## 2 The Basic Problem  
The analytical and finite element analysis (FEA) solutions to thin-wall buckling problems presented in previous works [12; 29; 30; 31; 32] assumed that the shell was uniform and free of defects. This assumption does not hold for additively-manufactured structures, as the nature of the AM process produces a patterned structure with natural inclusions and voids [32; 33]. Example geometry (single and double walled cylinders) produced by FDM from two common materials (acrylonitrile butadiene styrene (ABS) and polylactic acid (PLA)) is shown in Figure 1a-d. Figure 2a shows the basic parameters of this structure, while Figure 2b presents a series of parameter measurements from 50 sample cylinders made from both materials and wall thicknesses. The values of \(\rho\) and \(y\) (Figure 2) were measured using a PosiTector(r)(Defelsko Corp, Ogdensburg, New York, USA) surface profile gauge and calipers, while \(t=200\) um was set in the g-code when printing the samples. The parameters \(\rho\) and \(y\) were found to be close to the nominal values, as shown in Figure 2b and Table 1. After reviewing previous successful work (see [12]) on reinforcing thin-walled shells without greatly increasing the mass of the structure, two methods were selected for inclusion in this study: (1) light printed rectilinear infill (3% density with stacked-beam local geometry) (Figure 1e) and (2) polyurethane foam filling (Figure 1f). The infill structure was to be printed along with the shell, while a sprayed expanding polyurethane foam from Dow Chemical Company(r)(Midland, Michigan, USA) [34] was used for the foam infill samples. A series of cores were made with the foam (produced inside of open-ended PVC cylinders and allowed to cure for 24 h) to assess the uniformity of expansion and density; 10 randomly selected  
The wall thickness of 0.40 mm was produced using a single shell (Figures 1a,b and 2a), while the thicker wall was produced using two shells (Figures 1c,d and 2b). The specimens were produced using a CTC Creator FDM machine with a 0.40 mm brass extrusion nozzle, standard filament from Hatchbox, printing temperatures of 230 \({}^{\circ}\)C (ABS) and 200 \({}^{\circ}\)C (PLA), build plate temperatures of 100 \({}^{\circ}\)C (ABS) and 60 \({}^{\circ}\)C (PLA), print speeds of 50 mm/s, a layer thickness of 0.20 mm, and an acceleration and jerk of 300 and 8 mm/s\({}^{2}\)[16], respectively. The temperature and humidity during the print and testing were measured to be 23-24 \({}^{\circ}\)C and 48-54% throughout the several days needed to manufacture and test all the cases. All specimens were printed axially. The experiments were completed using a basic screw-driven press (Figure 4) designed for column buckling experiments with a 2500 N load cell providing output and a speed of 0.03 mm/s. The collected data for the experiments was the maximum load applied to the cylinder immediately before collapse; the full compressive curves for these cylinders were not collected, as only the critical load was needed for statistical analysis over a variety of sample configurations. A total of 32 samples were tested with no experimental failure nor repeated tests. In all cases, the wall collapse was dramatic and obvious, not simple local yielding; examples will be shown and discussed in Section 5.  
Figure 3: Example loadings (taken from [12] and reused under CC-by 4.0 license): (**a**) uniform axial (e.g., balanced windmill tower), (**b**) non-uniform axial (e.g., non-symmetric traffic light tower), (**c**) angled radial (full-length loading) (e.g., culvert under a road), and (**d**) angled radial (local loading) (e.g., grain bin transfer shaft with guide wires).