I'. . l'.A ö -v . = r e ;. ; o i o o C 3 Li -< cc C j cc «•< -! y t -J ï : o x E C L m l i 6 J MB — M
" [ " R I S K C R I T E R I A I N D E S I G N
S T A B I L I T Y O F S L O P I N G S T R U C T U R E S I N
R E L A T I O N TO £ = t a n a
Per Bruun,
A ! i Riza Gün bak
Trie Norwegian institute of Technology
Introduction
Existing design formulas for rubble-mound breakwaters under wave attack contain the wave height, H, and the slope angle, cc, as parameters of the wave
characteristics (for d/H > 3.0, d being the water depth in front or the structure) and structure characteristics (incl. permeability, roughness, interlocking-, etc.) excluding the effect of wave period. When compared, they are .mostly inconsistent with their definition of stability and generally used the doubtful method of counting blocks that left their original location in the slope. A theory was there-fore developed for the importance of the wave period, T , on the stability of rubble-mound breakwaters. Its validity is shown using the available laboratory data of Ref. (T) and by some experiments recently made in Norway (7, 15).
In connection wave run-up, run-down, breaking characteristics were summarized tan a
using the parameter £ = ^/^HTT"
-Flow in Sloping Structures General
The pertinent hydrodynamic and structural factors involved in the stability of sloping structures smooth as well as rubble-mounds include water depth in front of the structure, d, wave height, H, and period, T, the accompanying breaker, run-up, Ru, and run-down, Rd, characteristics, the geometry and design of the structure, including its slope, a, roughness, block size, permeability and pattern of placement of the armour. In the tests referred to below the d/H ratio w a s > 3 but the results and principles developed are qualitatively correct also for ratios 1.5 < d/H < 3.0.
Breaker Types
types of breakers occurring on a slope which could be a.-neach or any breakwater whether impermeable or permeable. The character of breaker depends upon the depth, the slope angle,cc.as well as on wave characteristics including H, and T and combinations of these factors.
Using Battjes (2) and Hunts (21) parameter £ = t a n a / ^ / H / L o
(earlier used by Iribarren as parameter for breaking criteria), one gets the following ranges for different breaker types as functions of £ :
Using the offshore surf parameter, £0 = tana/^/H^T^"
Breaker Type Limiting Criteria Surging or Collapsing if 3.3 < £0
Plunging if 0 . 5 < £o< 3 . 3 . .
SpiSiing if £o< 0 . 5
Using the breaker surf parameter, £b = t a n c / ^ / R7/ ^
Breaker " y e s Limiting Criteria Surging or Collapsing if 2.0 < £5
Plunging if 0 . 4 < £b< 2 . 0
Spilling if £b< 0 . 4
Iribarren and Nogales (42) gave the limiting criteria for wave breaking on a slope from their semi-theoretical analysis.
Their limit corresponding to
£c r = (tanix ) « 2.3
0 cr
defines the condition halfway between the limit of complete reflection and complete breaking.
Wave heights at breaking are e.g. dealt w i t h in.refs. (15, 44). Run-up, Run-down
Wave run-up, Ru, is defined as the height of the water table above SWL at the highest point of its advance on the slope.
Theoretical run-up calculations mainly investigate the behaviour of a bore on a slope. Most theories (12, 19, 3 1 , 42) describe the phenomena by a non-linear long 2
wave theory. In this :ect they use the method of characteristics which was first introduced by Stoker iJ40).
Tests mentioned in ref. (35) revealed that run-up is independent of the water depth in front of the structure, d, when the d/H ratio i s > 3. For d/H < 3 run-up increases until somewhere between 1 < d/H < 2 (23).
Hunt (21) using available experimental data on wave run-up gave an empirical relationship:
Ru 2.3 t a n a , ,
which, using the ^-parameter, may be w r i t t e n :
2 ü = £ f o r ê < gc r ~ 2.3 (2)
H
Battjes and Roos (3) conducted experiments on smooth slopes and found that the run-up time, t u , from the still water level up t o the run-up point may be expressed as:
tu = T • 0.7 • £ ~/ 2 f o r 0 . 5 < £ < 2 . 0 (3)
When slopes are rough or are built as composite slopes w i t h a berm the run-up time increases and run-up decreases (5, 10, 17, 36). So does overtopping.
Wave run-down, Rd, is defined as the height of the water level above or below SWL at the lowest point of its recession on the slope. Its vaiue is positive when it is above SWL and negative when penetrates below SWL.
Battjes and Roos(3) present the following formula for Rd on smooth slopes based on experiments:
Rd = Ru (1-0.41) 0 . 5 < £ < 2 . 0 0 (4)
if this formula is applicable for all ranges of wave breaking (£ < 2.3) on smooth slopes, then run-down does not extend to levels beiow SWL and is always going to interact with the running up water from the next wave. The existence of this condition is analysed assuming the movement of a water mass along the smooth slope from the maximum run-up point down to SWL under the action of the gravity only. With such an assumption, assuming (t) is the time spent by this particle during its travel down the slope starting f r o m the t = 0 sec. at the maximum run-up point, and using formula (1), one obtains:
v / B I f l T / C O S a = % • g - sina- t2
27T
From which;
With an error of at most 5% one may approximate O - ct «s 1.0 for Cotcc<3.G which will simplify formula (5) to formula (6);
T- = ° '5 6 4 (6)
Using formula (3) the relative time, t j / T , left for the wave front to retreat to SWL without interacting w i t h the new run-up is:
t , / T = 1 - t u / T = 1 - 0 . 7 r1 / s (7)
When t / T > t j / T run-up and run-down will always interact above SWL. Using formulas (6) and (7) one may see that on smooth slopes run-down water will always interact w i t h run-up water above SWL for £ < 1.60. Although pressure forces and friction forces will retard the run-down and will cause higher £ values than 1.60 for the run-down to penetrate beiow SWL it seems from the analysis that this may occur with waves breaking on the slope.
Concentrating the interest on permeable slopes Figs. 2, 3 and 4 show the variation of run-up and run-down with £ on a smooth quarrystone cover, a rough quarrystone cover and a rough quadripod cover respectively where data above the £ axis shew relative wave run-up values, Ru/H, and data below £ axis show relative run-down values, Rd/H. The data were obtained on three different model scales by r e t (11) and all of these are included in the above figures. Also, water depth to deep water wave height ratio ( d / h0) was not always > 3.0, and run-up and
run-down may, as mentioned above, be affected by depth. Fig. 5 gives similar results for Dolos blocks (18). Data are available only for £ > 1.4. Qualitatively the remarks made for run-up and run-down w i t h increasing £ values also hold for these data.
