7 D-R14l 545 STABILITY OF STONE- AND DOLOS-RORED RUBBLE-ROUND

/2BREAKWARTER TRUNKS SUB.(U) COASTAL ENGINEERINGRESERCH.CENTER VICKSBURG MS R D CARVER DEC 83

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In3.11)TECHNICAL REPORT CERC-83-5

* 0S _STABILITY OF STONE- ANDof EngneersDOLOS-ARMORED, RUBBLE-MOUND

BREAKWATER TRUNKS SUBJECTED TOBREAKING WAVES WITH NO OVERTOPPING

by

Robert 0. Carver

Coastal Engineering Research CenterU. S. Army Engineer Waterways Experiment Station

P. 0. Box 631, Vicksburg, Miss. 39180

December 1983Final Report

' -Approved For Public Rel".is- Distrtut Uri t L T

.5. PrearedDTIC

Washingto 0 C 20314

Ud2 Wor Unt3165.4 iB

C=3(aQ~ /

Prpae fo Ofie Chief~ of Engners U SArm

i Destroy this report when no longer needed. Do noti return it to the originator.

. ,' The findings in this report are not to be construed as an--' .'. official Department of the Army position unless so

=- '-%designated by other authorized documents.

The contents of this report are not to be used for" '.'.-*•"advertising, publication, or promotional purposes.

Citation of trade names does not constitute an." ""official endorsement or approval of the use of such

commercial products.

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UnclassifiedSECURITY CLASSIFICATION OF THIS PAGE (W7ein D ea Entered) _ _._-

READ INSTRUCTIONSREPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

Technical Report CERC-83-5 /py ~l 5 _

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

STABILITY OF STONE- AND DOLOS-ARMORED, Fia rprRUBBLE-MOUND BREAKWATER TRUNKS SUBJECTED Final report

TO BREAKING WAVES WITH NO OVERTOPPING 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(a) 8. CONTRACT OR GRANT NUMBER() -

Robert D. Carver

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASKAREA & WORK UNIT NUMBERS

U. S. Army Engineer Waterways Experiment StationCoastal Engineering Research Center Work Unit No. 31269P. 0. Box 631, Vicksburg, Miss. 39180

11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Office, Chief of Engineers, U. S. Amy December 1983 ...13. NUMBER OF PAGES

Washington, D. C. 20314 11314. MONITORING AGENCY NAME & ADDRESS(If different from Controlllng Office) IS. SECURITY CLASS. (of thin report)

Unclassified

Ie.. DECLASSIFICATION/DOWNGRADINGSCHEDULE

10. DISTRIBUTION STATEMENT (of thin Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract angered In Block 20, It dliferent from Report)

I. SUPPLEMENTARY NOTES

Available from National Technical Information Service, 5285 Port Royal Road,Springfield, Va. 22161

1S. KEY WORDS (Continue on reveres aide If neceesary nd identify by block number)

Breakers (water waves) StoneBreakwaters Water wave actionDolosse Water wavesRubble-mound breakwaters

20. AISTRACT (CentAu a reverse ffP f neceoam and dfldify by block numwber)

-The objective of the research presented is to furnish design informa- Otion for stone and dolos armor on nonovertopping breakwater trunks that are ..subjected to severe depth-limited breaking waves. Since it would be a mam-moth task to comprehensively investigate all the different types of existingarmor, this particular research effort concentrated on stone and dolosse.Stone is a natural and economical protection when it is of sufficient sizeand quality to meet design constraints; and dolosse, according to nonbreaking"

(Continued)

~pF 13 EDITION OF? NOV GS1 ISOBSOL ETEI JAN 73 oUnclassified

SECURITY CLASSIFICATION OF THIS PAE (Xher Dels Entered).4 ,-

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Unclassified

e SECURITY CLASSIFICATION OF THIS PAGE(Whirt Data Entorod)

20. ABSTRACT (Continued) l

wave data, is the best hydraulically stable concrete armor unit. Results ofthis research indicate that both stone stability and dolos stability are cri-tical relative to breaking waves, thus these breaking wave stability coeffi-cients for breakwater trunks should be on the order of 2 and 15, respectively. .'.

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PREFACE

Authority for the U. S. Army Engineer Waterways Experiment Station (WES)

to conduct this study, Work Unit No. 31269, "Stability of Breakwaters,"

Coastal Structure Evaluation and Design Program, Coastal Engineering Area of

-: Civil Works Research and Development, was contained in a letter from the

-' Office, Chief of Engineers (OCE), U. S. Army, dated 19 May 1972. OCE Tech-

nical Monitors for this research were Messrs. J. H. Lockhart, CDR USACE (DAEN-

CWH-D), J. G. Housley, CDR USACE (DAEN-CWP-P).

The study was conducted by personnel of the Hydraulics Laboratory, WES,

under the general direction of Messrs. H. B. Simmons and F. A. Heirmann, Jr.,

Chief and Assistant Chief of the Hydraulics Laboratory; Dr. R. W. Whalin and

Mr. C. E. Chatham, former and present Chief of the Wave Dynamics Division,

and Mr. D. D. Davidson, Chief of the Wave Research Branch. The Wave Dynamics

Division and its personnel were transferrL to the Coastal Engineering Re- O

search Center (CERC) of WES on 1 July 1983 under the direction of Dr. Whalin,

Chief of CERC. Tests were planned by Mr. R. D. Carver, 7coject Engineer, and

Mr. W. G. Dubose, Engineering Technician. The model was operated by Mr. Dubose

under the supervision of Mr. Carver. This report was prepared by Mr. Carver.

Commander and Direct)r of WES during the conduct of the study and the

preparation and publication of this report was COL Tilford C. Creel, CE.Technical Director was Mr. F. R. Brown.

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CONTENTS

N_

A ~ PREFACE. ................................

CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (SI)UNITS OF MEASUREMENT............................3

PART I: INTRODUCTION ............................. 4

Background. .............................. 4Purpose of Study.............................

PART II: DIMENSIONAL ANALYSIS........................6

4.Stability of Rubble-Mound Breakwaters. ................ 6Wave Runup...............................8Stability Scale Effects ......................... 9

PART III: TESTS .............................. 11

Selection of Test Conditions ..................... 11Test Procedures. ........................... 12

Test Equipment and Materials ..................... 15ki

PART IV: TEST RESULTS ........................... 16

Stability Tests. ........................... 16Wave Runup Tests ........................... 18

*PART V: CONCLUSIONS AND RECOMMENDATIONS..................20

REFERENCES ................................. 21

TABLES 1-6

PHOTOS 1-63

PLATES 1-17

4,APPENDIX A: NOTATION. ........................... Al

K2

* .i1

CONVERSION FACTORS, U. S. CUSTOMARY TO METRIC (SI)UNITS OF MEASUREMENT 0!

U. S. customary units of measurement used in this report can be converted to .

metric (SI) units as follows:

Multiply By To Obtainfeet 0.3048 metres

feet per second per second 0.3048 metres per second persecond

inches 25.4 millimetres

pounds (mass) 0.4535924 kilograms

pounds (mass) per cubicfoot 16.01846 kilograms per cubic metre

square feet 0.09290304 square metres

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STABILITY OF STONE- AND DOLOS-ARMORED, RUBBLE-MOUND BREAKWATER .--

TRUNKS SUBJECTED TO BREAKING WAVES WITH NO OVERTOPPING

PART 1: INTRODUCTION

Background

1. The experimental investigation described herein constitutes a por-

tion of a research effort to provide engineering data for the safe and eco-

nomical design of rubble-mound breakwaters. In this study, a rubble-mound

breakwater is defined as a protective structure constructed with a core of

quarry-run stone, sand, or slag and protected from wave action by one or more

stone underlayers and a cover layer composed of selected quarrystone or spe-

cially shaped concrete armor units.

2. Rubble-mound breakwaters are used extensively throughout the world

to provide protection from the destructive forces of storm waves for harbor

and port facilities. A proposed structure may necessarily be designed for

either nonbreaking or breaking waves depending upon positioning of the break-

water and severity of anticipated wave action during its economic life. Some

local wave conditions may be of such magnitude that the protective cover layer

must consist of specially shaped concrete armor units in order to provide eco-

nomic construction of a stable breakwater; however, many local design require-

ments are most advantageously met by quarrystone armor. This particular re-

port addresses the use of quarrystone and dolos armor on breakwater trunks

subjected to breaking waves.

3. Previous investigations have yielded a significant quantity of de-

sign information for quarrystone (Hudson 1958 and Carver 1980) tetrapods,

quadripods, tribars, modified cubes, hexapods, and modified tetrahedrons

(Jackson 1968), dolosse (Carver and Davidson 1977), and toskane (Carver 1978).

