Appendix: deriving material indices

This appendix describes how material indices are derived. You can find out more about them and their use in the first two texts listed under Further Reading.

The performance of a component is characterized by a performance equation called the objective function. The performance equation con­tains a group of material properties. This group is the material indices of the problem. Sometimes the "group" is a single property; thus if the per­formance of a beam is measured by its stiffness, the performance equation contains only one property, the elastic modulus E. More commonly, the performance equation contains a group of two or more properties. Familiar examples are the specific stiffness, E/p, (where E is Young’s modulus and p is the density), and the specific strength, ay/p (where ay is the yield strength or elastic limit), but there are many others. For reasons that will become apparent, we express the indices in a form for which a minimum, not a maximum, is sought.

Recall that the life energy and emissions for transport systems are domin­ated by the fuel consumed during use. The lighter the system is made, the less fuel it consumes and the less carbon it emits. So a good starting point is minimum weight design, subject, of course, to the other necessary con­straints of which the most important here have to do with stiffness and strength. We consider the generic components shown in Figure 8.17: ties, panels, and beams, loaded as shown. The derivation when the objective is that of minimizing embodied energy or material cost follows in a similar way.

Minimizing mass: a light, stiff tie rod. A material is sought for a cylindri­cal tie rod that must be as light as possible (Figure 8.17a). Its length Lo is specified and it must carry a tensile force F without extending elastically by more than S. Its stiffness must be at least S* = F/S. We are free to choose the cross-section area A and, of course, the material. The design require­ments, translated, are listed in Table 8.6.

We first seek an equation that describes the quantity to be minimized, here the mass m of the tie. This equation, the objective function, is

Generic components. (a) A tie, a tensile component; (b) a panel, loaded in

bending; (c) and (d) beams, loaded in bending.

 Table 8.6 Design requirements for the light, stiff tie rod Function Tie rod Constraints Stiffness S* specified (a functional constraint) Length L0 specified (a geometric constraint) Objective Minimize mass Free variables Choice of material Choice of cross-section area A

where p is the density of the material of which it is made. We can reduce the mass by reducing the cross-section, but there is a constraint: the section-area A must be sufficient to provide a stiffness of S*, which, for a tie, is:

S = > S* (8.14)

Lo

where E is Young’s modulus. If the material has a low modulus, a large A is needed to give the necessary stiffness; if E is high, a smaller A is needed. But which gives the lower mass? To find out, we eliminate the free variable A between these two equations, giving

( P_

, E,

 Table 8.7 Design requirements for the light, stiff panel Function Panel Constraints Stiffness S* specified (a functional constraint) Length L and width b specified (a geometric constraint) Objective Minimize mass Free variables Choice of material Choice of panel thickness h

Both S* and Lo are specified. The lightest tie that will provide a stiffness S* is that made of the material with the smallest value of the index

Mti = E (8.16a)

provided that they also meet all other constraints of the design. If the con­straint is not stiffness but strength, the index becomes

where ay is the yield strength. That means that the best choice of material for the lightest tie that can support a load F without yielding is that with the smallest value of this index.

The mode of loading that most commonly dominates in engineering is not tension but bending—think of floor joists of buildings, wing spars of aircraft, or shafts of golf clubs and racquets. The index for bending differs from that for tension, and this (significantly) changes the optimal choice of material. We start by modeling a panel, specifying stiffness and seeking to minimize its embodied energy.

Minimizing mass: a light, stiff panel. A panel is a flat slab, like a table­top. Its length L and width b are specified, but its thickness h is free. It is loaded in bending by a central load F (Figure 8.17b). The stiffness con­straint requires that it must not deflect more than S. The objective is to achieve this with minimum mass, m. Table 8.7 summarizes the design requirements.

The objective function for the mass of the panel is the same as that for the tie:

m = AL p = bhL p

Its bending stiffness S must be at least S*:

S = – S* (8-i7)

Here C1 is a constant that depends only on the distribution of the loads and I is the second moment of area, which, for a rectangular section, is

bh3

I = — (8.18)

12

We can reduce the mass by reducing h, but only so far that the stiffness constraint is no longer met. Using the last two equations to eliminate h in the objective function gives

The quantities S*, L, b, and C1 are all specified; the only freedom of choice left is that of the material. The best materials for a light, stiff panel are those with the smallest values of

M,1 = ~p3 (8-2°a)

Repeating the calculation with a constraint of strength rather than stiff­ness leads to the index

MP2 = P-2 (8.20b)

ay

These don’t look much different from the previous indices, p/E and p/ay but they are; they lead to different choices of material, as we shall see in a moment. For now, note the procedure. The in-plane dimensions of the panel were specified, but we were free to vary the thickness h. The objective is to minimize its mass, m. Use the stiffness constraint to eliminate the free variable, here h. Then read off the combination of material properties that appears in the objective function—the equation for the mass. It sounds easy, and it is—as long as you are clear from the start what the constraints are, what you are trying to maximize or minimize, and which parameters are specified and which are free.

