UFL/COEL2000/012
BEACH NOURISHMENT DESIGN: REDUCTION IN BEACH
WIDTH DUE TO PROFILE EQUILIBRATION
by
Robert G. Dean
September 15,2000
Prepared for:
Office of Beaches and Coastal Systems
Florida Department of Environmental Protection
Tallahassee, Florida 323993000
UFL/COEL2000/012
BEACH NOURISHMENT DESIGN:
REDUCTION IN BEACH WIDTH DUE TO PROFILE EQUILIBRATION
September 15, 2000
Prepared for:
Office of Beaches and Coastal Systems
Florida Department of Environmental Protection
Tallahassee, Florida 323993000
Prepared by:
Robert G. Dean
Civil and Coastal Engineering Department
University of Florida
345 Weil Hall, P. O. Box 116580
Gainesville, Florida 326116580
TABLE OF CONTENTS
LIST OF FIGURES ...................................... ....... ........... iii
LIST OF TABLES ....................................................... v
1 INTRODUCTION .................................................. 1
2 M ETHODS ............ .............................................. 1
2.1 General ......................................................... 1
2.2 CrossShore Equilibrium ................................ ......... 3
2.3 Planform Evolution ............................................... 8
2.4 Time Variation of Profile Equilibration ................................. 8
2.5 Consideration of Time Varying Shoreline Recession Due to Both Profile
Equilibration and Planform Evolution ................................ 10
3 EXAMPLES ILLUSTRATING APPLICATION OF THE RESULTS .......... 10
3.1 Example 1 ..................................................... 10
3.2 Example 2 ..................................................... 12
3.2.1 Initial Placed Beach W idth, Ay, ................................ 13
3.2.2 Equilibrium Beach Width, Ayo .............................. 13
3.3 Example 3 ...................... .............................. 14
3.4 Example 4 ..................................................... 15
4 SUM M ARY .................................... .................... 16
REFERENCES ...................... .............. .....................16
APPENDICES
A DEVELOPMENT OF SHORELINE ADVANCEMENT, Ay, FOR
NOURISHMENT PLACEMENT AT A UNIFORM SLOPE ................ A1
B PLOTS OF AyS/W. (TWO PLOTS) AND Ay/Ay1 (SIX PLOTS) .............. B1
ii
LIST OF FIGURES
FIGURE PAGE
1 Example of Beach Nourishment with DN = 0.2 mm, DF = 0.18 mm, Volume
Density = 100 yd3/ft, B = 6 ft, h. = 18 ft ..................................... 2
2 Variation of NonDimensional Volume Density with NonDimensional
Alongshore Distance and Time. No Background Erosion ......................... 3
3 NonDimensional Placed Shoreline Displacement, Ay,' (= Ay,/W.) Versus Non
Dimensional Volume Density, V' (= V/(B W)) and Slope Parameter h./(miW,).
h./B=2.0 .......................................................... 6
4 NonDimensional Placed Shoreline Displacement, Ayi (= Ay1/W.) Versus Non
Dimensional Volume Density, V' (= V/(B W)) and Slope Parameter h,/(miW ).
h /B = 3.0 ............................................................ 6
5 Recommended Distribution of h. and h/B Along the Sandy Shoreline of Florida.
Note h. Distribution is Solid Line ( ) and hJB is Dashed Line () ........... 7
6 Ratio of Equilibrium to Placed Beach Widths, Ay/Ayl, for NonDimensional
Volume Densities, V', AFtAN. Specific Values for This Plot: hJB = 2.0, hJ(miW.)
= 0.1 .................. ...................................... 8
7 Approximate Estimates of G(ft2/s) Around the Sandy Beach Shoreline of the State
of Florida (From Dean and Grant, 1989) ................................... 9
A1 Definition Sketch ................................................... A2
B1 NonDimensional Placed Shoreline Displacement, Ay,' (= Ay,/W.) Versus Non
Dimensional Volume Density, V' ( V/(B W)) and Slope Parameter h,/(miW.).
h /B=2.0 .................................................. ..... B3
B2 NonDimensional Placed Shoreline Displacement, Ay,l ( Ay,/W,) Versus Non
Dimensional Volume Density, V' (= V/(BW.)) and Slope Parameter h /(miW.).
