evaluating selected scour equations FOR bridge piers

in coarse streambeds in new york

**By L. J. Welch Jr., Graduate Assistant, U.S. Geological Survey, Troy,
New York; G.K. Butch, Hydrologist, U.S. Geological Survey, Troy, New
York**

**[Abstract]{.underline}:** The U.S. Geological Survey analyzed
cross-section data from 29 bridge sites in New York and developed a
best-fit, linear-regression equation to estimate the depth of scour that
would result at a bridge pier from a single peak flow in coarse
streambeds. The equation is based on 61 field measurements that are
associated with specific peak discharges and includes a stream-force
factor and a bed-material grain-size factor. The equation and three
other equations \-- the Federal Highway Administration (FHWA) equation,
the Mueller-modified FHWA equation, and the Froehlich equation were
applied to 116 field measurements made during 1943-96 for comparison.
The difference between estimated scour and measured scour ranged from
-0.8 meters to 0.5 meters for the best-fit equation, from 0.9 meters to
6.0 meters for the FHWA equation, from 0.1 meters to 2.9 meters for the
Mueller-modified FHWA equation, and from -0.2 meters to 2.3 meters for
the Froehlich equation.

# 

# Introduction

Many equations have been developed to estimate scour depth at bridge
piers. Because most equations were developed for sand-bed channels, they
overestimate scour depth in New York streams, which typically are
armored with gravel, cobbles, and (or) boulders. Failure of these
equations to account for this armor layer produces results that rarely
agree with field measurements.

In 1988, the U.S. Geological Survey (USGS), in cooperation with the New
York State Department of Transportation, analyzed 116 cross-section
measurements made during 1943-96 at 29 selected bridge sites in New York
(Butch 1993a). The criteria for site selection and methods of data
collection are described in Butch (1991). Streambed cross sections were
measured before, during, or after high flows. The depth of a scour hole
was defined as the difference between the ambient streambed elevation
and the minimum elevation in the scour hole. Ambient streambed
elevations were calculated from the 116 cross-section measurements
through a simple computer program to eliminate bias (Butch 1996).
Hydraulic values for factors such as water depth and flow velocity were
calculated from a step-backwater model calibrated to water-surface
elevations and discharges measured by the USGS (Butch 1993b).

A best-fit, linear-regression equation was developed to estimate the
depth of scour that would result at a bridge pier from a single peak
flow in coarse streambeds. The equation and three other equations \--
the Federal Highway Administration (FHWA) equation (Richardson and Davis
1995), the Mueller-modified FHWA equation (Mueller 1996), and the
Froehlich regression equation (Froehlich 1988) were applied to 116 field
measurements made during 1943-96 for comparison. This report describes
the methods that were used to develop the New York scour equation and
compares scour depths estimated from the New York equation with scour
depths obtained by the other three equations.

## 

## Database

The New York database (referred to herein as the "total New York
database") consists of 116 (55 historical and 61 discrete) scour
measurements made during 1943-96 at 29 bridge sites across the State.
Statistics for the total New York database, the historical data set, and
the discrete data set are given in Table 1; accuracy of scour-depth
measurements is about 0.1 m. The 55 historical scour measurements
(Table 1) were derived from cross-section data collected during 1943-89;
the measured scour depths are assumed to be the result of the maximum
peak flow from bridge construction. The 61 discrete scour measurements
(Table 1) represent data collected during 1972-96; the measured scour
depths are associated with a specific peak discharge. This association
is considered valid because the time between cross-section measurements
was generally less than 1 year, and most field observations and
measurements in the project confirmed that scour holes had not become
back filled (Butch 1994).

All bridge piers represented in the database are aligned with the
direction of flow. Scour depths that were affected by ice or debris were
omitted from the analysis. Many measurements in the total New York
database represent the deepening of previous scour holes, but neither
the effects of previous scour holes nor the length of time during which
a discharge exceeded a given hydraulic condition (such as mean velocity
or water depth) were analyzed, although the length of time may affect
the scour process (Butch and Lumia 1994).

The total New York database includes 27 zero-scour measurements. Only
the highest discharge was included for bridge piers at which multiple
high flows resulted in zero scour. Inclusion of zero-scour measurements
in the database decreased a bias of the equation to estimate scour for
every flow. Zero-scour measurements correspond to peak flows in which
hydraulic conditions did not produce scour, even though some peak flows
had recurrence intervals exceeding 100 years. Additional field data are
needed to define the onset of the scour process.