Although all of the above studies were important and filled a need, they were

limited in that test waves were always nonbreaking. A few breaking wave tests

are reported (Hudson 1961), but the breaking wave coefficients presented in

the Shore Protection Manual (U. S. Army CERC 1977) were primarily developed

from adjustments made to nonbreaking wave coefficients as supplemented by

site-specific studies. A comprehensive study to develop general stability co-

efficients for depth-limited breaking waves has not previously been conducted.

4

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Purpose of Study

4. The purpose of the present investigation was to obtain design infor-

mation for stone and dolos armor used on breakwater trunks and subjected to

breaking waves. More specifically, it was desired to determine the minimum

weight of individual armor units (with given specific weights) required for

stability as a function of:

a. Type of armor unit.

b. Sea-side slope of the structure.

c. Wave period.

d. Wave height.

e. Water depth.f. Sea-bottom slope on which the breakwater is constructed.

Also, it was desired to determine the magnitude of wave runup that can be

expected for a selected range of incident wave conditions.

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PART II: DIMENSIONAL ANALYSIS

Stability of Rubble-Mound Breakwaters

5. When short-period waves attack rubble-mound breakwaters, the inter- iI"." action of the dislodging forces induced by the water motion and the resistive

action of the armor units produce a complex dynamic phenomenon. Previous

attempts to analyze this phenomenon to ascertain the magnitude of the dynamic

forces involved by theoretical analyses have not been successful; however,

coastal scale models/experimental tests of breakwaters can yield accurate de-

sign information that relates the required weight of individual armor units to

breakwater geometry, local bathymetry, wave characteristics, etc.

6. An attempt will be made through the use of dimensional analysis to

develop functional relationships between the primary variables affecting armor

stability. Buckingham's 71 theorem can be used to determine the number of

dimensionless and independent quantities (pi terms) required to express a re-

-. ' lationship among the variables in any phenomenon. Dimensional analysis may

then be used to obtain a suitable set of pi terms.

7. Definitions and characteristic dimensions in terms of force (F),

length (L), and time (T)* of the primary variables affecting armor stability

are as follows:

3"a = specific weight of an armor unit, F/L

W = weight of an armor unit, F

A = shape factor of the armor unit, dimensionless

W = specific weight of water, F/L

H = wave height, L

L = wavelength, L

d = water depth, L

g = acceleration due to gravity, L/T

h = height of breakwater crown, L

= angle of wave attack, dimensionless2

v = kinematic viscosity, L /T

o= angle between the horizontal and the seaward face of thebreakwater, dimensionless

* For convenience, symbols and unusual abbreviations are listed and defined

in the Notation (Appendix A).

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'-- :,; X , * :- ,-4 ,< ,-'..: -,.'.:-:-..: ,.--'-.--.'.<.7-- .-. '..'--.'.--..-'-. . .---. . .. . ..- . -.-: -

7. 7.

0 = angle between the horizontal and the sea bottom on which the.. breakwater is constructed, dimensionless

PT = technique used to place armor units in the cover layer,dimensionless O

D = damage parameter, dimensionless

8. The present investigation addresses only waves normal to nonover-

topping breakwater sections. Therefore the variables and h are elim-

inated. Also, since a is directly related to the seaward slope of the

breaLwater, this variable can be replaced by cot u where cot t is the

reciprocal of breakwater slope. With these considerations, the list of

variables becomes

Ya = specific weight of an armor unit, F/L3

W = weight of an armor unit, FaA= shape factor of the armor unit, dimensionless

Yw = specific weight of water, F/L3

.-I H = wave height, L

L = wavelength, L

d water depth, Lg = acceleration due to gravity, LIT2

2g = kinematic viscosity, L/iT

cot a = reciprocal of breakwater slope, dimensionless

0 = angle between the horizontal and the sea bottom on whichS the breakwater is constructed, dimensionless

PT = technique used to place armor units in the cover layer,dimensionless

D = damage parameter, dimensionless

9. With 13 independent variables and 3 basic dimensions involved, Buck-

ingham's t theorem predicts that armor stability should be a function of 10

dimensionless pi terms. One possible set of pi terms is

Y1/3H

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"a~o a

= H/d (2)

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74 = a -(4)

4.--

TI cot o,-.-...

6 A (6)

n =0 (77

* (gl) Wa

81/3 (8

71 PT (9)

A 10 D (10)

Correlation of the test data will be attempted by the functional relationship

. ,= f( 72 , , n T 5 . 6, n7, n8, n9 , nlO) (n7 9

orx:. .1/3.a

a = H/d H/L cot , 0v:.:, a Y

(YWI\~/3 w

\Y ' a

(gH) l/2Wl/3 1a/3 PT D (12)

a

Wave Runup

10. Before a breakwater design can be optimized, it is necessary for

the designer to be able to accurately estimate wave runup (Ru ) for the anti-

cipated range of wave conditions to which the structure will be subjected.

4.

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- - ;'.'- -- --. --,.- .*- -" '.- '. --" -, .-' -'. .-' .--.. ' -'.- ".- :'. .-. .- v: - -v : • - - S.

Runup data are necessary for selecting a crown elevation that will prevent

excessive wave overtopping.

11. The primary variables affecting wave runup on sloping structures

are a , 6 , A , H , d , L , and the porosity (P) of the armor layer

and underlayers, i.e.,

R = f(a, A, H, d, L, P, , ) (13)

One possible set of pi terms is

n= Ru/H (14)

7=A (15) O

713 = H/L (16)

74 = H/d (17)

TI o (18)

4:. n6 = P (19I1 .--

71" =0 (20)

6

Correlation of the test data will be attempted by the functional relationship

T1 = f(2, 73 n4) 75 7T6' 717) (21) "'-'

or '., ,

R /H f A, H/L, H/d, a, P, (22)

Stability Scale Effects

12. If the absolute sizes of experimental breakwater materials and wave

dimensions become too small, flow around the armor units enters the laminar

regime; and the induced drag forces become a direct function of the Reynolds

number. Under these circumstances, prototype phenomena are not properly simu-

lated and stability scale effects are induced. Hudson (1975) presents a

9

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detailed discussion of the design requirements necessary to ensure the preclu-

sion of stability scale effects in small-scale breakwater tests and concludes

that scale effects will be negligible if the Reynolds stability number,1/2 1/2 4

%,~ = (g H k )1v , is equal to or greater than 3 x 10 . For all tests- reported herein, the sizes of experimental arinor and wave dimensions were

selected such that scale effects were insignificant (i.e., RN was greater4Nthan 3 x 10)

4

-V..

100

PART III: TESTS

Selection of Test Conditions

13. A review of past site-specific stability projects and hydrographic

data showed that typical prototype sea-bottom slopes could range from almost

flat to as steep as IV on 1OH. Realizing that wave deformation and severity

of breaking action increases as bottom slope increases and since time re-

straints would allow testing of only one foreslope, it was decided to use a

IV-on-IOH slope, thus ensuring severe depth-limited breaking wave action

(plunging breakers). This type of breaking wave normally causes the most dam-

age to rubble-mound structures.

14. By nondimensionalizing design conditions from site-specific proj-

ects, it was found that a d/L range of 0.04 to 0.14 should include most pro-

totype conditions encountered in breaking-wave stability designs. A review of

capabilities of the available flume and wave generator showed that this range

of d/L values could be achieved for a reasonable range of testing depths.

15. In planning a stability investigation, it is not possible to pre-

select exact values of H/L and Hid since the design-wave heights are un-

known at the outset of the study. However, the widest possible range of these

parameters can be ensured by using various armor weights that range from just

above the scale-effect regime at the lower limit up to the maximum weights

that the test facility is capable of displacing. For the present investiga-

tion, armor weights ranged from 0.23 to 0.71 lb.*

16. The wave flume was calibrated for depths from 0.40 to 0.95 ft in

0.05-ft increments at d/L values of 0.04, 0.06, 0.08, 0.10, 0.12, and 0.14.

This range of depths and consequently breaking wave heights proved to be com-

patible with the selected armor weights and sea-side breakwater slopes.

17. All stability and wave runup tests were conducted on sections of

the type shown in Plate 1 and Photos 1-16. Sea-side slopes of IV on 1.5H, IV

on 21, and IV on 31 were investigated while the beach-side slope was held con-

stant at IV on 1.5H. Structure heights varied from 1.0 to 1.6 ft. The height .-

* A table of factors for converting U. S. customary units of measurements to -Ometric (SI) units is presented on page 3.

-a,-

necessary to prevent wave overtopping was determined by the combination of

slope, armor type and weight, and water depth being investigated.

Test Procedures

Method of constructing "

test sections

18. All experimental breakwater sections were constructed to reproduce

" as closely as possible results of the usual methods of constructing full-scale

breakwaters. The core material was dampened as it was dumped by bucket or jshovel into the flume and was compacted with hand trowels to simulate natural J_

consolidation resulting from wave action during construction of the prototype

structure. Once the core material was in place, it was sprayed with a low-

velocity water hose to ensure adequate compaction of the material. The under-

layer stone was then added by shovel and smoothed to grade by hand or with

trowels. No excessive pressure or compaction was applied during placement of

the underlayer stone. Armor units used in the cover layers were placed in a

random manner corresponding to work performed by a general coastal contractor,

i.e., they were individually placed but were laid down without special orien-

tation or fitting. After each test the armor units were removed from the

breakwater, all of the underlayer stones were replaced to the grade of the

original test section, and the armor was replaced.