h /B = 3.0 ................................................... ........ B3
B3 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: hJB
= 2, hJ(mW.) = 0.1. No Volume Change in Profile .......................... B4
B4 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: hJB
= 2, hJ(miW.) = 0.3. No Volume Change in Profile ......................... B4
B5 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: hJB
= 2, hd(mW.) = 0.5. No Volume Change in Profile ........................... B5
B6 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: hJB
= 3, hJ(mW.) = 0.1. No Volume Change in Profile ........................... B5
B7 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: hJB
= 3, hJ(miW.) = 0.3. No Volume Change in Profile ........................... B6
B8 Ratio of Equilibrium to Placed Beach Widths for the Following Parameters: hJB
= 3, hJ(mW.) = 0.5. No Volume Change in Profile ......................... B6
LIST OF TABLES
TABLE PAGE
1 Summary of Recommended A Values (ft"3) for Diameters from 0.10 to 1.09 mm
(From Dean, 2000) ..................................................... 5
2 Parameters of Plots Provided in Figures B3 through B8 in Appendix B ............ 9
3 Characteristics of Examples Illustrating Application of Methodology .............. 11
Bl Parameters of Plots Provided in Figures B3 through B8 in Appendix B .......... B2
BEACH NOURISHMENT DESIGN:
REDUCTION IN BEACH WIDTH DUE TO PROFILE EQUILIBRATION
1 INTRODUCTION
Profiles placed as beach nourishment are usually steeper than equilibrium profiles. The
evolution of the project includes both profile equilibration and spreading losses, each of which
contribute to shoreline recession. Shoreline recession is a significant design and prediction issue
since unanticipated shoreline recession may lead to misinterpretation of the project performance.
Thus the subject of anticipated shoreline recession should be addressed in the design documents and
conveyed to the Project Sponsor(s).
This report considers nourished profiles to be placed at a uniform slope seaward of the berm
and to intersect with the native profile. Considering equilibrium beach profile concepts and idealized
beach profiles typical of the east and west coasts of Florida, and various nourishment volume
densities, the recession of the shoreline due to equilibration of the nourished profile is developed and
presented in graphical forms. The methods and design aids presented in this report are considered
appropriate for preliminary design and checking of and comparison with results from more detailed
methods which may be employed for the final design.
2 METHODS
2.1 General
Figures la and lb depict placed and equilibrated profiles considering conservation of sand
within the profile (Figure la) and a 25% reduction of sand volume due to longshore spreading
(Figure Ib), respectively. In addition to the recession associated with profile equilibration, the time
required for equilibration is a significant design and performance prediction parameter. Most present
design methodologies consider profile equilibration to occur instantaneously under the argument that
the crossshore equilibration times are considerably shorter than those for planform evolution.
Idealized planform evolution of an initially rectangular planform on an open coast is
schematized in Figure 2 where, considering compatible sand, in which even for the case of obliquely
incident waves the initially symmetric planform remains symmetric and the planform centroid
remains fixed.
The objectives of this report are to present simple approximate procedures for estimating the
initial and equilibrium shoreline advancements and shoreline recession versus time for the profile
equilibration component and to discuss procedures for including the contribution of both
components.
Ai 6 61.2 Placed Profile, m, = 0.05
> 0
0
o Equilibrated Placed Profile,
1 10 'D, =0.18 mm
n1
0.
u 20
w N tive Profile, DN= 0.2 mm
30
200 0 200 400 600 800 1000 1200 1400 1600 1800 2000
Distance From Original Shoreline (ft)
a) Example Native, Placed and Equilibrated Shorelines for Case of No
Volumetric Change in Profile.
30
200
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Distance From Original Shoreline (ft)
b) Example Native, Placed and Equilibrated Shorelines. Case of a 25%
Reduction in Volume. Note the Placed Profile Corresponding to the
25% Volume Reduction Is Not Shown in this Plot.
Example of Beach Nourishment with DN = 0.2 mm, DF = 0.18 mm, Volume Density
= 100 yd3/ft, B = 6 ft, h.= 18 ft.
Figure 1.
1.0 /..  .
> .F0 j0. t'= Gt/ 72
0.02 4.0 __
0. _
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x'= xl/
Figure 2. Variation of NonDimensional Volume Density with Non
Dimensional Alongshore Distance and Time. No Background
Erosion.
