Materials Selection Without
Shape
...when function is independent
of shape...
Selection Procedure
Performance Indices

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Component performance described by the
objective function
p = 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.
Example to illustrate the
Procedure - A Tie Rod
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Mass of rod, m = Al
P f
Tie rod must be able to carry a stress 
A Sf
  
P
P

m  S f P  
l  
 

 f 
F


G 
M
Sf: safety factor
The lightest tie rod without failing under
P is that with largest performance index,
M = f /
For a light stiff tie rod, M = E/
Example - A Light Stiff Column
P
A=r2
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Buckling load
2
2
4

Pcrit n EI n E r 
P


P


2
2 
1
Sf
Sfl
S f l  4 
4
2

1  l


2
Mass of column, m  Al  2S f P     1 2 
 n  E


 
For buckling, M = E1/2/
Note the changes in M due to changes in
loading direction while the geometry
remains unchanged
Common Features of the Steps
for the 2 Examples
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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
Procedure for Deriving a
Performance Index
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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.)
Procedure for Deriving a
Performance Index (cont.)
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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
Some
Commonly
Used
Performance
Indices
Procedure for Selecting Materials
(Primary Constraints)
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Some non-negotiable constraints exists,
e.g. operating temperature, conductivity,
etc.
Either P > Pcirt or P < Pcrit
These constraints appear as horizontal or
vertical lines on materials selection chart
Those satisfying the constraints are in the
viable search region
Procedure for Selecting Materials
(Primary Constraints)
Procedure - Performance
Maximizing Criteria
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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.
Procedure - Performance
Maximizing Criteria
Multiple Constraints
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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.
Multiple Constraints
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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.
Reduce the Need for Judgement for
Multiple Constraints Problems

E.g. one free variable, two constraints
 p  f1 F   f 2 G   f 3 M 
f 3 M  g1 F   g 2 G 



g 3 M  f1 F   f 2 G 
 p  g1 F   g 2 G   g 3 M 

The ratio of the two performance indices is
therefore fixed by the functional and
geometric requirements
Example for Multiple Constraints
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A tie loaded in tension
minimum weight
without failing or elastic
deformation less than u

  






m

S
F
l



f 



m   S f F  l 2   
 u  E


  

f
 
E /    l

f
 u
(specified!)
Multiple Design Goals
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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.
Summary
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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’