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B59EH2 Mechanical Engineering Science 8 Part 1: Mechanics of Materials Topic 2: Thermal stress, plasticity and creep Summary 2.1 Thermal stress, bolt loading and fatiguelimit design Assignment 2.2 Plastic deformation of metals

2.3 Creep of metals Linear thermal expansion z ly lz T lx T+TT

x y Constrained thermal expansion Stiffness of a rod A F Constrained thermal expansion L

T (a) RB T+TT (b) RA Constrained thermal expansion T+TT (c)

t RA T+TT (d) r Differential thermal expansion same material, different TT Structure Skin

TT2 TT1 2 1 2* 2 1* 1 Differential thermal expansion same material, different TT

2 1 Differential thermal expansion same material, different TT TT2 TT1 2 1

2* 2 1* 1 Differential thermal expansion same material, different TT Structure Skin TT2 TT1

2 1 2* 2 1* 1 Differential thermal expansion same material, different TT Structure

Skin TT2 TT1 2 1 Differential thermal expansion same material, different TT Differential thermal

expansion same material, different TT TT2 TT1 2 1 Differential thermal expansion different materials, same TT Differential thermal expansion different

materials, same TT Differential thermal expansion different materials, same TT Differential thermal expansion different materials, same TT Finally, the thermal stresses in the bolt and in the sleeve are: Bolt preloading W2

W2 Clamped members Wb Gasket W2 W2 Bolt preloading W2

W2 Clamped pieces Wb Gasket k1 k2 k3 W2 W2

kb Bolt preloading W2 W2 Bolt is preloaded (in tension) to W1 by the application of tightening torque. Clamped pieces

Wb Gasket W2 W2 An external load, W2, adds to the preload as well as reducing the compression in the clamped members. Sharing of load depends on the relative proportions of the parallel stiffnesses accounted for by the bolt and the clamped members:

Bolt preloading W2 W2 Clamped pieces Wb W2 Gasket W2

As the external load is increased, the bolt load will increase until it fails or until all of the elastic strain in the clamped members is taken up. At the separation load Wsep, the bolt is carrying all of the external load so the separation load can be found from: Worked example Ten M12 steel bolts are to be used to hold down the lid of a cast iron vessel through a zinc gasket and the bolts are preloaded to 22 kN at 21C with atmospheric pressure inside the vessel .

Worked example (a) Find the load in the bolts if the internal pressure is increased to 1.4 MPa (gauge) (b) Find the load in the bolts if the temperature is increased to 121C with the vessel under atmospheric pressure (c) Find the load in the bolts if the pressure in (b) is increased to 1.4 MPa (d) Find the internal pressure at which leakage would occur (i) at 21C and (ii) at 121C Worked example The bolt stiffness:

The clamped member stiffness: Stiffness ratio: Worked example (a) Find the load in the bolts if the internal pressure is increased to 1.4 MPa (gauge) Increasing the internal pressure applies a load W2 to each bolt equal to 1/10 pressure the internal area of the lid: So bolt load: Worked example

(b) Find the load in the bolts if the temperature is increased to 121C with the vessel under atmospheric pressure Without constraint, the temperature change would cause the bolt length to increase by an amount: and the clamped members by an amount: Worked example (b) Find the load in the bolts if the temperature is increased to 121C with the vessel under atmospheric pressure However, the lengths of the bolt and the clamped members must be the

same, so the bolt load must increase. If the new bolt load is : Similarly, the clamped members will be put into additional compression: Worked example (b) Find the load in the bolts if the temperature is increased to 121C with the vessel under atmospheric pressure The new lengths of the bolt and the clamped members must be the same, so : giving:

Worked example (c) Find the load in the bolts if the pressure in (b) is increased to 1.4 MPa Previously Applying the pressure to the new preload : Worked example (d) Find the internal pressure at which leakage would occur (i) at 21C and (ii) at 121C The separation pressure occurs when Wb = W2

which is 22.5kN at 21C and about 24kN at 121C. The corresponding pressures are 1.79MPa and 1.91MPa What is fatigue? Crack growth under alternating stress main parameters are stress range, S, the difference between maximum and minimum stress, and presence of concentrators. Stress S Time What is fatigue?