Selection of criticallybreaking waves

* ...r~19. For a given wave period and water depth, the most detrimental break-

ing wave (i.e., the most damaging wave) was determined by increasing the

stroke adjustment on the wave generator in small increments and observing

which wave produced the most severe breaking wave condition on the experimen-

tal structures. Wave heights of lower amplitude did not form the critical

breaking wave and wave heights of larger amplitude would break seaward of the

test structures and dissipate their energy so that they were less damaging

than the critically tuned wave.

20. A typical stability test consisted of subjecting the test section

to attack by waves of a given height and period until it was certain no damage ...

was going to occur or if damage was occurring, until all damage had abated or

the amount of damage exceeded an acceptable level. Test sections were

12

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subjected to wave attack in approximately 30-sec intervals between which the

wave generator was stopped and the waves were allowed to decay to zero height.

This procedure was necessary to prevent the structures from being subjected to

an undefined wave system created by reflections from the experimental break-

water and wave generator. Newly built test sections were subjected to a short

duration (five or six 30-sec intervals) of shakedown using a wave equal in

" height to about one-half of the estimated no-damage wave. This procedure pro-

vided a means of allowing consolidation and armor unit seating that would nor-

mally occur during prototype construction.

Method of determining damage

21. In order to evaluate and compare breakwater stability test results,

it is necessary to quantify the changes that have taken place in a given

structure during attack by waves of specified characteristics. The U. S. Army

Engineer Waterways Experiment Station (WES) developed a method of measuring

the percent damage incurred by a test section during the early 1950's. This

method has proven satisfactory and was used as a means for analyzing and com-

paring the stability tests delineated herein.

22. The WES damage-measurement technique requires that the cross-

sectional area occupied by armor units be determined for each stability test

section. Armor unit area is computed from elevations (soundings) taken at

closely spaced grid-point locations over the seaward face of the structure

before the armor is placed on the underlayer, after the armor has been placed

but before the section has been subjected to wave attack, and finally after

wave attack. Elevations are obtained with a sounding rod equipped with a cir- IT

cular spirit level for plumbing, a scale graduated in thousandths of a foot,

and a ball-and-socket foot for adjustment to the irregular surface of the

breakwater slope. The diameter (Diam) in inches of the circular foot of the

sounding rod was related to the size of the material being sounded by the

following equation:

Diam C (23)

where C = 6.8 and 13.7 for stone and dolosse, respectively. A series of

sounding tests in which both the weight of the armor and the diameter of the

sounding foot were varied indicated that the above relation would give a

1.3

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measured thickness which visually appeared to represent an acceptable two-

layer thickness.

23. Sounding data for each test section were obtained as follows:

after the underlayer was in place, soundings were taken on the sea-side slope

of the structure along rows beginning at and parallel to the longitudinal

center line of the structure and extending in 0.25-ft horizontal increments

until the interface between the structure's toe and the flume bottom was

reached. On each parallel row, 13 sounding points, spaced at 0.25-ft incre-

ments, were measured. This distance represented the middle 3 ft of a 5-ft-

wide test section; the 1 ft of structure next to each wall was not considered

because of the possibility of discontinuity effects between the armor units

and the flume walls. Soundings were taken at the same points once the armor

was in place and again after the structure had been subjected to wave attack.

24. Sounding data from each stability test were reduced in the follow-

ing manner. The individual sounding points obtained on each parallel row

were averaged to yield an average elevation at the bottom of the armor layer ,

before the armor was placed and then at the top of the armor layer before and

after testing. From these values, the cross-sectional armor area before test-

ing and the area from which armor uniLs were displaced (either downslope or

off the section) were calculated. Damage was then determined from the follow-

ing relation:

Percent damage = (100) (24)

where

A : area before testing, ft21 2

A = area from which armor units have been displaced, ft2 A~

The percentage given by the WES sounding technique is, therefore, a measure-

ment of an end area which converts to an average volume of armor material

that has been moved from its original location (either downslope or

off-structure).

Measurement of wave runup

25. Values of wave runup were obtained with a point gage calibrated inincrements of 0.001 ft and mounted on an aluminum framework that could be

% moved along and across the seaward breakwater slope. Due to slight height

7variations from wave to wave within a given wave train and the highly porous

14

-.. S'"

texture of the breakwater slope, five measurements of R were made for eachU

test wave condition. Photos 17-22 show breaking and runup for selected waveconditions. Runup values were taken as the upper edge of the solid water,

* -. not the foam or splash line.

Test Equipment- and Materials

. Equipment used26. All wave-action tests were conducted in a 5-ft-wide, 4-ft-deep, and

119-ft-long concrete wave flume with test sections installed about 90 ft from

a vertical displacement wave generator. The first 10 ft of flume bottom,

immediately seaward of the test sections, was molded on a IV-on-10H slope

s. while the remaining 80 ft was flat. The generator is capable of producing

sinusoidal waves of various periods and heights. Test waves of the required

characteristics were generated by varying the frequency and amplitude of the

* -. plunger motion. Changes in water-surface elevation as a function of time

(wave heights) were measured by electrical wave-height gages in the vicinity

of where the toe of the test sections was to be placed and recorded on chart

paper by an electrically operated oscillograph. The electrical output of the

wave gages was directly proportional to their submergence depth.

Materials used

27. Rough hand-shaped granitic stone (Wa) with an average length of

approximately two times its width, average weights of 0.38 lb (±0.02 lb),

0.55 lb (±0.025 lb), and 0.71 lb (±0.03 Ib), and a specific weight of 167 pcf

was used to armor the stone sections. Dolos sections were armored with the

following sizes of units.

W, lb Ya' pcfaa

0.234 137.7

0.276 142.2

0.589 141.1

Sieve-sized limestone (y = 165.0 pcf) was used for the underlayer (W1) and

core (W2 ).

2d

15O,

PART TV: TEST RESULTS

Stability Tests

28. Breaking wave stability test results for stone and dolos armor are

summarized in Tables I and 2, respectively. Presented therein are experi-

mentally determined design wave heights and corresponding stability numbers as

functions of relative depth, wave steepness, relative wave height, Ursell

Number (N ), and breakwater slope. All stability test results presented inU

Tables I and 2 were verified by at least one repeat test. Sea-side breakwater

slopes of IV on 1.5H, IV on 2H, IV on 3H were used for both armor types. The

following ranges of armor weights, water depths, wave periods and heights,

relative depths, wave steepness, Ursell Numbers, and relative wave heights

were investigated.

Range for Indicated Type of ArmorVariable Stone Dolosse

Armor weight, lb 0.38-0.71 0.234-0.589

Water depth, ft 0.40-0.75 0.45-0.95

Wave period, sec 1.04-2.82 1.30-2.32

Wave height, ft 0.33-0.55 0.45-0.77

Relative depth 0.04-0.14 0.06-0.14

Wave steepness 0.042-0.099 0.058-0.094

Relative wave height 0.64-1.05 0.64-1.02

Ursell Number 32.5-656.3 33.6-284.0

The number of armor units per given surface area, A , was N = 1.26 V-2 / 3

with n = 2 , ka = 1.00 , and P = 37 percent for stone armor, and

N = 0.83 V with n = 2 , kA = 0.94 , and P = 56 percent for dolos

armor. The variable, V , is defined as the volume of an individual armor

unit. It should be noted that k. = 1.00 obtained from numerous stone armor

tests is less than the present SPM (CERC 1977) value of 1.15. Photos 23-63

show the after-testing stability condition of the structures.

29. As previously discussed, it was hoped that stability test results

could be analyzed by the following functional relation for the stability num-

ber, N . where

16

%S,,.,

.,-4 ' .'.4:. .N '-". :, , " " '":'- " -- - - -, --- . ', '-.'.;""'.:-.,.- •.-' .,' -:--:-

-.. -.-. I..~~* -- - - - - .

