M 1795
A2 90.08
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opdrachtgever:
Rijkswaterstaat
Dienst Weg- en Waterbouwkunde TAW-A2
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taludbekleding van gezette steen
O O T) O
band 1
waterbeweging en golfbelasting op een glad talud
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deel XVII
maart 1990
O Oo
GRONDMECHANICA
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° B I B L l O P H E E K
Dienst Weg- en Waterbouwkunde Postbus 5044, 2600 GA DELFT
"
8M E l f 9 9 0
taludbekleding van gezette steen
band 1
waterbeweging en golfbelasting op een glad talud
M. Klein Breteler
INHOUD
BAND 1:
SEKTIE 1 : Overzicht van onderzoeksresultaten
SEKTIE 2 : Evaluation of measurements of the wave pressures on a slope. Large scale tests in Delta flume
SEKTIE 3 : Evaluation of measurements of the wave pressures on a slope. Small scale tests in Schelde flume
SEKTIE 4 : Evaluation of the wave pressure on a slope Re-evaluation of Schelde flume investigations
SEKTIE 5 : Eindverifikatie onderzoek Deltagoot Analyse van gemeten stijghoogte op talud
SEKTIE 6 : Invloed schaalfactor doorlatendheid, ruwheid en taludhelling op de golfdrukken op een talud
BAND 2:
SEKTIE 7 : Wave pressures on a slope generated by orthogonal and oblique wave attack
SEKTIE 8 : Scheve golfaanval op taluds bij regelmatige golven
SEKTIE 9 : Invloed van scheve golfinval op de golfoploop van steile taluds ten opzichte van loodrechte inval
SEKTIE 10; Grootschalige golfoploop bij voor reflektie gekompenseerde golfaanval
SEKTIE 11: Numerical simulation of gravity wave motion on steep slopes
SEKTIE 1
INHOUD
blz.
1. Inleidinfi 1
2. Stijghoogte op talud als gevolfi van loodrechte Rolfaanval 3
3. Invloed van scheve Rolfaanval 6
-1-SEKTIE 1: Overzicht van onderzoeksresultaten
1. Inleiding
De hydraulische belasting op een taludbekleding van gezette steen onder gol-faanval bestaat uit een in de plaats en in de tijd wisselende druk op de zetting en een stroming over het talud. In deel VIII uit deze verslagenreeks over steenzettingen [1] is gekonkludeerd dat de stabiliteit voornamelijk bedreigd wordt op het moment vóór de golfklap, als er sprake is van maximale golfneerloop. Deze konklusie is van toepassing op het in dit onderzoek cen-traal staande type steenzettingen, dat een relatief open oppervlak heeft van maximaal 15 a 20% en gefundeerd is op een granulair filter. Voor dit type konstrukties geldt dat de kwasi-stationaire druk op het talud vlak vóór de golfklap via het filter wordt doorgegeven naar het deel van de zetting met geringe druk op het talud, alwaar de blokken uit de zetting kunnen worden gedrukt:
.•' '. balts
BREKENDE GOLF
toplaag
3RTPLANTING HOGE DRUK DOOR HET FILTER
Schematische weergave van bezwijkmechanisme
De stroming over het talud heeft geen vat op de individuele stenen in de zetting en leidt daardoor niet tot bezwijken van de zetting.
Het onderhavige verslag geeft een gedetailleerde beschrijving van al het re-levante onderzoek betreffende de kwasi-stationaire druk op het talud (op het moment vlak vóór de golfklap) dat tot en met 1989 is uitgevoerd bij het Waterloopkundig Laboratorium (WL) en Grondmechanica Delft (GD). Dit onder-zoek is uitgevoerd in opdracht van de Dienst Weg- en Waterbouwkunde van
-2-Rijkswaterstaat in het kader van het meerjarige onderzoek naar de stabili-teit van taludbekledingen van gezette steen.
Het verslag is verdeeld in 12 sekties, die elk een afzonderlijk aspekt of een duidelijk te onderscheiden onderzoek behandelen.
In sektie 2 tot en met 5 zijn de verschillende onderzoeken naar de maatge-vende druk op het talud bij loodrecht invallende golven beschreven. In sek-tie 6 is een burostudie gerapporteerd naar de invloed van de modelschaal, de doorlatendheid van de toplaag, de ruwheid en de taludhelling.
De sekties 7 en 8 behandelen de invloed van de hoek van golfinval ten op-zichte van de normaal op de dijk.
Sektie 9 en 10 geven een overzicht van de onderzoeksresultaten met betrek-king tot de golfoploop op steenzettingen.
In sektie 11 is de stand van zaken beschreven van de ontwikkeling van een numeriek model voor het berekenen van de waterbeweging op een talud.
In sektie 12 is een raeetverslag toegevoegd van enkele metingen op een talud van 1:2.5, die nog niet zijn geanalyseerd.
In onderstaande hoofdstukken van sektie 1 is een overzicht gegeven van de konklusies uit de sekties 2 tot en met 12, voor zover ze van belang zijn voor het berekenen van de stabiliteit van een zetting.
-3-2. Stiiehoogte op talud als gevolg van loodrechte golfaanval
De druk op het talud wordt enerzijds bepaald door de waterbeweging, die een gevolg is van de golfaanval, en anderzijds door het nivo ten opzichte van de waterspiegel. Gezien het feit dat dit laatste minder relevant is voor de stabiliteit van de steenzetting en bovendien gemakkelijk tot verwarring kan leiden bij de interpretatie van het begrip "verschildruk over de zetting", wordt in het vervolg gewerkt met de stijghoogte op het talud:
4> = '2- + z (1) Pg
met:
<J) = stijghoogte (m) p = druk op nivo z ten opzichte van stilwaterlijn (Pa) p = soortelijke massa van water (kg/m3)
g = zwaartekrachtsversnelling (m/s2)
z = nivo ten opzichte van de stilwaterlijn (m)
De stijghoogte is uitsluitend afhankelijk van de waterbeweging en niet van de plaats ten opzichte van de waterspiegel.
De stijghoogte op het talud als gevolg van loodrecht invallende golven is onderzocht door middel van een omvangrijk proevenprograir^a, beschreven in sektie 2 en 3.
De analyse van deze proeven leverde bij het schrijven van deze sekties ech-ter grote problemen op, zodat dit in een laech-ter stadium van het onderzoek is overgedaan. Het resultaat hiervan is opgenomen als sektie 4. Met name een veel gedetailleerdere definitie van de relevante parameters en een kritische selektie van de proeven hebben ertoe bijgedragen dat de analyse leidde tot bruikbare empirische formules voor de stijghoogte op het talud.
De maatgevende stijghoogte op het talud is geschematiseerd tot een recht stijghoogte-front met een zekere hoogte <(> en een hellingshoek (ten opzichte van de vertikaal) •{$. De plaats van het front ten opzichte van de stilwater-lijn wordt weergegeven door d :
4
-brekende golf stijghoogte talud
Definitieschets
Voor een gedetailleerde definitie van <(>, , f$ en d wordt verwezen naar hoofd-D S
stuk 3 van sektie 4.
