The term A/P is the hydraulic radius R. If the channel slope angle is small,
sin θ ~ S o

(2.94)
and for steady uniform flow the average shear stress on the boundary is
τ o = γRSo
(2.95)
If the flow is gradually varied nonuniform flow, the average boundary shear stress is
τ o = γRS f
(2.96)
where Sf is the slope of the energy grade line.
2.4.5 Relation Between Shear Stress and Velocity
Measuring the average bottom shear stress directly in the field is tenuous. However, the average
bottom shear stress can be computed from the expression
τ o = γRS f
(2.97)
For steady uniform flow, the local and average shear stress on the bed can be estimated by
employing the velocity profile equations in Section 2.4.2. If the local velocity v1 at depth y1 is known
and X = 1 for hydraulically rough boundary then, from Equation 2.78 the local shear stress can be
determined. The equation is:
ρv 1
2
τo =
(2.98)
2
y
30.2 1
5.75 log
k s
This equation and the ones given below are valid for fully turbulent uniform flow over a hydraulically
rough boundary in wide channels with a plane bed. Alternatively, if two point velocities in a vertical
profile are known (preferably in the lower 15 percent of the depth) the local shear stress on the bed
can be determined from the following equation:
ρ(v 1  v 2 )2
τo =
(2.99)
2
y
5.75 log 1
y
2
If the depth of flow yo, the grain roughness ks and the average velocity in the vertical V are known,
then the average shear stress can be determined from Equation 2.79. The equation is:
2.27