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Wave slamming on a horizontal plate Bhat, Shankar Subraya 1994-02-26

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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 partialfulfillment of therequirements for an advanced degree atthe University of BritishColumbia, I agree that the Library shallmake it freely availablefor reference and study. I furtheragree that permission forextensive copying of this thesisfor scholarly purposes maybegranted by the head of my department orby his or herrepresentatives. It is understood thatcopying or publication ofthis thesis for financialgain shall not be allowed withoutmywritten 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 <t <t1; (c) completesubmergence of plate, t1 <t <t2; (d) submergence of downwave portion ofplate, t2 <t < t3; (e) wave detaching from plate, t = t3.Fig. 2.3 Sketch of ideal force components variation over a wave cycle. (a) free surfaceelevation; (b) wetted length;(c) proposed variation of iAJat as a velocity c’;(d) force components: inertia force,Fal;added mass force,Fa2;drag force,Fd; buoyancy force, Fb; (d) total vertical force: actual force, predicted force.Fig. 2.4 Variation of added mass functions with plate aspect ratio 2fb. (a)f1(Jb);(b)f2(Ib).Fig. 2.5 Variation of dimensionless added mass with plate aspect ratio ?Jb. (a) Eq. 2.7;(b) Eq. 2.26.Fig. 2.6 Definition sketch of a single degree of freedom (SDOF) system.Fig. 2.7 Sketch of an idealized force as a triangular pulse.Fig. 2.8 Dynamic amplification factor and relative rise-time as a function ofTr/Tnforapplied impulsive force withTd/Tr = 1. (Isaacson and Prasad, 1993).Fig. 3.1 Photographs of the plate assembly. (a) side view; (b) top view indicating thedetails of supports and load-cell arrangements.Fig. 3.2 Experimental setup showing the details of the plate and the load-cellarrangement. (a) elevation; (b) cross-section.Fig. 3.3 Sketch showing the wave flume and the test location.Fig. 3.4 Flow chart indicating the wave generation and the data control setup.Fig. 3.5 Response of load-cells A and B to a step load of 117.7 N (12 kg) tested in air.(a) time history; (b) spectral density.Fig. 3.6 Response of load-cells A and B to a step load of 58.9 N (6 kg) tested for asubmerged condition. (a) time history; (b) spectral density.Fig. 3.7 Time histories of the measured vertical force showing the effect of filtering(h = 1.4 cm, T = 2.02 see, H = 17.5 cm). (a) unfiltered force signal;(b) filtered force signal at 15 Hz cut-off.ixFig. 3.8 Time histories of total vertical force and its line of action showing the effect offiltering (h = 1.4 cm, T = 2.02 sec, H = 17.5 cm). (a) unfiltered force(b) filtered force with 15 Hz cut-off.Fig. 3.9 Time histories of (a) free surface elevation; (b) vertical force and its line ofaction. (h = 1.4 cm, T = 2.02 sec, H = 17.5 cm).Fig. 3.10 Flow chart indicating the sequence of analysis of the measured force and waveelevation time histories.Fig. 4.1 Time histories of free surface elevation, wetted length, vertical force measuredat the supports, total vertical force and the associated line of action during onewave cycle for h = 0.8 cm, T = 1.68 sec, H = 14.2 cm. (a) free surfaceelevation and wetted length; (b) vertical force; (c) total vertical force and line ofaction.Fig. 4.2 Time histories of free surface elevation, wetted length, vertical force measuredat the supports, total vertical force and the associated line of action during onewave cycle for h = 0.8 cm, T = 1.70 sec, H = 10.5 cm. (a) free surfaceelevation and wetted length; (b) vertical force; (c) total vertical force and line ofaction.Fig. 4.3 Time histories of free surface elevation, wetted length, vertical force measuredat the supports, total vertical force and the associated line of action during onewave cycle for h = 0.8 cm, T = 1.70 sec, H = 6.80 cm. (a) free surfaceelevation and wetted length; (b) vertical force; (c) total vertical force and line ofaction.Fig. 4.4 Time histories of free surface elevation, wetted length, vertical force measuredat the supports, total vertical force and the associated line of action during onewave cycle for h = 1.4 cm, T = 1.68 sec, H = 14.2 cm. (a) free surfaceelevation and wetted length; (b) vertical force; (c) total vertical force and line ofaction.Fig. 4.5 Time histories of free surface elevation, wetted length, vertical force measuredat the supports, total vertical force and the associated line of action during onewave cycle for h = 1.4 cm, T = 1.68 sec, H = 10.5 cm. (a) free surfaceelevation and wetted length; (b) vertical force; (c) total vertical force and line ofaction.Fig. 4.6 Time histories of free surface elevation, wetted length, vertical force measuredat the supports, total vertical force and the associated line of action during onewave cycle for h = 0 cm, T = 2.02 sec, H = 17.5 cm. (a) free surface elevationand wetted length; (b) vertical force; (c) total vertical force and line of action.Fig. 4.7 Time histories of free surface elevation, wetted length, vertical force measuredat the supports, total vertical force and the associated line of action during onewave cycle for h = 1.4 cm, T = 2.02 sec, H = 17.5 cm. (a) free surfaceelevation and wetted length; (b) vertical force; (c) total vertical force and line ofaction.xFig. 4.8 Time histories of free surface elevation, wetted length,vertical force measuredat the supports, total vertical force and the associated line of action during onewave cycle for h = 2.5 cm, T = 2.02 sec, H = 17.5 cm.(a) free surfaceelevation and wetted length; (b) vertical force; (c) total vertical force andline ofaction.Fig. 4.9 Maximum upward force coefficient F/pgbHe as a functionof relative plateclearance h/H and relative plate length VL. (a) 0.016< H/L <0.020; (b) 0.026<H/L < 0.030; (c) 0.035 <HJL < 0.041.Fig. 4.10 Maximum downward force coefficient Fm/pgbHL asa function of relative plateclearance h/H and relative plate length LJL.(a) 0.0 16 <H/L < 0.020; (b) 0.026<HIL < 0.030; (c) 0.035 <H/L < 0.041.Fig. 4.11 Non-dimensionalised duration of plate’s partialsubmergenceTIT as a functionof relative plate clearance h/H and relative plate lengthL/L. (a) 0.0 16 <H/L <0.020; (b) 0.026 < HJL < 0.030; (c) 0.035 <HJL <0.041.Fig. 4.12 Non-dimensionalised duration of plate’scomplete submergenceT/T as afunction of relative plate clearance h/H and relativeplate length L/L. (a) 0.016 <H/L <0.020; (b) 0.026 < HIL <0.030; (c) 0.035<HIL < 0.04 1.Fig. 4.13 Non-dimensionalised rise-time TIT ofpeak vertical force as a function ofrelative plate clearance h/H and relativeplate length L/L. (a) 0.016 <HIL <0.020; (b) 0.026 < HIL < 0.030; (c) 0.035 <H/L<0.041.Fig. 4.14 Non-dimensionalised time of occurrenceTmIf of maximum downward force asa function of relative plate clearance h/H and relativeplate length L1t. (a) 0.0 16<HJL <0.020; (b) 0.026 <H/L < 0.030; (c)0.035 <H/L <0.041.Fig. 4.15 Non-dimensionalised lineof action Spit of maximum upward force as afunction of relative plate clearance h/H and relativeplate length Lit. (a) 0.016 <H/L <0.020; (b) 0.026 < H/L <0.030;(c) 0.035 <Hit < 0.041.Fig. 4.16 Non-dimensionalised line of actionSmit ofmaximum downward force as afunction of relative plate clearance h/Hand relative plate length Lit. (a) 0.016 <HIL < 0.020; (b) 0.026 <H/L <0.030;(c) 0.035 <H/L < 0.041.Fig. 4.17 Non-dimensionalised wetted lengthfL corresponding to maximum upwardforce as a function of relative plate clearanceh/H and relative plate length Lit.(a) 0.016 <H/L < 0.020; (b) 0.026 <H/L < 0.030; (c)0.035 <HIL < 0.04 1.Fig. 4.18 Non-dimensionalisedwetted lengthm/Lcorresponding to maximumdownward force as a function of relativeplate clearance h/H and relative platelength Lit. (a) 0.0 16 <H/L <0.020;(b) 0.026 <FI/L < 0.030; (c) 0.035 <Hit < 0.041.Fig. 4.19 Photographs of waveimpact. (a) instant of complete submergence;(b) waterdrainage following the submergence.Fig. 4.20 Video images indicatingvarious stages of wave impact during one wave cycle.(a) partial submergence; (b) complete submergence;(c) wave recession;(d) water drainage.xFig. 4.21 Variation of normalized free surface elevationand predicted vertical force basedon Eq. 2.23 for H/L = 0.04, £/L = 0.146 with two different plate clearances.(a) free surface elevation; (b) vertical force for h/H = 0; (c) vertical forceforh/H = 0.15.Fig. 4.22 Variation of normalized free surface elevationand predicted vertical force basedon Eq. 2.27 for HIL 0.04, //L = 0.146 with two differentplate clearances.(a) free surface elevation; (b) vertical force for h/H= 0; (c) vertical force forh/H = 0.15.Fig. 4.23 Comparison of vertical force predictedby analytical models with experimentalobservation for h = 0 cm, T = 2.02 sec, H = 17.5cm. (a) free surface elevationand the wetted length; (b) vertical force.Fig. 4.24 Comparison of vertical forcepredicted by analytical models with experimentalobservation for h = 0 cm, T = 1.68 sec, H= 14.2 cm. (a) free surface elevationand the wetted length; (b) vertical force.Fig. 4.25 Comparison of vertical forcepredicted by analytical models with experimentalobservation for h = 1.4 cm, T = 2.02sec, H = 17.5 cm. (a) free surfaceelevation and the wetted length; (b) verticalforce.Fig. Al Free body diagram of theplate indicating the support forces and the line ofaction of the vertical force.xiiList of Symbolsb = width of the plateC = damping of the measuring systemc = wave celerityc’ = wave front velocityCd = drag coefficientCs = slamming coefficientD = cylinder diameterd = water depthF = vertical force due to wave actionon the plateFA = reaction measured at support A (see Fig. Al)Fa = hydrodynamic forceFai = inertia forceF = added mass forceFB = reaction measured at support B (see Fig. Al)Fd = drag forceF = peak downward forceF0 = peak forceF = peak upward forceFt = measured vertical forceg = gravitational constantH = wave heighth = plate clearance above the still water levelK = stiffnessk = wave number, 2nfLXIIIk1 = stiffness of support Ak2 = stiffness of support