SCOUR ANALYSIS OF THE lNTERSTATE35 AND
CIMARRON RIVER CROSSING USING THE
FESWMS2DH AND SMS
COMPUTER MODELS
By
MICHAEL T. BUECHTER
Bachelor of Science
University of Missouri at Rolla
Rolla, Missouri
1990
Submitted to tbe Faculty of the
Graduate College of the
Oklahoma Slate University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
May, 1997
SCOUR ANALYSIS OF THE INTERSTATE35 AND
CIMARRON RIVER CROSSING USING THE
FESWMS2DH AND SMS
COMPUTER MODELS
Thesis Approved:
(\,j .
____.!....t:rU::'..L..::....~~+~..:...' _
Thesis Advisor
~ C./17~
Dean of the Graduate College
PREFACE
This study uses the commonly used scour prediction methods to analyze the
October] 986 flood which damaged the bridges at the Interstate35 and Cimarron River
crossing. The hydraulic analysis of this site was completed using the FESWMS2DH
computer program. The FESWMS2DH computer program considers the dynamics of
flow in both directions and is well suited to analyze this complex site. The use of this
computer program was greatly simplified by using the SMS computer program which
allows data to be input and output in graphical environment. The results of this study
demonstrate the validity of the commonly used scour equations and the usefulness of the
FESWMS2DH and SMS computer program.
I wish to express my sincere gratitude to the individuals who assisted me in this
project and during my course work at Oklahoma State University. In particular J wish to
thank my major advisor Dr. A. K. Tyagi for his guidance. I am also grateful to Dr. Mast
and Dr. Oberiender, both for serving on my committee and their enlightening courses. [
would also like to thank Ms. Ramona Wheately for constant suppOIi and encouragement.
Special thanks are due to Dr. Alan Zundel and the Engineering Computer
Graphics Laboratory at Bingham Young University for both providing me with the SMS
computer software and for their assistance in solving the flow problem. 1 would also like
to thank Mr. Larry Arneson of the FHWA for his intelligent input and advice. Mr. Dale
III
I
."
I
Abernathje is also thanked for his valuable computer advice and his help in preparing the
manuscript.
Additional thanks are due to the Oklahoma Department of Transportation
Hydraulic Branch which provided the information needed for this study, along with an
original version of the FESWMS2DH computer program.
[ would especially like to thank my wife, Mrs. Rita Buechter, for her constant
support, encouragement and for typesetting the manuscript.
lV
CHAPTER
TABLE OF CONTENTS
PAGE
1. INTRODUCTION 1
Statement of the Problem 1
Crossing History I
Flood Events 2
Description of Watershed 2
Scope of the Investigation ' 3
II. FINITE ELEMENT ANALYSIS 4
General 4
Solution Technique , 4
Basic Concepts 5
Governing Equations 8
Steady State Solution 10
Time Derivatives 17
Boundary Conditions 21
III. SCOUR EQUATIONS 24
General , 24
Total Scour 24
Aggradation and Degradation 25
Contraction Scour 27
General 27
Live Bed Contraction Scour 28
Clear Water Contraction Scour 30
Local Scour 32
Pier Scour 32
Abutment Scour 38
ClearWater and LiveBed Scour 41
v
i
~ ,
IV. METHODOLOGY AND APPLICATION " 43
Modeling Systems Operations 43
Site Overview 44
Description of Site 44
Hydrologic Data 47
Soils Information 48
Recorded Scour Data 49
Modeling 52
Modeling Strategy 52
Hydraulic Modeling 53
Scour Modeling 57
V. RESULTS AND DISCUSSION 60
Sununary of Results 60
Discussion of Results 66
VI. CONCLUSIONS AND RECOMMENDATIONS 68
Conclusions '" 68
Recommendations 69
REFERENCES 71
APPENDICES , 73
APPENDIX A  SITE MAP , 73
APPENDIX B  HYDROLOGY DATA 75
APPENDIX C  SOILS DATA 77
APPENDIX D  SCOUR DATA 80
APPENDIX E  CALCULATED SCOUR DATA 100
VI
LIST OF TABLES
TABLE PAGE
I. Possible Boundary Specifications for Various Flow Conditions and
Boundary Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
II. Value ofK] and K2 ....•..•.................................... 29
III. Correction Factor K2 for Angle of Attack of the Flow .37
IV. Increase in Equilibrium Pier Scour Depths K] for Bed Condition 38
V. Shape Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
VI. Bridge Dimensions .45
VII. Maximum Scour Depths Near Overflow Structures at the 135 Bridge
on the Cimarron River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51
VIII. Comparison of Actual to Calculated Scour . . . . . . . . . . . . . . .. 60
Vll
FIGURE
LIST OF FIGURES
PAGE
I. Examples of Two Dimensional Elements 9
2. Diagram of Coordinate System Axes I I
3. Illustration of Depth Averaged Velocity 12
4. Open and Closed Boundaries 22
5. Definition Sketch of Sediment Continuity Concept Applied to a Gi ven
Channel Reach Over a Given Time Period 26
6. FaH Velocity of Sand Sized Particles 29
7. Flow Pattern at a Cylindrical Pier 33
8. Scour Depth for a Given Pier and Sediment Size as a Function of Time
and Approach Velocity 34
9. Comparison of Scour Formulas for Variable Depth Ratios (y/a) 35
10. Comparison of Scour Formulas with Field Scour Measurements , , 36
I I. Influence of Abutment Alignment on Scour Depth 41
12. Interstate35 and the Cimarron River Site Plan 46
13. Aerial Photos of the Scour Holes at Interstate35 and the Cimarron River .... 50
14. Site Element Network 54
15. Velocity Vectors for October 1986 Flood 58
16. Calculated Versus Actual Scour Upstream of Overflow Bridge] , 62
17. Calculated Versus Actual Scour Upstream of Overflow Bridge 3 63
18. Calculated Versus Actual Scour Upstream of Overflow Bridge 5 64
]9. Calculated Versus Actual Scour Upstream of Overflow Bridge 7 65
V111
NOMENCLATURE
Ae = an element surface
Dso = median diameter of the bed material in feet
FESWMS2DH = Finite Element Surface Water Modeling System: TwoDimensional
Flow in a Horizontal Plane
f = a known function
g = acceleration due to gravity
H = water depth
L = a differential operator
N; = the assumed interpolation functi.on
n = Manning's n
n & n = the direction cosines between outward nOlmal to the boundary and the positive x y
x and y directions, respectively
O.D.G.T. = Oklahoma Department of Transportation
O. F. = overflow
Qi = total source / sink flow attributed to node 1
Q5 = nmoff produced at a 5 year event
Q10 = runoff produced at a 10 year event
Q25 = runoff produced at a 25 year event
Q50 = runoff produced at a 50 year event
QS2 = runoff produced at 52 year event
IX
QJOO = runoff produced at a 100 year event
Qsoo = mnoff produced at a 500 yem event
SCS = Suil Conservation Service
SMS = surface water modeling system
se = an element boundary
subscripto = the known values at the start of a time step
[', t = the length of the time step
U = horizontal velocity in the x direction at a point along the vertical coordinate
USGS = United States Geological Survey
u = the unknown nodal variable
v = horizontal velocity in the y direction at a point along the vertical coordinate
vc = the critical velocity above which bed material of a size 0 50 and smaller will be
transported
WI = bottom width of upstream main channel
W? = bottom width of main channel in contracted section
y = Dow depth
YI = average depth in upstream channel
Y2 = average depth in contracted section
z = the vertical direction
Zb = the bed elevation
Zs = the water surface elevation
ax = arctan (02;) lox)
cxy = arctan (o~ loy)
x
Puu' Pvv & Pvv = momentum flux conection coefficients that account for the variation of
velocity in th.::: vertical direction
8 = a weighting coefficient ranging from 0.5 to 1
p = water mass density
1:bx & 1:by = bed shear stress acting in the x and y directions, respectively
1:sx & 1:sy = surface shear stress acting in the x and y di.rections, respectively
'(XX> 1: xy & 1:yy = shear stress caused by turbulence where, for example, 1:xy is the shear
stress acting in the x direction on a plane that is perpendicular to the y
direction
"[2 = average bed shear stress in the contracted section
Q = Coriolis parameter
Xl
CHAPTER I
INTRODUCTION
Statement of the Problem
Crossing History
Interstate35 crosses the Cimarron River at the border of Payne and Logan
Counties, north of Guthrie, Oklahoma. Prior to 1988 the four lanes of Interstate3S, two
lanes in each direction separated by a 40 foot median and shoulders, crossed the Cimarron
River and its floodplain at this location on two main bridges and a series of eight
overflow bridges. However, these bridges were severely damaged by a large flood in
October of 1986. As a result of this damage the previously existing bridges were replaced
by the existing stmctures in 1988. The existing structures include two main bridges over
the Cimarron River and two large overflow bridges on the floodplain.
As mentioned previously, prior to 1988 Interstate35 crossed the Cimarron River
on two main bridges and eight overflow bridges. The two main bridges were located in a
parallel installation on the southern edge of the river valley, over the river channel. The
eight overflow bridges were located on the floodplain in series of four parallel
installations. The overflow stmctures were placed at increments of 900 feet, 450 feet,
650 feet apar1. The main bridges were 805 feet long white the overflow bridges ranged in
length from 160 to 280 feet. The main bridges had a flowline of approximately 870.2 feet
while the flowlines of the overflow bridges ranged from 885 feet to 887 feet.
The existing main structures are atso located on the southern edge of the river
I,'
,L'
I ,
•
valley over the main channel. The two overflow structures are located in a parallel
installation on the northern edge of the river valley. The main bridges are 800 feet long
and have a flowline elevation of 870.5 feet. The overflow bridges are 1,360 feet long and
have a flowline elevation of 887.0 feet.
Flood Events
The October 1986 flood is one of the two most severe floods on record for this
site. The October 1986 flood had a recorded high water surface elevation of 898.0 feet
and a recorded peak flow of 156,000 cubic feet per second, approximately a QS2 event.
The other of the two most severe floods which occurred at this site, occurred in May of
1957. No discharge information is available for this event, however, a high water surface
elevation of 899.0 feet was recorded.
Description of Watershed
As mentioned previously, Interstate35 crosses the Cimarron River at the border
of Payne and Logan Counties in the state of Oklahoma. Before entering Oklahoma, the
Cimarron River originates in New Mexico. The river first enters and exits the state of
Oklahoma at Ciman'on County. Secondly, the river reenters the state at Beaver County
and exits at Harper County. Finally, the river enters the state for a third and final time
where it forms part ofthe eastern portion of the Harper County line. Then the river flows
in a southeasterly direction to its tennination at the Keystone Reservoir. Approximately
17,505 square miles of watershed contribute runoff to the river up to the crossing with
Interstate35. Of these 17,505 square miles, approximately 4,296 square miles are
2
controlled by SCS water detention structures.
Generally, the river valley varies in width from 0.8 to ].2 miles. The river valley is
approximately one mile i.n width with high banks at the Interstate35 crossing. The main
channel is 700 to 2,000 feet wide and, like many mature rivers, it is highly meandering.
Currently the river is located at the southern edge of the floodplain and makes a sharp
turn towards the east to go under the main structure. This condition existed at the time of
the October 1986 Hood and may have contributed to the large amount of scour which
occurred during this flood. History indicates that the meander just upstream of the
Interstate35 crossings is moving downstream causing the main channel to move to the
north of the floodplain.
Scope of the Investigation
The scope of this investigation is to apply advanced hydraulic and scour analysis
to the October 1986 flood and previously existing structures at the lnterstate35 and
Cimarron River crossing. From this analysis, scour depths can be calculated a1 the
overflow bridges. The commonly used scour equations are based on theoretical
assumptions, and studies of sand bed flumes, and correlated with little field collected
data. Therefore, the results of this study wi 11 allow comparison of collected data to
calculated scour values. This comparison should help validate the use of the scour
equations as a design tool.
3
CHAPTER II
FINITE ELEMENT ANALYSIS
General
The finite element method is a numerical procedure which can produce
approximate solutions to the initial boundary value partial differential equations common
to physics and engineering. This method was originally conceived by engineers to
analyze aircraft structural systems. However, the rapid development of the high speed
digital computer led to the continuous development and applicati.on of finite element
techniques to a wide range of engineering problems, including surface water flow
problems. Lee and Froehlich (I 986) published a detailed review of the literature
discussing the application of finite element solutions to the equations of two dimensional
surface water flow in a horizontal plane. Additionally, Finnie and Jeppson (1991 )
presented a method for solving turbulent flows with finite elements.
