# Elastica of Non-Prismatic and Nonlinear Elastic Cantilever Beams under Combined Loading
## 5 Results and Comparisons
In contrast, the bending moments of the aluminum alloy and annealed copper are very different. The \(E\) values of these two materials are nearly the same, but the \(n\) values differ significantly. From these results, we can conclude that both \(E\) and \(n\) have significant effects when calculating bending moments. Additionally, the maximum bending moments (\(\blacksquare\)) occur at the clamped ends, whereas the minimum bending moments (\(\bullet\)) occur at the free tips. These moments correspond to the tip couple loading \(c\) for all three materials. Figure 7: Elastica of steel, aluminum alloy, and annealed copper beams. Figure 9 presents diagrams of the strains \(\varepsilon\) and stresses \(\sigma\) along the beam height at the clamped ends for the three beam materials. The strain \(\varepsilon\) and stress \(\sigma\) at a height \(H\) are computed as follows:  
\[\varepsilon=\kappa H=\frac{d\theta}{ds}H, \tag{29}\]  
\[\sigma=E\left(\frac{d\theta}{ds}H\right)^{1/n}, \tag{30}\]  
where \(H\) is the distance measured from the centroid along the vertical beam height. By using the bending moments \(M_{c}\) (\(\blacksquare\) in Figure 8) at the clamped ends (\(s=0\) m) for the three beam materials, the strains \(\varepsilon\) and stresses \(\sigma\) were computed using Equations (29) and (30). The results for \(\varepsilon\) and \(\sigma\) are presented in Figure 9a, b, respectively. For the reader's convenience, the extreme values \(\varepsilon_{e}\) and \(\sigma_{e}\) on both sides of the beams (i.e., \(H(=H_{c}/2)=\pm 5\) cm) in Figure 9a, b are tabulated in Table 1 along with the corresponding values of \(M_{c}\), curvatures \(\kappa_{c}\), and mechanical properties \(E\) and \(n\). The strains \(\varepsilon\) along the height for the three beam materials are presented in Figure 9a. The values of \(\varepsilon\) decrease from the annealed copper to the aluminum alloy to the steel. Some unexpected patterns are visible in these results. For example, the strains of the steel and aluminum are nearly identical despite the two materials having very different \(E\) and \(n\) values. In contrast, the strains of the aluminum  
Figure 3:0 Bending-moment diagrams for steel, aluminum alloy, and annealed copper beams. Figure 3:1 Values of (**a**) strain \(\varepsilon\) and (**b**) stress \(\sigma\) as functions of the distance \(H\) from the neutral axis for steel, aluminum alloy, and annealed copper beams at clamped ends. and copper are very different despite both materials having similar \(E\) values. Figure 9b presents the stresses \(\sigma\) along the beam height for the three beam materials. The values of \(\sigma\) decrease from the steel to the aluminum to the copper. The stress distributions of the copper and aluminum near the neutral axis are relatively larger than that of the steel because their exponent constants are larger. However, one can see that the relationships between the stress \(\sigma\), and \(E\) and \(n\) are reversed compared to the strain relationships shown in Figure 9a, which is an unexpected result (see Table 1). From the results in Figure 9 and Table 1, we can conclude that the effects of \(E\) and \(n\) play important roles in computing the strain and stress of nonlinear elastic beams.