BL = wave length£ = length of the cylinder or plateM = mass of measuring systemm = added mass per unit width of the platePm= amplitude of the free vibration force tracemeasured after m cyclesq = half wetted length of the plates = line of action of F= line of action associated withFm= line of action associated with FT = wave period= timeto= instant at initial impactt1 = instant at complete submergencet2 = instant at which leadingedge of the horizontal plate located at aclearance h just emerges out of wavet3 = instant at which horizontal plate locatedat a clearance hcompletely out of water.T = duration for plate’s partial submergence(=ti - to)Td = decay-timeTm= time of occurrence associated withFmtm = average of times ti and t2Tr = rise-time associated with applied forceT = duration of plate’s complete submergence(=t2 - t1)Tt= measured rise-time associated with Fpu = displacement of the measuringsystemv = vertical velocity of the freesurfacexivw = plate thicknessx = horizontal coordinate in wave direction= vertical force per unit widthu = acceleration of the measuring system= velocity of the measuring system= vertical acceleration of the free surface profile= added mass correction factor= free surface elevationTI()= free surface elevation at x =0= free surface elevation at x =ij = vertical particle acceleration of the free surface profile= vertical particle velocity of the free surface profile£ = plate length= wetted length= wetted length associated with F= wetted length associated with Fpp = fluid density= wave angular frequency, 2it/T= damped frequency of the measuring system= natural frequency of the measuring system= damping ratioxvAcknowledgmentsThe author would like to thank his supervisor Dr. Michael Isaacson forhis guidance andencouragement throughout the preparation of this thesis. The author wouldlike to expresshis gratitude to Kurt Nielson for his help in buildingthe model and other facilitiesassociated with the experimental investigation; to JohnWong for his help relating toelectronic parts for the experiments; and to SundarPrasad for his help relating to theexperimental setup, computing and the wave generatorfacilities. Also, Amal Phadke, NealWhiteside and Henry Kandioh are thanked for their helpin running the experiments. Theauthor would also like to thank the Director, National Institute of Oceanography,Goa,India, (Council for Scientific and Industrial Research), forgranting a study leave to pursuehigher education, and also the Ministry of Human ResourcesDevelopment, Government ofIndia, for being selected for the Canadian CommonwealthScholarship Plan. Finally,financial support in the form ofa Scholarship from the Canadian CommonwealthFellowship Plan, Canadian Bureau for InternationalEducation, Government of Canada, isgratefully acknowledged.1Chapter 1Introduction1.1 GeneralThe design of coastal and offshore structures requiresa thorough understanding ofenvironmental loads which are primarilydue to waves. Structural elements such as decks,which are located in the splash zone (i.e. at elevations whichcause them to be intermittentlysubmerged), may be subjected to impulsive loads thatcan be several times larger than thoseexperienced by continuously submerged elements. Theseimpulsive forces may give rise tofatigue and to localized damage.There are various examples of structural damagedue to wave impact. For example,Denson and Priest (1971) described theinspection of structural damage due to hurricanesalong the Gulf Coast, which revealedthat horizontal floors, decks and platforms aresusceptible to severe damage by wave action.Da Costa and Scott (1988) reported that amoderate storm on Lake Michigan in1987 moved partially constructed concrete slabsat theJones Island East Dock. Another exampleis the case of the Ekofisk platform whose deckwas exposed to severe wave impact (Broughtonand Horn, 1987).Figure 1.1 provides a view ofa typical jetty with its deck structure above the mean waterlevel. Clearly, such structuresshould be designed for local stresses dueto wave impact inaddition to the design for overallloads. Even so, decks need to be sufficiently highabovethe water surface in order to avoidunduly severe wave impact. Besides thephenomenonof wave impact on decks, other situationssuch as seaplane landing, ship bow slamming,2platform bracings situated in the splash zone, and liquid sloshing intanks also requiredesign with respect to hydrodynamic impact. It is therefore importantto have a goodunderstanding of the wave slamming process on the basisof theoretical and/or experimentalinvestigations. Although previous work has contributed to the understandingof the waveimpact nature, there is still considerable uncertainty in the estimationof impact loads onstructural elements near the water surface. In thiscontext, the present study has beencarried out to address the problem of wave loads ona fixed horizontal plate located near thestill water level.1.2 Literature ReviewThe Morison equation is commonly usedto calculate the wave force on fully submergedslender structural members of different cross sections.However, structural memberswhich are located in the splash zone,such as the deck of a wharf, are intermittentlysubmerged and experience a large verticalforce which cannot be predicted by the Morisonequation. These vertical forces are highly dynamic andcharacterized by large magnitudeswith short duration.Although, problems relating to water entry andwave impact on flat bottom ships havebeen the subject of numerous theoretical and experimentalstudies for many years, waveimpact forces on horizontal decks has receivedattention only since the early sixties. In thefollowing sections, a brief review of availabletheoretical and experimental investigationsrelating to water impact and entry problemis presented.1.2.1 Water Entry ProblemHydrodynamic impact refers to the early stagesof the entry of a body into water.Approaches to describing this havegenerally been based on potential flow theory for anincompressible fluid with a free surface. The solutionof such problems may be related to3the determination of a variable added mass associated with the bodyas it enters the fluid.In a classical paper, von Kármán (1929) presented aphysical picture of the impact of awedge on a still water surface, intended to represent the impact processfor the case of aseaplane landing. On the basis of von Kármán’s approach, Wagner(1932) provided amathematical treatment of forces acting ona seaplane float. In this classical approach, noconsideration has been given to the effects of entrappedair or water compressibility.Nevertheless, Wagner’s theoretical values show reasonableagreement with experimentalresults obtained for two-dimensional models (Chuang,1967).The above approach is based on potential flow theoryand provides an estimate of theimpact force on the impacting member during the initialstage of water entry. Beyond thisstage, the member also experiences inertia and drag forces.An extensive review of thesubject has been given by Szebehely (1966), emphasizingthe principles involved indifferent kinds of wave impact.Also, Faltinsen (1990) has summarized recentdevelopments of the water impact and entry problem.A number of experimental studies ona flat bottom plate striking normal to a smoothwater surface have been reported. The magnitude ofthe maximum impact pressure istheoretically equal to the acousticpressure, which is the product of the velocity of soundinthe fluid, the fluid density and the velocityof the striking body. The results of drop testson models with flat bottoms have, however,shown that peak pressures are usually muchlower than the acoustic pressurebecause of the air entrapped between the body andthewater surface. Verhagen (1967)developed a theory to predict the impact pressures byconsidering the influence of the compressedair between the plate and water surface. Healso investigated the phenomenon experimentallyand showed that the predicted values arein good agreement with the observedvalues obtained from two-dimensional tests of a bodyhaving a completely flat bottom.Recently, Chan et al. (1991) emphasizedthe influence of4trapped air on impulsive pressure and examined the process of wave impact relating tovertical plates on the basis of a simplified one-dimensional model.In the following sections, the related situation of a horizontal circular cylinder isconsidered, followed by experimental investigations of wave action ona horizontal plate.1.2.2 Horizontal CylinderThe impact force on a cylinder is given by the rate of change of fluid momentum which is afunction of the cylinder’s added mass that varies with the submergence. From a number ofpast theoretical and experimental investigations on circular cylindrical horizontal members,the slamming forceFis considered to be proportional to the square of the wave impactvelocity and is expressed as:F5=(C5)(pv2)(De) (1.1)where Cs is a slamming coefficient, £ is the cylinder length, D is thecylinder diameter,p is the fluid density, and v is water particle velocity normal to the member surface. Therehas been considerable debate on the choice of a proper valueofC,typically ranging fromIt to 2n (e.g. Kaplan and Silbert, 1976, Sarpkaya, 1978, Sarpkayaand Isaacson, 1981,Armand and Cointe, 1987, Greenhow and Li, 1987,Chan and Melville, 1989,Chan, et at. 1991, Isaacson and Prasad, 1992). Althoughthe slamming force isassociated with the rate of change of momentum during theearly stages of impact,extending such a formulation beyond the initialstages gives rise to a number ofcomplications. These are mainly attributedto the water level variations in the vicinity of thepartially submerged member and the subsequent onsetof drag forces. In addition to theabove, the buoyancy force and the inertia force also form significantcomponents of thevertical force on the cylinder. The associatedforce coefficients also vary with thesubmergence, member size and flow kinematics.Also, the problem of wave action on a5cylinder may involve splashing and