wave in deck loads on exposed jetties

(2007) 657–679www.elsevier.com/locate/coastaleng

Coastal Engineering 54

Wave-in-deck loads on exposed jetties

Giovanni Cuomo a,⁎, Matteo Tirindelli b, William Allsop c

a Department of Civil Engineering, University of Roma TRE, Via Vito Volterra 62, Roma, 00146, Italyb Coast & Harbor Engineering, 155 Montgomery Street, Suite 608, San Francisco, CA 94104, USA

c Maritime Structures, HR Wallingford, Howbery Park, Wallingford, OX10 8BA, UK and Department of Civil and Environmental Engineering,University of Southampton, UK

Received 8 February 2006; received in revised form 4 January 2007; accepted 12 January 2007Available online 12 April 2007

Abstract

This paper presents results from research on the hydraulic loadings of exposed (unsheltered) jetties (open pile piers with decks and beams). Thework presented here focuses on results from physical model tests on wave-induced loads on deck and beam elements of exposed jetties and similarstructures. These tests investigated the physics of the loading process, and provided new guidance on wave-in-deck loads to be used in design.Wave forces and pressures were measured on a 1:25 scale model of a jetty head with projecting elements. Structure geometry and wave conditionstested were selected after an extensive literature review (summarised in the paper) and consultation with the project steering group. Differentconfigurations were tested to separate 2-d and 3-d effects, and to identify the effects of inundation and of down-standing beams.

Results presented in this paper have been obtained by re-analysing wave loads using wavelet analysis to remove corruption from the dynamicresponses of the instrumentation. Both quasi-static and impulsive components of the loading were identified. Previous methods to predict waveloading on jetty elements (decks and beams) were tested against these new data and clear inconsistencies and gaps were recognised. Newdimensionless equations have been produced to evaluate wave forces on deck and beam elements of suspended deck structures. The results areconsistent with the physics of the loading process and reduce uncertainties in previous predictions.© 2007 Elsevier B.V. All rights reserved.

Keywords: Wave-in-deck loads; Jetty; Pier; Wave impacts; Wavelet

1. Introduction

1.1. Definitions of a “jetty”

In this paper a “jetty” has been taken to be an open structurewith deck and perhaps beams, supported on piles. The deck (andbeams) are suspended well clear of normal water levels, so areonly at risk of direct wave effects under infrequent combinationsof surge and wave condition. Such jetties may be quite long(perhaps 0.5–5 km), orientated approximately normal to theshoreline/bed contours, and carry pipes or conveyors to load/unload gas, liquid/bulk granular cargoes from vessels moored atthe jetty head. Similar structures include leisure and passengerpiers, mooring dolphins and some highway bridges. Selectedresults of this work may possibly be applied to large culverts,

⁎ Corresponding author. Fax: +39 06 55173469.E-mail addresses: [email protected] (G. Cuomo),

[email protected] (M. Tirindelli).

0378-3839/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.coastaleng.2007.01.010

temporary decks (including falsework) over water, and otherdecks on beams that may be hit by waves. These studies were nothowever intended to address wave loads on deep water (offshore)structures termed “rigs”, “jackets” or “platforms”, nor do thewaves covered include tsunamis or other waves of periodN25 s.

1.2. Background

Marine trade between many coastal nations has often reliedon jetties or piers to berth vessels for loading or discharge ofcargo/passengers. These facilities were traditionally constructedin areas where wave-induced loads are relatively small,naturally sheltered locations and/or locations protected bybreakwaters. In the last 15–20 years there has been an increaseddemand for liquid natural and petroleum gas terminals (LNGand LPG), which require sheltered berths in deep water for largevessels, but may not need shelter to the approach trestlescarrying the delivery lines, see McConnell et al. (2003, 2004).This has led to some jetties being constructed with limited or no

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http://dx.doi.org/10.1016/j.coastaleng.2007.01.010

Notation

A area of the element exposed to wave actiona, b experimental coefficientbl length of structural elementbw width of structural elementbt thickness of structural elementc wave celerityC dampingcl clearance (i.e. vertical distance between an element and swl)Cd drag coefficientd water depthE errorF force (generic)F c force (corrupted component)F f force (filtered)F s force (smoothed)F tot force (response)Fmax maximum value reached by the signal within each eventFqs 1/250 quasi-static component of the force (at 1/250 level)Fqs+1/250 maximum value of the quasi-static component of the signal (at 1/250 level)Fqs−1/250 minimum value of the quasi-static component of the signal (at 1/250 level)F⁎ dimensionless force (generic)F1/250 force at 1/250 exceedance levelFD configuration flat deckg gravitational acceleration (=9.81 m/s2)G Fourier transform of wh vertical wetted lengthhd time derivative of hHmax max wave height during a stormHs significant wave heightK stiffnessL wave lengthL0 deepwater wave lengthl horizontal wetted lengthld time derivative of lM massNz number of wave within a storm eventNP configuration without side panelsNt number of testsP configuration with side panelsPqs1/250 quasi-static component of the pressure (at 1/250 level)s scale dilatation parameterse standard error of the estimatesm wave steepness (=Hs /L0) for T=Tmt timeT wave period (generic)Tn n-th resolved equivalent periodTn, min minimum period corrupted by the dynamics of measurement instrumentTn, max maximum period corrupted by the dynamics of measurement instrumentTsmooth cut-off period of low-pass filterT0 natural period of resonance of the structureTm mean wave periodtr rise time of the force signalu velocity vectoru˙ acceleration vector

658 G. Cuomo et al. / Coastal Engineering 54 (2007) 657–679

ux horizontal velocityuy vertical velocityx displacementx· velocityx·· accelerationyi measured load (generic)ŷi predicted load (generic)WT wavelet transform coefficientΨ mother waveletρw water densityη wave surface elevationη˙ time derivative of ηη·· time derivative of η˙ηmax max wave surface elevationτ translation parameter

659G. Cuomo et al. / Coastal Engineering 54 (2007) 657–679

Fig. 1. Wave-in-deck loads on an idealised section of a jetty platform supportedby piles.

breakwater protection, increasing wave exposure and potentialrisks of high wave loads on the structure. The higher probabilityof wave loading on jetties may be increased by processes suchas subsidence, decreasing the clearance (i.e. vertical distancebetween still water level (swl) and underside of the jetty deckstructural elements), as seen for the Ekofisk platform complex(Broughton and Horn, 1987).

Similar loadings can arisewhere other suspended structures aretoo close to the water surface. Key examples are temporary worksused in construction or refurbishment around harbour structures,or some low-lying transport bridges over coastal waterways.

Whilst a number of prediction methods have been developedfor wave-in-deck loads, gaps and weaknesses in availablemodels stimulated the “Exposed Jetties” research supported inthe UK by Department of Trade and Industry under PII Project39/5/130 cc2035 (see Tirindelli et al., 2002). Within this project,a series of 2-dimensional physical model tests measured waveloads on deck and beam elements, see Tirindelli et al. (2002)and McConnell et al. (2003, 2004). These measurements wereanalysed to explore the process of wave loading, with theobjective of developing improved predictions. Results from themain project were summarised by McConnell et al. (2003,2004) and Cuomo et al. (2004). Extending, but after the formalend of the Exposed Jetties project, a new method for theanalysis of non-stationary time-history loads was developed byCuomo et al. (2003) and Cuomo (2005), based on wavelettransform. Further analysis of the original data allowed a deeperunderstanding of the loading process and a new interpretation ofthe measured data.

