# Simple Approximate Formulas for Postbuckling Deflection of Heavy Elastic Columns
## 4 Discussion  
We determined that the phase boundary curve that separates the vertically standing phase (i.e., \(\theta_{0}=0\)) and deflection phase (\(\theta_{0}>0\)) in the \(\alpha\)-\(\beta\) plane can be approximately expressed by a simple linear equation  
Figure 4: (**a**,**b**) Power-law dependences of \(\theta_{0}\) on \(\alpha\) and \(\beta\). The fitting curves for data points in the \(0<\theta_{0}<1.0\) range are also shown. (**c**,**d**) Optimal values of coefficients \(A\) and \(B\), and those of exponents \(p_{\alpha}\) and \(p_{\beta}\).  
(Equation (21)). To determine the critical values of \(\alpha\) and \(\beta\) with higher accuracy, the following formula can be used [43]:  
\[J_{-\upsilon}(\zeta_{1})J_{-1+\upsilon}(\zeta_{0})+J_{\upsilon}(\zeta_{1})J_{1- \upsilon}(\zeta_{0})=0\ \ \text{with}\ \upsilon=1/3, \tag{25}\]  
where \(J_{\upsilon}\) is the \(\upsilon\)-th order Bessel function and  
\[\zeta_{m}=\frac{2}{3}\left(m+\frac{\alpha}{\beta}\right)^{\frac{3}{2}}\beta^{ \frac{1}{2}},\ \ \ m=0,1. \tag{26}\]  
Using this accurate formula of Equation (25), we can obtain the \(\alpha\)-intercept of the boundary curve at \(\alpha_{c}=\pi^{2}/4\), which is consistent with our approximate value of \(\alpha_{c}=2.46\). Similarly, the \(\beta\)-intercept derived from the accurate formula is \(\beta_{c}=7.83735\), which is again proximate to our result of \(\beta_{c}=7.82\). This comparison of the critical values confirms the practicality of our simple expression for the phase boundary, given by Equation (21).  
One of the interesting ways of using the approximate formulas derived in this study is to diagnose the mechanical stability of upright plants, such as large trees and tall bamboos. Understanding the stability of the tree trunk under self-weight and the applied loads caused by the leaf and branches is crucial when developing management strategies to reduce the risk of damage by abiotic agents. In addition, bamboo is a unique plant that does not break, exhibiting considerable flexure, even when subjected to load [44]; it is extensively used as building material in areas such as Southeast Asia. Note that when using mathematical expressions in actual stability assessment, rapid estimation of the stability using a simple formula is desired, rather than exact numerical data based on expensive numerical simulation. Our simple formula for heavy column deflection facilitates an approximate perspective of the stability analysis of plants in a postbuckling state. However, to investigate the mechanical response of a plant more quantitatively, it is necessary to consider the variation in both the mechanical and geometrical properties of the plant from the ground to the top as well as the effect of initial imperfection. In this context, we need to extend our formulation considering the spatial variation in the cross-sectional geometry [33; 45], that in Young's modulus [38], and the presence of cracks in the initial state [35]; I intended to address these interesting problems in future.  
The effect of boundary condition variation on the postbuckling behavior is another interesting topic. In this study, our focus was restricted to the deflection of a column in which one end is fixed and the other is free. Another important case to consider is a column in which both ends are fixed because it is frequently used in structural applications, in both macroscopic and nanoscopic [13; 14; 15] systems. Imposing different boundary conditions on a column can inherently change its mechanical response to the applied loads. Therefore, as in the case of this study, it is interesting to examine whether an approximate expression for the deflection can be derived from a column with fixed conditions at both ends. Our results can pave the way for exploring simple approximate formulas describing column deflection under various end conditions.