CHAPTER 5.1: WAVE IMPACT LOADS - PRESSURES AND FORCES -

Transcription

1 CHAPTER 5.1: WAVE IMPACT LOADS - PRESSURES AND FORCES - A. KORTENHAUS 1) ; H. OUMERACI 1) ; N.W.H. ALLSOP 2) ; K.J. MCCONNELL 2) ; P.H.A.J.M. VAN GELDER 3) ; P.J. HEWSON 4) ; M.WALKDEN 4) ; G. MÜLLER 5) ; M. CALABRESE 6) ; D. VICINANZA 6) 1) Leichtweiß-Institut, Technical University of Braunschweig, Beethovenstr. 51a, DE Braunschweig, Germany 2) 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 Netherlands 4) University of Plymouth, School of Civil and Structural Engineering, Palace Street, GB-Plymouth PL1 2DE, U.K. 5) Queens University of Belfast, Department of Civil Engineering, Stranmills Road, GB-Belfast BT7 1NN, Northern Ireland 6) Università degli Studi di Napoli 'Frederico II', Dipartimento di Idraulica, Via Claudio n. 21, IT 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 lever arms of the forces on the front face and the bottom of a vertical breakwater. For this purpose, (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 background and data information. Suggestions for estimating the forces on a caisson in feasibility studies are also given

2 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. 2. INTRODUCTION 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

3 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES 4. OVERVIEW OF RECENT WORK There are a number of formulae available for different types of waves breaking at the structure. These formulae generally include magnitudes of maximum pressures, their distributions and forces. In some cases, uplift pressures are given as well. All formulae are fully empirical 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 in the respective references. Tab. 1: Overview of design methods for wave loading Author Year Pressures Sainflou 1928 yes Miche-Rundgren Forces Uplift Comments Quasi-Static Waves yes, but difficult no vertical wall, no berm yes yes no design curves from SPM, 1984 Goda 1985 yes yes yes Impact Waves most-widely used design method Hiroi 1919 yes yes no vertical wall Bagnold conceptual model only Minikin 1963 yes yes no Ito 1971 yes yes yes Blackmore & Hewson 1984 yes yes no sometimes incorrect dimensions! Partenscky 1988 yes not given no air content of wave needed Kirkgöz yes yes no vertical wall only Takahashi 1994 yes yes yes extension of Goda model Allsop et al no yes yes Walkden et al no yes no relation of forces and rise time Oumeraci & Kortenhaus 1997 yes yes yes time-dependent approach! - 3 -

4 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. Author Year Pressures Forces Uplift Comments McConnell 1998 no yes no amendment of O&K, 1997 Hull & Müller 1998 yes yes no amendment of O&K, 1997 Vicinanza 1998 yes yes no amendment of O&K, 1997 Broken Waves SPM 1984 yes yes no vertical walls only Camfield 1991 yes yes no amendment of SPM, 1984 Jensen 1984 yes yes yes Crown walls Bradbury & Allsop 1988 yes yes yes Crown walls Pedersen 1997 yes yes yes Crown walls Martín et al yes yes yes Crown walls This paper is concentrated on calculation of pressure distribution and related forces under impact conditions. Furthermore, the dynamic characteristics of impact forces were considered essential 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 the statistical distributions of impact and uplift pressures, the vertical pressure distribution at the front face, and the relation between rise time and duration of impact forces. 6. OVERVIEW OF DATA SETS Different hydraulic model tests have been carried out and analysed to obtain the design method proposed in this paper. These tests are summarized in Tab. 2 where the most important information is given. Furthermore, references are added where more detailed information on these tests is available

5 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES Tab. 2: Overview of hydraulic model tests (random waves) Tests Year Co n- fig. 1) Scale f sam [Hz] Waves Slope No. tests WKS : : Imp. 2) Upl. 2) References Oumeraci et al., 1995 GWK 1993/94 1 1: : McConnell & Kortenhaus, HR : : PIV : : HR : :50 1:20 1:10 1:7 9 - QUB : : ) number of configurations tested; 2) number of transducers Oumeraci et al., 1995 McConnell & Allsop, 1998 Kortenhaus & Löffler, 1998 It may be assumed from the differences in the number of waves per test and the acquisition rate 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 other which will be explained in more details in the successive sections. 8. PREPARATORY STEPS - 5 -