1.3. Wave load definitions

Hydraulic loads applied by waves to the deck or otherprojecting elements (beams, fenders) can be defined as “wave-in-deck loads”. Those covered in this paper can be summarised as:

– uplift loads on decks;– uplift loads on beams or other projecting elements;– downward loads on decks (inundation and suction);

– horizontal loads (both seaward and shoreward) on beams orother projecting elements.

A sketch of wave-in-deck loads acting on a jetty is shown inFig. 1. The nature, occurrence and magnitude of these waveloadings vary significantly for different structures and waveconditions. Horizontal elements such as deck slabs may besubject to large vertical forces upward or downward (especiallyunder conditions that inundate the deck). Vertically faced ele-ments like beams and fenders can experience significant forcesboth horizontally and vertically (if of significant thickness).

2. Previous work

In the last fifty years, many prediction methods have beendeveloped to evaluate wave-in-deck loads on jetty structures. Aliterature review of the most important among these works isdescribed in the following.

El-Ghamry (1965) and Wang (1970) first performed physicalmodel tests to investigate wave loads on horizontal decks subjectto breaking and non-breaking wave attacks. The authors foundsubstantial similarity between the mechanisms of wave impact onhorizontal platforms and vertical barriers. Uplift pressures are

Fig. 2. Experimental setup: overall view.


characterised by an initial peak pressure of considerablemagnitude but of short duration, followed by a slowly-varyinguplift pressure of less magnitude but of considerable duration,typically first positive, then negative. For regular progressivewaves, Wang's method for prediction of uplift pressure P reads:

P ¼ k � H � qw � g2

� tanh 2 � k � dL

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−

4 � c2lH 2

rð1Þ

where ρw is the water density, g the acceleration due to gravity,H is the wave height at the structure, cl the deck clearance abovethe swl, d the water depth and L the incoming wave length.

Broughton and Horn (1987) proposed a simple approach forthe prediction of wave loads based on the assumption that theforce impulse is equal to the change in momentum when thewave crest hits the leading edge of the deck. This leads to thefollowing expressions for the vertical (FV) and horizontal (FH)force per unit length:

FV ¼ k2� qw � c � uy � bw � l ð2Þ

FH ¼ qw � c � ux � bw � h ð3Þ

where h and l are respectively the height and length of wavecrest in contact with the deck, bw the cellar deck breadth, c thewave celerity (according to Stoke's 5th order theory), and uyand ux respectively, the vertical and horizontal velocity of thewater particles.

Broughton and Horn performed physical model tests in awave basin (scale 1:50) on three different platform configura-tions. Observed force time histories were characterised bypositive (uplift) forces, followed by downward forces of thesame magnitude order of the uplift. Due to the low sampling rate(20 Hz) used during the experiments, no information wasretained on impulsive short-duration peaks.

Shih and Anastasiou (1992) and Toumazis et al. (1989)analysed wave-induced forces and pressures on horizontalplatform decks at small and very small scales. Forces weremeasured by means of strain gauges. Pressures were logged at asample rate of 500 Hz. A high speed video camera (up to 1000frames/s) was used to capture snapshots of the interactionbetween waves and the suspended structure.

Impact pressure peaks on horizontal plates were found toincrease with decreasing clearance and increasing wave height.

The following empirical relations for peak pressures PV weresuggested:

PV ¼ 1:8� 7:6ð Þ � qw � g � H ð4Þ
PV ¼ 4:0� 8:0ð Þ � qw � g � Hs ð5Þrespectively for regular and random waves (Hs = significantwave height).
The authors confirmed that air entrainment at impact affectsimpulsive pressures, generating scatter in data. Froude scalingdoes not account for difference in air compressibility at modeland prototype scales, potentially leading to distortion of mea-surement of impact maxima and rise times. Recorded slowly-varying positive pressures (Pqs+) were found to be always lowerthan the hydrostatic head. The following expression was sug-gested for both regular and random waves:

Pqsþ ¼ 0:65 � qw � g � g−clð Þ ð6Þwith η = wave crest elevation.

“Ventilated shocks” (Lundgren, 1969) were observed on thevertical plate, and the following relation was suggested for theevaluation of horizontal impact pressures PH,max:

PH;max¼ a � qw � c2 ð7Þwhere α is an empirical coefficient. For waves impacting beforethe breaking condition was reached, the authors indicate that αis always less than 1.3, for more violent impacts a value of 4.1has to be used. For deep water spilling breakers impinging on avertical suspended plate, Kjeldsen and Myrhaug (1979)suggested values of α between 1 and 2. In other test conditions,Chan and Melville (1988) measured values of α between 3 and10, whilst field measurements on seawalls by Blackmore andHewson (1984) suggest α to vary between 0.5 and 4.

Suchithra and Koola (1995) performed model tests to mea-sure vertical forces on a horizontal slab. They tested differentconfigurations (with and without stiffeners) to investigate theinfluence of down-stand beams on wave-induced loading. Theimpact vertical force on the slam was expressed in terms of theslamming coefficient Cs, as follows:

FS ¼ 12� Cs � qw � A � u2y ð8Þ

where A is the area of contact. Cs was found to vary between 2.5and 10.2 and to mainly depend on the frequency of the

Fig. 3. Model test device. a) Down-standing frame of cross and longitudinal beams with testing elements connected to force transducers housed into a longitudinalbeam (plan view); b) supporting structure with piles and tubular frame.


incoming wave and on the clearance of the element. Theparameter Cns=Cs ·d /L (average value=1.7) was found to beindependent from the wave frequency, and to increase withdecreasing values of clearance. The presence of longitudinalbeams was found to slightly increase applied forces, whilsttransversal beams, causing air pockets to be entrapped betweenthe wave and the structural boundaries, reduce slam forces.

Kaplan and Silbert (1976), Kaplan (1979, 1992), Kaplan et al.(1995) investigated wave forces on flat decks and horizontalbeams on offshore platforms. Moving from the original work byMorison et al. (1950), they developed a semi-analytical model forthe evaluation of wave-in-deck time-history loads on both verticaland horizontal members. According to Kaplan, as the wavetravels along the platform, it transfers its energy to the super-structure and the variation in time of wave-in-deck loads resultsfrom the combination of an inertia force (resulting from thevariation of the momentum in terms of structural acceleration andadded mass), and a drag force. Bolt (1999) reviewed the state-of-the-art for wave-in-deck time-history load calculations and foundthe model by Kaplan to provide the most sophisticated repre-sentation of wave-in-deck loads. The model also accounts forrelative location of element along the structure, providing themost appropriate tool for detailed comparisons with the new data.

Kaplan's model works through the following equations(Eq. (9) for vertical forces FV, Eq. (10) for horizontal forces FH):

FV ¼ qw � k8� bw � l2

1þ lbw

� �2� �1=2 � ̈gþ qw � k

4� bw � l � :l

�1þ 1

2 � lbw

� �2

1þ lbw

� �2� �3=2 � �gþ qw

2� bw � l � Cd � �g � j�gj þ qw

� g � g−clð Þ � bw � bl ð9Þ

Fig. 4. Model structure in the absorbing flume during a test. The testingelements, formed of metallic elements are visible as well as two of the waveprobes (one before and one after the model) used to monitor the wave field.