" 21/3... N = a f H dH Ya"" '1

J"'"-'.',L (S a lxW1 / 3 = /d /L cot ot, e, A,.-.I_

(gH) P2 Iv PT, (25)a

For tests described herein 0 , PT , and D were held constant and y a/yw

was essentially invariant for a given type of armor; therefore Equation 25

" . reduces to

Ns = [H/d, H/L, cot Y, A, (gH)l/2 / (26)

Also, the sizes of experimental armor units and wave dimensions were selected

such that turbulent flow was always obtained; therefore N was independen..S. _-." [(gH)1/2 aV

: .of Reynolds number ( k / and Equation 26 becomes

N = f(H/d, H/L, cot a, A) (27)s

30. Plots of N versus H/d and H/L are presented in Plates 2 andS

3, respectively. Effects of H/d and H/L are combined in the Ursell Number2 3

* - (L H/d ) and results are given in Plate 4. These data show a weak functional

2 3dependence of N on H/d , H/L , and L H/d with the dependence being5

more pronounced for dolos armor. For both armor types, it generally appears2 23that minimum stability occurs for the larger values of H/d and L H/d and

for the intermediate range of H/L (0.07 < H/L < 0.085) . Results of pre-

vious tests conducted on quarrystone by Hudson (1958) and Carver (1980) for

nonbreaking waves, H/d < 0.32 and 0.03 < H/L < 0.08 , do not show these a..

trends. Also the trends are absent from earlier nonbreaking wave tests on

dolosse (Carver and Davidson 1977). The tests of Carver and Davidson were

conducted with H/d < 0.37 and 0.031 < H/L < 0.083

31. Plate 5 presents a log-log plot of N versus cot a Average and

lower limit linear fits of the Hudson type, i.e., IV-on-3H slope linear fits, .Oe

are also shown. Even though there is some data spread for each distinct value2 3of cot a (due to variations of H/d , H/L , and L H/d ), the linear fits

generally give a reasonable approximation of N as a function of cot as

especially for stone armor. The lower limit lines correspond to stability

coefficient (KD) values of 2.0 and 15.0 for stone and dolosse, respectively.

17" -

.. ].

.:'h"

Wave Runup Tests

32. Runup, average runup, and the standard deviation are shown in

Tables 3 and 4 for all test conditions. Considering the small random varia-

tions inherent in test waves within a given wave train and small local varia-

. tions in the texture and porosity of the breakwater slope, test results appear

to be quite consistent.

33. As described in paragraph 11, it was hoped that runup test results

could be correlated by the following functional relation for relative runup

(Ru/H)u

Ru/H = fIA, H/L, H/d, ci, P, (22 bis)u o

For runup tests described herein P was constant for a given type of armor

and P/6 was invariant for all tests; therefore Equation 22 reduces to

R /H = f(A, H/L, H/d, a) (28)u

Calculated values of relative runup along with corresponding values of H/L

H/d , and cot a are presented in Tables 5 and 6 using the average runup from

Tables 3 and 4. Plates 6-11 present R /H as a function of H/L for con-U

stant values of cot a and Plates 12-17 present R /H as a function of H/du

for constant values of cot a . These data show relative runup to be a func-

tion of breakwater slope, wave steepness, relative wave height, and armor

type. It is very interesting to note that maximum values of R u/H are gener-

ally observed in the same H/L and H/d ranges that minimum stability was

obtained, i.e., Ns is a minimum and R u/H is a maximum when H/d > 0.90

and 0.06 < H/L < 0.085 . Consistent with nonbreaking wave data (Hudson 1958,

Jackson 1968, Carver and Davidson 1977. Carver 1980) flattening the slope from

IV on 1.5H to IV on 3H generally reduced runup. The general tendency for

runup to decrease at the milder slopes seems reasonable since as the slope

becomes flatter the wave has a longer travel distance to reach a given eleva-

tion and, therefore, a greater opportunity to dissipate energy.

4 4 34. Hudson (1958) found that for nonbreaking wave attack on stone

armor, plots of R u/H versus H/L gave concave curves; i.e., for small

values of H/L , R /H is relatively large and as H/L increases Ru/H'..'"u ..U

18t'.€.. .'.

'CO"

increases to a maximum value and then decreases as H/L continues to in-

crease. Data presented in Plates 6-8 show a trend very similar to that ob-

served by Hudson and the dolos trends evident in Plates 9-11 are similar to

those observed by Carver and Davidson (1977). Recent runup tests on stone

armor by Carver (1980) show a similar dependency between R /H and H/L tou..-that observed in the data presented herein. Maximum observed values of Ru/H

reported herein are generally 10 to 20 percent less than those obtained by

Hudson (1958), Carver and Davidson (1977), and Carver (1980) in nonbreaking

wave tests. This trend seems consistent since (for breaking waves) energy is

dissipated in the turbulence of breaking; therefore less energy is available

to produce runup. It should be noted that for very steep structure slopes,

such as those found in bulkheads and seawalls, breaking wave runup may be

greater than nonbreaking wave runup. This occurs when plunging breakers im-

pact directly on a near-vertical wall and the force of breaking causes a

shooting type of runup on the wall. The dependency of R /H on H/d , evi-u

dent in the present investigation, was not observed in the earlier nonbreaking

wave tests for the following reasons: (a) all nonbreaking wave tests were

conducted in an H/d range of 0.1 to 0.3; therefore H/d was sufficiently

low to preclude a significant effect on wave form, and (b) the range of H/d

was too limited (0.1 < H/d < 0.3) for a subtle trend (even if one were pres-

ent) to be detected.

.( 9..

.5. o

I.'

I!19 -.6,

- .- VXf$ *.%% . .e.

PART V: CONCLUSIONS AND RECOMMENDATIONS

35. Based on the tests and results described herein, in which stone and

dolos armor are used on breakwater trunks and subjected to breaking waves with

a direction of approach of 90 deg, it is concluded that:

a. Armor stability is influenced by wave steepness (H/L), Ursell

Number (L H/d ), relative wave height (H/d), and breakwater%%% slope.

Sb. Effects of H/d L H/d and H/L are more pronounced for'< "- dolos armor.

,- c. In general, minimum stability for each armor type occurred forthe larger values of H/d (H/d > 0.90) , intermediate values of

2 3H/L (0.06 < H/L < 0.085) , and larger values of L H/d

d. Linear Hudson-type data fits generally give a reasonable ap-proximation of N as a function of cot U ; however, the

influences of H/d , H/L , and L 2H/d 3 are strong enough tomerit their consideration in final selection of armor unit .O

7-. weight.

e. Relative wave runup (R /H) is a function of wave steepness(H/L), relative wave height (H/d), type of armor, and break-water slope.

f. Maximum values of R /H were generally observed in the same -UH/L and H/d ranges that minimum stability was obtained. K-.

g. Lower limit K values (i.e., values that will yield a stableDdesign for any combination of wave height, wave period, and

" water depth investigated herein) obtained in this study are2.0 and 15.0 for stone and dolosse, respectively.

36. Based on the above conclusions, it is recommended that armor sta-

bility for breaking waves be presented as a function of wave height, wave

period, and water depth (e.g., Ursell Number). If wave conditions vary sig-

nificantly or their precise combinations cannot be accurately defined, then

the lower limit KD coefficients given in item g above should be used. It

should be noted that the K values of 2.0 and 15.0 (lower limits of the_r -.-

stone and dolos data presented herein) are significantly different from those

presented in the SPM (CERC 1977) and EM 1110-2-2904 (USAOCE 1963).

- 37. It is presumed that conclusions reached as a result of these tests

can be extended to wave spectra, most probably using the period and energy

density associated with the spectral peak (especially for relatively narrow

spectra). Future testing programs are planned to address the more complex

questions of bimodal and directional spectra.

20

, %" .

REFERENCES

Carver, R. D. 1978 (Jun). "Hydraulic Model Tests of Toskane Armor Units,"ETL 1110-2-233, U. S. Army Engineer Waterways Experiment Station, CE, Vicks-burg, Miss.

1980 (Jan). "Effects of First Underlayer Weight on the Stability

of Stone-Armored Rubble-Mound Breakwater Trunks Subjected to Nonbreaking Waveswith No Overtopping; Hydraulic Model Investigation." Technical Report HL-80-1,U. S. Army Engineer Waterways Experiment Station, JE, Vicksburg, Miss. "0

Carver, R. D., and Davidson, D. D. 1977 (Nov). "Dolos Armor Units Used onRubble-Mound Breakwater Trunks Subjected to Nonbreaking Waves with No Over-

S ,topping," Technical Report H-77-19, U. S. Army Engineer Waterways ExperimentStation, CE, Vicksburg, Miss.

Hudson, R. Y. 1958 (Jul). "Design of Quarry-Stone Cover Layers for Rubble-Mound Breakwaters; Hydraulic Laboratory Investigation," Research Report No.

--'.. 2-2, U. S. Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.

____"____ 1961 (Sep). "Wave Forces on Rubble-Mound Breakwaters and Jet-

ties," Miscellaneous Paper No. 2-453, U. S. Army Engineer Waterways ExperimentStation, CE, Vicksburg, Miss.

__.._____ 1975 (Jun). "Reliability of Rubble-Mound Breakwater StabilityModels; Hydraulic Model Investigation," Miscellaneous Paper H-75-5, U. S. ArmyEngineer Waterways Experiment Station, CE, Vicksburg, Miss.

Jackson, R. A. 1968 (Jun). "Design of Cover Layers for Rubble-Mound Break-waters Subjected to Nonbreaking Waves; Hydraulic Laboratory Investigation,"Research Report No. 2-11, U. S. Army Engineer Waterways Experiment Station,CE, Vicksburg, Miss.

