## 1. Introduction

In the design of a structural beam, there are several geometric and material parameters that must be considered and optimized in order to yield the desired strength, lifespan, and reliability. For typical Euler–Bernoulli beams, the bending stress is much greater than the transverse shear stress. However, there are many cases where the beam is short enough, such that the transverse shear stress dominates. In cases where there is a strong possibility for beam failure in the transverse direction, the shape of a cross-section of a beam can be treated as a design variable. Uniaxial-reinforced composite beams, anisotropic beams, wooden beams, tree branches, bones of young mammals, and railroad ties are all prone to fracture due to their low transverse strength [

1,

2,

3]. In general, the practical approach to handling high transverse shear stresses is to use reinforcement stiffeners [

4,

5,

6]. However, there are limitations on the efficiency of these structures. For this methodology, the theoretical solution will be investigated with a perfectly efficient cross-section as the goal.

A structural beam can be subjected to multiple shear forces acting perpendicularly to its longitudinal axis, which could then be reduced to a resultant shear force

$V$ at any point along the length of the beam. Longitudinal and transverse shear stresses are developed; the latter of which will be discussed and optimized in this study. The longitudinal shear stress in a beam occurs along the longitudinal axis and is visualized by a slip in the layers of the beam, as shown in

Figure 1. The longitudinal shear strength of the beam acts as the agent by which the slip between layers is prevented [

7,

8].

The transverse shear stress

$\tau $ varies along the cross-section of a beam as shown in

Figure 2, where

${\tau}_{max}$ is the maximum transverse shear stress and

$NA$ is the neutral axis, or the centroid of the cross-section [

7]. For the rectangular cross-section shown in

Figure 2, the solution is easily derived as a parabolic transverse shear stress distribution which has zero shear stress at the top, peaks at the neutral axis and returns to zero at the bottom [

8].

The resultant shear force produces longitudinal and transverse shear stresses shown in

Figure 3. These stresses must be equal to each other to maintain the equilibrium of an infinitesimal stress element which is located at any theoretical point in the beam [

7].

It should be noted that

$\tau $ is equal to zero at the top and bottom of the beam, shown in

Figure 2. This is due to the fact that the top and bottom boundaries of the beam are exposed to air and therefore carry no longitudinal load [

7]. It can also be observed in

Figure 2 that the maximum transverse shear stress occurs at the neutral axis. Though this is not always the case, it can be derived that the maximum transverse shear stress will always occur at the neutral axis if the cross-section of the beam is thinnest at that point [

7].

For the optimization of the overall lifespan of a beam which may be prone to transverse shear failure, it is desirable to maintain an even transverse shear stress distribution along the cross-section in order to avoid shear stress concentrations within the part. In general, a part is only as strong as its highest stress value. Any material that is not carrying a stress close to that of the maximum is wasted and is therefore unnecessary. Optimization is obtained by altering the geometry of the cross-section.

The optimization of a beam for transverse shear stress efficiency can be solved as an inverse problem. The output (constant transverse shear stress along the cross-section) is known while the input (the geometry of the cross-section) is not. The determination of the shape of the cross-section of the beam to achieve maximum overall transverse shear stress efficiency can be completed using general analytical or numerical methods. Both methods are discussed, but only the solution using the latter method is identified and utilized in this study. A numerical method was chosen because of its proven effectiveness in a wide variety of problem-solving and parameter-optimization applications [

9,

10]. Numerical/iterative approaches, such as the use of Microsoft Excel 2013′s Solver and MATLAB 8.3, have been used before to solve and verify solutions to similarly-posed problems, such as in the optimization of cylindrical bar cross-sections [

10], and open beam geometry [

9]. Other approaches using the boundary element method have shown how the transverse shear stress can vary within an arbitrary cross-section [

11].

## 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.

## 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