Based on the above review it may be concluded that relative run-up, Ru/H, and run-down, Rd/H, on rubble mounds show a trend with £ for relative depth d/H > 3.0. Both increase w i t h increasing £ values as wave breaking goes from plunging towards collapsing and surging breakers and assume approximately a constant value for surging breakers of £ > 4.0.
With respect to overtopping the reader is referred to refs. (3, 5, 15, 17, 25, 33, 41 and 42)'.
Wave set-up and set-down have only a minor influence on wave run-up and run-down due to their limited magnitude. See refs. (4, 13, 22, 26, 30 and 32). Stability of Rubble Mounds
The most frequently used design-formula for breakwaters is the Irribaren formula, modified by Hudson (20) given by:
specific weight of stone specific weight of water stability coefficient
A great number of tables on K Q are given for regular waves, no overtopping and for certain specific breakwater profiles referring to different damage levels.
Ref. (43) states that different laboratories in the world list different K r j values for the initial damage. These differences are caused by lack of consideration to depth, friction and permeability and to the fact that tests were conducted at different ranges of these parameters. Refs. (6) and (7) give detailed analysis of the effect of friction and permeability on the stabiiity of rubble mound breakwaters.
It is, however, not logical to ignore the different flow characteristics occurring by assuming a constant stability coefficient Kpj for all wave periods. A hypothesis was therefore developed which includes the effect of period, using the knowledge of flow characteristics (6, 7).
The importance of wave period on the formation of beach (step and bar) profiles was discussed by Kemp (27, 28). His experiments showed that for low phase differences ( tu/ T < 0.3) a step profile and for high phase differences
( tu/ T > 1.0) a bar profile developed. A transition from step to bar profile exists
when t , / T is in between 1.0 and 0.3.
Bruun et a! (6) and (7) compare step profiles on the beaches to stabilized breakwater profiles. The stable breakwater profiles are cross-sections of some prototype breakwaters which obtained a stable cross-section after long duration wave action. The step beach profiles are taken from experimental data and converted to the prototype scale using model laws. It is concluded that a stable breakwater profile assumes a section similar to a beach step profile.
Sigurdsson (38), Sandstrom (34) and Hedar (16) mentioned the importance of run-down for the stability of rubble mound breakwaters. Carstens et al (9), from experiments w i t h irregular waves, found a relation between wave run-up and stability of rubble-mounds. Indeed run-up and run-down are closely related as explained below.
The Resonance Phenomenon
Thesocalled "resonance phenomenon" was first mentioned in ref. (6). On page 20 (6) the occurrence of this phenomenon is defined as: "Such a situation may-occur if the uprush-downrush period or what may be termed the down-rush period is equal to the wave period, assuming that down-rush is at its lowest position at the toe of the breaking wave so that every down-rush meets a breaking wave at the lowest position of the down-rush". The uprush-downrush period was defined as the time needed for the water front to travel f r o m the breaking point where the incoming wave front becomes vertical on the slope, up to the run-up level and back to the run-down level.
It was also indicated that, at resonance, the hydrostatic pressure from the core wnere 7 r
TW
K D
structure would be maximized causing uplift forces on 1 armour blocks. The
time history of the wave front along the slope above ST, water level (SWL) is drawn in Fig. 6. Three different conditions may occur in ail ranges of slope and wave characteristics.
Fig. 6a is the condition at which run-down will never go below SWL, and run-up and run-down always interact above SWL. On smooth slopes the above condition occurs for £ < 1.60. This range of £ values includes plunging and spilling breakers. On rubble mounds the condition described in Fig. 6a may only occur at £ values much lower than 1.60, due to permeability.
Fig. 6b presents the interaction of run-up and run-down at SWL. On smooth slopes this corresponds to a £ value of min. 1.60. On permeable slopes £ will be lower, (Figs. 2, 3, 4, 5).
Fig. 6c shows that run-down may reach the breaking point and be completed before the arrival of the next wave, or it may interact with the run-up below SWL. On smooth slopes this corresponds to £ > 1.60. Fig. 6c includes the above definition of resonance. Fig. 6d gives the description of resonance as defined above, in terms of time history where point " B " refers to the breaking point and the dotted line is the wave profile at maximum run-up. In this study, the breaking point refers to the point on the slope where the wave front becomes vertical.
From the above mentioned, it is known that breaking occurs approximately when £ < 2.3. This condition, together w i t h the above deductions restricts resonance t o the range 1.60 < £ < 2.30 for smooth and t o £ < 2.3 for permeable slopes.
Analysis of Figs. 2, 3, 4 and 5 show that run-down did not reach its maximum value at £ « 2.3 on rubble mounds. Therefore, the above definition of resonance had to be changed.
The importance of resonance comes from its relation to maximum forces on sloping structures due t o the kinematic conditions occurring below the breaker causing lift forces. Strong drag and inertia forces by high run-up and run-down values occur. Impact forces.also maximize around resonance. The mean water table elevation in the core builds up due to high run-up values which cause outward pressure on the armour.
Iversen (24) measured the velocities under a breaking wave. So did Kemp for beach profiles (27, 28, 29), Wiegel (45) calculated the horizontal accelerations occurring in a breaking wave using the data given in ref. (24). High horizontal forward velocities exist under the breaking wave crest. They interact w i t h run-down velocities causing rotational flow under the incoming breaking crest which will cause forces normal t o the slope directed outwards trying to pull the armour blocks out of their place (lift forces) as observed by Sandström (34) and Sigurdsson (38). The Hb/ db ratio increases w i t h £ causing higher breaker toe
velocities. Run-down velocities also increase w i t h £ in the range of breaking waves. As run-up increases, the build-up of hydrostatic forces inside the mound increases simultaneously. All these forces maximize at or close to — perhaps a littie earlier than — the above defined "resonance c o n d i t i o n " .