4. U. S. Army Coastal Engineering Research Center, CE. 1977. Shore Protection _.- -Manual, Fort Belvoir, Va.

U. S. Army, Office, Chief of Engineers. 1963 (Apr). "Engineering and Design;Design of Breakwaters and Jetties," EM 1110-2-2904, WashingLon, D. C.

Veo

VS

-e,-

. ,+,,-. ., : .... *-.., . ...-. .-..- **....*.*..,..............-- 1 -

Table I

.Vaues of H d/L, H/L, H/d, L2H/d 3, and N for Two Layers of

S

Stone Armor Randomly Placed on Breakwater Trunks and Subjected

to Breaking Waves with No Overtopping: W = 0.38, 0.55,a

and 0.71 lb; ya = 167 pcf; cot u = 1.5, 2, and 3

aS

W lb H ft 2 3 Na dft Tse D=' d/L H/L H/d L H/d s

cot U = 1.5

0.38 0.45 1.07 0.33 0.12 0.088 0.73 50.9 1.500.38 0.55 1.04 0.35 0.14 0.089 0.64 32.5 1.59

0.55 0.40 1.45 0.37 0.08 0.074 0.93 144.5 1.480.55 0.55 1.18 0.38 0.12 0.083 0.69 48.0 1.52

0.55 0.60 1.09 0.40 0.14 0.093 0.67 34.0 1.60

0.71 0.40 1.90 0.42 0.06 0.063 1.05 291.7 1.550.71 0.40 2.82 0.42 0.04 0.042 1.05 656.3 1.550.71 0.50 1.32 0.42 0.10 0.084 0.84 84.0 1.55

cot C( = 2.0

0.38 0.50 1.13 0.41 0.12 0.098 0.82 56.9 1.860.38 0.55 1.18 0.38 0.12 0.083 0.69 48.0 1.72

0.38 0.60 1.09 0.40 0.14 0.093 0.67 34.0 1.810.55 0.40 2.82 0.42 0.04 0.042 1.05 656.3 1.68

0.55 0.50 1.32 0.42 0.10 0.084 0.84 84.0 1.68

0.55 0.60 1.24 0.45 0.12 0.090 0.75 52.1 1.800.55 0.65 1.13 0.46 0.14 0.099 0.71 36.1 1.84

0.71 0.45 2.02 0.46 0.06 0.061 1.02 284.0 1 .69

0.71 0.65 1.29 0.51 0.12 0.094 0.78 54.5 1.88

cot u =3.0

0.38 0.40 2.82 0.42 0.04 0.042 1.05 656.3 1.90

0.38 0.60 1.24 0.45 0.12 0.090 0.75 52.1 2.040.38 0.65 1.13 0.46 0.14 0.099 0.71 36.1 2.09

-. 0.55 0.45 2.02 0.46 0.06 0.061 1.02 284.0 1.84

0.55 0.60 1.45 0.52 0.10 0.087 0.87 86.7 2.09

0.55 0.65 1.29 0.51 0.12 0.094 0.78 54.5 2.04

0.55 0.75 1.38 0.55 0.12 0.088 0.73 50.9 2.21

".1%~P.

.'&S.

' " " " %-2.:.--.. .... ..'2.-.2-,...-..---..-.............-"...,.......i...-...-.....................-.....,2. .-. -..

47,.

Table 2

H2 3Vaues of HD0 d/L, H/L, H/d, L H/d and N for Two Layers of

Dolos Armor Randomly Placed on Breakwater Trunks and Subjected .:

to Breaking Waves with No Overtopping: W = 0.234, 0.276,

and 0.589 lb; cot a = 1.5, 2, and 3

W-lb H ft 2 3 N

a) d, ft T, sec HD=O' d/L H/L H/d L H/d s

cot o' = 1.5

0.276 0.45 2.02 0.46 0.06 0.061 1.02 284.0 2.88 14O.

0.276 0.50 1.62 0.45 0.08 0.072 0.90 140.6 2.820.589 0.65 1.85 0.60 0.08 0.074 0.92 144.2 2.950.589 0.85 1.73 0.71 0.10 0.084 0.84 83.5 3.500.589 0.90 1.78 0.77 0.10 0.086 0.86 85.6 3.79

cot a 2.0

0.234 0.45 2.02 0.46 0.06 0.061 1.02 284.0 3.190.276 0.55 1.70 0.54 0.08 0.079 0.98 153.4 3.390.276 0.85 1.30 0.56 0.14 0.092 0.66 33.6 3.510.276 0.85 1.47 0.63 0.12 0.089 0.74 51.5 3.95 S0.276 0.95 1.37 0.61 0.14 0.090 0.64 32.8 3.82

cot a = 3.0

0.234 0.70 1.34 0.55 0.12 0.094 0.79 54.6 3.82 O0.234 0.80 1.43 0.55 0.12 0.083 0.69 47.7 3.82,0.234 0.85 1.30 0.56 0.14 0.092 0.66 33.6 3.890.276 0.60 2.32 0.58 0.06 0.058 0.97 268.5 3.640.276 0.65 1.85 0.60 0.08 0.074 0.92 144.2 3.76

0.276 0.90 1.52 0.64 0.12 0.085 0.71 49.4 4.010.276 0.95 1.56 0.66 0.12 0.083 0.69 48.2 4.14 ,

NZ.a -.. •_____•___ ___

,a -,.-.1% %' .S

*..7-. p* 7- ' 7-; ' -

r .D

Table 3

Wave Runup (Ru) Data for Quarrystone Armor Randomly Placed

on Breakwater Trunks and Subjected to Breaking Waves with

No Overtopping: cot U = 1.5, 2.0, and 3.0

Ru , ft, for Test. ¢ , Standard

d, ft T, sec H, ft 1 2 3 4 5 Average Deviation, ft

-'- 0.40cot u = 1.5

0.40 0.89 0.27 0.21 0.23 0.22 0.22 0.24 0.22 0.0120.40 1.01 0.33 0.31 0.27 0.29 0.30 0.29 0.29 0.0150.40 1.18 0.36 0.34 0.36 0.35 0.33 0.37 0.35 0.0160.40 1.45 0.37 0.37 0.40 0.40 0.42 0.40 0.40 0.0180.40 1.90 0.42 0.50 0.45 0.44 0.48 0.42 0.46 0.032

0.40 2.82 0.42 0.40 0.38 0.38 0.42 0.40 0.40 0.0170.45 0.94 0.31 0.31 0.26 0.22 0.28 0.25 0.26 0.0340.45 1.07 0.33 0.32 0.32 0.34 0.30 0.35 0.33 0.0200.45 1.26 0.39 0.33 0.36 0.36 0.37 0.35 0.35 0.0160.50 0.99 0.30 0.32 0.30 0.30 0.28 0.30 0.30 0.014

0.50 1.13 0.41 0.38 0.38 0.36 0.35 0.38 0.37 0.0140.50 1.32 0.42 0.38 0.38 0.42 0.42 0.38 0.40 0.0220.55 1.04 0.35 0.32 0.33 0.31 0.35 0.33 0.33 0.0150.55 1.18 0.38 0.42 0.42 0.42 0.38 0.42 0.41 0.0180.60 1.09 0.40 0.38 0.41 0.41 0.37 0.39 0.39 0.018

cot u = 2.0

0.40 0.89 0.27 0.21 0.18 0.17 0.20 0.17 0.19 0.0190.40 1.01 0.33 0.20 0.19 0.18 0.22 0.19 0.20 0.0160.40 1.18 0.36 0.23 0.24 0.23 0.26 0.24 0.24 0.0120.40 1.45 0.37 0.28 0.28 0.32 0.28 0.29 0.29 0.0170.40 1.90 0.42 0.36 0.29 0.34 0.36 0.39 0.35 0.037

0.40 2.82 0.42 0.28 0.32 0.29 0.30 0.31 0.30 0.0160.45 0.94 0.31 0.17 0.17 0.18 0.16 0.17 0.17 0.0070.45 1.07 0.33 0.21 0.23 0.23 0.25 0.19 0.22 0.0230.45 1.26 0.39 0.28 0.26 0.28 0.30 0.29 0.28 0.0150.45 1.54 0.44 0.36 0.35 0.37 0.33 0.34 0.35 0.016

0.45 2.02 0.46 0.32 0.34 0.39 0.37 0.40 0.36 0.034

0.50 0.99 0.30 0.15 0.18 0.20 0.21 0.18 0.18 0.0230.50 1.13 0.41 0.29 0.23 0.23 0.22 0.23 0.24 0.028

l ' 0.50 1.32 0.42 0.26 0.27 0.27 0.28 0.27 0.27 0.0070.50 1.62 0.45 0.38 0.37 0.39 0.39 0.38 0.38 0.009

0.55 1.04 0.35 0.22 0.20 0.22 0.24 0.23 0.22 0.015

0.55 1.18 0.38 0.23 0.27 0.26 0.27 0.26 0.26 0.0170.55 1.39 0.45 0.34 0.35 0.32 0.29 0.35 0.33 0.025

(Con t. i nued)

-- M ...... ..q~ V M ~.