FH ¼ 2

k� qw � h2 � :ux þ qw

2� bt � Cd � ux � juxj þ 4

k� qw � h �

:h � ux

� �� bw

ð10Þ

where l (horizontal wetted length) and l ˙ are determined from therelative degree of wetting of the flat deck underside on whichloading occurs, bl is deck length, Cd is the drag coefficient, h(vertical wetted length) and h˙ are determined from the relativedegree of wetting of the vertical face of the beam where loadingoccurs, bt is the thickness of the deck.

Further developments of Kaplan's model have beendiscussed, among the others, by Isaacson and Bhat (1994),Isaacson et al. (1994) and Cuomo (2005). An alternative semi-empirical method, also accounting for the dynamic amplifica-tion of slamming due to dynamic response of structural ele-ments has been proposed by Bea et al. (2001).

3. Experimental setup

3.1. Model design

Wave flume tests were carried out in a wave absorbing flumeat HRWallingford, UK. The flume was 40 m long, 1.5 m wide,with a maximum water depth of 1.2 m at the paddle. Waveswere generated by a piston-type paddle at one end of the flume

Fig. 5. Structural configurations tested: from left to right, no panels (NP), panels (P) and flat deck (FD).


and dissipated at the other end by a 1:5 slope beach coveredwith absorbing matting and rocks to minimize wave reflection.Behind the test structure, the absorbing beach extended inheight±0.15 m around swl, lying on a steeper 1:1.5 beach, seeFig. 2. A 1:25 Froude scale model of a jetty structure was boltedto the floor of the wave flume and instrumented to providedirect measurements of wave loading. The configurations of thetest structure and measurement elements were determined by ananalysis of existing jetties in UK, Oman, Kuwait, India andCaribbean. At 1:25 scale, model wave heights were above thoseused in studies by Allsop et al. (1995) and Howarth et al. (1996)to identify scale effects on wave pressures, where measurementsof impulsive loads were found to be slightly conservative.During the study design, it was also noted that the chosen scalealso simulates conditions at scales around 1:50–1:75 for“offshore platforms”, although the study had not been intendedto address such structures per se.

Fig. 6. Matrix of test conditions. Circles represent wave conditions (twodifferent water depths d ); solid lines are breaking limit and stroke limit; dashedand dotted lines represent three values of wave steepness sm, respectively twostorm and one swell conditions.

3.2. Model structure

The test structure consisted of a jetty deck (110 cm long,100 cm wide, 2 cm thick), on a down-standing frame of cross(100×6×6 cm3) and transverse (110×10×10 cm3) beams. Thesuspended structure was made of wood and was supported bysix steel piles (5 cm diameter) mounted on a tubular base framebolted to the floor, see Figs. 3 and 4.

Two beams (6.0×19.5×7.5 cm3) and two decks (19.5×19.5×2.0 cm3) of the jetty superstructure were replaced bymetal elements (aluminium and steel) in two different positionsalong the jetty (first and third spans, hereinafter respectivelynamed “external” and “internal”). Each of these elements wereconnected to a force transducer, housed in the wooden frame ofdown-standing beams, see Fig. 3. Each force transducer wasable to record forces in two normal directions (vertical andhorizontal). Two pressure transducers in the seaward beam ofthe platform at two opposite sides of the longitudinal axis ofthe jetty measured horizontal pressures. Force and pressure

Fig. 7. Idealised force time history superimposed on a typical force signalrecorded by a horizontal element.

Fig. 8. Horizontal time-history loads on vertical elements. From top to bottom:force on external beam, pressure on transducers A (solid line) and B (circles),force on internal beam.

Fig. 9. Force time history on the whole set of monitored elements during sameloading event. From top to bottom: horizontal force on external beam, verticalforce on external beam, vertical force on external deck, horizontal force oninternal beam, vertical force on internal beam, vertical force on internal deck.


transducers were logged at 200 Hz. The incoming wave fieldwas monitored by three wave gauges along the wave flume.One gauge was located well away from the structure in order torepresent the generated wave field. The remaining two gaugesmeasured waves before and after the model structure, to de-scribe the wave field at the structure and to provide a descriptionof the wave energy dissipation through the interaction with thejetty.

Three different configurations are shown in Fig. 5. Theoriginal configuration (no panels, NP) had the supporting beamsfacing downward, and no side plates to limit 3-dimensionaleffects. A second configuration (panels, P) used large sidepanels to limit 3-dimensional effects due to lateral inundation ofthe deck. The last configuration (flat deck, FD) inverted thedeck and beams to investigate wave loads on the (now) flatunderside.

Water depths at the jetty were either d=0.60 m or d=0.75 m.Four different values of static water clearance cl were tested:cl =0.01 m, 0.06 m, 0.11 m, 0.16 m, achieved by raising orlowering the deck assembly over the piles by means of spacers.Random sea states were defined by scaling typical wave con-

ditions for exposed jetty locations using JONSWAP spectra(γ=3.3). Chosen wave conditions were based on three wavesteepnesses (sm=0.065, 0.040, 0.010), which represented twostorm and one swell condition. The latter also representedwaves for a jetty sheltered by a breakwater. Wave conditions(model units) were in the range: Hs=0.10 m to 0.22 m; Tm=1 sto 3 s. The matrix of test conditions is represented in Fig. 6.

4. The loading process

4.1. General

Wave-induced vertical forces on horizontal decks or plat-forms may be considered in three phases in Fig. 7. At theinstance of contact between the wave crest and the element, theslam or impulsive force may be large in magnitude and short induration. This is followed by a longer duration (pulsating)positive force and then by a long-duration negative force(especially if the deck is frequently inundated).


The first contact between the water and the element causesan abrupt transfer of wave momentum from the water to thestructure, generating the impact force. Such high intensityforces acting on limited areas, even over a short time, may causesevere local damage, local yielding and fatigue failure.Impulsive loads vary substantially in both magnitude andduration even under nominally identical conditions, confirmingprevious observations from research on wave impacts. Acomprehensive review is given in Cuomo (2005).

As the wave propagates along the underside of a deck, jets ofwater may shoot out sideways as the contact area moves alongthe deck (unless otherwise restrained). These lateral jetsgenerally disappear as the free surface rises above soffit level.A difference between water levels under the deck and thatalongside the structure gives rise to the pulsating or quasi-staticpositive force. The magnitude of this force is consistently lowerthan any initial impact, but its duration is of order 0.25·Tm.

Finally, the wave surface falls below soffit level and movesinward below the deck, reducing the contact area with the wave(referred to as “wetted length”). A quasi-static negative force(suction) may then act on the deck. This may be substantiallyincreased when the wave inundates the deck, adding the weightof green water above the deck, sometimes leading to thedownward (negative) force reaching the same order of intensityas the quasi-static uplift (positive) force.