2.2 CrossShore Equilibration
Applying equilibrium beach profile concepts and considering the native and nourished sands
to be each characterized by a single size, the nondimensional additional dry beach width due to
beach nourishment, Ayo, has been shown to be a function of three nondimensional variables (Dean,
1991) as
Ayo V h AF
f (1)
W. BW. B A
where V is the nourishment volume density expressed in volume added per unit length of beach, B
is berm height, W., the distance to the depth of closure, h., on the native profile and AF and AN are
the profile scale parameters of the nourishment and native sands, respectively. It can be shown that
three types of nourished profiles (intersecting, nonintersecting and submerged) can occur. These
results are based on equilibrium beach profiles of the following simple form
h =Ay 23 (2)
in which h is the water depth at a distance y from the shoreline and A is the profile scale parameter
as mentioned above. The methods leading to Eq. (1) have been described previously in a report
(Dean, 2000) to the Office of Beaches and Coastal Systems (OBCS) and will not be repeated here.
Table 1 presents the profile scale parameters, A, for the range of sediment sizes 0.1 mm < D < 1.09
mm.
Referring to Figure la, the initial dry beach width, Ay1, of the placed profile with uniform
slope, mi, can be expressed in nondimensional form as
Ayr V h. h
f (3)
W. BW. B miW.
It is noted that the native profile characteristics are inherent in the term h/(mrW.) which can also be
3/2
N
written as  The nourishment scale factor Ap, does not appear in Eq. (3) as the nourished
mirh
profile is placed according to geometric rather than equilibrium profile considerations. The
development of Eq. (3) is presented in Appendix A. Thus, comparing Eq. (3) with Eq. (1), there is
one additional variable (hJmW.) which complicates the presentation in compact graphical form for
general conditions. Limiting the conditions to those appropriate for Florida's east and west coasts,
the ranges of variables are reduced. For this purpose, the following parameters will be considered
V
V = variable, ranging from 0.02 to 5.0
BW
h
= 2 and 3
B
AF (4)
=variable, ranging from 0.8 to 1.8
AN
h
= 0.1, 0.3 and 0.5
miW
Figures 3 and 4 present for h.B = 2.0 and 3.0, respectively, the variation of Ay,/W. versus V'
V
( )for three values of hd(miW.) ranging from 0.1 to 0.5. With the limited selection ,of
BW
variables above, the results are presented on these two plots.
Examining Eqs. (1) and (3), it is seen that the ratio of equilibrated to initial widths, Ayo/Ay,
depends on the four nondimensional parameters below
Ayo V h. AF h,
A f B (5)
Ay, BW,' B 'A miW.
11 ~N *
Table 1
Summary of Recommended A Values (ft /3) for Diameters from 0.10 to 1.09 mm (From Dean, 2000)
D(mm) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.1 0.0936 0.0999 0.1061 0.1123 0.1186 0.1248 0.1296 0.1343 0.1391 0.1438
0.2 0.1486 0.1531 0.1575 0.1620 0.1664 0.1709 0.1739 0.1768 0.1798 0.1828
0.3 0.1858 0.1887 0.1917 0.1947 0.1976 0.2006 0.2036 0.2066 0.2095 0.2125
0.4 0.2155 0.2178 0.2202 0.2226 0.2250 0.2274 0.2297 0.2321 0.2345 0.2369
0.5 0.2392 0.2410 0.2428 0.2446 0.2464 0.2482 0.2499 0.2517 0.2535 0.2553
0.6 0.2571 0.2589 0.2606 0.2624 0.2642 0.2660 0.2678 0.2696 0.2713 0.2731
0.7 0.2749 0.2762 0.2776 0.2789 0.2803 0.2816 0.2829 0.2843 0.2856 0.2869
0.8 0.2883 0.2895 0.2907 0.2919 0.2930 0.2942 0.2954 0.2966 0.2978 0.2990
0.9 0.3002 0.3014 0.3025 0.3037 0.3049 0.3061 0.3073 0.3085 0.3097 0.3109
1.0 0.3121 0.3132 0.3144 0.3156 0.3168 0.3180 0.3192 0.3204 0.3216 0.3228
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
/.
0.0
0.0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Nondimensional Volume, V = V / (B W.)
Figure 3. NonDimensional Placed Shoreline Displacement, Ay,'
Ayl/W,) Versus NonDimensional Volume Density, V'
( V/(BW,)) and Slope Parameter h./(miW.). h./B=2.0.