air entrapment, and partial and/or completesubmergence.1.2.3 Horizontal PlateThe following paragraphs give a brief account of previous studies relatingto wave actionon a horizontal plate.El Ghamry (1963) carried out an early experimental study on the vertical forcedue tonon-breaking and breaking regular waves slamming on a horizontal plate.He indicated thatthe vertical force is characterized by an initial peak of considerable magnitudeand shortduration, followed by a slowly varying force of smaller magnitude extendingover theremaining period of submergence. He proposed a theoreticaldescription of the force basedon a potential flow past a rigid fixed flat plate, incorporatingsuitable correction factorsrelating to the deck length, wave length and water depth.Furudoi and Murita (1966) studied experimentally thetotal vertical force on a horizontalplate extending seaward from a vertical wall and noteda sharp impulsive force as indicatedby El Ghamry, with the average pressure head on the platformranging from 1 to 8 timesthe incident wave height.Wang (1967) carried out experiments ona horizontal pier model subjected to slammingby progressive and standing waves. He derived simple theoretical valuesfor peakpressures, by adapting an approximate analysis basedon the fluid momentum principle,and related the peak pressure to the celerity of the wave andthe velocity of the fluid elementnear the wave front. The slowly varying pressure headwas simply taken as the pressure inthe undeformed wave at the deck elevation.French (1969) carried out an extensive laboratory studyand confirmed the nature of theimpact force to be similar to thatobserved by El Ghamry. He predicted the impact force6magnitude on the basis of a momentum conservationand energy equation. Denson andPriest (1971) carried out a laboratory studyto identify the influence of relative waveheight, relative plate clearance, relative platewidth and relative plate length on the pressuredistribution under a thick horizontal plate. Tanimotoand Takahashi (1979) reported on anexperimental investigation to obtain the horizontaland vertical forces on a rigid platformdue to periodic waves. The uplift pressure was expressedas the sum of a shock pressurecomponent and a static pressure component. Theydeveloped an empirical shock pressureterm as a function of the contact anglebetween the undisturbed wave surface and thebottom of the horizontal platform. More recently,Toumazis et al. (1989) investigatedexperimentally wave impact pressures onboth horizontal and vertical plates. Pressuremeasurements in conjunction with observationsusing video records were adoptedto studythe impact loading behaviour.Irajpanah (1983) studied wave upliftpressures on horizontal platforms and presentedafinite element method to investigate thehydrodynamic loads on a horizontalplatform.Also, Lai and Lee (1989) developed a potentialflow model using the finite elementmethod,and predicted the vertical forces of largeamplitude waves on docks. They used a Galerkinfinite element method and studiedthe interaction of finite amplitude nonlinear waterwaveswith platforms. Their results comparedreasonably well with the experimentalresults ofFrench (1969).Kaplan (1992) extended the hydrodynamictheory of ship slamming to study wave actionon a deck slab. He proposed the timevarying vertical force as a combinationof ahydrodynamic impact force anda drag force. The drag force was computed fromaconstant force coefficient and assumedto act over a complete slamming event.Although hedid not compare the predictedvertical force with experimental results,in his re-examinationof hydrodynamic impact theory,he briefly assessed the features of time historiesof the7predicted force with respect to field data. The time histories of force indicated that themagnitudes were comparable during the initial stages of impact. However, the variationshowed a large discontinuity at the instant of complete submergence of the structure.1.3 Scope of the Present InvestigationThe primary aim of the present investigation is to study experimentally hydrodynamicaspects of the vertical force on a fixed rigid horizontal plate. Despitethe considerableimportance of this problem, a literature review reveals little information regardingtheestimation of the slamming force. On the basisof the studies carried out byEl Gahmry (1963) and French (1969), an experimental investigationwas carried out on aninstrumented horizontal plate located above the stillwater level and subjected to waveaction. Vertical reactions at the two plate supportswere measured for differentcombinations of incident wave conditions and plate elevations, andthe vertical force and itsline of action was computed from the measuredsupport reactions. The results arepresented in the form of time series of the force and its lineof action. Also, an analysis ofthe force records is made in order to obtain the peak upwardand downward forces, theirtimes of occurrence and their lines of action. Videorecords of the experiments are studiedin order to identify the problems involved in the experimentalinvestigations. Finally, anattempt is also made to predict the slamming force on a theoreticalbasis.8Chapter 2Theoretical DevelopmentIn this chapter, important parameters influencing thevertical force on a fixed horizontalplate subjected to regular waves are identified anda theoretical description of the force ispresented.2.1 Dimensional AnalysisDimensional analysis provides an important preliminarystep to any experimentalinvestigation and may be usedto identify important dimensionless parameters of theproblem at hand. Figure 2.1 providesa simplified definition sketch indicating aunidirectional regular progressive wave trainin water of constant depth d propagating pasta horizontal plate of width b, thickness w and length£ located at a distance h above themean water level. The vertical force on theplate, denoted F, is of interest and is influencedby a number of variables which include thefollowing:• wave height, H• wave period, T• water depth, d• plate length, £• plate width, b• plate thickness, w• plate elevation, h• fluid density,p9• gravitational constant, g• fluid viscosity, j.L• time, tAdditional parameters such as surface roughness, surface tension, air and watercompressibility, and spray effects may also play significant role but are neglected here.The width of the plate may be important since it influences the escape of air belowtheplate as the wave advances. The deck thickness w is assumed to be small enoughso thatits influence may be neglected. And finally, the effect of the fluid viscosityji is alsoneglected.In view of the above simplifications, the vertical force maybe expressed in the form:F = f(p, g, H, T, d, h, £, b, t) (2.1)where is the vertical force per unit width. A dimensionalanalysis providesF fh H d£ bt’pgHe=i’ r L’‘i(2.2)In engineering applications, the maximum force F0,is of particular interestand is givenas:F0 (h H d b£‘,(2.3)The parameter h/H defines the relative clearance suchthat there is no wave contact forhJH> 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 theplate, 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 <h/H <0.5. The depth parameter d/L shouldhave a negligible influence on the force for deep water conditions corresponding to d/L>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)whereFa(t), Fd(t) and Fb(t) are hydrodynamic impact, drag and buoyancy forcecomponents respectively. These components are briefly discussed inthe sections tofollow. However, it is useful to consider initially the various stages of interaction duringawave cycle.Stages of Wave InteractionConsider a wave train interacting with the plate during the course of onecycle. 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 withthe leading edge of the plate. InFig. 2.2(b), the wave progresses further, partially wettingthe 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 wavefrom the rear edge of theplate.Defining the intersection locations of the downwave and upwavefree surfaces as x1 andx2respectively, along the length of the plate, with the originat 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<t<ti xi0, ?x2• Fort1<t<t2 xj=0,• Fort2<t <t30<xi <L, x= L,• Fort=t3 x1=L,In the present study, only the case ilL < 1 is consideredso that there is no more thanone region of water contact at any instant.Hydrodynamic ImpactA simplified hydrodynamic analysis similar to that for a cylinder,can be carried out toprovide an approximate formulation for the vertical force on theplate. It is assumed thatthe fluid is incompressible and inviscid, the flow is irrotational,the body is rigid, and thatthe surface tension is negligible. FollowingKaplan (1992), the force componentFa due to12hydrodynamic impact may be expressed in terms of the rate of change fluid momentumassociated with the submerged portion of the plate:Fa= a(mv)(2.5)atwhere m is the vertical added mass of the submerged portion of the plate, t is time, and v isthe vertical velocity of the fluid striking the plate. Equation 2.5 can be expanded to obtainam .Fa = m— +v— = my + v——— (2.6)at at aatwhere is the wetted length of the plate and v is the vertical acceleration.The primary difficulty with applying this equation relates to the use of a reliableexpression for the added mass, since the flow field around the plate is rather complex. Asuitable approach to estimating m is to equate this to the added mass of a fully submergedplate in an infinite fluid. Thus, the required added mass m when submerged plate length is? is considered to be equal to the added mass of a fully submerged plate length ?.However, because of the uncertainty associated with this equivalence, it is appropriate tointroduce an unknown factor a. Thus the expression for added mass m is given by:m = ap2b2f1(2Jb) (2.7)where a replaces t/4 in Eq. 2.7 and the functionf1(Jb) is described in Sarpkaya andIsaacson (1981). On the basis of Eq. 2.7, the hydrodynamic impact forceagiven byEq. 2.6 may be expressed as:Fa = a p b2f1(?Jb) ‘ + a p b2 v f2(2Jb) (2.8)where an overdot denotes a time derivative, the functionf2QJb) is given as:f2QJb) = f1QJb) + (XJb)fQJb) (2.9)13and a prime denotes a derivative with respectto the argument. The first term in Eq. 2.8,denotedFai,is an inertia force associated with the fluidacceleration, and the second term,denotedFa2,is an added mass force associated withthe rate of change of added mass.It is also possible to formulate the impactforce by considering instead the hydrodynamicforce,IS.Faon an element of length zSx and integratingthis over the instantaneous wettedlength of the plate.iFa= a(Lm v)(2.10)atwhere zm is an infinitesimaladded mass, given as zm= [4pvbq4q2- x2jsx(Lamb, 1932), x is the coordinate measuredfrom the centre of the plat&s wetted length,and q is half the wetted length ofthe plate. By integrating over the wetted length, thetotalimpact force may be obtainedas:Fa JAFadX (2.11)where x andx2 are the intersection locations of the downwave and upwave limits of thesubmerged plate length, measuredhere from the centre of this submerged plate length.Ananalytical solution of the aboveintegration is intractable, and this approach hasnot beenpursued since it may not leadto increased accuracy because of the various uncertaintiesinthe procedure.Drag ForceThe drag forceFd acting on the submerged potion of the plate is expressed in usual wayinterms of a drag coefficientCd and is given by:Fd=Cd1pbvIvI2.