Table 3 (Concluded)

Ru, ft, for Test Standard

d, ft T, sec H, ft 1 2 3 4 5 Average Deviation, ft .

cot a = 2.() (Continued)

0.60 1.09 0.40 0.25 0.28 0.26 0.29 0.28 0.27 0.0170.60 1.24 0.45 0.27 0.36 0.28 0.30 0.30 0.30 0.0350.65 1.13 0.46 0.28 0.31 0.34 0.31 0.28 0.30 0.0250.65 1.29 0.51 0.33 0.38 0.33 0.33 0.34 0.34 0.0220.70 1.18 0.46 0.31 0.32 0.33 0.32 0.35 0.33 0.016

cot a = 3.0

0.40 0.89 0.27 0.14 0.11 0.14 0.14 0.11 0.13 0.0170.40 1.01 0.33 0.18 0.19 0.18 0.18 0.19 0.18 0.0070.40 1.18 0.36 0.20 0.22 0.21 0.22 0.22 0.21 0.0100.40 1.45 0.37 0.23 0.22 0.24 0.23 0.21 0.23 0.0120.40 1.90 0.42 0.31 0.24 0.30 0.28 0.25 0.28 0.031

0.40 2.82 0.42 0.29 0.27 0.28 0.28 0.29 0.28 0.0090.45 0.94 0.31 0.14 0.16 0.14 0.16 0.16 0.15 0.0110.45 1.07 0.33 0.19 0.19 0.18 0.17 0.17 0.18 0.0100.45 1.26 0.39 0.23 0.25 0.22 0.18 0.19 0.21 0.029 --.0.45 1.54 0.44 0.26 0.27 0.27 0.26 0.24 0.26 0.012

0.45 2.02 0.46 0.32 0.32 0.35 0.33 0.33 0.33 0.0120.50 0.99 0.30 0.15 0.17 0.15 0.15 0.16 0.16 0.0100.50 1.13 0.41 0.21 0.19 0.21 0.20 0.20 0.20 0.0090.50 1.32 0.42 0.21 0.23 0.21 0.25 0.23 0.23 0.0170.50 1.62 0.45 0.28 0.28 0.27 0.25 0.29 0.27 0.016

0.55 1.04 0.35 0.19 0.18 0.15 0.20 0.17 0.18 0.0190.55 1.18 0.38 0.19 0.26 0.19 0.25 0.22 0.22 0.0330.55 1.39 0.45 0.30 0.31 0.34 0.30 0.31 0.31 0.0170.55 1.70 0.54 0.38 0.37 0.38 0.39 0.37 0.38 0.0090.0 . .0 •2 00.60 1.09 0.40 0.24 0.23 0.22 0.23 0.20 0.22 0.016 .- '

0.60 1.24 0.45 0.22 0.25 0.27 0.29 0.26 0.26 0.0260.60 1.45 0.52 0.34 0.35 0.33 0.34 0.33 0.34 0.0090.65 1.13 0.46 0.26 0.26 0.24 0.26 0.26 0.26 0.0100.65 1.29 0.51 0.29 0.30 0.28 0.31 0.30 0.30 0.0120.70 1.18 0.46 0.30 0.31 0.27 0.31 0.31 0.30 0.017

0.70 1.34 0.55 0.36 0.35 0.37 0.38 0.35 0.36 0.0130.75 1.22 0.44 0.28 0.26 0.27 0.28 0.28 0.27 0.010 pO0.75 1.38 0.55 0.35 0.33 0.36 0.37 0.34 0.35 0.016

C~."O

C',-,' ..- ,:.'.,.:.. -',-,.,,--,% ...- '.,.. ... . -.. ... ,:.-v ", ,: < -- '. . :

4.V.

Table 4

Wave Runup (Re) Data for Dolos Armor Randomly Placed on

Breakwater Trunks and Subjected to Breaking Waves

with No Overtopping: cot a = 1.5, 2.0, and 3.0

Ru , ft, for Test Standard

d, ft T, sec H, ft 1 2 3 4 5 Average Deviation, ft

cot o= 1.5

0.40 1.45 0.37 0.25 0.24 0.26 0.22 0.26 0.25 0.017

0.40 1.90 0.42 0.32 0.32 0.32 0.32 0.32 0.32 0.0000.40 2.82 0.42 0.30 0.30 0.27 0.30 0.31 0.30 0.016 '0.45 0.94 0.31 0.19 0.18 0.16 0.16 0.19 0.18 0.0160.45 1.26 0.39 0.27 0.27 0.24 0.28 0.27 0.27 0.016

0.45 1.54 0.44 0.33 0.34 0.34 0.32 0.35 0.34 0.0120.45 2.02 0.46 0.35 0.35 0.35 0.35 0.35 0.35 0.0000.50 0.99 0.30 0.20 0.20 0.17 0.21 0.22 0.20 0.0190.50 1.32 0.42 0.28 0.26 0.27 0.28 0.30 0.28 0.0150.50 1.62 0.45 0.38 0.35 0.37 0.38 0.39 0.37 0.016

0.55 1.04 0.35 0.23 0.23 0.23 0.24 0.24 0.23 0.0070.55 1.18 0.38 0.30 0.26 0.28 0.28 0.27 0.28 0.0150.55 1.70 0.54 0.44 0.44 0.43 0.37 0.40 0.42 0.0310.60 1.09 0.40 0.32 0.28 0.27 0.28 0.26 0.28 0.0230.65 1.13 0.46 0.32 0.34 0.30 0.31 0.30 0.31 0.017

0.65 1.29 0.51 0.41 0.40 0.38 0.38 0.40 0.39 0.0140.65 1.85 0.60 0.59 0.58 0.60 0.56 0.57 0.58 0.0160.70 1.18 0.46 0.33 0.32 0.33 0.34 0.33 0.33 0.0070.75 1.99 0.70 0.66 0.68 0.66 0.62 0.64 0.65 0.0230.80 1.67 0.66 0.53 0.55 0.59 0.58 0.56 0.56 0.024

0.85 1.73 0.71 0.64 0.60 0.62 0.60 0.60 0.61 0.018-- 0.90 1.78 0.77 0.58 0.62 0.61 0.60 0.62 0.61 0.017

cot c = 2.0

-4 0.40 1.45 0.37 0.26 0.24 0.24 0.23 0.26 0.25 0.0140.40 1.90 0.42 0.33 0.31 0.30 0.31 0.30 0.31 0.0120.40 2.82 0.42 0.26 0.30 0.27 0.29 0.29 0.28 0.0170.45 0.94 0.31 0.17 0.17 0.17 0.17 0.16 0.17 0.0050.45 1.26 0.39 0.25 0.26 0.25 0.23 0.23 0.24 0.014

0.45 1.54 0.44 0.33 0.30 0.31 0.30 0.32 0.31 0.0130.45 2.02 0.46 0.36 0.37 0.35 0.35 0.34 0.35 0.0120.50 0.99 0.30 0.21 0.18 0.20 0.18 0.20 0.19 0.0140.50 1.32 0.42 0.27 0.27 0.26 0.25 0.23 0.26 0.0170.50 1.62 0.45 0.34 0.37 0.34 0.37 0.33 0.35 0.019

(n u(Continued) ""

o% ,

,P . :' . '. : -'.-..'.'.':';.".' " " " "" .'-... ' .'.'.'_.''.' %'. ¢ '. . .'- .'-* " -"- -"-4.-

I

Table 4 (Concluded)

Ru, ft, for Test Standard

d, ft T H ft 1 2 3 4 5 Average Deviation, ft

cot o , 2.0 (Continued)

.4

0.55 1.04 0.35 0.23 0.24 0.22 0.22 0.23 0.23 0.0090.55 1.18 0.38 0.25 0.26 0.27 0.26 0.27 0.26 0.0090.55 1.70 0.54 0.36 0.39 0.38 0.39 0.36 0.38 0.016

. 0.60 1.09 0.40 0.25 0.27 0.25 0.23 0.27 0.25 0.0170.65 1.13 0.46 0.25 0.24 0.24 0.25 0.25 0.25 0.007

0.65 1.29 0.51 0.31 0.31 0.33 0.33 0.33 0.32 0.011

0.70 1.18 0.46 0.27 0.30 0.32 0.29 0.27 0.29 0.0210.80 1.43 0.55 0.40 0.41 0.43 0.42 0.45 0.42 0.0190.85 1.30 0.56 0.43 0.40 0.37 0.38 0.40 0.40 0.0230.85 1.47 0.44 0.47 0.47 0.48 0.45 0.45 0.46 0.017