Op basis van modelonderzoek met regelmatige golven en taludhellingen van 1:2, 1:3 en 1:4 zijn de volgende empirische relaties opgesteld (zie sektie 4 ) : ^ = 0 . 3 6 A a n a n , tana • _.,
als
mr -
37
(2) H 0.17 . tana _ _^ alsÏÏJT >
37 (3) (4) de H~ t a n o t/ ^)0-8 o , tana ^ _, alsÏÏ7ÏT -
26 (5)i r -
1-
5 als tanaUIT
26 (6) met:4>. = stijghoogte onder de aankomende golftop, ten opzichte van het punt waar het stijghoogte-front op het talud aansluit f$ = hellingshoek van het stijghoogte-f ront, ten opzichte van
de vertikaal
-5-d = snijpunt van het stijghoogte-front en het talu-5-d, ten opzichte s
van de stilwaterlijn (m) H = hoogte van de inkomende regelmatige golven (m) L = diep water golflengte = gT2/(2ir) (ra)
T = golfperiode (s)
a = taludhelling ten opzichte van de horizontaal (°)
De overgang van formule (2) naar (3) vindt ongeveer plaats als £ gelijk is aan 3 è 3.5. De overgang van formule (5) naar (6) vindt ongeveer plaats als
i is 2.5 è 3.
o
De formules zijn geldig voor de volgende omstandigheden: golfsteilheid : 0.01 < H/L < 0.07
relatieve waterdiepte: 0.05 < h/L < 0.2 en 2.5 < h/H < 10. taludhelling : 0.25 < tana < 0.5
Buiten deze intervallen is de geldigheid nog niet aangetoond.
In sektie 5 is de verifikatie van deze formules met behulp van grootschalig modelonderzoek in de Deltagoot beschreven. De taludhelling bij deze proeven was 1:3, terwijl de golfsteilheid is gevarieerd tussen 0.007 en 0.05.
Er is gekonkludeerd dat de formules goed voldoen. Het verschil tussen de ge-meten en de berekende $, en d blijft beperkt tot 10 a 15 % en voor f$ tot 15 a 20%.
In sektie 6 is op basis van het vergelijken van een groot aantal drukme-tingen bij regelmatige golven gekonkludeerd dat de invloed van de model-schaal, de ruwheid van het taludoppervlak en de doorlatendheid van de top-laag verwaarloosbaar is. Gebleken is dat de invloed van de relatieve water-diepte h/L (h = waterwater-diepte (ra)) niet verwaarloosbaar is, hoewel deze
invloed in het later uitgevoerde onderzoek, dat gerapporteerd is in sektie 2 tot en met 5, niet teruggevonden is. Voorlopig wordt daarom aangenomen dat de relatieve waterdiepte geen invloed heeft.
-6-3. Invloed van scheve golfaanval
In sektie 9 en 10 is aandacht besteed aan de invloed van de hoek van golf-inval op de parameters <{>,, p en d en op de grootte van de golfoploop en stroming over het talud.
Het modelonderzoek naar de stijghoogte op het talud bij scheve golfaanval (sektie 7) is veel beperkter van opzet geweest dan dat met loodrecht inval-lende golven. Er zijn 7 proeven met loodrecht invalinval-lende golven (y = 0°) en 10 met scheef invallende golven uitgevoerd:
taludhelling : tanct = 0.25
golfhoogte : 0.11 < H < 0.13 m golfsteilheid : 0.01 < H/L < 0.05
o
relatieve waterdiepte: 0.06 < h/L < 0.30 hoek van golfinval : 0 < y < 50°
Op basis van de meetresultaten zijn empirische formules opgesteld die opti-maal op de metingen aansluiten (zie hoofdstuk 6 van sektie 7 ) .
Er kan echter gekonkludeerd worden dat formules (2) tot en met (6) bruikbaar zijn zolang y < 40°. Als y = 50° dan moet de met formule (2) en (3) bereken-de waarbereken-de van <J>. verhoogd worbereken-den met 15%, terwijl bereken-de met formule (5) en (6) berekende waarde van d dan verhoogd moet worden met 20%. De grootte van tan(J is niet aantoonbaar afhankelijk van de invalshoek y.
In sektie 8 is gekonkludeerd dat de golfklap bij kleine golfsteilheid en y * 0 groter is dan bij loodrechte aanval (y = 0 ) . Bij grote golfsteilheid is geen invloed van y te verwachten. Opgemerkt wordt dat in sektie 8 de letter f3 is gebruikt voor de invalshoek van de golven.
In appendix E van sektie 8 is gekonkludeerd dat het stijghoogteverschil over de toplaag afneemt bij toenemende hoek y. Als er echter meerdere losse blok-ken naast elkaar liggen, zoals kan voorkomen bij een overgangskonstruktie, dan kan de sterkte beduidend lager zijn als gevolg van het pianola effekt
(appendix F van sektie 8 ) . Dit heeft geen invloed op de kritieke golfhoogte bij begin van beweging van de losse blokken, maar resulteert in een grotere blokbeweging als deze kritieke waarde wordt overschreden. Hoe groter y is, hoe groter de blokbeweging is.
REFERENTIES
1. A.M. Burger
Taludbekleding van gezette steen
Evaluatie Oesterdam onderzoek, hydraulische aspekten Waterloopkundig Laboratorium, Grondmechanica Delft
SEKTIE 2
Evaluation of measurements of the wave pressures on a slope Large scale tests in Delta flume
January 1987
Note: This is part 1 of 4 parts (part 2 = section 3; part 3 = section 4; part 4 = section 7 ) .
INVOCATION
The present study is directed towards an evaluation of wave pressure parameters, important for slope stability design. It is intended to express the
characteristics of the pressure wave front in terms of the slope angle and the surface wave condition.
The two series of a laboratory investigation regarding the characteristics of wave induced pressure waves on a steep slope were carried out at Delft Hydrau-lics at the De Voorst Laboratory. The Delta Flume and the Schelde Flume experi-ments are described in Part I and Part II of this study, respectively.
Part I of this report describes the analysis of experiments that were performed on fuil scale in the Delta Flume of Delft Hydraulics, in March 1986, dealing with measurements of soil and water pressures due to the surface gravity wave in a filter layer and on a Basalton top layer of the slope. Only one slope angle, s = 1/3, was tested. The measurements were carried out for regular and irregular surface waves. For more details of the experiments reference is made to the work of Lindenberg [1986]. The preliminary results of the measurements for regular waves show that:
i) the number of pressure gauges in a near breaking zone was not sufficient to evaluate the <5^ and p parameters exactly;
ii) the applied method of searching for the <£ and p values from pressure dis-b
tributions in time and in space is reasonably accurate;
iii) the 2 parameter relations of <£> and p were the best, however the 1 parame-ter relations of $ in parame-terms of £ , and of p in parame-terms of UR parameparame-ter give also a very high correlation coefficients;
iv) the conclusions regarding approximations of $, and p can not be extended to steeper or flatter slopes, the investigations for other slope angles are necessary.