The maximizatic: j impact forces on slops around resonance condition may be explained as follows: (Fig. 7)
Assuming that a water mass plunges from the crest of a breaking wave w i t h a velocity of Cb = y g fHfci + z) and travels a distance " X p " along the slope under
the action of gravity oniy. Therefore, for y « 0 the fall time " tb" for this mass
may be w r i t t e n : and 2 ( Hb+ z - X p S i n & ) y, ( —B— : 5 h (9) or XpCosct = Cb - tb Xp = ( Hb +z ) ( ( S i n^+ 2 c C o° ^> Y 2-S i n c C) (10)
If this " X p " reaches a higher elevation on the slope than the run-down (i.e. run-down penetrates below point " p " in Fig. 7), the plunging crest will hit the bare slope. Otherwise it will hit a layer of water remained from the previous run-down which will cause high damping of impact forces coming on to the slope. The former will always occur around resonance condition on smooth slopes for which run-down penetrates deeper and plunging crests can travel longer. On rubble mounds the impact may occur on to the bare stones due t o high permeability of the structure causing lower run-down. The impact velocity " V p " may be written as:
Vp = (cb2 + ( 3 XPC o s t X) 2 )% cb
Eqn. (11) shows that Vp > Cb. Therefore, maximum flow velocity occurs at
the point of impact. The orientation of the velocity vector " V p " at the impact may influence stability. As the magnitude of this component parallel to the slope is bigger, it causes larger over-turning moments on the blocks.
Tests on a Smooth Slope
Tests were conducted on slopes 1 in 3, 1 in 2 and 1 in 1.5. Wave heights we re between 1.6 in and 6.0 in (4 cm and 15cm) w i t h periods ranging from 0.8 to 2.43 sec. Wave steepness was H / L0 < 0 . 1 which corresponds to average prototype
conditions. Tests covered the 1.33 < J < 7.96 range which included all types of breakers except spilling. Curing the tests a constant water depth of 20 in (50 cm) was used. This corresponds to a d/H > 3.0 at which it is known that wave run-up is not affected by water depth. Results for relative run-up and run-down are shown in Fig. 8a and 3b.
Fig. 8a shows the variation of relative run-up ( Ru/ H ) w i t h £. In this figure all
data obtained during the experiments are included. It shows considerable scatter. Although tests were run with regular waves, the succeeding waves show deviations in run-upvalues of about 10%.
Fig. 8a indicates a trend w i t h (J. Relative run-up (F I) increases very sharply with £ reaching a maximum at 2.0 < £ < 3.0 and decreases again w i t h further increase of £. It attains approximately a constant level for high £ values (£ > 4.0). Types of breakers were also included in Fig. 8a, showing that plunging waves were observed until £ = 3.2 and collapsing breaking until £ = 3.4. G or 1 . 0 < £ < 2.5 only plunging breakers occurred. From £ = 2.5 to £ = 3.2 plunging and collapsing breakers mixed. A t 3.2 < £ < 3.4 collapsing and surging mixed and after £ = 3.4 only surging waves on the slope occurred. The observation of wave breaking was similar to the results given above except that wave breaking might occur up to £ ~ 3.4. Hunt's run-up prediction mentioned earlier ( Ru/ H = £) seems to hold for
these data, also at £ < 3.0. The above tests showed that at the range of 2.0 < £ < 3.0 wave run-up on smooth slopes is maximum.
Wave run-down data were less scattered than wave run-up data. Fig. 8b shows these data together with breaking data. It may be noted that wave run-down penetrates further down and increases continuously with £ and assumes approxi-mately a constant value for high £ (£ > 4.0). It crosses the SWL roughly at a £ value of 2.2. For £ < 2.2 run-down canrïot occur below SWL and run-up and run-down always interact above SWL. The model mentioned earlier gave £ = 1.50 for run-down penetrating to SWL. The difference between these two results is undoubtedly caused by neglect of the effects by friction and pressure forces on the flow.
Analysis of the breaking point data together w i t h run-down data showed that resonance as described above can never be achieved as there will always be a run-down tongue remaining from the previous wave. Therefore, the resonance condition defined earlier was changed t o "the condition that occurs when run-down is in a low position and wave breaking takes place simultaneously and repeatedly close to that location".
Analysis of Fig. 8b indicates that the distance between the breaking point and the maximum run-down point increased for decreasing £ values. Resonance occurred for £ close to 3.
Pressure recordings on the slope mentioned in detail in refs. (8) and (15) showed that the impact and suction pressures penetrate deeper on the slope w i t h
increasing £ values. Maximum suction occurred below the sloping front of the wave crest before the crest passed the point.
Fig. 9 demonstrates that suction pressures are minimum (numerically maximum) at resonance, that is close to £ = 3.0. Impact pressures also maximizes the same £ range. This is in accordance w i t h the Russian data (37,39). On fully impervious slopes the static head of the wave will tend to compensate the suction pressure. On rubble mounds, blocks which are submerged are subjected to buoyancy that means uplift which will increase by suction. As wave breaking and wave run-up/ run-down are similar on smooth impermeable and permeable slopes it may be assumed that force patterns show similar trends on permeable and on smooth impermeable slopes although high roughness may cause more scattering at the permeable slope. Based on the experiments on smooth slopes it may therefore be concluded that maximum lift forces on an armour block due to flow on the rubble 6
mound averagely wil , td to maximize around resonance. This was confirmed by the tests mentioned in the next section using an optical instrument, the OBDS, Ly recording the actual movements of rock in the mound.
Tests on a Rubfaie Mound
Tests were conducted on a rubble mound slope 1 in 1.5 water depth 20 in (50 cm), d/H > 3.0, H varying from 3.5 in (9 cm) to 5.5 in (13 cm), T = 1.0—2.4 sec. Other data are shown in Fig. 11. Two different kinds of core material, d5Q% (coarse) = 10 mm, d5fj% (tine) = 4 mm were used, filter was d50% = 20 mm. Pressure transducers were placed in the core at four points ail 8 in (20 cm) below SWL (16). Fig. 10 shows the wave run-up compared to results from smooth slopes and t o rubble mounds bv others.
Fig. 11 shows the maximum and minimum elevations that the water table assumes inside the filter for different £ values. Measurements in the core show similar but weaker trends. It may be seen that until £ « 3.0 the maximum water
level in the filter stays much lower than the maximum run-up on the slope. Afte* £ * 3.0 the filter water table follows the wave run-up. This means that inflow cannot be " c o m p l e t e d " for small £ values. The minimum water table elevation in the filter layer stays very close to SWL for £ < 2.5. For £ > 2.5 it goes below SWL and decreases w i t h increasing £ values. The trend of the water table fluctuations inside the core is similar. For £ < 3.0 it does not go below SWL in the core. This causes an extra head in the core. Details on the development of water tabie fluctuations in filter and core are given in refs. (8) and (15).
