Engineering Materials vol 1 Part 4 potx

Optimum combination of elastic properties for the mirror support Consider the selection of the material for the mirror backing of a 200-inch 5m diameter telescope.. We want to identify

Case studies of modulus-limited design 67

Fig 7.1 The British infrared telescope at Mauna Kea, Hawaii The picture shows the housing for the 3.8 m diameter mirror, the supporting frame, and the interior of the aluminium dome with its sliding 'window' (0 1979

by Photolabs, Royal Observatory, Edinburgh.)

Optimum combination of elastic properties for the mirror support

Consider the selection of the material for the mirror backing of a 200-inch (5m) diameter telescope We want to identify the material that gives a mirror which will distort by less than the wavelength of light when it is moved, and has minimum weight We will limit ourselves to these criteria alone for the moment - we will leave the problem of grinding the parabolic shape and getting an optically perfect surface to

the development research team

At its simplest, the mirror is a circular disc, of diameter 2a and mean thickness t,

simply supported at its periphery (Fig 7.2) When horizontal, it will deflect under its own weight M ; when vertical it will not deflect significantly We want this distortion (which changes the focal length and introduces aberrations into the mirror) to be small

enough that it does not significantly reduce the performance of the mirror In practice, this means that the deflection 6 of the mid-point of the mirror must be less than the

wavelength of light We shall require, therefore, that the mirror deflect less than = 1 km

at its centre This is an exceedingly stringent limitation Fortunately, it can be partially

overcome by engineering design without reference to the material used By using counterbalanced weights or hydraulic jacks, the mirror can be supported by distributed forces over its back surface which are made to vary automatically according to the attitude of the mirror (Fig 7.3) Nevertheless, the limitations of this compensating system still require that the mirror have a stiffness such that 6 be less than 10 km You will find the formulae for the elastic deflections of plates and beams under their own weight in standard texts on mechanics or structures (one is listed under 'Further Reading' at the end of this chapter) We need only one formula here: it is that for the deflection, 6, of the centre of a horizontal disc, simply supported at its

Fig 7.3 The distortion of the mirror under its own weight can be corrected by applying forces (shown as arrows) to the back surface

Case studies of modulus-limited design 69

periphery (meaning that it rests there but is not clamped) due to its own weight It is:

where p is the density of the material We can make it smaller by reducing the thickness t

-but there is a constraint: if we reduce it too much the deflection 6 of eqn (7.1) will be too

great So we solve eqn (7.1) for t (giving the t which is just big enough to keep the deflection down to 6) and we substitute this into eqn (7.2) giving

(7.3)

Clearly, the only variables left on the right-hand side of eqn (7.3) are the material

properties p and E To minimise M, we must choose a material having the smallest possible value of

where MI is called the ’material index’

Let us now examine its values for some materials Data for E we can take from Table 3.1 in Chapter 3; those for density, from Table 5.1 in Chapter 5 The resulting values of the index M , are as shown in Table 7.1

Table 7.1 Mirror bocking for 200-inch telescope

7.8 2.5 2.7 2.5

2.0

1.85 0.6 0.1 1.5

1.54 0.58 0.53 0.48 0.45 0.15 0.13 0.1 3

1 .o

0.4

1 1 6.8 0.36

70 Engineering Materials 1

Conclusions

The optimum material is CF'RP The next best is polyurethane foam Wood is obviously impractical, but beryllium is good Glass is better than steel, aluminium or concrete (that is why most mirrors are made of glass), but a lot less good :!-tan beryllium, which

is used for mirrors when cost is not a concern

We should, of course, examine other aspects of this choice The mass of the mirror can be calculated from eqn (7.3) for the various materials listed in Table 7.1 Note that the polyurethane foam and the CFRP mirrors are roughly one-fifth the weight of the glass one, and that the structure needed to support a CRFP mirror could thus be as much as 25 times less expensive than the structure needed to support an orthodox glass mirror

Now that we have the mass M , we can calculate the thickness t from eqn (7.2) Values of

f for various materials are given in Table 7.1 The glass mirrar has to be about 1 m thick (and real mirrors are about this thick); the CFRP-backed mirror need only be 0.38 m thick The polyurethane foam mirror has to be very thick - although there is no reason why one could not make a 6 m cube of such a foam

