The HIRE Equation

The HIRE equation is based on field data of scour at the end of spurs in the Mississippi River (obtained by the USACE). The HIRE equation is:

1)	$\begin{array}{l}\displaystyle \displaystyle y_s = 4y_1 \left( \frac{K_1}{0.55} \right) K_2 Fr^{0.33}_1\end{array}$

Symbol	Description	Units
$\begin{array}{l}y_s\end{array}$	Scour depth	ft (m)
$\begin{array}{l}y_1\end{array}$	Depth of flow at the toe of the abutment on the overbank or in the main channel, taken at the cross section just upstream of the bridge.	ft (m)
$\begin{array}{l}K_1\end{array}$	Correction factor for abutment shape, Table 10-4
$\begin{array}{l}K_2\end{array}$	Correction factor for angle of attack ( $\begin{array}{l}\theta\end{array}$ ) of flow with abutment. $\begin{array}{l}\theta\end{array}$ = 90 when abutments are perpendicular to the flow, $\begin{array}{l}\theta\end{array}$ < 90 if embankment points downstream, and $\begin{array}{l}\theta\end{array}$ > 90 if embankment points upstream. _{$\begin{array}{l}K_2 = (\theta / 90) ^{0.13}\end{array}$}
$\begin{array}{l}Fr_1\end{array}$	Froude number based on velocity and depth adjacent and just upstream of the abutment toe

Table 10-4 Correction Factor for Abutment Shape, K₁

Description	K1
Vertical-wall Abutment	1.00
Vertical-wall Abutment with wing walls	0.82
Spill-through Abutment	0.55

The correction factor, _{$\begin{array}{l}K_2\end{array}$}, for angle of attack can be taken from the figure below.
Correction Factor for Abutment Skew, K2