> **Table 1.** Statistics on eight variables that affect scour at bridge
> piers in New York. \[*D~xx~*~,~, grain diameter that exceeds *xx*
> percent of armor layer\].

  ------------------------ -------- --------- --------- --------- ---------
  Variable                 Min.     Max.      Std. dev. Mean      Median

  **A. Total database (116                                        
  measurements, 29                                                
  bridges, 1943-96)**                                             

  Flow velocity at pier    0.2      4.6       0.9       2.6       2.7
  (m/s)                                                           

  Water depth at pier (m)  1.4      9.7       1.8       4.1       3.8

  Discharge (m^3^/s)       59       6,600     1,380     1,290     816

  Pier width (m)           0.9      3.0       0.5       1.6       1.5

  *D~50~* armor layer (mm) 22       68        10        37        34

  *D~84~* armor layer (mm) 38       134       21        75        72

  Scour depth (m)          0.0      1.9       0.4       0.3       0.2

  Stream force (kg m/s^2^) 46       154,000   32,300    35,000    26,400

  **B. Historical data (55                                        
  measurements, 23                                                
  bridges, 1943-89, scour                                         
  depth assumed to be a                                           
  result of maximum flow                                          
  from bridge                                                     
  construction)**                                                 

  Flow velocity at pier    0.2      4.6       1.0       2.5       2.4
  (m/s)                                                           

  Water depth at pier (m)  1.4      7.7       1.3       3.6       3.7

  Discharge (m^3^/s)       114      6,600     982       1,080     818

  Pier width (m)           0.9      3.0       0.5       1.6       1.5

  *D~50~* armor layer (mm) 22       68        11        39        38

  *D~84~* armor layer (mm) 38       134       22        80        76

  Scour depth (m)          0.0      1.0       0.3       0.4       0.3

  Stream force (kg m/s^2^) 51       154,000   29,700    29,000    19,400

  **C. Discrete data (61                                          
  measurements, 20                                                
  bridges, 1972-96, scour                                         
  depth associated with a                                         
  specific peak                                                   
  discharge)**                                                    

  Flow velocity at pier    0.2      4.2       0.8       2.7       2.7
  (m/s)                                                           

  Water depth at pier (m)  1.4      9.7       2.0       4.6       4.4

  Discharge (m^3^/s)       59       5,350     1,640     1,480     637

  Pier width (m)           0.9      3.0       0.6       1.6       1.7

  *D~50~* armor layer (mm) 27       57        8.0       35        33

  *D~84~* armor layer (mm) 53       127       19        71        66

  Scour depth (m)          0.0      1.9       0.4       0.3       0.2

  Stream force (kg m/s^2^) 46       147,000   33,700    40,400    31,700
  ------------------------ -------- --------- --------- --------- ---------

## [Scour Equations]{.underline}: Scour is the result of work \-\-- the movement of bed material from one location to another. The force for this work is provided by flowing water that is redirected and accelerated as it flows around a pier. The equation developed in this study relates the magnitude of stream force directly upstream of a pier to the scour depth. Stream force is calculated from the water depth and flow velocity directly upstream from a pier and a flow width of 1 meter:

$\text{Sf} = \text{ρy}_{1}\text{wV}_{0^{2}}$, (1)

where *Sf* = stream force (kg m/s^2^);

= density of water (assumed 1,000 kg/m^3^);

*y~1~* = water depth directly upstream from pier (m);

*w* = flow width (1 m); and

*V~0~* = approach flow velocity directly upstream from pier (m/s).

The equation expresses estimated scour as a function of the ratio of
stream force near the pier to grain size:

$\text{scour} = f\left( \frac{\text{Sf}}{D_{\text{84}}} \right)$ , (2)

where *Sf* = stream force (kg m/s^2^); and

*D~84~* = grain size exceeding that of 84 percent of the armor layer
(mm).

Including *D~84~* as part of the stream-force regression term provided a
better correlation with local scour than *D~50~*, *D~90~*, or *D~95~*
grain sizes and was also less sensitive to sampling error than *D~90~*
or *D~9~*~5~. The *D~84~* grain size represented the armor layer at the
surface of the streambed. The armor layer is assumed to be present
throughout the reach over which the bridge was constructed and the
grain-size distribution is assumed to represent the gradation of the
armor layer.

