Materials Selection Without Shape

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Materials Selection Without Shape

• ...when function is independent of shape...

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Selection Procedure
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Performance Indices

• Component performance described by the objective
  functionp f(Functional requirement, F),
  (Geometric parameters, G), (Materials properties,
  M)
• p may mean mass, volume, cost or life, etc.
• If F, G and M are not inter-related, i.e. p
  f1(F)?f2(G)f3(M), then the choice of material is
  independent of geometric details of the design.
• p can then be optimized by optimizing f3, called
  performance index.

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Example to illustrate the Procedure - A Tie Rod

• Mass of rod, m Al?
• Tie rod must be able to carry a stress
• The lightest tie rod without failing under P is
  that with largest performance index, M ?f /?
• For a light stiff tie rod, M E/?

Sf safety factor
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Example - A Light Stiff Column
A?r2

• Buckling load
• Mass of column,
• For buckling, M E1/2/?
• Note the changes in M due to changes in loading
  direction while the geometry remains unchanged

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Common Features of the Steps for the 2 Examples

• The length, l, of the rod is specified
• The mass, m, of the rod is to be minimized
• Write the objective function, i.e. the equation
  for m.
• The constraints are either no yielding or no
  buckling under the prescribed load, P
• The free variables (geometric parameters in these
  cases) are eliminated

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Procedure for Deriving a Performance Index

• Identify the attribute to be maximized or
  minimized
• Develop equation for this attribute in terms of
  the functional requirements, the geometry and the
  material properties (the objective function)
• Identify the free (unspecified) variables
• Identify the constraints rank them in order of
  importance
• Develop equations for the constraints (no yield
  no fracture no buckling, maximum heat capacity,
  cost below target, etc.)

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Procedure for Deriving a Performance Index (cont.)

• Substitute for the free variables from the
  constraints into the objective function
• Group the variables into three groups functional
  requirements, F, G and M, thus ATTRIBUTE ?
  f(F,G,M)
• Read off the performance index, expressed as a
  quantity M, to be maximized
• Note that a full solution is not necessary in
  order to identify the material property group

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Some Commonly Used Performance Indices
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Procedure for Selecting Materials (Primary
Constraints)

• Some non-negotiable constraints exists, e.g.
  operating temperature, conductivity, etc.
• Either P gt Pcirt or P lt Pcrit
• These constraints appear as horizontal or
  vertical lines on materials selection chart
• Those satisfying the constraints are in the
  viable search region

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Procedure for Selecting Materials (Primary
Constraints)
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Procedure - Performance Maximizing Criteria

• To seek in the search region the materials which
  maximize the performance
• e.g. Performance Index for tie is E/? ( C)
• log E log ? log C represents a set of
  straight lines, known as design guidelines on MS
  chart for various C.
• All materials lying on the same guideline perform
  equally well those above are better and those
  below are worse.

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Procedure - Performance Maximizing Criteria
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Multiple Constraints

• Most materials selection problems are
  overconstrained, i.e. more constraints than free
  variables.
• For aircraft wing spar, weight must be minimized,
  but with constraints on stiffness, strength,
  toughness, etc.
• Performance maximization can be done in steps by
  considering the most important constraint first
  and apply the second constraint to the subset,
  and so on.

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Multiple Constraints

• The materials in the search region become the
  candidates for the next stage of the selection
  process
• Judgement is needed for prioritizing the
  constraints and the size of the subset in each
  stage.

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Reduce the Need for Judgement for Multiple
Constraints Problems

• E.g. one free variable, two constraints
• The ratio of the two performance indices is
  therefore fixed by the functional and geometric
  requirements

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Example for Multiple Constraints

• A tie loaded in tension
• minimum weight without failing or elastic
  deformation less than u
• (specified!)

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Multiple Design Goals

• There are always more than one quantity to be
  optimized, e.g. weight, cost, safety, etc.
• One possible way is to assign weighting factors
  to each goal, e.g. weight (10) and cost (6), etc.
• A more objective way is to convert all goals into
  the same currency, e.g. small weight can reduce
  transportation cost, and means less fuel cost,
  etc.

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Summary

• Fully constrained problem ? identify performance
  index to be maximized or minimized
• Over constrained problem ? optimize in stages or,
  preferably, derive the coupling equation for the
  performance indices
• Multiple goals problem ? convert the design goals
  into common currency