It is noted that pressure in the core particularly builds up for fine material. This build-up acts upon the permeable filter and armour due to high pressure gradients in the core close to filter layer (8) and (15). It may therefore be concluded that, the difference in pressure in the core will be maximum close to the filter layer. This may be seen from Fig. IS in ref. (15) where filter and armour are shown at -3 in (-20 cm) elevation. It was impossible to measure forces directly, but the importance in relation to the permeability of the filter layer is clearly
demonstrated by the decrease in stability for decreasing size of core material as depicted In Fig. 15 of refs. (5, 7). Set down increases the outward force further (3, 15). From the above mentioned it is known that run-up and run-down increase w i t h increasing £ and assume a constant level approximately for £ > 4.0. This may cause higher run-down velocities on the breakwater for increasing £ values acting on the armour as drag forces. For a breakwater attacked by constant wave heights, it is therefore iikely that damage will occur w i t h the longest period waves when £ < 4.0. The maximum impact forces on the armour averagely speaking also occur at resonance. Suction forces occurring under a breaker due t o the inter-action between run-up and run-down velocities undoubtedly also maximize at resonance although no direct proof can be given herefor. In other words: the forces described above all maximize at longer periods in the spectrum when £ is less than 4.0. Maximum impact and suction forces occur under breaking waves for 2.0 < £ < 3.0. The increase in core pressures and run-down drag forces is mild for £ > 3.0. In this range no wave breaking occurs. It may therefore be concluded
that the first dislocations of armour occur for J values'—veen 2 and 3 where waves break. A f t e r the dislocation of some stones they ay roll down due to run-down forces. This means that for the advancement of damage long period waves which surge up on the breakwater are as responsible as waves at resonance.
For continuation of damage waves occurring in the wave spectrum w i t h £ > 2.0 therefore are all critical.
Some large scale stability experiments were conducted at the Coastal
Engineering Research Center (1) during 1974 using regular waves. Tests were run in a 640 ft (193.5 m) long, 15 f t (4.57 rn) wide and 20 ft (6.1 m) deep wave flume. Wave heights of 1.8 f t (0.55 m) < H < 6 f t (1.33 m) w i t h periods 2.8 sec < T <
11.3 sec were tested. Tests were conducted at a constant water depth of 1 5 f t (4.57 m). A rip-rap slope w i t h slopes of 1 in 2.5, 1 in 3.5 and 1 in 5.0 was tested. Different tests were run with different armour stones in the range of 26 lbs (11.77 kg) < W50 < 117 lbs (52.32 kg). Specific weights w e r e 7r = 2.60 g r / c m3 and 7W = 1.00 g r / c r nJ. Core material of D5Q «* 2 mm and filter material of D50 a* 30-40 mm were used. Results were presented in terms of the zero damage wave height " H ^ D " . Zero damage wave height is defined as the highest wave which does not cause damage to the structure. Wave heights were measured in front of the structure. Data are plotted in Fig. 12 in terms of zero damage stability number " N ^ D " versus £ where N2D and £ were defined as,
FIZD NZ D = m (12) ,W50, 1/3 ( SM ) l-7r ana 7w V H Z D / L Q
When eq. (12) is compared t o eq. (8) N7Q is equivalent to (KfjCota)
Fig. 12 also includes zero damage heights for 1 in 1.5 slope by tests conducted in the same flume in 1959 (1). It clearly shows the effect of period on the stability. When the range of breakwater and wave characteristics covered are considered they verify the above conclusions for the stability of rubble-mounds which were based on hydrodynamic analyses.
Various data by the U.S. Waterways Experiment Station claim no changes in K Q for variation in T. If their data, however, are plotted in terms of £, they show a similar but weaker trend as Fig. 12 (8, 15).
Analysis of Fig. 12 show that the stability number N2D depends on the slope angle much more than expressed in £. In Fig. 12 curves were drawn from the lowest zero damage numbers of each slope. This shows that minimum stability shifts from a £ value around 2 to a £ value around 3 with increasing slope angle. It also shows the effect of £ on stability and demonstrates that minimum stability occurs f o r 2 . 0 < £ < 3.0. Due t o the difficulties involved in recording of the actual damage (movements of rock) a new instrument was devejoped for quantification
of damage in a reliel: 1 ^ay. It is a photographic instrument called the Optical Breakdown Sensor (OisOS) described in ref. 15.
Tests with a Rubble Mound
Stability tests were conducted on the rubble mound breakwater of 1/2.5
continuous slope. The damage was quantified using the OBDS. Uniformly graded stones of W5Q ~ 72 grams with yr = 173.53 l b / f t3 (2.78 g r / c m3) were used. Coarse core material applied as filter material during the tests. Tests were conducted for a constant water depth of 50 cm. Armour was placed pell-mell using randomly chosen black and white painted stones (a necessity for using OBDS). Waves of 5.0 cm were run for ten minutes for all periods before starting any recording. Tests were conducted for periods of T = 1.0 sec, T = 1.52 sec. and
T = 2.28 sec. The armour layer was removed and reconstructec' before the tests w i t h each new wave-period. No repairs were made between the tests w i t h a constant wave period.
Fig. 13 shows the amplifier readings w i t h each wave height. It includes data for all periods. Fig. 14 shows the variation of the amplifier readings with £ for constant incoming wave heights.
Discussion of Results by the O E D S and B C O M
The purpose of these tests was to observe the characteristics of the new instrumentation and show its applicability for model testing on the stability of the breakwater.
During the stability tests w i t h the OBDS a slope of 30 cm above SWL and 45 cm below SWL of 60 cm width was used for all experiments. All calibrations and test data refer to dry armour stones.
Fig. 13 shows the amplifier deflections w i t h different wave heights and periods. It shows that the movements of the stones increase w i t h increasing wave height. Il should -be emphasized that tilting of the rocks without causing any net movement of the centre of the mass may cause a shift on the amplifier. This defiection of the amplifier is due to the change of the sectional area of the stone perpendicular to the camera vision.
Analysis of Fig. 13 also indicate that the increase in the movement of rocks w i t h increasing wave height is more severe after H s i 11.0 cm.
