# Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading
## 5 Results and Comparisons  
To analyze typical elastica behaviors, three beam materials were considered in this study: steel, the NP8 aluminum alloy, and annealed copper. For these materials, the values of the [PERSON]'s modulus \(E\) and exponent constant \(n\) to be introduced into [PERSON]'s constitutive law \(\sigma=E\left|\varepsilon\right|^{1/n}\) are listed below. * Steel: \(E=207\) GPa and \(n=1.0\)
* NP8 aluminum alloy: \(E=455.8\) MPa and \(n=4.79\)
* Annealed copper: \(E=458.5\) MPa and \(n=2.16\)  
First, the results of this study are compared to those from the literature for degenerate cases, including non-prismatic, nonlinearly elastic, and combined loading effects, for validation purposes. The steel and annealed copper materials are considered in these comparisons. Figure 3 presents comparisons between the tip responses \(\left(\Delta_{h},\Delta_{v}\right)\) and \(\left(\delta_{h},\delta_{v}\right)\) from this study and those from [4; 5; 8; 16]. The beam parameters used for comparison are presented. The tip responses from this study (\(\blacksquare\)) and those from the literature (\(\bullet\)) are presented in the context of the elastica obtained in this study. Both sets of responses are in good agreement for the four degenerate cases. These comparisons validate the elastica theories and solution methods presented in this paper. Next, we present parametric analysis of the elastica for various combinations of \(w\), \(p\), and \(c\). The results for the annealed copper material (\(n=2.16\)) are presented in Figures 3-5. The beam parameters considered are also presented. Figure 4 presents the tip responses of \((\delta_{v},\delta_{h},\varphi)\) versus the load curves (i.e., equilibrium paths) for three loading cases: (a) uniform load \(w\), (b) tip point load \(p\), and (c) tip couple load \(c\). The trends appear as expected. Specifically, the relationships are strongly nonlinear and \((\delta_{v},\delta_{h},\varphi)\) all increase as the load increases. For the two loading cases of \(w\) and \(p\), the angle parameter \(\varphi\big{(}=\theta_{f}/(\pi/2)\big{)}\) converges to a value of \(\varphi=1\) because the tip rotation \(\theta_{f}\) cannot physically reach \(\pi/2\) based on the vertical loading direction. However, \(\varphi\) for load \(c\) can overcome this limitation and exceed \(\varphi=1\) as \(c\) increases. Additionally, the physical meaning of \(\varphi=4\) is one-cycle flip-over at the tip end, that of \(\varphi=8\) is two-cycle flip-over, etc. Typical examples of the elastica for a (a) uniform load \(w\), (b) tip point load \(p\), (c) tip couple load \(c\), and (d) combined load with various load magnitudes are presented in Figure 5. For all loading cases, the deflections of the elastica increase as the load increases. For the loading cases of (a) \(w\) and (b) \(p\), one can see that the angle parameter \(\varphi\) approaches one with an increasing load, which corresponds with the results in Figure 3a,b. For the loading case of \(c\) (i.e., pure bending), \(\varphi\) increases with an increasing \(c\). Tip rotation occurs at one-cycle flip-over (i.e., \(\varphi=4\)) when  
Figure 3: Comparisons of tip responses from this study (\(\blacksquare\)) and those from the literature (\(\bullet\)) for four degenerate cases: (**a**) prismatic nonlinear elastic beam with \(P\)[4], (**b**) prismatic nonlinear elastic beam with \(C\)[5], (**c**) prismatic nonlinear elastic beam with \(P\) and \(W(=P/l)\)[8], and (**d**) non-prismatic linear elastic beam with \(P\) and \(C\)[16]. and at two-cycle flip-over (i.e., \(\varphi=8\)) when \(c=4.69\times 10^{-5}\) (see loading combinations five and six). Additionally, the elastica under the negative couple load \(c=-1.79\times 10^{-5}5\) are presented and show upward deflection with a negative curvature \(\kappa\) (see loading combination seven). For the combined loading case (d), the deflections of the elastica for all seven loading combinations are presented. Load superposition increases the deflection compared to the sums of individual deflections.