WAVE SLAMMING ON AHORIZONTAL PLATEbySHANKAR SUBRAYA BHATB.E., Karnataka University, India, 1984M.Tech., Mangalore University, 1986A THESIS SUBMflED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF CIVIL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust, 1994© Shankar S. Bhat, 1994In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(Signature)Department of_________________The University of British ColumbiaVancouver, CanadaDate___________IIAbstractThe design of coastal and offshore structures requires a thorough understanding ofenvironmental loading, particularly due to waves. Structural elements such as deckslocated in the splash zone encounter intermittent contact with the water, and the loadsassociated with the water impact may be several times larger than those experienced byelements when fully submerged. These forces may give rise to localized damage and tofatigue problems.Such structures should clearly be designed to account for wave impact, in addition tomore general wave loading. Several studies have reported the related problems of shipbottom slamming, missile entry and sea plane landing. Although previous studies havecontributed to an improved understanding of wave impact, there is still considerableuncertainty in the estimation of impact loads on structural elements near the water surface.In this context, the present study has been carried out to examine the wave loads on a fixedhorizontal plate located near the still water level.Experiments were conducted in the wave flume of the Hydraulics Laboratory of theDepartment of Civil Engineering at the University of British Columbia. A plate, 60.0 cmlong, 20.0 cm wide and 6.25 mm thick, was instrumented with load-cells to measure thevertical force on the plate due to waves. The plate was supported by two vertical rodsthrough the load-cells which were connected to a cross shaft mounted on bearings at theends.Tests were conducted over a range of wave periods and wave heights in combinationwith different plate clearances above the still water level. The vertical reactions at the twosupports were measured, and the time histories of vertical force and its line of action areifithereby obtained. The wave surface elevations at the leading and rear end of the plate weremeasured with the plate absent. Results are presented in the form of force time histories,their lines of action and the associated water surface elevation. An analysis of these timehistories is carried out to obtain various parameters of wave impact which include, the peakupward and downward force, their lines of action and times of occurrence, and theassociated wetted lengths. The influence of incident wave parameters on these isinvestigated. Video images are studied to understand the impact process and to identify thedifficulties involved in the investigation. An attempt is also made to predict the verticalforce based on the hydrodynamic impact, drag and buoyancy forces.ivTable of ContentsPageAbstractTable of Contents.ivList of Tables viiList of Figures viiiList of Symbols xiiAcknowledgments xvChapter 1 Introduction 11.1 General 11.2 Literature Review 21.2.1 Water Entry Problem 21.2.2 Horizontal Cylinder 41.2.3 Horizontal Plate 51.3 Scope of the Present Investigation 7Chapter 2 Theoretical Development 82.1 Dimensional Analysis 8V2.2 Vertical Force Formulation 102.2.1 Wave Theory and Associated Kinematics 142.2.2 Superposition of Force Components 162.3 Dynamic Response of SDOF System 19Chapter 3 Experimental Investigation 213.1 Introduction 213.2 The Plate 213.3 Wave Flume and Generator 223.4 Control and Data Acquisition 233.5 Measurements 233.6 Experimental Procedures 243.7 Dynamic Characteristics of the Assembly 253.8 Data Processing 27Chapter 4 Results and Discussion 294.1 Vertical Force 304.2 Vertical Force and Incident Waves 324.3 Video Records 344.4 Force Predictions 35viChapter 5 Conclusions and Recommendations 38References 40Appendix A Static Analysis 42List of TablesTable 2.1 Added mass constant for a thin rectangular plate.Table 3.1 Regular wave parameters used in experiments.Table 4.1 Summary of test conditions and principal results.Table 4.2 Computed values of the factor a. in selected tests.viivifiList of FiguresFig. 1.1 Photographs of typical jetty facilities, Jericho beach, Vancouver.Fig. 2.1 Defmition sketch.Fig. 2.2 Stages of wave propagation past a horizontal plate. (a) initial contact, t = to;(b) submergence of upwave portion of plate, to 0.5 if waves are assumed to be sinusoidal. HIL is the wave steepness, dJL is adepth parameter, and £JL is a relative length parameter of the plate, analogous to a wavediffraction parameter.10It is illustrative to consider the typical ranges of some of these parameters. Intermittentsubmergence of the plate occurs for -0.5 0.5. Based on stability considerations, the wave steepness varies up to 0.142 in deepwater. Typical wave conditions may include wave periods ranging from 5 to 20 sec so thatilL may span a relatively wide range.2.2 Vertical Force FormulationThe vertical force on a horizontal member subjected to intermittent submergence in waves isgenerally taken to be made up of hydrodynamic impact, drag and buoyancy forcecomponents. (Although the buoyancy force was omitted in Section 2.1, in typicalexperiments it may not be negligible compared to the other force components.) Thus, thevertical force expressed as:F(t) = Fa(t) + Fd(t) + Fb(t) (2.4)where Fa(t), Fd(t) and Fb(t) are hydrodynamic impact, drag and buoyancy forcecomponents respectively. These components are briefly discussed in the sections tofollow. However, it is useful to consider initially the various stages of interaction during awave cycle.Stages of Wave InteractionConsider a wave train interacting with the plate during the course of one cycle. Figure 2.2shows the various stages of the plate submergence in terms of the positions of watersurface relative to the plate, and the corresponding times of occurrence. Figure 2.2(a)shows the instant at which a wave just makes contact with the leading edge of the plate. InFig. 2.2(b), the wave progresses further, partially wetting the plate. Figure 2.2(c) shows11the stage at which the plate is fully submerged. The plate remains submerged until thedownwave free surface reaches the leading edge of the plate. Figure 2.2(d) shows thewave recession stage, during which the leading edge of the plate is exposed. Finally,Fig. 2.2(e) shows the instant of complete detachment of the wave from the rear edge of theplate.Defining the intersection locations of the downwave and upwave free surfaces as x1 andx2 respectively, along the length of the plate, with the origin at the leading edge of theplate, and with the wetted length denoted as = x2 — xi, the above sequence for one cyclemay be summarized as follows.