(2.12)14whereCd is the drag coefficient. The choice of Cd depends on various factors such as thegeometry and shape of the plate, and its orientation relative to the wave propagationdirection. The drag coefficient value during the early stages of the impact is unclear, buthas been chosen to be equal to 2.0, as for the plate when completely submerged.Buoyancy ForceAlthough it is consistent to assume that the plate is very thin, in practice the submergedvolume of the plate may not be negligible, so that any comparison withthe experimentaldata should account for the buoyancy force associated with the finite volume of the plate,the stiffeners and the submerged portion of the support system. Thus, the buoyancy forceis given as:Fb = pgV (2.13)where V is the submerged volume of the plate and submerged portionof the supportsystem. In the present study,Fbis assumed to vary sinusoidally with the time.2.2.1 Wave Theory and AssociatedKinematicsThe hydrodynamic impact and drag forces discussed in the previoussections are expressedin terms of the particle velocity and acceleration evaluatedat the plate elevation. In thepresent study, linear wave theory is used to determinethe free surface elevation andassociated kinematics for application to these expressions.For a regular progressive wave train, the water surfaceelevationTIis expressed as:11 =cos (kx - 0t)(2.14)15where H is the wave height, k = 2icfL is the wave number, co = 2it/T is the wave angularfrequency, L is the wave length, and T is the wave period. The wave number and the wavefrequency are related by the linear dispersion relationship:= ‘.Jgk tanh(kd) (2.15)where d is the still water depth. On the basis of linear wave theory, the associatedkinematics are given ascoH sinh[k(d + z)]v= 2 sinh(kd)sin(kx - cot) (2.16)co2Hsinh[k(d+z)]v =- 2 sinh(kd)cos(kx - cot) (2.17)with z measured upwards from still water level. In the present case z is set to the deckelevation h. Several approaches have been proposed to evaluate water particle kinematicsnear the instantaneous free surface (e.g. Gudmestad and Connor, 1986). The simplest ofthese is to take z = 0, since this is consistent with the accuracy of linear wave theory. Thusby substituting the approximation into Eqs. 2.16 and 2.17 we getv r = sin(kx-cot) (2.18)=cos(kx-cot) (2.19)The wave kinematics vary along the submerged length of the plate, and are evaluatedhere at the centre of the instantaneous submerged length of the plate.162.2.2 Superposition of Force ComponentsWetted LengthThe wetted length and its time variation are important in the formulationof the verticalforce indicated by Eq. 2.8. Figure 2.3(a) is a sketch of the free surfaceelevation at theleading and rear edges of the plate, denotedfloand i respectively. And Fig. 2.3(b) is asketch of the wetted length variation for a given wave formand plate clearance. At time to(see also Fig. 2.2(a)), is zero and starts to increase as the wave progresses.At time t1,the deck is completely submerged and 2. reaches the fullplate length, ? = £. From t1 to t2(see also Fig. 2.2(c)), the plate remains fully submerged. For t >t2, the wave recedes sothat A. decreases; and finally at t = t3, = 0. On the basis of linear theory, thewetted lengthcan be obtained by equating the wave surface elevation rto 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 componentsmay 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 clearanceh is of the order of wave heightH, so that on the basis of linear wave theory the velocityv and acceleration v may be takenas v r and v ij. Substituting the expressionsfor the force components and the abovesimplification for wave kinematics inEq. 2.21, we obtain:F = cc p ? b2f1(2Jb) ij + ap 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 timetm.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 <t< (ti+t2)(i.e. ij 0)0 otherwiseA possible variation of the added mass force is sketched in Fig. 2.3(d) along with theother three components. The total vertical force predicted resulting from the superpositionis shown in Fig. 2.3(e) along with the actual total vertical force.In applying Eq. 2.23, some attention should be given to the determinationof thefunctions f1()andf2Q. Tabulated values of the added mass for the three-dimensional caseof different length to width ratios are given by Sarpkaya and Isaacson (1981) and arereported in Table 2.1. The two-dimensional limits of infinite widthor length correspond18respectively tof1(?/b) - (?Jb) (so that m = o it b) as (?Jb) -4 0; andf1QJb)—> 1 (sothat m = x it ? b2) as (2Jb) —ooA suitable function f1() is fitted to this data and used inEq. 2.23. Thus, the functionf1() andf2() 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)10 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 addedmass 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 theassociated uncertainty relating tothe use of Eq. 2.7 for the added mass, a simpler alternativebased on the two-dimensionalcase of infinite width may instead be adopted. The two-dimensional limitA/b —* 0corresponds tof1(AIb) —* (A/b) so that Eq. 2.7 wouldbe 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 variationof added mass with the plate aspect ratio.Fig. 2.5 indicates such a variation for arbitrary valuesof o, 1 and 3 for Eq. 2.23 and 0.5and 0.7 for the simpler added mass model givenby Eq. 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 forceis 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, stiffnessK and dampingcoefficient C as indicated schematically in Fig. 2.6. The stiffness Kis related to thestiffnesses of each of the load-cells placed at two supports. The equationof 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, velocityand accelerationrespectively. The natural frequency o and the damping ratioof the system are definedas o= JK/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)Mojwhere0dis the damped natural frequency of the assembly definedas0d =w[i2,and tis a variable of integration.The force F(t) is the applied vertical forceas given in either Eq. 2.23 or Eq. 2.27 and itis composed of force components given in Eq.2.21. However, the force transmittedFt(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-timeTrand then drops linearly to zero in a decay-timeTd. The completeresponsehistory 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 forceF0immediately after the impact and the associated rise timeT asimportant parameters, and presented a set of characteristic curves describing these asindicated in Fig. 2.8. In the figure,Ft0is 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 forceFt0andT 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 loadingto 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 modesof vibrationwhich may affect the force measurement; one corresponding to a first mode beam vibrationand the other to the whole assembly vibrating as a lumped masssupported 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 shouldbe sufficiently stiff.Bearing these considerations in mind, a plate model wasdesigned to be stiff enough toapproximate a rigid body, and at the same time tobe 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 longand6.25 mm thick. Two aluminum angles are fixed above the plate to increase its rigidity.Asmall gap between the plate and the angles allows overtopped waterto 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 rodto 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 section5 cmhigh x 7.5 cm wide. The box section is aligned in such a way that its longitudinalcentre-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 theplate wasadjusted by lengthening or shortening the part of the rods between thesupportinglongitudinal channel and the load-cell. By this arrangement, vertical forceswere 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 shownin 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 generatedby a single paddle wave actuatorlocated at the upwave end. The generator iscontrolled by a DEC VAXstation-3200minicomputer using the GEDAP software packagedeveloped by the National ResearchCouncil, Canada, (NRC). The generator is capableof 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 allstages ofthe experimental investigation. This software package is available for the analysis andmanagement of laboratory data, including real-time experimental control anddata-acquisition functions. GEDAP is a fully-integrated, modular system whichis linkedtogether by a common data file structure. GEDAP maintains a standard data file formatsothat 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 