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Wave forces on vertical cylinders upon shoals
COASTAL ENGINEERING ELSEVIER
Coastal Engineering 27 (1996) 263-286
Wave forces on vertical cylinders upon shoals Alexander Kyte a, Alf Torum bp* aBerdal Str@nme A/S, Consulting Engineers, Sand&a, Norway b SINTEF Norwegian Hydrotechnical Laboratory, Norwegian Institute of Technology, Trondheim, Norway Received 27 April 1995; accepted 20 December 1995
Abstract An experimental study has been carried out on the forces from plunging breaking regular and irregular waves on a vertical cylinder on a shoal. Total as well as local wave forces have been measured. Engineering formulae for the calculation of the horizontal forces and overturning moments have been derived. The duration of the impact forces have been measured and compares fairly well with theoretical values.
1. Introduction
There are plans in Norway to replace light buoys for navigation purposes with small fixed light houses. The location of many of these lights are on shoals, with water depths 5-10 m, in wave exposed locations such that the light houses will be exposed to breaking waves. The shoals are frequently very pesky. The depth increases rapidly to 50 m and more around the shoals. Lie and Tprrum (1991) investigated the wave conditions for long crested irregular waves around an idealized form of one of these shoals, the Arsgrunnen, in scale 1: 100. The water depth at the shallowest point corresponded to 5 m. For the largest waves the shoal went dry and the wave crest elevation was about three times the water depth Subsequently Hovden and Terrum (1991) investigated the wave forces on a vertical cylinder on top of the idealized shoal used by Lie and Torum (1991) for the wave studies.
’ Corresponding author. 0378-3839/96/$15.00 Science B.V.
Copyright 0 1996 Elsevier Science B.V. All rights reserved. Published by Elsevier
PII SO378-3839(96)00003-8
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27 (1996) 263-286
Based on the findings in these two studies the Norwegian Coast Directorate decided to sponsor a study on more details on the wave forces from breaking waves on vertical cylinders on shoals. It is the results of this study that are presented in this paper.
2. Previous
work on breaking
wave forces on vertical cylinders
A number of studies have been undertaken on breaking wave forces on vertical cylinders. Most of the studies are for vertical cylinders on uniformly sloping bottoms, Goda et al. (19661, Watanabe and Horikawa (1974) Sawaragi and Nochino (1984) Tanimoto et al. (1986a,b) and Apelt and Piorewiez (1987). Goda (1973) investigated the breaking wave forces on vertical cylinders upon circular and ledged reefs. Since Goda was not in this case interested in the short duration impact forces, his force measurement system was not designed to pick up the short duration high intensity impact forces on the cylinder (Goda, pers. commun., 1990). The planned Norwegian small lighthouses have natural periods of oscillations such that dynamic amplification of the structural response due to the short duration high intensity shock forces from the breaking waves may occur. In their study Hovden and Torum used a “stiff” measuring system. They obtained thus significantly higher wave forces than Goda (1973). In all the referenced studies, except by Tanimoto et al. (1986b), only the total forces on the cylinder were measured. Tanimoto et al. (1986b) measured also local forces in an area above the still water line. In order to get more details on breaking wave forces on a vertical cylinder on a shoal it was decided to carry out experiments and measure the local wave forces as well as total wave forces on a vertical cylinder on a shoal. A particular shoal “Hausene”, where a new lighthouse is planned, was installed in scale 1 : 100 in a wave flume as shown in Fig. 1. The shoal topography was modelled to a water depth of 50 m. The water depth on the shoal is 5 m. Strictly speaking directional waves should probably have been used for waves over three dimensional shoals. Vincent and Briggs (1989) showed that for non-breaking irregular waves there was a difference in wave heights on a shoal for long crested or short crested waves. However, since we did not have availability to laboratory short crested waves, we made tests with long crested waves. Nevertheless the results of our study are deemed to be very useful, also because most of the tests had to be run with regular waves (see below).
3. Some theoretical
considerations
Waves that enter shallow water will eventually break. The type of breaking (spilling, plunging or surging) is governed by the wave parameters and the slope of the bottom. It is the plunging type breaker which gives high impact forces on a vertical cylinder. The
I
WAVE ABSORBER ‘INSTRUMENTED
-
CYLINDER I
WG,-REFERENCE
Fig. I. Wave flume with the Hausene shoal (1: 300; dimensions in m).
I GAUGE
266
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B
A Fig. 2. Breaking
27 (1996) 263-286
waves: (A) Uniformly
sloping beach. (B) Shoal.
situation with a plunging wave hitting a vertical cylinder on a uniformly sloping bottom is shown schematically in Fig. 2A. The wave breaking on a shallow shoal is different and as shown schematically in Fig. 2B. When the waves are large the shoal goes dry under the wave through and the wave thereafter surges over the shoal. When the vertical wall of water hits a cylinder the situation is comparable with the situation when a falling cylinder hits a water surface. Several researchers have investigated the latter situation and the results have been compared with laboratory tests. One of the first attempt to derive a theoretical formulation was made by Von Karman (1929). He arrived at the following expression for the impact force on a falling cylinder when it hits the water: F = 0.5pC, Dlv,2 ( 1 - t/T)
where p = mass density of the water, C, = force coefficient, D = cylinder diameter, 1 = cylinder length, v, = velocity of the cylinder when it hits the water, t = time, and T = duration of the impact. The impact force has its maximum value at t = 0, e.g. in the beginning of the impact. Von Karman set the duration of the impact load to T = D/(2v,) or until the cylinder is half submerged into the water. He obtained theoretically, by some assumptions, a value C, = n. Experimental values of C, differ, however, from this value. Miller (1977) for example gives C, - 6.0 and Sarpkaya (1978) gives C, - 3.17 f 0.05. Sawaragi and Nochino (1984) show also that the value of C, varies and may be up to C, - 10 for waves breaking against a vertical cylinder and assuming v, = C, where C is the wave celerity when the wave breaks. The wave forces below the breaking region at the wave surface are calculated by using a Morison force formulation for a pile on a uniformly sloping bottom, Fig. 2A, (Goda et al., 1966; Watanabe and Horikawa, 1974). Apelt and Piorewiez (1987) found that in this case the force was depending on D/H,, H, = breaking wave height, where they considered D/H, as a kind of inverted Keulegan-Carpenter (KC) number (KC = u,,,T/D, where u,,,~~ is the horizontal max. particle velocity and T is the wave period).
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For a vertical cylinder on a shoal, Fig. 2B, the wave surges/plunges against the whole height of the cylinder and give a impact type load on the cylinder. A different approach was then used by Goda (19731, which we will revert to later. A special feature about impact forces from breaking waves on cylinders is the big scatter in the forces measured during laboratory tests, even when the waves are regular. This is probably due to small variation from wave to wave in the front of the breaking waves as the waves hit the cylinder.
Dz60
Fl
‘ti= F2
F3
Measures in mm Fig. 3. Instrumented
cylinder.
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4. Test set-up The wave flume with the model Hausene shoal is shown in Fig. 1. The water depth on top of the shoal is 5 m. Fig. 3 shows the instrumented vertical cylinder. The total forces are measured with the force transducers marked Fll and FlO. The FlO force transducer is shown in Fig. 4. It is developed and built by MAFCINTEK,SINTEF Group. The strain gauges measure the shear deformation and hence the shear force. Force transducer Fl 1 is a commercial available force transducer. The total force is equal to the sum of forces measured at FlO and Fl 1. Wave force moments at the base is set equal to M = 0.399 X Fll (m, N). This is based on the assumption that the shear force is constant over the height of the force transducer FlO. The mid point of the transducer, which is located at the shoal bottom level, will then have zero moment. This assumption may not be absolutely true, and there is apparently a small error in the moment values. However, this error is deemed to be insignificant compared with the uncertainty related to the scatter shown in the test results. The force transducers Fl-F9 are local force transducers that measure the wave force in the wave direction on a ring of the cylinder, 1 cm high. Fig. 5 shows the local force transducer. It measures the shear deformation and hence the shear force on a cantilever beam in the direction of wave travel. The first tests were run with irregular waves. The data were sampled at a rate of 500 per second per recording channel (sampling frequency of 500 Hz) and a run time of 160 s. However, during the analysis it was revealed that it was preferable to use a higher sampling frequency. It was therefore decided to run the majority of the tests with regular waves and a sampling frequency of 4000 Hz. However, the run time was then limited to 20 s due to data storage limitations, which then made it less meaningful to use irregular waves. Based on previous work some effort was made to make the measuring system stiff to avoid dynamic amplifications in the response of the force measurement system. By tapping the cylinder and the different force transducers and recording the force responses we obtained natural periods of oscillations. The system had generally complicated modes of oscillations. By tapping the bottom of the cylinder we recorded natural frequencies of oscillation on force transducer No. FlO in the range 167- 182 Hz. By tapping the top of the cylinder we recorded natural frequencies of oscillations on force
Fig. 4. Force transducer.