6 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL 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, Chapter 2.4.1) without any account for load duration can be used (Fig. 2a); F h F h H b F h 3.0 F h D g H b T=wave period t/t t/t t/t (a) Standing wave (b) Slightly Breaking wave (c) Plunging breaker 0.0 "Pulsating load" (Goda-formula applicable) Impact load (Goda-formula not applicable) Fig. 1: Pulsating and impact load - problem definition (d) (f) (h) slightly breaking wave loads which already consist of some breaking waves but not significantly exceeding the Goda loads (Fig. 2b); an impact load for which new formulae including impact duration are to be used (Fig. 2c); and 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 more detail described in Chapter 2.2 of Volume IIa. Input for this map are geometric and wave parameters which in combination yield an indication of a certain probability that one of the aforementioned breaker types will occur

7 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES "Vertical" Breakwater h * b < 0.3 Composite Breakwater 0.3 < h * b< 0.9 Crown Walls Rubble Mound Breakwater h * b > 0.9 SWL h s d Low Mound Breakwater 0.3 < h * b< 0.6 H si L h s d B eq h b High Mound Breakwater 0.6 < h * b< 0.9 SWL Small waves H * s< 0.35 Large waves 0.35 < H s * Small waves 0.1 < H * s< 0.2 Large waves 0.2 < H * s< 0.6 Small waves 0.1 < H * s< 0.2 Large waves 0.25 < H * s< 0.6 Narrow berm 0.08 < B < * 0.12 Moderate berm w < B * < 0.4 Wide berm B * > 0.4 F h Quasi-standing wave F h * Slightly breaking wave Impact loads Broken waves * * F h F h F hmax F hmax F hq F hmax Fhq F hq F hmax Fhq t/t t/t t/t t/t * h * with F = F h h b= b * H s * B eq ; H s = ; B = ; h h s h D g H 2 s L b Fig. 2: PROVERBS parameter map 8.4. Breaker height at the structure A breaking criterion which accounts for the reflection properties of the structure has been suggested 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): H bc ' L % @ 1& C r 1% C r (1) where L pi is the wave length in the water depth h s for the peak period T p which can be calculated iteratively by: L pi ' L tanh 2B@ h s L pi (2) - 7 -

8 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. where L 0 is the wave length in deep water which can be taken as: L 0 ' g T 2 p (3) or can be approximated using the method given in Fenton (1990): L pi ' L tanh 2B@ h s L 0 3 / 4 2 / 3 (4) The reflection coefficient C r in Eq. (1) may be estimated as follows (Calabrese and Allsop, 1998): C r = 0.95 for simple vertical walls and small mounds, high crest C r = @R c / H si for low crest walls (0.5 < R c / H si < 1.0) C r = 0.5 to 0.7 for composite walls, large mounds, and heavy breaking The empirical correction factor k b can be estimated as follows: k b ' @ B eq / d 2 & @ B eq / (5) where B eq is the equivalent berm width which is defined as: B eq ' B b % h b 2 tan" (6) and B b is the berm width in front of the structure. Further details on this approach are given in Section 2.3 of Volume IIa

9 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES 8.6. Probability of occurrence of impacts The parameter map as given in Fig. 3 results in different branches where the probability of the respective breaker type is not known in advance. The branch of 'impact breakers' proposes to use an impact loading formulae which generally yields much higher forces than any other approach for quasi-standing waves, slightly breaking waves or broken waves. Hence, it is necessary 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 non impulsive Goda forces). The aforementioned method by Calabrese and Allsop (1998) described in Chapter 2.3 of Volume IIa also gives a simple formula for the probability of broken waves P b based on the idea that every wave with a higher wave height than the breaking wave height H bc (as calculated in Section 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: P b ' exp & 2@ H bc / H si 100% (7) The maximum wave height H bs which describes the transition from impact breakers to already broken waves can be described by Eq. (1) where B eq /d and C r are set to zero which then yields: H bs ' @ L tanh 2B h s L pi (8) The proportion of impacts can then be derived from: P i ' exp & 2@ H bc / H si 2 & exp & 2@ H (9) The magnitude of the horizontal force itself is strongly related to the type of breakers at the wall which are essentially depth limited. It can be expected that the magnitude is related to the relative wave height at the wall H si /h s. Eq. (9) can be regarded as a filter in the 'impact' domain of the parameter map. For very low percentages of impacts (smaller than 1%) the problem can be reduced to the quasi-static problem 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 be used

