What are mechanical Properties of Wood

mechanical properties of wood-based composite materials and properties of manufactured woods
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Published Date:03-08-2017
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Special topic: wood 277 Chapter 26 Special topic: wood Introduction Wood is the oldest and still most widely used of structural materials. Its documented use in buildings and ships spans more than 5000 years. In the sixteenth century the demand for stout oaks for ship-building was so great that the population of suitable trees was seriously depleted, and in the seventeenth and eighteenth centuries much of Europe was deforested completely by the exponential growth in consumption of wood. Today the world production is about the same as that of iron and steel: roughly 9 10 tonnes per year. Much of this is used structurally: for beams, joists, flooring or supports which will bear load. Then the properties which interest the designer are the moduli, the yield or crushing strength, and the toughness. These properties, summarised in Table 26.1, vary considerably: oak is about 5 times stiffer, stronger and tougher than balsa, for instance. In this case study we examine the structure of wood and the way the mechanical properties depend on it, using many of the ideas developed in the preceding five chapters. Table 26.1 Mechanical properties of woods 1 1,2 1,3 1 Wood Density Young’s modulus Strength (MPa) Fracture toughness −3 1/2 (Mg m ) (GPa) to grain (MPa m ) to grain ⊥ to grain Tension Compression to grain ⊥ to grain Balsa 0.1–0.3 4 0.2 23 12 0.05 1.2 Mahogany 0.53 13.5 0.8 90 46 0.25 6.3 Douglas fir 0.55 16.4 1.1 70 42 0.34 6.2 Scots pine 0.55 16.3 0.8 89 47 0.35 6.1 Birch 0.62 16.3 0.9 – – 0.56 – Ash 0.67 15.8 1.1 116 53 0.61 9.0 Oak 0.69 16.6 1.0 97 52 0.51 4.0 Beech 0.75 16.7 1.5 – – 0.95 8.9 1 Densities and properties of wood vary considerably; allow ±20% on the data shown here. All properties vary with moisture content and temperature; see text. 2 Dynamic moduli; moduli in static tests are about two-thirds of these. 3 Anisotropy increases as the density decreases. The transverse strength is usually between 10% and 20% of the longitudinal.278 Engineering Materials 2 Fig. 26.1. The macrostructure of wood. Note the co-ordinate system (axial, radial, tangential). The structure of wood It is necessary to examine the structure of wood at three levels. At the macroscopic (unmagnified) level the important features are shown in Fig. 26.1. The main cells (“fibres” or “tracheids”) of the wood run axially up and down the tree: this is the direction in which the strength is greatest. The wood is divided radially by the growth rings: differences in density and cell size caused by rapid growth in the spring and summer, and sluggish growth in the autumn and winter. Most of the growing pro- cesses of the tree take place in the cambium, which is a thin layer just below the bark. The rest of the wood is more or less dead; its function is mechanical: to hold the tree 8 up. With 10 years in which to optimise its structure, it is perhaps not surprising that wood performs this function with remarkable efficiency. To see how it does so, one has to examine the structure at the microscopic (light microscope or scanning electron microscope) level. Figure 26.2 shows how the wood is made up of long hollow cells, squeezed together like straws, most of them parallel to the axis of the tree. The axial section shows roughly hexagonal cross-sections of the cells: the radial and the tangential sections show their long, thin shape. The structure appears complicated because of the fat tubular sap channels which run up the axis of the tree carrying fluids from the roots to the branches, and the strings of smaller cells called rays which run radially outwards from the centre of the tree to the bark. ButSpecial topic: wood 279 Fig. 26.2. The microstructure of wood. Woods are foams of relative densities between 0.07 and 0.5, with cell walls which are fibre-reinforced. The properties are very anisotropic, partly because of the cell shape and partly because the cell-wall fibres are aligned near the axial direction. Fig. 26.3. The molecular structure of a cell wall. It is a fibre-reinforced composite (cellulose fibres in a matrix of hemicellulose and lignin). neither is of much importance mechanically; it is the fibres or tracheids which give wood its stiffness, strength and toughness. The walls of these tracheid cells have a structure at a molecular level, which (but for the scale) is like a composite – like fibreglass, for example (Fig. 26.3). The role of the strong glass fibres is taken by fibres of crystalline cellulose, a