Functionally Graded Materials in Engineering - Structural Interfaces and Attachments in Biology - page 33

Fig. 2.2 Optimum fiber distribution in the facings of a square sandwich plate resulting in an
18.3% increase of the buckling load ( left ) and in an 8.7% increase of the fundamental frequency
( right ). Different colors correspond to different fiber volume fractions varying from 30 to 70%
resemble the case where the edges are supported by stringers of negligible out-of-
plane stiffness and high axial stiffness. The same in-plane boundary conditions
were adopted for the couple of unloaded edges. In the vibration problem, all edges
were prevented from both tangential and normal in-plane displacements. Such a
situation is encountered if the plate is supported by stringers and frames, while
being constrained by adjacent structures from expansion or contraction.
The plate was modeled with 1,089 four-node, reduced integration shell elements
using the ABAQUS commercial finite element code. Local volume fraction of
fibers in the facing layers of the modified FGM plate was varied using the “material
properties” module of ABAQUS [ 96 ]. A Constrained Minimization (CONMIN)
was performed to optimize the plate using the Method of Feasible Directions that is
built in the DAKOTA code [ 97 ]. The optimization conducted in this study was
limited, i.e., while the volume fractions of fibers varied over the surface of the
facings, all layers remained identical. Furthermore, the orientation of the layers was
not altered. Thus, it may be possible to further improve the outcome of the
procedure outlined below.
The optimum distribution of fibers in the layers of the facings is shown in Fig. 2.2
for the buckling problem where the panel is subject to uniaxial compression and for
the free vibration problemwhere we maximize the fundamental frequency. The fiber
distribution was changed piecewise in all layers, i.e., further improvements could
be possible using different fiber content in different layers, varying the fiber content
as a monotonous function of coordinate, or using ellipsoidal boundaries of the
regions with different fiber fractions. Nevertheless, even with the present optimiza-
tion approach, the buckling load was increased by 18.3% and the frequency by 8.7%,
without any weight penalty. Furthermore, while the optimization could be refined,
the present piecewise optimization complies with technological requirements to