10 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. 10. WAVE IMPACT LOADING 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 wave height H si /d has been found to most significantly influence the wave forces non dimensionalized by the water depth over the berm. All forces were given at a 1/250 level thus taking the mean out of the highest two waves (500 waves per test were measured). The magnitude of the horizontal impact force can then be estimated from: F h, 1/250 ' 15@D W g d H si / d (10) This formula has been derived from data sets with a 1:50 foreshore slope and checked against other slopes where it also seems to fit the data reasonably well. In Fig. 4 data from three different model tests have been plotted and compared to the prediction method given in Eq. (10) GWK 1993/94 WKS 1993 HR 1994 Random waves 30.0 A&V prediction H si /d [-] Fig. 3: Relative wave force F h,max /(Dgd 2 ) plotted vs. relative wave height H si /d and comparison to calculation method given by 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 H si /d in between 0.35 and 0.6 whereas here the graph in Fig. 4 shows that it can be used up to

11 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES relative wave heights H si /d = 1.3. It is not possible to take into account the structural response (dynamic load factor) as rise time and duration of the force is not calculated. Hence, Eq. (10) provides a quick estimation of the expected horizontal force to the structure but cannot predict the length of its duration. Furthermore, problems occur when tests were performed with a low water depth over the berm resulting in unreasonably high relative values for the forces and wave heights. For any occurrence probabilities of impacts higher than 1% it is therefore recommended to also use the method described in the sucessive section Statistics of relative wave forces A statistical distribution of the relative impact forces F * h,max is needed in order to allow for a choice of exceedance or non exceedance values for the relative impact forces. Following discussions 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): F ( x ) ' exp & 1 & (@ x 1 ( with or (11) exp & exp & x with

12 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL GEV Parameter (ME): n = 260 " = ß = ( = *) s = Kolm.-Sm. Test: Dn = " = 0.01 (%) " = (%) " = 0.05 (%) *) scales x-axis Non exceedance probability Fig. 4: Statistical distribution of relative impact forces observed in the WKS for random waves In Eq. (11) the standardized x-parameter x can be written as follows: x ' x & $ " (12) The probability density function (pdf) of the GEV in its standardized form can be given as: f ( x ) ' 1 & (@ x 1 ( & exp & 1 & ( (13) exp & exp & exp & x where parameters ", $ and ( can be taken from model tests similar to the structure to be designed or can generally be estimated as " = 3.97; $ = 7.86; and ( = The latter values were derived from large-scale tests with a 1:50 foreshore slope and almost non-overtopping conditions (GWK, 1994) and are used in PROVERBS for all probabilistic calculations of structures 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:

13 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES Tab. 3: Values of ", $, and ( for GEV distribution of relative horizontal force Bed slope no. waves scale " $ ( 1: :20 / HR : :20 / HR : :20 / HR : :20 / HR 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 less than about 250. The values given in Tab. 3 for bed slopes of 1:7 and 1:10 should therefore be considered 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 frequencies (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 total number 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: F ( h, max ' " ( 1 & & ln P ˆF ( h, max ( % ß (14) where P(F * h,max) is the probability of non exceedance of relative impact forces which generally may be taken as 90% and F * h,max is the relative horizontal force at the front face of the structure non dimensionalised by DgH b 2. The maximum horizontal force can then be calculated by : F h, max ' F ( h, max@dgh 2 b (15)

14 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. where H b is the wave height at breaking (Eq. (1)) and D is the density of the water Calculation of impact force history A full impact force history is needed to account for the temporal variation of the forces and pressures induced by the breaking wave. For practical reasons the typical impact force history may be reduced to a triangle 1) (Fig. 6) which is described by the rise time t r and the total duration of the force t d as discussed below 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 = F h,max /DgH b 2 and the rise time t rfh can be derived. It is proposed to use the following equation for t rfh : t rfh ' k@ 8.94 d eff / g F ( h, max (16) The effective water depth in front of the structure d eff can be assumed to be identical to the water depth in which the wave breaks and may be calculated as follows: d eff ' d % B m h s & d (17) 1) The triangular shape is derived from the actual force history based on the equivalence of breaking wave momentum and force impulse

15 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES F h F h F h,max I rfh I dfh I rfh I dfh t rfh t t r t t dfh t d Fig. 5: Substitution of actual force history by an equivalent triangular load where B rel is the part of the berm width which influences the effective water depth (B rel = 1 for no berm width): B rel ' 1 for smalle 1 & 0.5 Bb for larger L (18) and m rel is the part of the berm slope influencing the effective water depth (m rel = 0 for simple vertical wall): m rel ' 1 for steeper slopes 1 for flatter slopes m (19)

