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Wave impacts on vertical breakwaters are among the most severe and dangerous loads this type of structure can suffer. Whilst many design procedures for these structures are well established worldwide recent research in Europe has shown that some of those design methods are limited in their application and may over or underpredict the loading under important conditions. This will then lead to overdesigned and very expensive structures or, even more dangerous, to underdesign and consequently to danger to personnel and properties. Within PROVERBS engineering experience from various fields (hydrodynamic, foundation, structural aspects) concerned with vertical breakwaters has been brought together. Furthermore, data available from different hyraulic model tests, field surveys and experience from numerical modelling were collected and analysed to overcome the aforementioned limitations. Engineers from both universities and companies were working together to derive new methods for calculating forces and pressures under severe impact conditions taking into account the influence of salt water and aeration of the water. This new approach was then further optimized by taking into account the dynamic properties of the structure itself and the foundation of the breakwater (see Volume I, Chapter 3.4). The multidirectionality of the waves approaching the structure (Vol. I, Chapter 2.5.3) has also been considered. The intention of this paper is to describe a procedure to calculate both impact and uplift loadings under 2D conditions and to give references to more detailed work on the different aspects of the steps described in here. For sake of completeness and easier understanding of the whole method some parts had to be repeated from other sections within Vol. II of the PROVERBS report. This was considered to be more useful rather than giving too many references to other sections. Geometric dimensions and a sketch of a typical caisson breakwater are given in Fig. 1

A KORTENHAUS1); H OUMERACI1); N.W.H ALLSOP2); K.J MCCONNELL2);P.H.A.J.M VAN GELDER3); P.J HEWSON4); M.WALKDEN4); G MÜLLER5);

M CALABRESE6); D VICINANZA6)

1) Leichtweiß-Institut, Technical University of Braunschweig, Beethovenstr 51a, DE-38106

Braunschweig, Germany2) HR Wallingford, Howbery Park, GB-Wallingford OX10 8BA, U.K

3)

Delft University of Technology, Faculty of Civil Engineering, Stevinweg 1, NL-2628 CN

Delft, The Netherlands4) University of Plymouth, School of Civil and Structural Engineering, Palace Street, GB-Ply-

mouth PL1 2DE, U.K

5) Queens University of Belfast, Department of Civil Engineering, Stranmills Road,

GB-Bel-fast BT7 1NN, Northern Ireland6)

Università degli Studi di Napoli 'Frederico II', Dipartimento di Idraulica, Via Claudio n 21,

IT-80125 Naples, Italy

ABSTRACT

The tentative procedures for both impact and uplift loading proposed by Oumeraci and

Kortenhaus (1997) have been brought together and amended by many partners in

PROV-ERBS This paper proposes a procedure to calculate time-dependent pressures, forces and ver arms of the forces on the front face and the bottom of a vertical breakwater For this pur-pose, (i) the data sets on which this method is based are briefly described or referred to; and(ii) a stepwise procedure is introduced to calculate the wave loading supported by some back-ground and data information Suggestions for estimating the forces on a caisson in feasibilitystudies are also given

le-2 INTRODUCTION

Wave impacts on vertical breakwaters are among the most severe and dangerous loads thistype of structure can suffer Whilst many design procedures for these structures are well estab-lished worldwide recent research in Europe has shown that some of those design methods arelimited in their application and may over- or underpredict the loading under important condi-tions This will then lead to overdesigned and very expensive structures or, even more danger-ous, to underdesign and consequently to danger to personnel and properties

Within PROVERBS engineering experience from various fields (hydrodynamic, foundation,structural aspects) concerned with vertical breakwaters has been brought together Further-more, data available from different hyraulic model tests, field surveys and experience fromnumerical modelling were collected and analysed to overcome the aforementioned limitations.Engineers from both universities and companies were working together to derive new meth-ods for calculating forces and pressures under severe impact conditions taking into accountthe influence of salt water and aeration of the water This new approach was then furtheroptimized by taking into account the dynamic properties of the structure itself and the founda-tion of the breakwater (see Volume I, Chapter 3.4) The multidirectionality of the waves ap-proaching the structure (Vol I, Chapter 2.5.3) has also been considered

The intention of this paper is to describe a procedure to calculate both impact and uplift ings under 2D conditions and to give references to more detailed work on the different aspects

load-of the steps described in here For sake load-of completeness and easier understanding load-of the wholemethod some parts had to be repeated from other sections within Vol II of the PROVERBSreport This was considered to be more useful rather than giving too many references to othersections

Geometric dimensions and a sketch of a typical caisson breakwater are given in Fig 1

Tab 1: Overview of design methods for wave loading

There are a number of formulae available for different types of waves breaking at the ture These formulae generally include magnitudes of maximum pressures, their distributionsand forces In some cases, uplift pressures are given as well All formulae are fully empirical

struc-or semi-empirical as the process of wave breaking at the structure is still not fully explained.Tab 1 summarizes the most important methods in a chronological order, details are given inthe respective references

Quasi-Static Waves

method

Impact Waves

dimen-sions!

