## Structural Partitions Of

4.5 (B). A steel rotor disc of uniform thickness 50 mm has an outer rim of diameter 800 mm and a central hole of diameter 150 mm. There are 200 blades each of weight 2 N at an effective radius of 420 mm pitched evenly around the periphery. Determine the rotational speed at which yielding first occurs according to the maximum shear stress criterion.

Yield stress in simple tension = 750 MN/m2.

The basic equations for radial and hoop stresses given in Example 4.4 may be used without proof.

4.6 (B). A rod of constant cross-section and of length 2a rotates about its centre in its own plane so that each end of the rod describes a circle of radius a. Find the maximum stress in the rod as a function of the peripheral speed V. [j(pio2a2).]

4.7 (B). A turbine blade is to be designed for constant tensile stress a under the action of centrifugal force by varying the area A of the blade section. Consider the equilibrium of an element and show that the condition is

where Aand r/, are the cross-sectional area and radius at the hub (i.e. base of the blade).

4.8 (B). A steel turbine rotor of 800 mm outside diameter and 200 mm inside diameter is 50 mm thick. The rotor carries 100 blades each 200 mm long and of mass 0.5 kg. The rotor runs at 3000 rev/min. Assuming the shaft to be rigid, calculate the expansion of the inner bore of the disc due to rotation and hence the initial shrinkage allowance necessary. [0.14 mm.]

4.9 (B). A steel disc of 750 mm diameter is shrunk onto a steel shaft of 80 mm diameter. The interference on the diameter is 0.05 mm.

(a) Find the maximum tangential stress in the disc at standstill.

(b) Find the speed in rev/min at which the contact pressure is zero.

(c) What is the maximum tangential stress at the speed found in (b)? [65 MN/m ; 3725; 65 MN/m2.]

4.10 (B). A flat steel turbine disc of 600 mm outside 'diameter and 120 mm inside diameter rotates at 3000 rev/min at which speed the blades and shrouding cause a tensile rim loading of 5 MN/m2. The maximum stress at this speed is to be 120 MN/m2. Find the maximum shrinkage allowance on the diameter when the disc is put on the shaft. [0.097 mm.]

4.11 (B). Find the maximum permissible speed of rotation for a steel disc of outer and inner radii 150 mm and 70 mm respectively if the outer radius is not to increase in magnitude due to centrifugal force by more than 0.03 mm. [7900 rev/min.]

4.12 (B). The radial and hoop stresses at any radius r for a disc of uniform thickness rotating at an angular speed co rad/s are given respectively by

/"- o where A and B are constants, v is Poisson's ratio and p is the density of the material. Determine the greatest values of the radial and hoop stresses for a disc in which the outer and inner radii are 300 mm and 150 mm respectively. Take w = 150 rad/s, v = 0.304 and p = 7470 kg/nr\ [U.L.I [1.56, 13.2 MN/m2.]

4.13 (B). Derive an expression for the tangential stress set up when a thin hoop, made from material of density p kg/m3, rotates about its polar axis with a tangential velocity of r m/s.

What will be the greatest value of the mean radius of such a hoop, made from fiat mild-steel bar. if the maximum allowable tensile stress is 45 MN/m- and the hoop rotates at 300 rev/min?

4.14 (C). Determine the hoop stresses at the inside and outside surfaces of a long thick cylinder inside radius = 75 mm. outside radius = 225 mm. which is rotated at 4000 rev/min.

Take v = 0.3 and p = 7470 kg/m\ [57.9, 11.9 MN/m2.]

4.15 (C). Calculate the maximum principal stress and maximum shear stress set up in a thin disc when rotating at 12 000 rev/min. The disc is of 300 mm outside diameter and 75 mm inside diameter.

Take v = 0.3 and p = 7470 kg/m\ [221. I 10.5 MN/m2.]

4.16 (B). A thin-walled cylindrical shell made of material of density p has a mean radius r and rotates at a constant angular velocity of w rad/s. Assuming the formula for centrifugal force, establish a formula for the circumferential (hoop) stress induced in the cylindrical shell due to rotation about the longitudinal axis of the cylinder and, if necessary, adjust the derived expression to give the stress in MN/m2.