Part II of the study describes the second set of experiments performed on small scale in the Schelde Flume of Delft Hydraulics, in February 1987, dealing with measurements of the surface wave induced pressures on a concrete slope. The measurements were carried out for regular and irregular surface waves under the supervision of L. Banach, who also drew up this report. The description of the small scale experiments (model set-up, programme and results of the measurements together with mean pressure wave calculations) is given in Chapter 2. A small number of examples of pressure, run up and surface wave signals is presented as well as mean pressures, surface elevations and pressure potential distribution
over the slopes as a function of time and space. All sets of plots and signals recorded on magnetic tape during experiments and preliminary calculations are available at the De Voorst Laboratory under the project H 195.08. In Chapter 3, the analysis of the data is presented: (3.1) an evaluation of the pressure wave parameters $, and (3 from measurements; (3.2) the establisment of relations for $ and p in terms of slope angle and wave conditions. These relations are given
in a graphical and/or analytical form. The Schelde Flume models were equipped with more wave pressure gauges than the Delta Flume model, especially in the nearbreaking zone. Three slope angles were used, s = 1/2, 1/3 and 1/4.
The detailed small scale investigations give a possibility to rework the data from the Delta Flume experiments. In Chapter 3 the corrected relations have been presented. In the fuil and small scale investigation regular surface wave condi-tions and pressures acting on the top layer of the slope revetment are
considered. The results of the analysis are:
i) a best approximation of $, in terms of a 1 parameter relation with E,Q
parameter is given; the influence of a slope angle on the fit coefficients was detected;
ii) a best approximation of P in terms of a 2 parameter relation for
(C , D/H ) and (BL1, H /D) is given. The relation between a fit coëfficiënt and the slope angle is presented also.
CONTENTS
ABSTRACT
page 1. Introdution 1
2. Descrlption of the experlments 3 2.1 Model set-up 3 2.2 Programme of tests 4 2.3 Results of measurements 7
3. Analysis of the data 9 3.1 Evaluation of the mean pressure wave 9 3.2 Evaluation of $b and p values from measurements 13 3.3 Experimental formulae for $b and |3 values for regular waves 17
4. Conclusions 22
ACKNOWLEDGEMENTS
REFERENCES
LIST OF FIGURES
1. General layout of experiment (Lindenberg [1986]).
2. Location of gauges on a basalton's top layer (Lindenberg [1986]) 3. Pressure records - test 01.
4. Pressure, run up and surface wave records - test 01. 5. Pressure records - test 02.
6. Pressure, run up and surface wave records - test 02. 7. Pressure records - test 04.
8. Pressure, run up and surface wave records - test 04. 9. Pressure records - test 05.
10. Pressure, run up and surface wave records - test 05. 11. Pressure records - test 06.
12. Pressure, run up and surface wave records - test 06. 13. Pressure records - test 07.
14. Pressure, run up and surface wave records - test 07. 15. Pressure records - test 08.
16. Pressure, run up and surface wave records - test 08. 17. Pressure records - test 09.
18. Pressure, run up and surface wave records - test 09. 19. Pressure records - test 11.
20. Pressure, run up and surface wave records - test 11. 21. Pressure records - test 13.
22. Pressure, run up and surface wave records - test 13. 23. Pressure records - test 16.
24. Pressure, run up and surface wave records - test 16 25. Pressure records - test 17.
26. Pressure, run up and surface wave records - test 17 27. Pressure records - test 18.
28. Pressure, run up and surface wave records - test 18
29. Normalized mean pressure, surface waves and run up - test 01 30. Normalized mean pressure, surface waves and run up - test 02 31. Normalized mean pressure, surface waves and run up - test 04 32. Normalized mean pressure, surface waves and run up - test 05 33. Normalized mean pressure, surface waves and run up - test 06 34. Normalized mean pressure, surface waves and run up - test 07 35. Normalized mean pressure, surface waves and run up - test 08 36. Normalized mean pressure, surface waves and run up - test 09 37. Normalized mean pressure, surface waves and run up - test 11 38. Normalized mean pressure, surface waves and run up - test 13
LIST OF FIGURES (contlnued)
39. Normalized mean pressure, surface waves and run up - test 16 40. Normalized mean pressure, surface waves and run up - test 17 41. Normalized mean pressure, surface waves and run up - test 18
42. Pressure potential distribution along a slope 1:3 in time - test 01 43. Pressure potential distribution along a slope 1:3 in time - test 02 44. Pressure potential distribution along a slope 1:3 in time - test 04 45. Pressure potential distribution along a slope 1:3 in time - test 05 46. Pressure potential distribution along a slope 1:3 in time - test 06 47. Pressure potential distribution along a slope 1:3 in time - test 07 48. Pressure potential distribution along a slope 1:3 in time - test 08 49. Pressure potential distribution along a slope 1:3 in time - test 09 50. Pressure potential distribution along a slope 1:3 in time - test 11 51. Pressure potential distribution along a slope 1:3 in time - test 13 52. Pressure potential distribution along a slope 1:3 in time - test 16 53. Pressure potential distribution along a slope 1:3 in time - test 17 54. Pressure potential distribution along a slope 1:3 in time - test 18
55. Definition sketch of <£b and p\
56. Variations of (Tc-Tt)/T versus dimensionless wave lenght (k^x). a) for all tests
b) for the same periods.
57. Variations of (Tc-Tt)/T versus dimensionless wave lenght (k^x) for similar wave steepnesses.
58. Variations of Hp/(pgHi) versus dimensionless wave lenght (kjx). a) for all tests
b ) for the same periods.
59. Variations of Hp/(pgHi) versus dimensionless wave lenght (kjx) for similar wave steepnesses.
60. Measured values of <£b/Hi and (3 versus dimensionless parameters -1 parameter regression.
NOTATIONS
p = angle between pressure wave front and vertical axls BL1 = Hl/(g*T*T) = Hl/(2u Lo)
BL = BLl*Hi/D
D = depth of flume
d = depth along a slope (related to SWL)
<£>b = potential pressure wave height before breaking g = gravitational acceleration
H = designed wave height
Hi = incoming wave height [(Hmax+Hmin)/2] Hp = Pjmax+Pjmin - height of pressure wave
Hpi = Hp/(p gHi) - normalized potential pressure wave height k = wave number
Li = incoming wave length (from dispersion relation for d=D) Lo = deep water wave length
P = potential pressure ( [m] )
Pij = pressure for i periods and j sampling points Pj = pressure averaged over i periods
Ru = run up
SWL = still water level T = wave period
Te = pressure wave crest time Tt = pressure wave through time
Tct = (Tc-Tt)/T - . Ur = Ursell number (H*L*/d3)
Uro = Ur~(-1) = (D~3/[Hi*Lo*Lo]) Vi = surface wave velocity (V^Li/T) Vc = pressure wave crest velocity Vt = pressure wave through velocity x = horizontal axis
6 = standard deviation
Co = surf similarity parameter (tan 9//[H/Lo]) £io = surf similarity parameter (tan 9//[Hi/Lo]) £i = surf similarity parameter (tan 9//[Hi/Li]) w = wave frequency
p = water density 0 = slope angle
ABSTRACT
A laboratory study has been conducted to measure the regular wave induced pressure distribution on a constant slope as a function of time and space. The measurements were carried out in the Delta Flume of the Delft Hydraulics in march 1986.
