# FE Exam Concept – Max Transverse Shear Stress

When we look at a beam carrying loads transverse to its axis will experience

shearing forces, denoted by V. In the analysis of beams, it is typical to calculate the variation in shearing force across the entire length of the beam by drawing the shear force diagram. With the shearing force known, we can calculate the vertical shearing stress at any section by using the following equation as provided under the “Stresses in Beams” section **page. 136 in FE Handbook 10.1**.

\begin{equation}

\tau_{x y}=V Q /(I b)

\end{equation}

Where,

\begin{equation}

V=\text { shear force }

\end{equation}

\begin{equation}

Q=A^{\prime} \bar{y}^{\prime}=\text { first moment of area above or below the point where shear stress is to be determined }

\end{equation}

Where,

\begin{equation}

\begin{aligned}

A^{\prime} &=\text { area above the layer (or plane) upon which the desired transverse shear stress acts } \\

\overline{y^{\prime}} &=\text { distance from neutral axis to area centroid } \\

b &=\text { width or thickness or the cross-section }

\end{aligned}

\end{equation}

## Walkthrough Video

Watch this simple step by step walkthrough solution that uses the equation above to solve for the maximum transverse shear stress developed within a rectangular shaped beam:

## Max Shear Stress for Special Cases

The above equation can be cumbersome because we need to evaluate the moment of the area Q. Several commonly used cross sections have special, easy-to-use formulas for the maximum vertical shearing stress are given as follows:

Asset-65-1**We must note that for most beams, the vertical shearing stress is so small when comparted to the bending stress. For this reason, sometimes the vertical shearing stress is not calculated at all but there are some exceptions. These include: **

- The material of the beam has a relatively low shear strength (such as wood).
- The bending moment is zero or small (making the bending stress is small), for example, at the ends of simply supported beams and for short beams. Since the bending moment is so small, the shear force will typically dominate at that location.
- The thickness of the section carrying the shearing force is small, as in sections made from rolled sheet, some extruded shapes, and the web of rolled structural shapes such as wide-flange beams.

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