A. Kyte. A. T~t~ttt/C~ustul
Engineering
-
Plan
Strain
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gauge
view
Side view
Fig. 5. Local force transducer.
transducer No. Fl 1 in the range 40-100 Hz. By tapping the local force transducers No. Fl-F9 we recorded natural frequencies of oscillations in the range 100-150 Hz. In order to see what effect these natural frequencies might have had on the response of the measurement system we did some calculations of the response based on the recorded apparent force, assuming the system is a one-degree of freedom system. Fig. 6 shows the relative response for a recorded force on a force transducer with a natural frequency of 166 Hz. In this case the dynamic amplification factor is approximately 1.15. It should be noted that we for sake of simplicity took the recorded response as the input force in the calculations, while the correct procedure would be to calculate the force giving the recorded response. We did similar calculations for other situations. In most of the cases the dynamic amplification factor was in the range 1.0-1.3, with exceptions as high as 1.5. In the analysis we have not made any attempt to correct for any dynamic amplifications. Nevertheless we think the results we have obtained are very useful for the design
Calculated
response
0.6 -
Milliseconds Fig. 6. Calculated
and recorded
force.
A. Kyte, A. T@wn / Coastal Engineering 27 (1996) 263-286
210
of the planned lighthouses or other types of vertical cylindrical structures on shoals, because there are so few existing data on the total wave impact forces and no data on the wave impact force distribution on such structures. Three wave gauges of the resistance type was mounted in the flume in positions as shown in Fig. 1, WGl, WG2 and WG3. The wave gauge WGl was in “deep” water, 0.90 m, and was used as a reference gauge. The wave heights generally used in the analysis was the wave heights measured at wave gauge WGl. The waves were generated through a combined wave generator control and data acquisition system. The data were analyzed primarily through the data analysis program STARTIMES, (Hoen and Brathaug, 1987). Some statistical calculations were carried out using the computer program EXTPAR, Mathiesen (1992). The PC based computer program MATLAB, The MathWorks Inc, was used for some minor calculations and graphical presentation of data.
5. Test programme The tests were carried out with regular and irregular waves according to the programme shown in Table 1. The wave parameters refer to wave gauge WG 1, Fig. 1 (deep water).
6. Analysis of the test results 6.1. Analysis
of the irregular
wave tests
As stated previously the breaking wave impact forces tend to vary considerably in magnitude from wave to wave even if the waves are regular. This is because the impact force is probably sensitive to the form of the front of the breaking waves. The steepness of the front face may vary slightly from wave to wave. The variations of the forces we
Table 1 Test program Tp (s)
Irregular waves, JONSWAP, y = 2.0
T (s)
1.o, 1.2, 1.4, 1.6
H, (cm)
H, (cm)
2.8- 14.7; total 17
Water depth (cm)
Run duration (9
Sampling frequency (Hz)
5
160
500
5
20
4OQO
runs Regular waves
1.0, 1.2, 1.4, 1.6
4-22; 14 waveheights for each period
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Coastal Engineering
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.L~_
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!
I -5
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I
22
2+
_--_I
26
28
30
Seconds Fig. 7. Sample of time series for the total force -
irregular waves.
see from the tests with irregular waves do not thus necessarily reflect the variation due to variability of the wave heights only. To get a complete statistical picture we should have run several tests, may be 50, with the same input wave generator signal. We would thus obtained the statistical variability of the impact wave force as well as the statistical variability due to the irregular waves. Since we had a sampling frequency of only 500 Hz for the irregular waves and thus some inaccuracies in the recorded forces, we only made one run for each combination of significant wave height and peak period to obtain some information of the variability of the wave forces in irregular waves. Fig. 7 shows a sample of time series for the total wave force. Due to the oscillations of the force measuring system we have multiple peaks during one impact. Since we are interested in the peak forces we used a procedure in the analysis program to pick out only one and the largest value for each impact. Since we also were interested in distinct peak forces we set a lower limit for the peak forces to be included in the analysis. This lower limit varied somewhat from run to run and was set higher for higher waves than for lower waves. In the example shown in Fig. 7 the lower limit was set to 1 N. The peak force values for each irregular wave run was fitted to a three parameter Weibull distribution:
where P(F) = probability of not exceeding F, F. = location factor, F, = scale factor, and y = shape factor. Fig. 8 shows a Weibull fit of peak force data. The fitting is carried out according to the methods of moments. The diagram of Fig. 8 is based on 87 data points. The force data are collected in bins, each bin being 0.5 N wide. Each point (triangular) in the plot of Fig. 8 is located at the upper end of each bin.
272
A. Kyte, A. T@rum/Coastal I
:: _1
Engineering
I
I
A A A A tlaxcmo
0.9990
---
ii
d
27 (1996) 263-286 I
1
I
serces
WecbuLL-3 WeLbuLL-3 fLt
B 0.9900
0
.sooo
0.5000
0.~000
NEWT01 Fig. 8. Weibull distribution
x-Tp=
1.07sek.
of peak forces.
*-Tp=1.32sek.
o-Tp=
1.59sek.
20
15
x 0
. 0
10 x
x
0
0 l
* l
*
1
x
5
0 *
0
0
2
4
6
8
Significant waveheight,
10 I-&, cm
Fig. 9. “One per mille” force.
12
14
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27 (1996) 263-286
273
The location, scale and shape factors of the Weibull distribution are in this particular case 0.53 N, 3.15 N and 1.48 respectively. In order to condense the information we took the forces and moments corresponding to one per mille exceedence according to the Weibull distribution. (For the plot of Fig. 8 the one per mille force is 13.2 N). Figs. 9 and 10 show the “one per mille” force and moment for different significant wave heights and peak periods. It turned out that the target spectra with Tp = 1.2 s and 1.4 s came close to each other and are lumped together under Tp = 1.32 s. The general trend is that the force increases with increasing significant wave height, but with a slight tendency that the force and moments tend to flatten out for significant wave heights of 6 cm. When the significant wave height is 10 cm and higher the wave force tend to increase again. The shortest peak periods tend to give the highest forces, but this is not a very clear tendency. The maximum moment for each run was almost always recorded for the maximum force and at almost the same time point as the maximum forces.
6.2. Analysis
of total forces
and moments for regular waves
The tests with regular waves were carried out with a sampling frequency of 4000 Hz with a duration of each run of 20 s before the data storage was filled up. The tests were
x-Tp=l.Msek
* - Tp = 1.32 sek.
o-Tp=1.59sek
2.5 -
. 2
1.5
1
I
0.5
I 0
O_ U
2
4
6
Sign&ant
8
waveheight,
10
I-I,, cm
Fig. 10. “One per mille” moment.
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274
*T= 1.0s.
Engineering
XT= 1.2s.
27 (1996) 263-286
o T = 1.4 s.
+ T = 1.6 s. x
*
35 30 0
25 20 15 l
x 10 I 5-
0’ 0
l
l
x t 0 +
+ x
I x
*SF+ o
5
0
10
*
l
0
+. b x* Xx@+ xx +oo + + *+
x 0 + x+x
15
20
Wave height, H,, , cm. Fig. 11. Maximum horizontal
forces in regular waves vs. HO and T.
carried out with wave periods 1.0, 1.2, 1.4 and 1.6 s. The data logging system was switched on just before the first waves came to the cylinder in order to get the zero line for the force transducers recorded. Experience showed that there could be a slight shift in the zero-setting for the force transducers. This shift was then recorded and taken care of in the analysis. The valid recording period was then about 14 s or 8-14 waves for each run, depending on the wave period. Due to the limitation in the number of waves for each run the force data was not subject to any statistical analysis. Only the mean and maximum peak forces for each run were used. Figs. 11 and 12 show the maximum total force and the mean total force as a function of wave height and wave period. The wave height is the height of the wave at the reference gauge WGl, Fig. 1. Figs. 13 and 14 show the maximum and mean moment at the base of the cylinder. By comparing Figs. 11 and 12 we see that the maximum force is frequently twice the mean force or more, indicating the big scatter we have in the force data and which has been observed previously for similar tests on the impact forces from breaking waves. We see that the forces tend to increase up to a wave height of 10 cm. For higher wave heights the force flattens out until the wave height is approximately 15 cm. For still higher wave heights the force and moments increase significantly more again.
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A. Kyte, A. T@rum /
*T= 1.0s.
XT=
1.2s.
o T = 1.4 s.
+T=
275 1.6s.
I
1
I
10
5
15
20
Wave height, I-&, cm. Fig. 12. Mean horizontal
forces in regular waves vs. H,, and T.