On the basis of this experiment the relation between certain surface wave parameters are values of <&b and (3 angle are found. The formulae were found by means of a one and two parameters regression procedure. These relations can be applied further to the calculation of a pressure difference over the top layer and the gradients in a filter layer of the slope to enable a safe design. The validity of the results is, for the time being, limited to this experiment with one slope angle and one type of slope top layer.
-1-1. Introduction
The description of the of hydrodynamical loads on slope protection structures is very important for practical design of those structures. There is no use-full theory that describes the behaviour of a surface wave approaching a revetment structure with a steepness between 10° < 9 < 45°. The difficulties rise when the wave breaks on the slope.
The most popular manner to solve this problem is a parametric approach. It is possible to find a relation between incoming or deep water wave characteris-tics and revetments describing the wave behaviour on the slope which has to be evaluated. On the basis of experiments a usefull coefficients are searched for. This work presents similar ways of calculating relations for the deter-mination the parameters $b and (3.
Experiments were performed on full scale in the Delta Flume of the Delft Hydraulics, during march 1986, dealing with measurements of soil and water pressures due to surface gravity wave in a filter layer and a top layer of slope. The measurements were carried out for regular and irregular surface waves. In this work the regular surface wave conditions and pressures on a Basalton top layer are considered. For more details reference is made to the work of Lindenberg [1986].
Many experiments of regular wave induced pressures on a sloping bottom (e.g. DHL report [1985]) show significant changes of the measured pressure signals with decreasing water depth. The surface profile of the wave approaching a coast becomes asymmetrie with respect to the horizontal axis (i.e. higher crest elevation than through level) and with respect to the vertical axis, with a steeper wave crest front than the crest rear face (Svendsen, Buhr-Hansen [1978]; Funke, Mansard [1982]; Hwang [1984]). This fact is also reflec-ted in pressure signals recorded along a slope during Lindenberg's [1986] experiments. The experimental set-up and recorded pressure signals on the slope are described in Chapter 2.
Measured pressure signals show that design surface wave period remains con-stant over the slope and is equal to the pressure period. This allows the superposition of particular waves to obtain, for each measurement point, one mean pressure wave which is investigated instead of entire records consisting of a large number of waves. The procedure and the plots of the mean pressure waves are shown in Chapter 3.1 together with the distribution of the pressure along the slope.
-2-A very popular procedure to calculate conditions of slope stability under wave action is to evaluate a pressure gradients over the top layer of a slope
revetment. The wave approaching the coast reach maximum steepness just before the point of breaking. Wave crests front become vertical. This situation gives the maximum gradiënt of the pressure between two neighbouring pressure gauges on the slope. At this moment also the run down reachs maximum value and a part of the slope is dry, i.e. no pressures acting on the slope top layer. The pressure in the filter layer underneath the top layer, acting upward, has a value depending on phreatic (groundwater) level and the permeability of top and under layer. If the pressure difference between top layer and filter layer exceeds a level determined by weight of slope revetment blocks, firction
forces between blocks e.t.c. damage of the top layer can occur. The pressure gradients over the top layer can be calculated from theory (e.g. DHL report
[1985]). This theory needs values of the potential pressure height just before breaking of the surface wave, $b and p (the angle between the pressure wave front and vertical axis) as input data. Thè procedure of searching for values of <£b and P is described in Chapter 3.2. The relations between these values and certain surface wave parameters are presented in Chapter 3.3.
The Chapter 4 contains some remarks concerning the design of a model set-up for pressure recording with special attention for the determination of the location of the pressure gauges on the slope and conclusions.
-3-2. Description of the experiment
2.1 Model set-up
The measurements were performed in the "Delta Flume", a 233 m long, 5 m wide and 7 m deep wave flume of Delft Hydraulics at the De Voorst Laboratory. This flume is equipped, at one end, with a translating wave generator with a com-pensating system for wave re-reflection. The wave generator board is driven by an electronically controlled hydraulic actuator.
During all the tests the water depth was fixed at 4.5 n and only one slope angle was used of tan 9 = 1/3. The slope was constructed with 2 layers in the deeper part and 3 layers in the upper part of the flume. The top layer was constructed with irregular shaped Basalton blocks with dimension of approxi-mately 20 cm x 20 cm x 20 cm placed on an permeable geotextile material. Below, the filter layer with sand size of 0.14 mm covered, partially, the concrete slope and grain layer. The set-up of the cross section of the slope is shown in Fig. 1. The surface wave height was measured by means of two wire type wave gauges. The determination of the incoming wave height, Hi, is based on wave height measurements in two characteristics points in the flume. One of them is placed at the point of maximum wave height, the second one at the point of minimum wave height. The distance between these points is approxi-mately of 1/4 of the wave length (DHL reports [1983], [1984]).
The 13 pressure meters were installed along the slope in the Basalton blocks. The run up meter (wire type) was installed along a slope. Fig. 2 and Table 1 shows the location of the wave and pressure meters and reference system.
-4-Nr.
10
67
64
60
58
57
56
54
53
52
50
49
48
12
of gauge
(wave g.)
(pressure g.)
/
M\
( " )
f II V f n \ ( " ) * II V ( " ) ( " ) f II V ( " ) f t% \ (run up g. ) X(m)
184.00 9.00 6.14 4.86 4.12 3.79 3.39 2.69 2.41 1.950
-1.50 -3.00[ — ]
d
(m)
4,50 3.00 2.03 1.62 1.37 1.25 1.13 0.90 0.80 0.650
-0.50 -1.00[ — ]
Table 1 Location of the gauges
2.2 Programme of tests
The conditions for regular wave are selected for 13 tests. A list of the
designed (deep w a t e r ) and measured parameters of these tests is presented in
Table 2. The measured wave height was only sligtly different from that was
desinged, the wave period was reproduced exactly.