Blackmore &

Author Year

Broken Waves

Bradbury &

This paper is concentrated on calculation of pressure distribution and related forces underimpact conditions Furthermore, the dynamic characteristics of impact forces were consideredessential for the behaviour of the structure subject to this type of loading The design

procedure is therefore based on the approach by Oumeraci and Kortenhaus, 1997 which was

derived from solitary wave theory but amendments were made to many details like thestatistical distributions of impact and uplift pressures, the vertical pressure distribution at thefront face, and the relation between rise time and duration of impact forces

Different hydraulic model tests have been carried out and analysed to obtain the designmethod proposed in this paper These tests are summarized in Tab 2 where the mostimportant information is given Furthermore, references are added where more detailedinformation on these tests is available

Tab 2: Overview of hydraulic model tests (random waves)

Tests Year

Co n- fig 1)

-McConnell

& Allsop, 1998

Kortenhaus

& Löffler, 1998

1) number of configurations tested; 2) number of transducers

It may be assumed from the differences in the number of waves per test and the acquisitionrate that results of pressure distributions and forces might also differ significantly.Nevertheless, data analysis has confirmed that most of the data sets fit well to each otherwhich will be explained in more details in the successive sections

Impact load (Goda-formula not applicable) (Goda-formula applicable)

1.0 2.0 3.0

1.0 2.0

0.2 0.4 0.5 1.0

1.0

0.0 0.0

Fig 1: Pulsating and impact load - problem definition

8.2 Identification of wave impact loading

A simple method is needed to distinguish between:

(b) quasi-standing loads for which available formulae (e.g Goda, see Vol I, ter 2.4.1) without any account for load duration can be used (Fig 2a);

Chap-(d) slightly breaking wave loads which already consist of some breaking waves but notsignificantly exceeding the Goda loads (Fig 2b);

(f) an impact load for which new formulae including impact duration are to be used(Fig 2c); and

(h) broken wave loads, i.e the waves already broke before reaching the structure

For this purpose the PROVERBS parameter map (Fig 3) was developed which is in moredetail described in Chapter 2.2 of Volume IIa Input for this map are geometric and waveparameters which in combination yield an indication of a certain probability that one of theaforementioned breaker types will occur

Breakwater

h < 0.3b*

d h

b

B eq s

h

L

H si

Composite Breakwater 0.3 < h < 0.9 *b

Crown Walls Rubble Mound Breakwater

h > 0.9 *b

Low Mound Breakwater 0.3 < h < 0.6 *b

High Mound Breakwater 0.6 < h < 0.9 *b

Large waves 0.2 < H < 0.6s*

Small waves 0.1 < H < 0.2s*

Large waves 0.25 < H < 0.6*s

Small waves 0.1 < H < 0.2s*

Narrow berm 0.08 < B < 0.12 *

F = F

h h

6.0 8.0

0.1 0.2 0.0

0.0 4.0

Fhmax

Fhq

Fh*

t/T 2.0

6.0 8.0

0.1 0.2 0.0

6.0 8.0

0.1 0.2 0.0

0.0

4.0 Fhmax

Fhq

Fh*Slightly breaking wave Impact loads Broken waves

t/T 2.0

6.0

8.0

0.2 0.4 0.0

8.4 Breaker height at the structure

A breaking criterion which accounts for the reflection properties of the structure has been

sug-gested by Calabrese (1997) (see Chapter 2.3 of Volume IIa) based on extensive random wave tests in hydraulic model tests and previous theoretical works (Oumeraci et al., 1993):

where Lpi is the wave length in the water depth hs for the peak period Tp which can be lated iteratively by:

where L0 is the wave length in deep water which can be taken as:

or can be approximated using the method given in Fenton (1990):

The reflection coefficient Cr in Eq (1) may be estimated as follows (Calabrese and Allsop,

1998):

Cr = 0.8 + 0.1@Rc / Hsi for low crest walls (0.5 < Rc / Hsi < 1.0)