• For t = xi = 0, X2 = 0,• Forto t2, the wave recedes sothat A. decreases; and finally at t = t3, = 0. On the basis of linear theory, the wetted lengthcan be obtained by equating the wave surface elevation r to the plate clearance h andsolving for replacing x as:h = cos(k- cot) (2.20)CombinationsFollowing section 2.2 and Eq. 2.4, the force components may be superposed to obtain thetime variation of the vertical force:FFa1+Fa2+Fd+Fb (2.21)Also, as discussed in section 2.2.1, the plate clearance h is of the order of wave heightH, so that on the basis of linear wave theory the velocity v and acceleration v may be takenas v r and v ij. Substituting the expressions for the force components and the abovesimplification for wave kinematics in Eq. 2.21, we obtain:F = cc p ? b2f1(2Jb) ij + a p b2 i f2(AJb)+Cdpb?1l1I +pgV (2.22)17The magnitude of /at may be taken as the wave celerity c when the plate is partiallysubmerged, and zero when it is fully submerged or completely above the water surface.However, this highlights a difficulty with the force formulation that has been proposed, inthat the corresponding term falling abruptly to zero as the plate becomes fully submergedbecause of this abrupt change in /3t. In fact, the actual added mass is expected to varyso as to give rise to a more gradual variation of this term. Based on the foregoing, thevariation F in Eq. 2.8 is modified by replacing JAJ& with a velocity c’ which is constantand equal to wave celerity c during partial submergence (i.e. from to to t1); and thenassumed to fall linearly from c at time t1 to zero at time tm. This variation is sketched inFig. 2.3(c) and is introduced simply as a device to avoid the abrupt fall in F.Then Eq. 2.22 may be re-written asF = a p b2f1(X/b) T) + a p b2 i c’f2QJb)+pgV (2.23)wherec for to < t < t1; (i.e. ij > 0)= c[ t2 +t1- 2t] for ti 1 (sothat m = x it ? b2) as (2Jb) — oo A suitable function f1() is fitted to this data and used inEq. 2.23. Thus, the function f1() and f2() are given by:f1(X/b) = (XJb) exp [- 0.53 (AJb)088] + 0.1E[QJb) - 2.0] (2.24)f2(A/b) = 2.0 - [0.53 ()O.88] f1(A/b) + 0.25 v[ (A/b) - 1.5] (2.25)1 0 for (A/b) < 2.0whereL. 1 for (A/b) 2.0for(A/b)<1.5I. 1 for (A/b) 1.5.Figure 2.4 is a plot indicating the variation of the added mass functions based on thetabulated values of Table 2.1, and those based on Eqs. 2.24 and 2.25.Due to the complicated flow around the plate and the associated uncertainty relating tothe use of Eq. 2.7 for the added mass, a simpler alternative based on the two-dimensionalcase of infinite width may instead be adopted. The two-dimensional limit A/b —* 0corresponds tof1(AIb) —* (A/b) so that Eq. 2.7 would be replaced by:m = xpA2b (2.26)This leads to Eq. 2.23 being replaced by:F = cxp??bij + 2opAc’bi +1CdpAb1jIiI + pgV (2.27)It is illustrative to consider the variation of added mass with the plate aspect ratio.Fig. 2.5 indicates such a variation for arbitrary values of o, 1 and 3 for Eq. 2.23 and 0.5and 0.7 for the simpler added mass model given by Eq. 2.26.192.3 Dynamic Response of SDOF SystemA formulation of the vertical force on a horizontal plate has been presented in the previoussection assuming that the plate and the load-cell assembly act as a fixed structure.However, in almost all cases the dynamic response of the structure and its measuringsystem occur to some extent, and influence the force that is measured, particularly underimpulsive loading. An estimation of the effect of this on the measured force is of interestand may be made on the basis of a simple analysis of single degree of freedom (SDOF)system subjected to impact loading (e.g. Isaacson and Prasad, 1993). The plate and load-cell assembly is modelled as a SDOF system with a mass M, stiffness K and dampingcoefficient C as indicated schematically in Fig. 2.6. The stiffness K is related to thestiffnesses of each of the load-cells placed at two supports. The equation of motion of thesystem when subjected to a time varying load F(t), is given as:Mü+Cu+Ku=F(t) (2.28)where u, i.t and ü are the instantaneous plate displacement, velocity and accelerationrespectively. The natural frequency o and the damping ratio of the system are definedas o = J K/M and = C/2Mo. The response u(t) of the system starting from rest isgiven by:u(t) = $F(t) en(tt) sin[d(t-t)] dt (2.29)Mojwhere 0d is the damped natural frequency of the assembly defined as 0d = w[i2, and tis a variable of integration.The force F(t) is the applied vertical force as given in either Eq. 2.23 or Eq. 2.27 and itis composed of force components given in Eq. 2.21. However, the force transmitted Ft(t)20to the load-cell (i.e. measured by the load-cell) at the supports is proportional to thestiffness of the load-cell, (with damping neglected) and is given by:F(t) = K u(t) (2.30)The SDOF system is now considered to be subjected to an idealized load, F(t) given as atriangular function as indicated in Fig. 2.7. F(t) increases linearly to a peak value of F0 in arise-time Tr and then drops linearly to zero in a decay-time Td. The complete responsehistory of the system can be determined using the Eq. 2.29 for the given triangular pulseload (e.g. Humar, 1990).Isaacson and Prasad (1993) obtained a closed-form solution for the SDOF systemidentifying the peak force F0 immediately after the impact and the associated rise time T asimportant parameters, and presented a set of characteristic curves describing these asindicated in Fig. 2.8. In the figure, Ft0 is the peak measured force, F0 is the peak appliedforce, Tm is the rise-time associated with measured force, Tr is the rise-time associated withthe applied force, and T= 2ic/o1,is the natural period of the system. The influence ofdynamic characteristics on the peak measured force Ft0 and T can readily be estimatedfrom the figure.21Chapter 3Experimental Investigation3.1 IntroductionAs mentioned earlier, objectives of the present investigation are to study the hydrodynamicsof the impact process and to relate the important parameters of the loading to those thatgovern the process. To this end, a series of experiments was carried out at the HydraulicsLaboratory of the Civil Engineering Department, University of British Columbia, and thischapter gives a detailed account of the experimental investigation.3.2 The PlateAn important aspect of the force measurement relates to the requirement of providing anaccurate measurement of the external force acting on the plate, without including anyextraneous effects due to the dynamic response of the test setup. Because of thisrequirement, the plate and load-cell assembly should have a high natural frequency incomparison to dominant loading frequencies. There are two principal modes of vibrationwhich may affect the force measurement; one corresponding to a first mode beam vibrationand the other to the whole assembly vibrating as a lumped mass supported by the loadcells. Because of this, the beam should be as stiff as possible, the overall mass supportedby the load-cells should be minimized, and the load-cell should be sufficiently stiff.Bearing these considerations in mind, a plate model was designed to be stiff enough toapproximate a rigid body, and at the same time to be as thin as possible so as to simulate a22thin plate. Photographs of the test assembly are shown in Fig. 3.1, and a sketch of theexperimental set-up is provided in Fig. 3.2. The plate assembly was designed as a pinnedbeam supported at two points and with a large overhang on the upwave side