high polymer (C H O ) , 6 10 5 n made by the tree from glucose, C H O , by a condensation reaction, and with a DP of 6 12 6 4 about 10 . Cellulose is a linear polymer with no cumbersome side-groups, so it crystal- lises easily into microfibrils of great strength (these are the fibres you can sometimes see in coarse paper). Cellulose microfibrils account for about 45% of the cell wall. The role of the epoxy matrix is taken by the lignin, an amorphous polymer (like epoxy), and by hemicellulose, a partly crystalline polymer of glucose with a smaller DP than cellulose; between them they account for a further 40% of the weight of the wood. The remain- ing 10–15% is water and extractives: oils and salts which give wood its colour, its smell, and (in some instances) its resistance to beetles, bugs and bacteria. The chemistry of wood is summarised in Table 26.2.280 Engineering Materials 2 Table 26.2 Composition of cell wall of wood Material Structure Approx. wt% Fibres Cellulose (C H O ) Crystalline 45 6 10 5 n Matrix Lignin Amorphous 20 Hemicellulose Semi-crystalline 20 Water Dissolved in the matrix 10 Extractives Dispersed in the matrix 5 Table 26.3 Properties of cell wall Property Axial Transverse −3 Density, r (Mg m ) 1.5 s Modulus, E (GPa) 35 10 s Yield strength, s (MPa) 150 50 y It is a remarkable fact that, although woods differ enormously in appearance, the composition and structure of their cell walls do not. Woods as different as balsa and −3 beech have cell walls with a density ρ of 1.5 Mg m with the chemical make-up given s in Table 26.2 and with almost the same elaborate lay-up of cellulose fibres (Fig. 26.3). The lay-up is important because it accounts, in part, for the enormous anisotropy of wood – the difference in strength along and across the grain. The cell walls are helically wound, like the handle of a CFRP golf club, with the fibre direction nearer the cell axis rather than across it. This gives the cell wall a modulus and strength which are large parallel to the axis of the cell and smaller (by a factor of about 3) across it. The properties of the cell wall are summarised in Table 26.3; it is a little less stiff, but nearly as strong as an aluminium alloy. Wood, then, is a foamed fibrous composite. Both the foam cells and the cellulose fibres in the cell wall are aligned predominantly along the grain of the wood (i.e. parallel to the axis of the trunk). Not surprisingly, wood is mechanically very anisotropic: the properties along the grain are quite different from those across it. But if all woods are made of the same stuff, why do the properties range so widely from one sort of wood to another? The differences between woods are primarily due to the differences in their relative densities (see Table 26.1). This we now examine more closely. The mechanical properties of wood All the properties of wood depend to some extent on the amount of water it contains. Green wood can contain up to 50% water. Seasoning (for 2 to 10 years) or kiln drying (for a few days) reduces this to around 14%. The wood shrinks, and its modulus andSpecial topic: wood 281 strength increase (because the cellulose fibrils pack more closely). To prevent move- ment, wood should be dried to the value which is in equilibrium with the humidity where it will be used. In a centrally heated house (20°C, 65% humidity), for example, the equilibrium moisture content is 12%. Wood shows ordinary thermal expansion, of −1 −1 course, but its magnitude (α = 5 MK along the grain, 50 MK across the grain) is small compared to dimensional changes caused by drying. Elasticity Woods are visco-elastic solids: on loading they show an immediate elastic deformation followed by a further slow creep or “delayed” elasticity. In design with wood it is usually adequate to treat the material as elastic, taking a rather lower modulus for long-term loading than for short loading times (a factor of 3 is realistic) to allow for the creep. The modulus of a wood, for a given water content, then depends principally on its density, and on the angle between the loading direction and the grain. Figure 26.4 shows how Young’s modulus along the grain (“axial” loading) and across the grain (“radial” or “tangential” loading) varies with density. The axial modulus varies linearly with density and the others vary roughly as its square. This means that the anisotropy of the wood (the ratio of the modulus along the grain to that across the grain) increases as the density decreases: balsa woods are very anisotropic; oak or beech are less so. In structural applications, wood is usually loaded along the grain: then only the axial modulus is important. Occasionally it is loaded across the grain, and then it is important