Open-File Report 2003-295
Equation 1 - Original Froehlich equation without safety factor
Equation 1 is discussed on page 8 of OFR 2003-295.
where
ys is the local abutment-scour depth, in feet (without the contraction-scour component);
ya is the average depth of flow on the flood plain upstream from the abutment, in feet;
K1 is the dimensionless correction factor for abutment shape;
K2 is the dimensionless correction factor for the angle of embankment to flow;
L is the length of the embankment projected normal to flow, in feet; (Many laboratory investigations define the road embankment that blocks approaching flows as the abutment length. In this report, the term embankment length is used.) and
Fr is the Froude number of the flow upstream from the embankment, which is defined as:
where
g is the acceleration due to gravity, in feet per square second, and
Ve is the average flow velocity upstream from the embankment, in feet per second, and is defined as:
where
Qe is the flow obstructed by the embankment, in cubic feet per second; and
Ae is the flow area obstructed by the embankment, in square feet.
Equation 2 - Original Froehlich equation with safety factor
Equation 2 is discussed on page 8 of OFR 2003-295.
where all variables are defined in equation 1.
Equation 3 - Modified Froehlich equation with safety factor
Equation 3 is discussed on page 10 of OFR 2003-295.
where
L' is the embankment length blocking live flow, in feet, and all other variables are defined in equation 1.
Equation 4 - Laursen abutment-scour relation
Equation 4 is discussed on page 11 of OFR 2003-295.
where
ds is the abutment-scour depth, in feet;
r is some constant greater than one; and
dsc is the contraction-scour depth, in feet.
Equation 5 - Sturm clear-water contraction scour equation
Equation 5 is discussed on page 11 of OFR 2003-295.
where
dsc is the clear-water contraction-scour depth, in feet;
yf0 is the average undisturbed depth of flow on the flood plain at the approach, in feet; (The term “undisturbed” refers to flow conditions without the bridge.)
qf1 is the constricted unit-width flow rate on the flood plain, in cubic feet per second per unit width; (The term “constricted” refers to the flow conditions resulting from the constriction of flow forced by the bridge.)
Vc is the critical velocity of the median bed material on the flood plain for the undisturbed flow depth, yf0, in feet per second; and
M is the flow distribution factor defined as (Q1/2ch + Qfp – Qbf)/(Q1/2ch + Qfp),
where
Q1/2ch is half of the constricted flow in the main channel at the approach section (from the channel centerline to the channel bank), in cubic feet per second;
Qfp is the constricted flow on the left or right flood plain at the approach section, in cubic feet per second; and
Qbf is the constricted flood-plain flow blocked by the road embankment at the approach section, in cubic feet per second.
Equation 6 - Sturm general equation for clear-water abutment scour
Equation 6 is discussed on page 12 of OFR 2003-295.
where
ds is the clear-water abutment-scour depth, in feet; (This value reflects total scour depth at the abutment.)
C1 and C0 are coefficients to be determined by laboratory experiments; and all other variables are defined in equation 5.
Equation 7 - Sturm clear-water abutment-scour equation without safety factor
Equation 7 is discussed on page 12 of OFR 2003-295.
where
ds, yf0, qf1, M, and Vc are as defined in equations 5 and 6, and
Kst is Sturm's abutment shape correction factor that ranges from 0 to 1. For vertical abutments with or without wingwalls, Kst is set to 1. For spill-through abutments, Equation 8 is used.
Equation 8 - Sturm abutment shape correction factor equation
Equation 8 is discussed on page 12 of OFR 2003-295.
where
; and qf1, M, Vc , and yf0 are defined in equation 5.
Equation 9 - Sturm clear-water abutment-scour equation with safety factor
Equation 9 is discussed on page 12 of OFR 2003-295.
where all variables are as defined in equations 5, 6, and 7.
Equation 10 - Critical velocity equation published in HEC-18
Equation 10 is discussed on page 15 of OFR 2003-295.
where
Vc is the critical velocity above which bed material size D and smaller will be transported, in feet per second;
y is the average depth of flow, in feet (when applied to the Sturm equation at setback abutments, y is equal to yf0); and
D is the grain size of interest, in feet.
Equation 11 - Neill clear-water contraction-scour total flow-depth equation
Equation 11 is discussed on page 16 of OFR 2003-295.
where
d is the total flow depth in the contraction, including the clear-water scour depth, in feet;
Vc is the critical flow velocity for the median grain size of the bed material, in feet per second; and
q is the unit-width flow in the contraction, in cubic feet per second per unit width, and is defined as:
where
V is the average velocity in the contraction prior to scour, in feet per second; and
y is the average flow depth in the contraction prior to scour, in feet.