The flattening of the wave forces at a wave height of 10 cm and larger may be partly explained from Goda’s particle velocity measurements in waves breaking over a shoal (Goda, 1973). The increase of the forces from a wave height of 15 cm and on cannot so easily be explained from Goda’s measurements. We believe that as the waves become higher they break more violently in front of the shoal and thus create larger velocities and forces on the cylinder. 6.3. Comparison
with previous
work
The only known previous works on similar problems is by Goda (1973) and Hovden and Tot-urn (1991). Goda presented the mean and standard deviation of the forces for 203 consecutive waves while Hovden and Torum (1991) used the maximum force for 10 consecutive waves. Goda dit not record the high intensity short duration forces. Goda formulated the forces and moments at the base of the cylinder in the following way: Force F = p,pgDH,( Moment
d + q,,)
M= 0.5p’MpgDH,(
d + q,,)’
(3) (4)
rl max= 0.754 rl max is the crest height,
d is water depth and H is wave height in “deep”
water. PF
276
A. Kyte, A. Tpmam / Coasral Engineering *T=
1.0s.
XT=
1.2s.
oT=
27 (1996) 263-286 1.4s.
+T=
1.6s.
3.5 .
l
3-
*x L
0 0
5
10
15
Wave height, H, , cm. Fig. 13. Maximum
moment in regular waves vs. H,, and T.
10
Wave height, &,
cm.
Fig. 14. Mean moment in regular waves vs. H,, and T.
20
A. Kyte, A. T#rum / Coastal Engineering *-T=
1.0s.
x-T=
1.2s.
277
27 (1996) 263-286 o-T=1.4s.
+-T=1.6s.
3-
2.5 -
2-
. .
PFmU
I .5 -
O8
12
10
16
14
18
20
22
24
Wave height, H,, , cm. Fig. 15. PF -
*-T=
1.0s.
maximum force.
x-T=1.2s.
o-T=
1.4s.
+-T=
1.6s.
0.9 0.8 0.7-
x o
l
0.6 -
PFmun
+
0.5 -
o
C.
0.4 0.3 0.2 0.1 -8
IO
12
I4
16
Wave height, H,, , cm. Fig. 16. PF -
mean force.
18
20
22
24
278
A. Kyle, A. T@wn/ +-T=
Coastal Engineering 27 (1996) 263-286
1.0s.
x-T=1.2s.
12
14
o-T=1.4s.
+-T=
1.6s.
I
0’
8
10
16
18
20
22
24
22
24
Wave height, &, cm. Fig. 17. pw -
*-T=
1.0s.
maximum
moment.
x-T=1.2s.
o-T=
1.4s.
+-T=
1.6~.
)I_________ 2.5 -
2-
P Mrnun
.
1.5-
x
I l-
0
0.5 -
01
8
IO
12
14
16
18
Wave height, I-&,, cm. Fig. 18. PM -
mean moment.
20
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279
and PM are force and moment coefficients obtained from the mean forces. Goda obtained for a circular reef mean values of BF in the range 0.3-0.5 and @‘, = 0.3-0.55. We obtained PF coefficients based on the maximum force as shown in Fig. 15 and on the mean force as shown in Fig. 16. Based on our measurements we found that the arm of the force was approximately (d + 0.2Ha). Hence the moment becomes M=p,Dpg(d+0.75Ho)(d+0.2H,)H,
(5)
This differs slightly from Goda’s formulation but within the range of wave heights H,, = lo-20 cm the difference between (d + 0.2H,) and 0.5(d + 0.75H,) is not very large. To what extent there is a ‘ ‘Keulegan-Carpenter number” effect in this case, like Apelt and Piorewiez (1987) indicated, was not specifically looked at by Goda. However, the large wave forces occurred as the wave front passed the cylinder and before vortices were shed regularly behind the cylinder. Hence it may not be relevant to introduce a KC-number consideration, which is valid and normally used for the part of a cylinder which is always submerged under wave action. We found PM values based on Eq. (5) as shown in Figs. 17 and lg. The diagrams show that the maximum moment coefficients are large compared to the mean moment coefficients. Goda does only give the standard deviation of his coefficient. From the standard deviation, however, it is deemed that the biggest difference between Goda and our results are that the maximum forces we obtained are larger than the forces measured by Goda (1973). This may possibly be attributed to the differences in the measurement system. Hovden and Torum (1991) carried out their tests on a more elongated type of shoal and less steep in the front than we have used in this study. The scale of the shoal Hovden and Torum used was the same, 1: 100, and the water depths on the top of the shoal corresponded to 5 m as in this present study. For waves lower than No = 0.16 m Hovden and Torum obtained slightly higher forces than we have found on the Hausene shoal. However, the extreme forces for higher waves (H, > 0.16 m> found in this present study are larger than those found by Hovden and Torum (1991). The latter shows that the shape of the shoal has a significant influence on the wave forces. In general we will expect to find the maximum forces on the steepest shoal. For a less steep shoal the highest waves will break and loose their energy before arriving at the vertical cylinder. For a steeper shoal such as Hausene, the largest waves can break more directly on the cylinder and thus introducing larger forces.
6.4. Duration
of the wave impact load
Fig. 7 shows a time series of total wave forces. Generally the peak impact force is followed by a Morison type force. The time history of wave force can schematically be shown as in Fig. 19. ~~ is the rise time, or is duration of the peak and ~~~~is “total”
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A. Kyte, A. T@um / Coastal Engineering 27 (1996) 263-286
Fig. 19. Duration of impact force.
duration. From the dynamic amplification of the structural response TI, may have been a better choice than rP. There was a significant scatter of the duration or. As an example we found for regular waves in the range H, = 0.13-0.16 m, 7P = 0.004-0.016 s. The impact duration rP is related to the diameter and the water velocity when the wave hit the cylinder, or = AD/u). Different researchers give different relations, but normally in the range Tp = 0.25D/u - 0.5D/u. Tanimoto et al. (1986b) set the duration 7p = 0.25D/u. If we use the results of Goda’s water particle measurements we find for H, = 0.145 m the
a-20
-L
n
1
H, = 14.76 cm T,= 1.38s
-2 -3 -4 --
7
--
I 29-5
I
I
I
1
>
Newton per local force meter Fig. 20. Local force distribution.
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27 (1996) 263-286
I
281
y = 14.5 cm TP= 1.4s
1
z-20
2 --
3
--
6 --
=-
z-0 r--5
10
5
-6 -7 -8 -..9...-: . . . . .
I
I
SWL . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .
: I
t
I
I
I
0.5
I.0
IS
2D
>
Newton per local force meter Fig. 21. Local force distribution.
maximum velocity u = 1.3 m/s. According to Tanimoto et al.‘s formulation obtain 7r = 0.012 s. This is in the “ball park” of our observations.
we then
6.5. Force intensity along the cylinder Several diagrams showing the force intensity along the cylinder were prepared. There was generally scatter in these plots. Figs. 20-22 show examples of the measured force intensity when the total force on the cylinder was about its maximum value. The location of the local force transducers was based on the results by Tanimoto et al. (1986b), who measured impact forces on a vertical pile on a uniformly sloping bottom. Their measurements showed that the impact force occurred well above the still
a-20
+ 12-3
x-10
II
H, = 20.6 cm T,= 1.4s
--
6 -__ 5 --
z-Q z--5
Newton per local force meter Fig. 22. Local force distribution.
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27 (1996) 263-286
water line. Hence we thought that impact forces would occur only above the still water line in our case also and did not locate any local force transducers below the still water line. However, during the tests it was revealed that impact forces probably occurred below the still water line when the cylinder is located on a shallow shoal like the Hausene shoal. Hence, it would have been useful with local force transducers below the still water line also. Since we did not use local force transducers below the still water line we have assumed the force intensity of the lowest local force transducers to be valid at the stillwater line and set the impact force intensity to zero at the bottom (see later). The Goda (1973) force formulation is based on the assumption that the force is uniformly distributed along the cylinder. The measured force intensities show that this is not necessarily the case.
7. Engineering
formulas
to calculate the wave forces
We want to apply the obtained results for a tower with two diameters as shown in Fig. 23 and which is frequently used in Norway. The lower cylinder is extending beyond the smaller cylinder so much that an impact force on the lower cylinder may be over when the wave hit the smaller cylinder. To be a bit conservative we will assume the front of the wave as indicated in Fig. 23. In this case the impact load may be exerted on the upper smaller and lower larger cylinders at the same time. In order to calculate the wave force on the two cylinders we need an evaluation of the wave force intensity. The results showed, however, a large scatter. As a reasonable choice we used the force intensity distribution as shown in Fig. 24 for the highest waves, e.g. H, > 0.1 m. The fixing of the upper end at a distance 3.5 times the water depth above the top of the lower cylinder may seem a bit strange. But the results showed that the upper end of the force intensity diagram was not much influenced by the wave height (at the reference point) when the wave height was above H,=O.lO m. Once the force intensity diagram is given the force intensity f of the base is given such that the force at the base is equal to the force given by Eq. (3) with qma, = 0.75H,. f = 0.421BF(
D/d)pg(
d + 0.75H,)H,
Fig. 23. Two cylinders.