Horizontal loads on beam elements often exhibit differentcharacteristics from vertical loads. The magnitude of the first

Fig. 10. Vertical time-history load on deck element: recorded time history (left

impact load on an external beam (i.e. vertical element at the edgeof the jetty) is generally lower than the corresponding verticalimpact. Example time histories are mainly characterised byquasi-static (pulsating) components. For waves underneath anyplatform formed by beams and deck elements, interactions withthe protruding elements are complex, and wave crests and airmay be trapped between beam and deck. This may result in highhorizontal impulsive loads on the seaward face of internalelements and noticeable horizontal forces acting seaward on theshoreward face of the vertical elements. Example histories ofhorizontal forces and pressures on vertical elements are shown inFig. 8. Momentum transfer to the face of the external beam ismore gradual, giving quasi-static or pulsating forces (Fig. 8a)confirmed by time histories recorded by pressure transducers Aand B (Fig. 8b). High frequency oscillations in Fig. 8a are due tothe dynamic response of the instrument in the front beam. Waveinteraction with internal elements is more complicated, in somecircumstances resulting in high intensity impact loads (seeFig. 8c recorded as the wave slams against the seaward face ofthe internal beam).

4.2. Observation and parameterisation of time history ofwave-in-deck loads

Force histories recorded as the wave travels along the teststructure are plotted in Fig. 9. From top to bottom, the waveinitially hits the seaward beam (horizontal, 9a and vertical, 9b)

— model units) and wavelet transform in the time-period domain (right).

Fig. 11. Filtering out corruption of signal due to dynamic response of measuringinstruments. From top to bottom: time-history load on deck element (solid line)and reconstructed signal by mean of inverse wavelet transform (circles), inversewavelet transform using only energy components corresponding to resonanceperiod of measuring instrument, cleaned signal after filtering.


and external deck (vertical, 9c), then moves shoreward to theinternal beam (horizontal, 9d and vertical, 9e) and deck(vertical, 9f).

From the signals in Fig. 9, it is possible to derive the fol-lowing general observations:

– intensities of loads do not necessarily reduce as the wavetravels under the structure;

– internal elements are subjected to wave impacts as are theexternal elements;

Fig. 12. Finite element model of the test structure (right) and deck element (left),

– trapping wave crests underneath a soffit between down-standing beams may result in local amplification of appliedloads;

– phase differences between positive and negative loads mayincrease as the wave travels along the structure;

– rise time of wave loads (tr, see Fig. 7) may be comparable tothe characteristic periods of oscillation of structural ele-ments, therefore the resulting loading process dependsstrongly on the dynamic response of the structure.

Observing force histories during the experiments, largevariations in both magnitude and shape of the signal werenoticed, even for similar test conditions. Despite this variability,an idealised time history has been developed to represent thegeneral shape drawn by the force signal during the loading. Thesuggested idealised time history, superimposed on a measuredone for the external deck element in Fig. 7, consists of a short-duration triangular pulse (linear increase from zero to its peakvalue), followed by the quasi-static component. The proposedmodel is characterised by:

– Fmax = maximum value reached by the signal within eachevent — considered as representative of a typical impactforce, see Section 4.4;

– tr = rise time of the force signal;– Fqs+ = maximum value of the quasi-static component of thesignal within each event;

– Fqs− = minimum value of the quasi-static component of thesignal within each event.

Within this project, wave loads (pulsating and impulsive)have been parameterised at 1/250 level (average of the top 1/250values). This choice was made to reduce dependence on highlyvariable extreme loads; and to maintain consistency with thegeneral use of 1/250 values for wave loads on walls, see Goda(2000), Allsop (2000) and Oumeraci et al. (2001). In assessingthe exceedance level, the number of measured loads was takenas a proportion of the number of incoming zero-crossing waves,themselves calculated from the test duration and mean wave

shape deformed according to the first mode of oscillation (vertical direction).

Fig. 13. Filtering out long-duration components from signal recorded bymeasurement devices.


period, Tm. For tests of 1000 waves, the 1/250 value wastherefore evaluated by simply averaging the top 4 loads.

4.3. Impacts and quasi-static loads

Distinguishing between impulsive and quasi-static waveloads is not straightforward. Although much research has been

Fig. 14. Comparison of vertical quasi-static forces on

undertaken on short-duration wave loads on some classes ofmaritime structures, clear definitions of short-duration loads aswell as of quasi-static loads are still missing. As the loadingprocess depends on both incoming wave field and the dynamicresponse of the structure, the dynamic characteristics of thestructure should be taken into account when distinguishing loadtypes. In this paper, “impacts” or “short-duration” loads aredefined as those that act on the structure for durations shorter orcomparable with the resonance period of the structure.Conversely, “quasi-static” (also called slowly-varying or pul-sating) loads are those that act on the structure for longer. Infigures:

trV2 � T0N2 � T0

impactsquasi�static loads

where tr is the load rise time and T0 is the resonant period for themode corresponding to the applied load.

In analysing measurements in the model, dynamic char-acteristics of the instrument and of the jetty model must be takeninto account. In particular, defining any value of an impulsivewave load to be used later in design (either as a staticallyequivalent load for feasibility studies; or as time-history loadsfor dynamic analysis of more complex structures) requiresfiltering out corruptions from the dynamic response of themodel setup (see Cuomo et al., 2003 and Cuomo, 2005).

deck elements with existing prediction methods.


4.4. Wavelet analysis of time-history loads

Non-stationary signals are frequently encountered in wind,ocean and earthquake engineering. Force signals from theseexperiments show occasional high peaks marking an otherwiserelatively undisturbed signal. A standard Fourier analysis willnot represent correctly this signal because sine or cosinefunctions in the Fourier series are periodic. Spectral analysismethods are not therefore able to describe transient features ofthese short-duration phenomena.

The need to preserve time dependence and to describeevolving spectral characteristics of non-stationary processesrequires tools which allow localization of energy content in bothtime and frequency domains. Wavelet transformations retaintransient signal characteristics beyond the capabilities of Fouriermethods. Time and frequency analysis by wavelet transformsprovides insight into transient signals through time–frequencymaps of the time variant spectra missed by traditional ap-proaches. Analysis in this paper has used the Morlet wavelet,widely used to describe processes related to ocean waves(Massel, 2001).

An extract from the force time history for a jetty deckelement is shown in Fig. 10 together with its wavelet transformin the time/period domain. A peak can be recognised about0.25 s after the beginning of the event, with energy in almost allthe resolved scales/periods. Some energy can be recognised in

Fig. 15. Comparison of quasi-static forces

periods between 0.03 s and 0.07 s on the left hand side of theamplitude contour graph.

The oscillating signal after the impact in Fig. 11 (top panel)suggests that the measurement element was responding dynam-ically to the wave loading. De-noising of the recorded signal iseasily accomplished with the inverse wavelet transform by elimi-nating or reducing coefficients for components that are related tolow energy processes or noise. Editing components affected bythe dynamic response of instrumentation is possible, but moredifficult, as it requires identifying dynamic characteristics of thetest elements. This was assessed here by modelling the dynamicresponse of the instrument using finite element models (Fig. 12).The period of oscillation of the model structure (left hand side ofFig. 12) was found to be far enough from the characteristic periodsof the loading (the slowest mode in the vertical direction hasperiod equal to 0.005 s) and therefore not to significantly affect themeasurements. The model of the deck element is shown on theright hand side of Fig. 12, superimposed on the deformed shapecorresponding to its first mode of vertical oscillation,corresponding to a period of resonance of approximately 0.05 s.