16 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. b) Deterministic approach The k-factor used in Eq. (16) is dependent on the breaker type and the fluid mass directly involved in the impact process and can be expressed as follows: k ' M imp M tot ' I rfh I dfh (20) where M imp is the mass part involved in the impact; M tot is the total mass of the breaking wave moving towards the structure and I rfh and I dfh are the corresponding force impulses. This approach is based on solitary wave theory again and thus is regarded as preliminary for random waves. Detailed investigations on pressure impulses on vertical walls and breakwaters have been performed by Bristol University and references are given in Vol. I, Section 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 even though there is some scatter the k-parameters for waves breaking over low and high berms are in the same a mean value of k = ± 20% (standard deviation of 11%) can be assumed for all plunging 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 data sets have been re-analysed where the following filters were the highest 10 waves of each only breaking waves of the highest 10 waves (following the criteria as given by Kortenhaus and Oumeraci, the total duration t d is shorter than 6 times the rise time t r (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

17 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES 45.0 GWK 1993/94 WKS 1993 HR 1994 Random waves No. of tests = McConnell & Allsop, 1998 Eq. (16), k = Relative rise time t rfh /%(d eff /g) [-] Fig. 6: Comparison of prediction formula to large-scale measurements (random waves) The equivalent 'triangular' rise time t r (see Fig. 6) is assumed to be much shorter for breaking waves, especially when rise times are very short. For longer rise times it is expected that triangular rise time t r and measured rise time t rfh are in the same range. For non breaking waves it may be expected that the rise time is no longer than a 1/4 of the wave period whereas for breaking waves much shorter relative rise times can be assumed. It is, however, extremely difficult to derive a clear relationship between both values so that the ratio t rfh /t r was derived for the GWK, WKS and some of the HR94 data and a statistical distribution was plotted (Fig. 8)

18 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. 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 and a standard deviation of 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 to triangular 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 t rfh (left side) and the 'triangular' rise time t r (assuming a constant relationship). Eq. (16) will then read:

19 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES t r ' k d eff / g F ( h, max (21) From statistical analysis of the unfiltered data the factor k' can best be described by a LogNormal distribution with a mean value of and a standard deviation of (Van

20 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. 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 Total duration t d b) Deterministic approach The relationship between rise time and total duration of the impact is dependent on the breaker type. The aforementioned filter which was applied to the data sets has led to a new approach describing the total duration of impact forces: t d ' t 2.0 % 8.0@ exp &18@ t r / T p (22) In Fig. 9 this relation is examplarily plotted for random wave tests. According to what was expected the total duration is rarely smaller than 2.0, i.e. the decay time of the impact is usually longer than the rise time. For longer relative rise times the total duration is close to 2.0 whereas for shorter rise times the factor can increase significantly. When ignoring very sharp peaks (and thus very high ratios of t d /t r ) Eq. (22) gives a good estimate of the upper bound of these data Random waves No. of tests = 239 GWK 1993/94 WKS 1993 HR Eq. (22) t d = 2*t r Relative rise time t r /T p [-] Fig. 8: Triangular impact duration t d vs. relative triangular rise time t r for random waves

21 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES d) Probabilistic approach 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 t d can be calculated from t r by: t d ' &c ln ( t r ) (23) 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 values correspond to large-scale measurements without any filtering of the data. Different parameters were found from other (small-scale) tests which indicates that filtering of the data as indicated above would be useful for this statistical approach as well Force impulses I hr and I hd Force impulses are more relevant to the response of the overall structure than the impact forces and should therefore be calculated and probably used for the selection of the worst design situation of the brakwater. Since rise time (or total duration, respectively) and the maximum force are known (Eqs. (21) and (23)), the force impulse over the rise time I hr can be obtained from: I hr ' 1 F t r (24) and the horizontal force impulse over the total duration I hd can be calculated as follows: I hd ' 1 F t d (25)