Solution Technique
The FESWMS2DH computer program uses the Galerkin finite element method
to solve the governing system of differential equations which describe surface water flow.
Any finite element analysis solution begins by dividing the area of interest into a number
of elements. These elements are usually triangular or quadrangular in shape and can be
easily anangecl to fit complex boundaries. Elements are defined at a number of points
situated on the boundary and interior of the elements. These points are refelTed to as
nodes. Values of the dependent variables are then approximated at these node points
4
using a set of interpolation or shape functions. In the FESWMS2DH computer program,
mixed interpolation is used to help stabilize the solution. Quadratic interpolation
functions are used to interpolate depth averaged velocities and linear functions are used to
interpolate flow depth.
To form a set of equations for each element the method of weighted residuals is
applied to the goveming differential equations. Various approximations of the dependant
variables are then substituted into the governing equations which generally are not
satisfied exactly, resulting in residuals. These residuals are made to vanish, in an average
sense, when they are multiplied by a weighting function and summed at every point in the
solution domain. In Galerkin's method the weighting functions are the same as the
interpolation functions. By requiring the summation oftbe weighted residuals to equal
zero, the finite element equations take on an integral form. Coefficients of the equations
are integrated numerically and all the element equations are assembled to obtain a global
system of equations which are solved simultaneously. Because this system of equations
is nonlinear, Newton's iterative method as outlined by Zienkiewicz (1977) is used to solve
them.
Basic Concepts
The fundamental concept of the finite element method is to divide the problem
domain into a finite number of small regions called finite elements. Many convenient
shapes are available for this purpose including triangles and quadrilaterals. Within each
of these elements it is assumed that the value of a continuous quantity can be
approximated by a set of piecewisesmooth functions using the values of that quantity at a
5
N
finite number of points. The piecewisesmooth functions are known as interpolation or
shape functions. The points at which the continuous quantity is defined are called node
points. The behavior ofthe solution tllroughout the assemblage of elements is described
by the interpolation functions, once the unknown nodal quantities are found.
Once the elements and their interpolation functions have been chosen the
derivation of the element equations may be achieved by several methods. These methods
include direct methods, variational methods, or weighted residual methods. Although
these methods provide a means of forming the element equations they are not directly
related to the finite element method.
Weighted residual methods are general techniques for obtaining approximate
solutions to linear and nonlinear partial differential equations and include collation, least
squares, and Galerkin methods. In all these, the unknown solution is approximated by a
set of interpolation functions containing a~justable constants or [·unctions. The chosen
constants define the type of weighted residual method and attempt the "best"
approximation of the exact solution.
As mentioned previously the particular weighted residual methods differ from one
another in the choice of the weighting functions. In the method most used to derive
finiteelement equations, known as Galerkin's method, the weighting functions are
chosen to be the same as the interpolation functions of the trial solution. Therefore, in
Galerkin's method, Wi = N, for I = 1,1, ... , m. Thus Galerkin's method requires that:
where
6
I = 1,2, ... , m (21)
Nj = the assumed interpolation function
L = a differential operation
u = the unknown nodal variable
f =a known function
R = the domain
Additionally, the differential equation of a problem can be written as:
Luf=O (22)
The left hand side of Equation 21 can be written as the sum of expressions governing the
behavior of Equation 22 on individual elements. The variable u can be approximated
with respect to an element as:
n
:E N(e)u (e)
I I
i=l
(23)
Where the superscript (e) denotes the restriction of the relevant variable or function to an
element. Then the left hand side of Equation 21 can be written as the sum of expressions
of the form:
where,
R(c) = the element domain
fe) = the defined element function
[=1,2, ... ,11 (24)
A set of expressions like Equation 24 is written for each element in the network.
7

The assembly of element expressions results in a set of global algebraic equations, which
must be solved simultaneously. In a finite element solution, the values of a quantity at
the node points are the unknowns. The behavior of the solution within the entire
assemblage of elements is described by the element interpolation functions and the node
point values, after they have been found.
The basic idea of the finite element method is that a solution domain ofarbitrary
shape can be discretized by assumptions of elements in such a way that a sequence of
approximate solutions defined on successively more defined discretizations will converge
to the exact solution of the governing differential equations. The shapes of the elements
chosen to model a region, along with the order ofapproximation desired, will determine
the interpolation functions, Nj, which are used. AdditionaHy, the interpolation functions
need to satisfy certain criteria so that convergence of the numerical solution to an exact
solution of the governing differential equations can be achieved. Because of these
reasons most finite element networks consist of elements that arc geometrically fairly
simple. Common two dimensional elements are shown in Figure I, Examples of Two
Dimensional Elements. Although it is conceivable that many types of functiuns could be
used as interpolation functions, almost all finite element solutions use polynumials
because of their relative simpl.icity.
Governing Equations
In many surface water flow problems of practical engineering concern, the width
to depth ratio of the water body is very large. In these instances the three dimensional
8
y
':=_. x
c F
l .. x
w
A 0 )( ll~
n;aa
y y .
~~
;~
~~
~~II
~~
5~
:~
~\ ~ I
J( 41 :
B E S~ ,
~~ ,
y y
y
Figure 1. Examples of Two Dimensional Elements: (A) Threenode triangle; (B) Fournode
quadrilateral; (C) Sixnode triangle; (D) Eightnode quadrilateral; (E) Ninenode
quadrilateral; (F) Tennode triangle [Source: Lee and Froehlich, 1986, p.8]
9
nature of the flow may be ignored and a two dimensional flow application may be used.
Cases in which flows may be mostly two dimensional in character include shallow coastal
areas, harbors, estuaries, rivers and floodplains.
The FESWMS2DH computer program calculates depth averaged horizontal
velocities, flow depths, and the time derivatives of these quantities if a time dependant
flow is modeled. As with any numerical model, a fundamental requirement of the
FESWMS2DH program is that a satisfactory quantitative description of the physical
processes that are involved must be made. The equations that govern depth averaged
surface water flow account for the effects of friction, wind induced stresses at the water
surface, fluid stresses caused by turbulence, and the effect of the earth's rotation.
Steady State Solution
The equations that govern hydrodynamic behavior of a Newtonian fluid are based
on the concepts of conservation of mass (continuity) and momentum (motion). As
mentioned previously, for many practical surface water flow applications, knowledge of
the full three dimensional flow structure is not required, and it is sufficient to use mean
flow quantities in two perpendicular horizontal directions. By integrating the three
dimensional equations over the water depth and assuming a constant fluid density, a set of
three equations appropriate for modeling flow in shallow water bodies is obtained.
The coordinate system and variables used to obtain these equations are illustrated
in Figure 2, Diagram of Coordinates System Axes. Depthaveraged velocity is illustrated
in Figure 3, Illustration of Depth Averaged Velocities. Because the flow is assumed to be
in a horizontal direction, it is convenient to use a right hand Cmiesian coordinate system
10

lk',
H
l S
Zb
EXPLANATION
H DEPTH OF Fl.OW
u & ~ COMPONENT OF VELOCITY
v:: y COMPONENT OF VELOCITY
w= z COMPONENT OF VELOCITY
~ COORDINATE SYSTEM AXIS
y COORDINATE SYSTEM AXIS
z COORDINATE SYSTEM AXIS
'Z b GROUNDSURFACE AI..T!TUDE
z s WATERSURFACE ALTITUDE
~
Figure 2. Diagram of Coordinate System Axes [Source: Gilbert and Myers, 1989, p.6]
C.."l.r.rr,....".'.U'""'. .A."~"""""""'.A.. Q, ..._.. , .."'''' ............1"fJ'" ..._~~ ~~~ ~~= .'T..1"'"'X.~~~..L"
DepthAveraged Velocities
u
• ,  u ,.
U=
H
1/
~ ~I v=
y
1 ./ XIII '" NI /// '" '" ; / '" '"
Zb
Datum
zl:
Figure 3. Illustration of Depth Averaged Velocity [Source: Froehlich, 1992, p. 4.3]
12
with the x and y axes in the horizontal planes and the z axis directed upward. The depth
averaged velocity components in the horizontal x and y coordinate directions,
respectiveiy, are defined as follows:
u (25)
v
where
H = the water depth
Zs = the vertical direction
zlJ = the bed elevation
(26)
U = horizontal velocity in the x direction at a point along the vertical coordinate
v = horizontal velocity in the ydirection at a point along the vertical coordinat~
and
Zs = ~ + H = the water surface elevation
Chaudhry (1993) presents a through derivation of the depth averaged surface
water flow equations completed by integrating the three dimensional mass and
momentum transport equations with respect to the vertical coordinate from the oed to the
water surface and assuming that vertical velocities and accelerations are negligible. The
13
vertically integrated momentum equations are written as:
a(HU)
at + ~.(p HUU + (casa casa)2 ~gH2] + ~(A ax "" HUV) + x z 2 ay P Uy .
 !JHV + I [T;  T; O(HTx;x)  B(HT;)] xy = 0
P bx sx ax ay
BZb
caset~Hax
(27)
for flow in the x direction, and
a(HV)
at
a a( + _ax(A HVU) + _ A HVV P"y ay· P yV
I
+ QHU + L  L P by sy
+ (coset coset i ~ gH 2
] +
Y z 2
a(H\,) _ a(HT;y)] c 0
ax oy
for flow in the y direction, where
PULl' PUy, and Pvv = momentum flux correction coefficients that account for the
variation of velocity in the vertical direction
LXX = arctan (azb / ax)
0\ = arctan (azh / ay)
g = gravitational acceleration
Q = Coriolis parameter
p = water mass density, which is considered constant
Lbx and Lby = bed shear stress acting in the x and y directions, respectively
T;sx and T;sy = surface shear stress acting in the x and y directions, respectively
Lxx' T;xy' and T;yy = shear stresses caused by turbulence where, for example, LXI' is the
shear stress acting in the x direction on a plane that is
perpendicular to the y direction
14
c
The bottom friction coefficient, used to compute the bed shear stress, may be
computed as:
(29)
(210)
a(HV) ::: q ay
aH + a(RU) +
at ax
The vertically integrated mass transport, continuity equation is:
where
Boussinesq's eddy viscosity concept assumes that the turbulent stresses are proportional
C = the Chezy discharge coefficient
The effect of turbulence is modeled using Boussinesq's eddy viscosity concept.
cr= g n2 / 2.208 HI!) (21 1)
n = the Manning's roughness coefficient
where
or as
to the depth averaged velocity gradients. The eddy viscosity is defined so that when it is
multiplied by the mean velocity gradients, the appropriate depth averaged stresses due to
turbulence are obtained. Therefore, the eddy viscosity is not a true depth averaged
quantity in the mathematical sense.
Finite element formulations for the residuals of the depth averaged flow
equations, where the summation is with respect to all elements, written at node I are:
15

aZb 1
+ gHax  OHV + p(1"bx
 1" )] +
S.l
(212)
c
for flow in the xdirection, and
~ =Lf1THav +vaH +gH aZh +QHU+~(1" 1")]+
v, e rI at at ay p by sy
A,
aN;[_PHUV + VH[ au + av]] + aN; PHVV _~gH2 + 2VHavljdAe + (213) ax ay ax ay 2 ay
~fN{[PHUVTl +[PHVV+~gH2]Tl  VH[au+av]l1 +2vH
avTl jdS L I x 2 y a a x ::lye
e Y X oy
A, .
for flow in the y direction, where
Ae= an element surface
se = an element boundary
nx and Dy = the direction cosines between the outward normal to the boundary and
the positive x and y directions, respectively
All second order derivatives in the moment expressions have been integrated by
parts using the GreenGauss theorem. Reduction of the order of the expressions in this
way allows use of quadratic functions to interpolate velocities. Integration by parts of the
direction tenns simplifies the finite element equation formulation. Integration by parts of
16
the pressure terms facilitates application of normal stress boundary conditions. The last
boundary integral in square brackets ( [ ] ) in the two momentwll residual expressions
represents the lateral shear stresses resulting from the transport of momentum by
turbulence.