The parameters shown in Table 2 are defined as follows:
Lo = g*T*T/(2*-n) = 1.56*T*T
Li = g*(2*u/w)
2/(2*i0
Test NR i DGB01 DGB02 DGBO4 DGBO5 DGBO6 DGBO7 0GB08 DGBO9 DGB11 DGB13 DGB16 DGB17 DGB18 D [m] 4.5 •• •• •• •• 4.5 •• •• •• •• 4.5 •• T [81 3 . 1 1 •• •• » •• 3 . 7 9 •• •• M 5 . 3 7 •• M Lo [m] 15.09 •• •• » •• 22.43 •• •• •• •* 44.99 •• •• DESIGN PARAMETERS Li [ra] 14.49 •• •• •• •• 19.16 •• •• •• •• 32.0 •• H [m] 0.29 0.43 0.57 0.71 0.86 0.41 0.62 0.84 1.04 1.24 0.42 0.84 1.26 D/Lo [-] 0.30 •• •• •• " 0.20 •• •• •• " 0.10 •• " D/Li [-1 0.31 •• •• •• 0.23 •• •• •• " 0.14 •• H/Lo [%] 1.9 2.8 3.8 4.7 5.7 1.8 2.8 3.7 4.6 5.5 0.9 1.9 2.8 Ru [m] 0.7 0.8 1.0 1.1 1.2 1.0 1.2 1.4 1.5 1.7 0.8 2.0 2.4 Rd [m] 0.4 0.4 0.5 0.6 0.7 0.5 0.6 0.8 0.9 1.0 0.7 1.1 1.3 [-1 2.40 1.97 1.71 1.54 1.40 2.48 1.99 1.73 1.55 1.42 3.45 2.44 1.99 Hi [m] 0.29 0.41 0.54 0.65 0.84 0.40 0.58 0.79 1.05 1.23 0.47 0.94 1.38 MEASURED PARAMETERS H i / L o [%] 1.9 2.7 3.6 4.3 5.6 1.8 2.6 3.5 4.7 5.5 1.0 2.1 3.1 Hi/Li [m] 2.0 2.8 3.7 4.5 5.8 2.1 3.0 4.1 5.5 6.4 1.5 2.9 4.3 Ru [m] 0.52 0.74 0.87 0.95 1.07 0.91 1.11 1.19 0.96 1.50 0.87 1.95 1.52 Ho [-] 2.40 2.02 1.76 1.61 1.41 2.48 2.07 1.78 1.54 1.42 3.26 2.31 1.90 £i [-] 2.36 1.98 1.73 1.57 1.38 2.30 1.92 1.64 1.42 1.32 2.75 1.94 1.61
-7-Tests were divlded into 3 groups of tests on the basis of the water depth wave length ratio (D/Lo) which varies between 0.1 to 0.3, while the wave steepness parameter (H/Lo) between 0.010 to 0.057.
During each test signals from all gauges were recorded simultaneously on a HP computer hard disk with sampling time 0.04 s (25 samples per second). The test duration was 2 minutes which gave 3000 data points of the time series for each gauge. For wave periods of 3.11, 3.79, 5.37 seconds the number of recorded waves was of 39, 31, 22 respectively.
Table 2 shows a very wide range of investigated waves. The breaking wave conditions can be characterized by the surf similarity parameter, £• Small values of E, indicate a plunging type of breaking and large values of £ indi-cate collapsing up to surging type of surface wave breaking.
2.3 Results of measurements
The pressure acting on a slope can be split into three terms: the static pressures mainly due to gravity, the dynamic pressures due to flow forces generated by the waves and impact pressure due to the effect of wave breaking. During measurements the two latter types of pressures were recorded. Impact pressures observed on plots are not considered in a present study.
Moreover the data sampling interval was too long to investigate this pheno-menon that is assummed to have a minor influence on the slope stability.
The results of the experiments are presented in a graphical form. Figures 3-28 shows the recorded in time signal of the pressure gauges, one surface wave gauge and run up gauge. A time from 0 to 60 s is chosen, i.e. 1/2 of entire time serie (because of very high repetition of the signal over wave period). The type of gauge is known by it's number (see Table 1 ) . Values of the
pres-sure are in units of (kN/m2), wave and run up values in units of (m).
This plots shows that the wave period (T) remains constant for all measurement points, only the amplitude of the signal is changing in accordance with sur-face wave height. Generally the sursur-face wave shape and the pressure signals on deeper parts of the slope are more or less sinusoidal. From a certain point on the slope the smooth pressure signal shape is disturbed by additional smaller oscillations which become bigger towards the coast. These oscillations have several possibl-e origins: i) changes in surf ace wave transformation, ii) complicated flow pattern on the sloping bottom, iii) the impact pressure in breaking wave region.
-8-In the region before breaking (for steep slopes) surface elevation is a com-bination of surface wave approaching the coast and reflected waves from the slope, i.e. a standing or nearby standing waves can be formed. A second peak in the pressure signals appears due to this interaction (e.g. Fig. 11 - 2 8 ) , however for larger values of the surface wave height the wave instability and higher harmonie components of the wave can be cause of it as well.
The most interesting region with respect to stability of the slope is just before breaking point. In this zone the maximum pressure gradiënt over the slope can cause a damage of the slope. In the breaking region the impact
pressures disturb the recorded signal. The impact amplitude can be higher than main dynamic pressure amplitude (Fig. 26, 2 8 ) , lower than the main one (Fig. 24) or both. In this region the wave changes its character of motion from oscillatory to progressive one. After breaking the wave reaches the run-up/-down region where water velocities up and run-up/-down the slope determine the pres-sure signal. The latter two regions have minor importance for slope revetment stability conditions. This basic set of data is used to construct the average pressure and run up waves.
-9-3. Analysis of the data
3.1 Evaluation of the mean pressure wave
It was mentioned above that the period of the waves (for one group of test) is the same for each measurement point. Only the pressure amplitude changes with respect to the wave height. The first fact allows the construction of an average signal by summing up completely overlaping section of the signal and dividing by the number of waves. The mean pressure wave can be investigated easier than particular pressure waves or the entire record. Also insignificant minor pressure fluctuations which can disturb the analysis are filtered out. The wave period (T) was not an integer number of data sampling interval (0.04 s) therefore it was impossible to cut the recorded signal into n=20 parts with a length of T seconds exactly. The adopted wave period was found differ ±0.01 s only from that original one. The adopted periods for 3 groups of tests are: T=3.12 s corresponds to T=3.11 s (shift +0.01 s)
T=3.80 s corresponds to T=3.79 s (shift +0.01 s) T=5.36 s corresponds to T=5.37 s (shift -0.01 s)
Those new periods will be' applied to further analysis. This also means that during the averaging procedure of the recorded signals the time shift of n * ±.01 s (= ±0.2 s) will be present. To avoid this problem the following
correc-tion was introduced: every three periods the fourth wave was shifted of plus or minus 0.04 s (depending on sign of shift value). Thus, the entire time shift for each signal is equal ±0.03 s which is very small in comparison with the initial shift and is independent of the number of waves taken into ac-count. The standard statistics procedure to compute the mean wave was used. Denote Pij as a pressure value for i periods and j sampling points and Pj as a value of pressure averaged over i periods,
where i=l,2,...,n = number of waves taken into account (n = 2 0 ) ,
j=l,2,...,m : m=T/0.04 s = number of points in time over one period. Then
Pj = (l/n) l Pij (1) i=l
-10-m
(l/m) * l ój (2)
where n6j = { [l/(n)] * [ I (Pij-Pj)
2] } (3)
The same procedure was used to calculate surface wave and run up values. The pressure wave are normalized by pg to obtain potential pressure values (unit of [m]) to further analysis. The mean pressure waves, surface wave and run up are shown in Fig. 2^-41. It is clear that small fluctuations on the mean shape are no longer present. The wave assymmetry is changing towards the coast, i.e. the wave becomes more assymmetrical. The second peak observed in the plots, with the recorded signal is, in most cases, filtered out. It is visible only for higher surface waves, Hl > 0.7 m, and larger wave periods, T > 3.80 s. The impact peak appears for the wave condition. mentioned above while is not pre-sent in plots with shorter waves. The maximum potential pressure amplitudes, Pjmax, are in the range from 0.17 m for small waves (Hi<0.3 m) to 0.73 m for bigger waves (Hi>1.3 m ) . Values of Pjmax increase with increasing of D/L ratio for similar wave steepnesses. The absolute values of the minima of potential pressure waves, Pjmin, are a bit larger than the absolute values of the
maxima, Pjmax. Pjmin values change from -0.18 m for small waves (HK0.3 m) to -0.81 m for bigger waves (Hi>0.9 m ) . Just as Pjmax, absolute values of Pjmin increase with decreasing D/L ratio for the same wave steepness. Both values, Pjmax and Pjmin, tends to zero at the highest point of the slope.In general, the Standard deviation for all points is very small, in the range of 3% to 15% of the pressure wave amplitude. Standard deviations increase towards the
coast. In several cases the impact pressures increase this value up to 40%, particularly for the high waves (Hi>l m ) . This indicates that even in the case of regular waves the near breaking or breaking waves have a random character due to rapid wave transformation over the steep bottom and interaction between incoming and reflecting waves.