Crack growth under alternating stress main parameters are stress range and presence of concentrators. What is fatigue? Cracks grow with time until failure occurs when remaining section cannot resist maximum stress: Max Gaulliers tarty bike http://www.tartybikes.co.uk/product.php?product_id=10383&category_id=21 http://en.wikipedia.org/wiki/Mountain_bike_trials http://www.youtube.com/watch?v=Z19zFlPah-o http://www.youtube.com/watch?v=Cj6ho1-G6tw Three approaches to fatigue analysis

Lifetime design, based on S-N curves used for designing structures where fatigue is manageable (i.e. numbers of cycles <108 in lifetime). Crack growth rate design, based on fracture mechanics used to assess the time taken for a crack to propagate from a specified size to critical size. Fatigue limit design using a Goodman diagram, or similar used to avoid fatigue entirely. Three approaches to fatigue analysis Lifetime design, based on S-N curves used for designing structures where fatigue is manageable (i.e. numbers of cycles <108 in lifetime). Crack growth rate design, based on fracture

mechanics used to assess the time taken for a crack to propagate from a specified size to critical size. Fatigue limit design using a Goodman diagram, or similar used to avoid fatigue entirely. Fatigue limit design Can only be used with materials which show a fatigue limit: Fatigue limit design Stress a m Time

Fatigue limit design Stress m a Time log S 2a(e) log N

Fatigue limit design Stress amplitude y,t a,e Fatigue-safe region y,c Compression y,t Mean stress

Tension u Fatigue limit design Supposed to avoid this: Worked example on fatigue limit design A high strength steel is to be used in a machine bolted connection where the expected frequencies of vibration are such that it is necessary to design below the fatigue limit. A combination of testing and analysis has shown that the maximum and minimum bolt tensions expected as result of machine vibrations will vary with the preload as follows:

Pre-tension (kN) 0 8.4 16.8 25.2 33.6 42.0 Minimum tension (kN)

0 8.4 16.8 25.2 33.6 42.0 Maximum tension (kN)

50.4 50.4 50.4 50.4 58.8 67.2 Worked example on fatigue limit design The bolts are 20 mm diameter and the tensile and compressive yield

stresses of the bolt material are 650 MPa and 700 MPa respectively. The ultimate tensile strength of the bolt material is 800 MPa. Plot the above data on a Goodman Diagram and determine if it is possible to select a suitable pre-tension for fatigue-safe design. If not, comment on how you might achieve a fatigue-safe design. Pre-tension (kN) 0 8.4 16.8 25.2

33.6 42.0 Minimum tension (kN) 0 8.4 16.8 25.2 33.6

42.0 Maximum tension (kN) 50.4 50.4 50.4 50.4 58.8

67.2 Worked example on fatigue limit design Simply dividing load by cross-sectional area, the stresses can be determined as follows: Pre-tension (kN) 0 8.4 16.8 25.2

33.6 42.0 Minimum stress (MPa) 0 27 53.5 80.2

107 133.7 Maximum stress (MPa) 160 160 160 160

187 214 Stress amplitude (MPa) 80 66.5 53.3 39.9

40 40.1 Mean stress (MPa) 80 93.5 106.8 120.1

147 173.5 Worked example on fatigue limit design Worked example on fatigue limit design Plotting on Goodman Diagram: 100 Stress amplitude (MPa) 90

80 Yield 70 End. (Mean) 60 End. (-2SD) 50 Bolts

40 Linear (Yield) 30 Linear (End. (Mean)) 20 Linear (End. (-2SD)) 10 0

0 500 Mean stress (MPa) 1000

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