so as tominimize the interference of the load-cells and plate supports during the initialsubmergence. The plate is constructed of acrylic, and is 20 cm wide, 60 cm long and6.25 mm thick. Two aluminum angles are fixed above the plate to increase its rigidity. Asmall gap between the plate and the angles allows overtopped water to drain freely. Pinnedsupports are made of two bearings attached to the inner face of the angle. Aluminum shaftsthrough these bearings connect the load-cell and support rod to the stiffener angles. Thesesupports are at 30.0 cm and 54.5 cm away from the leading edge of the plate. Thesupporting rods are threaded and are fixed by nuts to a longitudinal steel box section 5 cmhigh x 7.5 cm wide. The box section is aligned in such a way that its longitudinal centre-line coincides with that of the flume. It is bolted to steel cross-channels which rested on thetop of the flume side-wall and was clamped to the flume walls. The level of the plate wasadjusted by lengthening or shortening the part of the rods between the supportinglongitudinal channel and the load-cell. By this arrangement, vertical forces were measuredby the load-cells at two supports. Since the plate is very thin, and the front end is beveled,the horizontal wave force on the plate during wave impact is neglected.3.3 Wave Flume and GeneratorA sketch of the Hydraulics Laboratory wave flume is shown in Figure 3.3. The waveflume measures 20 m long x 0.6 m wide x 0.75 m deep. An artificial beach is located atone end to reduce wave reflection. Waves are generated by a single paddle wave actuatorlocated at the upwave end. The generator is controlled by a DEC VAXstation-3200minicomputer using the GEDAP software package developed by the National ResearchCouncil, Canada, (NRC). The generator is capable of producing wave heights up to 30 cm23and wave periods as low as 0.5 sec. In the present study, regular waves of heights rangingfrom 3.0 to 17.5 cm and periods ranging from 0.8 to 2.0 sec. were used. During each ofthese tests, water depth was maintained at 0.55 m.3.4 Control and Data AcquisitionThe GEDAP general purpose software package was used extensively during all stages ofthe experimental investigation. This software package is available for the analysis andmanagement of laboratory data, including real-time experimental control and data-acquisition functions. GEDAP is a fully-integrated, modular system which is linkedtogether by a common data file structure. GEDAP maintains a standard data file format sothat any GEDAP program is able to process data generated by any other GEDAP program.This package also includes an extensive set of data analysis programs so that mostlaboratory projects can be handled with little or no project-specific programming. Anattractive feature is the fully-integrated interactive graphics capability, such that results canbe conveniently examined at any stage of the data analysis process. It also includes anextensive collection of utility packages, which consist of a data manipulation routine, afrequency domain analysis routine, and statistical and time-domain analysis routines. Inparticular, the program RTC_SIG generates the control signal necessary to drive the wavegenerator, and the routine RTC_DAS reads the data acquisition unit channels and stores theinformation in GEDAP binary format compatible with other GEDAP utility programs.3.5 MeasurementsThe water surface elevation and the associated vertical force were required to be measuredfor each test. The vertical forces at two supports were measured using load-cells. Theselection of the load-cells is based primarily on considerations of sensitivity, load range andstiffness requirements. Two axial ‘S’ type load-cells (Interface SSM 500) have been used24in the present study. These work on the principle of flexure of the central limb of an ‘S’sensed by precision strain gage circuitry. Each load-cell has a load capacity of 500 lb(2.2 kN) and an axial stiffness of 4.9 kN/mm, with a sensitivity of 15 iVfN for anexcitation of 10 V. The output voltage of the load-cells are amplified by a PacificInstruments Model 8255 Transducer Condition Amplifier. An amplification of 1000 wasfound to be adequate to obtain a reasonably good output to noise ratio. A low-pass filterwith a cut-off frequency of 1 kHz was also used. This amplified filtered signal wastransmitted to an analog-to-digital converter. Based on the Nyquist criterion, a samplingrate of 2.5 kHz was selected.Capacitance type ‘bow string’ probes were used to measure the water surface elevation.Each probe is made up of a taut loop of wire on a light metal ‘C’ frame and has a linearitybetter than 98.5% and a resolution better than 1 mm.3.6 Experimental ProceduresThe experiments were carried out in two parts. In the first part, waves were generated withthe plate absent, and water surface elevation measurements were carried out using probesplaced at two locations (x = 0 cm) and (x = 60 cm) along the centre-line of the plate. Dueto a limitation of the RTC_DAS package, a sampling rate of 2.5 kHz was used for bothforce as well as water surface elevation data measurements. Previous tests have indicatedthat the repeatability of a particular wave train over separate experiments is very good(Isaacson & Prasad, 1993), so that the surface elevation with the plate absent and the forceson the plate were measured in separate stages. In the second stage, the wave probes wereremoved and the plate assembly was installed in the flume. Waves corresponding to thesame stored wave signals were repeated and the vertical forces on the plate were recorded ata sampling rate of 2.5 kHz. A video record was also obtained for each experiment.Experiments were carried out for five different elevations of plate from 0 to 25 mm, in25combination with three wave steepness values and four wave periods. A constant waterdepth of 0.55 m was maintained during the experiment. Table 3.1 lists the differentincident wave conditions used in the investigation.During the second part of this investigation, video records were obtained using a normalspeed camcorder. When the force data sampling was initiated, a switch was also triggeredto light up the light emitting diode at the same time instant so as to synchronize the timebase of the film with that of vertical force and the free surface elevation records.The video records were used to study the impact process qualitatively and thereby toaccess the effects of the assumptions made, and in particular to assess the importance ofwave profile deformation, air entrapment, wave overtopping effects, and water curtainingproblems due to drainage between each event of wave impact.3.7 Dynamic Characteristics of the AssemblyAfter calibrating the load-cells for static loads, free vibration tests of the plate wereconducted both in air as well as for a fully submerged condition. In both cases, a step loadwas applied to the plate assembly by the sudden release of a load carried by a thin steelsingle stranded wire. This was achieved by cutting-off the wire using an acetylene torch.The free vibration traces were recorded for both the load-cells A and B indicated inFig. 