sothat mostlaboratory projects can be handled with little or no project-specific programming.Anattractive feature is the fully-integrated interactivegraphics capability, such that results canbe conveniently examined at any stage of the data analysis process. It also includesanextensive collection of utility packages, which consistof a data manipulation routine, afrequency domain analysis routine, and statistical and time-domain analysis routines.Inparticular, the program RTC_SIG generates the control signal necessaryto drive the wavegenerator, and the routine RTC_DAS reads the data acquisition unit channelsand stores theinformation in GEDAP binary format compatible withother GEDAP utility programs.3.5 MeasurementsThe water surface elevation and the associated vertical forcewere required to be measuredfor each test. The vertical forces at two supports weremeasured using load-cells. Theselection of the load-cells is based primarily onconsiderations 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. Previoustests 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 absentand the forceson the plate were measured in separate stages. In the secondstage, the wave probes wereremoved and the plate assembly was installed in the flume. Wavescorresponding to thesame stored wave signals were repeated and the vertical forceson the plate were recorded ata sampling rate of 2.5 kHz. A video record was alsoobtained for each experiment.Experiments were carried out for five different elevationsof 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 anormalspeed 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 thetimebase 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 therebytoaccess the effects of the assumptions made, and in particular to assess the importanceofwave profile deformation, air entrapment, wave overtoppingeffects, 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 platewereconducted 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 releaseof a load carried by a thin steelsingle stranded wire. This was achieved by cutting-off the wireusing an acetylene torch.The free vibration traces were recorded for both the load-cells A and B indicatedinFig. 3.2. The recorded force time histories and the corresponding spectral densitiesfor theplate in air and fully submerged condition are shown in Fig.3.5 and 3.6 respectively.Figure 3.5 indicates that for vibrationsin air the plate response has widely distributedfrequencies with a predominant frequency of approximately125 Hz for both load-cells. Itis informative to study the initial stages ofan impact event just after releasing the step load.Consider the free vibration trace for the load-cell Aas 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 andPm+nare 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 freevibrationtrace for load-cell B.Apart from free vibration tests in air, the plate and load-cell assemblywere also testedfor free vibrations under a fully submerged situationso 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 peakis observed at10 msec and the system comes to rest 1.0 sec after the releaseof the load. The averagedamping ratio for this submerged condition wasfound 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 reducesto approximately 25 Hz.As explained in Section 2.3, Fig. 2.8 may beused to estimate the influence of thesystem characteristics on the measured force andassociated rise-time. From thepreliminary experimental results, the minimum rise-timewas found to be not less than 100msec. And with the natural period of the systemtaken to be equal to 1/125 sec, thereappears to be no noticeable amplification in thepeak value of the measured force and therise-time. The natural period of the system is takento 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 removenoise in therecords. One of the causes of noise may be due to electromagneticfrequency 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 frequencyof 1000 Hz. Theforce data from two the load-cells were first plottedto observe the noise present in therecords. The GEDAP filtering program FILTW wasused to filter the data. Several levelsof cut-off frequency ranging from 500 to as low as 5 Hz were used toexamine the effect onthe peak force. Cut-off frequencies below 10 Hz show considerablesmoothening of theforce peak and loss of information, and it was decided toset the cut-off frequency at 15Hz. As an illustration of the low-pass filterthat was adopted, Fig. 3.7 showscorresponding unfiltered and filtered force records.A static analysis was applied to obtain the vertical forceand its line of action in themanner indicated in Appendix A. Figure Alis a sketch of free body diagram of the platerepresented as pinned rigid beam. Fromthis analysis, the total vertical force and its line ofaction were obtained and are illustrated in Fig. 3.8 forthe case of unfiltered signals andfiltered signals.In a similar way, the wave records were filteredand resampled at 10 Hz since such ahigh sampling rate is unnecessary fora slowly varying signal. The GEDAP routines,FILTW and RESAMPLE2 were usedto filter and resample the wave records. Once thevertical force from the filtered records are available alongwith the resampled wave profile,the remaining analysis was carriedout as explained below. It was decided to study onlyone slamming event for each test and therefore onlyone slamming cycle is selected from a284 sec record as shown in Fig. 3.9. The instant of slamming tois determined from themeasured water surface elevationoand 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 measuredfrom the leadingedge of the plate, as mentioned in the Section 2.2.1. The buoyancyforce was evaluatedbased on the displaced volume of the water by the plate and load—cellassembly as discussedin Section 2.2, and is assumed to vary sinusoidallyfor the duration of submergence withthe peak buoyancy force occurring at t = t. Variousother parameters such as peakupward and downward forces, their lines of action,times of occurrence and associatedwetted lengths were then determined. The vertical force waspredicted based on the linearwave theory for different wave conditionstested 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 thetests with regularnon-breaking waves of different heights and periods, and differentplate elevations. Theimportant parameters thereby estimated correspond to the maximumupward force, theassociated point of application, the rise-time, and the corresponding valueof the wettedlength. Similar quantities relating to maximum downwardforce 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 predictingthe verticalforce is discussed.The application of the two analytical models (Eqs. 2.23and 2.27) for estimating thevertical force is discussed, and associated discrepanciesare identified. Finally, a correctionfactor for the added mass associated with each of thesemodel is introduced, and its rangefor different incident wave conditions is presented.A total of 69 tests were carried out, correspondingto 5 plate elevations, 5 wave periods,and 3 wave steepness. (Some of the 75 combinationsof these three do not give rise toimmersion of the plate.) The plate elevationsranged from h = 0 to h = 2.5 cm; waveperiods ranged from 0.8 to 2.0sec; and the wave heights for each wave period wereselected to correspond to steepnessHit 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 ofthe vertical forces measured at two supportlocations FA andFB, the total vertical force F, and the corresponding water surfaceelevations‘rioand r, at the leading and rear edgesof the plate. The figures correspond toeight different test conditions characterizedby changes in wave steepness and plateclearance. In these figures, t is the timemeasured from an arbitrary origin. Asan aid tointerpreting results, these figures includehorizontal and vertical lines indicating theplateelevation and the corresponding instantsof impact and complete submergence.On thisbasis, the variation of wetted length 2(t)is also obtained.Figures 4.1, 4.2 and 4.3 correspondto a plate clearance h = 0.8 cm and waveswithT 1.70 sec and H = 14.2, 10.5 and 6.8cm respectively. As a wave advancespast theplate, the force increases quite graduallyfrom the instant of water contact, exhibitsa fairlysharp maximum, and then variesmore gradually over the remainder of the cycle,passingthrough a noticeable minimum duringthe later stages of the event. The noticeablemaximum downward (i.e. negative)force which is present during the later portionof thewave cycle is due to a suction associatedwith the water surface receding belowthe plate,together with the weight of some overtoppedwater remaining above the plate.Similarly, Figs. 4.4 and 4.5 correspondto a plate clearance h = 1.4 cm and waveswithT 1.70 sec and H = 14.2cm and H = 10.5 cm respectively. These figuresindicate asharper rise in force than for the caseh = 0.8 cm. Finally, Figs. 4.6, 4.7and 4.8correspond to different clearances,h = 0, 1.4 and 2.5 cm respectively, with the samewavecondition T = 2.02 sec and H = 17.5 cm.It is useful to study the slamming processwith respect to the free surface elevation.As mentioned already, horizontal linesin Figs. 4.1(a) - 4.8(a) indicate the plate elevation,and the vertical lines indicatethe instants of initial wave contact, completesubmergence, the31onset of wave recession and complete wave recession. The timebetween the two lowervertical lines indicate the duration over whicha wave is in contact with the plate. Extendingthese line to the force time histories F(t), the observedinstant of impact agrees reasonablywell with the instant at which the wetted length startsrising from zero. It may be noticedthat the total force F before and after these points isnon-zero even though the plate isentirely in air. This may be attributed to the overtoppedwater draining from the top of theplate. It can be observed that the waterdrains out completely for the wave of period T =2.02 sec, so that then the force F is nearlyzero as indicated in the Fig. 4.6(c). The timebetween the upper two vertical lines indicatethe duration for which the plate is completelysubmerged. The intersection of two wave profiles‘rioandieshows the symmetric waveprofile with respect to support A (mid span location)along the plate length for that instant.The total force F is positive indicatingthat 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 ins are associated with F(t) passing through zero.The force’s line of action initially movesaway from the leading edge of the plate asexpected, but does not span the wholewave cycle. It moves abruptly from the leadingedge to the rear edge just after the occurrence ofthe symmetric wave profile along thelength. And during the wave recessionera, 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 suctionassociated with the water surface receding below theplate, together with the weightof some water remaining above the plate. The negativeforce seems to start just before the wavesurface leaves the plate’s leading edge i.e.just32before timet2. 