0.0
0.0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Nondimensional Volume, V = V / (B W.)
Figure 4. NonDimensional Placed Shoreline Displacement, Ayi
Ay/W,) Versus NonDimensional Volume Density, V'
( V/(B W)) and Slope Parameter h./(miW,). h /B=3.0.
Approximate values of h. and h/B for the Florida sandy shoreline are presented in Figure 5. Note
in applying all results presented here, it is necessary to use consistent units.
20 3
S6I h./B  h. (Feet)
LL. 12 16 20 24
12 .
JA
h/B  2 3 MA
ST
cc ~
CL
VB
V !, WP
MI
12. 16. 20 24
h. (Feet) 2 3
" h,/B 
Figure 5. Recommended Distribution of h. and hJB Along the Sandy
Shoreline of Florida. Note h. Distribution is Solid Line ( ) and
hJB is Dashed Line ().
Figure 6 presents, as an example, the ratio Ayo/Ay, for ranges of the nondimensional
parameters, V' = V/(BW.) and Ap/AN for values of hJB = 2.0 and hJ(mi W.) = 0.1. There is a total
of six such plots presented in Appendix B with the characteristics listed in Table 1.
A discussion of Figure 6 and some of the other figures will assist in interpreting the results.
First consider the nondimensional volume, V = 0.2. For sediment sizes smaller than the native
(AF/AN< 0.88), there will be no additional equilibrium dry beach width, Ay,, i.e., the profile is
submerged. For very coarse nourishment sand (A/A, > = 1.8), the average slope of the equilibrium
profile is approximately equal to that of the placed profile (mi = 0.05) and thus there will be little
change in the position of the constructed shoreline during equilibration. Considering next the
variation of Ayo/Ay with nondimensional volume, V', for sediment which is finer than the native
(ArAN < 1), it is seen that the ratio Ayo/Ay increases with increasing nondimensional volumes. The
reason is that for small relative sediment sizes, small nondimensional volumes can result in
submerged profiles (Ayo = 0). However, it is interesting from Figure 6 that for the larger relative
nourishment sediment sizes, the ratio Ayr/Ay, is less for the larger nondimensional volumes. The
explanation is that whereas the placed profiles are intersecting (by definition), the equilibrium
profiles are nonintersecting and thus have a decreasing Ay/Ay, with increasing V.
0.0 
0.8
1.0 1.2 1.4 1.6 1.8 2.0
AF/AN
Figure 6. Ratio of Equilibrium to Placed Beach Widths, Ay/Ay,, for Non
Dimensional Volume Densities, V', AI/AN. Specific Values for This
Plot: hJB = 2.0, hJ(mW.) = 0.1.
2.3 Planform Evolution
For the case of nourishment with compatible sand forming an initially rectangular planform
on a straight shoreline, the volumetric density evolution is represented in nondimensional form as
presented in Figure 2. The nondimensional time parameter t' = 16 Gt/Q2, in which t is time, Q is the
project length and G is the socalled longshoree diffusivity". Recommended values of G are
presented in Figure 7 for the sandy beach shorelines of Florida. With an estimate of the local volume
density, V(x,t), available, this value can be employed in Eq. (3) or Figures 3 and 4 to determine Ay,
and Ay/Ay, in the six plots presented in Appendix B for the characteristics presented in Table 2.
2.4 Time Variation of Profile Equilibration
Although relatively little is known regarding profile equilibration time scales, the following
is recommended
Ay(t) = Ayo + (AYIAYo)e Kt
i.e., the profile approaches equilibrium exponentially. In nondimensional form
S.lo I I t
.06. G(ft2/s)
 0.02 0. 0.1 0.14
0.02
I I
0.02 0.06 0.10 0.14
G(ft2/s)
Figure 7. Approximate Estimates of G(ft2/s) Around the Sandy Beach
Shoreline of the State of Florida (From Dean and Grant, 1989).
Table 2
Parameters of Plots Provided in Figures B3 through B8
in Appendix B
Case Reference Figure No. hJB (hJmiW.)