A drum rotor is to be used for a speed of 3000 rev/min. The material is steel with an elastic limit stress of 248 MN/m- and a density of 7.8 Mg/m3. Determine the mean diameter allowable if a factor of safety of 2.5 on the elastic limit stress is desired. Calculate also the expansion of this diameter (in millimetres) when the shell is rotating.

For steel, E = 207 GN/m2. [I.Mech.E.] [0.718 m; 0.344 mm.]

4.17 (B). A forged steel drum. 0.524 m outside diameter and 19 mm wall thickness, has to be mounted in a machine and spun about its longitudinal axis. The centrifugal (hoop) stress induced in the cylindrical shell is not to exceed 83 MN/m2. Determine the maximum speed (in rev/min) at which the drum can be rotated.

4.18 (B). A cylinder, which can be considered as a thin-walled shell, is made of steel plate 16 mm thick and is 2.14 m internal diameter. The cylinder is subjected to an internal fluid pressure of 0.55 MN/m2 gauge and, at the same time, rotated about its longitudinal axis at 3000 rev/min. Determine:

(a) the hoop stress induced in the wall of the cylinder due to rotation;

(b) the hoop stress induced in the wall of the cylinder due to the internal pressure;

(c) the factor of safety based on an ultimate stress of the material in simple tension of 456 MN/m2.

Steel has a density of 7.8 Mg/m3. [89.5, 36.8 MN/m2; 3.6]

4.19 (B). The "bursting" speed of a cast-iron flywheel rim. 3 m mean diameter, is 850 rev/min. Neglecting the effects of the spokes and boss, and assuming that the flywheel rim can be considered as a thin rotating hoop, determine the ultimate tensile strength of the cast iron. Cast iron has a density of 7.3 Mg/m3.

A flywheel rim is to be made of the-same material and is required to rotate at 400 rev/min. Determine the maximum permissible mean diameter using a factor of safety of 8. [U.L.C.I.] [2.25 mm]

4.20 (B). An internal combustion engine has a cast-iron llywheel that can be considered to be a uniform thickness disc of 230 mm outside diameter and 50 mm inside diameter. Given that the ultimate tensile stress and density of cast iron are 200 N/mirr and 7180 kg/m3 respectively, calculate the speed at which the flywheel would burst. Ignore any stress concentration effects and assume Poisson's ratio for cast iron to be 0.25.

4.21 (B). A thin steel circular disc of uniform thickness and having a central hole rotates at a uniform speed about an axis through its centre and perpendicular to its plane. The outside diameter of the disc is 500 mm and its speed of rotation is 81 rev/s. If the maximum allowable direct stress in the steel is not to exceed 110 MN/m2 (I 1.00 h bar), determine the diameter of the central hole.

For steel, density p = 7800 kg/m3 and Poisson's ratio v = 0.3.

Sketch diagrams show ing the circumferential and radial stress distribution across the plane of the disc indicating the peak values and state the radius at which the maximum radial stress occurs. [B.P.] [264 mm.]

4.22 (B). (a) Prove that the differential equation for radial equilibrium in cylindrical coordinates of an element in a uniform thin disc rotating at w rad/s and subjected to principal direct stresses oy and o» is given by the following expression:

(b) A thin solid circular disc of uniform thickness has an outside diameter of 300 mm. Using the maximum shear strain energy per unit volume theory of elastic failure, calculate the rotational speed of the disc to just cause initiation of plastic yielding if the yield stress of the material of the disc is 300 MN/m2, the density of the material is 7800 kg/m3 and Poisson's ratio for the material is 0.3. [B.P.] [324 rev/s.]

### Thermal gradients

4.23 (C). Determine expressions for the stresses developed in a hollow disc subjected to a temperature gradient of the form 7" = Kr. What arc the maximum stresses for such a case if the internal and external diameters of the cylinder are 80 mm and 160 mm respectively: a = 12 x Kr6 per °C and E = 206.8 GN/m2.

The temperature at the outside radius is -50CC. [-34.5, 27.6 MN/m2.]

4.24 (C). Calculate the maximum stress in a solid magnesium alloy disc 60 mm diameter when the temperature rise is linear from 60°C at the centre to 90°C at the outside.

a = 7 x 10~6 per °C and E = 105 GN/m2. [7.4 MN/m2.]