The expression for the weighted residual of the continuity equation is:
c
where
is the total source/sink flow attributed to node I
Time Derivatives
dA e
(214)
(215)
The residuals expressions 212, 213, and 2 J 4, given above, apply to a particular
instant in time. For a steady state solution all the time derivatives are equal to zero and
do not need to be evaluated. However, if the solution is time dependent, the residuals
need to be integrated with respect to both time and space. Time integration is
accomplished by using an implicit finite difference representation of the time derivatives.
For example, the derivative ofU with respect to time at the end of a time step is:
au
at
::: _1_ (u _ u)  ()  8) (au)o
GLlt 0 (J at
17
(216)
where
8 = a weighting coefficient ranging from 0.5 to 1
~t = the length of the time step
subscript 0 = the known values at the stmt of a time step
For 8 = 0, the time integration scheme is explicit (forward Euler), for 8 = ], the
time integration scheme is implicit (backward Euler), and for 8 = 0.5, a trapezoidal
(CrankNicholson) time integration scheme results. Setting 8 equal to 0.67 can provide
an accurate and stable solution for even relatively large time steps. The expressions for
au /at can be rearranged as:
au
at
where
]
ct =
8~t
and
PI = ctU
o
I (l  8) (au)
8 at 0
where
PI = only quantities that are known at the start of a time step
In a similar manner, time derivatives ofV and M are defined as:
(217)
(218)
(2 19)
av
at (220)
and
18
where
P2 = aVo + (I  8) (av)
8 at 0
(22] )
(222)
c
Using the procedure just outlined the expressions for derivatives of residuals arc
written for node I with respect to variables at node j. The derivative expressions for the
residual of the conservation of momentum equation in the x direction are:
(224)
and
19
where
(225)
y direction are:
Derivative expressions for the residual of the conservation of momentum equation in the
and
1
0 > if Chezy discharge coefficents are used
¢gn 2 > If Manning roughness coefficients are used
H 4/3
<I> = 0.151 for u.s. customary units, or 0.333 for Sl units
(226)
aJv ~kN,IQH thy  I  + U ] + a'NN ·[PHV] + ,IaN'x.'a[N]/]ldA au p u 2 + v 2 aX' ay v e
) (227)
+ L Jr,Nj[PHUllJ  Ni~~[VHlly]}dSe
e S.
(228)
20
3fVi
:= L f~M[ex.V+ oV + Qv + g GZh + 't:by JCf ] aH) e I) at ay pcfaH
A.
aN, rl au] aN [ ( au avJ ]} + aMx ) puv  gH + 2vax + _o'yM) PUV + v oy + ax dA e
+ L f~M[PUU + gH + 2'V au~ + [PUV  v[ au + av) ~ }dS
I) a x a a v e
e X y X s.
(229)
The derivative expressions for the equation of continuity residuals arc:
(231)
(230)
dQ,
au) J{ M.aNJ[H] + MN. [aH]) dA
I ax ') ax e
A.
e
=L
L JfaM.! au av] oM. aM} aQ j) ex. +  +  + Mj)[U] + Mj)[V] dA e
 .r (232)
e ax ax ay ax ax oH. A. )
Boundary Conditions
A physical region modeled in a surface water flow problem will have either
closed, or no flux boundaries, or open boundaries. These boundary conditions are shown
in Figure 4, Open and Closed Boundaries. The type of boundary and the flow condition
will determine the needed boundary information.
21
Open boundary
(outflow)
Open boundary
(Inflow)
Closed boundary
(slip)
Boundary Specifications for Various Flow Conditions and Boundary Types.
possible boundary conditions which may be specified are given in Table I, Possible
tangential shear stress at all points on the boundary. The types and combinations of
Figure 4. Open and Closed Boundaries [Source: Froehlich, 1996, p. 41 5J
Boundary conditions are specified around the perimeter of the area being modeled
for the entire duration of the simulation. Boundary condition specification consist of
either the normal flow or the normal stress, in addition to either the tangential flow or the
A closed boundary defines a geometric feature such as a natural shoreline, an
embankment, ajetty, or a seawall. Flow across a closed boundary generally equals zero.
An open boundary defines an area along the boundary of a finite element network where
flow is allowed to enter or leave the network.
The Galerkin finite element formulation allows complex conditions to be
automatically satisfied as natural conditions of the problem. These natural boundary
conditions are implicitly impressed in the problem statement and require no further
treatment. Those boundary conditions imposed explicitly are known as forced, or
essential, conditions. These boundary values are prescribed by modifying the finite
22
element equation goveming that variable. Additionally, special boundary conditions
imposed by one dimensional flow can also be applied.
TABLE I
POSSIBLE BOUNDARY SPECIFICATJONSa FOR VARIOUS FLOW CONDITIONS
AND BOUNDARY TYPES. (FROEHLICH, 1996, P. 418)
Type of boundary Row condition
Subcritical I Supercritical
Closed, U.;; 0 U. = V.' (usually U.' ;; 0)
Inflow, Un < 0 U. = U.· and V, = U,', or Un:;:; U;, V, = V;, and H = H', or q. = q:.
q. = q,: and q, = q,' , or q, = q,', and H = H' (usually V, = q, = 0).
H = !rand U. = D,', or
H=!r andq, = q,
(usually V, = q, = 0).
Outflow, V. > 0 H=!r !lothing
Weaklyreflecting V ~ 2{iii =  Vft
_ ~ 2JgH. n
·U = outward normal velocity. V, = tangential velocity, qn = Vjf = outward Donnal unit flow rate, qs = V,H =
~gential unit flow rate, U•• =outward norma.1 velocity in a fictitious river far upstream from tlie boundary, H. =
depth in a fictitious river far upstream from the boundary.
23
CHAPTER III
SCOUR EQUATIONS
General
Scour is the result of the erosive forces of flowing streams. These forces carry
material from one area of the stream downstream to another area ofthe stream. Scour,
such as occurs at highway bridge crossings, is generally differentiated Ii'om general bed
degradation and plan changes in a liver as being localized in nature.
Obviously, different materials will scour at different rates. Loose, uncemented
materials such as sand, may scour quickly in rapidly flowing water. Cemented soils, such
as clays, may scour much more slowly. However, Richardson, Hanison, Richardson, and
Davis (1993) state that the ultimate scour in cohesive or cemented soils can be as deep as
scour in sand bed streams. Foundations placed in massive homogeneous rock formations
are likely to be highly resistant to scour during the lifetime of a highway bridge.
Many different researchers have developed equations for predicting contraction
and local scour. AU of these equations are based upon theoretical assumptions and
laboratory experiments with little or no field verification. Additionally, many of these
equations do not account for site specific or subsurface conditions.
Total Scour
Total Scour at a highway bridge is generally considered to be made up of three
components. All three of these components together comprise the total scour at a
24
! '
II
'
I

highway bridge. These are:
1. Aggradation and Degradation
2. Contraction Scour
3. Local Scour
In addition to the three types of scour mentioned above lateral, shifting of the stream
within its floodplain can often damage a highway bridge.
Aggradation and Degradation
Aggradation and degradation, sometimes referred to as gradation changes. refer to
long term, general changes of the slope and elevation of the stream bed. These changes
generally occur over a large segment of the stream. Aggradation involves the raising of
the stream bed as a result of deposition of sediment. Degradation involves the lowering
of the stream bed as a result of the removal of sediment.
Gradation changes can be caused by both natural and man made factors. Some of
the man made factors which can cause these changes are: channel alterations, stream bed
mining, construction of dams and reservoirs, and land use changes. Natural causes of
stream gradient instability are primarily natural channel alterations, earthquake, tectonic
and volcanic activities, climatic changes, fire and channel bed and bank material
erodibility. Additionally, a long term trend in bed elevation may change over the life ofa
highway structure.
The long term stability of a stream can be described by the sediment continuity
concept. According to Lagasse, Schall, Johnson, Richardson, Richardson and Chang
(1991) the sediment inflow minus the sediment outflow equals the time rate of change of
25

sediment volume in a given reach. In simpler terms this means that the amount of
sediment entering a reach minus the amowlt of sediment leaving that same reach equals
the change in the amount of sediment stored in that reach of the stream. This concept is
demonstrated in Figure 5, Definition Sketch of Sediment Continuity Concept Applied to a
Given Channel Reach Over a Given Time Period.
SedIment 'Inflow
(Vokme )
Change In VoIurn4t" Inflow  Outftow
[
If negative • ltI'a.ion wII OCCU' ]
" poaitive • aedlmentation wII occur
Figure 5. Definition Sketch of Sediment Continuity Concept Applied to a Given Channel
Reach Over a Given Time Period [Source: Lagasse et aI., 1991, p. 28.1
The problem facing the engineer is to predict the change in the stream bed
elevation which will occur over the life of a highway structure. A quantitative estimate of
change in the stream bed elevations can be made by using the Federal Highway
Administrations HEC20 Stream Stability at Highway Structures. A sediment continuity
computer program such as BRISTARS or the Corps of Engineers HEC6 can be used to
make a quantitative estimate of the change in the stream bed elevation. Also, data
documenting the long term changes in stream bed elevations is available from the U.S.
Army Corps of Engineers and other agencies.
26
Contraction Scour
General Contraction Scour occurs at bridges because the flow area of the stream
is reduced by either a natural decrease in flow area or by abutments or piers blocking a
portion of the flow area. This reduction in waterway area results in an increase in
velocity at the bridge. Increased velocity results in an increase in shear stress which
causes the removal of sediment from the area of the bridge. This removal of sediment
results in a lowering of the natural stream elevation and a subsequent increase in the
stream cross section. This increase of the stream cross section continues, in the riverine
situation, until the velocity and shear stress are reduced to a point where equilibrium is
reached.
There are two types of contraction scour. Livebed scour occurs when there is
sediment being transported into the constricted channel section from the unconstricted
area upstream. Clearwater scour occurs when there is negligible transport of sediment
from the unconstricted section to the constricted sections. Typically, hoth types of scour
are cyclic. That is, scour increases during the rising stage of a runoff event and fills at
least partially on the falling stage.
Contraction scour equations are based upon the principle of conservation of
sediment transp011. In the case of livebed scour, maximum scour occurs when the shear
stress reduces to the point that the sediment transported into the constricted section equals
the sediment transported out of the constricted section. At this point the conditions for
sediment continuity are in balance. During clearwater scour the maximum scour occurs
when the shear stress reduces to the critical shear stress of the material.
Contraction scour will also depend upon whether the bridge is a relief bridge or a
27
bridge over a main channel. According to Laursen (1963) a secondary bridge placed on
the floodplain will divert a part of the flow from the main channel crossing: the "relief"
thus obtained presumably permits a reduction in the length of the bridge over the main
channel and in the height of fills. Therefore. to calculate contraction scour at a bridge it is
necessary to detelmine if the flow upstream of the bridge is transporting sediment or not
and whether the bridge is a relief bridge or a bridge over the main channel.
Live Bed Contraction Scour Laursen (1960) derived a clearwater contraction
scour equation based upon a: long contraction and a simplified transport flillction.