Cr = 0.5 to 0.7 for composite walls, large mounds, and heavy breaking

The empirical correction factor kb can be estimated as follows:

where Beq is the equivalent berm width which is defined as:

and Bb is the berm width in front of the structure Further details on this approach are given inSection 2.3 of Volume IIa

Pb ' exp & 2@ Hbc/ Hsi 2 @ 100% (7)

Hbs ' 0.1242@ Lpi@ tanh 2 B hs

8.6 Probability of occurrence of impacts

The parameter map as given in Fig 3 results in different branches where the probability of therespective breaker type is not known in advance The branch of 'impact breakers' proposes touse an impact loading formulae which generally yields much higher forces than any otherapproach for quasi-standing waves, slightly breaking waves or broken waves Hence, it isnecessary to know how many of the waves approaching the structure will break at the wall(thus causing impulsive forces) and how many will not break at the wall (inducing nonimpulsive Goda forces)

The aforementioned method by Calabrese and Allsop (1998) described in Chapter 2.3 of

Vol-ume IIa also gives a simple formula for the probability of broken waves Pb based on the ideathat every wave with a higher wave height than the breaking wave height Hbc (as calculated inSection 4.2) is already broken or will break as an impact breaker at the wall The probability

of occurrence of breaking and broken waves can therefore be calculated as follows:

The maximum wave height Hbs which describes the transition from impact breakers to alreadybroken waves can be described by Eq (1) where Beq/d and Cr are set to zero which then yields:

The proportion of impacts can then be derived from:

The magnitude of the horizontal force itself is strongly related to the type of breakers at thewall which are essentially depth limited It can be expected that the magnitude is related to therelative wave height at the wall Hsi/hs

Eq (9) can be regarded as a filter in the 'impact' domain of the parameter map For very lowpercentages of impacts (smaller than 1%) the problem can be reduced to the quasi-staticproblem and the Goda method can be used to calculate pressures and forces (see Chapter 4.1

of Volume IIa) In all other cases the method as described in the successive sections has to beused

HR 1994

Fig 3: Relative wave force Fh,max/(Dgd2) plotted vs relative wave height Hsi/d and

comparison to calculation method given by Eq (10)

10 WAVE IMPACT LOADING

10.2 Initial calculation of impact forces

Allsop et al (1996) have investigated a large data set (10 different structure geometries, see

Tab 2) to predict horizontal wave forces on vertical breakwaters The relative waveheight Hsi/d has been found to most significantly influence the wave forces nondimensionalized by the water depth over the berm All forces were given at a 1/250 level thustaking the mean out of the highest two waves (500 waves per test were measured) Themagnitude of the horizontal impact force can then be estimated from:

This formula has been derived from data sets with a 1:50 foreshore slope and checked againstother slopes where it also seems to fit the data reasonably well In Fig 4 data from three dif-ferent model tests have been plotted and compared to the prediction method given in Eq (10)

It can be seen from the graph that Eq (10) gives an upper bound to the data and is only

exceeded by some data points Allsop et al (1996) give the validity range of the method as

Hsi/d in between 0.35 and 0.6 whereas here the graph in Fig 4 shows that it can be used up to

Furthermore, problems occur when tests were performed with a low water depth over theberm resulting in unreasonably high relative values for the forces and wave heights For anyoccurrence probabilities of impacts higher than 1% it is therefore recommended to also usethe method described in the sucessive section.

10.4 Statistics of relative wave forces

A statistical distribution of the relative impact forces F*

h,max is needed in order to allow for achoice of exceedance or non exceedance values for the relative impact forces Following dis-cussions and exchange of data within PROVERBS, a Generalized Extreme Value distribution(GEV) is proposed (see example for small-scale data in Fig 5) The cumulative distribution

function (cdf) of the GEV distribution can be written in its standard form as follows (Johnson

and Kotz, 1995):

0.1 10.0 50.0 80.0 90.0 95.0 97.5 99.0 99.5 99.9 99.99 4.0

n = 260 " = 3.7816

ß = 8.7819 ( = 0.0381 *)

s = 0.0917 Kolm.-Sm Test:

Dn = 0.000 " = 0.01 (%) " = 0.025 (%) " = 0.05 (%)

exp & ˜x @ exp & exp & ˜x

(13)

In Eq (11) the standardized x-parameter ˜x can be written as follows:

The probability density function (pdf) of the GEV in its standardized form can be given as:

where parameters ", $ and ( can be taken from model tests similar to the structure to bedesigned or can generally be estimated as " = 3.97; $ = 7.86; and ( = -0.32 The latter valueswere derived from large-scale tests with a 1:50 foreshore slope and almost non-overtoppingconditions (GWK, 1994) and are used in PROVERBS for all probabilistic calculations ofstructures where impact waves are considered Further advice for different bed slopes have

been given by McConnell and Allsop (1998) so that the following parameters for further use

can be suggested:

Tab 3: Values of ", $, and ( for GEV distribution of relative horizontal force

For details on HR94 and HR97 tests see Tab 2

In Kortenhaus (1998) the influence of the number of waves on statistical parameters of the

various distributions available has shown that the number of data points should be not lessthan about 250 The values given in Tab 3 for bed slopes of 1:7 and 1:10 should therefore beconsidered carefully More details on the influence of geometric and wave parameters on

statistics of wave impact forces can also be found in Kortenhaus (1998).

Furthermore, Eq (11) has been used to plot data from other wave flumes where significant

differences were observed resulting in much higher relative forces (McConnell and Allsop,

1998) These differences are assumed to be mainly due to the differences in logging

fre-quencies (GWK: 100 Hz; WKS: 600 Hz; HR94: 400; HR97: 1000 Hz), the different number

of waves per test (GWK: 100; WKS: 100; HR94: 500; HR97: 1000) and the different totalnumber of impacts as given in Tab 3

Eq (11) can be transformed using Eq (12), yielding the force as a function of the probability

of the horizontal impact force:

where P(F*h,max) is the probability of non exceedance of relative impact forces which generallymay be taken as 90% and F*

h,max is the relative horizontal force at the front face of the structurenon dimensionalised by DgHb2 The maximum horizontal force can then be calculated by :

1) The triangular shape is derived from the actual force history based on the equivalence ofbreaking wave momentum and force impulse.

trFh ' k@ 8.94 deff/ g

where Hb is the wave height at breaking (Eq (1)) and D is the density of the water

10.6 Calculation of impact force history

A full impact force history is needed to account for the temporal variation of the forces andpressures induced by the breaking wave For practical reasons the typical impact force historymay be reduced to a triangle1) (Fig 6) which is described by the rise time tr and the total dur-ation of the force td as discussed below

10.6.2 Rise time t r

Following considerations derived from solitary wave theory (Oumeraci and Kortenhaus,

1997) a relationship between relative impact force peak F*

h,max = Fh,max/DgHb2 and the rise time

trFh can be derived It is proposed to use the following equation for trFh:

The effective water depth in front of the structure deff can be assumed to be identical to thewater depth in which the wave breaks and may be calculated as follows:

dFh I dFh

where Brel is the part of the berm width which influences the effective water depth (Brel = 1 for

no berm width):

and mrel is the part of the berm slope influencing the effective water depth (mrel = 0 for simplevertical wall):

Specific details on the most relevant parameters involved can be found in Cooker and

Peregrine (1990).

Preliminary analysis by early PIV measurements in 1994 (Oumeraci et al., 1995) and more detailed analyses of breaker types at University of Naples (Vicinanza, 1998) together with

University of Edinburgh derived from random wave trains have shown that

@ even though there is some scatter the k-parameters for waves breaking over low andhigh berms are in the same range;

@ a mean value of k = 0.205 ± 20% (standard deviation of 11%) can be assumed for allplunging breakers regardless the relative height of the berm

Eq (16) is compared to random wave data from three different model tests in Fig 7 All datasets have been re-analysed where the following filters were applied:

@ the highest 10 waves of each test;

@ only breaking waves of the highest 10 waves (following the criteria as given by

Kortenhaus and Oumeraci, 1997);

@ the total duration td is shorter than 6 times the rise time tr (to avoid unreasonable results)These filters reduce the scatter considerably and will lead to some other formulae than

indicated by previous papers (Oumeraci and Kortenhaus, 1997) Fig 7 shows that the

proposed formula represents a curve fitted to most of the random data sets whereas an even

better fit is obtained for solitary waves (Oumeraci and Kortenhaus, 1997) Data from

McConnell and Allsop (1998) have not yet been plotted using the aforementioned filters but it

is assumed that the high values found in these data are reduced considerably as well

0.0 0.5 1.0 1.5 2.0 2.5 15.0

HR 1994

No of tests = 239

McConnell & Allsop, 1998

Eq (16), k = 0.205

Fig 6: Comparison of prediction formula to large-scale measurements (random waves)

