Each panel of the wingbox is also constrained not to buckle for specified load conditions. This is approximated through a simplified panel-level buckling analysis of the stiffened panels applied on every component in the wingbox. This buckling analysis considers both longitudinal and shear buckling modes through several different buckling mechanisms, in- cluding: inter-stiffener panel buckling, stiffener buckling, and global panel buckling. The first step of the procedure is to compute the panels’ critical buckling loads for each mech- anism, both in compression (N1,cr) and shear (N12,cr). These values are dependent on the panels’ local stiffness as well as geometric properties. Once these critical values have been obtained the constraint is then applied to every element in the panel as a conservative en- velope of the form:
B(N1, N12) = N 2 12 N2 12,cr + N1 N1,cr ≤ 1, (4.9)
for all three buckling mechanisms. This method has the benefit of being cheap to compute relative to performing a full buckling eigenvalue analysis for each panel. Like the failure constraints, the buckling constraints are evaluated at the centroid of every element and then aggregated over each component group using aKSfunction. More details of this approach
CHAPTER 5
Manufacturing considerations
In this chapter, the consideration ofAFP-specific manufacturing constraints for tow-steered layups are introduced and formulated. Since part of the goal of this research is to provide a scaled tow-steered wingbox structure for manufacturing through gradient-based optimiza- tions, these constraints are critical to ensuring that subsequent designs could be realistically produced with modern-dayAFPmachines.
In particular, three constraints are considered to ensure that the tow in the laminate varies smoothly and is thus realizable with anAFPmachine. The first of these constraints is on the minimum turning radius of the tow path,Rmin. This constraint is defined by the manufacturer to prevent the prepreg tow from twisting over on itself or puckering out of plane as the tow is laid down by the head of theAFPmachine. An example of tow puck- ering can be seen in Figure 5.1a. More aggressive values of minimum turning radii can be achieved by laying down paths with narrower prepreg tow, but for the same layup area this leads to an increase in manufacturing time and cost. The next constraint considered is related to the presence of gaps and overlaps of the prepreg tow. As the prepreg tow is laid down there will be regions in the layup that will either be void or where two portions of ad- jacent tow overlap due to fixed width of tow. Depending on the structural application, gaps and overlaps may not be desirable, as these regions can potentially lead to unwanted thick- ness variations if they are allowed to build up during the layup process, see Figure 5.1b.
AFPmachines typically have preprogrammed rules for cutting and adding tow to prevent gaps and overlaps from becoming excessively large. However, if these regions happen fre- quently in the layup, they can lead to an unacceptable increase in manufacturing time and effort. For this reason, we also wish to develop a constraint to bound how frequently tows need to be cut and added by theAFPmachine. The final constraint on the tow-steered de- sign is on the ply drop rate of the design. If plies are dropped too quickly spatially through the layup of the laminate, undesired inter-laminar stress concentrations can occur, leading to a reduction in strength. This restricts how quickly the panel thickness is allowed to vary across the surface.
(a) Tow puckering for curved tow path1 (b) Thickness build up due to overlaps [103]
Figure 5.1: Example of tow-steering defects due to AFP layup process
The manufacturing constraints derived throughout the remainder of this chapter will be for a general 2D tow pattern. All of these results can be easily extended to curved 2D surfaces by replacing the coordinatesx and y with arc-length parametric coordinates u and v. These new parametric coordinates differ slightly from the non-dimensional parametric coordinates,ξ and η, introduced in Chapter4, in thatu and v will now have units of length. We will now continue the derivation with the 2D Cartesian coordinates,x and y.
The ply drop rate is the most straightforward of the three manufacturing constraints. This can be enforced by constraining the magnitude of the thickness gradient distribution, ||∇tp(x, y)||. The remaining two constraints, tow-path curvature and tow cut/add rates, require further work to derive. In the remainder of this chapter, it will be shown that an arbitrary tow-steered pattern can be defined as a 2D unit vector field. This definition allows for the manufacturing constraints on tow-path curvature and gaps/overlaps to be related to the curl and divergence of the vector field, respectively. Figure5.2 provides a preview of how these two quantities can be used to identify regions of manufacturing difficulty in a layup.
1
(a) Curl contour (b) Divergence contour
Figure 5.2: The curl, κ, and divergence, ψ, contours can be used to identify regions of manufacturing difficulty.