The equation of motion for a dynamic system can be gen-eralised as:

M tð Þ � ̈x tð Þ þ C tð Þ � �x tð Þ þ K tð Þ � x tð Þ ¼ F tð Þ ð11Þwhere M, C and K represent respectively the mass, thedamping and the stiffness of the system (in our case the

with prediction by Kaplan's model.


experimental setup), including added mass and hydrodynamicdamping.

Reading the displacement from the transducer (x) will lead tothe measurement of:

x tð Þ ¼ F tð Þ− M tð Þ � ̈x tð Þ þ C tð Þ � �x tð Þ½ �f g=K tð Þ: ð12Þ

Hanssen and Tørum (1999) filtered out corruptions due todynamic response of measurement instruments by numericallyevaluating the terms in squared brackets in Eq. (12), that is,solving Duhamel's integral in time for a single degree offreedom system. For our type of structure, this procedure is notstraightforward, since the level of inundation of the structuralelements changes with time, together with the added mass andthe hydrodynamic damping terms in Eqs. (11) and (12).

For this reason, an alternative procedure for filtering outcorruption from dynamic response of measurement instrumentshas been developed based on the wavelet transform of recordedsignals. The procedure is described in the following.

Let the wavelet transform of the signal F(t) be given byEmery and Thomson (2001):

WT s;sð Þ ¼Z l

−lF tð Þ � ws;b t;s;sð Þ � ds: ð13Þ

In Eq. (13), the wavelet coefficient: WT(τ, s) represents thecorrelation between the wavelet and a localized section of the

Fig. 16. Comparison of vertical impact forces on de

signal, τ is the translation parameter (localizing the positionof the wavelet in time), s is the scale dilation parameter(determining the width of the wavelet) and ψ(t) is the “motherwavelet” defined as:

ws;b t;s;sð Þ ¼ 1ffiffis

p � w t−ss

� �: ð14Þ

In the analysis, we adopt the Morlet wavelet, defined as:

ws;b t;s;sð Þ ¼ 1ffiffis

p � exp −12

t−ss

� �2� �

� exp i2ks� t−sð Þ

� �: ð15Þ

From Eq. (15), it is possible to appreciate the similaritybetween the scale (s) and the more familiar Fourier period (T ).For this reason, we will use the expression “equivalent periods”meaning the “scales” resolved by the analysis.

We can also define the following:

– a range of (equivalent) periods [Tn,minbTnbTn,max] affectedby the n-th mode of vibration having period Tn;

– the cut-off (equivalent) period Tsmooth for low-pass filter.

Assuming linear behaviour for the measurement instrumentbeing analysed (generally valid for instruments within theirprincipal range), it is possible to obtain a “filtered” signal bysumming contributions from every resolved frequency, but

ck elements with existing prediction methods.

Fig. 17. Comparison of impact forces with prediction by Kaplan's model.


removing those affected by the resonance. The followingsignals can thus be obtained from the original records:

– the reconstructed signal F tot(t) or “response”, that is theinverse transform using all resolved (equivalent) periods Ti;

F tð Þ ¼ 1C

Z l

−l

Z l

−l

WT t;s;bð Þs2

� ws;b t;s;sð Þ � ds � ds ð16Þ

where C −1 ¼ Rl−l G xð Þ=x � dx and G(ω) is the Fourier

transform of ψ.– the filtered signal Ff(t), that is the sum of components fromall resolved (equivalent) periods Ti but those affected by thedynamics of measurement instrument;

– the smoothed signalF s(t), that is the sum of components having(equivalent) periods larger than Tsmooth (this can be easily seenas a low-pass filter). The choice of Tsmooth depends on both thedynamic characteristics of the measurement instrument and thetime-history loads; an initial estimate for Tsmooth is 2·T0.

The suitability of Morlet wavelet as a base for the transfor-mation of signals recorded during physical model tests is shownin Fig. 11, with the original time-series (top) superimposed to itsinverse transform derived using Eq. (16) and extending theintegral operator to cover respectively the whole range ofresolved (equivalent) periods. The filtered signal Ff(t), obtainedby integrating Eq. (16) over all resolved periods but those in therange [Tn,minbTnbTn,max] is also shown in the bottom panel ofFig. 11. In this case, filtering out the corrupted component doesnot affect significantly the impact load magnitude or duration.

For the sake of completeness, Fig. 11 also shows the part ofthe signal affected by the dynamics of the measurementinstrument F c(t)=F tot(t)−F f(t) (central panel), confirmingthat the dynamic response of the measurement element hascorrupted the recorded signal.

It is worth noticing that since the maximum duration ofwave-in-deck loads is comparable with the incident waveperiod, the whole loading process develop within a relativelylimited range of (short) periods, and thus the cone of influenceof the wavelet transform (Torrence and Compo, 1998) isextremely narrow and almost no information is lost.

To minimize corruption due to dynamic response of mea-surement instruments on the extraction of meaningful para-meters defined in Section 4.2 from recorded signals, thefollowing procedure has been adopted:

– Fqs+ has been taken as the maximum value reached by thesmoothed signal within each loading event;

– Fqs− has been taken as the minimum value reached by thesmoothed signal within each loading event;

– Fmax has been taken as: max {F tot(tmax); F f(tmax)}, themaximum between the reconstructed (F tot) and the filtered(Ff) signals at time t= tmax (time at which the filtered signalreaches the maximum value within each loading event).

Values of Fmax extracted by means of the aforementionedtechnique might still be affected by the dynamics of the

measurement instruments as Fmax might be larger then the realmaximum force (max[F(t)]) acting on the measurement instru-ment during the loading. Indeed, F tot(tmax) in general differsfrom Ff(tmax), the difference between the two being givenby the dynamic response of the measurement instrument andthe following relation is valid: min{Ftot(tmax); Ff(tmax)}≤max[F(t)]≤max {Ftot(tmax); F

f(tmax)}.Nevertheless, the choice of the aforementioned method for

the extraction of Fmax has been made based on the followingfacts:

1. Ff might underestimate the maximum impact force, as theenergetic contribution to the real force acting on the fre-quencies affected by dynamic of the resonance instruments isindeed neglected in the filtering process;

2. for pulse-shapes similar to those observed in time-historyloads recorded during physical model tests, the maximumamplification due to dynamic response of measurementinstruments is less than 1.5 (Cuomo, 2005). In such cases,the maximum displacement occurs at time tN tmax (Fig. 11)and thus extracting Fmax at time t= tmax significantlycompensates for such potential over-estimation.

In our experience, the use of the aforementioned techniquereduces corruption due to dynamics of measurement instru-ments more effectively than using Eq. (12) with potentiallyerroneous estimation of mass and damping coefficients. Fur-thermore, when a difference exists between the real load and the


value extracted by the analysis program, the latter is always(slightly) larger than the former.

Applying a high-pass filter to the inverse wavelet transformalso eliminates long-duration distortions like drift or non-lineardisplacement of the instruments, without affecting the descrip-tion of the main processes (Fig. 13). Further details on thisanalysis, as well as on the dynamics of the experimental setupused in the physical model tests, can be found in Cuomo (2005).

5. Comparison with existing prediction methods

Tirindelli (2004) and Cuomo (2005) reviewed predictionmethods for wave-induced forces on beam and deck elements ofexposed jetties and offshore platforms, including most recentworks. In this section predictions by selected methods are

Fig. 18. Relative importance of Hs on wave-in-deck loads. From left to right: horiz(uplift) forces on decks. From top to bottom: quasi-static forces (NP), quasi-static fowith d=0.75 m.

compared with wave-in-deck loads measured during physicalmodel tests.