The equivalent 'triangular' rise time tr (see Fig 6) is assumed to be much shorter for breakingwaves, especially when rise times are very short For longer rise times it is expected that trian-gular rise time tr and measured rise time trFh are in the same range For non breaking waves itmay be expected that the rise time is no longer than a 1/4 of the wave period whereas forbreaking waves much shorter relative rise times can be assumed It is, however, extremelydifficult to derive a clear relationship between both values so that the ratio trFh/tr was derivedfor the GWK, WKS and some of the HR94 data and a statistical distribution was plotted(Fig 8)

Fig 7: Probability distribution for triangular rise time ratio for horizontal forces

This best fit was achieved by a log-Normal distribution with a mean value of 1.487 and a

stan-dard deviation of 0.667 The mean value is higher than what was found in Oumeraci and

Kortenhaus (1997) which is most probably due to that only breaking waves are included in the

present analysis

d) Probabilistic approach

For probabilistic calculations the aforementioned uncertainties in the relations of rise time totriangular rise time and triangular rise time to relative impact force were considered together.This was achieved by defining a factor k' which summarizes k*8.94 (right side of Eq (16))and the relation of the measured rise time trFh (left side) and the 'triangular' rise time tr (assum-ing a constant relationship) Eq (16) will then read:

tr ' k)@ deff/ g

From statistical analysis of the unfiltered data the factor k' can best be described by a

LogNormal distribution with a mean value of 0.086 and a standard deviation of 0.084 (Van

HR 1994

No of tests = 239

t = 2*td r

Eq (22)

Fig 8: Triangular impact duration td vs relative triangular rise time tr for random waves

Gelder, 1998) These values are again based on results from large-scale hydraulic model tests

which are believed to best represent the situation under prototype conditions

10.6.4 Total duration t d

b) Deterministic approach

The relationship between rise time and total duration of the impact is dependent on thebreaker type The aforementioned filter which was applied to the data sets has led to a newapproach describing the total duration of impact forces:

In Fig 9 this relation is examplarily plotted for random wave tests According to what wasexpected the total duration is rarely smaller than 2.0, i.e the decay time of the impact isusually longer than the rise time For longer relative rise times the total duration is close to 2.0whereas for shorter rise times the factor can increase significantly When ignoring very sharppeaks (and thus very high ratios of td/tr) Eq (22) gives a good estimate of the upper bound ofthese data

Upper bound approaches as given in the previous section are not applicable for probabilistic

design Therefore, a relation of rise time and total duration was derived by Van Gelder (1998)

where td can be calculated from tr by:

Eq (23) is dimensionally incorrect but has given the best correlation of the data In Eq (23) c

is a random variable (dimension: [-s@ln(s)]) with a Gaussian distribution which can be given

by its mean value (c = 2.17) and its standard deviation (F = 1.08) Again, these valuescorrespond to large-scale measurements without any filtering of the data Different parameterswere found from other (small-scale) tests which indicates that filtering of the data as indicatedabove would be useful for this statistical approach as well

10.6.6 Force impulses I hr and I hd

Force impulses are more relevant to the response of the overall structure than the impactforces and should therefore be calculated and probably used for the selection of the worstdesign situation of the brakwater Since rise time (or total duration, respectively) and themaximum force are known (Eqs (21) and (23)), the force impulse over the rise time Ihr can beobtained from:

and the horizontal force impulse over the total duration Ihd can be calculated as follows:

Fig 9: Simplified Pressure Distribution at a Vertical Wall

10.8 Pressure distributions at the wall

Two types of pressure distributions were proposed and discussed within PROVERBS one ofwhich is based on extensive large-scale testing of waves breaking at a vertical wall whereasthe other is derived from small-scale tests of a composite breakwater with extensive variation

of geometric and wave conditions Both distributions start with the maximum force at the wall

as the dominant input parameter so that the overall loading of the structure is identical in bothcases It should be noted that results from these pressure distributions are not used forprobabilistic calculations Both approaches are described in the following

10.8.2 Distributions from vertical wall tests

Based on the analysis of almost 1000 breakers of different types hitting a vertical wall, thesimplified distribution of impact pressure just at the time where the maximum impact forceoccurs, can tentatively be determinated according to Fig 10 Three or four parameters need to

be calculated in order to describe the pressure distribution: (a) the elevation of the pressuredistribution 0* above design water level; (b) the bottom pressure p3; (c) the maximum impactpressure p1 which is considered to occur at the design water level; and (d) the pressure at thecrest of the structure if overtopping occurs