3.2. The recorded force time histories and the corresponding spectral densities for theplate in air and fully submerged condition are shown in Fig. 3.5 and 3.6 respectively.Figure 3.5 indicates that for vibrations in air the plate response has widely distributedfrequencies with a predominant frequency of approximately 125 Hz for both load-cells. Itis informative to study the initial stages of an impact event just after releasing the step load.Consider the free vibration trace for the load-cell A as shown in Fig. 3.5(a). The first peakis observed to occur 5 msec after the step load release, indicating the transient response26ability of the assembly. The high frequency component then disappears and lowervibration amplitudes with a frequency of approximately 62.5 Hz become dominant, andultimately the load-cell stops vibrating after 0.6 sec since the release of the step load. Thedamping ratio can be evaluated from the above trace using the following relation:= D+42n2 (3.1)where Df = ifl[Pm/Pm+nl, and p and Pm+n are the amplitudes of the free vibration forcetrace measured after n and (m+n) cycles respectively. The average damping ratio wasfound to be 2.7% in air. Similar features were found from the analysis of the free vibrationtrace for load-cell B.Apart from free vibration tests in air, the plate and load-cell assembly were also testedfor free vibrations under a fully submerged situation so as to obtain similar characteristicsfor step load releases (Fig. 3.6(a)). For both load-cells A and B, the traces are more or lesssmooth and indicate a single dominant oscillation frequency. The first peak is observed at10 msec and the system comes to rest 1.0 sec after the release of the load. The averagedamping ratio for this submerged condition was found to be 5.3%. Figure 3.6(b) showsthe variance spectral density of the free vibration trace. It indicates that the naturalfrequency for the submerged condition reduces to approximately 25 Hz.As explained in Section 2.3, Fig. 2.8 may be used to estimate the influence of thesystem characteristics on the measured force and associated rise-time. From thepreliminary experimental results, the minimum rise-time was found to be not less than 100msec. And with the natural period of the system taken to be equal to 1/125 sec, thereappears to be no noticeable amplification in the peak value of the measured force and therise-time. The natural period of the system is taken to be the one that corresponds to the27tests conducted in air since the preliminary tests indicated that the impact occurs just beforecomplete submergence of the plate.3.8 Data ProcessingThe first step in data analysis involves filtering the measured data to remove noise in therecords. One of the causes of noise may be due to electromagnetic frequency at 60 Hz.Another would be due to the system response itself, as discussed in sections 3.2 and 3.7.On the amplifier, a low-pass filter was used with a cut-off frequency of 1000 Hz. Theforce data from two the load-cells were first plotted to observe the noise present in therecords. The GEDAP filtering program FILTW was used to filter the data. Several levelsof cut-off frequency ranging from 500 to as low as 5 Hz were used to examine the effect onthe peak force. Cut-off frequencies below 10 Hz show considerable smoothening of theforce peak and loss of information, and it was decided to set the cut-off frequency at 15Hz. As an illustration of the low-pass filter that was adopted, Fig. 3.7 showscorresponding unfiltered and filtered force records.A static analysis was applied to obtain the vertical force and its line of action in themanner indicated in Appendix A. Figure Al is a sketch of free body diagram of the platerepresented as pinned rigid beam. From this analysis, the total vertical force and its line ofaction were obtained and are illustrated in Fig. 3.8 for the case of unfiltered signals andfiltered signals.In a similar way, the wave records were filtered and resampled at 10 Hz since such ahigh sampling rate is unnecessary for a slowly varying signal. The GEDAP routines,FILTW and RESAMPLE2 were used to filter and resample the wave records. Once thevertical force from the filtered records are available along with the resampled wave profile,the remaining analysis was carried out as explained below. It was decided to study onlyone slamming event for each test and therefore only one slamming cycle is selected from a284 sec record as shown in Fig. 3.9. The instant of slamming to is determined from themeasured water surface elevation o and the known plate clearance. This method wasfound to be consistent and reliable for all the experimental results. The wave kinematicswere evaluated half way along the instantaneous wetted length measured from the leadingedge of the plate, as mentioned in the Section 2.2.1. The buoyancy force was evaluatedbased on the displaced volume of the water by the plate and load—cell assembly as discussedin Section 2.2, and is assumed to vary sinusoidally for the duration of submergence withthe peak buoyancy force occurring at t = t. Various other parameters such as peakupward and downward forces, their lines of action, times of occurrence and associatedwetted lengths were then determined. The vertical force was predicted based on the linearwave theory for different wave conditions tested and a comparison of the measured andobserved force was carried out. A flow chart of the procedure is given in Fig. 3.10.29Chapter 4Results and DiscussionThe results of the experimental and analytical investigation are presented and discussedin this chapter. The first part primarily describes the results from the tests with regularnon-breaking waves of different heights and periods, and different plate elevations. Theimportant parameters thereby estimated correspond to the maximum upward force, theassociated point of application, the rise-time, and the corresponding value of the wettedlength. Similar quantities relating to maximum downward force are also presented. A briefqualitative analysis of the video records included is also presented, with a focus on thephysical process involved and the related departure of the theory in predicting the verticalforce is discussed.The application of the two analytical models (Eqs. 2.23 and 2.27) for estimating thevertical force is discussed, and associated discrepancies are identified. Finally, a correctionfactor for the added mass associated with each of these model is introduced, and its rangefor different incident wave conditions is presented.A total of 69 tests were carried out, corresponding to 5 plate elevations, 5 wave periods,and 3 wave steepness. (Some of the 75 combinations of these three do not give rise toimmersion of the plate.) The plate elevations ranged from h = 0 to h = 2.5 cm; waveperiods ranged from 0.8 to 2.0 sec; and the wave heights for each wave period wereselected to correspond to steepness Hit 0.02, 0.03, and 0.04, such that the wave heightsare in the range of 3 to 17 cm.304.1 Vertical ForceFigures 4.1 to 4.8 show the time histories of the vertical forces measured at two supportlocations