Quasi-static vertical (upwards) loads (at exceedance levelF1/250) measured during these new tests are compared withpredictions by some of the reviewed methods of Section 2 inFig. 14 for both external (○) and internal (⁎) deck elements.The scatter is large over the range of measurements for mostmethods used in the comparison.

Quasi-static vertical loads on external and internal elementsare compared with predictions by Kaplan in Fig. 15. Accordingto what was suggested by Kaplan et al. (1995) wave kinematicsin Eqs. (9) and (10) has been evaluated using linear theory butassuming the wave amplitude to be equal to ηmax; drag coef-ficient Cd in Eqs. (9) and (10) has been taken respectively equalto 2 and 1. The scatter is relatively large, with most predicted

ontal (seaward) forces on beams, vertical (uplift) forces on beams and verticalrces (P), impacts (NP) and impacts (P). All data refer to experiments carried out

Fig. 20. Comparison between non-dimensional quasi-static horizontal (shoreward)forces and pressures on external beam, solid line has an equation: P=F /A.


values falling within the “unsafe” regions (upper left corner forpositive loads – on the left hand side – and lower right corner fornegative loads – on the right hand side). This is particularly truefor horizontal loads on internal elements, where Kaplan suggestsassuming only the drag component of the hydrodynamic force toact, resulting in a general under-estimation of total loads.

Vertical (upwards) impulsive loads measured during thesetests are compared with predictions by the existing methodsin Fig. 16 for both external and internal deck elements. Scatteraround predictions is large over the range of measurementsfor almost all methods in the comparison. Predictions of im-pact loads have improved significantly in recent times, at leastpartially due to improvements in measurement and data ac-quisition methods.

Impacts on external and internal elements are compared withpredictions by Kaplan in Fig. 17 for both vertical and horizontalloads. In general, slam forces on suspended elements are under-estimated by Kaplan's model, but predictions of slam (bothvertical and horizontal) were compared satisfactorily withmeasurements (at 1/250 level) on the seaward face of theexternal beam, where severe impacts were rarely recorded duringthe experiments. Differences between predictions and measure-ments are greater for internal elements, probably because themodel assumes wave flows not to be affected by the presence ofthe structure. Kaplan's simple method cannot therefore includelocal amplification of pressures by trapping wave crestsunderneath the structure, or by 3-dimensional flows above.

6. New prediction method

6.1. General

Guidance for evaluating wave loads for decks and beamswas derived within the “Exposed Jetties” research project, seeMcConnell et al. (2003, 2004) and Tirindelli et al. (2003a). Newequations/coefficients were developed to provide designerswith safe and user-friendly prediction methods. The “ExposedJetties” data on wave loads did however suffer from some

Fig. 19. Quasi-static horizontal (seaward) pressures on external beam, solid lineobeys Eq. (19) with a=1.186 and b=0.429.

corruption of the wave load measurements (described above),and from having to represent too many variations in the loadingprocess. This paper therefore describes methods to refine andextend such predictions. Further non-dimensional analysis wascarried out to reduce scatter around predictions, providingphysically-based prediction equations less influenced byspurious correlations. Wave force results used in the followingsections were extracted from signals filtered using wavelettransforms as in Section 4.

6.2. Parametric analysis

Through the review of previous prediction methods for waveloading on platforms/decks, a series of geometric andhydrodynamic variables influencing forces F applied to aplate are identified in Eq. (17).

F ¼ f Hmax;Hs;d;gmax;Tm;L;sm;c;u;:u;bl;bw;bt;cl;h;

:h;l;

:l

� �ð17Þ

where terms not previously defined are Hmax = maximum waveheight, ηmax = maximum crest elevation, u = local water ve-locity vector, u ˙ = local water acceleration vector.

Among the variables in Eq. (17), the most informative appearto be Hs, ηmax, c, cl, Tm, and sm. Analysis of the new dataconfirmed the importance of these variables. Clear trends werefound between vertical and horizontal loads (at 1/250exceedance level) and Hs, ηmax, and cl (see Tirindelli et al.,2003b, Tirindelli, 2004 and Cuomo, 2005). Regardless of otherwave and geometry parameters, the magnitude of forcesdepends strongly on wave height. The new data show clearincreases of wave loads with Hs, see Fig. 18 (top panel: quasistatic load, bottom panel: impacts).

The variable most closely linked to Hs is ηmax. Althoughboth wave height and maximum crest elevation represent theenergy of the wave field approaching the jetty, ηmax is morestrongly linked to the loading process as it takes into account thenon-linear dependence of the wave profile from both waveperiod and water depth. This is particularly true for structures in

Fig. 21. Horizontal (seaward) forces on external beam, solid lines obey Eq. (18) with coefficients a and b given in Table 2.


shallow and intermediate water depths. For these structures, theprobability of occurrence of wave loading itself variesaccording to the probability distribution of ηmax (see Cuomo,2005 and Bentiba et al., 2004).

Fig. 22. Quasi-static vertical (upward) forces on deck elements solid

Unfortunately, although being more informative than Hs,values of ηmax are not always easily available. With this inmind, a simple method is used to evaluate ηmax as follows. First,for a given Hs, Tm, d and Nz (number of waves within a test),

lines obey Eq. (18) with coefficients a and b given in Table 1.

Fig. 24. Vertical force time-history loads on external (dashed line) and internal(solid line) deck element (no-panel configuration).


the maximum wave height Hmax is calculated assuming themost appropriate local wave height distribution. The value ofηmax is then taken as the maximum crest elevation for a wavehaving H=Hmax, T=Tm and propagating in water of constantdepth d. Results presented here have been obtained bydescribing the incoming wave profile according to Fenton'sFourier Transform, which is valid for regular waves in the rangeof parameter tested. Forces and pressures measured during theseexperiments show strong dependence on ηmax, evaluated asabove, with wave loads increasing with increasing ηmax.

The parameter that best synthesizes the geometricalinformation that must be taken into account for wave loadingon exposed jetties is the clearance cl. The composite variable(ηmax−cl) provides an effective measure of water that inundatesthe deck or beam. The variable (ηmax−cl) can therefore beconsidered as the most useful variable for calculations of waveloadings on exposed jetties.

6.3. Dimensionless analysis

Bearing in mind that the most important variables forproviding prediction of wave loading on structures as jetties areHs, ηmax and cl, and considering that a dimensionless approachis required for generalisation of results, Eq. (18) (forces) andEq. (19) (pressures) have been fitted to the new dataset.

F⁎1=250 ¼

Fqs 1=250

qw � g � Hs � A ¼ a � gmax−cld

� �þ b ð18Þ

P⁎1=250 ¼

Pqs 1=250

qw � g � Hs¼ a � gmax−cl

d

� �þ b ð19Þ

where Fqs 1/250 is quasi-static force (at 1/250 significance level),Pqs 1/250 is quasi-static pressure (at 1/250 significance level), Ais the area of the element (orthogonal to the direction ofapplication of the load), and a and b are empirical fittingcoefficients.