22 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL Pressure distributions at the wall Two types of pressure distributions were proposed and discussed within PROVERBS one of which is based on extensive large-scale testing of waves breaking at a vertical wall whereas the 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 both cases. It should be noted that results from these pressure distributions are not used for probabilistic calculations. Both approaches are described in the following Distributions from vertical wall tests Based on the analysis of almost 1000 breakers of different types hitting a vertical wall, the simplified distribution of impact pressure just at the time where the maximum impact force occurs, 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 pressure distribution 0 * above design water level; (b) the bottom pressure p 3 ; (c) the maximum impact pressure p 1 which is considered to occur at the design water level; and (d) the pressure at the crest of the structure if overtopping occurs. F h (t) DWL = Design Water Level F h,max R c = Freeboard p 4 0 = 0.8@ H b * p 1 R c DWL F h,max F h (t) l Fh d d c t r t t p 3 = 0.45 p 1 t d Fig. 9: Simplified Pressure Distribution at a Vertical Wall

23 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES b) Elevation of pressure distribution 0 * The elevation of pressure distribution 0 * may be calculated from the following tentative formula (see Fig. 10): 0 ( ' 0.8 H b (26) d) Bottom pressure p 3 The bottom pressure p 3 may be derived as a function of the maximum pressure at the height of the still water level as follows (see Fig. 10): p 3 ' 0.45 p 1 (27) f) Maximum pressure p 1 The maximum impact pressure p 1 can be calculated directly from the equivalent force history (Fig. 10), since F h (t) represents the area of the pressure figure at any time of the history (assuming an infinitely high wall): F h ( t ) ' 1 2 p 1 ( t )@ 0.8@ H b % d % d c (28) Substituting Eq. (27) in Eq. (28) yields: p 1 ( t ) ' F h ( t ) 0.4@ H b % 0.7@ d % d c (29) h) Pressure at the crest of the structure p 4 If the waves in front of the structure are high enough, overtopping is expected to occur. This will reduce the total impact force as parts of the energy of the wave will get lost. This effect can be taken into account by cutting off the pressure distribution at the top of the structure (Fig. 10) so that the pressure at the crest of the structure can be described as follows: 0 für 0 ( < R c p4 ' 0 ( & R c 0 ( p1 für 0 ( $ R c (30)

24 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. The horizontal impact force 1) is then reduced to: F h,max,ov ' F h,max & 1 0 ( & R p 4 (31) Pressure distributions from breakwater tests Very recently, Hull et al. (1998) proposed a different pressure distribution based on the HR94 data set (see Tab. 2). This distribution is given in a non dimensionalized form only dependent on the maximum pressure observed at still water level, p max, the height of the berm, h b, and the height of still water level at the toe of the foundation, h s. Fig. 11 shows this pressure distribution which is separated into three areas: pulsating, impact, and wave runup. This distribution has been derived from all composite type breakwaters tested (10 different configurations) and can therefore be used for any structural configuration. The z- coordinate at the wall is obtained for a very high wall (no overtopping) as follows: (0 / 1.2) Wave run-up (8 / 0.4) DWL Pulsating (0 / -0.9) (40 / 0.17) Impact (40 / -0.25) (100 / 0) z ' y@ h b h 2 s (32) Fig. 10: P / P max [%] Vertical pressure distribution after Hull et al. (1998) where h b is the height of the berm, h s is the water depth at the toe of the foundation, and y is the vertical distance above or below still water level (positive upwards). All points of the distribution may be calculated using the coordinates given in the graph. The relative pressures in this graph are only dependent on the maximum pressure p max, and the vertical coordinate is only dependent on two known variables. Hence, the integration of this distribution is straightforward for a very high wall with no overtopping and relatively deep water over the berm. The summation of areas of the relative pressure distribution then yields: 1) Consequently, the statistical distribution given in Fig. 5 is no longer valid for this reduced force as the statistical distribution was determined for relatively high caisson structures and almost no overtopping

25 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES F h,max ' @ p h 2 s h b if R ( c > (33) where R c * is defined as relative crest height (= R b /h s 2 ) and d * is the relative water depth at the structure (= d'@h b /h s 2 ), d' being the distance from the bottom of the caisson to the still water level. The maximum pressure p max and the lever arm of the resultant force z can graphically be estimated from Fig. 12 using as input the relative water depth at the wall. For calculation, a distinction of cases will be necessary, if all types of wall shall be considered. The maximum pressure at the wall, p max, has to be calculated from F hmax to obtain the full pressure distribution. Therefore, the following steps are needed: Fig. 11: Relative maximum pressure at SWL and lever arm of forces as a function of relative water depth at the structure (Hull et al., 1998) 2) find equations to describe the integration of pressure distribution to give the maximum horizontal force dependent crest height R c, water depth d'; 4) rearrange these equations such that the maximum pressures can be derived from the maximum horizontal force. Regarding the first point Eq. (33) can be expanded to three equations where the first gives a reduction due to low crest level (still assuming that d * > 0.9):