1 Standard 1 6, B3 2 0.1
2 Standard 1, Except hJmrW. = 0.3 B4 2 0.3
3 Standard 1, Except hJmW. = 0.5 B5 2 0.5
4 Standard 2 B6 3 0.1
5 Standard 2, Except hJmiW. = 0.3 B7 3 0.3
6 Standard 2, Except hJmiW. = 0.5 B8 3 0.5
Ay(t) Ay0 AY Kt(7)
+ 1 e (7)
Ay, Ay, Ay,
in which K' is a time scale. For present purposes, based on analysis of a limited number of
nourishment projects, the following range of values is recommended
0.1/yr < K <0.3/yr (8)
consistent with a profile response of 63% to equilibrium in 10 years and 3.3 years respectively, for
K = 0.1/yr and 0.3/yr.
2.5 Consideration of Time Varying Shoreline Recession Due to Both Profile Equilibration
and Planform Evolution
The shoreline position due to both profile equilibration and volumetric spreading is
determined by first calculating the volume density remaining, V(x,t), at a particular location from
the volumetric evolution model (Figure 2) or a numerical model. With V(x,t) available, non
dimensional placed shoreline displacement, Ayi/ (Ay,/W.) is determined from Figures 3 and 4 (also
Figures B1 and B2, respectively in Appendix B) and the ratio of equilibrium to placed shoreline
displacements determined from Figures B3 to B8 in Appendix B. The time varying value of Ay is
calculated from Eq. (6) using K values of 0.1/yr and 0.3/yr to bracket the results.
3 EXAMPLES ILLUSTRATING APPLICATION OF THE RESULTS
Four examples are presented below which illustrate application of the methodology. The
characteristics of these examples are summarized in Table 3.
3.1 Example 1
This is a hypothetical example, i.e., it does not apply to any particular location. The following
nondimensional quantities are calculated
h 18 ft
S 3
B 6ft
DN =0.2 mm, A = 0.149 ft"13, Table 1
DF =0.18 mm, A= 0.139 ft'13, Table 1
AF/AN = 0.93
Table 3
Characteristics of Examples Illustrating Application of Methodology
Parameters Specified by Designer
Example Location Dp Volume Placed Project Objective of Example
(mm) Density Slope Length
(yd'/ft) mi (miles)
1 Hypothetical 0.18 100 1:20 NA For a Native Sediment Size of 0.20 mm, h.=18
ft, B=6 ft, Calculate the Initial (Ay,) and
Equilibrium (Ayo) Shoreline Position
2 Martin County, FL 0.21 120 1:30 2 For Martin County, FL Conditions, Calculate
Ay, and Ay,
3 Martin County, FL 0.21 120 1:30 2 Extend Example 2 to Include Ay at Project
Centerline and at Onehalf Mile from Centerline
After 3 Years, Accounting for Profile
Equilibration
4 Martin County, FL 0.21 120 1:30 2 Extend Example 3 to Account for a Background
Erosion Rate of 2 ft/year
W. = (h./A )'5 = (18/0.149)15 = 1328 ft.
h,
mWi.
18
(0.05)(1328)
= 0.271
V/ V (100)(27) = 0.339
BW. (6)(1328)
The ratio Ayi/W. is determined from Figure 4 (which applies for h/B = 3.0) as
AY 0.18
W
yielding a value of the placed shoreline additional width, Ay, = (1328)(0.18)=239 ft and the values
of Ayo/Ay from Figures B6 and B7 for AF/AN = 0.93 are both approximately 0.25, which results
in an equilibrium value, Ayo, without any volume density change of
Ayo = (0.25)(239) = 60 ft
The actual values are presented in Figure la.
A second part to this example would be to consider the equilibrium shoreline position after
a reduction of 25% of the nourishment volume due to transport to adjacent beaches,
V = 2700 x 0.75 = 2025 ft3/ft
2025
V 2025 = 0.254
(6)(1328)
Referring to Figure 4 (applicable for hJB = 3.0), the nondimensional placed width, Ay/W. is
approximately 0.14. The associated value Ay, is thus 186 ft. From Figures B6 and B7, for AI/AN,
the values of Ayo/Ay, are the approximate same value: Ayo/AyI = 0.18. Thus the equilibrated
additional beach width, Ayo, at this time is approximately
Ayo = (0.18)(186) = 33.5 ft
The actual value based on the equations is 34.9 ft as presented in Figure lb.
