# Simple Approximate Formulas for Postbuckling Deflection of Heavy Elastic Columns
## 3 Result  
### Relationship Between the Combined Loads and Tip-Tilt Angle  
Figure 2 depicts the typical deflection behaviors of the column for various \(\alpha\) and \(\beta\). When \(\alpha\) is fixed at \(\alpha=1.0\), as presented in Figure 2a, the column remains vertical until \(\beta\) reaches a critical value of \(\beta_{c}=4.75\). When \(\beta\) exceeds \(\beta_{c}\), the column begins to deflect and the tip-tilt angle \(\theta_{0}\) increases monotonically with \(\beta\). Note that the threshold value \(\beta_{c}\) changes, if \(\alpha\) is fixed at a different value, as demonstrated later. A similar deflection process is observed when \(\beta\) is fixed; see Figure 2b. For instance, when \(\beta\) is fixed at \(\beta=1.0\), the critical value of \(\alpha\) is \(\alpha_{c}=2.16\) beyond which \(\theta_{0}\) increases with \(\alpha\). Again, the value of \(\alpha_{c}\) depends on the choice of \(\beta\).  
The contour map in Figure 3 illustrates the variation of the tip-tilt angle \(\theta_{0}\) with the changes in the values of \(\alpha\) and \(\beta\). The upper limit of the contour line is set to 1.6 (unit: radian); data above this limit are omitted in the plot. It can be clearly observed that \(\theta_{0}=0\) within the left-bottom triangular region (gray colored). This indicates that for relatively small \(\alpha\) and \(\beta\), the vertically standing state is energetically favored; hence, the column is not deflected but merely compressed in the axial direction. In the rainbow-colored region, in contrast, \(\theta_{0}>0\), implying that deflection will occur. The boundary curve that separates the two equilibrium states \(\theta_{0}=0\) and \(\theta_{0}>0\) is found to be a nearly linear slanted line in the \(\alpha\)-\(\beta\) plane, which has intersections with the \(\alpha\)-axis and \(\beta\)-axis, respectively, at \(\beta_{c}=7.815\) and \(\alpha_{c}=2.468\). The latter value is close to the normalized Euler buckling load \(\pi^{2}/4\) with deviation of 0.024%. If the series expansion in Equation (7) were truncated at the seventh or fifth degree, for example, the deviation from the Euler value would be enlarged to 0.097% or 2.78%, respectively. It is also noted that our values of \(\alpha_{c}\) and \(\beta_{c}\) obtained by the ninth-degree expansion are nearly equal to those evaluated in the existing work with differences less than 1% [29; 31].  
Figure 2: Deflection curves for various \(\alpha\) and \(\beta\) parameter conditions: (**a**) \(\alpha=1.0\) (fixed) and (**b**) \(\beta=1.0\) (fixed).  
An important observation in Figure 3 is that all the contour lines are nearly parallel to each other, approximately obeying the following relationship:  
\[\alpha+\frac{10}{3}\beta=c, \tag{21}\]  
with a \(\theta_{0}\)-dependent parameter \(c\). The phase boundary, for instance, is nearly equivalent to the line corresponding to \(c=2.47\). Similarly, the dashed-dotted line in the magnified view of the inset of Figure 3 (red colored) coincides with the line for \(c=2.89\). The physical meaning of Equation (21) is that the effect of the self-weight \(\rho L\) on the slender column's deflection is nearly 0.3 times that of the concentrated load \(F\) applied at the tip. Hence, as long as the values of \(\alpha\) and \(\beta\) satisfy Equation (21), the tip-tilt angle \(\theta_{0}\) will remain constant irrespective of the values of \(\alpha\) and \(\beta\). This 0.3-rule represented by Equation (21), which describes the balance of the contributions by the concentrated load and distributed load to the tip-tilt angle, is the first main finding of this study.