26 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. F h,rc p h 2 s h ' p h 2 s h % 0.04 R ( c p h 2 s h % 0.24 R ( c (34) The second equation calculates a further reduction due to lower relative water depths 0.25 < d * # 0.9: p h 2 s h % 0.2 d ( F h,d ) ' p h 2 s h % 0.04 R ( c % p h 2 s h % 0.24 R ( c % (35) and the third gives the force for low water levels over the berm (d * < 0.25): p h 2 s h % 0.7 d ( F h,d ) ' p h 2 s h % 0.04 R ( c % p h 2 s h % 0.24 R ( c % (36) The case where R c * is smaller than 0.17 is not considered here as this is very unlikely to happen. For this case it can be assumed that no impacts will occur as the crest height is too low (waves will heavily overtop the structure). From Eqs. (34) to (36) the maximum pressure can be easily calculated by solving these equations for p max. For all cases the pressure distribution can then be calculated using the relative coordinates of Fig. 11 and considering the relative crest height and the relative water depth, respectively

27 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES Lever arm of horizontal force Finally, the lever arm for the horizontal impact force can be calculated from the pressure distribution as given in Fig. 10 at the wall as follows: l Fh (t) ' p ov% 3@p ov % 3@p ) 6@F h ( (37) In Eq. (37) 0 ov is defined in relation to the height of the wave crest to the wall (see Fig. 10): 0 ov ' min 0 ( ; R c (38) and d' is defined as: d ) ' d % d c (39) In case of the second pressure distribution described under Section the analytic calculation of the lever arm of the horizontal force is rather complicated and very much dependent on the relative water depth and crest height. For simplicity reasons it may be either assumed that the force attacks at the height of the still water level or the height may be taken from Fig WAVE UPLIFT LOADING The procedure for calculating the uplift pressures and forces is similar to the procedure for calculation of horizontal components in Section 5. A Generalized Extreme Value (GEV) distribution was also found to fit the data and is consistent to the method used for impact forces (see example for small-scale test in Fig. 12). The GEV distribution function has already been defined in Eqs. (11) and (13). Parameters for application of the GEV to relative uplift forces can be taken from large-scale tests (for quasi-non overtopping conditions " = 2.17; $ = 4.384; ( = -0.11). The same procedure as already used for horizontal forces can be applied to derive the relative uplift force resulting in:

28 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. F ( u, max ' " ( 1 & & ln P ˆF ( u, max ( % ß (40) GEV Parameter (ME): n = 255 " = ß = ( = *) s = Kolm.-Sm. Test: Dn = " = 0.01 (%) " = (%) " = 0.05 (%) *) scales x-axis Fig. 12: Non exceedance probability Statistical distribution of relative uplift forces observed in the WKS for random waves Calculation of uplift force history Similar parameters are needed to describe the uplift force history than already used for the impact forces (see Figs. 6 and 17): (b) t ru = triangular rise time; (d) t du = triangular total duration of maximum uplift force; It should be stressed that the times listed above are not necessarily identical to the times for the impact loads. If it is assumed that t 0 is identical for both impact and uplift loading the time difference between the peaks can be calculated as follows (McConnell and Kortenhaus, 1997): dt Fu ' t rfu & t r (41)

29 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES Rise time t ru b) Deterministic approach The following relationship between relative uplift force peak F * u,max = F u,max /(DgH b 2 ) and rise time t r was derived from Fig. 14 following the procedure for impact loads: t rfu ' k 8.94 d eff / g F ( u, max (42) 45.0 GWK 1993/94 WKS 1993 HR 1994 Random waves No. of tests = Eq. (42), k = Fig. 13: Relative uplift force versus relative rise time (GWK data) Relative rise time t rfu /%(d eff /g) [-] The effective water depth d eff can be assumed to be identical to the water depth in which the wave breaks and may be estimated by the procedure described for impact loads (Eq. (17)). The k u -factor used here is dependent on the breaker type and the part of the mass taking part in the impact process, and will be set to k u = k = ± 20% identical than for the impact forces. From Eq. (42) the total duration t rfu can then be calculated as follows: t rfu ' 1.83@ d eff / g F u, g@ H 2 b (43)

