crown of a gas engine piston may be treated as a cast-iron diaphragm 300 mm diameter and 10 mm thick, clamped at its edges. If the gas pressure is 3 MN/m2, determine the maximum principal stresses and the central deflection.

i> = 0.3 and E = 100 GN/m2. [506, 329 MN/m2; 2.59 mm.]

7.9 (B/C). How would the values for Problem 7.8 change if the edges are released from clamping and freely supported? [835,835 MN/m2; 10.6 mm.]

7.10 (B/C). A circular flat plate of diameter 305 mm and thickness 6.35 mm is clamped at the edges and subjected to a uniform lateral pressure of 345 kN/m2.

Evaluate: (a) the central deflection, (b) the position and magnitude of the maximum radial stress.

7.11 (B/C). The plate in Problem 7.10 is subjected to the same load, but simply supported round the edges. Evaluate the central deflection. [24.7 x 10-4 m.]

7.12 (B/C). The flat end-plate of a 2 m diameter container can be regarded as clamped around its edge. Under operating conditions the plate will be subjected to a uniformly distributed pressure of 0.02 MN/m2. Calculate from first principles the required thickness of the end plate if the bending stress in the plate should not exceed 150 MN/m2. For the plate material E = 200 GN/m2 and v = 0.3. [C.E.I.] [10 mm.]

7.13 (C). A cylinder head valve of diameter 38 mm is subjected to a gas pressure of 1.4 MN/m2. It may be regarded as a uniform thin circular plate simply supported around the periphery. Assuming that the valve stem applies a concentrated force at the centre of the plate, calculate the movement of the stem necessary to lift the valve from its seat. The flexural rigidity of the valve is 260 Nm and Poisson's ratio for the material is 0.3.

7.14 <C). A diaphragm of light alloy is 200 mm diameter, 2 mm thick and firmly clamped around its periphery before and after loading. Calculate the maximum deflection of the diaphragm due to the application of a uniform pressure of 20 kN/m2 normal to the surface of the plate.

Determine also the value of the maximum radial stress set up in the material of the diaphragm.

Assume E = 70 GN/m2 and Poisson's ratio v = 0.3. [B.P.] [0.61 mm; 37.5 MN/m2.]

7.15 (C). A thin plate of light alloy and 200 mm diameter is firmly clamped around its periphery. Under service conditions the plate is to be subjected to a uniform pressure p of 20 kN/m2 acting normally over its whole surface area.

Determine the required minimum thickness t of the plate if the following design criteria apply;

(a) the maximum deflection is not to exceed 6 mm;

(b) the maximum radial stress is not to exceed 50 MN/m2.

7.16 (C). Determine equations for the maximum deflection and maximum radial stress for a circular plate, radius R, subjected to a distributed pressure of the form q = K/r. Assume simply supported edge conditions:

7.17 (C). The cover of the access hole for a large steel pressure vessel may be considered as a circular plate of 500 mm diameter which is firmly clamped around its periphery. Under service conditions the vessel operates with an internal pressure of 0.65 MN/m2.

Determine the minimum thickness of plate required in order to achieve the following design criteria:

(a) the maximum deflection is limited to 5 mm;

(b) the maximum radial stress is limited to 200 MN/m2. For the steel, E = 208 GN/m2 and v = 0.3.

You may commence your solution on the assumption that the deflection y at radius r for a uniform circular plate under the action of a uniform pressure q is given by:

0 0

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