C". 0.95 1.37 0.61 0.42 0.42 0.41 0.42 0.40 0.41 0.0100.95 1.56 0.66 0.50 0.51 0.50 0.50 0.51 0.50 0.007

4. cot a = 3.0

0.40 1.45 0.37 0.18 0.20 0.19 0.22 0.20 0.20 0.0150.40 1.90 0.42 0.29 0.28 0.28 0.28 0.28 0.28 0.0050.40 2.82 0.42 0.27 0.27 0.25 0.28 0.28 0.27 0.0120.45 0.94 0.31 0.14 0.15 0.12 0.15 0.15 0.14 0.0130.45 1.26 0.39 0.22 0.23 0.19 0.20 0.22 0.21 0.017

0.45 1.54 0.44 0.27 0.26 0.25 0.24 0.26 0.26 0.0120.45 2.02 0.46 0.28 0.28 0.26 0.28 0.28 0.28 0.0100.50 0.99 0.30 0.18 0.17 0.16 0.17 0.17 0.17 0.0070.50 1.32 0.42 0.24 0.23 0.23 0.22 0.24 0.23 0.0090.50 1.62 0.45 0.32 0.30 0.28 0.31 0.30 0.30 0.015

0.55 1.04 0.35 0.20 0.21 0.21 0.20 0.21 0.21 0.0070.55 1.18 0.38 0.25 0.25 0.23 0.22 0.24 0.24 0.0130.55 1.70 0.54 0.31 0.32 0.31 0.31 0.32 0.31 0.0070.60 1.09 0.40 0.22 0.23 0.23 0.22 0.21 0.22 0.0090.65 1.13 0.46 0.27 0.25 0.24 0.26 0.27 0.26 0.013

0.65 1.29 0.51 0.28 0.30 0.27 0.27 0.28 0.28 0.0120.65 1.51 0.57 0.30 0.36 0.34 0.36 0.37 0.35 0.0280.65 1.85 0.60 0.38 0.41 0.40 0.40 0.42 0.40 0.0150.70 1.18 0.46 0.29 0.27 0.28 0.28 0.26 0.28 0.0120.70 1.57 0.63 0.34 0.38 0.39 0.37 0.35 0.37 0.021

0.80 1.43 0.55 0.34 0.36 0.34 0.35 0.33 0.34 0.0120.8 1 • 0 0..0.85 1.30 0.56 0.33 0.34 0.34 0.35 0.33 0.34 0.0090.85 1.47 0.63 0.38 0.39 0.39 0.37 0.39 0.38 0.0100.90 1.52 0.64 0.38 0.40 0.40 0.37 0.36 0.38 0.0180.95 1.37 0.61 0.31 0.35 0.35 0.35 0.33 0.34 0.018,Oo

0.95 1.56 0.66 0.35 0.38 0.40 0.40 0.40 0.39 0.022

44 4%

* - - --U -*3 b77 7 . 7- 5717 .

Table 5

Values of Relative Runup (R /H) for Quarrystone Armor Randomly

Placed on Breakwater Trunks and Subjected to Breaking

Waves with No Overtopping: cot a = 1.5, 2.0, and 3.0

R t R/ Standard "2id, ft d/L T, sec H, ft H/L H/d Ru ft R / DevStnr

.,- ___ ___ __ u'u Deviation, ft .O

cot u = 1.5

0.40 0.14 0.89 0.27 0.094 0.68 0.22 0.81 0.04 j0.40 0.12 1.01 0.33 0.099 0.83 0.29 0.88 0.050.40 0.10 1.18 0.36 0.090 0.90 0.35 0.97 0.040.40 0.08 1.45 0.37 0.074 0.93 0.40 1.08 0.050.40 0.06 1.90 0.42 0.063 1.05 0.46 1.10 0.08

0.40 0.04 2.82 0.42 0.042 1.05 0.40 0.95 0.04

0.45 0.14 0.94 0.31 0.097 0.69 0.26 0.84 0.110.45 0.12 1.07 0.33 0.088 0.73 0.33 1.00 0.06

, 0.45 0.10 1.26 0.39 0.087 0.87 0.35 0.90 0.04

0.50 0.14 0.99 0.30 0.084 0.60 0.30 1.00 0.05

0.50 0.12 1.13 0.41 0.098 0.82 0.37 0.90 0.03

0.50 0.10 1.32 0.42 0.084 0.84 0.40 0.95 0.050.55 0.14 1.04 0.35 0.089 0.64 0.33 0.94 0.040.55 0.12 1.18 0.38 0.083 0.69 0.41 1.08 0.050.60 0.14 1.09 0.40 0.093 0.67 0.39 0.98 0.05

cot a = 2.0

0.40 0.14 0.89 0.27 0.094 0.68 0.19 0.70 0.07

0.40 0.12 1.01 0.33 0.099 0.83 0.20 0.61 0.050.40 0.10 1.18 0.36 0.090 0.90 0.24 0.67 0.030.40 0.08 1.45 0.37 0.074 0.93 0.29 0.78 0.050.40 0.06 1.90 0.42 0.063 1.05 0.35 0.83 0.09

0.40 0.04 2.82 0.42 0.042 1.05 0.30 0.71 0.04

- ' 0.45 0.14 0.94 0.31 0.097 0.69 0.17 0.55 0.02

0.45 0.12 1.07 0.33 0.088 0.73 0.22 0.67 0.07

" 0.45 0.10 1.26 0.39 0.087 0.87 0.28 0.72 0.04

0.45 0.08 1.54 0.44 0.078 0.98 0.35 0.80 0.04

0.45 0.06 2.02 0.46 0.061 1.02 0.36 0.78 0.070.50 0.14 0.99 0.30 0.084 0.60 0.18 0.60 0.080.50 0.12 1.13 0.41 0.098 0.82 0.24 0.59 0.07

0.50 0.10 1.32 0.42 0.084 0.84 0.27 0.64 0.02

0.50 0.08 1.62 0.45 0.072 0.90 0.38 0.84 0.02

0.55 0.14 1.04 0.35 0.089 0.64 0.22 0.63 0.04

0.55 0.12 1.18 0.38 0.083 0.69 0.26 0.68 0.040.55 0.10 1.39 0.45 0.082 0.82 0.33 0.73 0.06

0.60 0.14 1.09 0.40 0.093 0.67 0.27 0.68 0.04

0.60 0.12 1.24 0.45 0.090 0.75 0.30 0.67 0.08

(Continued)

..

-:, ..-- %%%~ $. %

..-. :::1:

Table 5 (Concluded)

R t" / Standard"-",

d, ft d/L sec H ft H/L H/d Ru, ft R /H DStn____ ___ ____ u Deviation, ft 0O

cot u = 2.0 (Continued)

0.65 0.14 1.13 0.46 0.099 0.71 0.30 0.65 0.050.65 0.12 1.29 0.51 0.094 0.78 0.34 0.67 0.040.70 0.14 1.18 0.46 0.092 0.66 0.33 0.72 0.03 O

cot cu = 3.0

0.40 0.14 0.89 0.27 0.094 0.68 0.13 0.48 0.060.40 0.12 1.01 0.33 0.099 0.83 0.18 0.55 0.020.40 0.10 1.18 0.36 0.090 0.90 0.21 0.58 0.030.40 0.08 1.45 0.37 0.074 0.93 0.23 0.62 0.030.40 0.06 1.90 0.42 0.063 1.05 0.28 0.67 0.07

0.40 0.04 2.82 0.42 0.042 1.05 0.28 0.67 0.020.45 0.14 0.94 0.31 0.097 0.69 0.15 0.48 0.04

-. 0.45 0.12 1.07 0.33 0.088 0.73 0.18 0.55 0.030.45 0.10 1.26 0.39 0.087 0.87 0.21 0.54 0.070.45 0.08 1.54 0.44 0.078 0.98 0.26 0.59 0.03

0.45 0.06 2.02 0.46 0.061 1.02 0.33 0.72 0.030.50 0.14 0.99 0.30 O.OR4 0.60 0.16 0.53 0.030.50 0.12 1.13 0.41 0.098 0.82 0.20 0.49 0.020.50 0.10 1.32 0.42 0.084 0.84 0.23 0.55 0.040.50 0.08 1.62 0.45 0.072 0.90 0.27 0.60 0.04

- 0.55 0.14 1.04 0.35 0.089 0.64 0.18 0.51 0.05" 0.55 0.12 1.18 0.38 0.083 0.69 0.22 0.58 0.09

0.55 0.10 1.39 0.45 0.082 0.82 0.31 0.69 0.040.55 0.08 1.70 0.54 0.078 0.98 0.38 0.70 0.020.60 0.14 1.09 0.40 0.093 0.67 0.22 0.55 0.04

0.60 0.12 1.24 0.45 0.090 0.75 0.26 0.58 0.060.60 0.10 1.45 0.52 0.087 0.87 0.34 0.65 0.02 "..0.65 0.14 1.13 0.46 0.099 0.71 0.26 0.57 0.020.65 0.12 1.29 0.51 0.094 0.78 0.30 0.59 0.020.70 0.14 1.18 0.46 0.092 0.66 0.30 0.65 0.04

0.70 0.12 1.34 0.55 0.094 0.79 0.36 0.65 0.020.75 0.14 1.22 0.44 0.082 0.59 0.27 0.61 0.02

- 0.75 0.12 1.38 0.55 0.088 0.73 0.35 0.64 0.03

'..°

..

