FLOOD08704

• County: Statewide
• Basin: Statewide
• Title: Hydraulics of Bridge Waterways
• Date: 3/1/1978
• Prepared For: US
• Prepared By: USDOT
• Floodplain - Doc Type: Educational/Technical/Reference Information

Chapter II.-COMPUTATION OF BACKWATER

2.1 Expression for backwater, Bridge back-
water analysis is far from simple regardless of the
method employed, Many minor as well as major
variables are involved in any single waterway
problem, For the model which was installed in a
rectangular flume and operated with uniform rough-
ness, minor variables such as type and geometry of
abutments, width of abutments, slope of embank-
ments, roadway widths and width to depth ratio
could be evaluated in relation to the Froude Number
as was done in the comprehensive model study report
(18). In the case of bridges in the field where rough-
ness of flood plain and main channel differ materially
and channel cross sections are irregular, the Froude
Number is no longer a meaningful parameter and
minor variables lose their significance. This is es-
pecially true as bridge length is increased. Fortu-
nately, reasonable accuracy is acceptable in most
bridge backwater solutions, thus, a practical method,
utilizing the dominant variables, is presented in thi8
chapter for computing backwater produced by
bridge constrictions.
A practical expression for backwater has been
formulated by applying the principle of conserva-
tion of energy between the point of maximum back-
water upstream from the bridge, section I, and a
point downstream from the bridge at which normal
stage has been reestablished, section 4 (fig, 2A), The
expression is reasonably valid if the channel in the
vicinity of the bridge i8 essentially straight, the
cross sectional area of the stream is fairly uniform,
the gradient of the bottom is approximately con-
stant between sections 1 and 4, the flow is free to
contract and expand, there is no appreciable scour
of the bed in the constriction and the flow is in the
subcritical range.
The expression for computation of backwater up-
stream from a bridge constricting the flow, which is
developed in the comprehensive report (18), is as
follows:

h,' = K'", ~'; + '" [(~:')' - e7)'] ~~' (4)

Where

h,' = total backwater (ft.).
K' ~ total backwater coefficient.
'" & '" = as defined in expressions 3a and 3b (sec,
1.11).
An2 = gross water area in constriction measured
below normal stage (sq, ft,),
Vn, = average velocity in constriction' or Q/ An'
(f,p,s,),
A, = water area at section 4 where normal stage
is reestablished (sq, ft.),
A, = total water area at section 1, including
that produced by the backwater (sq. ft,),

To compute backwater, it is necessary to obtain
the approximate value of h,' by using the first part
of expression (4) :

V2n2
h1* = K*ot.2-
2g

(4a)

The value of A dn the second part of expression (4),
which depends on h,', can then be determined and
the second term of the expression evaluated:

'" [e:')' - e:')'] v;;' (4b)

This part of the expression represents the difference
in kinetic energy between sections 4 and 1, expressed
in terms of the velocity head, V'n2/2g, Expression
(4) may appear cumbersome, but this is not the case,
Since the comprehensive report (18) is generally
not available, a concise explanation regarding the
development of the above backwater expression and
the losses involved is included in appendix A of this
bulletin under type I flow,
2.2 Baekwater coefficient. Two symbols are
interchangeably used throughout the text and both
are backwater coefficients. The symbol K. is the
backwater coefficient for a bridge in which only the
bridge opening ratio, M, is considered, This is known

* The velocity, Vn,l. is not an actual mea6urable velocity, but represents a
reference velocity readily computed for both model and field structurelI.

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