Equation 12 - Neill clear-water contraction-scour depth equation
Equation 12 is discussed on page 16 of OFR 2003-295.
where
dsc is the contraction-scour depth, in feet; and
d and y are as defined in equation 11.
Equation 13 - Maryland clear-water contraction-scour total flow-depth equation
Equation 13 is discussed on page 16 of OFR 2003-295.
where
D50 is the median grain size of the bed material, in feet;
;
and d and q are defined in equation 11.
NOTE: Chang and Davis (1999) specified methods for computing the unit-width flow, q, in equation 13 based on the distance that the bridge abutment is set back from the main channel. (See Maryland State Highway Administration (2005) for details on computing the unit-width flow.)
Equation 14 - Maryland clear-water abutment-scour total flow-depth equation
Equation 14 is discussed on page 16-17 of OFR 2003-295.
where
dab is the total flow depth at the abutment, including the total clear-water abutment-scour depth, in feet;
kv is a coefficient to account for the increase in flow velocity at the abutment that ranges from 1.0 to 1.8 and is equal to 0.8(q1/q2)1.5 +1,
where
q1 and q2 are the unit-width flows for the approach and contracted sections, respectively;
kf is a coefficient to account for turbulence at the abutment that ranges from 1.00 to 4.0, and is equal to 0.13 + 5.8 F1,
where
F1 is the Froude number for the approach flow velocity, defined as:
where
g is the acceleration due to gravity, in feet per square second;
ya is defined in equation 1; and
Ve is the average flow velocity upstream from the embankment, in feet per second, and is defined as:
where
Qe is the flow obstructed by the embankment, in cubic feet per second; and
Ae is the flow area obstructed by the embankment, in square feet;
kp is a coefficient to account for the effect of pressure flow and ranges from 1.0 to 1.15 and is equal to 0.66 F1 -0.45, where F1 is defined above; and
d is as defined in equation 11.
Equation 15 - Maryland clear-water abutment-scour depth equation
Equation 15 is discussed on page 17 of OFR 2003-295.
where
ds is the total clear-water abutment-scour depth, in feet (local abutment scour and contraction scour);
kt is a coefficient for abutment shape;
ke is a coefficient for abutment skew; and
dab and y are defined in equations 14 and 11, respectively. (Refer to Maryland State Highway Administration (2006) for the equations of these coefficients.)
Equation 16 - HIRE abutment-scour equation
Equation 16 is discussed on page 19 of OFR 2003-295.
where
ys is the abutment-scour depth, in feet;
y1 is the depth of flow upstream and adjacent to the abutment, in feet;
K1 is the dimensionless correction factor for abutment shape;
Fr is the Froude number of the flow upstream and adjacent to the abutment, and is defined as:
where
Ve is the flow velocity upstream and adjacent to the abutment, in feet per second; and
g is the acceleration due to gravity, in feet per square second.
Equation 17 - Young contraction-scour equation
Equation 17 is discussed on page 20 of OFR 2003-295.
where
x is the total flow depth at the abutment, including the depth of scour, in meters;
y is the depth of flow over the flood plain prior to scour, in meters;
n is Manning's coefficient of roughness, defined as n = 0.0185(y)1/6 for sand (Blodgett, 1986);
S is the dimensionless Shields number (ratio of inertial to gravitational force) and is equal to 0.047 for sand motion;
SG is the specific gravity and is equal to 2.65 for sand;
D50 is the mean sediment size, in meters; and
VR is the resultant velocity at an abutment due to flow contraction, in meters per second.
Equation 18 - Young resultant-velocity equation
Equation 18 is discussed on page 20 of OFR 2003-295.
where
A is the cross-section area from the flood-plain edge to the center of the channel, in square meters;
V is the average flow velocity over area A, in meters per second; and
a is the flow area blocked by the embankment, in square meters.
Equation 19 - Young abutment-scour adjustment factor equation
Equation 19 is discussed on page 21 of OFR 2003-295.
where
K is the adjustment factor and is equal to the ratio of total scour at the abutment to contraction scour (K is truncated to 10 when it exceeds 10); and
a and y are as defined in equations 18 and 17, respectively.
Equation 20 - Young abutment-scour equation
Equation 20 is discussed on page 21 of OFR 2003-295.
where all variables are defined in equations 17 and 19.
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