Maximum and minimum amplitudes of Pj and Standard deviations for all tests and measurement points have been presented in Table 3.
One can conclude that the applied averaging procedure filters out small oscil-lations but does not change the pressure wave shape. The mean pressure wave is easier to analyse and is a basis for further investigations of variations of it characteristics. Using the values of Pj it is possible to reconstruct the
-11-development of pressures along the slope for different time steps. Figures 42-54 shows the potential pressure distribution along the slope. The x axis is chosen horizontally, positive towards the coast, with zero reference in the point of the deepest pressure gauge. The pressure is normalized as:
P = Pj/(pg) + z (4)
where z is the depth from SWL vertically (only positive values upwards SWL). Plots show changes of the pressures in space for several moments of time, i.e. one line corresponds to the one time step. The time step (0.16 s) was chosen equal for all tests. In one plot the characteristical phases of P are presen-ted. These phases of increasing and decreasing P reflect changes between consecutive time steps. All phases in one plot are presented also to give the impression of the character of the measured pressure waves.
Detailed figures of the potential pressure distribution enable the determi-nation of the maximum potential gradiënt in space. The values of the potential pressure wave height, §>b, can be eveluated to calculate the pressure diffe-rence on the bottom as well as the angle of (3. This procedure will be presen-ted in the Chapter 3.2.
-13-The general plots of P show that in one case a pure-standing wave was recorded (Fig. 5 2 ) . Other general plots indicate that the interaction between progres-sive and reflecting waves was too complicated to form an exact standing wave. In general a tendency towards a the standing wave pattern with decreasing surface wave steepness can be observed. However for long waves (T=5.36 s and Hi>0.9m) the number and the positions of the pressure gauges is not fit to reconfirm this observations.
3.2 Evaluation of <£b and |3 values from measurements
Values of $b and p are defined as follows (see Fig. 5 5 ) : $b is a potential pressure height, (P), just before breaking point for a certain moment in time when the maximum pressure gradiënt between two neighbouring points in space is reached. The angle of (3 is the angle between the vertical axis and the line of maximum pressure gradiënt.
In certain cases, the waves break on certain depth (db) when the slope is not entirely dry. Theory presented in DHL report [1985] enables the calculations of the pressure difference over the top layer under. restriction that the slope is dry just before the wave breaking. A thickness of a water layer on the slope however, can be significant. Then the theory can still be applied when value of db is known. Therefore the empirical relation between db and £io parameter was found (Pilarczyk (1976)):
db/Hi = 0.63 * l±o for tan9=l/3 (5)
Singamsetti, Wind (1980) determine $b with respect to db value in terms of £ parameter:
$b/db = 1.16 * £io~0.22 (6)
These formulae indicate that the waves break on a certain depth and always the water layer on the slope is present. From DHL report [1985, juni] it is clear
that in most cases the slope is almost dry just before breaking. The thin water layer on the slope is negligible in comparison with the wave height. Consequently the problem is divided into three parts:
- to find the place where the maximum pressure gradiënt is present, - to find the values for $b and (3,
-14-The place of the maximum gradiënt and its value can be found using Fig.42-54. Fig. 29-41 allows the elimination of the impact pressure influence on the pressure gradiënt. Generally, gradiënt values tend to infinity when the sur-face wave front becomes vertical. This also means that the wave breaks. This assumption is valid not for all types of breaking. During collapsing and surging breaking (£ > 3) surface wave front never becomes vertical (Bruun [1985]).
It is also impossible to calculate pressure gradiënt accurately, on the basis of measurements, because of the distance between pressure gauges. However, it is possible to find the variations of the height of the pressure wave and shape of this wave along the slope, only in measurement points.
The pressure wave shape changes can be characterized by time difference be-tween the moments of the passage of the wave crest (Te) and the preceeding trough (Tt):Tc-Tt. This value indicates, by normalization to the wave period, T, the wave pressure deformation in space, however information about wave assymmetry is limited.
If Tct = (Tc-Tt)/T then
Tct = 0.5 indicates the time between crest and trough is equal T/2 Tct > 0.5 indicates a more gentle wave front than the wave crest rear, Tct < 0.5 indicates a steeper wave front than the wave crest rear.
Determination of the pressure wave assymmetry according to e.g. Funke, Mansard [1982], Hwang [1984] is not possible. The bottom slope induces a rapid change of the wave propagation velocity. Therefore it is difficult to determine the wave length from the measurements. Fig. 56,57 show the time difference of the normalized pressure wave crest-trough (Tct). The horizontal axis is the dimen-sionless wave length, k x. The reference zero point is located at the inter-secting point of SWL and the slope. Minus value of k x correspond to the part of the slope above SWL while plus values correspond to the distance towards the wave maker compared with the wave number, k. Values of k x equal
n indicate half of wave length. Experimental short waves (T =3.12 s) are in
the range up to 4*k x, waves of T=3.80 s in the range up to 3*k.x while long
waves (T=5.36 s) in the range up to 1.8*k1x. One can conclude that for a
pressure gauge location the most accurate information, independent of wave period or wave length, is obtained from long waves. Therefore in the breaking zone (the most interesting) the measurement points were not close enough in comparison with the wave length for short waves. Consequently, lack of in-formation exist there while more exact inin-formation is available for long waves.
-15-Fig. 56a) shows a general view of all values of Tct. It is clear that Tct has a similar tendency for all tests". Values of Tct .fluctuate between 0.4-0.5 for 4 > kjx > 1.1 and decreases to 0.1 for 1.1 > kjX > 0 then Tct increases from 0.1 to 0.6 for k-jx < 0 again. For small waves (Hi<0.5 m) the highest point on the slope (k^x < -0.5) was difficult to establish because of half-wave recti-fier type of signal. Therefore interpretation of Tct in this region is not unique.
In Fig. 56 b) Tct is split into 3 subplots with the same wave period. It is seen that the most accurate measurement are for long waves while for short waves the accuracy is less. Values of Tct reach maximum near kjx = 1.0, i.e. x/Li = 0.159. From this point, Tct decreases rapidly up a value of 0.1 for kjX = 0, except tests 16 to 18 (Tct = 0.1 in 0.5 > kjx > 0 ) . The same trend can be observed for long and short waves. Near the point of kjx = 1 values of Tct decrease with increasing wave height or wave steepness. In k^x a 1.75 there is
characteristic cross point for curves with the same wave period,
Tct « 0.4*0.5. In this point the value of Tct depends on the wave period (or ratio D/Li), i.e. Tct increases from 0.4 to 0.5 when D/Li decrease from 0.3 to 0.1.