Now for another bending problem, in which the freedom to choose shape is rather greater than for the panel.

 Table 8.8 Design requirements for the light, stiff beam Function Beam Constraints Stiffness S* specified (a functional constraint) Length L (geometric constraints) Section shape square Objective Minimize mass Free variables Choice of material Area A of cross-section

Minimizing mass: a light, stiff beam. Beams come in many shapes: solid rectangles, cylindrical tubes, I-beams, and more. Some of these have too many free geometric variables to directly apply the previous method. However, if we constrain the shape to be self-similar (such that all dimen­sions change in proportion as we vary the overall size), the problem becomes tractable again. We therefore consider beams in two stages: first, to identify the optimum materials for a light, stiff beam of a prescribed simple shape (a square section); second, we explore how much lighter it could be made, for the same stiffness, by using a more efficient shape.

Consider a beam of square section A = b x b that may vary in size, but the square shape is retained. It is loaded in bending over a span of fixed length L with a central load F (Figure 8.17c ) . The stiffness constraint is again that it must not deflect more than S under the load F, with the object­ive that the beam should again be as light as possible. Table 8.8 summar­izes the design requirements.

Proceeding as before, the objective function for the embodied energy is:

m = AL p = b2 L p

The bending stiffness S of the beam must be at least S*:

S = > S* (8.21)

L3

where C is a constant; we don’t need is value. The second moment of area, I, for a square section beam is

= A2 12 12

For a given length L, the stiffness S* is achieved by adjusting the size of the square section. Now eliminating b (or A) in the objective function for the mass gives

The quantities S*, L, and C1 are all specified or constant; the best mater­ials for a light, stiff beam are those with the smallest values of the index Mb, where

Mh = EP-2 (8.24a)

Repeating the calculation with a constraint of strength rather than stiff­ness leads to the index

Mb2 273 (8.24b)

° y

This analysis was for a square beam, but the result in fact holds for any shape, so long as the shape is held constant. This is a consequence of Equation 8.21: for a given shape, the second moment of area I can always be expressed as a constant times A2, so changing the shape merely changes the constant C1 in Equation 8.23, not the resulting index.

As noted, real beams have section shapes that improve their efficiency in bending, requiring less material to get the same stiffness. By shaping the cross-section, it is possible to increase I without changing A. This is achieved by locating the material of the beam as far from the neutral axis as possible, as in thin-walled tubes or I-beams (Figure 8.17d). Some mater­ials are more amenable than others to being made into efficient shapes. Comparing materials on the basis of the index in Mb therefore requires some caution: materials with lower values of the index may "catch up" by being made into more efficient shapes. So we need to get an idea of the effect of shape on bending performance.

Figure 8.18 shows a solid square beam of cross-section area A. If we turn the same area into a tube, as shown on the right of the figure, the mass of the beam is unchanged. The second moment of area, I, however, is now much greater—and so is the stiffness (Equation 8.21). We define the ratio of I for the shaped section to that for a solid square section with the same area (and thus mass) as the shape factor Ф. The more slender the shape,

FIGURE 8.18

The effect of section shape on bending stiffness EI: a square section beam compared, left, with a tube of the same area (but 2.5 times stiffer) and, right, a tube with the same stiffness (but 4 times lighter).

 Table 8.9 The effect of shaping on stiffness and mass of beams in different structural materials Material Typical maximum shape factor (stiffness relative to that of a solid square beam) Typical mass ratio by shaping (relative to that of a solid square beam) Steels 64 1/8 Al alloys 49 1/7 Composites (GFRP, CFRP) 36 1/6 Wood 9 1/3

the larger is Ф, but there is a limit—make it too thin and the flanges will buckle—so there is a maximum shape factor for each material that depends on its properties. Table 8.9 lists some typical values.

Shaping is used to make structures lighter; it is a way to get the same stiffness with less material. The mass ratio is given by the reciprocal of the square root of the maximum shape factor, Ф~1/2 (because C1, which propor­tional to the shape factor, appears as (C1)-1/2 in Equation 8.23). Table 8.9 lists the factors by which a beam can be made lighter, for the same stiffness, by shaping. Metals and composites can all be improved significantly (though the metals do a little better), but wood has more limited potential because it is more difficult to shape it into efficient, thin-walled shapes. So, when we compare materials for light, stiff beams using the index in Equation 8.24, we find that the performance of wood is not as good as it looks, because other

materials can be made into more efficient shape. Composites (particularly CFRP) have attractive (i. e., low) values of all the indices Mt, Mp, and Mb, but this advantage relative to metals is reduced a little by the effect of shape.