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 accordingto the dimensionalanalysis indicated earlier and are listed in Table 4.1. These include themaximum upwardforce Fp, its time of occurrence T, its point of applications,and the correspondingwetted length X. The table also includes the maximum downwardforceFm, itstime ofoccurrence T, and point of application5mas 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 thepeak upward force coefficient F/pgHbe as afunction of relative clearance h/H for various valuesof the relative plate length £/L and forvarious ranges of wave steepness. As therelative clearance h/H approaches 0.5 the forcecoefficient F/pgHb/ approaches zero. Also, the forcecoefficient increases as the relativeplate length £/L increases. Although,there is no significant change in the force coefficientfor the range of wave steepness used in thetests (HJL 0.0 16 0.041), it may be seen thatthe trend of the which shows F/pgHb.eto decrease linearly with h/H is no longer observedfor the case of steeper waves.Figure 4.10 indicates similar plots forthe peak downward force coefficient Fm/pgHbe.Figures 4.10(a), 4.10(b) and 4.10(c)indicate no systematic relationship, althoughFm/pgHb apparently remainsconstant for any change in h/H.33Duration of SubmergenceFigure 4.11 is a plot of a dimensionless duration of partial submergence,TITwhereT =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 meanwater level, andconsequently influences the occurrence of the peak upward force predictedby the analyticalmodels presented. The figure shows a fairly constant valueofT/Tfor 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 ofa dimensionless duration relating to completesubmergenceT/T, where T = t2 - ti. This parameter indicates the duration for which theplate remains submerged and influences the span ofgradual variation of the added massforceFa2. 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 associatedwith the peak upward force, andFig. 4.14 shows corresponding results relating toTm/T. Figure 4.13 exhibits a fair degreeof scatter so that there are no particularly noticeable trendin there results. However, thereis some tendency for T/T to decreasewith increasing h/H. On the other hand Fig. 4.14shows a fairly clear trend forTm/T to decrease with increasing h/H and with decreasingilL. The scatter in the rise-times may be attributedto 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-dimensionalisedline of action of the peak upward force sILplotted against the relative clearance h/H for variousvalues of relative plate length and34wave steepness. There is considerable scatter in the results oncemore such that the relativeplate length ilL and wave steepness H/L appear to haveno significant influence on sfL,except that s/L increases with an increasing h/H.Figure 4.16 indicates corresponding results forSm/L. In this case, it is interesting tonote that the for all ranges of steepness tested, thereseems to be a linear trend betweenSmIL and h/H.Wetted LengthIn Fig. 4.17, the dimensionless wetted length 2/Lassociated with the peak upward forceis plotted as a function of relative clearance hJH, withHIL and ilL as parameters. It isobserved that for lower value of the ilL, theseems to reduce as h/H increases.Corresponding results form/Lare shown in Fig. 4.18 and indicate no particularcorrelation.From the above, it can be seen that ingeneral as h/H increases all impact parametersdecrease. The effect of steepness is notnoticeable, possibly because of the small rangeconsidered in the tests.4.3 Video RecordsIn order to complement the resultspresented in the previous section, a qualitativestudy ofthe impact process of the flowpast the plate has also been made on thebasis of the videorecords.Figure 4.19 shows photographs indicatingthe wave flow past the plate associated with adistortion in the shape of the wave profile.Figure 4.19(a) was takenat the instant ofcomplete submergence and shows someair entrapment at the rear end as theplate35completely 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 penetrationof theplate below the water surface. The wave progresses over the plate withouttoo 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 recessionstage, duringwhich water drains from the plate including some disturbance to the wave profile. Airentrainment between the plate and the water surface during this stageis 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 predictionsof the two analytical models of thevertical force given by Eq. 2.23 and Eq. 2.27 with the experimentalmeasurements ispresented and discussed.The steps involved in the computational aspects of the analytical modelsare as follows.Eq. 2.23 is evaluated form the instant of initial contactuntil the instant of completesubmergence. This is followed by a computation ofthe force during completesubmergence. Finally, Eq. 2.23 is evaluatedfor the period of wave recession. Parametersrelating to this formulation, which are computed at each timestep include the wetted length2 and the wave kinematics i and ij. 2. is obtained byevaluating Eq. 2.20, and r and j aredetermined at a point half way along the instantaneouswetted length on the basis of36Eqs. 2.18 and 2.19. A similar procedure is adopted forthe simpler analytical model givenby Eq. 2.27.Figures 4.21 and 4.22 show the variation of the dimensionlessvertical 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 analyticalmodels given by Eqs. 2.23 and2.27. In evaluating these equations arange of suitable a values ranging from 1 to3 forEq. 2.23 and 0.5 to 0.7 for Eq. 2.27 wereassumed. From these figures it can be observedthat a noticeable difference of the forcepredicted by two models occur in the partiallysubmerged stage, with the model basedon Eq. 2.23 indicating a faster rise than Eq. 2.27.This is due to the different assumptionsmade in added mass variations with respect tosubmergence [see also Fig.2.51.It is also of interest to considerthe suitability of the analytical model in predicting thevertical force variation, and a comparisonof the force predicted on the basis of Eqs. 2.23and 2.27 with a measured forcerecord is made in following paragraphs.The added massconstant a is selected so as to match themeasured and predicted maximum forces.Figure 4.23 provides a comparison of themeasured and predicted force time historiesforce the case h = 0 cm, T= 2.02 sec and H = 17.5 cm. The predicted forcebased onEq. 2.23 rises to a peak much earlier anddrops to negative values faster then the observedforce. On the other hand,the force predicted on the basis of Eq. 2.27 reachesa maximumduring the partial submergence with a rise-timealmost equal to the measured forceand itseems to follow the measuredforce reasonably well until the maximumforce is reached.Figure 4.24 compares the predictedand measured force for incidentwaves 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 forcebased on Eq. 2.27 seems to be muchsmaller 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 occursslightly laterthan that for the measured force, it follows the observed force reasonably well.Theprediction based on Eq. 2.23 does not show any improvement comparedto that based onEq. 2.27.Form all the three figures, it is observed that the predicted forcesdeviate significantlyfrom the observed force during the stage of complete submergencefollowed by waverecession.As mentioned earlier, the factoro of Eqs. 2.23 and 2.27 has been evaluated bymatching the measured and predicted maximumupward force. The results for a selectednumber of tests are listed in Table 4.2. These correspondto 9 tests with h = 0, two testswith h = 1.4 cm and one test with h = 2.5cm. 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. 