Some of the above solutions - such as the use of Polyurethane foam for mirrors - may

at first seem ridiculously impractical But the potential cost-saving (Urn5 m or US$7.5 m per telescope in place of Urn120 m or US$180 m) is so attractive that they are worth examining closely There are ways of casting a thin film of silicone rubber, or of epoxy, onto the surface of the mirror-backing (the polyurethane or the CFRP) to give an optically smooth surface which could be silvered The most obvious obstacle is the lack

of stability of polymers - they change dimensions with age, humidity, temperature and

so on But glass itself can be foamed to give a material with a density not much larger than polyurethane foam, and the same stability as solid glass, so a study of this sort can suggest radically new solutions to design problems by showing how new classes of materials might be used

CASE STUDY 2: MATERIALS SELECTION TO GIVE A BEAM OF A GIVEN STIFFNESS WITH MINIMUM WEIGHT

Introduction

Many structures require that a beam sustain a certain force F without deflecting more

than a given amount, 6 If, in addition, the beam forms part of a transport system - a plane or rocket, or a train - or something which has to be carried or moved - a rucksack for instance - then it is desirable, also, to minimise the weight

In the following, we shall consider a single cantilever beam, of square section, and will analyse the material requirements to minimise the weight for a given stiffness The results are quite general in that they apply equally to any sort of beam of square section, and can easily be modified to deal with beams of other sections: tubes, I-beams, box-sections and so on

Case studies of modulus-limited design 71

Fig 7.4 The elastic deflection 8 of a cantilever beam of length I under an externally imposed force F:

Ana I y si s

The square-section beam of length 1 (determined by the design of the structure, and

thus fixed) and thickness t (a variable) is held rigidly at one end while a force F (the maximum service force) is applied to the other, as shown in Fig 7.4 The same texts that list the deflection of discs give equations for the elastic deflection of beams The formula we want is

(7.7)

The mass of the beam, for given stiffness F / S , is minimised by selecting a material with

the minimum value of the material index

(7.8)

The second column of numbers in Table 7.2 gives values for M2

72 Engineering Materials 1

Table 7.2 Data for beam of given stiffness

The table shows that wood is one of the best materials for stiff beams - that is why it

is so widely used in small-scale building, for the handles of rackets and shafts of golf- clubs, for vaulting poles, even for building aircraft Polyurethane foam is no good at all -the criteria here are quite different from those of the first case study The only material which is clearly superior to wood is CFRP - and it would reduce the mass of the beam very substantially: by the factor 0.17/0.09, or very nearly a factor of 2 That is why CFRP is used when weight-saving is the overriding design criterion But as we shall see

in a moment, it is very expensive

Why, then, are bicycles not made of wood? (There was a time when they were.) That

is because metals, and polymers, too, can readily be made in tubes; with wood it is more difficult The formula for the bending of a tube depends on the mass of the tube

in a different way than does that of a solid beam, and the optimisation we have just performed - which is easy enough to redo - favours the tube

Case studies of modulus-limited design 73

(although this may neglect certain aspects of manufacture) Thus

The beam of minimum price is therefore the one with the lowest value of the index

z

(7.9)

(7.10) Values for M 3 are given in Table 7.2, with prices taken from the table in Chapter 2

Conclusions

Concrete and wood are the cheapest materials to use for a beam having a given stiffness Steel costs more; but it can be rolled to give I-section beams which have a much better stiffness-to-weight ratio than the solid square-section beam we have been analysing here This compensates for steel's rather high cost, and accounts for the interchangeable use of steel, wood and concrete that we talked about in bridge

construction in Chapter 1 Finally, the lightest beam (CFRP) costs more than 100 times that of a wooden one - and this cost at present rules out CFRP for all but the most specialised applications like aircraft components or sophisticated sporting equipment But the cost of CFRP falls as the market for it expands If (as now seems possible) its market continues to grow, its price could fall to a level at which it would compete with metals in many applications

Further reading

M E Ashby, Materials Selection in Mechanical Design, Pergamon Press, Oxford, 1992 (for material

M E Ashby and D Cebon, Case Studies in Materials Selection, Granta Design, Cambridge, 1996

S Marx and W Pfau, Observatories of the World, Blandford Press, Poole, Dorset, 1982 (for

Roawle's Formulas for Stress and Strain, 6th edn, McGraw-Hill, 1989

indices)

telescopes)