(6)
A. Kyte, A. T@rum / Coastal Engineering 27 (1996) 263-286
Fig. 24. One cylinder
and the total force and moment F = pFDpg( M=
-
force distribution.
at the bottom
d + 0.75H,)H,
2PFDpgd(
283
(7)
d + 0.75H,)H,
(8)
When we extend this approach to the two cylinder structures of Fig. 23 we arrive at a force intensity diagram as shown in Fig. 25. The discontinuity of the force intensity diagram is due to the discontinuity of the diameters. The force intensities f, and f2 are given by j-, = 0.421p,(
D,/d)pg(
d + 0.75H,)H,
(9)
f2 = 0.421p,(
D,/d)pg(
d + 0.75H,)H,
(IO)
The force F, and moment
moment
M, at the base of the upper cylinder M2 at the base of the larger cylinder are given by
F, = 0.737p,D,pg(
d + 0.75H,)H,,
M, = 0.860P,D,pgd(
(11)
d + 0.75H,)H,
F2 = pFpgHO( d + 0.75H,,)(O.7370, M, = PFpgHod(
d + 0.75&)(
and the force F2 and
(12) + 0.2630,)
(13)
1.781 D, + 0.2190,)
(14)
The choice of the characteristic value of the PF coefficient should be based on the wave climate at the specific site and the life time of the structure. For design purposes
3.5 d 7l
1.25
d
Y ,I
I u) N .i
fi
Fig. 25. Two cylinders
-
force distribution.
d
A. Kyte, A. T@rum/ Coastal Engineering
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27 (1996) 263-286
*e
should be used. Each point in the diagram, Fig. 15, represents the maximum PFmax value for 8-14 waves. If more than 8-14 waves of the specific wave height is expected at the site one probably should use a higher PF value than given in the diagram. If less than 8-14 waves are expected one may consider a lower value than given in the diagram. A practical load safety factor is not included when the factor PFmax is used. The formulas are applicable when the wave height H, is above H, > 0.10 m or HOP > 10.0 m). The tests have been carried out for d = 0.05 m (prototype depth - 5.0 m). One should be cautious about using the results for waterdepths or waterdepth/waveheight ratios that differ significantly from the ones that have been used during the testing.
8. Estimates
of the impact force coefficient
C,
The evaluation of the force coefficient C,, Eq. (l), from force measurements requires the knowledge of the water particle velocity, which we did not obtain in our study. We will as an approximation use the results of Goda’s (1973) measurements of the water particle velocities on a circular reef. For waves in the range H, = 0.13-0.16 m the maximum measured force on one of the local force transducers was approximately 1 N. The water particle velocity that can be derived from Goda (1973) results is then approximately u = u, = 1.32 m/s. This gives C, = 1.91. The maximum forces on a local force transducer for H, - 0.2 m was approximately 4.0 N. Goda (1973) has no measurements for this wave height. However, if we set u = ( gH0>o.5 as suggested by Goda, we arrive at C, = 6.8. Although the evaluated C, values are uncertain, the calculations show that there is apparently a big scatter in the results, a scatter that has also been found by others, as previously mentioned.
9. Conclusions The study has given more insight into the very complex phenomena of plunging breaking wave forces on a vertical cylinder on a peaky shoal. We have been able to derive formulae for practical use to arrive at the wave forces and moments on a vertical cylinder on a reef. A warning is however, raised that the formulae apply for large plunging breaking waves and for conditions not too different from those during the model tests.
10. NOTATION
cs d D
Impact force coefficient water depth with reference cylinder diameter
to still water level
A. Kyte, A. T@rum / Coastal Engineering
27 (1996) 263-286
285
total wave force acceleration of gravity wave height in “deep” water significant wave height length over the cylinder the impact force acts time wave period peak wave period (peak of spectrum) water particle velocities velocity of cylinder at the time of impact total force coefficient maximum force coefficient moment coefficient for moment around the bottom level maximum moment coefficient around the bottom level water elevation above the still water level crest level above the still water line mass density of water duration of peak, Fig. 21 rise time, Fig. 21 “total” duration time, Fig. 21 Acknowledgements The paper is based on the first author’s M.Sc. thesis at the University of Trondheim, the Norwegian Institute of Technology, while the second author was his supervisor. The financial support from the Norwegian Coast Directorate made the experiments possible. We are grateful for this support. The authors are also grateful to Mr. Gustav Jakobsen, Norwegian Institute of Technology, Mr. Anders Storler and Mr. Jon Eggen, SINTEF NHL, for their assistance with mechanical and instrumentation work. References Apclt, C.J. and Piorewiez, J., 1987. Laboratory studies of breaking wave forces acting on vertical cylinders in shallow water. Coastal Eng., 1l(3): 241-262. Goda, Y., 1973. Wave forces on circular cylinders erected upon reefs. Coastal Eng. Jpn., 16. Goda, Y., 1985. Random Seas and Design of Maritime Structures. University of Tokyo Press. Goda, Y., Haranaka, S. and Kitahato, M., 1966. Study on impulsive breaking wave forces on piles. Rep. Port Harbour Res. Inst., 6(5): l-30 (in Japanese). Hoen, C. and Brathaug, H.P., 1987. STARTIMES.SINTEF rapport STSF71 A87047, Trondheim. Hovden, S.I., 1990. Wave forces on a vertical cylinder on a shoal. M.Sc. equivalent thesis, Faculty of Civil Engineering, The Norwegian Institute of Technology, University of Trondheim, Trondheim, Norway (in Norwegian). Hovden, S.I. and Torum, A., 1991. Wave forces on a vertical cylinder on a reef. Proc. III Conference on Port and Coastal Engineering for Developing Countries (COPEDEC), Mombasa, Kenya, 16-20 Sep 1991. Kyte, A., 1992a. Wave forces on a light house on the Hausene shoal. Student project work, Faculty of Civil Engineering, Norwegian Institute of Technology, University of Trondheim, Trondheim, Norway (in Norwegian).
286
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/ Coastal Engineering
27 (1996) 263-286
Kyte, A., 1992b. Wave forces on a light house on a shoal. M.Sc. equivalent thesis, Faculty of Civil Engineering, Norwegian Institute of Technology, University of Trondheim, Trondheim, Norway (in Norwegian). Lie, V., 1989. Waves over shoals. M.Sc. equivalent thesis, Faculty of Civil Engineering, The Norwegian Institute of Technology, Univ. Trondheim, Trondheim, Norway (in Norwegian). Lie, V. and Torum, A., 1991. Ocean waves over shoals. Coastal Eng., 15: 545-562. Mathiesen, M., 1992. EXTPAR: A computer program for statistical analysis of extreme values. SINTEF NHL rapport STF60 F92 124, Trondheim. Miller, B.L., 1977. Wave loads on horizontal circular elements of offshore structures. Paper presented at a meeting of the Royal Institution of Naval Architecture, London. Morison, J.R., Johnson, J.W. and O’Brien, M., 1953. Experimental studies on wave forces on piles. Proc. 1st Conf. Coastal Engineering. Sarpkaya, T., 1978. Wave impact loads on cylinders. In: Proceedings Offshore Technology Conference, Houston, TX. Sawaragi, T. and Nochino, M., 1984. Impact forces of nearly breaking waves on a vertical circular cylinder. Coastal Eng. Jpn., 27. Tanimoto, K., Takahachi, S., Kaneko, T. and Shiota, K., 1986a. Impact forces of breaking waves on inclined piles. In: Proc. 5th International Offshore Mechanics and Arctic Engineering COMAE) Symposium. Tanimoto, K., Takahachi, S., Kaneko, T. and Shiota, K., 1986b. Irregular breaking wave forces on an inclined pile. In: Proc. International Conference on Coastal Engineering Conference, Houston, TX. Vincent, C.L. and Briggs, M.J., 1989. Refraction-diffraction of irregular waves over a mound. J. Waterw. Port Coastal Ocean Eng. ASCE, 115(2). Von Karman, T.H., 1929. The impact force on seaplane float during landing NACA TN 321. Wagner, H., 1932. ijber stoss- und gleitvorgange an der Uberflasche von Flbssigkeiten. Zeitschtift t%r Angewandte Mathematik und Mechanik, Band 12, Heft 4, pp. 193-215. Watanabe, A. and Horikawa, K., 1974. Breaking wave forces on a large diameter cell. In: Proc. 14th International Conference on Coastal Engineering, Copenhagen.
