## 3 Results  
Microsoft Excel and Matlab were both used extensively in this study to model and solve for a beam cross-section which will exhibit a nearly constant transverse shear stress distribution when subjected to a resultant shear force acting perpendicularly to its longitudinal axis. A Matlab script that implemented the equations in Section 2 was developed in order to formulate the above-described cross-section, and Excel's Solver was used to further optimize certain parameters. The methodology derived here is novel and defines the shape of a cross-section in order to obtain a constant transverse shear stress distribution. It will now be investigated if the shape converges to an exact solution as the number of elements \(n\) increases. The constant transverse shear stress solution using the numerical method is compared to the transverse shear stress of a rectangular cross-section in Figure 7. The rectangular cross-section transverse shear stress distribution is a parabola that is derived from Equation (7) and has an efficiency of 66.7% using Equation (10) [8]. The reason that the rectangular cross-section is not efficient is that the maximum transverse shear stress is well above the average transverse shear stress. The \"perfect shape\"derived here has an efficiency that approaches 100% as the number of elements approaches infinity from Equation (32). The shapes of the cross-sections were dependent on the number of elements chosen. A general observation is that the cross-section is very narrow at the top and extremely wide at the neutral axis. The flare-out point \(f_{y}\) is defined as the vertical position on the plot of the cross-section where the shape rapidly becomes much wider. This was defined to be the first \(y\) value where the thickness \(z\) was calculated to be greater than 0.2% of \(z_{1}\) while moving from the top down. The produced Matlab script required some user-determined values including \(n\) and \(z_{1}\). From these values, \(z_{n}\), \(z_{R}\), and \(\tau_{eff}\) were then calculated using Equations (30)-(32). The overall width and height were chosen to have a value of one in order to generate a clear-cut plot showing the effect that the chosen \(n\) value had on the shape of the cross-section. Starting from 25, \(n\) was increased by ascending increments to 700 using Matlab, after which the program failed for \(z_{1}=0.5\). Excel was used to calculate the shape up to the flare-out point for 4,000,000 elements but could not handle the extremely small values closer to the top of the cross-section. A representation of the cross-sections calculated is shown in Figure 8. As the number of elements increased, the cross-section did not converge on a particular shape. Through extrapolation, the cross-section for an infinite number of elements is infinitely thin at the top, and has a flare-out point approaching \(f_{y}=0\). To bypass the programming constraints, \(z_{1}\) was increased to \(10^{305}\) (the upper bound for integers in Matlab) and \(n\) was increased until the program failed, which was at 1400 elements. For this number of elements, the value of \(z_{n}\) was approximately \(10^{-305}\) (the lower bound for integers in Matlab.) The log of the \(z_{R}\) ratio obtained from each trial was taken to yield a nearly-linear relationship with \(n\), as shown in Figure 9. These values were then optimized using Excel's Solver paired with the least squares method to generate an equation and a linear trend line with \(R^{2}=1.0000\). The values of \(f_{y}\) and \(\tau_{eff}\) were also recorded for each value of \(n\). The data obtained from Matlab and Excel is shown in Table 1.  
The linear trend line on the semi-log plot shown in Figure 9 is extrapolated to predict the ratio of the widest to thinnest parts of the cross-section for higher numbers of total elements. It is shown in Table 2 that \(z_{R}\) quickly becomes large. With only \(n=25\), the ratio is one billion with other larger named numbers shown.