## 4 Conclusions  
The methodology developed here utilized a numerical solution to solve the inverse problem of determining a structural beam cross-section with a constant transverse shear stress when subjected to a resultant shear force. The cross-section is extremely thin at the top and extremely wide at the neutral axis and does not converge to a specific shape. As the number of elements approaches infinity, the ratio of the widest to the thinnest element will also approach infinity and the point of flare-out approaches the neutral axis. It is therefore determined that this inverse problem is ill-posed.  
The cross-section proved to be impractical for use in an engineering application. Due to the divergent nature of the numerical solution, these results cannot be verified through physical experimentation. For future work, this solution can be refined and applied to a more practical beam shape where there are certain constraints in place such as a minimum and maximum thickness to the cross-section. Such a solution would sacrifice efficiency for practicality  
**Author Contributions:** Conceptualization, E.T.B.III; methodology, E.T.B.III and R.R.M.; software, E.T.B.III and R.R.M.; validation, E.T.B.III and R.R.M.; formal analysis, E.T.B.III and R.R.M.; investigation, E.T.B.III and R.R.M.; resources, E.T.B.III and R.R.M.; data curation, E.T.B.III and R.R.M.; writing--original draft preparation, E.T.B.III and R.R.M.; writing--review and editing, E.T.B.III and R.R.M.; visualization, E.T.B.III and R.R.M.; supervision, E.T.B.III; project administration, E.T.B.III. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Acknowledgments:** The authors would like to thank [PERSON], [PERSON], and [PERSON] for their contributions to this research.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Nomenclature**  
\(A\): Area  
\(A_{j}\): Area of Discrete Element  
\(A_{Q}\): Area Used to Calculate \(Q\)  
\(a\): Numerical Index Where Stress Is Calculated  
\(C\): Constant Relating \(Q\) and \(t\)  
\(d_{j}\): Centroid of Discrete Element  
\(f_{y}\): Flare-out Point  
\(H\): Height of Discrete Element  
\(h\): Cross-section Height  
\(I\): Area Moment of Inertia  
\(I_{j}\): Area Moment of Inertia of Discrete Element  
\(j\): Element Index  
\(NA\): Neutral Axis of Cross-section  
\(n\): Total Number of Discrete Elements for Top Half of Cross-section  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**n** & **log(n)** & \(z_{R}\) & **log(\(z_{R}\))** & **Ratio Name** & \(\tau_{\textit{eff}}\) \\ \hline
7 & 0.84 & 1.0 & 1 & Ten & 85.51706\% \\
9 & 0.96 & 100 & 2 & Hundred & 89.13779\% \\
12 & 1.06 & 1000 & 3 & Thousand & 91.31023\% \\
18 & 1.27 & \(1.0\times 10^{6}\) & 6 & Million & 94.56880\% \\
25 & 1.40 & \(1.0\times 10^{9}\) & 9 & Billion & 96.05011\% \\
32 & 1.51 & \(1.0\times 10^{12}\) & 12 & Trillion & 96.89651\% \\
235 & 2.37 & \(1.0\times 10^{100}\) & 100 & Googol & 99.57403\% \\
702 & 2.85 & \(1.0\times 10^{303}\) & 303 & Centillion & 99.85754\% \\
1400 & 3.15 & \(1.0\times 10^{606}\) & 606 & & 99.92857\% \\
2319 & 3.37 & \(1.0\times 10^{1000}\) & 1000 & & 99.95687\% \\
2,314,103 & 6.36 & \(1.0\times 10^{1,000,000}\) & \(1.0\times 10^{6}\) & & 99.99995\% \\ \(2.31\times 10^{9}\) & 9.36 & \(1.0\times 10^{1,000,000,000}\) & \(1.0\times 10^{9}\) & & 99.99999\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Extrapolated data.