## 4 Design Method
### Local Buckling of Elliptical Sections  
The elastic critical local buckling stress \(\sigma_{\text{cr}}\) of an elliptical section can be found via an equivalent diameter concept [44; 45] whereby the elliptical section is represented by a circular section with a radius equal to the radius of curvature at the point of initiation of buckling. Note that, although PLA and ABS are certainly less ductile than structural steel in terms of ultimate fracture, the elastic ranges of these polymer materials are considerably  
Figure 16: Longitudinal strains measured at the mid-height cross-section, with location of strain gauges indicated, for specimens **(a)** EHS10-PLA2-90-60-1.5, **(b)** EHS01-PLA1-100-50-3.0, **(c)** EHS16-PLA2-105-35-2.5. greater, and thus local elastic failure modes are much more likely to occur in these materials, in contrast to steel specimens where associated plasticity is encountered in practice. The elastic nature of these failure modes thus lends well to the use of critical buckling formulae derived from elastic theory. As indicated in Figure 11b, in the case of an elliptical tube in pure compression, the point of initiation of buckling is at the extreme of the minimum diameter where the equivalent diameter \(D_{\text{eq}}=2(a^{2}/b)\)[44; 45]. Adapting the equivalent expression for a circular section, the critical buckling stress \(\sigma_{\text{cr}}\) for an elliptical tube in compression is given by:  
\[\sigma_{\text{cr}}=\frac{E_{\text{c}}}{\sqrt{3(1-\
u^{2})}}\frac{2t}{D_{\text{ eq}}} \tag{2}\]  
For slender elliptical tubes, i.e., those that are susceptible to local failure of their tube walls ahead of yielding the full cross-section or global member buckling, the compressive resistance of the member is largely a function of its slenderness. In the present study, the local slenderness parameter \(\overline{\lambda}_{\ell}\) is defined as:  
\[\overline{\lambda}_{\ell}=\sqrt{\frac{\sigma_{\text{c}}}{\sigma_{\text{cr}}}} \tag{3}\]  
where \(\sigma_{\text{c}}\) is the reference compressive strength of the material. When designing structural steel elliptical tubes, sections can be considered fully effective in compression, i.e., non-slender, if the following criterion is satisfied [46]:  
\[\frac{D_{\text{eq}}}{t\varepsilon^{2}}\leq 90 \tag{4}\]  
where the strength modification factor \(\varepsilon\) is adapted for use with materials other than carbon steel as follows [47]:  
\[\varepsilon=\sqrt{\left(\frac{235}{\sigma_{\text{c}}}\right)\left(\frac{E_{ \text{c}}}{210000}\right)} \tag{5}\]  
As shown by the values of \(D_{\text{eq}}/t\varepsilon^{2}\) in Table 8, the relative slenderness of the specimens examined in the present study far exceeds the limit of Equation (4), indicating that these are highly slender sections. When considering the value of compressive strength to adopt in the design method, the results from Section 2.2 suggest that although the strength can, overall, be considered isotropic, there is noticeable orthotropy observed in the results for compressive strength within PLA batch 1. Thus, failure of the tube wall material is based on a Tsai-Hill criterion [48] that accounts for anisotropic strength. It is assumed the walls of the elliptical tube are in a biaxial plane stress state imposed by the applied longitudinal axial compressive stress \(\sigma_{\text{long}}\) and circumferential stresses arising from the associated [PERSON]'s effect. Given the relative thickness of the tube walls compared to the outer diameters, it is assumed that a thin-walled condition prevails and that radial stresses are negligible. Since the compressive testing discussed in Section 2.2 indicates that the elastic moduli of the printed materials are isotropic, these circumferential stresses can be assumed equal to \(-\
u\sigma_{\text{long}}\) directly. Considering then the interaction of [PERSON]'s effects between adjacent elements around a closed section, these stresses can be taken as reduced to \(\sigma_{\text{long}}/(1+\
u^{2})\) and \(-\
u\sigma_{\text{long}}/(1+\
u^{2})\), respectively.