Minimizing embodied energy. When the objective is to minimize embod­ied energy rather than mass, the indices change. If the embodied energy of the material is Hm MJ/kg, the energy embodied in a component of mass m is just mHm. The objective function for the energy H embodied in the tie, panel, or beam then becomes

H = mHm = ALHm p (8.25)

Proceeding along the same steps as for minimum mass then leads to indices that have the form of Equations 8.16, 8.20, and 8.24, with p replaced by Hmp, as in Table 8.3.

Minimizing material cost. When, instead, the objective is to minimize cost rather than mass, the indices change again. If the material price is Cm \$/kg, the cost of the material to make a component of mass m is just mCm. The objective function for the material cost C of the tie, panel, or beam then becomes

C = mCm = ALCm p (8.26)

Proceeding as before then leads to indices that have the form of Equations 8.16, 8.20, and 8.24 with p replaced by Cm p, as in Table 8.3. (It must be remembered that the material cost is only part of the cost of a shaped component; there is also the manufacturing cost—the cost to shape, join, and finish it.)

8.9 Exercises

E.8.1. What is meant by an objective and what by a constraint in the requirements for a design? How do they differ?

E.8.2. Describe and illustrate the Translation step of the material selec­tion strategy.

E.8.3. Bikes come in many forms, each aimed at a particular sector of the market:

■ Sprint bikes

■ Touring bikes

■ Mountain bikes

■ Shopping bikes

■ Children’s bikes

■ Folding bikes

Use your judgment to identify the primary objective and the constraints that must be met for each of these.

E.8.4. You are asked to design a fuel-saving cooking pan with the goal of wasting as little heat as possible while cooking. What objective would you choose, and what constraints would you recommend must be met?

E.8.5. Formulate the constraints and objective you would associate with the choice of material to make the forks of a racing bicycle.

E.8.6. What is meant by a material index?

E.8.7. The objective in selecting a material for a panel of given in-plane dimensions for the lid casing of an ultrathin portable computer is that of minimizing the panel thickness h while meeting a constraint on bend­ing stiffness, S*, to prevent damage to the screen. What is the appropri­ate material index?

E.8.8. Plot the index for a light, stiff panel on a copy of the modulus/ density chart of Figure 8.11, positioning the line such that six materials are left above it, excluding ceramics because of their brittleness. Which six do you find? To what material classes do they belong?

E.8.9. Panels are needed to board up the windows of an unused build­ing. The panels should have the lowest possible embodied energy but be strong enough to deter an intruder who, in attempting to break in, will load the panels in bending. Which index would you choose to guide choice?

Plot the index on the strength/embodied energy chart of Figure 8.14 , positioning the line to find the best choice, excluding ceramics because of their brittleness. Which six do you find? To what material classes do they belong?

E.8.10. A material is required for a disposable fork for a fast-food chain. List the objective and the constraints that you would see as important in this application.

E.8.11. A designer seeks a material for a disposable drinking cup, with a goal of minimizing the embodied energy. When held between fingers and thumb, the cylindrical cup deflects like a panel in bending. Plot the

appropriate index onto the modulus/embodied energy and the strength/ embodied energy chart, using common sense to apply other necessary constraints, and make a selection.

E.8.12. Show that the index for selecting materials for a strong panel with the dimensions shown in Figure 8.17c, loaded in bending, with the minimum embodied energy content, is

Hmp

1/2

To do so, rework the panel derivation in Section 8.10, replacing the stiffness constraint with a constraint on failure load F requiring that it exceed a chosen value F*, where where C2 is a constant and the other symbols have the meaning used in the text.

E.8.13. Use the chart E-Hmp of Figure 8.13 to find the metal with a modulus E greater than 100 GPa and the lowest embodied energy per unit volume.

E.8.14. A maker of polypropylene (PP) garden furniture is concerned that the competition is stealing part of his market by claiming that the "traditional" material for garden furniture, cast iron, is much less energy and CO2 intensive than the PP. A typical PP chair weighs 1.6 kg; one made of cast iron weighs 11 kg. Use the data sheets for these two mate­rials in Chapter 12 of the book to find out who is right. Remember the warning about precision at the start of Chapter 12.

If the PP chair lasts five years and the cast iron chair lasts 25 years, does the conclusion change?

Updated: October 2, 2015 — 8:36 pm