Pier width and pier shape did not correlate well with scour depth within
the limited range of bridge geometry of this study (Butch 1993b) and
therefore were not included in the regression analysis. About 90 percent
of the total-database values are derived from piers ranging from 0.9 to
2.1 m in width; 71 percent of the total database represents round-nosed
piers, 26 percent represents sharp-nosed piers, and 3 percent represents
square-nosed piers. Using a pier-width factor to calculate stream force
rather than the 1-m flow width gave no significant improvement in scour
estimates.

## 

## [Regression Analysis]{.underline}: The initial linear regression analysis used all data collected during 1943-95 (97 measurements). The equation for the best-fit line, herein called the 1995a equation, was:

> $\text{scour }(m) = 0\text{.}\text{09} + \left\lbrack 6\text{.}\text{01} \times \text{10}^{- 4} \times \frac{\text{Sf}}{D_{\text{84}}} \right\rbrack$*.*
> (3)

The 1995a equation was then used to calculate scour corresponding to 19
scour measurements made in 1996. The mean error (estimated value minus
measured value) was 0.3 m. One explanation for the positive mean error
(overestimation) could be the greater percentage of zero-scour values in
the 1996 data set \-\-- 53 percent (10 of the 19 measurements) were
zero, compared to 18 percent (17 of the 97 values) for 1943-95. The mean
error decreased to 0.1 m when the estimates were compared with the
measurements made at the nine cross sections where scour was greater
than zero. The reason for the large number of zero-scour measurements in
1996 is unknown; the recurrence intervals of the peak flows ranged from
2 years to greater than 100 years.

An alternative explanation for the positive mean error between the 1995a
equation estimates and the 1996 measured scour values could be that the
assumption that historical scour depths (Table 1) resulted from a single
peak flow may be incorrect. The effect of the historical scour data was
evaluated through a second regression equation, herein called the 1995b
equation (eq. 4), which was developed only from discrete scour-data
collected during 1943-95 (42 values).

$\text{scour }(m) = - 0\text{.}\text{03} + \left\lbrack 6\text{.}\text{44} \times \text{10}^{- 4} \times \frac{\text{Sf}}{D_{\text{84}}} \right\rbrack$
. (4)

Applying the 1995b equation to the 1996 data resulted in a mean error of
0.2 m for all 19 measurements and 0.0 m for scour measurements greater
than zero. These improved results indicate that the historical scour
data, which could be affected by multiple high flows, ice, or debris,
affected the accuracy of the equation estimates for 1996.

The final linear regression equation, herein called the New York
equation (eq. 5), is based on 61 discrete scour measurements (1972-96)
that include 18 zero-scour values (Table 1). Applying this equation to
the 1943-96 data gave a standard error of 0.3 m and an R^2^ of 0.66.

The New York equation is:

> $\text{scour }(m) = - 0\text{.}\text{07} + \left\lbrack 6\text{.}\text{21} \times \text{10}^{- 4} \times \frac{\text{Sf}}{D_{\text{84}}} \right\rbrack$
> . (5)

The regression line for the New York equation is shown in figure 1.
Regression residuals were well distributed, although they indicate a
tendency to overestimate measured scour at, or near, zero. The user
needs to consider the standard error of equation 5 (0.3 m) when
evaluating scour estimates because the --0.07 intercept can result in
slightly negative scour values (aggradation). The intercept is not set
equal to zero because several regression statistics (such as R^2^ and
t-ratio) lose their usual meaning when the intercept term is removed
from the equation (Helsel and Hirsch 1992).

A sensitivity analysis of the New York scour equation indicated that the
variable to which the equation is most sensitive is flow velocity (the
squared term in the stream-force factor). The percent change in
estimated scour when one variable is varied (water depth, *D~84~*, or
flow velocity) while the others are held constant at their median values
is plotted in figure 2. The equation also is sensitive to grain size
(*D~84~*) at values less than --20 percent of the median (about the
minimum *D~84~* in the discrete database). The sensitivity to water
depth was nearly linear \-\-- a 50-percent change in water depth
resulted in about a 70-percent change in estimated scour depth.

![](media/image1.wmf){width="6.495833333333334in"
height="2.9097222222222223in"}

## Comparison of Equation Results

## 

## The New York scour equation and three other scour equations were applied to the total New York database through 1943-96 for comparison. These were the FHWA equation, the Mueller-modified FHWA equation, and the Froehlich equation. The total New York database was selected, rather than the discrete database from which the New York equation was derived, because the other equations are intended to estimate maximum scour. Using the total New York database also would test the New York equation when applied to other data, which include scour depths possibly resulting from multiple high flows, ice, or debris.