Fig. 57 shows the distribution of• Tct along k^x for similar wave steepnesses, separately. The dependence of Tct and wave steepness in point kjX =» 1 is easy to see and it is clear that the maximum value of Tct moves from k^x » 0.8 for small (long waves) to k^x » 1.3 for large wave steepness (short waves).
Confronting Fig. 29-41 and Fig. 56,57 it is obvious that breaking of surface wave is in progress when Tct < 0.3 for 0.8 > kjx > 0. In the region of k^x < 0
the swash velocities, parallel the slope, are predominant and conditions given above is no longer valid. However, the wave set-up effect (the mean water level is higher than SWL) moves point where k^x = 0 towards the slope. Therefore the boundary of the end of breaking and beginning of swash zone might be evaluated also by this criterium, i.e. if Tct < 0.3 the wave still breaks.
This conclusion is confirmed by the analysis of Hpi- distribution along k^x, where Hpi is the height of the pressure wave normalized by Hi:
Hpi = Hp/Hi (5)
-16-If the value of the pressure impact was greater than the main pressure crest, the latter value was used for the analysis. Therefore the influence of the impact effect was eliminated. If Hpi = 1 then the potential pressure value is equal to the surface wave height. Fig. 58,59 demonstrate the distribution of Hpi along k^x. Fig. 58 shows a general view of the Hpi distribution for all tests. The changes of Hpi along k^x are similar to those for Tct.
Values of Hpi increase from Hpi = 0.5 for k^x => 4 up to Hpi = 0.6 for kjx « 2.6. Afterwards Hpi decreases or remains constant to k x » 2. A decrease of Hpi
indicates standing wave type oscillations. Only few cases when near standing wave appear had to be present. The standing wave pattern is more probable when wave steepness is small. From k^x » 2 the value of Hpi increases again to a maximum of 0.8 -5- 1.3 (k^x « 1.25 * 0.8) and then decreases to zero (k^x < -0.6) high on the slope. Test 16 is the exception which shows a pure standing wave and a maximum value of Hpi = 1 . 7 for k^x » 0.4, which contradicts the general tendency. Fig. 58b) shows variations of Hpi for the same wave periods and different wave steepnesses. The maximum value of Hpi is almost in the same position, however a weak shift to bigger values of k^x can be observed, except tests 16, 17 where the maximum Hpi is in different position due to standing wave conditions. Also the changes of maximum magnitude of Hpi can be observed but Fig. 59 illustrates it in more detail.
The maximum value of Hpi is almost constant for similar wave steepnesses: for Hi/Li » 0.2 the value of Hpi is equal 1.2 * 1.3,
for Hi/Li « 0.3 the value of Hpi is equal 1.0 * 1.1, for Hi/Li « 0.4 the value of Hpi is equal 0.9 * 1.0, for Hi/Li * 0.6 the value of Hpi is equal 0.8 * 0.9.
In the most cases Hpi decreases from maximum value to zero for 1 > k^x > - 1 . A decrease of the Hpi value does not mean that the wave starts to break. From
Fig. 29-41 it can be observed that the wave breaks further, when Hpi is smal-ler than the maximum. This fact is also presented in DHL report [1985,
february]. In the case of short waves the breaking point was very difficult to establish because of lack of pressure gauges in the breaking zone, but it is possible that the breaking occurs when Tct < 0.3. In this way, the zones of breaking and no breaking were found. The accuracy of defining the breaking zone for shorter waves is less than for longer one.
The value of Hpi in the point before breaking is taken into account. In this place:
-17-$b/Hi = Hpi (8)
For steep long waves the breaking process starts probably on a water depth equal or deeper than Hp, but in the case of these measurements there is no tracé of impact pressures in the plots (fig. 29-41). Large impact peaks are -present on the depth approximately equal Hp/2. In this case the point just
before the impact is considered.
A (3 angle was found as the maximum pressure gradiënt from that measuring point to the next one shoreward. Fig. 29-41 enables the elimination of the gradients disturbed by the impact pressures.
Carefull analysis of <£b values shows that in most cases 3>b/2 is equal to the local water depth, which is contradictory to conclusions of Pilarczyk [1976] and Singamsetti, Wind [1980].
3.3 Experimental formulae for $b and ft values for regular waves
When a train of regular waves reach a slope phenomena of breaking, run up, run down, reflection, up/down rush occur. These phenomena mainly depend on the following physical- and structural parameters:
1) physical: - water depth at the toe of the slope, D - incoming wave height, Hi
- wave approach angle, <x - specific weight of water, p - acceleration of gravity, g 2) structural: - slope angle, 6
- characteristic of roughness and permeability - characteristic of sublayer(s).
Regarding the conditions of performed experiments the roughness and permeabi-lity only depend on the type of armour units, their geometry and their size. These phenomena have not been included in this analysis. The roughness can be neglected because of the geometry and size of the Basalton blocks, also the permeability tends to zero due to geotextile material present between top and sublayer. The wave approach angle, o, is equal zero.
Based on the dimensional considerations, the load parameters $b and P are a function of the following 6 variables:
-18-By dimensional analysis it is possible to express the parameters as:
($b/HI),p = f[Hi/(gT2),Hi/D,9] (10)
This function can be reduced to a simpler expression with a single variable:
p =-f(T tane//(2u Hi/g)) (11) or («b/Hl),p = f(D~3/(Hi Lo2)) (12) or («b/Hl),P = f(Hi2/(gT2D)) (13) where Lo=gT2/(2n)•
The first variable is known as the surf similarity parameter, £> the second as the Ursell number, Uro, the third one represents the relation between wave steepnes and the relative wave height, BL.
In the case of eq. (11) the influence of the water depth is not included. Eqs. (12 and (13) do not include the slope angle, but regarding conditions of this experiment only one slope angle was used, consequently this influence can not be found. The importance of this factor will be examined in the future experiments for different slope angles.
Now the following 1 parameter models can be introduced:
Z = B * exp (X * A) (14)
Z = B * (X~A) . (15)
Z = B + A * ln(X) (16) where X expresses the variables defined by dimensional analysis from eqs.
(11)-(13), and A and B are fit coefficients. Next a 2 parameter model can be formulated:
Z = A * ln(Xl) + B * ln(Yl) + C (17)
where XI = £io; Y1=D/Hi; and A,B,C, are the best fit coefficients.
Fit coefficients and the correlation coefficients, R, for eqs. (14)-(17) calculated by the least squared method and based on experimental data, are
-19-given in table 4, in which R denotes the correlation between the experimental
values and the formula.