FA and FB, the total vertical force F, and the corresponding water surfaceelevations ‘rio and r, at the leading and rear edges of the plate. The figures correspond toeight different test conditions characterized by changes in wave steepness and plateclearance. In these figures, t is the time measured from an arbitrary origin. As an aid tointerpreting results, these figures include horizontal and vertical lines indicating the plateelevation and the corresponding instants of impact and complete submergence. On thisbasis, the variation of wetted length 2(t) is also obtained.Figures 4.1, 4.2 and 4.3 correspond to a plate clearance h = 0.8 cm and waves withT 1.70 sec and H = 14.2, 10.5 and 6.8 cm respectively. As a wave advances past theplate, the force increases quite gradually from the instant of water contact, exhibits a fairlysharp maximum, and then varies more gradually over the remainder of the cycle, passingthrough a noticeable minimum during the later stages of the event. The noticeablemaximum downward (i.e. negative) force which is present during the later portion of thewave cycle is due to a suction associated with the water surface receding below the plate,together with the weight of some overtopped water remaining above the plate.Similarly, Figs. 4.4 and 4.5 correspond to a plate clearance h = 1.4 cm and waves withT 1.70 sec and H = 14.2 cm and H = 10.5 cm respectively. These figures indicate asharper rise in force than for the case h = 0.8 cm. Finally, Figs. 4.6, 4.7 and 4.8correspond to different clearances, h = 0, 1.4 and 2.5 cm respectively, with the same wavecondition T = 2.02 sec and H = 17.5 cm.It is useful to study the slamming process with respect to the free surface elevation.As mentioned already, horizontal lines in Figs. 4.1(a)- 4.8(a) indicate the plate elevation,and the vertical lines indicate the instants of initial wave contact, complete submergence, the31onset of wave recession and complete wave recession. The time between the two lowervertical lines indicate the duration over which a wave is in contact with the plate. Extendingthese line to the force time histories F(t), the observed instant of impact agrees reasonablywell with the instant at which the wetted length starts rising from zero. It may be noticedthat the total force F before and after these points is non-zero even though the plate isentirely in air. This may be attributed to the overtopped water draining from the top of theplate. It can be observed that the water drains out completely for the wave of period T =2.02 sec, so that then the force F is nearly zero as indicated in the Fig. 4.6(c). The timebetween the upper two vertical lines indicate the duration for which the plate is completelysubmerged. The intersection of two wave profiles ‘rio and ie shows the symmetric waveprofile with respect to support A (mid span location) along the plate length for that instant.The total force F is positive indicating that the plate experiences an upward force whichexceeds the weight of the water above the plate.Another interesting feature included in Figs. 4.1 - 4.8 relates to the time histories of theline of action, s(t). The sharp changes in s are associated with F(t) passing through zero.The force’s line of action initially moves away from the leading edge of the plate asexpected, but does not span the whole wave cycle. It moves abruptly from the leadingedge to the rear edge just after the occurrence of the symmetric wave profile along thelength. And during the wave recession era, s(t) again commences at the leading edge andtravels smoothly up to the mid span approximately.The noticeable negative (i.e. downward) force, which is present during the later portionof the wave cycle, is due to a suction associated with the water surface receding below theplate, together with the weight of some water remaining above the plate. The negativeforce seems to start just before the wave surface leaves the plate’s leading edge i.e. just32before time t2. As the wave recession progresses, the force F reaches a negative maximumjust before time t3. After time t3, the plate is in air and the force F drops gradually to zero.4.2 Vertical Force and Incident WavesResults relating to the force parameters have been grouped according to the dimensionalanalysis indicated earlier and are listed in Table 4.1. These include the maximum upwardforce Fp, its time of occurrence T, its point of application s, and the correspondingwetted length X. The table also includes the maximum downward force Fm, its time ofoccurrence T, and point of application 5m as well as the wetted length 2. The variationsof these values are studied as functions of relative clearance h/H, for various values ofwave steepness HIL and relative plate length £fL.Maximum ForceFigure 4.9 indicates the variation of the peak upward force coefficient F/pgHbe as afunction of relative clearance h/H for various values of the relative plate length £/L and forvarious ranges of wave steepness. As the relative clearance h/H approaches 0.5 the forcecoefficient F/pgHb/ approaches zero. Also, the force coefficient increases as the relativeplate length £/L increases. Although, there is no significant change in the force coefficientfor the range of wave steepness used in the tests (HJL 0.0 16 0.041), it may be seen thatthe trend of the which shows F/pgHb.e to decrease linearly with h/H is no longer observedfor the case of steeper waves.Figure 4.10 indicates similar plots for the peak downward force coefficient Fm/pgHbe.Figures 4.10(a), 4.10(b) and 4.10(c) indicate no systematic relationship, althoughFm/pgHb apparently remains constant for any change in h/H.33Duration of SubmergenceFigure 4.11 is a plot of a dimensionless duration of partial submergence, TIT whereT =t1-t0. This parameter is important since it indicates the time required for a wave tocompletely submerge the plate at a known elevation above the mean water level, andconsequently influences the occurrence of the peak upward force predicted by the analyticalmodels presented. The figure shows a fairly constant value of T/T for increasing relativeplate clearance h/H. As the steepness decreases, (Figs. 4.11(a) - 4.11(c)), T)T decreasesfor higher values of h/H.Similarly, Fig. 4.12 is plot of a dimensionless duration relating to completesubmergence T/T, where T = t2 - ti. This parameter indicates the duration for which theplate remains submerged and influences the span of gradual variation of the added massforce Fa2. As expected, an increasing relative clearance h/H leads to a decreasing value ofT/T.Times of OccurrenceFigure 4.13 shows the relative rise-time T/T associated with the peak upward force, andFig. 