The FHWA equation, used by many State Departments of Transportation
(Richardson and Davis 1995) is:

> ![](media/image2.wmf) , (6)

where *y~s~* = scour depth;

*a* = pier width (m);

*y~1\ ~*= water depth directly upstream from pier (m);

*Fr~1~* = Froude number directly upstream from pier;

*K~1~* = correction factor for pier-nose shape;

> *K~2~* = correction factor based on ratio of pier length to pier width
> and the alignment of flow to pier;

*K~3~* = correction factor for streambed condition; and

*K~4~* = correction factor for armoring by bed-material size.

The K~4~ coefficient was added in 1995 to reduce scour estimates for
streambeds containing large bed material. The K~4~ coefficient is
applied when the median grain size, *D~50~*, of the armor layer is 60
mm. The K~4~ coefficient ranges from 0.7 to 1.0 and is calculated from
equations from HEC-18 (Richardson and Davis 1995). The *D~50~* of the
armor layer at the bridge sites evaluated in this study ranges from 22
mm to 68 mm; only one site (4 measurements) had a *D~50~* 60 mm. Results
of the FHWA equation as applied to the total New York database are given
in Table 2. Two sets of values are listed in Table 2 for this equation:
(1) K~4~ for sites where *D~50~* 60 mm (4 measurements, mean K~4~ =
0.997), and (2) K~4~ applied to all 116 measurements (no grain-size
restriction, mean K~4~ = 0.898). The resulting FHWA scour values (K~4~
based on *D~50~* 60 mm) are plotted as a function of measured scour
(mean error 2.8 m) in figure 3A. Calculating K~4~ for all measurements
(no grain-size restriction) reduced the mean error to 2.6 m (figure 3B).

> **Table 2.** Statistics for scour equations applied to total New York
> database
>
> (116 measurements). \[All values in meters. FHWA, Federal Highway
> Admin.\]

+-----------+----------+----------+---------+-----------+------------+
| ###       | ######   |          |         |           |            |
|           | Equation |          |         |           |            |
+-----------+----------+----------+---------+-----------+------------+
| ###       | FHWA^1^  | FHWA^2^  | Mueller | Froehlich | New York   |
| Statistic |          |          |         |           |            |
+-----------+----------+----------+---------+-----------+------------+
| #### **E  |          |          |         |           |            |
| rror** (e |          |          |         |           |            |
| stimated  |          |          |         |           |            |
| value min |          |          |         |           |            |
| us measur |          |          |         |           |            |
| ed value) |          |          |         |           |            |
+-----------+----------+----------+---------+-----------+------------+
| Max.      | 6.0      | 6.0      | 2.9     | 2.3       | 0.5        |
+-----------+----------+----------+---------+-----------+------------+
| Min.      | 0.9      | 0.6      | 0.1     | -0.2      | -0.8       |
+-----------+----------+----------+---------+-----------+------------+
| Median    | 2.6      | 2.4      | 1.1     | 0.7       | -0.1       |
+-----------+----------+----------+---------+-----------+------------+
| Mean      | 2.8      | 2.6      | 1.2     | 0.8       | -0.1       |
+-----------+----------+----------+---------+-----------+------------+
| Std. dev. | 1.0      | 1.1      | 0.6     | 0.4       | 0.3        |
+-----------+----------+----------+---------+-----------+------------+
| ##        |          |          |         |           |            |
| ### Scour |          |          |         |           |            |
|  estimate |          |          |         |           |            |
+-----------+----------+----------+---------+-----------+------------+
| Max.      | 7.0      | 7.0      | 3.5     | 2.7       | 1.5        |
+-----------+----------+----------+---------+-----------+------------+
| Min.      | 0.9      | 0.6      | 0.8     | 0.4       | -0.1       |
+-----------+----------+----------+---------+-----------+------------+
| Median    | 3.0      | 2.7      | 1.4     | 1.1       | 0.2        |
+-----------+----------+----------+---------+-----------+------------+
| Mean      | 3.1      | 2.9      | 1.5     | 1.1       | 0.2        |
+-----------+----------+----------+---------+-----------+------------+
| Std. dev. | 1.1      | 1.2      | 0.6     | 0.4       | 0.3        |
+-----------+----------+----------+---------+-----------+------------+

^1^ K~4~ calculated for measurements where *D~50~* 60 mm (4
measurements)