2.27is approximately equal to irI4, whichcorresponds to the theoretical value of a plate in an infiniteflow situation, so that there is norequirement of any reduction factor. On the otherhand, 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 thevertical force due toregular non-breaking waves interacting with a fixedhorizontal plate located near the stillwater level. Force time histories were analyzed toobtain peak upward and downwardforces, their times of occurrence, their linesof action and the associated wetted length ofthe plate. Also, the free surface elevationat the leading and rear edges of the plate wererecorded for different wave conditions withthe plate absent. Two analytical models basedon a varying added mass of the submergedportion of the plate together with a drag andbuoyancy forces were used to predict the force on theplate, and a comparison of thesepredictions with the experimental results wasmade.The dependence of the various characteristics of themaximum and minimum forces onthe relative plate clearance h/H, wave steepnessH/L and relative plate length £/L have beenexamined and are indicated. Even thoughthere is a considerable degree of scatter in theresults, some general conclusionsmay be made. The maximum upward force decreaseswith increasing relative clearance of the plateh/H. And the peak upward force is higher forlarger value of relative plate lengthLit. In the present study the wave steepness HIL was inthe range of 0.016 to 0.041 andis observed not to influence the peak upward forcenoticeably. The maximum downwardforce seems to remain constant for increasingrelative clearance h/H. However,it shows similar variation as that of the peak upwardforce for changes in relative platelength Lit and in wave steepness Hit. The rise-timeassociated with the peak upward force shows considerablescatter. However, it isobserved that three is some tendencyfor Tjf to decrease with increasing h/H.On the39other hand, the dimensionless time of occurrenceTmIT 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 theF and Fm indicatedincreasing trend for steeper waves and remained constantfor less steep waves.Summarily, the analysis indicated that the peak forces Fand F and associated times ofoccurrence decrease some what linearly with the increasingplate clearance.The maximum upward force and associated rise-timecan be predicted reasonably wellon the basis of the simpler hydrodynamic model givenby Eq. 2.27, with the factor cxranging from 0.5 to 0.7. However, the predictedforce deviates significantly from themeasured force after the occurrence of the peak upwardforce. This indicates the need for aclearer understanding of the wave interaction with theplate during complete submergenceand recession stages.Several avenues of further study may be suggested. Improvementsmay be made toanalytical I numerical models to predict the verticalforce; and experiments relating toirregular and breaking waves also need to be carried out.40ReferencesArmand J. L., and R. Cointe, (1987), “Hydrodynamic impact analysisof 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 onsurface-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 loadson horizontalstructures in the splash zone,” Proc. 1st InternationalOffshore and Polar EngineeringConference, Vol. 3,pp. 203 - 209Chuang, S. L., (1967), “Experiments on slamming of wedge-shaped bodieswith variabledead rise angle,” J. Ship Research, SNAME, Vol. 11, No. 4,pp. 190-198.De Costa, S. L., and J. L. Scott, (1988), “Wave impactforces on the Jones Island eastdock, Milwaukee, Wisconsin,” Oceans, IEEE,pp. 1231-1238.Denson, K. H., and M. S. Priest, (1971), “Wave pressureon the underside of a horizontalplatform,” Proc. Offshore TechnologyConference, Houston, Texas, Paper No.OTC 1385, Vol. 1,pp. 555 - 570.El Ghamry, 0. A., (1963), “Wave forceson a dock,” Report no. HEL-9-1, Inst. of Engg.Research, Hydraulic Engg. Lab., Universityof California, Berkeley, California.Faltinsen 0., (1990), “Sea loads on shipsand offshore structures,” Cambridge OceanTechnology Series, Cambridge University Press, Cambridge,UK.French, J. A., (1969), “Wave uplift pressureson 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 induceduplift 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 forcircular 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), “Engineeringapproximation to nonlineardeep water waves,” Applied Ocean Research, Vol.8, No. 2,pp.76 - 88.Humar, J. L., (1990), “Dynamics ofstructures,” Prentice Hall, Englewood, New Jersey.Irajpanah, K., (1983), “Wave uplift forceon 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 AssociationofHydraulicResearch, Ottawa, Canada, Vol. C,pp.209 - 216.Verhagen, J. H. G., (1967), “The impact of flat plate ona water surface,” J. ShipResearch, SNAME, Vol. 11, No. 4,pp.211-223.von Kármán, T., (1929), “The impact on seaplane floats during landing,” NationalAdvisory CommitteeforAeronautics, 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, TechnicalReport No. R-546.42Appendix AStatic AnalysisThe load-cells placed at the two supports measure thesupport reactions due to the waveaction on the plate. The plate can be assumed to be analogousto a simply supported beamwith an over-hang from its supports and can be analyzed by the simple principlesof staticsin order to provide the unknown vertical force along with its lineof action at any instantfrom the recorded reactions at the two supports. The plate isassumed to be thin enough forany horizontal force to be neglected, and no measurementof the horizontal force has beencarried out.Referring to the Fig. Al, letFA(t) and FB(t) be the support reactions recorded at anytime t, F(t) the unknown force acting on the plate ands(t) its line of action measured fromthe leading edge of the plate. F(t) can be obtained by summing the support reactionsFA(t)andFB(t).F(t) = FA(t)+ FB(t) (A-i)The line of action s(t) can be obtained by taking momentsof the forces acting on theplate about the plate’s leading edge. This gives— 0.30FA(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 0.947001.000Flow 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.,570.£pgHbl?FFm‘rcrn(s)(m)LHL(N)pgHbpgHbeTTTTLLLL171.,3235.10.24-0.180.330.33-0.570.,160.2210.220.170.590.’Table4.1(contd.)Summaryoftestconditionsandprincipalresults.h=1.4cmRun#THHh£pgHbFFm‘r(s)(m)LHL(N)pgHbpgHbTTTTLLL321.,,24-,1780.30.21-‘riuJ2.iU(s)(m)LHL(N)pgHbpgHbtTTTTLLLL461.,230.500.,10.24-,090.00661.380.080.030.330.2388.50.15-,1594.80.16- 4.2 Computed values of the factor a in selected tests.RWave WaveFa 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<t<t2Fig. 2.2 Stages of wave propagation past a horizontal plate. (a)initial contact, t = to;(b) submergence of upwave portion of plate, t0<t <ti; (c) completesubmergence of plate, t1 <t < t; (d) submergence of downwaveportion ofplate, t2 < t <t3; (e) wave detaching from plate, t = t3.z(b):t0<t<t1x Thzhz(d):t2<t<t3(e): t=t30x52r(t)h0Cc’(t)F(t)F(t)(b)ttFig. 2.3 Sketch of ideal force components variation over a wave cycle. (a) freesurfaceelevation; (b) wetted length X; (c) proposed variation of AJat asa velocity c’;(d) force components: inertia force, Fai; added mass force,Fa2; drag force,Fd; buoyancy force, Fb; (d) total vertical force: actual force, predicted force.11x=(a)—----- 11x=—tttt tflt2t3(c)t(d)Fbt(e)531.0.75--00.50 -fiOIb)0.25 -__________________________— Sarpkaya and Isaacson (1981)-- Eq. 2.24(a)I I II I II IIII—075- -z0.50 -f2O/b)0.25 -___________________________— Sarpkayaandlsaacson(1981)—— Eq. 2.25(b)I I II I0.00 0.501.00 1.50 2.002.50 3.00Fig. 2.4 Variation of added massfunctions with plate aspectratio ?./b. (a) f1 (k/b);(b)f2(Ib).54[.0II I II(a)— cLl.O0.8 - -- c=3.O0.6pb20.00.5 1.0 1.5 2.0 2.5 3.0?Ib1.0 I III I— cz=05(b)0.8 --0.60MOi 1.0 i 10 2 3.0Fig. 2.5 Variation of dimensionless added mass with plate aspect ratio 2Jb. (a) Eq. 2.7;(b) Eq. 2.26.55CF(t)Fig. 2.6 Definition sketch of a single degree of freedom (SDOF) system.F(t)F00 tFig. 2.7 Sketch of an idealized force as a triangular pulse.561.6 . • • • • • •• • •• • • • . . • • • • • • • . ._________(a)1.4:-=O.O3• .O6F00.6-0.40.2 --0.0• • I i . . . I . . . . I . . . I . . . . I i . i . I . i . . I . • . .I . . . . I . i i14.0 . . . • . . • • • • • •• • • • . • • • • • • • • • • • • • •(b)12.010.0T8.0-T6.04.0. liii..., 1.0 2.0 3.0 4.0 5.0 6.07.0 8.0 9.0 10.0TrITFig. 2.8 Dynamic amplification factor andrelative rise-time as a function ofTr/Tnfor applied impulsive force withTd/Tr= 1. (Isaacson and Prasad, 1993).57Fig. 3.1 Photographs of the plate assembly. (a) side view; (b) top view indicatingthe details of supports and load-cell arrangements.(a)rrCDi-CDCDCDH0_fCD CD 0 CD I0059Wavegenerator60cml0.35m20 mCFig. 3.3 Sketch showing the wave flume and the test location.VAXstation 3200 computer forwave generation & data control(GEDAP software)D________ICRTCDASDAnalog to digitalconverter__________CRTCSIGDDigital to analogconverterLow pass filterWave generator4Amplifiersand low-passfilterI__4I II IWaveIPlate-loadprobes Icell__________assemblyFig. 3.4 Flow chart indicating the wave generation and the data control setup.300.0F (tO0.0F (N)—300.0200.0Load ccli A1(a)60‘Load cell ALoad cell BFI.00.0 100.0 200.0300.0Fig. 3.5 Responseof load-cells A and B to a step load of 117.7 N (12 kg) tested in air.(a) time histozy; (b) spectral density.I ILoad cell B—200.0o.ojiw—1.5 1.752.0 2.25 2.5t (sec)75.050.0S (N2!!!)25.00.G20.0S (N’,/Hz)10.00.f (Hz)400.0(b)500.061150.0 — •Load cell AF (N);.:150.0Load cell BF (N)0.0(a)—150.011.5 2.0 ‘2.6 3.0t (aec)300.0 •Load cell A200.0 -S (Ne/Hz)100.0 -0.0I I ILod cell B10.0S (N2/Hz)::.d1L0.0 25.0 50.0 76.0 100.0((Hz)Fig. 3.6 Response of load-cells A and B to a step load of 58.9 N (6 kg) tested for asubmerged condition. (a) time histoly; (b) spectral density.62100.075.50.0F (N)25.00.—25.0—50.0100.075.050.0F (N)25.00.—25.0—50.0Fig. 3.7 Time histories of the measured vertical force showing the effect of filtering(h = 1.4 cm, T = 2.02 see, H = 17.5 cm). (a) unfiltered force signal;(b) filtered force signal at 15 Hz cut-off.I (sec)63II I(a)]A1s (m)F (N)/ /.It’/- 41/ -.1--‘--/.:•Jo.25• ..‘:_50.0’iII • I I ••I•‘0.0100.0‘ i •• i ••• 0.75(b)Ft (eec)Fig. 3.8 Time histories of total vertical force and its line of action showing theeffect offiltering (h = 1.4 cm, T = 2.02 sec, H = 17.5 cm).(a) unfiltered force(b) filtered force with 15 Hz cut-off.i (m)—0.04—0.08I I I I.Ft (see)Fig. 3.9 Timehistories of (a) free surface elevation; (b) vertical force and its line ofaction. (h = 1.4cm, T = 2.02 see, H = 17.5 cm).64I I I I I////--\\c\\\\(a)-‘,,.....‘x2F (N)— 1.6 2.0 2.4 2.8 of surfaceelevations,110and r at2500 Hz.Wave records filtered at 10 Hz.Force time historiescompared and parameters .