## Theory  
### Transverse Shear Stress Distribution Within a Cross-Section--Analytical Formulation  
For a beam experiencing a resultant shear force \(V\) through external loads, a transverse shear stress distribution exists within the cross-section having a height of \(h\) as shown in Figure 4. The cross-section is defined as \(z\) (horizontal) as a function of \(y\) (vertical) in the first quadrant, with the shaped assumed to be symmetric about both the \(y\) and \(z\) axes. For the cross-section, three geometric properties can be calculated analytically [7]. The area \(A\) is calculated in Equation (1) by setting up a differential element with a height of \(dy\) and width of \(2z\). The area of the differential element is \(dA=2 zdy\).  
\[A=\int_{Aran}dA \tag{1}\]  
By using the symmetry about the \(z\) axis, the limits of \(y\) are from \(0\) to \(h/2\) and this value can be doubled. The area will be constant for a given cross-section as shown in Equation (2).  
\[A=4\int_{0}^{h/2}z\ dy \tag{2}\]  
The parameter \(Q\) is the first moment of area. The formula is similar to Equation (1) but with the addition of a \(y\) term within the integral. \(Q\) varies within the cross-section and is calculated from the top down to the point where the stress is formulated for any positive \(y\) value as shown in Equation (3).  
\[Q=\int_{Aran}y\ dA \tag{3}\]  
If the cross-section can be expressed as a simple geometric shape, an alternate formula for \(Q\) is used as shown in Equation (4). The portion of the cross-section is drawn from the top down to where  
Figure 4: Cross-section of a beam.  
\(Q\) is calculated, \(A_{Q}\) is the area and \(\overline{y}\) is the centroid. Furthermore, if the portion of the cross-section can be drawn from a series of simple geometric shapes, \(Q\) can be expressed from the components of the centroid. In this case, \(A_{j}\) is the area of each component and \(\overline{y_{j}}\) is its corresponding centroid, where \(j\) is an index used for the summation.  
\[Q=\overline{y}A_{Q}=\sum\overline{y_{j}}A_{j} \tag{4}\]  
The last property of the cross-section is the second moment of area, or area moment of inertia \(I\). Similar to area, it can also be doubled for symmetry and will be constant for the cross-section as shown in Equations (5) and (6).  
\[I=\int_{Area}y^{2}\;dA \tag{5}\]  
\[I=4\int_{0}^{h/2}y^{2}z\;dy \tag{6}\]  
The transverse shear stress as a function of vertical position \(\tau(y)\) is calculated for any point along the cross-section using the thickness across where the stress is measured \(t=2z\) as calculated in Equation (7) [7].  
\[\tau(y)=\frac{VQ(y)}{It(y)} \tag{7}\]  
The maximum transverse shear stress can occur at any point along the cross-section. If the thinnest part of the material is at \(y=0\), \(\tau_{max}\) will definitely occur there since \(Q\) is at a maximum and \(t\) is at a minimum. Otherwise, the derivative of \(\tau(y)\) is set equal to zero to find the points where local maxima occur as shown in Equation (8) [12]. Since there could be several local maxima for complex shapes, several points may have to be compared to find the global maximum.  
\[\frac{d\tau}{dy}=0 \tag{8}\]  
The average transverse shear stress \(\tau_{avg}\) is calculated in Equation (9) [12].  
\[\tau_{avg}=\frac{2}{h}\int_{0}^{h/2}\tau\;dy \tag{9}\]  
The efficiency of the cross-section \(\tau_{eff}\) is defined in Equation (10). It is expressed as a percentage, with 100% denoting a constant transverse shear stress distribution.  
\[\tau_{eff}=\frac{\tau_{avg}}{\tau_{max}}(100\%) \tag{10}\]