# Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading
## 5 Results and Comparisons
However, the superposition principle is not satisfied for the nonlinear elastic beams considered in this study. When comparing loading combinations four (\(w+p\)), seven (\(w+p+c\)), and eight (\(w+p-1.45c\)), one can see that a positive \(c\) increases the deflection, and a negative \(c\) considerably reduces the deflections caused by the vertical loads \(w\) and \(p\). Therefore, the elastica for loading combination eight show upward deflection beyond the horizontal axis. The effects of the cross-sectional ratio \(r_{B}(=r_{H})\) on the three tip responses of \((\delta_{v},\delta_{h},\varphi)\) and the elastica are presented in Figure 6 for the combined loading case with all of \(w+p+c\). The beam parameters are included in the figure. In Figure 6a, all the values of \((\delta_{v},\delta_{h},\varphi)\) decrease as \(r_{B}(=r_{H})\) increases. The reduction rate is larger than that with the smaller cross-sectional ratio \(r_{B}(=r_{H})\). From Figure 6a, one can clearly see that the relationship between deflection and the cross-sectional ratio is strongly nonlinear, as indicated by the equilibrium paths in Figure 4. Additionally, in Figure 6b, the deflections of the elastica become deeper as \(r_{B}(=r_{H})\) decreases. Figure 4: Tip responses of the beam under (**a**) uniform loading, (**b**) tip point loading, and (**c**) tip couple loading. Figure 5: Elastic of beams subjected to (**a**) uniform loading, (**b**) tip point loading, (**c**) tip couple loading, and (**d**) combined loading. Figure 6: Effects of cross-sectional ratio on the (**a**) tip responses and (**b**) elastica of beams under combined loading. In a third series of experiments, the effects of beam materials on elastica were investigated. Three different beam materials, namely steel, the NP8 aluminum alloy, and annealed copper, were considered. The mechanical properties of \(E\) and \(n\) for these materials were provided above. For these parametric experiments, the numerical results are presented in dimensional units, rather than the nondimensional forms, because the load parameters of \((w,\,p,\,c)\) are different for the same load magnitudes with the same beam geometry for different materials. The consistent beam geometry and loading conditions for all three beam materials are listed below. * Beam geometry: \(l=\)1 m, \(B_{c}=0.2\) m, \(B_{f}=0.1\) m, \(H_{c}=0.1\) m, and \(H_{f}=0.04\) m
* Loading conditions: \(W=50\) kN/m, \(P=40\) kN, and \(C=3\) kNm  
For the reader's convenience, the nondimensional forms of the beam parameters are listed below. * Beam geometry: \(b_{c}=0.2\), \(h_{c}=0.1\), \(r_{B}=0.5\), and \(r_{H}=0.4\)
* Loading conditions: \(w=2.42\times 10^{-7}\), \(p=1.93\times 10^{-7}\), and \(c=1.45\times 10^{-8}\) for steel; \(w=1.10\times 10^{-4}\), \(p=8.78\times 10^{-5}\), and \(c=6.58\times 10^{-6}\) for the NP8 aluminum alloy; and \(w=1.09\times 10^{-4}\), \(p=8.72\times 10^{-5}\), and \(c=6.54\times 10^{-6}\) for annealed copper. Based on the beam parameters listed above, the elastica, bending moments, strains, and stresses of the clamped beam ends \((s=0\) m\()\) were computed in the dimensional forms. The results are compared in Figures 7-9. Figure 7 presents the deflections of the elastica for the three beam materials. One can see that deflections increase in the order of the annealed copper to the aluminum alloy to the steel. In particular, the deflections of the copper and aluminum beams are very different, despite the \(E\) values of both materials being nearly identical. From these results, we can conclude that the property of \(n\), rather than \(E\), has a dominant effect on the elastica. Bending-moment diagrams along the beam axis for the three beam materials are presented in Figure 8. One can see that the bending moments decrease from the steel to the aluminum to the copper. It is noteworthy that the bending moments of the steel and aluminum alloy are nearly identical, despite the two materials having very different values of \(E\) and \(n\).