If the smallest zero damage height for the above structure is computed using eq. (12) and Fig. 12, one has:
H2D = 9.3 cm
It may therefore be said that this change In the trend of the data of Fig. 13 demonstrates the start of dislocation of stones. The analysis of the pictures taken after H > 11.0 cm confirm this.
The effect of the £ value on the movement is analysed in Fig. 14. It shows that for H = 11.3 cm more movement takes place for £ 2.30 than for £ « 1.45 and
i ~ 3.32. This may be explained by The maximization ' damaging forces at resonance.
After each test with wave height H > 11.0 cm a picture was taken. Pictures were compared to the reference picture taken after the test with H = 5.0 cm using the BCOM. The data obtained from these comparisons show that, in the range of measurements, most of the movements occurred around SWL and decreased further clown the slope. It indicates more moving stones for T = 1.52 sec waves, than for waves with periods T = 1.0 sec and T = 2.26 sec.
This result was obtained in the critical £ range, 2.0 < £ < 3.0, with period T = 1.52 sec. It may therefore be concluded that the resonance condition as defined generally present the most critical condition to the stability of rubble mounds.
Risk of Damage Based On £ Distribution:
Assuming that the flow conditions and the stability of breakwaters are described by %, it may be interesting to know the distribution of £. A n abstract of
introductory work on the distribution of £ which may be used for calculating the design risk is mentioned below. A more extensive report on this topic will follow in the near future.
The short-term steepness distribution is given by Overvik and Houmb on the distribution of wave steepness (paper in print at the Norwegian Institute of Technology) for e = o as:
P (s) 4s exp - y2 4S"
and the cum. prob. function Pc (s),
Pc(s> =o / P (s) ds = 1 - exp where, H L - y2 4s2 (1) (2) -2 Hs T , 2Z[ g T,2 2 v rN/ m0/ m2 m 2 1 -mum 0M4 .12 tn y rrij = the i = mome>.-L of the energy spectrum, and m£- is the mean square wave
amplitude. The validity of the above distributions needs checking w i t h prototype data. A comparison of the mean steepness and variance of the mode! w i t h the prototype data from a region of Tromsdflaket for 20 storm records of 20 min. duration each obtained during December 1976 gave satisfactory results.
From the prob. function (1) and (2), using £0 :
may be obtained as
-. _Ë.Ë prob. functions for £ \ A0 p(s)ds = p(f)d£ P(£) exp - j % ( 4 £d 4£ "4> Pc( | ) = 1 - e x p •2 I d " (4) where £d tg «. v ^ k 9_ 2k 50 tg a N / H Q T L O
-deep water wave height
The difference between £ used in the previous presentations and only slight due to the inclusion of the wave height into the expression with a square
root power. Therefore, at this point we assume that £0 is as representative as £ for
the previous conclusions of stability and run-up/run-down.
Egn's (3) and (4) together with Figs. (10) and (12) may be used for choosing an armour weight and the crest elevation of the breakwater at a risk level which is defined as the probability of occurence of mere destructive conditions than the design is based on. With reference to Fig. (12) the stabiiity formula (12) may be rewritten as:
w =
H3 7r
Nz D 3 ( Sr1 ) ( Sr1 )
(5)
The £ value " £m" which will give the maximum stor
aw =
as the solution of the eqn. be based on this £m value,
based on £j value other than £m
where cl- w <
: weight may be expressed A conservative design will ut an economical design including the risk, may be
From the analysis of Fig. (12) it may easily be seen that there will always be two £j values on the two side of £m on the curves
which will give the same stone weight for no damage. If they are £ji and £ j2 then
from eqn.(4) the probability (r) of having wave conditions during a storm which may create some damage on the structure may be calculated as,
r = P ( I i i ) - P ( l j2)
The input Hs and T2 to eqn. (4) must be supplied from the representative wave
spectrum. This r value may be used in deciding £j value and therefore the economic stone weight.
The cumulative long term distribution of £, ? C[ _ (if), may be obtained using the
short term distribution, eqn. 3, and the long term distribution of S2, P|_ (s2), given
by Ref. (Overvik and Houmb, "Parameterization of Wave Spectra and Long Term Joint Distribution of Wave Height and Period", Conf. on Behaviour of Offshore Structures, BOSS, 1976, Trondheim, pp. 144-169) as follows:
C O O C
PCL (4) = ƒ ƒ P i£> pL (s2> d sz d | O 0
The above described procedure for calculating the risk from short term distribution, may then be followed.
The above procedure assumes that the slope # o f the breakwater was pre-decided. I ndeed this decision may be based initially, according to the wave characteristics existing at the site, together w i t h the use of eqn. (4). It is known that the most critical condition for the armour stability occurs in the range 2.0 < fj < 3.0.' If one chooses a slope such that he can be out of this range w i t h acceptable level of risk, it is possible to decrease the armour weight appreciably (Fig. 12).
A similar argument may be made for the design of crest elevation. From Fig. (10) it can be seen that a conservative crest elevation h j will correspond to a £ value of around 5.0. If the design crest elevation is chosen as h2 < h j , then the
risk of overtopping during a storm may be calculated in terms of the prob. of occurence of higher run-up values. That is,
the risk of ovei topping during a storm = Pc( £2) - Pc ( )
where £2 corresponds to the run-up elevation h2 in Fig. 10.
14
Conclusion
The above mentioned is a result of many years of research and analyses of the behavior of rubble mound structures in the laboratory as well as in the fieid. The interesting aspect of the problem is that it has been, known for decades by the practical engineers and crews working on coastal protection in the North Sea (Denmark, England, Holland, Norway) that dikes and breakwaters were not over-run or destroyed during the peak of storms but rather at the end of the storm when periods increased and wave heights decreased. The same was observed in the Outer Banks, North Carolina and on the East Coast of Florida, e.g. at the Deerfield Beach during the March storm in 1961. This has not been explained.
The conclusions only cover sloping faced structures at a relative water depth of d/H > 3.0 where waves reach the structure w i t h o u t breaking. The results, however, may be transferred qualitatively to ratios 1.5 < d/H < 3.0. They are mostly applicable t o wave-protection structures w i t h steep continuous sloping faces, permeable as well as impermeable. They refer to monochromatic wave conditions.
