Metal Foams: A Design Guide

132 Metal Foams: A Design Guide
1. Form a rectangular grid of points (t/ℓ, c/ℓ) covering the potential design
space, such as that in Figure 10.11(a).
2. Determine whether each point (t/ℓ, c/ℓ) satisfies all the constraints (10.27):
if it does not, reject it; if it does, evaluate .
3. The point which produces the minimum will be close to the optimum
design. The grid of points can be further refined if necessary.
Dependence on load index and core density
The above procedures can be used to bring out the regimes within which the
design is limited by stiffness or strength. Some assistance is provided by reorganizing
equation (10.27) and replotting the design diagram (Figure 10.11(b)).
The constraints are:
⊲face yield⊳ ⊲1/5⊳⊲c/ℓ⊳ � ⊲1/8⊳⊲t/ℓ⊳ 1
⊲core shear⊳
�
fy
⊲1/5⊳⊲c/ℓ⊳ � ⊲1/2⊳ / c �
y
⊲deflection⊳
�
f f
2 y /Ef y
υ/ℓ � 5
C 2
B1⊲t/ℓ⊳⊲c/ℓ⊳ /Gf
�
B2⊲c/ℓ⊳
⊲10.28a⊳
⊲10.28b⊳
⊲10.28c⊳
By plotting ⊲1/5⊳⊲c/ℓ⊳ against t/ℓ using logarithmic axes (Figure 10.11(b))
the respective roles of the relative properties for the core, f y / c y and the stiffness
constraint, υ/ℓ, become apparent. The minimum weight design lies along
DCE. The precise location depends on the specifics of the core properties and
the allowable stiffnesses. As the core properties deteriorate, at given allowable
stiffness, line (3) moves up and the minimum weight is now likely to reside
along segment CE, being controled by the core. Correspondingly, for a given
core material, as the stiffness allowable increases, line (1) displaces downward
towards the origin, again causing the minimum weight to reside along CE and
be core-controled. Conversely, improvements in the core properties and/or a
lower allowable stiffness cause the minimum weight design to reside along
DC, such that the panel is stiffness-controled.
Specific results are plotted in Figure 10.12 for four values of the relative
core density, with υ/ℓ D 0.01. Again, the face sheet material and cellular
core are taken to be aluminum with ˛2 D 1. Note that the designs are indeed
stiffness-limited at low values of the load index and strength-limited at high
load indices. The lower (solid) portion of each curve, below the change in
slope, coincides with the former; the two strength constraints being inactive.
Moreover, for c/ s D 0.3, the deflection constraint is active over the entire
range. At higher load indices (dotted above the change in slope), either or both
strength constraints (10.27(a)) and (10.27(b)) are active, with the deflection at
the design load being less than υ.