FLOOD06232

• County: Statewide
• Basin: Statewide
• Title: Hydraulic Design of Flood Control Channels
• Date: 7/1/1991
• Prepared By: US Army Corps of Engineers
• Floodplain - Doc Type: Educational/Technical/Reference Information

e

,

,

e

.

,

e

bridge piers, Koch and Carstanjen (Koch 1926) fOWld that
the upstream momentum force had to be reduced by
(AJA))(1QZfgAt) to balance the total force in the
constrIctIOn.

(I) Equating the summation of the external forces
above and below the structures with those within the con-
tracted section yields

yQZ [(A J (yQZ]
ml + _ - .....!.. _
gAl Al gAl

= mz + mp

'YQZ
+-
gAz

and

m2 + mp

Z
+ 'YQ = m3
gAz

yQZ
+-
gA3

Combining these equations results in

ml + 'YQZ - [ ( Ap J ( yQZJ] = mz
gAl lAI 19AI

+ mp

Z
+ yQ = "':J
gAz

'YQZ
+-
gA3

This reduces to the Koch-Carstanjen equation

Z Z
ml - m + yQ (AI - A ) = mz + yQ
p . Z p gA
gAl Z
yQZ
= "':J - mp + -
gA3

where

'Y = specific weight of water, pOWlds
per cubic foot (pcl)

(2.18)

(2-19)

(2-20)

(2-21)

EM 1110-2.1601
1 Jul 91

Q = total discharge. cfs

m I = total hydrostatic force of water in
upstream section, Ib

mZ = total hydrostatic force of water in
pier section, Ib

m3 = total hydrostatic force of water in
downstream section, Ib

~ = total hydrostatic force of water on
pier ends. lb

Al = cross-sectional area of upstream
channel. square feel, f(2

AZ = cross.sectional area of channel
within pier section, f(2

A3 = cross-sectional area of downstream
channel. f(2

'\ = cross.sectional area of pier
obstruction. f(2

(g) Curves based on the Koch-Carstanjen equation
(Equation 2-21) are illustrated in Plate 12a. The resulting
flow profIles are shown in Plate 12b. The necessary
computations for developing the curves are shown in
Plate 13. The downstream depth is usually known for
tranquil-flow channels and is greater than critical depth.
It therefore plots on the upper branch of curve III in
Plate 12a. If this depth A is to the right of (greater force
than) the minimum force value B of curve II, the flow is
class A and the upstream design depth C is read on curve
I immediately above point A. In this case. the upstream
depth is controlled by the downstream depth A plus the
pier contraction and expansion losses. However. if the
downstream depth D plots on the upper branch of curve
III to the left of (less force than) point B, the upstream
design depth E is that of curve I immediately above point
B, and critical depth within the pier section B is the con-
trol. The downstream design depth F now is that given
by curve III immediately below point E. A varied flow
computation in a downstream direction is required to
determine the location where downstream channel condi.
tions effect the depth D.

(h) In rapid-flow channels. the flow depth upstream
of any pier effect is usually known. This depth is less
than critical depth and therefore plots on the lower branch
of curve 1. If this depth G is located on curve I to the

2-7