Composite Materials Research Progress

82
Michaël Bruyneel
design variables, while plies thicknesses are kept constant. The optimization consists in
minimizing the laminate’s strain energy density, i.e. maximizing its stiffness. The evolution
of this objective function with respect to the 2 angles θ1 and θ2 is reported in Figure 8.1, with
the initial and optimal design points. A restriction is imposed on the relative variation of the 2
design variables. The optimization problem writes :
1 T T 1 T
min ε 0 Aε 0+
κ Dκ
θ 2 2
θ 2 − θ1
≤ 45
(8.1)
0. 001 θ ≤180
i = 1,
2
≤ i
where the stiffness matrices A, B and D, and the laminate’s strain and curvature were
previously defined in Section 3.
Strain energy
density (N/mm)
θ2
θ1
Initial design
Optimal design
Solution
Figure 8.1. Variation of the strain energy density in the symmetric laminate subjected to the load case
of Table 8.1.
In-plane load case
( N 1,
N 2 , N6
)
in N/mm
Table 8.1. Problem’s definition: load case and initial design
Out-of-plane load case
( M 1,
M 2 , M 6 )
in N
Initial orientations
θ = ( θ1,
θ 2 )
in degrees
23.3°
22.3°
Initial thicknesses
t = ( t1,
t2
)
in mm
(2000,0,1000) (0,500,0) (45,135) (1,2)
In this application the laminate is subjected not only to in-plane but also to out-of-plane
loadings. Since the plies thicknesses are not identical (Table 8.1) the objective function is not
symmetric with regards to the axis θ 1 = θ 2 (Figure 8.2).