Coastal Engineering 27 (1996) 263-286
Wave forces on vertical cylinders upon shoals Alexander Kyte a, Alf Torum bp* aBerdal Str@nme A/S, Consulting Engineers, Sand&a, Norway b SINTEF Norwegian Hydrotechnical Laboratory, Norwegian Institute of Technology, Trondheim, Norway Received 27 April 1995; accepted 20 December 1995
Abstract An experimental study has been carried out on the forces from plunging breaking regular and irregular waves on a vertical cylinder on a shoal. Total as well as local wave forces have been measured. Engineering formulae for the calculation of the horizontal forces and overturning moments have been derived. The duration of the impact forces have been measured and compares fairly well with theoretical values.
1. Introduction
There are plans in Norway to replace light buoys for navigation purposes with small fixed light houses. The location of many of these lights are on shoals, with water depths 5-10 m, in wave exposed locations such that the light houses will be exposed to breaking waves. The shoals are frequently very pesky. The depth increases rapidly to 50 m and more around the shoals. Lie and Tprrum (1991) investigated the wave conditions for long crested irregular waves around an idealized form of one of these shoals, the Arsgrunnen, in scale 1: 100. The water depth at the shallowest point corresponded to 5 m. For the largest waves the shoal went dry and the wave crest elevation was about three times the water depth Subsequently Hovden and Terrum (1991) investigated the wave forces on a vertical cylinder on top of the idealized shoal used by Lie and Torum (1991) for the wave studies.
’ Corresponding author. 0378-3839/96/$15.00 Science B.V.
Copyright 0 1996 Elsevier Science B.V. All rights reserved. Published by Elsevier
PII SO378-3839(96)00003-8
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27 (1996) 263-286
Based on the findings in these two studies the Norwegian Coast Directorate decided to sponsor a study on more details on the wave forces from breaking waves on vertical cylinders on shoals. It is the results of this study that are presented in this paper.
2. Previous
work on breaking
wave forces on vertical cylinders
A number of studies have been undertaken on breaking wave forces on vertical cylinders. Most of the studies are for vertical cylinders on uniformly sloping bottoms, Goda et al. (19661, Watanabe and Horikawa (1974) Sawaragi and Nochino (1984) Tanimoto et al. (1986a,b) and Apelt and Piorewiez (1987). Goda (1973) investigated the breaking wave forces on vertical cylinders upon circular and ledged reefs. Since Goda was not in this case interested in the short duration impact forces, his force measurement system was not designed to pick up the short duration high intensity impact forces on the cylinder (Goda, pers. commun., 1990). The planned Norwegian small lighthouses have natural periods of oscillations such that dynamic amplification of the structural response due to the short duration high intensity shock forces from the breaking waves may occur. In their study Hovden and Torum used a “stiff” measuring system. They obtained thus significantly higher wave forces than Goda (1973). In all the referenced studies, except by Tanimoto et al. (1986b), only the total forces on the cylinder were measured. Tanimoto et al. (1986b) measured also local forces in an area above the still water line. In order to get more details on breaking wave forces on a vertical cylinder on a shoal it was decided to carry out experiments and measure the local wave forces as well as total wave forces on a vertical cylinder on a shoal. A particular shoal “Hausene”, where a new lighthouse is planned, was installed in scale 1 : 100 in a wave flume as shown in Fig. 1. The shoal topography was modelled to a water depth of 50 m. The water depth on the shoal is 5 m. Strictly speaking directional waves should probably have been used for waves over three dimensional shoals. Vincent and Briggs (1989) showed that for non-breaking irregular waves there was a difference in wave heights on a shoal for long crested or short crested waves. However, since we did not have availability to laboratory short crested waves, we made tests with long crested waves. Nevertheless the results of our study are deemed to be very useful, also because most of the tests had to be run with regular waves (see below).
3. Some theoretical
considerations
Waves that enter shallow water will eventually break. The type of breaking (spilling, plunging or surging) is governed by the wave parameters and the slope of the bottom. It is the plunging type breaker which gives high impact forces on a vertical cylinder. The
I
WAVE ABSORBER ‘INSTRUMENTED
-
CYLINDER I
WG,-REFERENCE
Fig. I. Wave flume with the Hausene shoal (1: 300; dimensions in m).
I GAUGE
266
A. Kyte, A. T@xm / Coastal Engineering
B
A Fig. 2. Breaking
27 (1996) 263-286
waves: (A) Uniformly
sloping beach. (B) Shoal.
situation with a plunging wave hitting a vertical cylinder on a uniformly sloping bottom is shown schematically in Fig. 2A. The wave breaking on a shallow shoal is different and as shown schematically in Fig. 2B. When the waves are large the shoal goes dry under the wave through and the wave thereafter surges over the shoal. When the vertical wall of water hits a cylinder the situation is comparable with the situation when a falling cylinder hits a water surface. Several researchers have investigated the latter situation and the results have been compared with laboratory tests. One of the first attempt to derive a theoretical formulation was made by Von Karman (1929). He arrived at the following expression for the impact force on a falling cylinder when it hits the water: F = 0.5pC, Dlv,2 ( 1 - t/T)
where p = mass density of the water, C, = force coefficient, D = cylinder diameter, 1 = cylinder length, v, = velocity of the cylinder when it hits the water, t = time, and T = duration of the impact. The impact force has its maximum value at t = 0, e.g. in the beginning of the impact. Von Karman set the duration of the impact load to T = D/(2v,) or until the cylinder is half submerged into the water. He obtained theoretically, by some assumptions, a value C, = n. Experimental values of C, differ, however, from this value. Miller (1977) for example gives C, - 6.0 and Sarpkaya (1978) gives C, - 3.17 f 0.05. Sawaragi and Nochino (1984) show also that the value of C, varies and may be up to C, - 10 for waves breaking against a vertical cylinder and assuming v, = C, where C is the wave celerity when the wave breaks. The wave forces below the breaking region at the wave surface are calculated by using a Morison force formulation for a pile on a uniformly sloping bottom, Fig. 2A, (Goda et al., 1966; Watanabe and Horikawa, 1974). Apelt and Piorewiez (1987) found that in this case the force was depending on D/H,, H, = breaking wave height, where they considered D/H, as a kind of inverted Keulegan-Carpenter (KC) number (KC = u,,,T/D, where u,,,~~ is the horizontal max. particle velocity and T is the wave period).
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267
For a vertical cylinder on a shoal, Fig. 2B, the wave surges/plunges against the whole height of the cylinder and give a impact type load on the cylinder. A different approach was then used by Goda (19731, which we will revert to later. A special feature about impact forces from breaking waves on cylinders is the big scatter in the forces measured during laboratory tests, even when the waves are regular. This is probably due to small variation from wave to wave in the front of the breaking waves as the waves hit the cylinder.
Dz60
Fl
‘ti= F2
F3
Measures in mm Fig. 3. Instrumented
cylinder.
268
A. Kyte, A. Tg?rm / Coastal Engineering 27 (1996) 263-286
4. Test set-up The wave flume with the model Hausene shoal is shown in Fig. 1. The water depth on top of the shoal is 5 m. Fig. 3 shows the instrumented vertical cylinder. The total forces are measured with the force transducers marked Fll and FlO. The FlO force transducer is shown in Fig. 4. It is developed and built by MAFCINTEK,SINTEF Group. The strain gauges measure the shear deformation and hence the shear force. Force transducer Fl 1 is a commercial available force transducer. The total force is equal to the sum of forces measured at FlO and Fl 1. Wave force moments at the base is set equal to M = 0.399 X Fll (m, N). This is based on the assumption that the shear force is constant over the height of the force transducer FlO. The mid point of the transducer, which is located at the shoal bottom level, will then have zero moment. This assumption may not be absolutely true, and there is apparently a small error in the moment values. However, this error is deemed to be insignificant compared with the uncertainty related to the scatter shown in the test results. The force transducers Fl-F9 are local force transducers that measure the wave force in the wave direction on a ring of the cylinder, 1 cm high. Fig. 5 shows the local force transducer. It measures the shear deformation and hence the shear force on a cantilever beam in the direction of wave travel. The first tests were run with irregular waves. The data were sampled at a rate of 500 per second per recording channel (sampling frequency of 500 Hz) and a run time of 160 s. However, during the analysis it was revealed that it was preferable to use a higher sampling frequency. It was therefore decided to run the majority of the tests with regular waves and a sampling frequency of 4000 Hz. However, the run time was then limited to 20 s due to data storage limitations, which then made it less meaningful to use irregular waves. Based on previous work some effort was made to make the measuring system stiff to avoid dynamic amplifications in the response of the force measurement system. By tapping the cylinder and the different force transducers and recording the force responses we obtained natural periods of oscillations. The system had generally complicated modes of oscillations. By tapping the bottom of the cylinder we recorded natural frequencies of oscillation on force transducer No. FlO in the range 167- 182 Hz. By tapping the top of the cylinder we recorded natural frequencies of oscillations on force
Fig. 4. Force transducer.