Richardson et a1. (1993) presented Laursen's equation as:
where
YI = average depth in upstream channel, in feet
Y2 = average depth in contracted section, in feet
WI = bottom width of upstream main channel, in feet
W2 = bottom width of main channel in the contracted section. in teet
(31)
QI = flow in the upstream channel transporting sediments, in cubic feet per second
Q2 = flow in the contracted channel, in cubic feet per second
n, = Manning's n for the upstream main channel
n2 = Manning's n for the contracted section
K1 and K2 = exponents determined from Table II below, based upon the mode of
material bed transport
28
TABLE II
VALUE OF k , AND kz
Vo/w K, Kz Mode of Bed Material Transport
<0.50 0.59 0.066 Mostly contact bed material
0.50 to 2.0 0.64 0.21 Some suspended bed material
discharge
>2.0 0.69 0.37 Mostly sllspended bed material
discharge
in Table II
Vo = (g y, SI)I/Z shear velocity in the upstream section, in feel per second
s, = slope of energy grade line of main channel, in feet per feet
0 50 = median diameter of the bed material, in feet
w = median fall velocity of the bed material based upon the Dso , see Figure 6
10'
.... 03 ;,
o
10 fa 10
I
,I
I
~
/
'I
T"32OF
"""
60~F
i'""
100°F
v
V
2 3 2 I 100
EE
.. o
W 1 fps
Figure 6. Fall Velocity of Sand Sized Particles [Source: Richardson et aI., 1993, p. 34]
29
In Equation 31 the material scoured during a flood is deposited over a large area so that:
(32)
where
Ys = average scour depth, in feet
ClearWater Contraction Scour Laursen (1963) also presented an equation
conunonly used to predict clearwater scour. This equation is based upon the assumption,
stated earlier, that clearwater scour is the greatest when the shear stress in the contracted
section equaJs the critical shear stress, or as:
(33)
where
't"2 = average bed shear stress in the contracted section, in pounds per square foot
L C = critical bed shear stress at incipient motion, in pounds per square foot
The shear stress may be stated in terms of the Manning's equation as:
where
y = the unit weight of water, 62.4 pounds per cubic foot
Sr = slope of the energy grade line, in feet per foot
V2 = average velocity in the contracted section, in feet per second
30
(34)
Richardson et a1. (L 993) make use of the previous relationships, Stricklers's
approximation and continuity to present Laursen's clearwater contraction scour equations
as:
v2 3
___'__]7
1 2
3 3
120 Yl Dso
(35)
Froehlich (1996) presented equation 34 in terms of Sl units and twodimensional
where
VI = average velocity in the upstream main chmmel; in feet per second
and again:
flow as:
d
sc [p g ; 2 q 2r7
 H
c
where,
dsc = clearwater contraction scour depth
p = density of water
H = water depth
and
(36)
(37)
:"'0« '.. ; ,,(
j I)
~ I~
.:~
• ,I>
.....~.
:)
" ~
:· "I I~· , "I
~ I" I
~ ')
• I~ ;J
:1 )'. q~
·':1'
~ :~ ··",~
q unit flow rate
31
(38)
where
U and V = depthaveraged velocities in the x and y directions respectively
Local Scour
Local scour is the result of the formation of vortices caused by obstructions to the
flow. In the case ofa highway bridge the flow obstruction can be either a pier or
abutment. These obstructions accelerate the flow in their immediate area and create
vortices that remove the material around them. Local scour, like contraction scour, may
be either clearwater or livebed scour.
Pier Scour As mentioned previously, a bridge pier will cause a system of vortices
which are responsible for local scour. These vortex systems are well understood and are
described in detail by Molinas (1990) and Chiew (1992). Depending upon the bridge pier
and the flow conditions the vortex system will be made tip of a horseshoevortex, a wakevortex,
and a trailing vortex. These vortex systems are shown in Figure 7, Flow Pattern aL
a Cylindrical Pier.
As shown in Figure 7, the vertical velocity distribution in an open channel is
characterized by the no slip condition at the bottom of the channel. As flow approaches a
pier a stagnation plane is formed on the upstream face of the pier. That is the flow
velocity in a vertical plane, approaches zero on the upstream face of the pier. Because of
the vertical velocity profile a downward pressure gradient, and therefore downward flow,
are formed on the upstream face of the pier. This downward flow will cause a threedimensional
separation of the boundary layer which rolls up ahead of the pier creating a
horseshoevortex system. The horseshoevortex system is very efficient at dislodging and
32
,,01'
·ot
·ot
I)
I~
. 1
. ,..
.,.~ ..4
J
" ~
Figure 7. Flow Pattem at a Cylindrical Pier [Molinas, 1990, p. 31]
removing soil particles from the base of the pier, therefore, causing local scour.
The formation of a stagnation plane on the upstream face of the pier also causes a
lateral acceleration of flow past the pier. This acceleration causes the formation of
vertical vortices downstream of the pier which are referred to as the wakevortex. This
vortex system also causes removal sediment from the base of the pier. A bl.untnosed pier
will cause the formation of a strong wake vortex system, whereas, a more streamlined
pier shape, referred to as a sharpnosed pier, will create a weaker vortex system.
A trailingvortex system is composed of one or more discrete vortices beginning
at the top of the pier and extending downstream. These vOl1ices are created when a finite
pressure difference exists between two flow surfaces moving at a corner. This type of
vortex system usually only occurs on a completely submerged pier.
Figure 8, Scour Depth for a Given Pier & Sediment Size as a Function of Time &
33
.'1,4. .,.
I)
i..
. 1
Approach Velocity, describes the progression of pier scour as a function of time and
approach velocity for a given pier and sediment size. Clearwater scour will cease when
the shear stress caused by the horseshoevortex systems equals the critical shear stress of
the sediment particles at the bottom of the hole. Livebed scour will fluctuate abollt an
equilibrium scour depth in response to the formation of varying bed fonns. This
equilibrium scour depth will be reached when the amount of sediment leaving tbe scour
hole is equal to the amount of sediment supplied to the scour hole.
I; ",·0.4
; "~
I')
•" ,':J~ .. ~
E~ilitJ"tt!m
Scour ~fJfh j d"...
Vrfoci'y
(bJ
Figure 8. Scour Depth for a Given Pier and Sediment Size as a Function of Time
and Approach Velocity. [Molinas, 1990, p.30]
According to Molinas (1990) and Richardson et al. (1993) the following factors
influence scour around bridge piers:
1. Pier shape, width, orientation, and presence of ice and debris.
2. Approach flow depth and velocity.
3. Bed configuration and sediment diameter and density.
4. Density and kinematic viscosity of the fluid.
34
By incorporating the above parameters several researchers have developed equations to
predict local scour at bridge piers. Most of these equations have been based on laboratory
tests and many yield different estimates of scour depth for a given set of scom data.
Richardson et al. (1993) presented Figure 9, Comparision of Scour Formulas for
Variable Depth Ratios (y/a) and Figure 10, Comparison of Scour Formulas with Field
Scour Measurements, which were prepared by the Federal Highway Administration and
compare many of the more common scour equations. Both Molinas (1990) and Becker
(1994) presented several of the more commonly used scour equations.
6
Fr=O.3
5 .c.
+
"0
~ 4
'
(l)
0
."c. 3 0
Q)
0
' 2
::J
00
(j) Bruesers
2 3 4 5 6
y/a (Flow Oepfh/P\er Width)
8
Figure 9. Comparison of Scour Formulas for Variable Depth Ratios (y/a) [Source:
Richardson et al., 1993, p.37]
35
2.5 rrr...,.,r..~
As can be seen from Figure lO the Colorado State University Equation envelopes
, 114
·• "4 ,,~
·· I] \:~
~
,;..
....~.
)
~
"I
"~
'1
• ,j
'J
1,1
:1
II' I "I ·\..:.;,,
...:.:,1 .,
, "
~ ::,
)
3.0 3.5
Melville 6 Sutherland
Laursen
Fr=O.1
Comparison of Scour Formulas with Field Scour Measurements. [Source:
Richardson et al., 1993, p.3?]
Bruesers
=.JOinaRscher
/ CSU ,/ / _A.58
~6·0~Shen
.09 /.29;:i~~*.Fr"0.13
~_~_~.68%.~L~
~.t' ..06. __f£1.91'Is P  oorlo Chitale
l~fj~..~..~~~~==~~=:S~~;[==~~ __...1...___o~_·  ··..Ahmod J
o 0.5 1.0 1.5 2.0 2.5
y/a (Flow Depth/P,er Width)
1.0
~
Q) 1.5
CL
.......
.I:: 0
Q) o
.I::
2.0 "0
3:
~
::J o
o
!:!1. 0.5
Figure 10.
all of the data points, but gives lower values than many of the other equations. Therefore,
the Colorado State University Equation is recommended by Richardson and others (1993)
for use in predicting pier scour. Richardson and others (1993) presented the Colorado
State University equation for pier scour as:
Ys = 2.0 K) K
2
K
3
(~)O.35 Fr.0.43
q a
(39)
where
ys = scour depth, in feet
36
a = pier width, in feet
K, = a coefficient based on pier nose shape, 1.\ for square nosed piers, 0.9 for
sharp nosed piers, 1.0 for round or circular nosed piers and 1.0 for a group of
cylinders
K2 = conection factor for angle of attack offlow from Table III
K3 = correction factor for bed condition from Table IV
y, = flow depth directly upstream of pier, in feet
VI = mean velocity of flow directly upstream of the pier, in feet per second
TABLE III
CORRECTION FACTOR KzFOR ANGLE OF ATTACK OF THE FLOW
[Souce: Richardson et aI., 1993, p. 40]
Angle L/a1 = 4 Lla=8 Lla = 12
0 1.0 1.0 1.0
15 1.5 2.0 2.5
30 2.0 2.5 3.5
45 2.3 3.3 4.3
90 2.5 3.9 5.0
Angle = skew angle of flow
L = length of pier
37
I 114 ... ..~ I]
,,~
~
:". • ,. ,,;
J
~
.,j
,,,\
"\
'''1
']
" ~
,:~
, ~~.
")
I'::·
, ::1
·'::1
, ""I
;)
TABLE IV
INCREASE IN EQUILIBRIUM PIER SCOUR DEPTHS K)
FOR BED CONDITION
[Source: Richardson et a1., 1993, p. 40]
Bed Condition Dune Height H (ft) KJ
Clear Water Scour N/A 1.1
Plane Bed & Antidune Flow N/A 1.1
Small Dunes 10>H>2 1.1
Medium Dunes 30>H>10 1.1 to 1.2
Large Dunes .II>3 1.3
Debris lodged on piers have the effect of increasing local scour at a pier. This
occurs because the effective pier width is increased and a greater amount of the flow is
directed downward. However, increasing flow depth tends to decrease the effect of
debris on piers. Melville and Dongal (1992) have made recommendations concerning the
treatment of debris on piers.
Abutment Scour The mechanism of local scour at abutments is identical to that at
pIers. The same system of v0l1ices is formed. These vortices, as at piers, remove
sediment from the stream bed in the vicinity of the abutment. Therefore, the same
considerations apply to abutment scour, as apply to pier scour.
As with the pier scour many equations have been developed to predict local scour
at abutments. These equations are based entirely upon laboratory data and tend to predict
excessively conservative scour depths for the field situation. This happens because the
38
, ~
,~
,,~
)
~
"'j
..I
"I
"1
'J
I'~
":.l ·'4 :>
"I
::1
"'1
.,
""
"I
)
length of the abutment obstmcting flow is easily measured in the laboratory, whereas, in
the field this value proves to be more elusive. Another problem is that little field data on
abutment scour exists.
Melville (1992) presented a procedure to calculate abutment scour based upon
laboratory data. This procedure accounts for abutment length, flow depth, and abutment
shape and alignment. The procedure presents equations in terms of the abutment length
to flow depth. In summary the abutment scour equations are:
where
d 2 K L s s
ds = depth of scour, in feet
Ks = shape factor from Table V
L
<
Y
s; L :;25
y
L
 > 25
y
(310)
(311)
(312)
Ko= factor accounting for abutment alignment from Figure 11
L = length of the abutment including the bridge approach measured perpendicular
to flow
y = flow depth in feet
The factor Ks accounts for abutment shape. However, as the abutment and bridge
approach become longer the effect of abutment shape diminishes. Therefore, the value Ks
5hould be adjusted as follows:
39
L
~ 10
y
(313)
Ks' = K + (l  K) (0.1 L  1.5) s s y
L
~ 25
y
L
10 <  < 25
y
(314)
(315)
TABLE V
SHAPE FACTORS [Melville, 1992, p. 617]
," .".'
Abutment Shape
(1)
Ve11ical plate or narrow vel1ical wall
Vel1ical wall abutment with semicircular end
45° wing wa1l
Spillthrough (H : V):
0.5 : 1
1 : 1
1.5 : I
Shape Factor, Ks
(2)
1.0
0.75
0.75
0.60
0.50
0.45
The factor Ke accounts for abutment alignment. However, the effect of abutment
alignment diminishes as the abutment and bridge approach become shorter. Therefore,
the value Ke should be adjusted as follows:
40
L
2 3
y
(316)
Ke' = Ke + (1  Ke) (1.5  0.5 1:....)
y
L
I < < 3
Y
(317)
K" L
0  s
y
(318)
....;
",",
.f
, I
,~
, t
''I
'J
':i ~I.