Fig. 23. Quasi-static vertical (downward) forces on deck elements, sol

These equations make forces or pressures dimensionlessthrough the use of Hs, and identify linear trends betweendimensionless forces and (ηmax−cl) /d including the maindependences in Section 4.1. Results of the experimental fittingto Eqs. (18) and (19) are reported in Section 7.

7. Results

7.1. Horizontal quasi-static loads on beams

7.1.1. Positive (shoreward) loadsThemain components of horizontal loads on external beams are

quasi-static, and for this case coefficients a and b in Eq. (19) arederived for pressure data (for which the sampling rate at 200 Hz ishigh enough to cover the quasi-static component of the signal).Responses of the pressure transducers will not have been corruptedby their dynamic response as were the force measuring devices.Dimensionless pressures P⁎qs 1/250=Pqs 1/250 / (ρw·g·Hs) on externalbeam are plotted against (ηmax−cl) /d in Fig. 19. The line given by

id lines obey Eq. (18) with coefficients a and b given in Table 2.

Fig. 25. Comparison of measured quasi-static forces and pressures with proposed new prediction method.


Eq. (19) provides good estimates of the pressures for the threeconfigurations. The relatively high scatter is due to inherent spatialvariability; the pressure transducers only give a local view ratherthan spatially averaged. For configurations without side panels,strong 3-dimensional effects increased spatial effects. Where wave

Table 1Coefficients a and b for fit lines and values of R2 for Eqs. (18) and (19), positive lo

Parameter Direction Element Position

Pressure Horizontal Beam Pa and Pb ExtPressure Horizontal Beam Pa and Pb ExtPressure Horizontal Beam Pa and Pb ExtForce Horizontal Beam IntForce Vertical Beam ExtForce Vertical Beam ExtForce Vertical Beam ExtForce Vertical Beam IntForce Vertical Deck ExtForce Vertical Deck ExtForce Vertical Deck ExtForce Vertical Deck IntForce Vertical Deck IntForce Vertical Deck Int

effects are constrained to be primarily 2-dimensional, the sidepanels channel waves towards the structure, limiting dispersion andgenerating the highest pressures of all configurations studied.

The assumption that horizontal loading is constant across thewidth of the flume is well-supported. For all three configurations,

ads; sε in model units: pressure [kPa] and force [N]

Configuration a b R2 se

FD 1.19 0.43 0.90 0.34P 1.19 0.43 0.87 0.22NP 1.19 0.43 0.96 0.17NP 0.56 0.75 0.90 6.84FD 1.74 0.14 0.96 1.68P 0.71 0.57 0.97 1.24NP 1.10 0.46 0.96 1.61NP 1.36 0.46 0.89 2.27FD 2.31 0.05 0.95 6.78P 1.23 0.51 0.96 7.28NP 1.57 0.52 0.84 7.64FD 0.83 0.13 0.69 9.80P 0.58 0.19 0.67 6.57NP 1.57 0.73 0.95 11.21

Table 2Coefficients a and b for fit lines and values of R2 for Eq. (18), negative loads; se in model units: pressure [kPa] and force [N]

Parameter Direction Element Position Configuration a b R2 se

Force Horizontal Beam Ext FD −0.77 0.00 0.87 2.15Force Horizontal Beam Ext P −0.56 −0.04 0.91 0.86Force Horizontal Beam Ext NP −0.84 0.04 0.75 1.08Force Horizontal Beam Int NP 0.00 −0.22 0.47 1.35Force Vertical Beam Ext FD −1.89 −0.12 0.86 1.90Force Vertical Beam Ext P 0.00 −0.49 0.69 1.74Force Vertical Beam Ext NP −0.04 −0.48 0.69 2.25Force Vertical Beam Int NP −0.23 −0.29 0.87 0.76Force Vertical Deck Ext FD −1.95 0.03 0.94 7.80Force Vertical Deck Ext P 0.00 −0.51 0.72 7.57Force Vertical Deck Ext NP −0.66 −0.36 0.61 8.74Force Vertical Deck Int FD −0.52 −0.05 0.89 2.24Force Vertical Deck Int P −0.08 −0.06 0.94 0.81Force Vertical Deck Int NP −1.35 −0.29 0.89 4.90


the magnitude of horizontal dimensionless pressures on externalbeam is slightly higher (∼20%) than that of horizontal forces, seeFig. 20, suggesting that predicted pressures will generally giveconservative results for design purposes.

7.1.2. Negative (seaward) loadsAs the wave travels through the structure, it may apply a

reverse or seaward force on the shoreward face of the verticalelements, giving a net negative (seaward) force, whose mag-nitude may be comparable to the positive (shoreward) quasi-

Fig. 26. Peak forces versus quasi-static vertical (upward) forces on deck e

static component if the element is still immersed. Pressuresignals measured only on the external face of the element do nottherefore give a reliable picture of the overall process, so valuesfor negative loads are extracted only from force time histories.

Dimensionless negative horizontal forces F⁎qs−1/250=Fqs− 1/250 /(ρw·g·Hs·A) on the external beam are plotted against (ηmax−cl) /din Fig. 21. The linear fits provide good estimates of the forcesfor the three different configurations, confirming that horizontalloads on external elements are not significantly affected by thestructural configuration.

lements, solid lines obey Eq. (22) with coefficient a given in Table 3.

Table 3Coefficient a for fit lines and values of R2 for Eq. (22); se in model units:pressure [kPa] and force [N]

Parameter Direction Element Position Configuration a R2 se

Force Horizontal Beam Ext All 2.45 0.90 1.10Force Horizontal Beam Int NP 3.35 0.89 25.62Force Vertical Beam Ext FD 2.87 0.94 5.38Force Vertical Beam Ext P 1.74 0.48 1.24Force Vertical Beam Ext NP 2.28 0.32 3.41Force Vertical Deck Ext FD 2.35 0.93 15.81Force Vertical Deck Ext P 1.99 0.64 7.28Force Vertical Deck Ext NP 2.22 0.85 20.41Force Vertical Beam Int NP 2.59 0.69 8.36Force Vertical Deck Int FD 2.35 0.98 21.32Force Vertical Deck Int P 1.84 0.88 6.57Force Vertical Deck Int NP 2.29 0.96 26.60


7.2. Vertical quasi-static loads on beams and decks

7.2.1. Positive (upward) loadsThe most complete data from the experiments are for

vertical uplift forces on horizontal elements. Deck elements(and to a lesser degree beam elements) expose significanthorizontal areas to wave action and may therefore be subjectto important quasi-static uplift loads. Dimensionless upliftforces F⁎qs+1/250=Fqs+1/250 / (ρw·g·Hs·A) on external beams andexternal and internal decks are plotted in Fig. 22 against(ηmax−cl) /d for each of the configurations tested. Data forthe internal beam element are not presented here because someforce transducer results may have been corrupted during theexperiments.

The linear response of forces with (ηmax−cl) /d is clear fromall the plots, particularly for external elements (first two rows inFig. 22), which are less influenced by the different configura-

Fig. 27. Peak forces versus quasi-static forces on beams: horizontal (left) and v

tions, particularly for the most seaward facing element (externalbeam, first row of Fig. 22), where lines for the three differentconfigurations are very similar. The configurations with sidepanels (P) seem to provide the highest loads for externalelements. Without the side panels (NP), 3-dimensional waveinteractions with the complex structure make forces on theinternal deck less predictable (see third row of Fig. 22). Forces oninternal elements are more sensitive to any 3-dimensional effects,see Section 7.2.2. Even for these elements, however, simplelinear equations still give reasonable predictions for uplift loads.