A. Kyte. A. T~t~ttt/C~ustul
Engineering
-
Plan
Strain
27 (1996) 263-286
269
gauge
view
Side view
Fig. 5. Local force transducer.
transducer No. Fl 1 in the range 40-100 Hz. By tapping the local force transducers No. Fl-F9 we recorded natural frequencies of oscillations in the range 100-150 Hz. In order to see what effect these natural frequencies might have had on the response of the measurement system we did some calculations of the response based on the recorded apparent force, assuming the system is a one-degree of freedom system. Fig. 6 shows the relative response for a recorded force on a force transducer with a natural frequency of 166 Hz. In this case the dynamic amplification factor is approximately 1.15. It should be noted that we for sake of simplicity took the recorded response as the input force in the calculations, while the correct procedure would be to calculate the force giving the recorded response. We did similar calculations for other situations. In most of the cases the dynamic amplification factor was in the range 1.0-1.3, with exceptions as high as 1.5. In the analysis we have not made any attempt to correct for any dynamic amplifications. Nevertheless we think the results we have obtained are very useful for the design
Calculated
response
0.6 -
Milliseconds Fig. 6. Calculated
and recorded
force.
A. Kyte, A. T@wn / Coastal Engineering 27 (1996) 263-286
210
of the planned lighthouses or other types of vertical cylindrical structures on shoals, because there are so few existing data on the total wave impact forces and no data on the wave impact force distribution on such structures. Three wave gauges of the resistance type was mounted in the flume in positions as shown in Fig. 1, WGl, WG2 and WG3. The wave gauge WGl was in “deep” water, 0.90 m, and was used as a reference gauge. The wave heights generally used in the analysis was the wave heights measured at wave gauge WGl. The waves were generated through a combined wave generator control and data acquisition system. The data were analyzed primarily through the data analysis program STARTIMES, (Hoen and Brathaug, 1987). Some statistical calculations were carried out using the computer program EXTPAR, Mathiesen (1992). The PC based computer program MATLAB, The MathWorks Inc, was used for some minor calculations and graphical presentation of data.
5. Test programme The tests were carried out with regular and irregular waves according to the programme shown in Table 1. The wave parameters refer to wave gauge WG 1, Fig. 1 (deep water).
6. Analysis of the test results 6.1. Analysis
of the irregular
wave tests
As stated previously the breaking wave impact forces tend to vary considerably in magnitude from wave to wave even if the waves are regular. This is because the impact force is probably sensitive to the form of the front of the breaking waves. The steepness of the front face may vary slightly from wave to wave. The variations of the forces we
Table 1 Test program Tp (s)
Irregular waves, JONSWAP, y = 2.0
T (s)
1.o, 1.2, 1.4, 1.6
H, (cm)
H, (cm)
2.8- 14.7; total 17
Water depth (cm)
Run duration (9
Sampling frequency (Hz)
5
160
500
5
20
4OQO
runs Regular waves
1.0, 1.2, 1.4, 1.6
4-22; 14 waveheights for each period
A. Kyte, A. Thrum/
Coastal Engineering
I
.L~_
20
271
!
I -5
27 (1996) 263-286
I
22
2+
_--_I
26
28
30
Seconds Fig. 7. Sample of time series for the total force -
irregular waves.
see from the tests with irregular waves do not thus necessarily reflect the variation due to variability of the wave heights only. To get a complete statistical picture we should have run several tests, may be 50, with the same input wave generator signal. We would thus obtained the statistical variability of the impact wave force as well as the statistical variability due to the irregular waves. Since we had a sampling frequency of only 500 Hz for the irregular waves and thus some inaccuracies in the recorded forces, we only made one run for each combination of significant wave height and peak period to obtain some information of the variability of the wave forces in irregular waves. Fig. 7 shows a sample of time series for the total wave force. Due to the oscillations of the force measuring system we have multiple peaks during one impact. Since we are interested in the peak forces we used a procedure in the analysis program to pick out only one and the largest value for each impact. Since we also were interested in distinct peak forces we set a lower limit for the peak forces to be included in the analysis. This lower limit varied somewhat from run to run and was set higher for higher waves than for lower waves. In the example shown in Fig. 7 the lower limit was set to 1 N. The peak force values for each irregular wave run was fitted to a three parameter Weibull distribution:
where P(F) = probability of not exceeding F, F. = location factor, F, = scale factor, and y = shape factor. Fig. 8 shows a Weibull fit of peak force data. The fitting is carried out according to the methods of moments. The diagram of Fig. 8 is based on 87 data points. The force data are collected in bins, each bin being 0.5 N wide. Each point (triangular) in the plot of Fig. 8 is located at the upper end of each bin.
272
A. Kyte, A. T@rum/Coastal I
:: _1
Engineering
I
I
A A A A tlaxcmo
0.9990
---
ii
d
27 (1996) 263-286 I
1
I
serces
WecbuLL-3 WeLbuLL-3 fLt
B 0.9900
0
.sooo
0.5000
0.~000
NEWT01 Fig. 8. Weibull distribution
x-Tp=
1.07sek.
of peak forces.
*-Tp=1.32sek.
o-Tp=
1.59sek.
20
15
x 0
. 0
10 x
x
0
0 l
* l
*
1
x
5
0 *
0
0
2
4
6
8
Significant waveheight,
10 I-&, cm
Fig. 9. “One per mille” force.
12
14
A. Kyte, A. T#rwn / Coastal Engineering
27 (1996) 263-286
273
The location, scale and shape factors of the Weibull distribution are in this particular case 0.53 N, 3.15 N and 1.48 respectively. In order to condense the information we took the forces and moments corresponding to one per mille exceedence according to the Weibull distribution. (For the plot of Fig. 8 the one per mille force is 13.2 N). Figs. 9 and 10 show the “one per mille” force and moment for different significant wave heights and peak periods. It turned out that the target spectra with Tp = 1.2 s and 1.4 s came close to each other and are lumped together under Tp = 1.32 s. The general trend is that the force increases with increasing significant wave height, but with a slight tendency that the force and moments tend to flatten out for significant wave heights of 6 cm. When the significant wave height is 10 cm and higher the wave force tend to increase again. The shortest peak periods tend to give the highest forces, but this is not a very clear tendency. The maximum moment for each run was almost always recorded for the maximum force and at almost the same time point as the maximum forces.
6.2. Analysis
of total forces
and moments for regular waves
The tests with regular waves were carried out with a sampling frequency of 4000 Hz with a duration of each run of 20 s before the data storage was filled up. The tests were
x-Tp=l.Msek
* - Tp = 1.32 sek.
o-Tp=1.59sek
2.5 -
. 2
1.5
1
I
0.5
I 0
O_ U
2
4
6
Sign&ant
8
waveheight,
10
I-I,, cm
Fig. 10. “One per mille” moment.
12
14
A. Kyte, A. T~rum/Coastal
274
*T= 1.0s.
Engineering
XT= 1.2s.
27 (1996) 263-286
o T = 1.4 s.
+ T = 1.6 s. x
*
35 30 0
25 20 15 l
x 10 I 5-
0’ 0
l
l
x t 0 +
+ x
I x
*SF+ o
5
0
10
*
l
0
+. b x* Xx@+ xx +oo + + *+
x 0 + x+x
15
20
Wave height, H,, , cm. Fig. 11. Maximum horizontal
forces in regular waves vs. HO and T.
carried out with wave periods 1.0, 1.2, 1.4 and 1.6 s. The data logging system was switched on just before the first waves came to the cylinder in order to get the zero line for the force transducers recorded. Experience showed that there could be a slight shift in the zero-setting for the force transducers. This shift was then recorded and taken care of in the analysis. The valid recording period was then about 14 s or 8-14 waves for each run, depending on the wave period. Due to the limitation in the number of waves for each run the force data was not subject to any statistical analysis. Only the mean and maximum peak forces for each run were used. Figs. 11 and 12 show the maximum total force and the mean total force as a function of wave height and wave period. The wave height is the height of the wave at the reference gauge WGl, Fig. 1. Figs. 13 and 14 show the maximum and mean moment at the base of the cylinder. By comparing Figs. 11 and 12 we see that the maximum force is frequently twice the mean force or more, indicating the big scatter we have in the force data and which has been observed previously for similar tests on the impact forces from breaking waves. We see that the forces tend to increase up to a wave height of 10 cm. For higher wave heights the force flattens out until the wave height is approximately 15 cm. For still higher wave heights the force and moments increase significantly more again.
CoastalEngineering 27 (1996) 263-286
A. Kyte, A. T@rum /
*T= 1.0s.
XT=
1.2s.
o T = 1.4 s.
+T=
275 1.6s.
I
1
I
10
5
15
20
Wave height, I-&, cm. Fig. 12. Mean horizontal
forces in regular waves vs. H,, and T.