Most of the overall flow characteristics like breaking, run-up, run-down may be defined by single parameter £ = t g « - /v /H / L0
-On smooth slopes in the range of 0.5 < £ < 2.0 wave run-up may be predicted using Hunt's formula Ru/H = %.
On smooth slopes maximum wave run-up occurs for waves breaking on the slope in the range of collapsing — plunging breakers. This corresponds approximately to 2.0 < J < 3.0.
On rubble mound breakwaters wave run-up increases continuously with £ until £ approximately equals to 4. From there on run-up assumes a constant level.
Wave run-down on slopes (permeable as well as impermeable) increases with increasing £ values until £ approximately equals 4.0. From there on it assumes a constant level.
Wave run-down on smooth slopes cannot penetrate below SWL for £ < 2.20 and run-up and run-down always interact above SWL.
Maximum impact pressures on smooth slopes occur at 2.0 < £ < 3.0 where the breaking wave hits the bare slope.
A build-up of hydrostatic pressure takes place inside the mound due to the wave run-up. It increases with decreasing permeability and with increasing values for £ < 4.0.
Stabiiity of rubble mound breakwaters depends upon the wave period. Forces trying to dislocate the armour maximize w i t h deep run-down occurring simultaneously and repeatedly w i t h collapsing — plunging wave breaking. This corresponds to 2.0 < % < 3.0 at which the initial stability of the rubble mound is most critical.
As expressed in refs. (7), (8), (15):
"the significance of wave period is clearly demonstrated. This underlines the necessity — demonstrated w i t h much pain in many practical mishaps — of designing rubble mounds and other sloping structures based on design
criteria which includes wave period. It is not eno' ~ to select a "design wave" and a " p r o p e r " Kn-value based on some m u e or less realistic laboratory experiments, it is also not enough to select a "design s t o r m " or a specific "design spectrum". The design wave or the design spectrum gives a " l o a d " which is sometimes regarded as the maximum exposure that can occur. This could be far from the t r u t h , however. A much more reliable scientifically as well as practically better reasoned design procedure is first to select one from a technical and economical view attractive design. The next step is to examine a number of actual wave spectra from the site including analyses of extreme events (7) and trains of approximately regular waves with special reference to the correlation between succeeding waves. Tests should then concentrate on irregular waves and on combin-ations of certain waves and periods that occur in the actual spectra w i t h particular reference to conditions that produce the most dangerous resonance phenomenons. This confirms actual experiences from a great number of actual observations in the North and Arctic seas and also the inadequacy of design-formulas that ignore wave period and spectral characteristics as w e l l . "
xi) A summary of Rvalues for wave action on smooth impermeable as well as on rubble mound slopes is given in refs. (8) and (15). They include expressions on breaker types, breaker point, breaker index, reflection, wave run-up and run-down, imoact pressures and max set-up all as function o f ? .
xii) To secure reliable data on stability from tests it is reeommendabie to use instruments which are able to quantify the damage by recording of the actual movements. Two instruments, the OBDS and the BOOM, have been developed. Preliminary results on their function are promising. Research is being continued.
xiii) The distribution of £ is obtained from the steepness distribution of waves. In connection w i t h the previous paragraphs it may be used to determine the most economical design in terms of probability of occurences of damaging situations.
References
1. Ahrens, P.J.: Large Wave Tank Tests of Riprap Stability, Tech.Mem. No. 5 1 , May 1975, U.S. A r m y Corps of Engineers.
2. Battjes, J.A.: "Surf Similarity", Proc. 14thConf. on Coastal Eng., Copen-hagen 1974, pp. 466-480.
3. Battjes, JLA. and Roos, A.: Characteristics of Flow in Run-Up of Periodic Waves, Delft Publ. 1974.
4. Bowen et a!.: Wave Set-Down and Set-Up, Journal of Geophysical Research, 74 (8),April 15, 1968.
5. Bruun, P.: Breakwaters for Coastal Protection, 18th Int. Nav. Congress, Rome, Section II, Question I, 1963.
16
6. Bruun, P. and Joi. ;esson, P.: A Critical Review of Hydraulics of Rubble Mound Structures; Division of Port and Ocean Eng., Univ. of Trondheim Publ., Inst. Rep. R3-1974.
7. Bruun, P. and Johannesson, P.; Parameters affecting the Stability of Rubble Mounds, Proc.ASCE, Journal of the Waterways, Harbors and Coastal Engineering Division, V o l . 102, No. WW2, 1976.
8. Bruun, P. and Günbak, A.R.: New Design Principles for Rubble Mounds, Proc. of the 15th International Conference on Coastal Engineering, Honululu, Hawaii 1976 (in print by the ASCE).
9. Carstens et al.: The Stability of Rjbble Mound Breakwaters against Irregular Waves, Proc. of 10th Conf. on Coastal Eng., Tokyo, Japan, Sept. 1966, Vol. 11. 10. Coastal Protection Manual: U.S. A r m y Coastal Eng. Research Center, Dept.
of the A r m y Corps of Engineers, 1973.
11. Dai, Y.B. and Kamel, A . M . : Scale Effect Tests for Rubble Mound Breakwaters. Research Report H-69-2 of U.S. A r m y Eng. Waterways Experiment Station, Corps of Engineers, Vicksburg, Mississippi, Dec. 1969.
12. Daubert, A. and Warluzei, A.: Modéle Mathematique non lineaire de la propagation d'une houle et de sa reflexion sur une plage, I.A.H.R., Voi. 4, Sept. 1967.
13. Dorrestein, R.: Wave Set-Up on a Beach, Univ. of Florida, Publication 1962. 14. Galvin, C.J.: Breaker Type Classification on Three Laboratory Beaches,
Journal of Geophysical Research, Vol. 73, Mo. 12, 1968.
15. Günbak, A.R.: The Stability of Rubble-Mound Breakwaters in Relation to Wave Breaking and Run-Down Characteristics and to the £ ~ tg T / y ' H Number. Thesis for the Dr. Eng. Degree. Report No. Rl-1976 by the Div. of Port and Ocean Engineering of the Norwegian institute of Technology. 16. Hedar, P.A.: Stability of Rock-fill Breakwaters, Doktors-avhandlingar vid
Chalmers Tekniska Hdgskile, No. 26, 1960.
17. Herbich, J.B., Sórensen, R.M. and Willenbrock, J.H.: Effect of Berm on Wave Run-Up on Composite Beaches, Proc. of American Society of Civil
Engineers, Journal of Waterways and Harbors Division, V o i . 89, WW2, May 1953.