C Yield strength, tensile strength, hardness and ductility

- both so that we can design structures which will withstand normal service loads

without any permanent deformation, and so that we can design rolling mills, sheet

presses, and forging machinery which will be strong enough to impose the desired deformation onto materials we wish to form To study this, we pull carefully prepared samples in a tensile-testing machine, or compress them in a compression machine (which we will describe in a moment), and record the stress required to produce a given

strain

Linear and non-linear elasticity; anelastic behaviour

Figure 8.1 shows the stress-strain curve of a material exhibiting perfectly linear elastic

behaviour This is the behaviour characterised by Hooke's Law (Chapter 3) All solids

are linear elastic at small strains - by which we usually mean less than 0.001, or 0.1% The slope of the stress-strain line, which is the same in compression as in tension, is of

.- Area = elastic energy, Ue' stored

78 Engineering Materials 1

- 1

Fig 8.2 Stress-strain behoviour for a non-linear elastic solid The axes are calibrated for a material such as rubber

course Young’s Modulus, E The area (shaded) is the elastic energy stored, per unit

volume: since it is an elastic solid, we can get it all back if we unload the solid, which behaves like a linear spring

Figure 8.2 shows a non-linear elastic solid Rubbers have a stress-strain curve like this, extending to very large strains (of order 5) The material is still elastic: if unloaded, it follows the same path down as it did up, and all the energy stored, per unit volume, during loading is recovered on unloading - that is why catapults can be as lethal as they are

Finally, Fig 8.3 shows a third form of elastic behaviour found in certain materials This

is called anelastic behaviour All solids are anelastic to a small extent: even in the rkgime where they are nominally elastic, the loading curve does not exactly follow the unloading

curve, and energy is dissipated (equal to the shaded area) when the solid is cycled Sometimes this is useful - if you wish to damp out vibrations or noise, for example; you

*

Area = energy dissipated

per cycle as heat,

I/ ‘6 irde

Fig 8.3 Stress-strain behaviour for an anelastic solid The axes are calibrated for fibreglass

The yield strength, tensile strength, hardness and ductility 79

can do so with polymers or with soft metals (like lead) which have a high damping capacity (high anelastic loss) But often such damping is undesirable - springs and bells, for instance, are made of materials with the lowest possible damping capacity (spring steel, bronze, glass)

Load-extension curves for non-elastic (plastic) behaviour

Rubbers are exceptional in behaving reversibly, or almost reversibly, to high strains; as

we said, almost all materials, when strained by more than about 0.001 (0.1%), do something irreversible: and most engineering materials deform plastically to change their shape

permanently If we load a piece of ductile metal (like copper), for example in tension, we get the following relationship between the load and the extension (Fig 8.4) This can be

F = O

Fig 8.4 Load-extension curve for a bar of ductile metal (e.g annealed copper) pulled in tension

demonstrated nicely by pulling a piece of plasticine (a ductile non-metallic material) Initially, the plasticine deforms elastically, but at a small strain begins to deform plastically, so that if the load is removed, the piece of plasticine is permanently longer than it was at the beginning of the test: it has undergone plastic deformation (Fig 8.5)

If you continue to pull, it continues to get longer, at the same time getting thinner because in plastic deformation volume is conserved (matter is just flowing from place to

place) Eventually, the plasticine becomes unstable and begins to neck at the maximum

load point in the force-extension curve (Fig 8.4) Necking is an instability which we shall look at in more detail in Chapter 11 The neck then grows quite rapidly, and the load that the specimen can bear through the neck decreases until breakage takes place The two pieces produced after breakage have a total length that is slightly less than the length jusf before breakage by the amount of the elastic extension produced by the terminal load

Why this great difference in behaviour? After all, we are dealing with the same material in either case

Fig 8.6

The yield strength, tensile strength, hardness and ductility 81

True stress-strain curves for plastic flow

The apparent difference between the curves for tension and compression is due solely

to the geometry of testing If, instead of plotting load, we plot load divided by the actual area of the specimen, A, at any particular elongation or compression, the two curves become much more like one another In other words, we simply plot true stress (see Chapter 3)

as our vertical co-ordinate (Fig 8.7) This method of plotting allows for the thinning of the material when pulled in tension, or the fattening of the material when compressed

strains; it represents a drawing out of the tensile specimen from lo to 1.5 lo, but a

squashing down of the compressive specimen from lo to 0.510 The material of the compressive specimen has thus undergone much more plastic deformation than the material in the tensile specimen, and can hardly be expected to be in the same state, or

to show the same resistance to plastic deformation The two conditions can be compared properly by taking small strain increments