,) ,,' :f
J,,' ,)
160 180
x
140
• ~d (lg53)
+ 1A.u.neD (HI58)
v Sutry (U~~2)
() ZqbJoul (UIB3)
A K:wan (UO,)
X Kandua:my (lgas)
80 100 120
9. degrees
20 ...0 60
x
•
o
0.8
0.7
1.2
¢=o
flow
1.1
Kg  ~ +
v
1
x
A
0.9
Figure II. Influence of Abutment Alignment of Scour Depth [Source: Melville,
1992, p. 623]
ClearWater and Live Bed Scour
As mentioned previously both clearwater, and livebed scour can occur during
contraction and local scour. Clearwater scour occurs when there is no movement ofthe
bed material upstream of the crossing. However, in this case, the acceleration of flow and
41
vortices created by local obstructions cause scour. Livebed scour occurs when the bed
material upstream of the crossing is moving.
Richardson et al. (1993) suggested using Neill's equation for determining the
velocity associated with the initiation of movement to determine if either dearwater or
livebed scour is occurring. This equation is:
vc
1
1.58 leSs  1) g Dso]2 (319)
where
vc = the critical velocity above which bed material of a size Dso and
smaller will be transported, in feet per second
Ss = the specific gravity of the bed material
A value of2.65 is conunon for most bed material. Therefore, Equation 319
.•.~.
)
reduces to:
vc
1 I
11.52 y 6 Ds~ (320)
Also according to Richardson et al. (l 993) Laursen presented this equation as:
vc
1 I
10.95 y 6 Ds~ (321 )
The only difference between these two equations are the coefficients 11.52 and 10.95.
Realistically either equation can be used to decide whether clearwater or livebed scour
will occur.
42
CHAPTER IV
METHODOLOGY AND APPLICATION
Modeling Systems Operations
The following steps are generally followed in any hydraulic or numerical
modeling application:
Therefore, these steps should also be followed in the application of the FESWMS2DH
2. Network design
3. Model calibration
4. Model testing
5. Model application
1. Data collection
~
,r'
)
.{
j
'.It
J
,oj
:1
computer model.
After a surface water now problem has been defined, the first step in the
construction of a hydraulic model consists of gathering adequate topographic and
hydraulic data. This data might incl ude things such as topographic maps, aerial photos,
and gage records. When applying the FESWMS2DH computer model network design is
accomplished by subdividing the area being modeled into an assemblage of finite
elements. The goal of network design is to create a representation of the area being
modeled that provides an adequate approximation of the true solution of the governing
equations at a reasonable cost.
The FESWMS2DH computer program provides a numerical approximation to
complex surface water now problems. This is accomplished by describing the physics of
43
surface water flow in a series of equations in which several empirical coefficients appear.
Therefore, when enough data are available, the dimensions of the simplified geometric
elements and empirical hydraulic coefficients need to be adjusted so that values computed
by the model reproduce as closely as possible measured values. This process is referred
to as model calibration.
Model testing is accomplished by applying a calibrated model to other flow
situations for which measured values are available. This is an important. but not always
possible step. If a model reproduces reasonable results on flow conditions outside the
range of which it was calibrated, it can be used to simulate conditions outside of that
range with more confidence than if no testing were carried out.
Model application consists of applying the model to simulate a variety of flow
conditions. Model application is only attempted after the previous steps have been
carried out in one form or another. Models still need to be applied carefuHy, especially to
model conditions outside of the range for which they were calibrated. However, a well
constructed. calibrated and tested model can be used to answer a variety of surface water
flow problems.
Site Overview
Description of Site
As mentioned previously, prior to 1988, the Interstate35 and Cimarron River
crossing consisted of two main bridges and a series of eight overflow bridges all placed in
a parallel arrangement. This arrangement utilized a main structure over the river channel
and four groups of overflow bridges on the floodplain. The overflow structures were
44
,.,
I
;
",
1
J
,1
J
placed at increments of 900 feet, 450 feet, and 650 feet apart. The flowline of the main
bridges were approximately 870.2 feet while the flowlines of the overflow bridges ranged
from 885 feet to 887 feet. This alTangement is shown in Figure 12, Interstate35 and the
Cimarron River Site Plan. Additional information concerning the length and elevation of
each bridge is given below in Table VI, Bridge Dimensions. A portion of the United
States Geological Survey quadrangle map describing the site is included in Appendix A.
TABLE VI "I
~
~
BRIDGE DIMENSIONS J
<I
~
..
Bridge Length floor Elevation Low Steel Elevation
,.
,J
(feet) (feet)
Main Bridge (Left) 805' 93
// 916.9
Main Bridge (Right) 805' 9 3// 905.9 1
:!
Overflow I & 2 282'6" 904.7 901.4
Overflow 3 & 4 200'6" 904.8 901.5
Overflow 5 & 6 280'6" 904.9 901.5
Overflow 7 & 8 160' 6" 904.8 901.4
According to Yalin (1992) a stream may be considered as meandering when the
deformation of a meandering stream exhibits a traceable periodicity along the general
flow direction and this deformation is induced by the stream itself: it should not be
"forced" upon the stream by its environment. Meandering is a phenomena which happens
to many mature streams and is not fully understood. Further, according to Strongylis
(1988), comparison of aerial photos of the site taken in 1937, 1939, 1957 and 1990
reveals that the Cimarron River exhibits a fair degree of meandering. Currently, and in
45
/
/
Clt.AARRON RIVER 
r
a.F. = OVERFLOW (REUEF) BRIDGE
PLAN
5C.ALE: 1=1000'
Figure 12. Interstate35 and the Cimarron River Site Plan
46
1987, the main channel crosses under the main bridge on the south side of the floodplain.
However, immediately before crossing under the bridge the channel makes a sharp curve
to go from running perpendicular to the axis of the floodplain to crossing undemeath the
main bridge parallel to the axis of the flood plain. This occurs because over the years the
meander curves in the river have moved downstream to the immediate vicinity of the
bridge.
Hydrologic Data
As mentioned previously a large flood passed under these previously existing
bridges in October of 1986. This flood caused a large amount of scour damage especially
to the overflow bridges which led to the replacement of the prev1iousl.y existing bridges in
1987. Information describing this event was available from Oklahoma Department of
Transportation study files and photographs.
The O.D.a.T. study files contained information concerning not only the lnterstatc
3S and Cimarron River crossing, but also data from the United States Geological Survey
gage number 07161000. This gage is located at Perkins, Oklahoma, approximately 17
miles downstream of the Cimarron River crossing. By projecting the data from this gage
upstream a flow rate of 156,000 cubic feet per second, approximately a Q52 event, was
determined for the Cimarron River crossing for the October 1986 flood. Additionally, the
water surface approximately 5000 feet downstream of the main bridge was determined to
be 898.0 feet for the same event. This downstream water surface corresponded to a water
surface upstream of the bridges of approximately 900.95 feet which would indicate a
lack of pressure flow at the overflow bridges.
47
In 1987, while designing the existing bridges, the Hydraulics Branch of the
a.D.O.T. Bridge Division developed discharge information for this site. This was
accomplished by using existing gage data and performing a statistical analysis using Log
Pearson Type III distribution. The results of this analysis are included in Appendix Band
below.
Q5 = 63,805 cfs
QlO = 88,650 cfs
Q25 = 125,040 cfs
Q50 = 154,600 cfs
QIOO = 185,800 cfs
Q500 = 264,600 cfs
Soils Information
Soils information for the overflow bridges was taken hom the Soil Survey oj'
Payne County Oklahoma completed by the Soil Conservation Service of the U.S.
Department of Agriculture. Applicable portions of this survey are included in
Appendix C. Additional information was taken from the set of construction plans,
completed by the O.D.a.T. in 1959, used to construct the previously existing bridges.
From the available soils information it was determined that the soil which was
present below overf10w bridges 1,2,3 and 4 had the soils name Yahola. This soil ranged
in texture from a fime sandy loam near the surface to a stratified loam to loamy fine sand
at a depth of approximately five feet. The soil which existed below overflow bridges 5,6,
7 and 8 had the soils name Hawley. This soil ranged in texture from a fine sand loam
48
near the surface to a stratified loamy fine sand to silty clay loam at a depth of
approximately five feet. The Cimarron River at this point has a wide floodplain and
coarse sand in its bed.
From the construction plans, it was determined that the overflow bridge piers
were 16 inch square piles driven to a layer of material labeled as "Red Bed". This
material was present at an elevation of approximately 854.0 feet to 856.9 feet. The label
"Red Bed" denotes a shale layer.
Recorded Scour Data
The October 1986 flood of the Cimarron River resulted in severe scour at all of
the previously existing bridges. However, the damage was particularly severe at the eight
overflow bridges. This occurred because a large amount of flow was directed through
these structures and the meander located upstream of the bridges lead to skewed flow on
the floodplain. Aerial photos ofthe scour holes taken from O.D.O.T. study files are
shown in Figure 13, Aerial Photos of the Scour Holes at Interstate35 and the Cimarron
River. These photographs were taken on 111186 approximately two weeks after the
flood. A considerable amount of scour damage also occurred to the main bridges during
this flood, however, in Figure 13 this damage is obscured by water in the main channel.
49
Match
Match
Figure 13. Aerial Photos of the Scour Holes at Interstate35 and the Cimarron River
50
Tyagi (1988) presented a summary and analysis of the scour holes located at the
eight overflow bridges. A summary of this study is presented in Table VII, Maximum
Scour Depths Near Overflow Structures at the 135 Bridge on the Cimarron River.
Additional parts of this study are located in Appendix D.
TABLE VII
MAXIMUM SCOUR DEPTHS NEAR OVERFLOW STRUCTURES AT THE 135
BRlDGE ON THE CIMARRON RIVER [Source: Tyagi, 1988, pA]
Overflow Structure
2
3
4
5
6
7
8
Maximum Scour Depth Scour
Location (feet)
Upstream 10.2
Downstream 27.0
Upstream 22.7
Downstream 12.2
Upstream 15.4
Downstream 1104
Upstream 30.0
Downstream 10.7
The data contained in Tyagi' s (1988) study were collected using an Electronic
Distance Meter and a small boat some time after the flood had receded. This analysis
revealed that the maximum scour depth, some time after the flood had receded, ranged
from 10 to 30 feet. As can be seen from Table VII most of the deep scour holes were
located on the upstream side of the structures where velocities could be expected 10 be the
highest.
51
Modeling
Modeling Strategy
Strongylis (1988) demonstrated that due to the complex nature of the flow at this
site it is suited for a two dimensional flow analysis. AdditionaUy, given the incorporation
of scour calculation capabilities into two dimensional modeling software the same
software used for a hydraulic analysis may also be used for a scour analysis. Therefore,
in this instance a hydraulic analysis of this site was completed and the results from this
stlJdy used to complete the scour analysis. To complete the hydraulic and scour analysis
the following resources were utilized:
1. The Surface Water Modeling System, Version 4.0, (SMS) developed by
the Engineering Computer Graphics Laboratory at Brigham Young
University was utilized for processing the data used in the hydraulic study.
2. To complete the hydraulic analysis, the Finite Element SurfaceWater
Modeling System: TwoDimensional Flow in a Horizontal Plane, Version
2 (FESWMS2DH), developed by the U.S. Department of Transportation
Federal Highway Administration was used.
3. Additional information concerning procedure for conducting the scour
analysis was gained from the U.s. Department of Transportation Federal
Highway Administration's publication HEC18, Evaluating Scour al
Bridges, Second Edition.
52
Hydraulic Modeling
As mentioned previously the first step in constructing any hydraulic model
consists of collecting all appropriate data. The data which was collected for use in
modeling this site includes:
]. A 3.5 foot by 3.5 foot aerial photo (scale] :200) of the site taken from an
altitude of 2900 feet on 63090.
2. Aerial photos (scale 1:200) of the site taken on 111186 showing the
scour damage done to the overflow bridges.
3. A contour map of the site made by G.F.M. & Associates.
4. a.D.a.T. constmction plans dated from 1957 for the previously existing
bridges.
5. a.D.QT. study files and photographs.
6. The S.C.S. Soil Survey Soil Survey ofPayne COlln()J Oklahoma.
7. Tyagi's 1988 Report No. 881 Scour Around Bridge Piers ofOvel:flow
Structures at 135 Bridge on the Cimarron River.
8. Strongylis' 1988 report "Water Surface Projiles Using FESWMS2DH
Model. "
The information from items 1, 2, 3, 4 and 8, was used to design a finite element
network representing the site. Information from items 3 and 4 was used to construct a
contour map, for use in determining elevations, accurately representing the site prior to
the 1986 flood. Hems I, 2 and 8 were used to determine roughness values for the element
network. The resulting element network is included in Figure 14, Site Element Network.