18. High Island Water Scheme - Hong Kong: Wallingford Publ. Oct. 1970 EX 532.
19. Ho, D.V. and Meyer, R.E.: Climb of a Bore on a Sloping Beach, Part 1, Journal of Fluid Mech., Vol. 14, 1962.
20. Hudson, R.Y.: Laboratory Investigation of Rubble Mound Breakwaters, Proc. of the Am.Soc. of Civ. Eng., Journal of the Waterways and Harbors Division, V o l . 85, No. WW3, Sept. 1959.
2 1 . Hunt, I.A.: Design of Seawalls and Breakwaters, Proc. of A m . Soc. of Civ. Eng., Journal of Waterways and Harbors Division, V o l . 85, WW3, Sept. 1959. 22. Hwang Li-San: Wave Set-Up of Nonperiodic Wave Train and Its Associated
Shelf Oscillation, Journal of Geophysical Research, Vol. 75, No. 2 1 , 1970. 23. Inoue, M.: Effects of Wave Height and Sea Water Level on Wave
Over-topping and Wave Run-up, Coastal Eng. in Japan, V o l . 8, 1965.
24. Iversen, H.W.: Discussion of Results from Studies Wave Transformation in Shoaling Water, Including Breaking, Univ. of Ca.... Publ., Series No. 3, Issue No. 3 3 1 , March 1952.
25. Iwagaki et al.: On the Effect of Wind on Wave Overtopping on Vertical Seawalls, Bull. Dis. Prev. Res. Inst., Kyoto Univ., 16, No. 104, Part 1, Sept. 1966.
26. Jonsson, I.G. and Jacobsen, T.S.: Set-Down and Set-Up In a Refraction Zone, Prog. Rep. 29, pp. 13-22, Aug. 1973, Inst, of Hydrodyn. and Hydraulic Eng., Tech.Univ. of Denmark.
27. Kemp, P.H.: The Relation Between Wave Action and Beacii Profile Characteristics, Proc. of 7th Int.Conf. on Coastal Eng., The Hague, 1960, Vol. 1.
28. Kemp, P.H. and Plinston, D.T.: Beaches Produced by low Phase Difference, Proc. of American Society of Civ. Eng., Journal of Hydraulics Division, Vol. 94, HY 5, Sept. 1968.
29. Kemp, P.H. and Plinston, D.T.: Internal Velocities in the Uprush and Back-wash Zone, Proc. of the 14th Int.Conf. on Coastal Engineering, Copenhagen
1974, V o l . 1.
30. LeMéhauté, B. and Webb, L.M.: Wave Set-up and the Mass Transport of Cnoidal Waves, National Eng. Science Co., Interim Report Mo. 3, 1966. 3 1 . LeMéhauté, 8.: A Synthesis on Wave Run-up, Journal of Am.Soc. of Civil
Engineers, Febr. 1968, WW1.
32. Longuet-Higgins, M.S. and Stewart, R.W.: Radiation Stress and Mass Transport in Gravity Waves,with application to "Surf Beats", Journal of
Fluid Mechanics, V o l . 13, 1962.
33. Rottinghaus et al.: Shore Protection Study for a Selection of U.S. Interstate Highway 35 in Duluth, Minnesota, Proc. of the First Int. Conf. on Port and Ocean Eng. Under Arctic Condition, Vol. I I , 1971, Norway.
34. Sandström, Ake: Wave Forces on Blocks of Rubble Mound Breakwaters, Bulletin No. 83, Hydraulics Lag., Royal Inst, of Tech., Stockholm, Sweden 1974.
35; Seville, T.: Wave Run-Up on Shore Structures, Transactions of the American Society of Civil Engineers, V o l . 123, 1953.
35. Saville, T.: Wave Run-Up on Composite Slopes, Proc. of 6th Int.Conf. on Coastal Eng., 1957.
37. Selivanov, L.V.: Discussion of Construction Norms: Determination of Wave Loads on Sloping Structures, Hydro-technical Construction, No. 5, May 1972. 38. Sigurdsson, Gunnar: Wave Forces on Breakwater Capstones, Proc. of the
A m . Soc. of Civ. Eng., Journal of the Waterways and Harbors Division, V o l . 88, WW3.
39. Skladnev and Popov, I.Ya.: Studies on Wave Loads on Concrete Siope Protections of Earth Dams, Proc. of the Symposium Research on Wave A c t i o n , Vol. I I , Delft, The Netherlands, July 1969.
40. Stoker: Water Waves, Pure and Applied Mathematics V o l . IV, Interscience Publ., 1957.
4 1 . Straumsnes, A . < Bruun, P.: Comparison Between Spray and Splash at Some Typical Permeable Coastal Structures and the Influence of Ice Floes Deposited on These Structures on Spray and Splash Quantities, First Int. Conf. on Part and Ocean Engineering Under Arctic Conditons, V o l . 1, 1971, Norway.
42. Technicai Advisory Committee on Protection Against Inundation. Wave Run-Up and Overtopping, The Hague, 1974.
43. Wang Hsiang: Synthesis of Breakwater Design and Design Review Procedures, Ocean Eng.Rep.No. 1, Oct. 1974, Dept. of Civil Engineering, Univ. of
Delaware.
44. Weggel, C.R.: Maximum Breaker Height, Proc. of the Am.Soc. of Civ. Eng., Journal of the Waterways and Harbors Division, V o l . 98, WW4, Nov. 1972. 45. Wiegel, R.L. and Skjei, R.E.: Breaking Wave Force Prediction, Proc. of the Am.Soc. of Civ. Eng., Journal of Waterways and Harbors Division, V o l . 84, No. WW2, March 1958.