The flattening of the wave forces at a wave height of 10 cm and larger may be partly explained from Goda’s particle velocity measurements in waves breaking over a shoal (Goda, 1973). The increase of the forces from a wave height of 15 cm and on cannot so easily be explained from Goda’s measurements. We believe that as the waves become higher they break more violently in front of the shoal and thus create larger velocities and forces on the cylinder. 6.3. Comparison
with previous
work
The only known previous works on similar problems is by Goda (1973) and Hovden and Tot-urn (1991). Goda presented the mean and standard deviation of the forces for 203 consecutive waves while Hovden and Torum (1991) used the maximum force for 10 consecutive waves. Goda dit not record the high intensity short duration forces. Goda formulated the forces and moments at the base of the cylinder in the following way: Force F = p,pgDH,( Moment
d + q,,)
M= 0.5p’MpgDH,(
d + q,,)’
(3) (4)
rl max= 0.754 rl max is the crest height,
d is water depth and H is wave height in “deep”
water. PF
276
A. Kyte, A. Tpmam / Coasral Engineering *T=
1.0s.
XT=
1.2s.
oT=
27 (1996) 263-286 1.4s.
+T=
1.6s.
3.5 .
l
3-
*x L
0 0
5
10
15
Wave height, H, , cm. Fig. 13. Maximum
moment in regular waves vs. H,, and T.
10
Wave height, &,
cm.
Fig. 14. Mean moment in regular waves vs. H,, and T.
20
A. Kyte, A. T#rum / Coastal Engineering *-T=
1.0s.
x-T=
1.2s.
277
27 (1996) 263-286 o-T=1.4s.
+-T=1.6s.
3-
2.5 -
2-
. .
PFmU
I .5 -
O8
12
10
16
14
18
20
22
24
Wave height, H,, , cm. Fig. 15. PF -
*-T=
1.0s.
maximum force.
x-T=1.2s.
o-T=
1.4s.
+-T=
1.6s.
0.9 0.8 0.7-
x o
l
0.6 -
PFmun
+
0.5 -
o
C.
0.4 0.3 0.2 0.1 -8
IO
12
I4
16
Wave height, H,, , cm. Fig. 16. PF -
mean force.
18
20
22
24
278
A. Kyle, A. T@wn/ +-T=
Coastal Engineering 27 (1996) 263-286
1.0s.
x-T=1.2s.
12
14
o-T=1.4s.
+-T=
1.6s.
I
0’
8
10
16
18
20
22
24
22
24
Wave height, &, cm. Fig. 17. pw -
*-T=
1.0s.
maximum
moment.
x-T=1.2s.
o-T=
1.4s.
+-T=
1.6~.
)I_________ 2.5 -
2-
P Mrnun
.
1.5-
x
I l-
0
0.5 -
01
8
IO
12
14
16
18
Wave height, I-&,, cm. Fig. 18. PM -
mean moment.
20
A. Kyte, A. T@rum / Coastal Engineering 27 (1996) 263-286
279
and PM are force and moment coefficients obtained from the mean forces. Goda obtained for a circular reef mean values of BF in the range 0.3-0.5 and @‘, = 0.3-0.55. We obtained PF coefficients based on the maximum force as shown in Fig. 15 and on the mean force as shown in Fig. 16. Based on our measurements we found that the arm of the force was approximately (d + 0.2Ha). Hence the moment becomes M=p,Dpg(d+0.75Ho)(d+0.2H,)H,
(5)
This differs slightly from Goda’s formulation but within the range of wave heights H,, = lo-20 cm the difference between (d + 0.2H,) and 0.5(d + 0.75H,) is not very large. To what extent there is a ‘ ‘Keulegan-Carpenter number” effect in this case, like Apelt and Piorewiez (1987) indicated, was not specifically looked at by Goda. However, the large wave forces occurred as the wave front passed the cylinder and before vortices were shed regularly behind the cylinder. Hence it may not be relevant to introduce a KC-number consideration, which is valid and normally used for the part of a cylinder which is always submerged under wave action. We found PM values based on Eq. (5) as shown in Figs. 17 and lg. The diagrams show that the maximum moment coefficients are large compared to the mean moment coefficients. Goda does only give the standard deviation of his coefficient. From the standard deviation, however, it is deemed that the biggest difference between Goda and our results are that the maximum forces we obtained are larger than the forces measured by Goda (1973). This may possibly be attributed to the differences in the measurement system. Hovden and Torum (1991) carried out their tests on a more elongated type of shoal and less steep in the front than we have used in this study. The scale of the shoal Hovden and Torum used was the same, 1: 100, and the water depths on the top of the shoal corresponded to 5 m as in this present study. For waves lower than No = 0.16 m Hovden and Torum obtained slightly higher forces than we have found on the Hausene shoal. However, the extreme forces for higher waves (H, > 0.16 m> found in this present study are larger than those found by Hovden and Torum (1991). The latter shows that the shape of the shoal has a significant influence on the wave forces. In general we will expect to find the maximum forces on the steepest shoal. For a less steep shoal the highest waves will break and loose their energy before arriving at the vertical cylinder. For a steeper shoal such as Hausene, the largest waves can break more directly on the cylinder and thus introducing larger forces.
6.4. Duration
of the wave impact load
Fig. 7 shows a time series of total wave forces. Generally the peak impact force is followed by a Morison type force. The time history of wave force can schematically be shown as in Fig. 19. ~~ is the rise time, or is duration of the peak and ~~~~is “total”
280
A. Kyte, A. T@um / Coastal Engineering 27 (1996) 263-286
Fig. 19. Duration of impact force.
duration. From the dynamic amplification of the structural response TI, may have been a better choice than rP. There was a significant scatter of the duration or. As an example we found for regular waves in the range H, = 0.13-0.16 m, 7P = 0.004-0.016 s. The impact duration rP is related to the diameter and the water velocity when the wave hit the cylinder, or = AD/u). Different researchers give different relations, but normally in the range Tp = 0.25D/u - 0.5D/u. Tanimoto et al. (1986b) set the duration 7p = 0.25D/u. If we use the results of Goda’s water particle measurements we find for H, = 0.145 m the
a-20
-L
n
1
H, = 14.76 cm T,= 1.38s
-2 -3 -4 --
7
--
I 29-5
I
I
I
1
>
Newton per local force meter Fig. 20. Local force distribution.
A. Kyre, A. T@rum/ Coastal Engineering
-I
27 (1996) 263-286
I
281
y = 14.5 cm TP= 1.4s
1
z-20
2 --
3
--
6 --
=-
z-0 r--5
10
5
-6 -7 -8 -..9...-: . . . . .
I
I
SWL . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .
: I
t
I
I
I
0.5
I.0
IS
2D
>
Newton per local force meter Fig. 21. Local force distribution.
maximum velocity u = 1.3 m/s. According to Tanimoto et al.‘s formulation obtain 7r = 0.012 s. This is in the “ball park” of our observations.
we then
6.5. Force intensity along the cylinder Several diagrams showing the force intensity along the cylinder were prepared. There was generally scatter in these plots. Figs. 20-22 show examples of the measured force intensity when the total force on the cylinder was about its maximum value. The location of the local force transducers was based on the results by Tanimoto et al. (1986b), who measured impact forces on a vertical pile on a uniformly sloping bottom. Their measurements showed that the impact force occurred well above the still
a-20
+ 12-3
x-10
II
H, = 20.6 cm T,= 1.4s
--
6 -__ 5 --
z-Q z--5
Newton per local force meter Fig. 22. Local force distribution.
A. Kyfe. A. T@rutn/ Coastal Engineering
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27 (1996) 263-286
water line. Hence we thought that impact forces would occur only above the still water line in our case also and did not locate any local force transducers below the still water line. However, during the tests it was revealed that impact forces probably occurred below the still water line when the cylinder is located on a shallow shoal like the Hausene shoal. Hence, it would have been useful with local force transducers below the still water line also. Since we did not use local force transducers below the still water line we have assumed the force intensity of the lowest local force transducers to be valid at the stillwater line and set the impact force intensity to zero at the bottom (see later). The Goda (1973) force formulation is based on the assumption that the force is uniformly distributed along the cylinder. The measured force intensities show that this is not necessarily the case.
7. Engineering
formulas
to calculate the wave forces
We want to apply the obtained results for a tower with two diameters as shown in Fig. 23 and which is frequently used in Norway. The lower cylinder is extending beyond the smaller cylinder so much that an impact force on the lower cylinder may be over when the wave hit the smaller cylinder. To be a bit conservative we will assume the front of the wave as indicated in Fig. 23. In this case the impact load may be exerted on the upper smaller and lower larger cylinders at the same time. In order to calculate the wave force on the two cylinders we need an evaluation of the wave force intensity. The results showed, however, a large scatter. As a reasonable choice we used the force intensity distribution as shown in Fig. 24 for the highest waves, e.g. H, > 0.1 m. The fixing of the upper end at a distance 3.5 times the water depth above the top of the lower cylinder may seem a bit strange. But the results showed that the upper end of the force intensity diagram was not much influenced by the wave height (at the reference point) when the wave height was above H,=O.lO m. Once the force intensity diagram is given the force intensity f of the base is given such that the force at the base is equal to the force given by Eq. (3) with qma, = 0.75H,. f = 0.421BF(
D/d)pg(
d + 0.75H,)H,
Fig. 23. Two cylinders.