to know that the stiffness can be a factor of 10 or more smaller (Table 26.1). Fig. 26.4. Young’s modulus for wood depends mainly on the relative density r/r . That along the grain s 2 varies as r/r ; that across the grain varies roughly as (r/r ) , like polymer foams. s s282 Engineering Materials 2 Fig. 26.5. (a) When wood is loaded along the grain most of the cell walls are compressed axially; (b) when loaded across the grain, the cell walls bend like those in the foams described in Chapter 25. The moduli of wood can be understood in terms of the structure. When loaded along the grain, the cell walls are extended or compressed (Fig. 26.5a). The modulus E of the wood is that of the cell wall, E , scaled down by the fraction of the section w s occupied by cell wall. Doubling the density obviously doubles this section, and there- fore doubles the modulus. It follows immediately that  ρ EE = (26.1) ws   ρ  s where ρ is the density of the solid cell wall (Table 26.3). s The transverse modulus E is lower partly because the cell wall is less stiff in this w⊥ direction, but partly because the foam structure is intrinsically anisotropic because of the cell shape. When wood is loaded across the grain, the cell walls bend (Fig. 26.5b,c). It behaves like a foam (Chapter 25) for which 2  ρ EE = . (26.2) ws ⊥   ρ  s The elastic anisotropySpecial topic: wood 283 E /E = (ρ /ρ). (26.3) w w⊥ s Clearly, the lower the density the greater is the elastic anisotropy. The tensile and compressive strengths The axial tensile strength of many woods is around 100 MPa – about the same as that of strong polymers like the epoxies. The ductility is low – typically 1% strain to failure. Compression along the grain causes the kinking of cell walls, in much the same way that composites fail in compression (Chapter 25, Fig. 25.5). The kink usually initiates at points where the cells bend to make room for a ray, and the kink band forms at an angle of 45° to 60°. Because of this kinking, the compressive strength is less (by a factor of about 2 – see Table 26.1) than the tensile strength, a characteristic of composites. Like the modulus, the tensile and compressive strengths depend mainly on the density (Fig. 26.6). The strength parallel to the grain varies linearly with density, for the same reason that the axial modulus does: it measures the strength of the cell wall, scaled by the fraction of the section it occupies, giving  ρ σσ = , (26.4) s  ρ  s where σ is the yield strength of the solid cell wall. s Figure 26.6 shows that the transverse crushing strength σ varies roughly as ⊥ Fig. 26.6. The compressive strength of wood depends, like the modulus, mainly on the relative density r/r . s 2 That along the grain varies as r/r ; that across the grain varies as (r/r ) . s s284 Engineering Materials 2 Fig. 26.7. The fracture toughness of wood, like its other properties, depends primarily on relative density. 3/2 That across the grain is roughly ten times larger than that along the grain. Both vary as (r/r ) . s 2  ρ σσ = . (26.5) ⊥ s   ρ  s The explanation is almost the same as that for the transverse modulus: the cell walls bend like beams, and collapse occurs when these beams reach their plastic collapse load. As with the moduli, moisture and temperature influence the crushing strength. The toughness The toughness of wood is important in design for exactly the same reasons that that of steel is: it determines whether a structure (a frame building, a pit prop, the mast of a yacht) will fail suddenly and unexpectedly by the propagation of a fast crack. In a steel structure the initial crack is that of a defective weld, or is formed by corrosion or fatigue; in a wooden structure the initial defect may be a knot, or a saw cut, or cell damage caused by severe mishandling. Recognising its importance, various tests have been devised to measure wood tough- ness. A typical static test involves loading square section beams in three-point bending until they fail; the “toughness” is measured by the area under the load–deflection curve. A typical dynamic test involves dropping, from an increasingly great height, a weight of 1.5 kg; the height which breaks the beam is a “toughness” – of a sort. Such tests (still universally used) are good for ranking different batches or species of wood; but they do not measure a property which can be used sensibly in design.Special topic: wood 285 The obvious parameter to use is the fracture toughness, K . Not unexpectedly, it IC 3/2 depends on density (Fig. 26.7), varying as (ρ/ρ ) . It is a familiar observation that s wood splits easily along the grain, but with difficulty across the grain. Figure 26.7 shows why: the fracture toughness is more than a factor