4.14 shows corresponding results relating to Tm/T. Figure 4.13 exhibits a fair degreeof scatter so that there are no particularly noticeable trend in there results. However, thereis some tendency for T/T to decrease with increasing h/H. On the other hand Fig. 4.14shows a fairly clear trend for Tm/T to decrease with increasing h/H and with decreasingilL. The scatter in the rise-times may be attributed to factors such as air-entrapment, sprayand splash effects at the instant of impact, plate surface roughness and structural vibration.Line of ActionFigure 4.15 shows the non-dimensionalised line of action of the peak upward force sILplotted against the relative clearance h/H for various values of relative plate length and34wave steepness. There is considerable scatter in the results once more such that the relativeplate length ilL and wave steepness H/L appear to have no significant influence on sfL,except that s/L increases with an increasing h/H.Figure 4.16 indicates corresponding results for Sm/L. In this case, it is interesting tonote that the for all ranges of steepness tested, there seems to be a linear trend betweenSmIL and h/H.Wetted LengthIn Fig. 4.17, the dimensionless wetted length 2/L associated with the peak upward forceis plotted as a function of relative clearance hJH, with HIL and ilL as parameters. It isobserved that for lower value of the ilL, the seems to reduce as h/H increases.Corresponding results for m/L are shown in Fig. 4.18 and indicate no particularcorrelation.From the above, it can be seen that in general as h/H increases all impact parametersdecrease. The effect of steepness is not noticeable, possibly because of the small rangeconsidered in the tests.4.3 Video RecordsIn order to complement the results presented in the previous section, a qualitative study ofthe impact process of the flow past the plate has also been made on the basis of the videorecords.Figure 4.19 shows photographs indicating the wave flow past the plate associated with adistortion in the shape of the wave profile. Figure 4.19(a) was taken at the instant ofcomplete submergence and shows some air entrapment at the rear end as the plate35completely submerges, and Figure 4.19(b) was taken just after the wave passes the plateand shows the overtopped water draining from the plate.The video records have been examined in order to assess further the interaction process,and Fig. 20 shows several frames corresponding to various stages of slamming process.In Fig. 4.20(a), the wave profile deforms considerably during the initial penetration of theplate below the water surface. The wave progresses over the plate without too muchfurther disturbance to its profile until the plate is completely submerged [see Fig. 4.20(b)].Figures 4.20(c) and 4.20(d) show successive views during the recession stage, duringwhich water drains from the plate including some disturbance to the wave profile. Airentrainment between the plate and the water surface during this stage is noticeable.The irregularities highlighted above in conjunction with uncontrollable parameters suchas the plate roughness, plate orientation, and structural vibrations, influence the forcemeasured and contribute to the scatter observed in the force results.4.4 Force PredictionsIn the present section, a comparison of the predictions of the two analytical models of thevertical force given by Eq. 2.23 and Eq. 2.27 with the experimental measurements ispresented and discussed.The steps involved in the computational aspects of the analytical models are as follows.Eq. 2.23 is evaluated form the instant of initial contact until the instant of completesubmergence. This is followed by a computation of the force during completesubmergence. Finally, Eq. 2.23 is evaluated for the period of wave recession. Parametersrelating to this formulation, which are computed at each time step include the wetted length2 and the wave kinematics i and ij. 2. is obtained by evaluating Eq. 2.20, and r and j aredetermined at a point half way along the instantaneous wetted length on the basis of36Eqs. 2.18 and 2.19. A similar procedure is adopted for the simpler analytical model givenby Eq. 2.27.Figures 4.21 and 4.22 show the variation of the dimensionless vertical force F/pgbHeover one wave cycle for HJL = 0.04; ilL = 0.146 and for 2 two plate clearances, h/H = 0and 0.15, and compare the predictions of the analytical models given by Eqs. 2.23 and2.27. In evaluating these equations a range of suitable a values ranging from 1 to 3 forEq. 2.23 and 0.5 to 0.7 for Eq. 2.27 were assumed. From these figures it can be observedthat a noticeable difference of the force predicted by two models occur in the partiallysubmerged stage, with the model based on Eq. 2.23 indicating a faster rise than Eq. 2.27.This is due to the different assumptions made in added mass variations with respect tosubmergence [see also Fig. 2.51.It is also of interest to consider the suitability of the analytical model in predicting thevertical force variation, and a comparison of the force predicted on the basis of Eqs. 2.23and 2.27 with a measured force record is made in following paragraphs. The added massconstant a is selected so as to match the measured and predicted maximum forces.Figure 4.23 provides a comparison of the measured and predicted force time historiesforce the case h = 0 cm, T = 2.02 sec and H = 17.5 cm. The predicted force based onEq. 2.23 rises to a peak much earlier and drops to negative values faster then the observedforce. On the other hand, the force predicted on the basis of Eq. 2.27 reaches a maximumduring the partial submergence with a rise-time almost equal to the measured force and itseems to follow the measured force reasonably well until the maximum force is reached.Figure 4.24 compares the predicted and measured force for incident waves withT = 1.68 sec, and H = 14.2 cm with the plate located at h = 0 cm. In this case, the risetime of the predicted force based on Eq. 2.27 seems to be much smaller and significantly37deviates from the observed force. However, prediction based on Eq. 2.23 seems to deviatemore than that by Eq. 2.27.Figure 4.25 is similar plot for the incident wave with T = 2.02 sec, H = 17.5 cm withh = 1.4 cm. For the case of Eq. 2.27, although the predicted peak occurs slightly laterthan that for the measured force, it follows the observed force reasonably well. Theprediction based on Eq. 2.23 does not show any improvement compared to that based onEq. 2.27.Form all the three figures, it is observed that the predicted forces deviate significantlyfrom the observed force during the stage of complete submergence followed by waverecession.As mentioned earlier, the factor o of Eqs. 2.23 and 2.27 has been evaluated bymatching the measured and predicted maximum upward force. The results for a selectednumber of tests are listed in Table 4.2. These correspond to 9 tests with h = 0, two testswith h = 1.4 cm and one test with h = 2.5 cm. For all the 12 selected tests, the factor a isin the range of 1.2 to 3.1 on the basis of Eq. 2.23, and is in the range of 0.5 to 0.7 on thebasis of Eq. 2.27. Thus, a based on Eq. 2.27 is approximately equal to irI4, whichcorresponds to the theoretical value of a plate in an infinite flow situation, so that there is norequirement of any reduction factor. On the other hand, the values of a based on Eq. 2.23seem to exhibit considerable scatter.38Chapter 5Conclusions and RecommendationsThe primary objective of this study was to examine experimentally the vertical force due toregular non-breaking waves interacting with a fixed horizontal