3.2 Example 2
Example 2 pertains to Martin County, FL. For purposes of this example, the following will
be considered:
V = 120 yd3/ft
DN = 0.21 mm,
DF = 0.23 mm,
h. = 15.6 ft
h./B = 2.6
AN = 0.153 ft1
AF/AA = 1.06
AF = 0.162 ft3 F N
Figure 5
.. B = 15.6/2.6 = 6.0 ft
G = 0.075 ft2/s, Figure 7
Placed Beach Slope = 1:30
Background Erosion Rate = 2 ft/year
Project Length, Q = 2 miles
3.2.1 Initial Placed Beach Width, Ay,
Calculate, W.
W. =(h./AN)1'5 = 1030 ft
Calculate, h.(m/W.) and NonDimensional Volume, V/(BW.)
h 15.6
S 156 0.454
miW. (.0333)(1030)
V (120)(27) 0.523
B W (15.6/2.6)(1030)
Determine Ay, From Figures 3 and 4
From Figure 3 (hJB = 2.0), Ay/W, = 0.27
From Figure 4, (hJB = 3.0), Ayl/W = 0.22
Interpolating, we find AyI/W. = 0.24
Ay, = 0.24 (1030) = 247 ft
3.2.2 Equilibrium Beach Width, Ayo
Determine Ayo by interpolation from Figure B4 (hJB = 2.0, hJ(mW.) = 0.3) and Figure B5
(h./B = 2.0, hJ(miW.) = 0.5) and Figure B7 (hJB = 3.0, hJ(miW.) = 0.3) and Figure B8 (hJB =
3.0 and hJ(mW.) = 0.5).
First, interpolating between Figures B3 and B5.
Figure B4 (hJB = 2.0, hJ(mW.) = 0.3): Ay/Ay, = 0.69
Figure B5 (hJB = 2.0, hJ(miW.) = 0.5): Ay/Ayi = 0.81
The interpolated value from the above for hJ(mW.) = 0.45 is AyjAy, = 0.78.
Second, interpolating between Figures B7 and B8
Figure B7 (hJB = 3.0, hJ(miW.) = 0.3): Ay/Ay, = 0.65
Figure B8 (hJB = 3.0, hJ(miW.) = 0.5): Ay/AyI = 0.80
The interpolated value for hJ(mW.) = 0.45 is Ay/Ay, = 0.76. Thus the interpolated value for h./B
= 2.6 and h/(miW.) = 0.45 is Ay0/Ay, = 0.77. Thus Ay, = (0.77)(247) = 190 ft.
3.3 Example 3
The basic conditions for this example are the same as for Example 2; however, now we wish
to calculate the actual shoreline displacement after 3 years at the project centerline (x/0 = 0) and at
a distance of 0.5 miles (x/Q = 0.25) from the centerline. Using consistent units, the value of non
dimensional time t' is
0.075 ft 2 (
s
Gt
t'
42
365 days 24 hrs 3600 s 2 i 5280 ft .
years x 2 miles = 0.064
year day hr mile
Referring to Figure 2,
V(0,0.64)
V
V(0.25,0.064) _
0.8
0.72
Thus
V(x'=0, t =0.06) = 0.8(120x27) = 2,592 ft 3/ft
V(x =0.25, t'=0.06) = 0.72(120x27) = 2,333 ft 3/ft
V 2,592
V (x '=0,t=0.06) = 2,592 = 0.42
BW. (6)(1030)
V (x =0.25, t '=0.06) = 2,333 0.38
BW (6)(1030)
Repeating the same double interpolation procedure as in Example 2, the initial and equilibrium
values of shoreline displacement are
x'=0. t'=0.06
Ay, = 195 ft
Ay, = 148 ft
x'=0.5. t=0.06
Ay = 185 ft
Ay, = 128 ft
The ranges of anticipated shoreline advancement at t = 3 years are (from Eq. (6))
Ay,(0,3 years) = 148 + (195148)e (0.)(3) = 183 ft
Ay2(0,3 years) = 148 + (195148) e(0.3)(3) = 167 ft
and the subscripts 1 and 2 apply for ki = 0.1 yearI and k2 = 0.3 year', respectively. Similarly for x'
= 0.5 (x = 0.5 miles)
Ay, (0.5 miles, 3 years) = 170 ft
Ay2 (0.5 miles, 3 years) = 151 ft
3.4 Example 4
This is an extension of Example 3 to include the effect of a background erosion rate of 2 ft/yr.