Notation
c Wave celerity (L/T) c Run-up front velocity (L/T) Cg Wave group velocity (L/T) D Characteristic length of stone (L)
Dm Equivalent diameter of stone or grain of which m percent of the weight is
contributed by stones or grains of lesser weight (L) d Water depth (L)
dD Water depth at the breaking point (L)
dg Water depth at the berm section (L) F Force (F)
f Bottom friction
g. Gravitational acceleration ( L / T2)
H Wave height in front of the structure (L) Hb Wave height at the breaking point (L)
H0 Deep water wave height (L)
H7D Zero damage wave height (L) hc Structure crest elevation (L)
L Wave length in front of the structure (L) L0 Deep water wave length (L)
m Water mass ( F T2 / L )
n Mannin's coefficient P Porosity (%)
p Pressure ( F / L2)
p Mean pressure averaged for one wave period ( F / L2 ) r Reflection coefficient
Ru Wave run-up ( L ) Wave run-down ( L )
Ru^ Maximum filter water table elevation ( L ) RQ^ Minimum filter water table elevation ( L ) R,i Maximum core water table elevation ( L )
uc
R ( jc Minimum core water table elevation ( L ) T Wave period ( T )
u Water particle velocity under a wave ( L / T )
v Water particle velocity on the slope ( L / T ) ' • W Averaqe armour stone weight ( F )
W50 Median armour stone weight at which m percent of the total weight o armour gradation is contributed by stones of lesser weight ( F )
20 Surging collapsing plungi ng spilLina F i g . 1 B r e a k e r T y p e s (3) 1.6- 1.2- 0.8- 0.4-S W L •0.4--0.8 -1.2-H . -1.2-H xx x XX i i X . . X X XX X , 1 c t g « = M . 5 Ü A „ - RELA-A RELA-A TIVE
RELATIVE ARMOR SPECIFIC S Y M - M O D E L WATER DEPTH WEIGHT WEIGHT B O L S C A L E D E P T H
3.0 <B/H Wl 0 0.3ib. yr=!76 l b / ! ;3 A i . O : l 2.e4<D/H W0 5 O.SIb yr=171 I M l3 X 0.5:1 2.0 4.0 6.0 X x< x<x. s.o 10.0 x X 12.0 F i g . 2 Run-Up/Run-Down on Smooth Q u a r r y S t o n e S l o p e (24) 21
1.6- 1.2- 0.8- 0.4-SWL -C.4-H . Rd H o XX XX X X , XX i Cl X A RELATIVE DEPTH RELA-TIVE A R M O R SPECIFIC S Y M - M O D E L W A T E R W E I G H T W E I G H T B O L S C A L E DEPTH W7 5» 161.51b ft-IGS.ilh/!!3 3.30 < D/H W1 0- 0.331b 2.60 < D/H W^5>C.C<!Sib • • 7.5:1 - i . 1.0:1 X 0.5:1 (1ST ( 2 ' ) ( 1 ' ) 2.0 4.0 X s.o s.o 10.0 tgcs
vWLT
-0.8--i jX x A , A X X X r i g . 3 Run-Up/Run-Down on Rough Q u a r r y s t o n e C o v e r B r e a k w a t e r 1.6- 1.2- 0.8- 0.4-S W L -0.4--0.8-1 Ra Rd H , H U.3 C J O.y RELA-TIVE RELATIVE A R M O R SPECIFIC S Y M - M O D E L W A T E R DEPTH W E I G H T W E I G H T B O L S C A L E DEPTH 3.58 < D/H W y ^ T C lb y. =150 Ib/1t3 (fc 7.5:1 4 2 5 < D / H Wl 0 = 0.191b 7r- 1 3 9 . 2 lb/11» A 1.0:1 2.0 — , [ -4.0 5.0 8.0 10.0 -.2.0 iq a.-1.2H
F i g . 4 Run-Up/Run-Down o n D o l a s C o v e r B r e a k w a t e r 1.25- 1.00- 0.75- 0.50- 0.25-S W L -0.25- -0.50--0.75-! H , Rd H <x o O o o • X * X' o x © —*- cot CL - 3.0 x —•- cct a = 2.C o —*-coi a =1.5 30 1.0 2.0 3.0 4.0 5.0 ° So „X X WU xx*x X0 O XX x ° O 0 ° Xc x ° ° o • F i g . 5 Run-Up/Run-Down on D o l o s C o v e r B r e a k w a t e r 23H , H-sin os 0.30- 0.40-0.00 -0.40-1 -0.80 -1.20--1.60-j -2.00 - / ^ = ( 1 - 0 . 4 5 l ) i
BREAKER T Y P E X — COLLAPSING o . - r - PLUNGING j p — SURGING
F i g . 8b Run-Down and B r e a k i n g P o i n t on Smooth Slor.es
-10'
9 Max. and M i n . Dynamic Pressures on Smooth Slopes 1 i n 3 H 3.0H 2.0- 10-O — — Smoulh Slopes of
Data from r*f 18) o r j ref.{24) Si'.ir.O'.h Slopes of 0<P„ t. A ° ° 93" cA^ " 0 ° 0 o0« <S> ° o
•as
A O O O 4 * Bo a a . a — - Bubble - Mound Brjjkvvjlcr of 1 ui 1.5 Slope Wsó = 2 6 3 . 9 2 gr yr - 2.70Qf/cm1 •PcP i . s l Hod:an's Data oo 2.0 Rubble- MNMI 2.5 I Breakwaters 3.0 J Mole' H0 is assumed * . o equil to H H - a H 0.0 1.0 ao 4.0la*
' f / L0F i g . 10 Wave Run-Up Spectrum (d/H 3.0) 1.2 1.0 0.8 0.6- 0.4- 0.2-0.0 -0.2- -0.4-Rgf Rdf H . H A A A A . A A A A A A S . A A A . A FILTER c o t a = 2 . 5 . A A A A A A . A A A A ^ A A , Ruf H 2.0 . 4.0 V ,.4* O e 5.0 tg a HTL^
F i g . 11 Water T a b l e Movement between F i l t e r and Core
•ZD
DATA FOR C O T a = 2.5.3.5 AMD 5.0 ARE (AKEN FROM AHSEKS 11274) [CERC TESTS) DATA FOR C 0 T a = 1.5 IS TAKEN FROM 3.E.B. TESTS ( 1 3 6 8 - 1 9 5 3 )
MOTE : CURVES ARE DRAWN FOR LOWEST STABILITY NUMBERS COT.- >.0 o cor<*= 3.5 A
-COT== 2.5 l •
C O T a = 1.5 V
( * ) « , - «
H7 [ )= 2 E R 0 DAMAGE WAVE HEIGHT MEASURED IN FRONT OF THE STRUCTURE.
0.0 3.0 4.0 5.0
t _ t g a / tg«-T\
5 ÏH/ra " ÏH I
F i g . 12 Slope Dependence o f Zero Damage S t a b i l i t y Number
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