Shear Stress in Beams: General Formula

Understanding shear stress—the internal force parallel to a material's cross-section—is critical for ensuring beams don't fail in a sliding or tearing mode. While bending stress often gets primary attention, shear becomes the governing design factor in short, heavily loaded beams, beams with thin webs, or those made from materials weak in shear. The general shear formula provides the essential tool to calculate this stress at any point within a beam's cross-section, revealing a distribution that is far from uniform.

The Derivation and Meaning of the Shear Formula

The formula for calculating transverse shear stress at a horizontal plane within a beam is derived by considering equilibrium of a small beam element. It is not derived from first principles of stress transformation but from the need to balance the changes in bending stress on either side of an element. The result is the Jourawski formula, more commonly known as the shear formula:

$$\tau =\frac{VQ}{It}$$

Here, $\tau$ represents the average transverse shear stress at a specific horizontal cut line through the cross-section. It's crucial to recognize that this formula gives the average stress across the width $t$, as the actual stress can vary. For most practical engineering purposes in standard beams, this average is an excellent and accepted approximation.

Deconstructing the Variables: V, Q, I, and t

Each variable in the shear formula has a specific, non-negotiable meaning. Misunderstanding any one of them leads to incorrect results.

• V (Shear Force): This is the internal shear force at the longitudinal beam location where you are calculating stress. You must obtain $V$ from the shear force diagram for the beam. Using the maximum bending moment instead of the maximum shear force is a frequent conceptual error.
• Q (First Moment of Area): This is the most nuanced term. $Q$ is formally the first moment of the area above (or below) the horizontal cut line where you want to find the shear stress. It is calculated as $Q=A^{\prime }{\bar{y}}^{\prime }$, where $A^{\prime }$ is the area of the cross-section above the cut, and ${\bar{y}}^{\prime }$ is the vertical distance from the neutral axis (NA) to the centroid of area $A^{\prime }$. $Q$ is a maximum at the neutral axis and zero at the top and bottom fibers, directly driving the shear stress distribution.
• I (Moment of Inertia): This is the entire beam cross-section's area moment of inertia about the neutral axis. It is a constant geometric property for a given cross-section. Use the correct $I$; for an I-beam, you typically use the value about the strong axis (x-axis).
• t (Width): This is the width of the cross-section at the exact horizontal line where you are calculating shear stress. For a rectangle, $t$ is constant (the base $b$). For an I-beam, $t$ is the web thickness when calculating stress in the web, and it becomes the flange width if calculating stress at a horizontal cut within the flange.

Applying the Formula: Stress Distribution in Common Shapes

The power of the formula is revealed when we calculate $\tau$ at multiple points to plot its distribution.

For a Rectangular Cross-Section: Let's find the shear stress at a distance $y_1$ above the neutral axis. The width $t=b$. The area above the cut is $A^{\prime }=b(\frac{h}{2}-y_1)$. The distance from the NA to the centroid of $A^{\prime }$ is ${\bar{y}}^{\prime }=y_1+\frac{1}{2}(\frac{h}{2}-y_1)=\frac{1}{2}(\frac{h}{2}+y_1)$. The moment of inertia for a rectangle is $I=\frac{b{h}^3}{12}$. Therefore: $$Q=A^{\prime }{\bar{y}}^{\prime }=b\left(\frac{h}{2}-y_1\right)\cdot \frac{1}{2}\left(\frac{h}{2}+y_1\right)=\frac{b}{2}\left(\frac{h^2}{4}-y_1^2\right)$$ Substituting into the shear formula: $$\tau =\frac{V\cdot \frac{b}{2}\left(\frac{h^2}{4}-y_1^2\right)}{\left(\frac{b{h}^3}{12}\right)\cdot b}=\frac{V}{2I}\left(\frac{h^2}{4}-y_1^2\right)$$ This is a parabolic equation. The maximum shear stress occurs at the neutral axis where $y_1=0$: $${\tau }_{max}=\frac{V{h}^2}{8I}=\frac{V\cdot h^2}{8\cdot \frac{b{h}^3}{12}}=\frac{3V}{2bh}=\frac{3V}{2A}$$ Thus, for a rectangle, the maximum shear stress is 50% greater than the average shear stress $V/A$.

For an I-Beam (Wide-Flange Section): The distribution is key. The flanges carry very little shear stress because their large width $t$ (the flange width) appears in the denominator, making $\tau$ small. Almost all the shear force is carried by the web. Consequently, the shear stress in the web is nearly uniform and very high compared to the flanges. The maximum shear stress again occurs at the neutral axis of the entire section. A common and accurate simplification is ${\tau }_{max,web}\approx \frac{V}{A_{web}}$, where $A_{web}$ is the area of the web (depth times thickness).

Common Pitfalls

1. Using the Wrong 't': The most common computational error is using a constant width. You must use the width at the specific cut line. For an I-beam, using the flange width to calculate web stress will drastically under-predict the actual stress.
2. Misunderstanding Q: $Q$ is not the first moment of the entire area. It is the first moment of the area beyond the cut, taken about the neutral axis of the full section. A related mistake is calculating ${\bar{y}}^{\prime }$ from the cut line instead of from the neutral axis.
3. Applying the Formula to Inappropriate Sections: The standard formula assumes shear stress is constant across the width $t$. This is valid for rectangles and I-beam webs but breaks down for non-prismatic sections or very irregular shapes. For a circular cross-section, the formula can be applied, but one must recognize the result is only an approximation; the actual maximum stress is slightly higher.
4. Confusing Maximum Bending and Shear Locations: The point of maximum bending moment (where bending stress is highest) is usually where the shear force $V$ is zero. The maximum shear stress typically occurs where $V$ is largest, which is often at the supports. You must analyze the correct beam section.

Summary

• The general shear formula $\tau =\frac{VQ}{It}$ calculates the average transverse shear stress at any horizontal cut within a beam's cross-section.
• $Q$, the first moment of area of the section above (or below) the cut, is the driver of the shear stress distribution, varying parabolically in a rectangular section.
• For common shapes like rectangles and I-beams, the maximum shear stress occurs at the neutral axis.
• In a rectangular beam, ${\tau }_{max}=\frac{3V}{2A}$, which is 1.5 times the average shear stress.
• In an I-beam, the web carries nearly all the shear force, and the maximum web shear stress can be accurately approximated by dividing the shear force by the web's cross-sectional area.