plate located near the stillwater level. Force time histories were analyzed to obtain peak upward and downwardforces, their times of occurrence, their lines of action and the associated wetted length ofthe plate. Also, the free surface elevation at the leading and rear edges of the plate wererecorded for different wave conditions with the plate absent. Two analytical models basedon a varying added mass of the submerged portion of the plate together with a drag andbuoyancy forces were used to predict the force on the plate, and a comparison of thesepredictions with the experimental results was made.The dependence of the various characteristics of the maximum and minimum forces onthe relative plate clearance h/H, wave steepness H/L and relative plate length £/L have beenexamined and are indicated. Even though there is a considerable degree of scatter in theresults, some general conclusions may be made. The maximum upward force decreaseswith increasing relative clearance of the plate h/H. And the peak upward force is higher forlarger value of relative plate length Lit. In the present study the wave steepness HIL was inthe range of 0.016 to 0.041 and is observed not to influence the peak upward forcenoticeably. The maximum downward force seems to remain constant for increasingrelative clearance h/H. However, it shows similar variation as that of the peak upwardforce for changes in relative plate length Lit and in wave steepness Hit. The rise-timeassociated with the peak upward force shows considerable scatter. However, it isobserved that three is some tendency for Tjf to decrease with increasing h/H. On the39other hand, the dimensionless time of occurrence TmIT showed a decreasing trend forincreasing values of plate clearance. Also, for higher relative lengths, £/L, a higher Tm/Twas observed. For relative line of action, associated with both the F and Fm indicatedincreasing trend for steeper waves and remained constant for less steep waves.Summarily, the analysis indicated that the peak forces F and F and associated times ofoccurrence decrease some what linearly with the increasing plate clearance.The maximum upward force and associated rise-time can be predicted reasonably wellon the basis of the simpler hydrodynamic model given by Eq. 2.27, with the factor cxranging from 0.5 to 0.7. However, the predicted force deviates significantly from themeasured force after the occurrence of the peak upward force. This indicates the need for aclearer understanding of the wave interaction with the plate during complete submergenceand recession stages.Several avenues of further study may be suggested. Improvements may be made toanalytical I numerical models to predict the vertical force; and experiments relating toirregular and breaking waves also need to be carried out.40ReferencesArmand J. L., and R. Cointe, (1987), “Hydrodynamic impact analysis of a cylinder,”J. Offshore Mechanics and Arctic Engineering, ASME, Vol. 109, pp. 237 - 243.Broughton, P., and E. Horn, (1987), “Ekofisk platform 214C: re-analysis due tosubsidence,” Proc. Institution of Civil Engineers, Part 1, Vol. 82, pp. 949 - 979.Chan, E. S., and W. K. Melville, (1989), “Plunging wave forces on surface-piercingstructures,” J. Offshore Mechanics and Arctic Engineering, ASME, Vol. 111, pp. 93-130.Chan, E. S., H. F. Cheong, and K. Y. H. Gin, (1991), “Wave impact loads on horizontalstructures in the splash zone,” Proc. 1st International Offshore and Polar EngineeringConference, Vol. 3, pp. 203 - 209Chuang, S. L., (1967), “Experiments on slamming of wedge-shaped bodies with variabledead rise angle,” J. Ship Research, SNAME, Vol. 11, No. 4, pp. 190-198.De Costa, S. L., and J. L. Scott, (1988), “Wave impact forces on the Jones Island eastdock, Milwaukee, Wisconsin,” Oceans, IEEE, pp. 1231-1238.Denson, K. H., and M. S. Priest, (1971), “Wave pressure on the underside of a horizontalplatform,” Proc. Offshore Technology Conference, Houston, Texas, Paper No.OTC 1385, Vol. 1, pp. 555 - 570.El Ghamry, 0. A., (1963), “Wave forces on a dock,” Report no. HEL-9-1, Inst. of Engg.Research, Hydraulic Engg. Lab., University of California, Berkeley, California.Faltinsen 0., (1990), “Sea loads on ships and offshore structures,” Cambridge OceanTechnology Series, Cambridge University Press, Cambridge, UK.French, J. A., (1969), “Wave uplift pressures on horizontal platforms,” Report no. KH-R19, Division of Engineering and Applied Science, California Institute of Technology,Pasadena, California.Furudoi, T., and A. Murota, (1966), “Wave induced uplift forces acting on quay-aprons,”Technology Reports of Osaka University, Vol. 16, No. 734, pp. 605 - 616.Greenhow, M., and Y. Li, (1987), “Added mass for circular cylinders near or penetratingfluid boundaries- review extension, and application to water-entry, -exit, and slamming,”Ocean Engineering, Vol. 14, No. 4, pp. 325 - 348.Gudmestad, 0. T., and J. J. Connor, (1986), “Engineering approximation to nonlineardeep water waves,” Applied Ocean Research, Vol. 8, No. 2, pp. 76 - 88.Humar, J. L., (1990), “Dynamics of structures,” Prentice Hall, Englewood, New Jersey.Irajpanah, K., (1983), “Wave uplift force on horizontal platform,” Ph. D. Thesis, Univ. ofSouthern California, Los Angeles, California.41Isaacson, M., and S. Prasad, (1992), “Wave slamming on a horizontal circular cylinder,”Proc. Civil Engineering in the Oceans V, ASCE, College Station, Texas, pp. 652 - 666.Isaacson, M., and S. Prasad, (1993), “Wave slamming on a horizontal circular cylinder,”Proc. 3rd International Offshore and Polar Engineering Conference, Singapore, Vol. 3,pp. 274-281.Kaplan, P., and N. Silbert, (1976), “Impact on platform horizontal members in the splashzone,” Proc. Offshore Technology Conference, Houston, Texas, Paper No. OTC 2498,Vol. 1, pp. 749 - 758.Kaplan, P., (1992), “Wave impact forces on offshore structures: re-examination and newinterpretations,” Proc. Offshore Technology Conference, Houston, Texas, PaperNo. OTC 6814, Vol. 1, pp. 79-88.Lai, C. P., and J. J. Lee, (1989), “Interaction of finite amplitude waves with platforms ordocks,” J. Waterways, Port, Coastal and Ocean Engineering, ASCE, Vol. 115, No. 1,pp. 19-39.Lamb, H., (1932), “Hydrodynamics,” 6th edition, Dover Publications, Inc., New York,N.Y.Sarpkaya, T., (1978), “Wave impact loads on cylinders,” Proc. Offshore TechnologyConference, Houston, Texas, Paper No. OTC 3065, pp. 169-176.Sarpkaya, T., and M. Isaacson, (1981), “Mechanics of wave forces on offshorestructures,” Van Nostrand Reinhold, New York, N.Y.Szebehely, V. G., and M. K. Ochi, (1966), “Hydrodynamic impact and water entry,”Applied Mechanics Survey, ed. H. N. Abramson, et al., Spartan Books, MacMillan &Co. Ltd., pp. 951-957.Tanimoto, K., and S. Takahashi, (1979), “Wave forces on a horizontal platform,” Proc.5th International Ocean Development Conference, Tokyo, Japan, Vol. Dl, pp. 29 - 38.Toumazis, A. D., W. K. Shih, and K. A. Anastasiou, (1989), “Wave impact loading onhorizontal and vertical plates,” Proc. 23rd Congress, International Association ofHydraulicResearch, Ottawa, Canada, Vol. C, pp. 209 - 216.Verhagen, J. H. G., (1967), “The impact of flat