Z = B * exp (X * A) Z = B * (X~A) Z = B + A l n ( X ) Z=$b/Hi B A R B A R B A R
X=£io 0.72 0.13 0.46 0.77 0.29 0.51
X=Uro 0.81 0.27 0.73 1.03 0.08 0.65 —
X=BL 1.07 -158 0.75 — — — 0.03 -0.12 0.79
Z=8
44.7 15.0 0.21
72.4 14.5 0.92
-37.2 -12.6 0.60
X=£io
X=Uro
X=BL
—
• —67.8
——
—
-297
0
——
—
.57
44.
74.
0
1
0
0
.24
.29
——
0
0
.17
.94
Z = A * ln(X) + B * ln(Y) + C
X=£io, Y=D/Hi A B C R
Z=$b/Hi
Z=B
-0.
-36
02
.8
0.
45
26
.6
0
-9
.45
.05
0
0
.897
.977
Table 4 Fit and generalized correlation coefficients of the models
defined in eqs. (14)-(17).
Table 4, Fig. 60 and Fig. 61- show that a good approxlmation of the 8 angle is
given by Uro parameter using the 1 parameter model, eq. (15), R=0.94, and by
combination of £io and D/Hi variables using 2 parameter logarithmic model,
eq. (17), (R=O.977). The 8 values calculated from the £io relation, eqs. (15)
and (16), show poor agreement with experimental data. The correlation
coëffi-ciënt is small, R=0.17 and 0.21 respectively. A better results are given by BL
-20-relation, eqs. (14), (16), with R=0.57 and 0.60, but still these approxima-tions are worse than the Uro relation. Therefore it is proposed to use eq. (16) with Uro variable or eq. (17) with £io and D/Hi variables to calculated the (3 angle:
p = 74.1 * (Uro * 0.29) (18)
p = -36.8 * ln(^io) + 45.6 * ln(D/Hi) - 9.05 (19)
In the case of evaluation of *b/Hi values it can be seen from Tab. 4 and Figs. 60,61 that the best approximation is given by the 2 parameter equation (17) in terms of £io and D/Hi variables, R=0.897. The best 1 parameter approximation of measured $b/Hi values is given by relation with BL parameter, eq. (16), R=O.79, which tak.es into account the ratio of Hi to the water depth at the toe of slope. Relation $b/Hi neither to £io nor to Uro parameter is not satisfac-tory because of small correlation coefficients. It is recommendëd to use the following equations to calculate <£b/Hi values:
$b/Hi = 0.03 - 0.12 * ln(BL) (20) or
$b/Hi = -0.02 * ln(Sio) + 0.26 * ln(D/Hi) + 0.45 (21)
The £io parameter was used by Pilarczyk [1976] and Singamsetti, Wind [1980] to establish $b/Hi experimentally. Substituting eq. (5) into eq. (6) gives:
$b/Hi = 0.73 * £io~1.22 (22)
Table 5 shows measured values of $b/Hi and the values calculated by eqs. (20), (21) and (22) for the test program variables.
-21-test nr Hi £io
meas. eq. (20) eq. (21) eq. (22)
[*] ["] [~] ["] [-1 ["] 01 02 04 05 06 07 08 09 11 13 16 17 18 0.29 0.41 0.54 0.65 0.84 0.40 0.58 0.79 1.05 1.23 0.47 0.94 1.38 2.40 2.02 1.76 1.61 1.41 2.48 2.07 1.78 1.54 1.42 3.26 2.31 1.90 1.17 1.09 0.99 0.94 0.80 1.07 1.01 0.94 0.74 0.82 0.98 0.75 0.86 1.06 0.97 0.91 0.86 0.80 1.03 0.94 0.86 0.79 0.76 1.07 0.90 0.81 1.15 1.06 0.99 0.94 0.88 1.06 0.97 0.89 0.82 0.78 1.01 0.84 0.74 2.12 1.72 1.45 1.31 1.11 2.21 1.77 1.47 1.23 1.12 3.09 2.03 1.60
Table 5 Measured and calculated values of $b/Hi of eqs. (20), (21), (22).
It can be seen that the best approximation of measured values of $b/Hi is given by the 2 parameter equation (21). Comparing the values given by eq. (21) and (22) it can be seen that the latter overestimtes the values of <$b/Hi. The difference between these values is large for large values of 5io parameter
(factor 2) and decreses with decreasing of £io (factor 1.5). This difference is due to the fact that in eq. (22) the water depth just before breaking of the wave is taken into account while in eq. (21) there is no influence of this factor. In both cases one can observe a decrease value of $b/Hi with decrea-sing £io parameter.
In general, a good agreement between measured <$b/Hi, p values and proposed formulae is found. Nevertheless it is necess.ary to confirm the results of this experiment for different slope angles. The problem of the breaking depth (db) at steep slopes should be examined more carefully. The assumption that during wave breaking the slope is almost dry is reasonable. If the waves break on a structure and plunging crest may hit the slope, the slope is exposed to the most damaging situation (Bruun [1985]).
-22-4. Conclusions
This study leads to characteristics for $b/Hi and p angle for pressures in-duced by regular wave on a slope of 1:3. On the basis of experiments in the Delta Flume of Delft Hydraulics at De Voorst Laboratory the empirical formulae have been proposed to calculate these values. The main results of this work can be summarized as follows:
1) The analysis of the data showed that the distance between gauges during experiment was too large, especially for short waves. It is found that the gauges should be located along a slope with the distance of
Al = 0.02 * L/cos8 starting from intersecting point of SWL and the slope up to 1 = 0.2 * L/cos9.
2) Existing theoretical models based on the standing wave solution, which consider assymmetrical wave profile, non breaking waves and no impact pressures can not be applied directly to the problem of slope stability. The interaction between progressive and reflected waves on a steep slope in certain specific circumstances can form a pure standing wave pattern. Therefore the experimental formulae are used to calculated relations between surface wave parameters and values of <$b and p.
3) An averaging procedure is used to compose a mean pressure wave. This wave represents the pressure characteristics in the measurement points on the slope. The application of an averaged wave is acceptable, however, the standard deviation of the signal increases shoreward.
4) Distribution of the pressure in time and in space shows that if in certain points Tct < 0.3 breaking is in progress. The value of <5b is found as the potential pressure height in a previous point on the slope. The (3 values is evaluated from pressure gradients, excluding influence of the impact pressures.
5) The analysis shows that the slope is almost dry during the wave breaking. The $b value is smaller that the maximum potential pressure height.
6) The relations between surface wave parameters and <$b and p have been determined. The best approximation of these values if given by a 2 para-meter logarithmic model in terms of £io and D/Hi parapara-meters:
$b/Hi = -0.02 * ln(£io) + 0.26 * ln(D/Hi) + 0.45
-23-The $b/Hi and 0 value decrease with decreasing £io parameter. The formulae show a significant influence of the D/Hi ratio.
7) These approximations should also be examined for different slope angles and different surface wave approach angles to the coast.
ACKNOWLEDGEMENTS
The author wishes to thank Andre Burger and Mark Klein Breteler for the hours they spent to discuss this problem and their usefull comments. I am gratefull aïso to Albert Scheer for his kind help in making available the experimental data used in this work.
This work has been carried out as a part of the research programme of slope revetments under contract of Dutch Publics Work Department executed at the Delft Hydraulics at the De Voorst Laboratory.
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