(6)
A. Kyte, A. T@rum / Coastal Engineering 27 (1996) 263-286
Fig. 24. One cylinder
and the total force and moment F = pFDpg( M=
-
force distribution.
at the bottom
d + 0.75H,)H,
2PFDpgd(
283
(7)
d + 0.75H,)H,
(8)
When we extend this approach to the two cylinder structures of Fig. 23 we arrive at a force intensity diagram as shown in Fig. 25. The discontinuity of the force intensity diagram is due to the discontinuity of the diameters. The force intensities f, and f2 are given by j-, = 0.421p,(
D,/d)pg(
d + 0.75H,)H,
(9)
f2 = 0.421p,(
D,/d)pg(
d + 0.75H,)H,
(IO)
The force F, and moment
moment
M, at the base of the upper cylinder M2 at the base of the larger cylinder are given by
F, = 0.737p,D,pg(
d + 0.75H,)H,,
M, = 0.860P,D,pgd(
(11)
d + 0.75H,)H,
F2 = pFpgHO( d + 0.75H,,)(O.7370, M, = PFpgHod(
d + 0.75&)(
and the force F2 and
(12) + 0.2630,)
(13)
1.781 D, + 0.2190,)
(14)
The choice of the characteristic value of the PF coefficient should be based on the wave climate at the specific site and the life time of the structure. For design purposes
3.5 d 7l
1.25
d
Y ,I
I u) N .i
fi
Fig. 25. Two cylinders
-
force distribution.
d
A. Kyte, A. T@rum/ Coastal Engineering
284
27 (1996) 263-286
*e
should be used. Each point in the diagram, Fig. 15, represents the maximum PFmax value for 8-14 waves. If more than 8-14 waves of the specific wave height is expected at the site one probably should use a higher PF value than given in the diagram. If less than 8-14 waves are expected one may consider a lower value than given in the diagram. A practical load safety factor is not included when the factor PFmax is used. The formulas are applicable when the wave height H, is above H, > 0.10 m or HOP > 10.0 m). The tests have been carried out for d = 0.05 m (prototype depth - 5.0 m). One should be cautious about using the results for waterdepths or waterdepth/waveheight ratios that differ significantly from the ones that have been used during the testing.
8. Estimates
of the impact force coefficient
C,
The evaluation of the force coefficient C,, Eq. (l), from force measurements requires the knowledge of the water particle velocity, which we did not obtain in our study. We will as an approximation use the results of Goda’s (1973) measurements of the water particle velocities on a circular reef. For waves in the range H, = 0.13-0.16 m the maximum measured force on one of the local force transducers was approximately 1 N. The water particle velocity that can be derived from Goda (1973) results is then approximately u = u, = 1.32 m/s. This gives C, = 1.91. The maximum forces on a local force transducer for H, - 0.2 m was approximately 4.0 N. Goda (1973) has no measurements for this wave height. However, if we set u = ( gH0>o.5 as suggested by Goda, we arrive at C, = 6.8. Although the evaluated C, values are uncertain, the calculations show that there is apparently a big scatter in the results, a scatter that has also been found by others, as previously mentioned.
9. Conclusions The study has given more insight into the very complex phenomena of plunging breaking wave forces on a vertical cylinder on a peaky shoal. We have been able to derive formulae for practical use to arrive at the wave forces and moments on a vertical cylinder on a reef. A warning is however, raised that the formulae apply for large plunging breaking waves and for conditions not too different from those during the model tests.
10. NOTATION
cs d D
Impact force coefficient water depth with reference cylinder diameter
to still water level
A. Kyte, A. T@rum / Coastal Engineering
27 (1996) 263-286
285
total wave force acceleration of gravity wave height in “deep” water significant wave height length over the cylinder the impact force acts time wave period peak wave period (peak of spectrum) water particle velocities velocity of cylinder at the time of impact total force coefficient maximum force coefficient moment coefficient for moment around the bottom level maximum moment coefficient around the bottom level water elevation above the still water level crest level above the still water line mass density of water duration of peak, Fig. 21 rise time, Fig. 21 “total” duration time, Fig. 21 Acknowledgements The paper is based on the first author’s M.Sc. thesis at the University of Trondheim, the Norwegian Institute of Technology, while the second author was his supervisor. The financial support from the Norwegian Coast Directorate made the experiments possible. We are grateful for this support. The authors are also grateful to Mr. Gustav Jakobsen, Norwegian Institute of Technology, Mr. Anders Storler and Mr. Jon Eggen, SINTEF NHL, for their assistance with mechanical and instrumentation work. References Apclt, C.J. and Piorewiez, J., 1987. Laboratory studies of breaking wave forces acting on vertical cylinders in shallow water. Coastal Eng., 1l(3): 241-262. Goda, Y., 1973. Wave forces on circular cylinders erected upon reefs. Coastal Eng. Jpn., 16. Goda, Y., 1985. Random Seas and Design of Maritime Structures. University of Tokyo Press. Goda, Y., Haranaka, S. and Kitahato, M., 1966. Study on impulsive breaking wave forces on piles. Rep. Port Harbour Res. Inst., 6(5): l-30 (in Japanese). Hoen, C. and Brathaug, H.P., 1987. STARTIMES.SINTEF rapport STSF71 A87047, Trondheim. Hovden, S.I., 1990. Wave forces on a vertical cylinder on a shoal. M.Sc. equivalent thesis, Faculty of Civil Engineering, The Norwegian Institute of Technology, University of Trondheim, Trondheim, Norway (in Norwegian). Hovden, S.I. and Torum, A., 1991. Wave forces on a vertical cylinder on a reef. Proc. III Conference on Port and Coastal Engineering for Developing Countries (COPEDEC), Mombasa, Kenya, 16-20 Sep 1991. Kyte, A., 1992a. Wave forces on a light house on the Hausene shoal. Student project work, Faculty of Civil Engineering, Norwegian Institute of Technology, University of Trondheim, Trondheim, Norway (in Norwegian).
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/ Coastal Engineering
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Kyte, A., 1992b. Wave forces on a light house on a shoal. M.Sc. equivalent thesis, Faculty of Civil Engineering, Norwegian Institute of Technology, University of Trondheim, Trondheim, Norway (in Norwegian). Lie, V., 1989. Waves over shoals. M.Sc. equivalent thesis, Faculty of Civil Engineering, The Norwegian Institute of Technology, Univ. Trondheim, Trondheim, Norway (in Norwegian). Lie, V. and Torum, A., 1991. Ocean waves over shoals. Coastal Eng., 15: 545-562. Mathiesen, M., 1992. EXTPAR: A computer program for statistical analysis of extreme values. SINTEF NHL rapport STF60 F92 124, Trondheim. Miller, B.L., 1977. Wave loads on horizontal circular elements of offshore structures. Paper presented at a meeting of the Royal Institution of Naval Architecture, London. Morison, J.R., Johnson, J.W. and O’Brien, M., 1953. Experimental studies on wave forces on piles. Proc. 1st Conf. Coastal Engineering. Sarpkaya, T., 1978. Wave impact loads on cylinders. In: Proceedings Offshore Technology Conference, Houston, TX. Sawaragi, T. and Nochino, M., 1984. Impact forces of nearly breaking waves on a vertical circular cylinder. Coastal Eng. Jpn., 27. Tanimoto, K., Takahachi, S., Kaneko, T. and Shiota, K., 1986a. Impact forces of breaking waves on inclined piles. In: Proc. 5th International Offshore Mechanics and Arctic Engineering COMAE) Symposium. Tanimoto, K., Takahachi, S., Kaneko, T. and Shiota, K., 1986b. Irregular breaking wave forces on an inclined pile. In: Proc. International Conference on Coastal Engineering Conference, Houston, TX. Vincent, C.L. and Briggs, M.J., 1989. Refraction-diffraction of irregular waves over a mound. J. Waterw. Port Coastal Ocean Eng. ASCE, 115(2). Von Karman, T.H., 1929. The impact force on seaplane float during landing NACA TN 321. Wagner, H., 1932. ijber stoss- und gleitvorgange an der Uberflasche von Flbssigkeiten. Zeitschtift t%r Angewandte Mathematik und Mechanik, Band 12, Heft 4, pp. 193-215. Watanabe, A. and Horikawa, K., 1974. Breaking wave forces on a large diameter cell. In: Proc. 14th International Conference on Coastal Engineering, Copenhagen.