This effect is accounted for by a simple reduction in the shoreline position by an amount equal to the
product of the shoreline change rate and time, in this case a total of six feet [(2 ft/yr)(3 years) = 6 ft)].
Thus, the values of Ay, and Ay2 for x = 0 and 0.5 mile at 3 years are
x=0. t=3 years
Ay, = 177 ft
Ay2 = 161 ft
x = 0.5 mile. t = 3 years
Ay, = 164 ft
Ay; = 145 ft
4 SUMMARY
Beach nourishment projects are usually constructed with approximately uniform sloped
profiles, the forms of which are relatively unrelated to the equilibrium profiles associated with the
nourishment sediment. Subsequent to the nourishment, wave action commences the equilibration
process which is accompanied by beach width reduction. Significant parameters in the design of
beach nourishment projects are the anticipated amount of beach width reduction and the associated
time scales.
This report has presented methodology, based on equilibrium beach profile concepts, for
determining the initial beach width, Ay,, and the equilibrium beach width, Ayo. A series of graphs
is provided which allows these results to be approximated with a hand calculator. A method is also
recommended for calculating the timevarying additional beach width from the time of placement
to the equilibrated state. The effects of evolution of an initially rectangular planform and background
erosion, both of which reduce equilibrium beach width can be taken into consideration using
graphical forms presented for the sandy beaches of Florida.
Application of the methodology is illustrated by examples. The methods and design aids
presented here are considered appropriate for preliminary design and checking of and comparison
with results from more detailed methods which may be employed for the final designs.
REFERENCES
Dean, R.G., and J. Grant (1989) "Development of Methodology for ThirtyYear Shoreline
Projections in the Vicinity of Beach Nourishment Projects," Report No. UFL/COEL89/026,
Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, FL.
Dean, R. G. (1991) "Equilibrium Beach Profiles: Characteristics and Applications," Journal of
Coastal Research, Vol. 7, No. 1, Winter, 1991, pp. 5384.
Dean, R. G. (2000) "Beach Nourishment Design: Consideration of Sediment Characteristics,"
Report No. UFL/COEL2000/002, Coastal and Oceanographic Engineering Program of the
Department of Civil and Coastal Engineering, University of Florida, Gainesville, FL.
APPENDIX A
DEVELOPMENT OF SHORELINE
ADVANCEMENT, Ay, FOR NOURISHMENT
PLACEMENT AT A UNIFORM SLOPE
APPENDIX A
DEVELOPMENT OF SHORELINE
ADVANCEMENT, Ay, FOR NOURISHMENT
PLACEMENT AT A UNIFORM SLOPE
Introduction
The purpose of this appendix is to present the development of the equations for the initial
shoreline displacement due to placement of a volume density, V, at a uniform slope, mi. Equilibrium
beach profile concepts are used. The geometry of interest is presented in Figure A1.
10
 Ay,
B
> 0
z h,
) I h.
10
20
a 2 0 
LIU
30
200 0 200 400 600 800 1000 1200 1400 1600
Distance From Original Shoreline (ft)
Figure Ai. Definition Sketch.
Development
Referring to Figure A1, the volume density added is expressed as
V = BAy + fY ANy 3dy (yIAyI)mi
= BAy + A Ay1)2
5 2
A2
(A2)
(A2)
in which B is the berm height, AN, is the profile scale parameter for the native sediment, y, is the
distance from the original shoreline to the intersection point of the original and placed profiles and
mi is the slope of the placed profile.
There are two unknowns in Eq. (Al). An auxiliary equation is obtained by equating the
intersection depth of the two profiles.
h, = ANYim = mi(lAyi) (A3)
Substituting Eq. (A3) into (A2) to eliminate Ay,
SNYi 3 5/3 1 2 4/3
V = B Y + +ANyl ANy, (A4)
m. 5 2 m.
which contains only one unknown. Defining the following nondimensional quantities
y' = yi/W.
V '= V/(BW.) (A6)
Ay1/= Ay/W.
the following can be shown
=,/ / h. /2/3 3 h. /5/3 1 h. h. /4/3
V y Yi + Yi  Yi (A6)
mW. 5 B 2 B miW.
which can be solved by iteration. Once y,' is determined, the nondimensional value of Ay,' can be
determined from the nondimensional form of Eq. (A3) as
AyI = y, yr (A7)
It is noted that Ayl andy,' are functions of three nondimensional variables
/=fl V/', 2 (A8)
YI f V ) (A8)
mi B
AYI=f2V B (A9)
The nondimensional equilibrium shoreline displacement, Ay0o, is also a function of three non
dimensional variables
/ AF h.
