Shear stress in rectangular and circular beams

Rectangular section ($b\times h$)

At a level $y$ above the NA, the area above is $b\left(\frac{h}{2}-y\right)$, with its centroid at $\frac12\left(\frac{h}{2}+y\right)$ from the NA:

$$Q = b\left(\frac h2 - y\right)\cdot\frac12\left(\frac h2+y\right) = \frac b2\left(\frac{h^2}{4}-y^2\right)$$

With $I = bh^3/12$ and width $b$:

$$\tau = \frac{VQ}{Ib} = \frac{6V}{bh^3}\left(\frac{h^2}{4} - y^2\right)$$

This is a parabola: $\tau = 0$ at $y = \pm h/2$, and it is maximum at $y = 0$:

$$\tau_{max} = \frac{6V}{bh^3}\cdot\frac{h^2}{4} = \frac{3V}{2bh} = 1.5\,\tau_{avg}, \qquad \tau_{avg} = \frac{V}{bh}$$

Solid circular section (diameter $d$, radius $r$)

At a level $y$ above the NA, the chord has width $b = 2\sqrt{r^2-y^2}$. The first moment of the segment above $y$ is

$$Q = \int_y^r 2\sqrt{r^2-y_1^2}\;y_1\,dy_1 = \frac23\,(r^2-y^2)^{3/2}$$

With $I = \pi r^4/4$:

$$\tau = \frac{VQ}{Ib} = \frac{V\cdot\frac23(r^2-y^2)^{3/2}}{\frac{\pi r^4}{4}\cdot2\sqrt{r^2-y^2}} = \frac{4V}{3\pi r^4}(r^2-y^2)$$

Maximum at $y = 0$:

$$\tau_{max} = \frac{4V}{3\pi r^2} = \frac43\cdot\frac{V}{\pi r^2} = \frac43\,\tau_{avg}$$

Distributions

   Rectangle                    Circle
  +--------+  0                 .--.    0
  |        |\                  /    \   |\
  |        | \                |      |  | \
  |   NA   |  > 1.5 avg       |  NA  |  |  > 1.33 avg
  |        | /                |      |  | /
  |        |/                  \    /   |/
  +--------+  0                 '--'    0

Both distributions are parabolic, zero at the top and bottom edges and maximum at the neutral axis.