^2^ K~4~ calculated for all measurements

Mueller (1996) proposed an alternative K~4~ coefficient for the FHWA
equation. If *D~50~* \< 2 mm, or if D~95~ \< 20 mm, then K~4~=1; if
*D~50~* 2 mm and if D~95~ 20 mm and if:

> ![](media/image3.wmf) , (7)

then

![](media/image4.wmf) , (8)

otherwise *K~4~* = 1;

where *D~50~* = median grain size of the armor layer;

*D~95\ ~* = grain size exceeding that of 95 percent of the armor layer;

*V~0\ ~*= approach-flow velocity just upstream from pier;

> ![](media/image5.wmf) = approach velocity corresponding to critical
> velocity and incipient scour in accelerated flow region at pier for
> grain size *D~50~*;
>
> ![](media/image6.wmf) = approach velocity corresponding to critical
> velocity and incipient scour for grain size *D~95~*; and

![](media/image7.wmf) = critical velocity for incipient motion for grain
size *D~50~*.

![](media/image8.wmf){width="6.097222222222222in" height="7.125in"}

The Mueller-K~4~ coefficient ranged from 0.27 to 1.0 in the total New
York database, and had a mean value of 0.508. The estimated values from
the Mueller-modified FHWA equation are plotted as a function of the
measured scour values (1943-96) in figure 3C. As indicated in Table 2,
the Mueller equation overestimated the measured scour (mean error 1.2 m)
but gave lower values than the FHWA equation. The Mueller equation also
did not underestimate scour for any measurement in the total New York
database.

The Froehlich scour equation is a linear regression equation developed
from field measurements (Froehlich 1988). For piers aligned with the
flow, the Froehlich equation is:

> ![](media/image9.wmf), (9)

where = coefficient for pier-nose shape (0.7 for sharp nose, 1.0 for
round nose, and 1.3 for square nose);

*g* = gravitational constant;

*V~0~* = approach velocity just upstream from pier;

*y~0~* = water depth just upstream from pier;

*b* = pier width; and

*D~50~* = median grain size of bed material.

As a regression equation, the Froehlich equation would be expected to
overestimate scour for half of the database from which it was developed
and to have a mean error of 0.0 m. When applied to the total New York
database, the equation gave a mean error of 0.8 m and overestimated 113
of the 116 New York scour measurements. A plot of the Froehlich scour
estimates as a function of measured scour is given in figure 3D.

The New York scour equation underestimated 74 (64 percent) of the 116
scour measurements and gave a mean error of -0.1 m (Table 2). The
negative error is a result of the historical data in the total database
(the mean error for historical data only was -0.2 m). The New York
equation would not be reliable for streams whose variables exceed the
range of values from which the equation was derived. A plot of the New
York equation scour estimates in relation to the measured values is
given in figure 3E, and statistics on the New York equation's results
are included in Table 2. The slightly negative scour estimates shown in
Figure 3E are a result of the equation's negative intercept (eq. 5).

The range and distribution of values resulting from the four scour
equations are depicted as box plots in Figure 4. Scour estimates from
each equation and measured scour are plotted in figure 4A, and the error
for each equation (estimated value minus measured scour) is plotted in
figure 4B.

![](media/image10.wmf){width="6.513888888888889in"
height="3.111111111111111in"}

## Summary and Conclusions

The New York scour equation is a linear regression equation based on 61
field measurements made at selected bridge piers in New York streams
during 1972-96. The equation was developed from measurements associated
with specific peak discharges in coarse streambeds. The scour-depth
estimate is proportional to the magnitude of stream force near a pier
and inversely proportional to grain size.

The New York equation and three others \-\-- the FHWA equation, the
Mueller-modified FHWA equation, and the Froehlich equation \-\-- were
applied to the 116 measurements in the total New York database
(1943-96). The total database consisted of 61 field measurements that
are associated with specific peak discharges and 55 historical scour
measurements that could be affected by multiple flows, ice, or debris.
Results from the New York scour equation ranged from an underestimate of
-0.8 m to an overestimate of 0.5 m (mean error -0.1 m).

The FHWA equation's K~4~ coefficient, as modified by Mueller, reduced
most FHWA scour estimates; whereas the unmodified K~4~ coefficient
reduced only 4 of the 116 scour estimates (where *D~50~* 60 mm). The
FHWA and the Mueller equations did not underestimate scour for any
measurement in the database. The FHWA equation overestimated scour by
0.9 to 6.0 m (mean error 2.8 m). Removing the grain-size restriction
from the K~4~ coefficient in the FHWA equation reduced the mean error to
2.6 m. The Mueller-modified FHWA equation overestimated scour by 0.1 to
2.9 m (mean error 1.2 m). The Froehlich equation results ranged from an
underestimate of -0.2 m, to an overestimate of 2.3 m (mean error 0.8 m).