30 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. The relation between t ru and t rfu was derived as a probability distribution again (see section for horizontal forces). For uplift forces, however, the distribution is no longer a log- Normal distribution but better described by a Generalized Extreme Value distribution (Fig. 15). The mean value of this ratio is 1.11, thus again larger than the value proposed by Oumeraci and Kortenhaus (1997). This is again due to the filtered values where only breaking waves are taken into account. Fig. 14: Probability distribution for triangular rise time ratio for uplift forces d) Probabilistic approach For probabilistic design the same equation than for horizontal forces (Eq. (21)) was used for uplift forces as well. The k' u parameter (for uplift forces) corresponds to k' (for impact forces) and has a mean of 0.16 and a standard deviation of

31 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES Total duration t du b) Deterministic approach The triangular total duration of the uplift force t du can be obtained from the rise time t ru by using the following relationship similar to Eq. (22) for horizontal forces: t du ' t 2.0 % 8.0@ exp &18@ t ru / T p (44) This relation was obtained from all data sets again (Fig. 16) and shows a large scatter. Eq. (44) gives the upper bound of this relation which might be used as a conservative approach GWK 1993/94 WKS 1993 HR 1994 Random waves No. of tests = Eq. (44) t d = 2*t r Relative rise time t ru /T p [-] Fig. 15: Relative triangular total duration versus relative triangular rise time (GWK data) d) Probabilistic approach

32 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. Again, Van Gelder (1998) has proposed the same formula than for impact duration and rise time (see Eq. (23)). t du ' &c u ln ( t ru ) (45) where c u is again normally distributed with a mean value of 1.88 for large-scale tests and a standard deviation of Given the rise time of the uplift force and the total duration, the force history can be calculated for each time step by interpolating the times between t 0 and the uplift force maximum at the time t ru and the total duration t du (Fig. 17) Pressure distributions Uplift pressures underneath vertical breakwaters should generally be calculated using the approach described in Section where the instantaneous pore pressure development underneath the breakwater is described. However, a very simple approach was derived empirically from the data available and is based on hydraulic model test data using 'upper bound' envelopes which may lead to conservative estimates. Therefore, all results should be compared to the pressures derived by using the Goda model (see Vol. I, Section 2.4.1) and are expected to be larger than those. It was observed from hydraulic model tests that the uplift pressure distribution (Fig. 17) should at least be digitized in three points (Kortenhaus and Oumeraci, pressure at the seaward edge of the structure p u pressure at the shoreward edge of the structure p ru (as it is not necessarily zero); pressure at about 25% of the structure width from the seaward edge p mu (as the maximum of the uplift force was observed to occur when the shock wave traveling underneath the structure has reached this point)

33 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES F u (t) DWL = Design Water Level R = Freeboard c B c F u,max R c DWL F u (t) d d c t ru t t p u p ru p mu t du l Fu 0.25@B c F u Fig. 16: Approximate pressure distribution for temporal development of pressure distribution For impact forces the pressures were given as functions of the impact force itself and the pressure distribution was assumed to remain constant in itself. A similar principle will be followed here but the number of points to form the pressure distribution is reduced to only two for simplicity reasons. Furthermore, the pressure underneath the landward side of the structure will be calculated using relative wave parameters. The pressure distribution is assumed to be constant in itself over the time Pressure at the shoreward edge of the structure p ru Different pressure heads underneath the shoreward edge of the structure were reported from model tests and prototype conditions in PROVERBS where many times pressures up to the same magnitude than at the seaward side were measured (rectangular pressure distribution). Explanation of the processes involved have been provided by Van Hoven (1997), Hölscher et al. (1998), and Peregrine (1997) which indicates that the most relevant parameters responsible for non zero pressures are the exit area, the foundation material and the water depth behind the structure. However, most of these parameters were kept constant over the tests so that analysis of the data did not take this into account

34 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. The inital analysis of available data for pressures underneath the shoreward edge of the breakwater related these pressures to the wave parameters by applying an upper envelope to the data for the time of the maximum uplift force (Fig. 18). From this the following empirical formula was achieved: p ru D@ g@ H b ' H b h s & 0.1 (46) 0.90 GWK H b /h s [-] Fig. 17: Uplift pressure underneath the shoreward edge of the breakwaters in GWK and WKS which can easily be transformed to: p ru ' D@ g@ H H b h s & 0.1 (47) For all times 0 # t ru # t du the respective values may be interpolated from the above following the same principle than already used for the uplift force history (Fig. 17)

