## 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  
Figure 1: Longitudinal shear stress in a beam. Figure 3: Infinitesimal shear stress element. Figure 2: Transverse shear stress distribution for a rectangular cross-section. (the geometry of the cross-section) is not.