53
54
Information describing the site was first entered in a "DINMOD" file of
FESWMS2DH Version 2.0. This information was then refined and corrected using the
SMS computer software. SMS is a pre and postprocessor for two dimensional finite
element and finite difference models. The SMS computer software greatly simplifies the
inputting of a large amount of data and aids in checking it, by allowing these activities to
be done in a graphic manner. SMS also provides the ability to check a finite element
network to ensure the "colTectness" of the network. This is done by locating elements
with large aspect ratios, adverse grades, gaps in the finite element network, or other
geometry type problems. Additionally, the "user friendly" environment allows a visual
overview of the finite element network to ensure that the network constructed at:curately
represents the area being modeled.
For this application a mixture of sixnode triangular, and ninenode quadrilateral
elements were used. The element size was varied depending upon the hydraulic
significance and geometric complexity of the area. Therefore, smaller elements were
used near bridges and larger elements in the flood plain. Both quadrilateral and triangular
elements were constructed by having their longer side parallel to the smaller gradient.
Using the SMS software element resequencing was performed to obtain a direct
solution of the equation which results from the application of the finite element method,
resequencing was performed in both the forward and backward direction, in relation to
the site, using a variety of means. The smallest front width was obtained by using the
minimum frontgrowth method in a backwards direction.
Often times use of the FESWMS2DH software and "FLOMOD" module
requires the use of a "cold start," "hot start" procedure. However, when modeling the
55
October 1986 flood with this model convergence ofthe residual equations could be
obtained in one run. Completing the model in one mn required the use of 15 iterations.
Once the finite element network was completed it was possible to calibrate the
model to ensure the validity of the results. Some criticaJ aspects of a finite element
network include shape, size and placement of the elements, selection of manning n
values, and selection of the kinematic viscosity. As mentioned previously, information
from G.D.G.T.'s files showed that the October 1986 flood had a flow rate of 156,000
cubic feet per second and a downstream water surface elevation of 898.0 feet. The
upstream flow rate of 156,000 cubic £eet per second was used as an upstream boundary
condition and the downstream water surface of 898.0 feet was used as a downstream
boundary condition. The finite element network was changed and refined until the model
yielded results which agreed with the available data which showed a lack of pressure flow
at the bridges.
Several finite element networks were constructed, progressively refining the area
of the overflow bridges, until a result was obtained from the FESWMS2DH. Difficulty
was found in modeling the high banks of the f:1oodplain. As FESWMS2DH tried to
arrive at a solution these elements were successfully "wetted" and "dried" leading to
instability in the solution. This problem was solved by eliminating all of the unnecessary
"dry" elements from the network.
Mannings n values were chosen for the floodplain and channel areas according to
standard engineering practice and text. These values were then varied, especially in the
floodplain area, by up to 50%. This variance proved to have a small effect upon the
FESWMS2DH output, therefore, the n values originally assumed were Llsed.
56
Kinematic eddy viscosity was varied from 10 to 100. A larger value, such as 100,
helps lend numerical stability to the model, whereas, a smaller value, such as 10, is likdy
to be more accurate. A value of 10 was used for the kinematic eddy viscosity and this
lead to numerical instability in the model. A high value of kinematic eddy viscosity
resulted in an unrealistically high water surface. Therefore, a value of J5 was llsed to
model the kinematic eddy viscosity.
Finally, after an accurate network had been built and calibrated it was used to
model the flood in question and obtain scom results. As mentioned previously, the flood
being studied had a flow rate of 156,000 cubic feet per second and a downstream water
surface of 898.0 feet. Analysis and presentation ofthe output was also greatly simplified
by use of the SMS computer software. A summary of the velocities resulting from the
October 1986 flood are contained in Figure 15, Velocity Vectors for October 1986 Flood.
An upstream water surface of 90 I .8 feet was determined to correspond with a
downstream water surface of 898.0 feet. This yields a water surface slope of 0.00036 feet
per feet, while the flow line slope is 0.00045 feet per feet. The water surface slope is
shallower than the flow line slope because the bridges tend to back up the water and
flatten the water surface slope.
Scour Modeling
Using Version 2.0 ofFESWMS2DH allows scour calculations to be completed
along with a hydraulic analysis. Clearwater contraction scour may be completed using a
version of Laursen's clearwater scour equation given in Chapter 3. Pier scour may be
completed by using either the Colorado State University equation, given in Chapter 3, or
57
~
"'0
00
"i
G:
~
/
\D
;'
00
~
0\ 
/
I<
/
Qj
/
..0
/
0
/
.....
u
/
0
/
I<
I I
/
r.8
/
(/)
I
/
.....
(
I
/
0.....
I
u
I
I
<lJ
I
I
>
I
{ /
?;.
J I I
.u
I
J
I
..9
<lJ
I
I I
>
I
In<lJ
3
.bJj
58
~
\ II ttl t \ \ \\ I
Froehlich's pier scour equation. A summary and analysis of the calculated scour results
are given in the next chapter. A complete lIsting of the calculated scour data for each pier
is included in Appendix E.
In this instance the values of velocity upstream of the overflow bridges were
larger than the critical velocity. This condition would normally indicate that clearwater
scour was occuning at the overflow bridges. However, in this case, as with many bridges
on floodplains, the assumption of clear water scour was maintained. The assumption was
maintained because:
1. There is vegetation growing on the floodplain.
2. The velocities are large enough that the fine bed material would probably
go into suspension at the bridge and not influence the contraction scour.
Computation of contraction scour was accomplished by inputting the correct n
value information and the critical shear value, Le, for the elements where contraction
scour was to be modeled. This consisted of all elements in the vicinity of the overflow
bridges. According to the recommendations inHEC18 the shcar value was chosen based
upon the value of 1.25(Dso). Again, this procedure was greatly simplified and verified by
using the SMS computer software.
Pier data was entered in SMS not only for the scour modeling but to improve the
accuracy of the hydraulic modeling. In this instance the Colorado State University
Equation was used to predict pier scour. According to the methods outl ined in BECI8
the five piles in the pile bent were entered as one pier having a width of 6.66 feet.
AdditionalJy, according to HEC18 since the Froude numbers at the bridge sites were less
than 0.8 the value of y/a was limited to 2.4, or to a maximum scour depth of 3.2 feet.
59
CHAPTER V
RESULTS AND DISCUSSION
Summary of Results
Calculated scour amounts for all of the overflow bridges are given in Table VIII,
Comparison of Actual to Calculated Scour. Table VIII lists not only the actual scour
recorded at each bridge but also the scour calculated at each bridge. The calculated scour
given in Table VIn represents the calculated contraction scour only. Scour values given
in Table VIII do not account for the depth, or width, of piers, which limit pier scour, or
the presence of "Red Bed" which may have also limited the contraction scour.
TABLE VIII
COMPARISON OF ACTUAL TO CALCULATED SCOUR
Overflow Structure
2
3
4
5
6
7
8
Recorded Scour Calculated
(Tyagi, 1988) Contraction Scour
(Feet) (Feet)
10.2 33.3
27.0 29.6
22.7 32.9
12.2 28.6
15.4 26.6
1].4 28.6
30.0 27.2
10.7 28.2
60
As can be seen from Table VIII the maximum scour at all oftlle overtlow bridges
except numbers 1 and 2 occurred at the upstream bridges, the odd numbered bridges. At
bridges 1 and 2 the maximum scour occurred at the downstream bridge, number 2.
Inspection of Figure 15, Velocity Vectors for October] 986 Flood, given previously,
shows that the velocity vectors upstream of bridges 1 and 2 appear more jumbled and less
streamlined than those upstream of the other overflow bridges. Therefore, the large scour
probably did not occur upstream of bridges] and 2 because the flow was less defined in
this area when compared to the other bridges.
Figures 16, ]7, 18, and 19 compare the actual scour to the calculated scour at the
upstre:un faces of overflow bridges ],3,5, and 7, respectively. The deepest actual scour
along with the scour occurring at these locations is shown in these figures. Similar
figures were not included for overflow bridges 2, 4, 6 and 8, the downstream bridges,
because by the time the actual sour was recorded fill had been placed around these
bridges to add support to their piers. Figures 16, 17, 18 and 19 show a breakdown of the
calculated contraction and pier scour and limit the pier scour to 2.4(yja).
The actual scour occurred over a large area, as shown in Appendix D, not just
under the bridges. However calculated contraction and pier scour can only be applied at
the bridge sections or piers. Additionally, when calculating contraction and pier scour
obtaining the actual limits which occurred in this case would be difficult. The reason
why scour occurred over this large area is unknown but probably has to do with the
increased velocities resulting from the bridges at these points.
61
Back of Backwall Back of Backwall
r
r
~
I
277+00
" "'V..1:2.
I
276+00
I
275+00
.c:.
LSta. 274+58.75 Sta. 277+41.25 J vUU
~ .... ..... ..... ..... .....
Q Q ~ l=l ~ ~
Il) Il) Il) Il) Il) Il) c:c c:c c:c ~ ~ c:c
~ ~ f;;ll ~ ~ ~
890
 , f  1 r_:.:.~    _/
\ ",,,._.
\ "
880 \ ,,'
\
,,'
<::' ,.,. .... 1 rlo • \ ' ....... ,," . ...................._......
............. "" " 1" =50' Horizontal
1" = 10' Vertical
Le2end
870 , • "Red Bed" \
'~"  Existing Ground Line
, I ·····r····· Aotual Scour I " , Maxim~ I  ...  Ca.lculated General Soou
, Record ~d /'\' _ .. . Calculated Pier Scour " Scour 860 ,, . ,>;IW,W;Q;Q;J; Maximum Recorded SCO\
~"""' \1/ &. ,/   
..,C56.9 "\'f'.", ~ IP,. .....
.
0'\
N
Figure 16. Calculated Versus Actual Scour L~pstrearn of Overflow Bridge 1
\ \.._ \\ _..__._.._ ___.._._ _ j' / ,.. . ;.I ...._~ ...""........._........_ _ ,'!
\ ·7'..·..·...1 ,:,/
I ,
• I
: / / ,
.. I
.. / /I
.'
I ,, " / ,
Back of Backwall
Sta. 283+81.24
+J
f::
~
III
~  
+J
~
~
t::jl
c:l
~
+J
I:l
Le~end
__ "Red Bed"
 Existing Ground Line
.. Actual Scour
+J
r::::l
~
III
t;l
 ,    t
\
,  1 l ... ~ 
'\       ....             ,.,.
\ , \,
Back of Backwall
Sta. 281 +78.75
870
890
880
90
0\ w
ilarlmUIn
Recorded
Scour
860
/:855.4
L~55.6
I
282+00
\\..,,
\',, 
""'

""'""'
...............
,,/
,,/'
,/
,,/
,/'
I
283+00
"/,,,
I
I
855.2
SQa~
1" =25' Horizontal
1" =50' Vertical
854.0~
85::l,l.
Figme 17. Calculated Versus Actual Scour Upstream of Overflow Bridge 3
Back of Backwall Back of Backwall·
c:.
C Sta. 287+88.75 Sta. 29U+71.25 J
...,l .., .., ..... ..... ...,l
Cl Cl Cl l::l d d
v v v v v v
CD l'Il CD c:o l'Il c:o
~ ~ ~ ~ ~ ~
890
   1 1 .
\ I
\ /
I
880 1\ ,,I
\ \ ....,. ilrnm~~ i ................ Scalst;
.................._......_00;  ._~~~&~~ 1" =50' Horizontal
\
......._.__._... _~ _....._.....J 1"=10' Vertical 870 ~\ I .L~ienci \
\ II! • "Red Bed"
\  Existing Ground Line \
\ ,
\ I .. Actual Scour \
\ \ ''\' ' , ,,/ _ ...  Calculated General Scour
860 \ ry·····r iT···_··""tt···._ l.1' " . Calculated Pier Scour I
\ I • I , ' .... , .... Maximum Recorded Scour II
I I
.L~54.9 ' lC55
.
0
0+>
I
289+00
I
290+00
I
291+00
Figme 18. Calculated Versus Actual Scour Upstream of Overflow Bridge 5
Baok of Backwall Back of. Backwall
299+00
:J..c: .c
I
298+00
L £.
