Critical stress for Re > 400 and critical diameter

Critical shear stress for $Re_* > 400$

From the Shields diagram, the dimensionless critical shear stress is

$$\tau_* = \frac{\tau_c}{(\gamma_s - \gamma)\,d}$$

For $Re_* = \dfrac{u_*d}{\nu} > 400$ (fully rough, fully developed turbulent flow) the viscous sublayer is negligible and $\tau_*$ becomes independent of $Re_*$ and equal to a constant:

$$\tau_* \approx 0.056 \approx 0.06$$

Therefore the critical shear stress is

$$\boxed{\tau_c = 0.06\,(\gamma_s - \gamma)\,d}$$

Critical diameter

The average boundary shear stress in uniform flow is

$$\tau_0 = \gamma R S_0$$

At the limit of stability $\tau_0 = \tau_c$ for the critical diameter $d_{cr}$:

$$\gamma R S_0 = 0.06\,(\gamma_s - \gamma)\,d_{cr}$$ $$d_{cr} = \frac{\gamma R S_0}{0.06\,(\gamma_s - \gamma)} = \frac{R S_0}{0.06\,(s - 1)}$$

where $s = \gamma_s/\gamma$ is the specific gravity of the sediment. For quartz sand and gravel, $s = 2.65$, so $s - 1 = 1.65$:

$$d_{cr} = \frac{R S_0}{0.06 \times 1.65} = 10.1\,R S_0$$ $$\boxed{d_{cr} \approx 10\,R\,S_0}$$

with $d_{cr}$, $R$ in the same units (for example metres). Grains larger than $d_{cr}$ remain at rest; smaller grains move.