Table 6

Values of Relative Runup (R u/H) for Dolos Armor Randomly Placed

on Breakwater Trunks and Subjected to Breaking Waves 0

with No Overtopping: cot a = 1.5, 2.0, and 3.0

. ft-R/HStandard

d, ft d/L T, sec H, ft H/L H/d Ru u Deviation, f

.%,cot U¢ = 1.5

- 0.40 0.08 1.45 0.37 0.074 0.93 0.25 0.68 0.050.40 0.06 1.90 0.42 0.063 1.05 0.32 0.76 0.000.40 0.04 2.82 0.42 0.042 1.05 0.30 0.71 0.040.45 0.14 0.94 0.31 0.097 0.69 0.18 0.58 0.050.45 0.10 1.26 0.39 0.087 0.87 0.27 0.69 0.04

N 0.45 0.08 1.54 0.44 0.078 0.98 0.34 0.77 0.030.45 0.06 2.02 0.46 0.061 1.02 0.35 0.76 0.000.50 0.14 0.99 0.30 0.084 0.60 0.20 0.67 0.060.50 0.10 1.32 0.42 0.084 0.84 0.28 0.67 0.040.50 0.08 1.62 0.45 0.072 0.90 0.37 0.82 0.04

0.55 0.14 1.04 0.35 0.089 0.64 0.23 0.66 0.020.55 0.12 1.18 0.38 0.083 0.69 0.28 0.74 0.040.55 0.08 1.70 0.54 0.078 0.98 0.42 0.78 0.060.60 0.14 1.09 0.40 0.093 0.67 0.28 0.70 0.060.65 0.14 1.13 0.46 0.099 0.71 0.31 0.67 0.04

0.65 0.12 1.29 0.51 0.094 0.78 0.39 0.76 0.030.65 0.08 1.85 0.60 0.074 0.92 0.58 0.97 0.030.70 0.14 1.18 0.46 0.092 0.66 0.33 0.72 0.020.75 0.08 1.99 0.70 0.075 0.93 0.65 0.93 0.030.80 0.10 1.67 0.66 0.083 0.83 0.56 0.85 0.040.85 0.10 1.73 0.71 0.084 0.84 0.61 0.86 0.03

% 0.90 0.10 1.78 0.77 0.086 0.86 0.61 0.79 0.02

cot u = 2.0

0.40 0.08 1.45 0.37 0.074 0.93 0.25 0.68 0.040.40 0.06 1.90 0.42 0.063 1.05 0.31 0.74 0.030.40 0.04 2.82 0.42 0.042 1.05 0.28 0.67 0.04

.- 0.45 0.14 0.94 0.31 0.097 0.69 0.17 0.55 0.020.45 0.10 1.26 0.39 0.087 0.87 0.24 0.62 0.04

0.45 0.08 1.54 0.44 0.078 0.98 0.31 0.70 0.030.45 0.06 2.02 0.46 0.061 1.02 0.35 0.76 0.030.50 0.14 0.99 0.30 0.084 0.60 0.19 0.63 0.050.50 0.10 1.32 0.42 0.084 0.84 0.26 0.62 0.040.50 0.08 1.62 0.45 0.072 0.90 0.35 0.78 0.04

(Continued)

"0,!

4,.,% -,. ,% ..... ,., . . .. . . -... , . .. . .. . ¢ -.. : .- -.....-. .... ,... . - . - .. - . . ...

Table 6 (Concluded)

Standardd, ft d/L T, sec H, ft H/L H/d Ruft R Deviation, ft

cot a = 2.0 (Continued)

* . 0.55 0.14 1.04 0.35 0.089 0.64 0.23 0.66 0.03. 0.55 0.12 1.18 0.38 0.083 0.69 0.26 0.68 0.02

0.55 0.08 1.70 0.54 0.078 0.98 0.38 0.70 0.03 0

0.60 0.14 1.09 0.40 0.093 0.67 0.25 0.63 0.040.65 0.14 1.13 0.46 0.099 0.71 0.25 0.54 0.02

0.65 0.12 1.29 0.51 0.094 0.78 0.32 0.63 0.020.70 0.14 1.18 0.46 0.092 0.66 0.29 0.63 0.050.80 0.12 1.43 0.55 0.082 0.69 0.42 0.76 0.030.85 0.14 1.30 0.56 0.092 0.66 0.40 0.71 0.04 O0.85 0.12 1.47 0.63 0.089 0.74 0.46 0.73 0.03

0.95 0.14 1.37 0.61 0.090 0.64 0.41 0.67 0.020.95 0.12 1.56 0.66 0.083 0.69 0.50 0.76 0.01

cot c = 3.0

0.40 0.08 1.45 0.37 0.074 0.93 0.20 0.54 0.040.40 0.06 1.90 0.42 0.063 1.05 0.28 0.67 0.01

, 0.40 0.04 2.82 0.42 0.042 1.05 0.27 0.64 0.030.45 0.14 0.94 0.31 0.097 0.69 0.14 0.45 0.040.45 0.10 1.26 0.39 0.087 0.87 0.21 0.54 0.04

0.45 0.08 1.54 0.44 0.078 0.98 0.26 0.59 0.03 "°-V., 0.45 0.06 2.02 0.46 0.061 1.02 0.28 0.61 0.02

0.50 0.14 0.99 0.30 0.084 0.60 0.17 0.57 0.02 'a-"0.50 0.10 1.32 0.42 0.084 0.84 0.23 0.55 0.020.50 0.08 1.62 0.45 0.072 0.90 0.30 0.67 0.03

0.55 0.14 1.04 0.35 0.089 0.64 0.21 0.60 0.020.55 0.12 1.18 0.38 0.083 0.69 0.24 0.63 0.030.55 0.08 1.70 0.54 0.078 0.98 0.31 0.57 0.010.60 0.14 1.09 0.40 0.093 0.67 0.22 0.55 0.020.65 0.14 1.13 0.46 0.099 0.71 0.26 0.57 0.03

0.65 0.12 1.29 0.51 0.094 0.78 0.28 0.55 0.020.65 0.10 1.51 0.57 0.088 0.88 0.35 0.61 0.050.65 0.08 1.85 0.60 0.074 0.92 0.40 0.67 0.030.70 0.14 1.18 0.46 0.092 0.66 0.28 0.61 0.030.70 0.10 1.57 0.63 0.090 0.90 0.37 0.59 0.03

0.80 0.12 1.43 0.55 0.082 0.69 0.34 0.62 0.020.85 0.14 1.30 0.56 0.092 0.66 0.34 0.61 0.02

A 0.85 0.12 1.47 0.63 0.089 0.74 0.38 0.60 0.020.90 0.12 1.52 0.64 0.085 0.71 0.38 0.59 0.03

. 0.95 0.14 1.37 0.61 0.090 0.64 0.34 0.56 0.03 -.

gD 0.95 0.12 1.56 0.66 0.083 0.69 0.39 0.59 0.03

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55 BITE OFSTONE- AND DOLOS-ARMORED RUBBLE-HOUNDEEN-14iF 54 REASANBT ER UNKS U S U AA ENIERING

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PLATE 17 £ell

APPENDIX A: NOTATION

A Surface area, ft2

d Water depth, ft

d/L Relative depth

D Damage parameter

Reads "function of"

F Force

g Acceleration due to gravity, ft/sec2

h Height of breakwater crown, ft

H Wave height, ft

H/d Relative wave height

H/L Wave steepness

kA Shape coefficient

KD Stability coefficient

.9 a Characteristic length of armor unit, ft

L Length, wavelength, ft

n Number of layers of armor units

N Number of armor units2 3N Ursell Number = (L H/d )

P Porosity of breakwater material, percent

PT Placement techniqueRN Reynolds stability number g 1/2H1/2 )/vL'

R /H Relative runupU

Ru Wave runup measured vertically above swl, ft

S Specific gravity of an armor unit relative to water in which thea breakwater is constructed

T Wave period, sec; time

W Weight, lb

a Angle of breakwater slope, measured from horizontal, deg

' cot a Reciprocal of breakwater slope

Angle of wave attack, deg 0

y Specific weight, pcf

Specific weight of an armor unit, pcfA Shape of armor unit or underlayer material

AlA1 ,"

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0 Angle between the horizontal and the sea bottom on which the

breakwater is constructed

V Kinematic viscosity

* Subscripts

a Refers to armor unit

s Refers to stability

w Refers to water in which the structure is located

I and 2 Refers to underlayer and core, respectively

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