of 10 smaller along the grain than across it. Loaded along the grain, timber is remarkably tough: much tougher than any simple polymer, and comparable in toughness with fibre-reinforced composites. There appear to be two contributions to the toughness. One is the very large fracture area due to cracks spreading, at right angles to the break, along the cell interfaces, giving a ragged fracture surface. The other is more important, and it is exactly what you would expect in a composite: fibre pull-out (Chapter 25, Fig. 25.6). As a crack passes across a cell the cellulose fibres in the cell wall are unravelled, like pulling thread off the end of a bobbin. In doing so, the fibres must be pulled out of the hemicellulose matrix, and a lot of work is done in separating them. It is this work which makes the wood tough. For some uses, the anisotropy of timber and its variability due to knots and other defects are particularly undesirable. Greater uniformity is possible by converting the timber into board such as laminated plywood, chipboard and fibre-building board. Summary: wood compared to other materials The mechanical properties of wood (a structural material of first importance because of the enormous scale on which it is used) relate directly to the shape and size of its cells, and to the properties of the composite-like cell walls. Loaded along the grain, the cell walls are loaded in simple tension or compression, and the properties scale as the density. But loaded across the grain, the cell walls bend, and then the properties depend on a power (3/2 or 2) of the density. That, plus the considerable anisotropy of the cell wall material (which is a directional composite of cellulose fibres in a hemicellulose/lignin matrix), explain the enormous difference between the modulus, strength and toughness along the grain and across it. The properties of wood are generally inferior to those of metals. But the properties per unit weight are a different matter. Table 26.4 shows that the specific properties of wood are better than mild steel, and as good as many aluminium alloys (that is why, for years, aircraft were made of wood). And, of course, it is much cheaper. Table 26.4 Specific strength of structural materials s E K y IC Material r r r Woods 20–30 120–170 1–12 Al-alloy 25 179 8–16 Mild steel 26 30 18 Concrete 15 3 0.08286 Engineering Materials 2 Further reading J. Bodig and B. A. Jayne, Mechanics of Wood and Wood Composites, Van Nostrand Reinhold, 1982. J. M. Dinwoodie, Timber, its Nature and Behaviour, Van Nostrand Reinhold, 1981. B. A. Meylan and B. G. Butterfield, The Three-dimensional Structure of Wood, Chapman and Hall, 1971. H. E. Desch, Timber, its Structure, Properties and Utilisation, 6th edition, Macmillan, 1985. Problems 26.1 Explain how the complex structure of wood results in large differences between the along-grain and across-grain values of Young’s modulus, tensile strength and fracture toughness. 26.2 What functions do the polymers cellulose, lignin and hemicellulose play in the construction of the cells in wood? 26.3 Discuss, giving specific examples, how the anisotropic properties of wood are exploited in the practical applications of this material.Design with materials 287 D. Designing with metals, ceramics, polymers and composites288 Engineering Materials 2Design with materials 289 Chapter 27 Design with materials Introduction Design is an iterative process. You start with the definition of a function (a pen, a hairdryer, a fuel pin for a nuclear reactor) and draw on your knowledge (the contents of this book, for instance) and experience (your successes and failures) in formulating a tentative design. You then refine this by a systematic process that we shall look at later. Materials selection is an integral part of design. And because the principles of me- chanics, dynamics and so forth are all well established and not changing much, whereas new materials are appearing all the time, innovation in design is frequently made possible by the use of new materials. Designers have at their disposal the range of materials that we have discussed in this book: metals, ceramics, polymers, and com- binations of them to form composites. Each class of material has its own strengths and limitations, which the designer must be fully aware of. Table 27.1 summarises these. Table 27.1. Design-limiting properties of materials Material Good Poor Metals Stiff (E ≈ 100 GPa) Yield (pure, s ≈ 1 MPa) → alloy y High E, K Ductile (e ≈ 20%) – formable Hardness (H ≈ 3s ) → alloy IC f y 1/2 1 Low s Tough (K 50 MPa m ) Fatigue strength (s = –s ) y IC e y 2 High MP (T ≈ 1000°C) Corrosion resistance → coatings m T-shock (DT 500°C) 1/2 Ceramics Stiff (E ≈ 200 