7.2.2. Negative (downward) loadsDownward forces may be by suction where sideways flows

are restrained by continuous beams, but will be substantiallyincreased when the deck is inundated by waves. Even for thistype of loading, dimensionless forces F⁎qs− 1/250 are still cor-related with (ηmax−cl) /d, see Fig. 23. Downward forces onboth external and internal deck elements are shown in Fig. 23afor the flat deck (FD). Inundation of the deck is limitedand downward forces are mainly due to suction, reducing as thewave travels along the jetty from external to internal elements.The relative importance of inundation on internal deck ele-ments is shown in Fig. 23b. Negative vertical loads are sub-stantially higher, and longer-lasting, when 3-dimensionaleffects can act.

Effects of inundation on downward forces are furtherillustrated in Fig. 24, showing example vertical force timehistories recorded by external and internal deck elements under3-dimensional attack (NP). Green water overtopping signifi-cantly increases downward loads; when lateral inundationof the deck is not prevented, downward loads on internalelements might be larger and longer-lasting than on externalelements.

ertical (right), solid lines obey Eq. (22) with coefficient a given in Table 3.

Table 4Comparison of error E between predictions and data for existing methods and new formulae for quasi-static uplift forces on the external deck element

El-Ghamry (1965) Wang (1970) Broughton andHorn (1987)

Shih andAnastasiou (1992)

Suchithra andKoola (1995)

Kaplan et al.(1995)

Present formulae

(min) (max)

N 0.120 0.130 0.080 0.104 0.135 0.344 0.131 0.050P 0.144 0.149 0.095 0.087 0.157 0.364 0.137 0.058FD 0.192 0.153 0.584 0.252 0.592 1.362 0.116 0.074All 0.084 0.082 0.155 0.083 0.169 0.393 0.094 0.034


7.3. Prediction method for quasi-static loads on beams and decks

Once F⁎qs 1/250 has been evaluated according to Eqs. (18) and(19), design forces (quasi-static) can be evaluated from:

Fqs 1=250 ¼ F⁎qs 1=250 � qw � g � Hs � Að Þ: ð20Þ

Quasi-static loads on structural elements of exposed jetties asmeasured during the physical model tests are compared withpredictions by Eq. (20) in Fig. 25. The comparison showsreasonable agreement, with values of a and b listed in Tables 1and 2, together with the corresponding estimates of goodnessof fit in terms of R2 (taken as the ratio of the sum of the squaresof the regressions and the total sum of the squares, andevaluated using robust fitting) and the standard error of theestimate, that is:

se ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNt

i¼1yi− ̂yið Þ2

Nt−2ð Þ

vuuut ð21Þ

where yi and ŷi are respectively measured and predicted loadsand Nt is the number of tests used in each fit. It should be notedthat values of sϵ are given in model units, that is pressure inkiloPascal [kPa] and force in Newton [N].

Compared with predictions by other methods in Figs. 14and 15, the equations and coefficients derived here showsignificant improvement, considering that a single uniquerelationship has been used to predict somewhat variable loadson different elements in different locations along the deck.

7.4. Impulsive loads on beams and decks

The main results of this paper have been related to quasi-staticloads. Nevertheless, some important information about impulsiveloads has also been derived from the revised data. Using definitionsdiscussed earlier, impulsive loads on beams and decks can reach 3times the value of corresponding quasi-static loads. Ratios betweendimensionless impact F⁎max 1/250=Fmax 1/250/ (ρw·g·Hs·A) (at 1/250level) and quasi-static positive forces F⁎qs+ (at 1/250 level) areshown in Fig. 26 for both external and internal deck elements.Fitting lines in Fig. 26 have the following expression:

F⁎max 1=250 ¼ a � F⁎

qsþ1=250 ð22Þwhere a is an empirical coefficient, given in Table 3.

Similar trends are observed in Fig. 27 for horizontal (a) andvertical (b) loads on the external beams (FD). Horizontalimpacts are on average 2.5 times the corresponding quasi-staticloads, whereas for vertical impacts the ratio is higher (2.9). Thehighest impact ratio (a=3.4 times quasi-static) are recorded onthe internal elements, where the complex geometry of thestructure may trap and amplify wave effects.

It must be stressed that impulsive loads measured duringphysical model tests have relatively short rise times (trb0.1·Tm)that might fall within the range of the natural periods ofvibration of prototype structures. This might result inamplification and/or reduction of actual loading (Oumeraciand Kortenhaus, 1994).

Indeed, when significant impulsive loads are expected to acton the suspended deck structure, the evaluation of the impactload to be used in design analysis must account for thedynamics of the prototype structure. Guidance for theevaluation of the effective load to be adopted in early feasibilitystudies can be found in Cuomo (2005).

7.5. Comparison with existing methods

Comparison of the new method with models reviewedpreviously confirms the improvement in prediction. Values ofrelative error E defined in Eq. (23) for a series of existingmethods and for the new method are shown in Table 4.

E ¼ 1Nt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNt

1

yi− ̂yiyi

� �2

:

sð23Þ

As most present methods were developed to predict onlyvertical wave loads on horizontal slabs, values in Table 4 onlyrefer to quasi-static uplift forces on external deck.

8. Summary and conclusions

An extensive review of previous work has identified gapsand inconsistencies in prediction methods for wave-in-deckloads. The need for guidance for wave loads on suspended deckstructures in exposed locations originally motivated researchwithin the “Exposed Jetties” project, including a series of flumetests to measure wave-induced loads on deck and beamelements, and that work has been extended here.

Time histories of vertical and horizontal forces have been re-analysed using wavelet transform, providing important insightinto physical mechanisms of the loading process. An improved


prediction method has been derived to include the key variablesthat influence the loading process, but retaining simplicity in theprediction formulae.

Results from the experimental fitting of the new data havebeen fully described above. Each prediction line is parame-terised by Eqs. (18) and (19) and slope and intercept coefficientsa and b in Tables 1 and 2 with corresponding regression fit R2

and standard error se. Similarly, the values of coefficient a inEq. (22) are listed in Table 3, together with their regression fitR2 and standard error se.

By selecting appropriate values of the empirical coefficientsa and b in Tables 1–3, the new methodology succeeds inaccounting for different structural configurations of the jetty andfor relative location of structural element along the jetty deck,fitting coefficients to each sub-set.

Comparison with experimental data shows good agreementbetween measured and predicted forces. When compared toprevious models, variability around predicted values is sig-nificantly reduced, see Table 4.

Acknowledgements

This analysis has been supported by Universities of RomaTRE and Bologna, and HR Wallingford. The authors gratefullyacknowledge contributions from the DTI PII Project on ExposedJetties (39/5/130 cc2035), Prof. Leopoldo Franco, University ofRoma TRE, Prof. Alberto Lamberti, University of Bologna; TerryHedges, University of Liverpool, Kirsty McConnell and IanCruickshank, HR Wallingford; and visiting researchers at HRWAmjad-Mohammed Saleem and Oliver de Rooij. The authorswish to thank the very valuable comments to an early versionof this manuscript provided by Andreas Kortenhaus and AlfTørum.

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