# Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading
## 6 Conclusions  
This paper presented the computation of the elastica of non-prismatic and nonlinear elastic cantilever beams under combined loading. The tested beams had solid rectangular cross-sections with widths and heights varying linearly along the beam axes. The beam materials obey [PERSON]'s constitutive law. The loading system used for testing can provide uniform loading, tip point loading, and tip couple loading individually or in combination, allowing for a total of seven loading combinations that have not been covered in the literature. We derived ordinary differential equations that govern the elastica of such beams and solved them numerically using the Runge-Kutta and Regula-Falsi methods. To analyze typical elastica behaviors, three beam materials were considered: steel, the NP8 aluminum alloy, and annealed copper. From our parametric study of elastic behaviors, the following conclusions can be drawn:  
1. The tip response of \((\delta_{v},\delta_{h},\varphi)\) increases with an increasing load along the equilibrium path. Regarding the relationship between tip response and cross-sectional ratio, the tip responses of \((\delta_{v},\delta_{h},\varphi)\) decrease as the cross-sectional ratio \(r_{B}(=r_{H})\) increases. The tip responses in this study are in good agreement with those in the literature.
2. The mechanical properties of \(E\) and \(n\) were both observed to have important effects when calculating bending moments. The exponential constant \(n\), rather than the Young's modulus \(E\), dominates elastica calculations.
3. For two loads of \(w\) and \(p\), the angle parameter \(\varphi\) converges to \(\varphi=1\) as \(w\), \(p\), and both increase. However, \(\varphi\) for a load \(c\) can exceed a value of one. The relationship between deflection and cross-sectional ratio is strongly nonlinear, as indicated by the equilibrium paths.
4. The stress values of copper and aluminum near the neutral axis are relatively larger than that of steel. The stress near the neutral axis increases as the exponent constant \(n\) increases.  
[PERSON] proposed the idea, derived the governing equations, and drafted the paper; [PERSON] coded the computer program, obtained the calculations, and assisted the writing of the paper.  
This work was supported by the National Research Foundation of Korea (Grant number NRF-2017R1C1B5015371).  
**Conflicts of Interest:** The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Beam Material**} & \multicolumn{2}{c}{**Extreme \(\varepsilon_{e}\)and \(\sigma_{e}\)**} & \multicolumn{2}{c}{\(M_{c}\) and \(\kappa_{c}\)**} & \multicolumn{2}{c}{**Mechanical Property**} \\ \cline{2-7}  & \(\varepsilon_{e}\) & \(\sigma_{e}\) (MPa) & \(M_{c}\) (kNm) & \(\kappa_{c}\)(m\({}^{-1}\)) & \(E\) (MPa) & \(n\) \\ \hline Steel & 0.000986 & 203 & 68.0 & 0.01971 & 207,000 & 1.0 \\ Aluminum alloy & 0.00481 & 149 & 67.6 & 0.09612 & 455.8 & 4.79 \\ Annealed copper & 0.0657 & 130 & 52.8 & 1.314 & 458.5 & 2.16 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparisons of extreme strain \(\varepsilon_{e}\) and stress \(\sigma_{e}\) for the three beam materials.