Ayo'= V', (A10)
SB
It is noted that two of the three nondimensional variables are common for Ay,' and Ayo.
APPENDIX B
PLOTS OF Ay/W. (TWO PLOTS) AND Ay/Ayi (SIX PLOTS)
APPENDIX B
PLOTS OF Ay/W. (TWO PLOTS AND Ay/Ay, (SIX PLOTS)
Introduction
Two types of graphs are presented in this appendix. The first type presents the non
dimensional values of the initial beach width, Ay1/W. for three values of hJ(miW.). These graphs
were presented earlier as Figures 3 and 4 in the main body of the report and are repeated here as
Figures B1 and B2 for convenience for hJB values of 2.0 and 3.0, respectively. The second type
of graph depicts the ratio of equilibrium dry beach Ayo to the initial beach width, Ay,. Each of these
graphs includes a range of ratios AlAN and nondimensional volumes, V/(BW.). A total of six plots
is presented which include combinations of two values of hJB (= 2.0 and 3.0) and three values of
hJ(mW.) (= 0.1, 0.3 and 0.5). The characteristics of these six plots are presented in Table 2 in the
main body of this report which is repeated in this appendix as Table B1 for convenience and the six
plots are presented as Figures B3 through B8.
Table Bl
Parameters of Plots Provided in Figures B3
through B8 in Appendix B
B2
Case Reference Figure No. hJB (hJmW.)
1 Standard 1 6, B3 2 0.1
2 Standard 1, Except hJmiW. = 0.3 B4 2 0.3
3 Standard 1, Except hJmiW. = 0.5 B5 2 0.5
4 Standard 2 B6 3 0.1
5 Standard 2, Except hJmiW. = 0.3 B7 3 0.3
6 Standard 2, Except hJmiW. = 0.5 B8 3 0.5
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
Figure B1.
0.0
0.0
Figure B2.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Nondimensional Volume, V = V / (B W.)
NonDimensional Placed Shoreline Displacement, Ay,'
(= AYI/W.) Versus NonDimensional Volume Density, V'
(= V/(B W)) and Slope Parameter h./(miW.). h /B =2.0.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Nondimensional Volume, V = V / (B W.)
1.8 2.0
NonDimensional Placed Shoreline Displacement, Ay,
( Ay,/W.) Versus NonDimensional Volume Density, V'
( V/(BW.)) and Slope Parameter h./(miW,). h./B=3.0.
B3
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.8
1.0 1.2 1.4 1.6 1.8 2.0
AF/AN
Figure B3. Ratio of Equilibrium to Placed Beach Widths for the Following
Parameters: hJB = 2, hJ(miW.) = 0.1. No Volume Change in
Profile.
0.0
0.8
1.0 1.2 1.4 1.6 1.8 2.0
AF/AN
Figure B4.
Ratio of Equilibrium to Placed Beach Widths for the Following
Parameters: hJB = 2, hJ(miW.) = 0.3. No Volume Change in
Profile.
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2 /
/
0.0
0.8
Ratio of Equilibrium to Placed Beach Widths for the Following
Parameters: hJB = 2, hJ(miW.) = 0.5. No Volume Change in
Profile.
1.0 1.2 1.4 1.6 1.8 2.0
AF/AN
Figure B6. Ratio of Equilibrium to Placed Beach Widths for the Following
Parameters: hJB = 3, hJ(miW.) = 0.1. No Volume Change in
Profile.
1.0 1.2 1.4 1.6 1.8
AF/AN
Figure B5.
0.0
0.8
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.8
Ratio of Equilibrium to Placed Beach Widths for the Following
Parameters: hJB = 3, hJ(mW.) = 0.3. No Volume Change in
Profile.
1.0 1.2 1.4 1.6 1.8 2.0
AF/AN
Figure B8. Ratio of Equilibrium to Placed Beach Widths for the Following
Parameters: hJB = 3, hJ(miW.) = 0.5. No Volume Change in
Profile.
1.0 1.2 1.4 1.6 1.8
AF/AN
Figure B7.
0.2 L
0.0
0.8