plate on a water surface,” J. ShipResearch, SNAME, Vol. 11, No. 4, pp. 211-223.von Kármán, T., (1929), “The impact on seaplane floats during landing,” NationalAdvisory Committee forAeronautics, Technical Note No. 321.Wagner, H., (1932), “Landing of Seaplanes,” National Advisory Committee forAeronautics, Technical Note No. 622.Wang, H., (1967), “Estimating wave pressures on a horizontal pier,” Naval CivilEngineering Laboratory, Port Heneme, California, Technical Report No. R-546.42Appendix AStatic AnalysisThe load-cells placed at the two supports measure the support reactions due to the waveaction on the plate. The plate can be assumed to be analogous to a simply supported beamwith an over-hang from its supports and can be analyzed by the simple principles of staticsin order to provide the unknown vertical force along with its line of action at any instantfrom the recorded reactions at the two supports. The plate is assumed to be thin enough forany horizontal force to be neglected, and no measurement of the horizontal force has beencarried out.Referring to the Fig. Al, let FA(t) and FB(t) be the support reactions recorded at anytime t, F(t) the unknown force acting on the plate and s(t) its line of action measured fromthe leading edge of the plate. F(t) can be obtained by summing the support reactions FA(t)and FB(t).F(t) = FA(t) + FB(t) (A-i)The line of action s(t) can be obtained by taking moments of the forces acting on theplate about the plate’s leading edge. This gives— 0.30 FA(t) + 0.545 FB(t) A 2so— F(t) —where s(t) in m.43Table 2.1 Added mass constant for a thin rectangular plate.(Sarpkaya and Isaacson, 1981).b/2. 131.00 0.5791.25 0.6421.59 0.7042.00 0.7572.50 0.8014.00 0.8725.00 0.8978.00 0.93410.00 0.94700 1.000Flow past a thinrectangular plateb -m =44Table 3.1 Wave parameters used in the experiments.Set Wave Wave Wave Waveno. period height steepness celerityT(s) h(m) H/L c(mls)1.1 1.08 0.062 0.035 1.6401.2 1.38 0.101 0.039 1.8771.3 1.68 0.142 0.042 2.0121.4 2.02 0.175 0.041 2.1132.1 1.12 0.048 0.026 1.6482.2 1.38 0.075 0.029 1.8742.3 1.70 0.105 0.030 2.0592.4 2.02 0.128 0.030 2.1123.1 1.12 0.030 0.016 1.6743.2 1.40 0.050 0.019 1.8803.3 1.70 0.068 0.020 2.0003.4 1.96 0.081 0.020 2.066Table4.1Summaryoftestconditionsandprincipalresults.h=0cmRun#THHh£pgHbtFFm‘r‘r-j-Sflj(s)(m)LHL(N)pgHbpgHbTTTTLLLL21.080.060.040.000.3472.80.32-0.140.330.170.250.700.170.160.270.1231.380.100.040.000.23119.00.24-0.210.230.270.120.640.100.140.090.2041.680.140.040.000.18167.70.25-0.160.170.340.210.520.100.090.180.1452.020.180.040.000.14205.70.27-0.150.140.370.140.510.070.080.140.0071.120.050.030.000.3256.50.36-0.130.320.180.240.710.170.180.250.1081.380.080.030.000.2388.50.26-0.230.240.270.130.640.090.140.130.0991.700.110.030.000.17123.10.27-0.190.170.330.220.550.090.090.170.12102.020.130.030.000.14150.90.32-0.170.140.360.140.540.070.080.140.10121.120.030.020.000.3235.10.41-0.140.320.180.300.700.170.140.310.12131.400.050.020.000.2359.00.26-0.230.230.280.310.650.110.130.230.08141.700.070.020.000.1780.30.31-0.210.170.330.250.570.090.090.180.08151.960.080.020.000.1594.80.34-0.190.150.350.160,570.070.080.150.08h=0.8cmRun#THHh£pgHbl?FFm‘rcrn(s)(m)LHL(N)pgHbpgHbeTTTTLLLL171.080.060.040.130.3472.80.24-0.140.3440.340.080.660.200.180.260.06181.380.100.040.080.23119.00.19-0.220.2320.230.220.600.120.140.150.18191.680.140.040.060.18167.70.23-0.180.1710.170.290.540.090.110.180.09202.020.180.040.050.14205.70.24-0.170.1440.140.330.480.070.080.140.00221.120.050.030.170.3256.50.24-0.140.3210.320.080.590.170.160.180.10231.380.080.030.110.2388.50.19-0.200.2250.220.200.600.110.140.190.05241.700.110.030.080.17123.10.24-0.200.1760.180.290.540.090.100.170.08252.020.130.030.060.14150.90.29-0.190.1390.140.320.500.070.080.120.08271.120.030.020.270,3235.10.24-0.180.330.33-0.570.170.160.190.08281.400.050.020.160.2359.00.21-0,160.2210.220.170.590.090.130.120.04291.700.070.020.120.1780.30.25-0.190.1650.160.260.520.090.090.180.05301.960.080.020.100.1594.80.30-0.230.1430.140.300.530.080.090.150.04U’Table4.1(contd.)Summaryof testconditionsandprincipalresults.h=1.4cmRun#THHh£pgHbFFm‘r(s)(m)LHL(N)pgHbpgHbTTTTLLL321.080.060.040.230.3472.80.20-0.110.330.020.160.600.170.170.180.06331.380.100.040.140.23119.00.19-0.180.220.180.190.570.120.150.140.16341.680.140.040.100.18167.70.25-0.170.180.260.150.520.090.110.160.11352.020.180.040.080.14205.70.24-0.170.140.310.110.460.070.080.140.00371.120.050.030.290.3256.50.19-0.120.29-0.150.540.180.170.150.09381.380.080.030.190.2388.50.15-0.160.220.090.300.510.140.110.210.10391.700,110.030.130.17123.10,24-0.180.180.240.150.490.080.090.150.09402.020.130.030.110.14150.90.24-0.190.140.300.120.490.070.080.120.08421.400.050.020.280.2359.00.13-0.130.230.090.280.520.130.140.230.00431.700.070.020.210,1780.30.21-0.150.170.190.120.480.080.090.130.07441.960.080.020.170.1594.80.23-0.210.150.240.130.500.080.090.140.04h=1.8cmRun#THHhpgHbiFFm‘riuJ2.iU(s)(m)LHL(N)pgHbpgHbtTTTTLLLL461.080.060.040.290.3472.80.18-0.090.31-0.150.580.190.190.080.03471.380.100.040.180.23119.00.15-0.170.230.150.210.550.120.150.150.16492.020.180.040.100.14205.70.21-0.170.130.300.110.460.070.080.140.00511.120.050.030.380.3256.50.10-0.120.23-0.120.430.190.170.110.11521.380.080.030.240.2388.50.14-0.140.250.090,230.500.130.130.220.08531.700.110.030.170.17123,10.24-0.160.160.230.150.480.090.090.150.07542.020.130.030.140.14150.90.22-0.200.120.270.070.470.050.080.080.08561.400.050.020.360.2359.00.09-0.130.240.010.260.450.180.120.21-571.700.070.020.260.1780.30.21-0.130.180.160.120.430.090.090.140.06581.960.080.020.220.1594.80.20-0.190.150.210.050.440.050.080.050.05Table4.1(contd.)Summaryoftest conditionsandprincipal results.h=2.5cmRun#THHhpgHbFFmIrn(s)(m)LHL(N)pgHbpgHbeTTTTLLLL601.080.060.040.400.3472.80.07-0.080.20-0.020.39-0.060.170.00611.380.100.040.250.23119.00.18-0.140.220.100.220.470.160.150.140.19621.680.140.040.180.18167.70.28-0.150.170.230.150.460.160.100.100.10632.020.180.040.140.14205.70.20-0.170.130.280.050.450.110.040,090.00661.380.080.030.330.2388.50.15-0.120.230.040.250.460.230.170.130.06671.700.110.030.240.17123.10.21-0.130.180.160.140.440.130.090.090.08682.020.130.030.200.14150.90.20-0.190.130.250.060.440.050.040.080.07691.960.080.020.310,1594.80.16-0.140.150.150.050.410.020.050.080.0548Table 4.2 Computed values of the factor a in selected tests.R Wave Wave F a aNo. (? (Eq. 2.23) (Eq. 2.27)h = 0.0 cm3 1.38 0.101 28.20 1.225 0.558 1.38 0.075 22.92 1.750 0.6113 1.40 0.050 15.22 1.600 0.634 1.68 0.142 42.09 2.050 0.499 1.70 0.105 32.99 2.200 0.5314 1.70 0.068 24.89 2.500 0.615 2.02 0.175 54.92 2.500 0.4810 2.02 0.128 47.83 3.050 0.6115 1.96 0.081 32.61 3.100 0.69h= 1.4cm35 2.02 0.175 48.54 2.400 0.5334 1.68 0.142 41.59 2.350 0.68h=2.5cm63 2.02 0.175 41.55 2.200 0.57•rI C C) CD CD C)50wave profiledFig. 2.1 Definition sketch.51wavethrecIx0Th(a): t=t00z0Th(C):t1”:. ,. .I I I I I I I I I I I I I I I I I I I I I I I I I I(1D)(c)-0.1 0.0 0.1 0.2 0.3 0.4 0.5LLL0.40.30.20.10.00.40.30.20.10.00.40.30.20.10.0h/HFig. 4.17 Non-dimensionalised wetted length fL corresponding to maximum upward forceas a function of relative plate clearance h/Fl and relative plate length ilL. (a) 0.016