## References

Butch, G.K. (1991). "Measurement of bridge scour at selected sites in
New York, excluding Long Island." Water Resources Investigations Report
91-4083, U.S. Geological Survey, 17 p.

Butch, G.K. (1993a). "Estimating bridge scour in New York from
historical U.S. Geological Survey streamflow measurements." ASCE
Hydraulic Engineering '93, Proceedings of 1993 Conference, v.2,
1866-1871.

Butch, G.K. (1993b). "Relation of local scour to hydraulic properties at
selected bridges in New York." ASCE Hydraulic Engineering '93,
Proceedings of 1993 Conference, v.2, 1872-1875.

Butch, G.K., and Lumia, R., (1994). "Effects of flow duration on local
scour at bridge piers in New York." ASCE Hydraulic Engineering '94,
Proceedings of 1994 Conference, v.1, 46-50.

Butch, G.K. (1996). "Scour-hole dimensions at selected bridge piers in
New York." ASCE Hydraulic Engineering '96, Proceedings of 1996
Conference, (cdrom).

Helsel, D.R., and Hirsch, R.M., (1992). "Statistical methods in water
resources." Amsterdam, \[The Netherlands\], Elsevier Science B.V.
Publishers, 522 p.

Froehlich, D.C. (1988). "Analysis of onsite measurements of scour at
piers." ASCE Hydraulic Engineering, Proceedings of 1988 Conference,
534-539.

Mueller, D.S. (1996). "Local scour at bridge piers in nonuniform
sediment under dynamic conditions." Ph.D. dissertation, Colorado State
University, 212 p.

Richardson, E.V., and Davis, S.R. (1995). "Evaluating scour at bridges."
Federal Highway Administration Hydraulic Engineering Circular 18,
Publication No. FHWA-IP-90-017, 204 p.

## SYMBOLS

+-----------------------------------+-----------------------------------+
| *D~50~* = median grain size of    | *V~0~* = approach-flow velocity   |
| bed material;                     | just upstream from pier;          |
|                                   |                                   |
| *D~84~* = grain size exceeding    | *V~c~D~50~* = critical velocity   |
| that of 84 percent of the armor   | for incipient motion for grain    |
| layer;                            | size *D~50~*;                     |
|                                   |                                   |
| *D~90~* = grain size exceeding    | *V\'~c~D~50~* = approach velocity |
| that of 90 percent of the armor   | corresponding to critical         |
| layer;                            | velocity and incipient scour in   |
|                                   | accelerated flow region at pier   |
| *D~95~* = grain size exceeding    | for grain size *D~50~*;           |
| that of 95 percent of the armor   |                                   |
| layer;                            | *V\'~c~D~95~* = approach velocity |
|                                   | corresponding to critical         |
| *Fr~1~* = Froude number directly  | velocity and incipient scour in   |
| upstream from pier;               | accelerated flow region at pier   |
|                                   | for grain size *D~95~*;           |
| *K~1~* = correction factor for    |                                   |
| pier-nose shape;                  | *y~0~* = water depth just         |
|                                   | upstream from pier (Froehlich     |
| *K~2~* = correction factor based  | equation);                        |
| on ratio of pier length to pier   |                                   |
| width and the alignment of flow   | *y~1~* = water depth directly     |
| to pier;                          | upstream from the pier (FHWA and  |
|                                   | New York equation);               |
| *K~3~* = correction factor for    |                                   |
| bed condition;                    | *y~s~* = scour depth;             |
|                                   |                                   |
| *K~4~* = correction factor for    | = Froehlich coefficient for       |
| armoring by bed-material size;    | pier-nose shape;                  |
|                                   |                                   |
| *R^2^* = coefficient of           | = density of water.               |
| determination;                    |                                   |
|                                   |                                   |
| *Sf* = stream force;              |                                   |
|                                   |                                   |
| *a* = pier width (FHWA equation); |                                   |
|                                   |                                   |
| *b* = pier width (Froehlich       |                                   |
| equation);                        |                                   |
|                                   |                                   |
| *g* = gravitational constant;     |                                   |
|                                   |                                   |
| *w* = unit width of flow;         |                                   |
+-----------------------------------+-----------------------------------+