Sta. 297+38.75 Sta. 299+01.25
900 'I ,.
...,J ...,J ...,J
Q Q = Q) Q) Q)
l:Q c:l l:Q
890 ~ ~ ~
\;, I         
1 .,.. I
 !!
, ,"" , "
\~\ (
880 .I,.I
\\ II Scale:
\ '\ ,I
, 1" =25' Horizontal
~\~ ~........_" / 1" =50' Vertical I
"
,
I
870 \ j).. "', / 1eieng " I
"'" '" " "Red Bed"
'. /,
"'" ' //  Existing Ground Une
"" t  Actual Scour
"" Maximum ".I  Calculated General Scour
"'" Recorded
860 Scour 1//  Calculated Pier Scour
7
...  .... I Maximum Recorded Scour .. J ;:;:,
" ,
~
0\
Vl
Figure 19. Calculated Versus Actual Scour Upstream of Overflow Bridge 7
Discussion of Results
As can be seen from Table vm the cakulated contraction scour ranged from 26.6
feet to 33.3 feet, whereas, the recorded scour ranged from 10.2 feet to 30.0 feet. The
largest actual scour of 30.0 feet corresponded to a caJculated contraction scour of 27.2
feet. The smallest actual scour of 10.2 feet corresponded to a calculated scour of33.2
feet, the largest calculated contraction scour.
As can be seen from Figures 16 to 19 no difference between contraction and pie]"
scour was apparent in the recorded scour. Pier scour may have occurred only not to be
recorded because it was obscured during measurement of the scour by water in the holes.
Additionally, review of Table VIn and Figures 16 to 19 reveals that the "Red Bed" layer
may have limited the actual scour which occurred.
In all instances, except at overflow structure number 7, the calculated contraction
scour was larger than the recorded scour. The maximum calculated scour, since it was
calculated using the clearwater scour and Colorado State University pier scour equations,
varied little from bridge to bridge since the maximum velocities and water depths were
similar at each bridge. However, actual scour varied from bridge to bridge with
maximum scour values generally being upstream of the bridges.
It should also be noted that the scour equations, mentioned above, are generally
used to yield a "design" value and not a maximum predicted value. Therefure, a
comparison of calculated to actual scour should yield a calculated scour near to or greater
than the actual scour.
66
As mentioned above the calculated contraction scour is generally larger that the
actual scour. These differences may be due to the following reasons:
1. The actual scour was recorded some time after the actual flood had
occurred, therefore, some filling of the scour holes should have occurred
during the receding portion of the food.
2. From reviewing the aerial photos it appears that there may have been
significant movement of soil particles into and out of the scour holes,
therefore, the scour may have been clearwater scour and not livebed
scour. However, for the reasons mentioned previously, this is unlikely.
3. The actual scour may have been limited in depth by the "Red Bed" layer.
4. The scour equations tend to over predict scour and are intended for use as
a design tooJ and not to predict actual scour depths.
67
.,.......
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
Based upon the results of this study the following conclusions may be made:
1. Microcomputer applications of the FESWMS2DH computer program
may be used to successfully perform the hydraulic analysis of complex
river crossings such as the Cimarron River and Interstate35. FESWMS2DH
reports depth averaged point velocities, direction and point water
surface elevations. FESWMS2DH is a powerful two dimensional
surface water flow analysis program which may correctly analyze complex
flow problems much more readily than traditional one dimensional
methods.
2. The SMS computer program greatly enhances the pre and postprocessing
of the data used in a two dimensional flow analysis. The SMS computer
program also aids on checking the validity of a model by providing "user
friendly" viewing, checking and updating of that model. In short the SMS
computer program is a powerful graphical users interface for use by
engineers performing two dimensional surface water flow analysis.
3. The results from the hydraulic analysis performed in this study appear
reasonable and correct. The calculated water surface values provide close
agreement with information contained in O.D.O.T. 's files. O.D.G.T. files
68
made:
described a flow rate of 156,000 cubic feet per second, a downstream
water surface elevation of 898.0 feet and a lack of pressure flow at the
bridges.
4. The scour values calculated in this study are generally larger than the
maximum recorded scour values. Some of this over prediction is
expected, and help ensure a valid design tool. Several reasons may have
contributed to this over prediction and these are outlined in the previous
chapter.
5. The scour equations provide values useful in design but not necessarily
useful in the prediction ofactual scour values. Had an analysis of this type
been performed as a pOliion of the design of the previously existing
bridges, the flaws in their design and their susceptibility to scour type
problems would have been apparent.
Recommendations
Based upon the results ofthis study the following recommendations may be
1. The scour equations outlined in HEC18 Evaluating Scour at Bridges, and
contained in the FESWMS2DH computer program, tend to over predict
actual scour values. These equations are based upon theoretical
assumptions and laboratory data. Little attempt to calibrate these
equations to actual field data has been made. More work needs to be
performed to correlate the scour equations to actual scour data.
69
2.' Although the scour equations tend to be conservative, the combination of
scour analysis techniques and two dimensional flow analysis provides the
engineer with a useful and powerful tool for predicting scour, and
analyzing complex river crossings.
70
REFERENCES
Becker, L.D. (1994). Investigation of bridge scour at selected sites on Missouri streams.
Waterresources investigations report 944200. Denver, CO: U.S. Geological
Survey.
Chaudhry, M.F. (1993). Openchannel flow. Englewood Cliffs, N1.: PrenticeHall.
Chiew, Y. (1992). Scour protection at bridge piers. Journal of Hydraulic Engineering.
American Society of Civil Engineers, Vol. 118, No.9, 1260  1269.
Finnie, J.1., & Jeppson, R.W. (1991). Solving turbulent flows using finite elements.
Journal of Hydraulic Engineering. American Society of Civil Engineers,
Hydraulic Division, Vol. 117, No. 11,1513 1530.
FroeWich, D.C. (1996). Finite element surface water modeling system: Twodimensional
flow in a horizontal plane, Version 2, Draft user's manual.
McLean Virginia: Federal Highway Administration.
Froehlich, D.C. (1992). Finite element surface water modeling system: Twodimensional
flow in a horizontal plane, Version 2, User's manual preprint. McLean Virginia:
Federal Highway Administration.
Gilbelt, 1.1., & Myers, D.R. (1989). Analysis of water surface and flow distribution for
the design flood at a proposed highway crossing at the Sabine River near Tatum,
Texas. Waterresources investigations report 884231. Denver, CO: U.S.
Geological Survey.
Hydrologic Engineering Center. (1993). HEC6 scour and deposition in rivers &
reservoirs user's manual. Davis, CA: U.S. Army Corps of Engineers.
Lagasse, P.F., Schall, J.D., Johnson, F., Richardson, E.V., Richardson, J.R., & Chang, F.
(1991). HEC20 stream stability at highway structures. Washington, D.C.:
Federal Highway Administration.
Laursen, E.M. (1960). Scour at bridge crossings. Journal of the Hvdraulics Division.
Proceedings of the American Society of Civil Engineers, Vol. 86, No. HY2, 39 54.
71
Laursen, E.M. (1963). An analysis of relief bridge scour. Journal of the Hydraulics
Division. Proceedings of the American Society of Civil Engineers, Vol 89, No.
HY3, 93  118.
Lee, 1. K., & FroeWich, D.C. (1986). Review ofliterature on the finite element solution
of the equations oftwodimensional surface water flow in the horizontal plane.
U.S. Geological Survey circular 1009. Denver, CO.: U.S. Geological Survey.
Melville, B.W. and Dongol, D.M. (1992). Blidge pier scour with debris accumulation.
Journal of Hydraulic Engineering. American Society of Civil Engineers,
Hydraulic Division, Vol. 118, No.9, 1306  1310.
Melville, B.W. (I 992). Local scour at bridge abutments. Journal of Hydraulic
Engineering. American Society of Civil Engineers, Hydraulic Division, Vol.
118, No.4, 615  631.
Molioas, A. (1990). User's manual for BRISTARS (bridge steam tube model for alluvial
river simulation). National Cooperative Highway Research Program, Project No.
MR ISIi. Fort Collins, CO.: HydrauTech, Inc..
Richardson, E.V., Harrison, C.J., Richardson, J.R., & Davis, S.R. (1993). HEC18
evaluating scour at bridges, Second edition. Washington, D.C.: Federal Highway
Administration.
Strongylis, D.G. (] 988). Water surface profiles using FESWMS2DH model. Norman,
OK: The University of Oklahoma. (Unpublished master's thesis).
Tyagi, A.K. (1988). Scour around bridge piers of the overflow structures at 135 bridge
on the Cimanon River. Stillwater, OK: School of Civil Engineedng, Oklahoma
State Uoiversity .
United States Geological Survey (I 989). Soil survey of Payne County, Oklahoma.
Washington D.C.: U.S. Department of the Interior.
Yalin, M.S. (1992). River mechanics. Oxford, England: Pergamon Press.
Zienkiewicz, O.c. (I 977). The finite element method (3rd ed.). Maidenhead, England:
McGraw Hill.
72
APPENDIX A
SITE MAP
N.
 N.
.. ,
I
/
~~n~  30 °b.M.:~
 1\ /"~
~~rlh:::::"'; ,' 1.1'  /, "~~'  " ~ y. ~ ..~..... /__~rll
:i . .....
(
\.
L .
. . ... X~ :.. .....:.... ."b ...::.. ....
" .y......;~~
\
'.
Site Map. Pre198? Conditions. Langston
7 1/2 mi~ Quandrangle Map Photorevised
1983. Scale: 1"=2000 feet
74
APPENDIX B
HYDROLOGY DATA
1 1~ 1
...........  "' ".. . _....... .. :. ~.:"':
~ 1 ~ l ill:l.....T..... ·.~~"1. l. ~ Hn :l. l.l. .
~ .. ~~~i~· . ::..:r:I:[:::rI!:::::I::tl:I1::III::..:.~.:::.:rI1:t! .. ...... ...... ...... .. .. .. .. .. .. ::.: ~ : : : ::::: : :::::::. :: : :~::
j : 1j .j ~ ~: ~;' . 1. j .l~~~
... " ';.. ."r.. .;..T.. 1.. "r .. .. _ .. ~ ~"""f" T.r " ~ ~ ~ ~ 1~ ~ n
:: :: :: ::::.:. :.. :...:. :.. :...:..:.:.:..:..:...:.
.. ~ ;~~~.~ ~ ~ .. :.. ;; ;;~~~;
....... ; : .. ":.~.~.:. : : .. ":.:.:'":: : : :.. ~ .. : :.: =. .. .. .. .. .. .. .. .. .. . .. .......... .. ..
~ .~~~~ .. ;~;;; ;1 j; ~
......::.. :.. ::.:: : :: : ::.. ::.:.:..:.:..=:.. :.. : ::.. ::.. .:.. .::':":::..
: ::: :: ::: : : :::: :
.. .... : : ::: : : :: :
.. :: ::: :: ... :: :
T'j"1" IIIIi'...."[..·l· .rj'nI!!.....t..'1' .nnI
280000
240000
290000:
lJl
\4l
0 16HOO9 
r....
r.... ll~OO8 0z~
0:
8CGOO
400ee
e
1
RECURRENCE INTERVAL ( years )
Runoff versus Recurrence Interval Curve
[Source: Strongylis (1992)]
76
APPENDIX C
SOILS DATA
lSource: Soi\ Survey of Payne County, Oklahoma
1
7& 
<30
I I
681 08 IFioe san<1y loam
Yahola I I I8461Fine san<1y loam,
1 I loam, very fiDe
san~ loaIll.
'I 4664IStratified loam
I to loamy fine
sand.
I I
ISM, liL, IA4
I CLMI., I
SMSC
ISM, MI., ct1A4
I SC I
ISM, HL, CLIA2, A4
I SC I
I I
o
a
a
100
100
100
I I ' I
195 100 9010013660 I II ... I
I I , I
195100190100'3685
I I I I I I I
195100'9010011585
I" I II II I I I
<26 I
tfPi
I
<30 I NFIO II
I NFIO II
...J
\D 84._p
__/ OlOIFine sandy loam I'KL, SM
Rawley lO32/Flne sandy loam, Sli, sc,
I 109. I SMSC,
CLML
1
3260,IStratif1ed loamy lmi, KL,
fine sand. to SMSC,
I I s11ty clay loam. I CLKL
I I I
1
"4 A(
I
1"2, A.(
II
oo
o
100
100
100
19810019410013660 'I
98100 90100 (575
981001 90 100 3070 I
I I I
I I I I
<26
<30
<30
NP4
NPI0
NP7
APPENDIX D
SCOUR DATA
1,
j
This appendix contains applicable portions of the report, Scour Around Bridge Piers of
Overflow Structures at 135 Bridge on the Cimarron River, by A.K. Tyagi (1988).