GPa) Very low toughness (K ≈ 2 MPa m ) IC High E, s Very high yield, hardness (s 3 GPa) T-shock (DT ≈ 200°C) y y Low K High MP (Tm ≈ 2000°C) Formability → powder methods IC Corrosion resistant Moderate density Polymers Ductile and formable Low stiffness (E ≈ 2 GPa) Adequate s , K Corrosion resistant Yield (s = 2–100 MPa) y IC y Low E Low density Low glass temp (T ≈ 100°C) → creep g 1/2 Toughness often low (1 MPa m ) Composites Stiff (E 50 GPa) Formability High E, s , K Strong (s ≈ 200 MPa) Cost y IC y 1/2 but cost Tough (K 20 MPa m ) Creep (polymer matrices) IC Fatigue resistant Corrosion resistant Low density290 Engineering Materials 2 At and near room temperature, metals have well-defined, almost constant, moduli and yield strengths (in contrast to polymers, which do not). And most metallic alloys have a ductility of 20% or better. Certain high-strength alloys (spring steel, for in- stance) and components made by powder methods, have less – as little as 2%. But even this is enough to ensure that an unnotched component yields before it fractures, and that fracture, when it occurs, is of a tough, ductile, type. But – partly because of their ductility – metals are prey to cyclic fatigue and, of all the classes of materials, they are the least resistant to corrosion and oxidation. Historically, design with ceramics has been empirical. The great gothic cathedrals, still the most impressive of all ceramic designs, have an aura of stable permanence. But many collapsed during construction; the designs we know evolved from these failures. Most ceramic design is like that. Only recently, and because of more demanding struc- tural applications, have design methods evolved. In designing with ductile materials, a safety-factor approach is used. Metals can be used under static loads within a small margin of their ultimate strength with confid- ence that they will not fail prematurely. Ceramics cannot. As we saw earlier, brittle materials always have a wide scatter in strength, and the strength itself depends on the time of loading and the volume of material under stress. The use of a single, constant, safety factor is no longer adequate, and the statistical approach of Chapter 18 must be used instead. We have seen that the “strength” of a ceramic means, almost always, the fracture or crushing strength. Then (unlike metals) the compressive strength is 10 to 20 times larger than the tensile strength. And because ceramics have no ductility, they have a low tolerance for stress concentrations (such as holes and flaws) or for high contact stresses (at clamping or loading points, for instance). If the pin of a pin-jointed frame, made of metal, fits poorly, then the metal deforms locally, and the pin beds down, redistribut- ing the load. But if the pin and frame are made of a brittle material, the local contact stresses nucleate cracks which then propagate, causing sudden collapse. Obviously, the process of design with ceramics differs in detail from that of design with metals. That for polymers is different again. When polymers first became available to the engineer, it was common to find them misused. A “cheap plastic” product was one which, more than likely, would break the first time you picked it up. Almost always this happened because the designer used a polymer to replace a metal component, without redesign to allow for the totally different properties of the polymer. Briefly, there are three: (a) Polymers have much lower moduli than metals – roughly 100 times lower. So elastic deflections may be large. (b) The deflection of polymers depends on the time of loading: they creep at room temperature. A polymer component under load may, with time, acquire a perman- ent set. (c) The strengths of polymers change rapidly with temperature near room temper- ature. A polymer which is tough and flexible at 20°C may be brittle at the temper- ature of a household refrigerator, 4°C. With all these problems, why use polymers at all? Well, complicated parts perform- ing several functions can be moulded in a single operation. Polymer components canDesign with materials 291 be designed to snap together, making assembly fast and cheap. And by accurately sizing the mould, and using pre-coloured polymer, no finishing operations are neces- sary. So great economies of manufacture are possible: polymer parts really can be cheap. But are they inferior? Not necessarily. Polymer densities are low (all are near −3 1 Mg m ); they are corrosion-resistant; they have abnormally low coefficients of fric- tion; and the low modulus and high strength allows very large elastic deformations. Because of these special properties, polymer parts may be distinctly superior. Composites overcome many of the