4.25 (C). Calculate the maximum compressive and tensile stresses in a hollow steel disc. 100 mm outer diameter and 20 mm inner diameter when the temperature rise is linear from 100°C at the inner surface to 50 C at the outer surface.

a = 10 x 10~6 per °C and E = 206.8 GN/m2. [-62.9. +40.3 MN/m2.]

4.26 (C). Calculate the maximum tensile and compressive stresses in a hollow copper cylinder 20 mm outer diameter and 10 mm inner diameter when the temperature rise is linear from 0°C at the inner surface to 100°C at the outer surface.

a - 16 x 10"6 per °C and E = 104 GN/m2. [142. -114 MN/m2.]

4.27 (C). A hollow steel disc has internal and external diameters of 0.2 m and 0.4 m respectively. Determine the circumferential thermal stresses set up at the inner and outer surfaces when the temperature at the outside surface is 100°C. A temperature distribution through the cylinder walls of the form T = Kr may be assumed, i.e. when r = zero, T = zero.

For steel, E = 207 GN/m2 and a = 11 x 10"6 per °C.

What is the significance of (i) the first two terms of the stress eqns. (4.29) and (4.30), (ii) the remaining terms?

Hence comment on the relative magnitude of the maximum hoop stresses obtained in a high pressure vessel which is used for (iii) a chemical action which is exothermic, i.e. generating heat, (iv) a chemical reaction which is endothermic, i.e. absorbing heat. [63.2, —50.5 MN/m-.]

4.28 (C). In the previous problem sketch the thermal hoop and radial stress variation diagrams across the wall thickness of the disc inserting the numerical value of the hoop stresses at the inner, mean and outer radii, and also the maximum radial stress, inserting the radius at which it occurs.

["mean = "2.78 MN/m2, <xrmal = 9.65 MN/m2.]

4.29 (C). A thin uniform steel disc, 254 mm outside diameter with a central hole 50 mm diameter, rotates at 10000 rev/min. The temperature gradient varies linearly such that the difference of temperature between the inner and outer (hotter) edges of the plate is 46°C. For the material of the disc, E = 205 GN/m2, Poisson's ratio = 0.3 and the coefficient of linear expansion = 11 x 10~6 per °C. The density of the material is 7700 kg/m1.

Calculate the hoop stresses induced at the inner and outer surfaces. [176-12.1 MN/m-.]

4JO (C). An unloaded steel cylinder has internal and external diameters of 204 mm and 304 mm respectively. Determine the circumferential thermal stresses at the inner and outer surfaces where the steady temperatures are 200°C and 100°C respectively.

Take E = 207 GN/m2, a = 11 x 10"6 per °C and Poisson's ratio = 0.29.

The temperature distribution through the wall thickness may be regarded as follows:

With this form of temperature distribution, the radial and circumferential thermal stresses at radius r where the temperature is T are obtained from

B EaT B EaT Eab ar = A----and <rw = A H—r-----

4.31 (C). Determine the hoop stresses at the inside and outside surfaces of a long thick cylinder which is rotated at 4000 rev/min. The cylinder has an internal radius of 80 mm and an external radius of 250 mm and is constructed from steel, the relevant properties of which are given above.

How would these values be modified if, under service conditions, the temperatures of the inside and outside surfaces reached maximum levels of 40°C and 90°C respectively?

A linear thermal gradient may be assumed.

432 (C). .(a) Determine the wall thickness required for a high pressure cylindrical vessel. 800 mm diameter, in order that yielding shall be prevented according to the Tresca criterion of elastic failure when the vessel is subjected to an internal pressure of 450 bar.

(b) Such a vessel is now required to form part of a chemical plant and to contain exothermic reactions which produce a maximum internal temperature of 120°C at a reaction pressure of 450 bar, the outer surface being cooled to an "ambient" temperature of 20°C. In the knowledge that such a thermal gradient condition will introduce additional stresses to those calculated in part (a) the designer proposes to increase the wall thickness by 20% in order that, once again, yielding shall be prevented according to the Tresca theory. Is this a valid proposal?

You may assume that the thermal gradient is of the form T = a + br2 and that the modifying terms to the Lamé expressions to cover thermal gradient conditions are aE

for radial stress: --j / Trdr f aE f for hoop stress: — / Trdr — aET.

For the material of the vessel, ay = 280 MN/m2. a = 12 x 10~6 per °C and E = 208 GN/ra2.

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