81
TABLE I
Maximum Scour Depths Near StnJctures C lhrough .1 at 135
Bridge on the Ci~River
Maximum
Span ~urhole ScourOepth
Structure (feet) Location (feet)
p 281.33 Upstream 30.0
0 281.:n Downstream 10.7
N 201.33 Upstream 15.4
M 201.33 Downstream 11.4
L 261.33 Upstream 22.7
K 281.33 Downstream 12.2
C 161.33 Upstream 10.2
D 161.33 Downstream 27.0
82
\,
900.8
ABOVE
WATER
886.8
Sta297+39.75
Sta299+00.25
• 67t6
8119 •
865.2 •
STRUCTURE P
870.2 •
PROFILE AXIS ""
8712 ~ ..... 
.... 865.2
• 864..9
J;
20 FT
STEEP SLOPE
FLOW
~
Figure 2. Location of Scour Hole Upstream of. Structure P.
83
FLOW
00 +>
'
ill
[ij 920
'
«
ill
C/) 900 z«
w~
~
Zo~
>w
860
w'
865.0 865.2
890.0
, ,BOTTOM
853' OF PIER
o 60 120 180 240
DISTANCE FROM WEST POINT OF SCOUR, FEET
Figure 3. Profile of Scour Hole Upstrea.m of Structure P.
300
BRIDG:
SEAT
900.84
Sta299+00.25
PROFILE AXIS
8918
690.8
STRUCTURE 0
8911
FLOW •
l1
20 FT
\
Figure 4. Location of Scour Hole Downstream
of Structure O.
85
FLOW
887.7
_ ....' 890.9 ~ .
880.8 87i8
883.5
...J
W>ill
...J
<t:
w
Cf)'
Z
<C w~
z..
o
~
>W·
...J
W
OQ
0\
870' I I I , I.. I ., , I ' .. o 40 80 120 . 
DISTANCE FROM WEST POINT OF SCOUR, FEET
Figure 5. Profile. of Scour Hole' Downstream of Structure O.
PROFILE
AXIS
1
FLOW•
PROFILE
AXIS
2
fI
20 FT
Sta290+7025
Figure 6. Location of Scour Hole Upstream
of Structure N.
87
J w 910
>UJ
J
« w 900 en
z«
8'9'0\w 891.0 FLOW
:E 890 .. I I /890.8
a
Z
0 00 00 ~
:w> I ........... 873.8 ..J
UJ
87°0
DISTANCE FROM WEST POINT OF SCOUR, FEET
Figure 7a. Profile of North Scour Hole Upstream of Structure N.
40 80 120
DISTANCE FROM WEST POINT OF SCOUR, FEET
FLOW... 890.5
BOTTOM
87q.8 I/'I OF PIER
853'
r ~/
oJ
W>W
..J
<C
W
C/)
Z
<: w~.. z
0
00 \D ~
>W
oJ w
870
0
Figure 7b. Profile of South Scour Hole Upstream of Structure N.
Sta290+70.25
• 878.1
PROFILE
AXtS
3
8616 •
8.90.7
STRUCTURE M
...,....~ 886.2
FLOW
~
~
20 FT
Figure 8. Location of Scour Hole Downstream
of Structure M.
90
870' I I ! I I I 1_ 1 .. o 60 120 180
DISTANCE FROM WEST POINT OF SCOUR, FEET
...J
W>w 910
...J
w~ I PROFILE AXIS 3
en' 900
z
~
w
::;
. 890~888.6 FLOW.. z O· / 889.3
'0 ..... ~
> w 880
...J
W , ., "". I._II 877.6
Figure 9. Profile of Scour Hole Downstream of Structure M.
Sta281+79.75
ssas
86a6
8912
BRIDGE
SEAT
900.64
878.7.
900.5
Sta.283+80.25
672.7 •
e
872.6
STRUCTURE L
871.7 •
i~
6712 e
8712
e ti
873.7
•
S714 •
e·
868.2
11
20 FT
Figure 10. Location of Scour Hole Upstream
of Structure L.
92
..J 920 w>
W
..J
« 9001 FLOW ~  w
en
z I\. I I 7888.8 «w
::E..
Z
w'D 0 I 868.8 ~
I I I BOTTOM >W
853' OF PIER ...J w
840
0 80 160 240
DISTANCE FROM WEST POINT OF SCOUR, FEET
Figure 11. Profile of Scour Hole Upstream of Structure L.
887.2
FLOW...
• 876.7
889.4
PROFILE AXIS
876.5 ~884'7 887.7
875.6. • ....... 
• '" 688..5
67•7.2 .... .......676.2 ___.......'876.5
879.2 ...,....~76.7
.".......
617.2 .677.2 •
668.9
Sta283+80.25
BRIOGE
SEAT
900.84
STRUCTURE K
..l
2{) FT
Figure 12. Location of Scour Hole Downstream
of Structure K.
94
888,6
876.5 876.2
FLOW 
876.7
120 180 240
DISTANCE FROM WEST POINT OF SCOUR, FEET
J 910 w>W
J
«
ill
(j)
Z«w
~
~ z
0
\D .~ V1 >W
J w
870
0
Figure 13. Profile of Scour Hole Downstream of Structure K.
Sta277+4t25
PROFILE AXIS
FLOW..
STRUCTURE C
BSERAITDGE'L
900.76
Sta274+5a75
11
20 FT
~STRUCTURE0
Figure 14. Location of Scour Hole Upstream
of Structure C.
96
I
UJ >W
l
<t:
w,
(J)
Z
<:
w
:E.. z
0
'0 ~ ..l >UJ
I
W
*DUE 10 SCOUR SHAPE PROFILE WAS
DRAWN FROM N & S DIRECTla4S
910
FLOW OUT OF pAGE
886.1
883...8
878.1 I 877.3
. BOTTOM
1853' T 1853' 1853' 1853' OF PIER
870' I I I , I I I ... o 60 120 '180
DISTANCE FROM SOUTH POINT OF SCOUR, FEET
Figure 15. Profile of Scour Hole at Structure C.
=
BRIDGE
SEAT
900.76
890.5
PROFILE AXIS
FLOW..
STRUCTURE 0
11
40 FT
Figure 16. Location of Scour Hole Downstream
of Structure D.
98
890.0
FLOW...
872.5
862.1 ~
8601 I I I I :;::= , I , I I I I .. I I. I I I .. o 80 160 240 320 u_   _. _.:
....J
UJ >W
....J
<C w
CJ)
Z
<C
W
::E
za
o
~
Gj
....J
W
'0
'0
DISTANCE FROM SOUTHWEST POINT OF SCOUR, FEET
FIgure 17. Profile of Scour Hole Downstream of Structure D.
APPENDIX E
CALCULATED SCOUR DATA
••• PIER SCOUR REPORT *. __ • __ c ~ ._•• ~. •• _. __ ~ •• • ·_·. •••••• _
 Pier   Approach Flow   Scour Oepths  Riprop
No. Width Lngth Nose Vel Depth Angle Locol Oanrl Total D50
(ft) (ft) shope (ft/s) (ft) (deg) (ft) (ft) (ft) (ft)
1
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
S9
60
61
6.2
63
64
65
66
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1.33
1. 33
1. 33
1. 33
1. 33
1. 33
1.33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
1. 33
6.65 Squo.re
6.65 Squo.re
6.65 Square
6.65 Squo.re
6.65 Square
6.65 Squore
6.65 Square
6.65 Square
6.65 Square
6.65 Square
6.65 Squore
6.65 Squore
6.65 Square
6.65 Squore
6.65 Square
6.65 Square
6.65 Square
6.65 Square
6.65 Square
6.65 Square
6.65 Square
6.65 Squore
6.65 Square
6.65 Square
6.65 Square
6.65 Square
6.65 Square
6.65 Square
6.65 Squore
6.65 Square
6.65 Square
6.65 Square
6.65 Squore
6.65 Squore
6.65 Squore
6.65 Squore
6.65 Square
6.65 Square
8.82
6.08
4.04
8.31
6.70
7.74
7.09
8.19
B.61
9.54
8.63
6.85
5.11
6.92
5.94
9.54
9.61
B.49
9.15
B.45
5.78
5.56
8.34
S.14
S.20
8.66
B.19
9.14
B.61
8.55
7.B8
6.74
6.25
5.59
8.83
9.17
8.97
9.56
9.34
11. 93
12.43
12.42
12.71
12.92
12.99
13.12
13.09
12.97
13.07
12.83
13.03
12.17
12.55
12.79
12.64
11.62
11. 43
8.60
10.26
10.90
11. 20
11.18
11. 74
11. 51
11. 96
11. 44
11. 43
11.00
l(J.58
10.70
9.80
HI.63
11. 37
11.17
11. 02
10.34
76.1}
83.3
86.2
7B.7
86.3
77 .0
81.6
79.3
80.0
62.4
79.4
85.5
73.2
65.3
65.6
73.5
79.9
76.4
77.0
84.6
88.8
85.6
81.1
84.8
81.0
81.8
82.5
81.5
85.0
79.2
79.5
72.3
82.1
81.3
77 .4
84.7
81.2
83.5
10.4S
9.20
7.73
10.59
9.64
to.33
9.95
10.60
10.83
11.29
10.95
9.75
8.63
9.61
9.14
11.26
11.30
10.59
10.91
10.12
EL 75
8.72
10.46
10.32
10.45
10.67
10.46
10.91
10.60
10.55
10.13
9.45
9.07
8.74
10.74
10.86
10.77
10.96
23.30
17.35
.00
27.16
20.68
25.63
22.93
27.66
29.63
33.28
30.48
21. 66
.00
21. 10
17.39
32.91
32.89
26.49
28.61
20.70
14.37
14.16
25. IS
24.37
25.57
26.94
25.91
28.62
26.60
25.55
22.39
18.42
15.51
14.04
27.29
28.15
27.13
27.87
33.75
26.54
7.73
37.75
30.52
3S.96
32.87
38.46
40.47
44.58
41. 43
31. 41
8.63
30.71
26.53
44.17
44.19
37.08
39.53
30.62
23.12
22.89
35.61
34.69
36.01
37.60
36.37
39.53
37.20
36.10
32.52
27.87
24.58
22.78
38.03
39.01
37.90
38.83
1. 47
.70
.31
1. 30
.85
1. 13
.95
1. 26
1. 40
1. 71
1.'17
.88
.49
.90
.66
1. 71
1. 74
1. 36
1. 58
1. 34
.63
.58
1. 31
1. 25
1. 27
1.41
1. 26
1. 57
1. 40
1. 38
1. 17
.86
.74
.59
1. 47
1. 58
1. 52
1.73
Note  Pier scour colculoted using CSU equdtion.
101
VITA
Michael T. Buechter
Candidate for the Degree of
Master of Science
Thesis: SCOUR ANALYSIS OF THE INTERSTATE3 5 AND CIMARRON
RIVER CROSSINGS USING THE FESWMS2DH AND SMS
COMPUTER MODELS
Majer Field: Civil Engineering
Biographical:
Personal Data: Born in S1. Louis, Missouri, on August 20, 1966, the son of Emil
and Estelle Buechter.
Education: Graduated from Kemper Military School and College, Boonville,
Missouri in May 1984. Received Associates of Arts Degrees in
Mathematics, Engineering Science, and General Transfer Studies from
Florissant Valley Community College in December 1988. Received a
Bachelor of Science Degree in Civil Engineering from the University of
Missouri at Rolla, Missouri, in May 1990. Completed the requirements
for the Master of Science degree with a major in Civil Engineering at
Oklahoma State University in May, 1997.
Experience: Employed by the Oklahoma Depm1ment of Transportation as an
Engineering Intern from 1990 to 1995. Employed by Booker Associates,
Inc., an architectural and engineering firm, from 1995 to the present.
Professional Memberships: American Society of Civil Engineers.