## 52 Formulating a property profile

The steps are as follows:

1. Function. Identify the primary function of the component for which a material is sought. A beam carries bending moments; a heat-exchanger tube transmits heat; a bus-bar transmits electric current.

2. Objective. Identify the objective. This is the first and most important quantity you wish to minimize or maximize. Commonly, it is weight or cost; but it could be energy absorbed per unit volume (a compact crash barrier); or heat transfer per unit weight (a light heat exchanger) - it depends on the application.

3. Constraints. Identify the constraints. These are performance goals that must be met, and which therefore limit the optimization process of step 2. Commonly these are: a required value for stiffness, S; for the load, F, or moment, M, or torque, T, or pressure, p, that must be safely supported; a given operating temperature implying a lower limit for the maximum use temperature, Tmax, of the material; or a requirement that the component be electrically insulating, implying a limit on its resistivity, R.

It is essential to distinguish between objectives and constraints. As an example, in the design of a racing bicycle, minimizing weight might be the objective with stiffness, toughness, strength and cost as constraints ('as light as possible without costing more than \$500'). But in the design of a shopping bicycle, minimizing cost becomes the objective, and weight becomes a constraint ('as cheap as possible, without weighing more than 25 kg').

4. Free variables. The first constraint is one of geometry: the length, I, and the width, b, of the panel are specified above but the thickness, t, is not - it is a free variable.

### 5. Lay out this information as in Table 5.1.

6. Property limits. The next three constraints impose simple property limits; these are met by choosing materials with adequate safe working temperature, which are electrical insulators and are non-magnetic.

7. Material indices. The final constraint on strength ('plastic yielding') is more complicated. Strength can be achieved in several ways: by choice of material, by choice of area of the cross-section, and by choice of cross-section shape (rib-stiffened or sandwich panels are examples), all of which

Table 5.1 Design requirements

Function

Panel to support electronic signal-generating equipment (thus carries bending moments) Minimize weight

Must have length, I, and width, b,

Must operate between -20°C and 120°C

Must be electrically insulating

Must be non-magnetic

Must not fail by plastic yielding

The thickness, t, of the panel

Objective Constraints

Free variable

Table 5.2 Deriving material indices: the recipe (Ashby, 1999)

(a) Identify the aspect of performance, P (mass, cost, energy, etc.) to be maximized or minimized, defining the objective (mass, in the example of Table 5.1)

(b) Develop an equation for P (called the objective function)

(c) Identify the free variables in the objective function. These are the variables not specified by the design requirement (the thickness, t, of the panel in the example of Table 5.1)

(d) Identify the constraints. Identify those that are simple (independent of the variables in the objective function) and those which are dependent (they depend on the free variables in the objective function)

(e) Develop equations for the dependent constraints (no yield; no buckling, etc.)

(f) Substitute for the free variables from the equations for the dependent constraints into the objective function, eliminating the free variable(s)

(g) Group the variables into three groups: functional requirements, F, geometry, G, and material properties, M, thus:

(h) Read off the material index, M, to be maximized or minimized impact the objective since they influence weight. This constraint must be coupled to the objective. To do this, we identify one or more material indices appropriate to the function, objective and constraints. The material index allows the optimization step of the selection. The method, in three stages, is as follows:

(i) Write down an equation for the objective

(ii) Eliminate the free variable(s) in this equation by using the constraints

(iii) Read off the grouping of material properties (called the material index) which maximize or minimize the objective.

A more detailed recipe is given in Table 5.2. Indices for numerous standard specifications are listed in the Appendix at the end of this Design Guide.

5.3 Two examples of single-objective optimization

### Panel of specified stiffness and minimum mass

The mode of loading which most commonly dominates in engineering is not tension, but bending. Consider the performance metric for a panel of specified length, i, and width, b (Figure 5.1), and specified stiffness, with the objective of minimizing its mass, m. The mass is m = btp (5.1)

F/unit width

F/unit width t

Figure 5.1 A panel of length, i, width, b, and thickness, t, loaded in bending by a force, F, per unit width

Figure 5.1 A panel of length, i, width, b, and thickness, t, loaded in bending by a force, F, per unit width where t is the thickness of the panel and p is the density of the material of which it is made. The length, i, width, b, and force, F, per unit width are specified; the thickness, t, is free. We can reduce the mass by reducing t, but there is a lower limit set by the requirement that the panel must meet the constraint on its bending stiffness, S, meaning that it must not deflect more than U under a load Fb. To achieve this we require that

where S* is the desired bending stiffness, E is Young's modulus, B1 is a constant which depends on the distribution of load (tabulated in Chapter 6, Section 6.3) and I is the second moment of the area of the section. This, for a panel of section b x t, is

bJL 12

Using equations (5.2) and (5.3) to eliminate t in equation (5.1) gives the performance equation for the performance metric, m:

This equation for the performance metric, m, is the objective function - it is the quantity we wish to minimize.

All the quantities in equation (5.4) are specified by the design except the group of material properties in the last bracket, p/E1/3. This is the material index for the problem. The values of the performance metric for competing materials scale with this term. Taking material M0 as the reference (the incumbent in an established design, or a convenient standard in a new one), the performance metric of a competing material M1 differs from that of M0 by the factor

where the subscript '0' refers to M0 and the '1' to M1. Panel of specified strength and minimum mass

If, for the panel of Figure 5.1, the constraint were that of bending strength rather than stiffness, the constraining equation becomes that for failure load, Ff, per unit width, meaning the onset of yielding:

where B2, like B1, is a constant that depends only on the distribution of the load; it is tabulated in Chapter 6, Section 6.4. The performance metric, again, is the mass, m:

where oy the yield strength of the material of which the panel is made and Ff b is the desired minimum failure load. Here the material index is p/oly/2. Taking material M0 as the reference again, the performance metric of a competing material M1 differs from that of M0 by the factor m1 (p1/o\/12)

More generally, if the performance metrics for a reference material M0 are known, those for competing materials are found by scaling those of M0 by the ratio of their material indices. There are many such indices. A few of those that appear most commonly are listed in Table 5.3. More are listed in the Appendix.

Table 5.3 Material indicesa

Function, objective and constraint (and example)

Index

Tie, minimum weight, stiffness prescribed

(cable support of a lightweight stiffness-limited tensile structure)

Tie, minimum weight, stiffness prescribed

(cable support of a lightweight strength-limited tensile structure)

Beam, minimum weight, stiffness prescribed (aircraft wing spar, golf club shaft)

Beam, minimum weight, strength prescribed (suspension arm of car)

Panel, minimum weight, stiffness prescribed (car door panel)

Panel, minimum weight, strength prescribed (table top)

Column, minimum weight, buckling load prescribed (push-rod of aircraft hydraulic system)

Spring, minimum weight for given energy storage (return springs in space applications)

Precision device, minimum distortion, temperature gradients prescribed

(gyroscopes; hard-disk drives; precision measurement systems)

Heat sinks, maximum thermal flux, thermal expansion prescribed (heat sinks for electronic systems)

Electromagnet, maximum field, temperature rise and strength prescribed

(ultra-high-field magnets; very high-speed electric motors)

pE'o2

a'X 1'KCpP

Note: p = density; E = Young's modulus; oy = elastic limit; X = thermal conductivity; a = thermal expansion coefficient; k = electrical conductivity; Cp = specific heat.

aThe derivation of these and many other indices can be found in Ashby (1999).

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