35 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES Pressure at the seaward edge of the structure p u Since the pressure at the shoreward side of the structure is known the pressure underneath the seaward edge p u can be calculated as follows: p u ' 2@ F u,max B c & p ru (48) where B c is the structure width, F umax is the maximum uplift force and p ru is the pressure at the shoreward side of the structure Lever arm of uplift force Finally, the lever arm for the uplift force can be calculated from the pressure distribution underneath the structure for each time step as follows: l Fu ( t ) ' B 2 p ru % 2@ p u 6@ F u,max (49) 14. AERATION OF IMPACT WAVES All results discussed so far have been achieved from model tests at different scale using fresh water. These results are very difficult to directly apply to prototype conditions as there are a lot of additional factors which should be taken into account. One of these phenomena which is most difficult to account for is the aeration of both non breaking and impact waves. Crawford et al. (1997) have reported field and model measurements performed with both fresh and sea water. The different behaviour of air bubbles in the breaking process of waves hitting a (almost) vertical breakwater is described, as well as effects of this for non breaking waves, similar results were obtained for fresh and sea water when generally very little aeration was observed in the reflected for impacting waves, field measurements have shown that high aeration levels coincide with long rise times and lower pressures whereas short duration high peaked pressures were also observed, but do occur at lower aeration comparing results with fresh and sea water under laboratory conditions has shown that pressures are generally higher with longer rise times and vice versa, thus following the observations made in the field;

36 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL. These results have led to some simple guidance on estimating the aeration of impact waves and its influence on the impact force (Hewson et al., 1998) which might be adopted for design purposes. The aeration in impact waves can be calculated to: P a ' 2.0 % 5.3@ N i (50) where N i is the number of breaking waves per minute and P a is the percentage of aeration in the breaking wave given in percent. The number of breaking waves per minute, however, is not known in advance but may be estimated for model tests using the percentage of breaking waves in a test P i (Eq. (7)): N i ' P N W t tot (51) where N W is the number of waves in a test, and t tot is the total length of the test given in minutes. Under prototype conditions N W may be replaced by the number of the waves in a storm whereas t tot is the duration of the design storm. From the aeration percentage obtained by Eq. (50) a force reduction factor k fa according to Hewson et al. (1998) can be calculated as follows: k fa ' % P a 97.5 & P a (52) It has also been shown that the total force impulses seemed to remain independent from the aeration level of the breaking wave. Assuming this impulse to be more or less equal to the triangular impulse as given by Eq. (24) longer rise times due to aeration can be calculated as the inverse of the force reduction factor: k ta ' 1 k fa (53) Since uplift and impact loading are strongly coupled it can be expected that the uplift force will be dependent on the aeration in the same way though no data supporting this are yet available. The working assumption therefore is to use Eqs. (50) to (53) for uplift forces as well

37 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES 16. CONCLUDING REMARKS The extremely complex phenomena of wave breaking at a vertical wall has been investigated in PROVERBS. Hydraulic model tests, numerical modelling, field measurements and desk studies have been performed to improve the physical knowledge of the phenomena involved. Significant progress has been achieved related to this problem and eventually have led to improved design procedures for impact loading which are summarized in this and other sections of Chapter 5 in Volume IIa ('Breaking Wave Loads'). The calculation process within this section is graphically summarized in Fig

38 LWI / HR / DUT / UE / UoN / UoP / QUB A. KORTENHAUS ET AL

39 CHAPTER 5.1 WAVE IMPACT LOADS: FORCES AND PRESSURES Geometric Conditions Fig. 1 Wave Conditions Vol. IIa, Chapter 2 Parameter Map Fig. 3 H bc Eq. (1) Impact Filter Section 4.3 P i Eq. (9) Quasi-Static Loading Impact Loading Volume IIa Chapter 4.1 Horiz. initial force calc. (Eq. statistics of rel. force (Fig. 5, Eq. calc. of force history (Eqs. (21), reduction by aeration (Eqs. (50) to (53)) Uplift statistics of rel. force (Fig. 11, Eq. calc. of force history (Eqs. (21), reduction by aeration (Eqs. (50) to (53)) Press. Distr. Vert. Wall (Eqs. (26) to (30)) Press. Distr. Breakwater (Eqs. (32) to (36)) Press. Distr. Uplift (Eqs. (46) to (48)) Parameter Output p 1(t) to p (t) 4, p u (t), p (t) ru, F h(t), F (t) u, M h(t), M (t) u, l Fh, l Fu Fig. 18: Overview of calculation scheme for impact loading (probabilistic approach)