‘stored.Fig. 3.10 Flow chart indicating the sequence of analysis of the measured force and waveelevation time histories.Measurement of vertical forcesFAandFBat 2500 Hz.Force records filtered at 15 Hz.Total vertical force F(t) andit’s line of action s(t)computed.Impact parameters obtainedfrom time histories ofvertical force.Time t0,t1, t2 and t3 andwetted length (t) variationobtained.Force prediction based onanalytical models and a’sdetermined.HFig. 4.1 Timehistories of free surface elevation, wettedlength, vertical force measuredat thesupports, total vertical force andthe associated line of action during onewavecycle for h = 0.8 cm, T= 1.68 sec, H = 14.2 cm. (a) free surfaceelevation and wettedlength; (b) vertical force; (c) totalvertical force and line ofaction.0.30.1(m)0.0— (N)0.0— (N) (m)s(m)—30.01.0 1.5 2.0 2.5t (sec)0.03.067Fig. 4.2 Time histories of free surface elevation, wetted length, vertical force measuredat the supports, total vertical force and the associated line of action during onewave cyclefor h = 0.8 cm, T = 1.70 sec, H = 10.5 cm. (a) free surfaceelevation and wetted length; (b) verticalforce; (c) total vertical force and line ofaction.0.06i (in) 0.0—0.06F (N)F (N)—25.00.6X (m)— 2.02.5t (sec)‘(rn)0.0F (N)F (N)---::___t (see)68A (m)s(rn)Fig. 4.3 Timehistories of free surface elevation,wetted length, vertical force measuredat the supports,total vertical force and the associated lineof action during onewave cyclefor h = 0.8 cm, T = 1.70sec, H = 6.80 cm. (a) free surfaceelevation and wetted length;(b) vertical force; (c) total verticalforce and line ofaction.0.040.6—0.0440.020.00.—— 2.0 2.50.1—0.160.030——30.01.0• / 4.4 Time historiesof free surface elevation, wettedlength, vertical force measuredat the supports,total vertical force and theassociated line of action during onewave cyclefor h = 1.4 cm, T= 1.68 sec, H = 14.2 cm. (a) free surfaceelevation and wettedlength; (b) vertical force; (c) total verticalforce and line ofaction.69— —— 7’x,.A(a)i (rn)00F (N)0x (in)(in)F (N).421.4 1.6 2.2 2.6t (see)70F (N)Fig. 4.5 Time histories of free surface elevation, wettedlength, vertical forcemeasuredat the supports, total vertical force and theassociated line of actionduring onewave cycle for h = 1.4 cm, T =1.68 sec, H = 10.5 cm.(a) free surfaceelevation and wetted length; (b) verticalforce; (c) total verticalforce and line ofaction.0.017 (m)0.0—0.0860.030.00.0— (m) (N)30.00.— 1.8 2.22.6t (sec)710.120.0—0.120.6A (rn) (m)F (N)F (N)3.2Fig. 4.6 Time historiesof free surface elevation, wetted length, vertical force measuredat the supports, total vertical force and the associated line of action during onewave cycle for h = 0 cm, T = 2.02 sec, H = 17.5 cm. (a) free surface elevationand wetted length; (b) vertical force; (c) total vertical force and line of action.—40.01.2 1.62.0 2.4 2.8t (sec)0.072F (N)Fig. 4.7 Time historiesof free surface elevation, wetted length, verticalforce measuredat the supports, total verticalforce and the associated line of actionduring onewave cycle for h = 1.4cm, T = 2.02 sec, H = 17.5 cm.(a) free surfaceelevation and wetted length;(b) vertical force; (c) total vertical forceand line ofaction.0.12(in)0.0—0.12F (N)80.00.6A (m) (m)——40.01.2 1.6 2.02.4 2.8t (see)Fig. 4.8 Time historiesof free surface elevation, wettedlength, vertical force measuredat the supports,total vertical force and the associatedline of action during onewave cycle for h= 2.5 cm, T = 2.02 sec, H = 17.5cm. (a) free surfaceelevation and wettedlength; (b) vertical force; (c) total verticalforce and line ofaction.0.300.12ii (m)0.0—0.1280.040.0F (N)0.0— (N)0.0—40.01.2730.6X (m)s (m)6420.03.21.6 2.02.4 2.8t (sec)I I I II• I III IIII1 I I I I(a):zzE.A.. . ............••...-I I I I I • • I • i i I • • • II I I i • iI I I I I I III I I III I I III I I III I:(b)(c)I I I I .. .-0.1 0.0 0.1 0.2 0.30.4 0.5Fig. 4.9 Maximum upward force coefficientF1,/pgbHe as a function of relative plate clearanceh/H and relative plate lengthilL. (a) 0.0 16 <HIL <0.020; (b) 0.026< HIL <0.030; (c) 0.035 <HIL < 0.041.74FpgHbeFpgHbt?FpgHbe0. 4.10 Maximum downward force coefficient Fm/pgbH.e as a functionof relative plateclearance h/H and relative plate length £/L. (a) 0.0 16 <HIL <0.020; (b)0.026 <HIL <0.030; (c) 0.035 <H/L <0.041.FmpgHbeFmpgHbeFmpgHb.e0.• •(a): ...-L:..I I I I I • • • • I • i • • I • • • • I • • • • I i •. •I I I II• • • I I II• • • • I I III I;::i3:;—.I I I I I . i • • I • • • • I • i i • I • i • • I • i • •I I I III I I III I I III I I III I III I 1 I(c)- .......k -—--I I I I I i • • • I • • • • I . • I • • . . I • • • •-0.1 0.0 0.1 0.2 0.3 0.4 0.5h/H76I I I I I I I II• I I III I I III I I III I I(a).--. H..- -—-.——.——.—- —.I I I I I . . i . I . i . i I . . i . I . i • i I . i I II I I I I III I I III III I I III I I I(b):EZ..E+--.4--.I I I I I . I I I I • • • i I • • i • I I I I I I i I I II I I II‘ III I III I III I I I I I I(c)EEZ7.:4—•—•—•--.I I I I I • • • • I i i I • i • • I r I I I I • i-0.1 0.0 0.1 0.2 0.3 0.4 4.11 Non-dimensionalised duration of plate’s partialsubmergence T/T as a function ofrelative plate clearance h/H and relative plate lengthilL. (a) 0.0 16 <H/L < 0.020;(b) 0.026 <H/L <0.030; (c) 0.035 <HIL <0.041.77• • • I I II• • • • I I III I-•••••.AiI-•.............-. . • . . . . . . . . . . • .. . . .I I I III I I I I I I III I I III I I I,*(b)‘- /S..Th•......... .1...,I I I II• I I III I III I I I I(c)A*...%A..:-0.1 0.0 0.1 0.2 0.30.4 0.5Fig. 4.12 Non-dimensionalised duration of plate’s completesubmergenceT/Tas a function ofrelative plate clearance h/H and relativeplate length ilL. (a) 0.016 <H/L <0.020;(b) 0.026 <H/L < 0.030; (c) 0.035 <HIL < 0.041.TTTTTT0.— 0.2T0. I I III 1 1 III I 1I‘ ‘I‘II I I(a):: -\______.I I I I I I I I I • • • • I • • • • I • i • I I • •• •I I I I I I I III I I I I I(c).I I IIII I I • • • . . I I-0.1 0.0 0.1 0.20.3 0.4 0.5h/HFig. 4.13 Non-dimensionalised rise-timeTIT of peak vertical force as a function of relativeplate clearance b/H and relative plate length £/L. (a) 0.016< HJL <0.020; (b) 0.026<H/L < 0.030; (c) 0.035 <HJL <0.041., ..IuIIIIIII I III I I III I IIIII.(a)I I I I I . . . . I a • , I I ii i . I i • . I . . rII I I I I I I I I I I I I II III I •..............-7I I I I I . I I I I i . i . I . i ., I . i • . I i , i iI I I III III I I II•II I I I I(c):I I I I I r I I I I • i . I I ,i r I I i . I I I I I I I-0.1 0.0 0.1 0.20.3 0.4 0.5Fig. 4.14 Non-dimensionalised timeof occurrenceTm/T of maximum downward force as afunction of relative plate clearanceh/H and relative plate length £/L. (a)0.016 <HJL<0.020; (b) 0.026 <HJL<0.030; (c) 0.035 <H!L < 0.04 1.790.80.70.6TmT0.•....e..ek —— ——A:— -+——E, I I I I I . . . . I . . • I . . . i Ii , . I . , •I I I • I I I III I I I I I I III(b)N\A//(c)/\-0.1 0.0 0.1 0.20.3 0.4 0.580-pLL-pL0. 4.15 Non-dimensionalised lineof action sIL of maximum upward force as a functionofrelative plate clearance h/H and relative plate length£fL. (a) 0.016 < H/L <0.020;(b) 0.026 <H/L < 0.030; (c) 0.035 <HIL<0.041.I I I • I I I II• I I I I IpI I I I(a):.••k -A4——A- - -. 4——•. I I I I I i . i i I i I I I I I I I I I •• I I I • • I II I I IJI I I III I I III I I I I I I II• • I I(b)H-, . . . . . . . . I I I I I I I • I I II I I III I I I I I I I I I I I I I I III I I I(c)...•.....A- -Ak4—•—•--••I I I I I . . , i I . i I I I • I I I I i • • •I i i i iFig. 4.16 Non-dimensionalised line of actionSm/L of maximum downward force as a functionof relative plate clearance hJH and relative plate length£/L. (a) 0.0 16 <HJL <0.020;(b) 0.026 <HIL <0.030; (c) 0.035 <HIL < 0.041.810.200.15- 0.0 0.1 0.2h/H0.3 0.4 0.582I I I III I I III I IIIII I III I I I: (a)H::A....>”:. ,. .I I I III I I III I I III I I I I I I III I(1D)(c)-0.1 0.0 0.10.2 0.3 0.4 0.5LLL0. 4.17 Non-dimensionalised wettedlength fL corresponding to maximum upward forceas a function of relative plate clearance h/Fl andrelative plate length ilL. (a) 0.016 <H/L <0.020; (b) 0.026 <H/L <0.030; (c) 0.035 <HIL<0.041.83:• I I III I I IjI I I III I I I I IE— —.—4+k_I•4C., I I I I I . . . . I. . , . I . I II . . , ,I I I I I I I IJI I I I I I I III I I III I....- A •I I I I I . . i . I . i .i I . . i I i . . . I . . • .I I I III I I II• • I III I IIIII I I I(c)S..- .........•.•.....— .-0.1 0.0 0.10.2 0.3 0.40.5Fig. 4.18 Non-dimensionalisedwetted length?mfL corresponding to maximum downwardforce as a function ofrelative plate clearance h/H and relativeplate length £/L. (a)0.016 <HIL <0.020;(b) 0.026 <H/L <0.030; (c) 0.035<H/L < 4.19 Photographs of wave impact. (a) instantof complete submergence;(b) water drainage following submergence.85Fig. 4.20Video images indicating various stages of wave impact during onewave cycle. (a) partial submergence; (b) complete submergence;(c) wave recession; (d) water drainage.860.60.30.0H—0.3——0.1 -—0.2 -—0.3 -0.4 -0.30.2____________0.1FpgbH.e—0.1 -—0.2 -—0.3 —0.5 0.75 1.0 1.25 1.5 1.75Fig. 4.21 Variation of normalizedfree surface elevation and predicted vertical force basedon Eq. 2.23 for HIL = 0.04, £/L = 0.146 withtwo different plate clearances.(a) free surface elevation; (b) verticalforce for h/H = 0; (c) vertical force forh/H = 0.15.I • I • Ia1.Oa3.0I I —(b)• I • I • I • I____(c)a0.7I • I •0.68700—0— 4.22 Variation of normalized free surface elevation and predicted vertical force basedon Eq. 2.27 for HJL = 0.04, £/1. = 0.146 with two different plate clearances.(a) free surface elevation; (b) vertical force for h/H = 0; (c) vertical force forh/H = 0.15.0—0.1—0.2——0.1—0.2—0.30.5 0.75 1.0 1.25 1.5t/TFig. 4.23 Comparisonof vertical force predictedby analytical modeis with experimentalobservation forh = 0 cm, T = 2.02 sec, H = 17.5 cm. (a) freesurface elevationand the wettedlength; (b) vertical force.0.120.0880.60.3A(m)0.0‘1(m)F (N)—0. 18040.00.0—40.01.2 1.6 2.0 2.4 2.8t (see)3.289Fig. 4.24 Comparisonof vertical force predicted by analytical models with experimentalobservation for h = 0 cm, T = 1.68 sec. H = 14.2cm. (a) free surface elevationand the wetted length; (b) vertical force.0.6A(m)0.120.0—0.1280.40.00.3(m)F (N)0.00.0—40.01.2 1.6 2.0 2.4 2.8t (see)3.290Fig. 4.25 Comparison of vertical force predicted by analytical models with experimentalobservation for h = 1.4 cm, T = 2.02 sec, H = 17.5 cm. (a) free surfaceelevation and the wetted length; (b) vertical force..6A (m)0.12r(m) 0.—0.12F (N)0.30.0—40.01.2 1.6 2.0 2.4 2.8t (see)3.291Load cell A Load cell BFig. AlFree body diagram of the plate indicating support forces and the line ofaction of the vertical force.30cm 24.5 cmSVertical force


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