remaining deficiencies. They are stiff, strong and tough. Their problem lies in their cost: composite components are usually expensive, and they are difficult and expensive to form and join. So, despite their attractive properties, the designer will use them only when the added performance offsets the added expense. New materials are appearing all the time. New polymers with greater stiffness and toughness appear every year; composites are becoming cheaper as the volume of their production increases. Ceramics with enough toughness to be used in conventional design are becoming available, and even in the metals field, which is a slowly devel- oping one, better quality control, and better understanding of alloying, leads to materials with reliably better properties. All of these offer new opportunities to the designer who can frequently redesign an established product, making use of the prop- erties of new materials, to reduce its cost or its size and improve its performance and appearance. Design methodology Books on design often strike the reader as vague and qualitative; there is an implica- tion that the ability to design is like the ability to write music: a gift given to few. And it is true that there is an element of creative thinking (as opposed to logical reasoning or analysis) in good design. But a design methodology can be formulated, and when followed, it will lead to a practical solution to the design problem. Figure 27.1 summarises the methodology for designing a component which must carry load. At the start there are two parallel streams: materials selection and com- ponent design. A tentative material is chosen and data for it are assembled from data sheets like the ones given in this book or from data books (referred to at the end of this chapter). At the same time, a tentative component design is drawn up, able to fill the function (which must be carefully defined at the start); and an approximate stress analysis is carried out to assess the stresses, moments, and stress concentrations to which it will be subjected. The two streams merge in an assessment of the material performance in the tentat- ive design. If the material can bear the loads, moments, concentrated stresses (etc.) without deflecting too much, collapsing or failing in some other way, then the design can proceed. If the material cannot perform adequately, the first iteration takes place: either a new material is chosen, or the component design is changed (or both) to overcome the failing. The next step is a detailed specification of the design and of the material. This may require a detailed stress analysis, analysis of the dynamics of the system, its response292 Engineering Materials 2 Fig. 27.1. Design methodology.Design with materials 293 to temperature and environment, and a detailed consideration of the appearance and feel (the aesthetics of the product). And it will require better material data: at this point it may be necessary to get detailed material properties from possible suppliers, or to conduct tests yourself. The design is viable only if it can be produced economically. The choice of produc- tion and fabrication method is largely determined by the choice of material. But the production route will also be influenced by the size of the production run, and how the component will be finished and joined to other components; each class of material has its own special problems here; they were discussed in Chapters 14, 19, 24 and 25. The choice of material and production route will, ultimately, determine the price of the product, so a second major iteration may be required if the costing shows the price to be too high. Then a new choice of material or component design, allowing an altern- ative production path, may have to be considered. At this stage a prototype product is produced, and its performance in the market is assessed. If this is satisfactory, full-scale production is established. But the designer’s role does not end at this point. Continuous analysis of the performance of a compon- ent usually reveals weaknesses or ways in which it could be improved or made more cheaply. And there is always scope for further innovation: for a radically new design, or for a radical change in the material which the component is made from. Successful designs evolve continuously, and only in this way does the product retain a competit- ive position in the market place. Further reading (a) Design G. Pahl and W. Beitz, Engineering Design, The Design Council, 1984. V. Papanek, Design for the Real World, Random House, 1971. (b) Metals ASM Metals Handbook, 8th edition, American Society for Metals, 1973. Smithells’ Metals Reference Book, 7th edition, Butterworth-Heinemann, 1992. (c) Ceramics W. E. C. Creyke, I. E. J. Sainsbury, and R. Morrell, Design with Non-Ductile Materials, Applied Science Publishers, 1982. D. W. Richardson, Modern Ceramic Engineering, Marcel Dekker, 1982. (d) Polymers DuPont Design Handbooks, DuPont de Nemours and Co., Polymer Products Department, Wilmington, Delaware 19898, USA, 1981. ICI Technical Services Notes, ICI Plastics Division, Engineering Plastics Group, Welwyn Garden City, Herts., England, 1981.294 Engineering Materials 2 (e) Materials selection J. A. Charles and F. A. A. Crane, Selection and Use of Engineering Materials, 2nd edition, Butterworth- Heinemann, 1989. M. F. Ashby, Materials Selection in Mechanical Design, Pergamon, 1992. M. F. Ashby and D. Cebon, Case Studies in Materials Selection, Granta Design, 1996. Problems 27.1 You have been asked to prepare an outline design for the pressure hull of a deep- sea submersible vehicle capable of descending to the bottom of the Mariana Trench in the Pacific Ocean. The external pressure at this depth is approximately 100 MPa, and the design pressure is to be taken as 200 MPa. The pressure hull is to have the form of a thin-walled sphere with a specified radius r of 1 m and a uniform thickness t. The sphere can fail in one of two ways: external-pressure buckling at a pressure p given by b 2 t   pE = 03.,   b   r where E is Young’s modulus; yield or compressive failure at a pressure p given f by t   p = 2σ ,   ff   r where σ is the yield stress or the compressive failure stress as appropriate. f The basic design requirement is that the pressure hull shall have the min- imum possible mass compatible with surviving the design pressure. By eliminating t from the equations, show that the minimum mass of the hull is given by the expressions ρ   30.5 mr = 22., 9p   b b 05 .   E for external-pressure buckling, and  ρ  3 mr = 2πp , ff  σ  f for yield or brittle compressive failure. Hence obtain a merit index to meet the design requirement for each of the two failure mechanisms. You may assume 2 that the surface area of the sphere is 4πr . 0.5 Answers: E /ρ for external-pressure buckling; σ /ρ for yield or brittle compressive f failure. 27.2 For each material listed in the following table, calculate the minimum mass and wall thickness of the pressure hull of Problem 27.1 for both failure mechanisms at the design pressure.Design with materials 295 −3 Material E (GPa) s (MPa) Density, r (kg m ) f Alumina 390 5000 3900 Glass 70 2000 2600 Alloy steel 210 2000 7800 Titanium alloy 120 1200 4700 Aluminium alloy 70 500 2700 Hence determine the limiting failure mechanism for each material. Hint: this is the failure mechanism which gives the larger of the two values of t. What is the optimum material for the pressure hull? What are the mass, wall thickness and limiting failure mechanism of the optimum pressure hull? Answers: Material m (tonne) t (mm) m (tonne) t (mm) Limiting failure mechanism b b f f Alumina 2.02 41 0.98 20 Buckling Glass 3.18 97 1.63 50 Buckling Alloy steel 5.51 56 4.90 50 Buckling Titanium alloy 4.39 74 4.92 83 Yielding Aluminium alloy 3.30 97 6.79 200 Yielding The optimum material is alumina, with a mass of 2.02 tonne, a wall thickness of 41 mm and a limiting failure mechanism of external-pressure buckling. 27.3 Briefly describe the processing route which you would specify for making the pressure hull of Problem 27.2 from each of the materials listed in the table. Com- ment on any particular problems which might be encountered. You may assume that the detailed design will call for a number of apertures in the wall of the pressure hull.296 Engineering Materials 2 Chapter 28 Case studies in design 1. DESIGNING WITH METALS: CONVEYOR DRUMS FOR AN IRON ORE TERINAL Introduction The conveyor belt is one of the most efficient devices available for moving goods over short distances. Billions of tons of minerals, foodstuffs and consumer goods are handled in this way every year. Figure 28.1 shows the essentials of a typical conveyor system. The following data are typical of the largest conveyors, which are used for handling coal, iron ore and other heavy minerals. −1 Capacity: 5000 tonne h −1 Belt speed: 4 m s Belt tension: 5 tonne Motor rating: 250 k W Belt section: 1.5 m wide × 11 mm thick Distance between centres of tail drum and drive drum: 200 m Fig. 28.1. Schematic of a typical conveyor system. Because the belt tends to sag between the support rollers it must be kept under a constant tension T. This is done by hanging a large weight on the tension drum. The drive is supplied by coupling a large electric motor to the shaft of the drive drum via a suitable gearbox and overload clutch.

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