Wave-in-deck loads on exposed jetties

Coastal Engineering 54 (2007) 657 – 679 www.elsevier.com/locate/coastaleng

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, Italy Coast & Harbor Engineering, 155 Montgomery Street, Suite 608, San Francisco, CA 94104, USA Maritime Structures, HR Wallingford, Howbery Park, Wallingford, OX10 8BA, UK and Department of Civil and Environmental Engineering, University of Southampton, UK b

c

Received 8 February 2006; received in revised form 4 January 2007; accepted 12 January 2007 Available 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). The work presented here focuses on results from physical model tests on wave-induced loads on deck and beam elements of exposed jetties and similar structures. 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 conditions tested were selected after an extensive literature review (summarised in the paper) and consultation with the project steering group. Different configurations 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 dynamic responses of the instrumentation. Both quasi-static and impulsive components of the loading were identified. Previous methods to predict wave loading on jetty elements (decks and beams) were tested against these new data and clear inconsistencies and gaps were recognised. New dimensionless equations have been produced to evaluate wave forces on deck and beam elements of suspended deck structures. The results are consistent 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 structure with deck and perhaps beams, supported on piles. The deck (and beams) are suspended well clear of normal water levels, so are only at risk of direct wave effects under infrequent combinations of surge and wave condition. Such jetties may be quite long (perhaps 0.5–5 km), orientated approximately normal to the shoreline/bed contours, and carry pipes or conveyors to load/ unload gas, liquid/bulk granular cargoes from vessels moored at the jetty head. Similar structures include leisure and passenger piers, mooring dolphins and some highway bridges. Selected results 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 other decks on beams that may be hit by waves. These studies were not however intended to address wave loads on deep water (offshore) structures termed “rigs”, “jackets” or “platforms”, nor do the waves covered include tsunamis or other waves of period N 25 s. 1.2. Background Marine trade between many coastal nations has often relied on jetties or piers to berth vessels for loading or discharge of cargo/passengers. These facilities were traditionally constructed in areas where wave-induced loads are relatively small, naturally sheltered locations and/or locations protected by breakwaters. In the last 15–20 years there has been an increased demand for liquid natural and petroleum gas terminals (LNG and LPG), which require sheltered berths in deep water for large vessels, but may not need shelter to the approach trestles carrying 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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Notation A area of the element exposed to wave action a, b experimental coefficient bl length of structural element bw width of structural element bt thickness of structural element c wave celerity C damping cl clearance (i.e. vertical distance between an element and swl) Cd drag coefficient d water depth E error F force (generic) Fc force (corrupted component) Ff force (filtered) Fs force (smoothed) F tot force (response) Fmax maximum value reached by the signal within each event Fqs 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 level FD configuration flat deck g gravitational acceleration (= 9.81 m/s2) G Fourier transform of w h vertical wetted length d h time derivative of h Hmax max wave height during a storm Hs significant wave height K stiffness L wave length L0 deepwater wave length l horizontal wetted length d l time derivative of l M mass Nz number of wave within a storm event NP configuration without side panels Nt number of tests P configuration with side panels Pqs1/250 quasi-static component of the pressure (at 1/250 level) s scale dilatation parameter se standard error of the estimate sm wave steepness (= Hs / L0) for T = Tm t time T wave period (generic) Tn n-th resolved equivalent period Tn, min minimum period corrupted by the dynamics of measurement instrument Tn, max maximum period corrupted by the dynamics of measurement instrument Tsmooth cut-off period of low-pass filter T0 natural period of resonance of the structure Tm mean wave period tr rise time of the force signal u velocity vector u˙ acceleration vector

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ux uy x ·x ··x yi ŷi WT Ψ ρw η η˙ ··η ηmax τ

659

horizontal velocity vertical velocity displacement velocity acceleration measured load (generic) predicted load (generic) wavelet transform coefficient mother wavelet water density wave surface elevation time derivative of η time derivative of η˙ max wave surface elevation translation parameter

breakwater protection, increasing wave exposure and potential risks of high wave loads on the structure. The higher probability of wave loading on jetties may be increased by processes such as subsidence, decreasing the clearance (i.e. vertical distance between still water level (swl) and underside of the jetty deck structural elements), as seen for the Ekofisk platform complex (Broughton and Horn, 1987). Similar loadings can arise where other suspended structures are too close to the water surface. Key examples are temporary works used in construction or refurbishment around harbour structures, or some low-lying transport bridges over coastal waterways. Whilst a number of prediction methods have been developed for wave-in-deck loads, gaps and weaknesses in available models stimulated the “Exposed Jetties” research supported in the UK by Department of Trade and Industry under PII Project 39/5/130 cc2035 (see Tirindelli et al., 2002). Within this project, a series of 2-dimensional physical model tests measured wave loads on deck and beam elements, see Tirindelli et al. (2002) and McConnell et al. (2003, 2004). These measurements were analysed to explore the process of wave loading, with the objective of developing improved predictions. Results from the main project were summarised by McConnell et al. (2003, 2004) and Cuomo et al. (2004). Extending, but after the formal end of the Exposed Jetties project, a new method for the analysis of non-stationary time-history loads was developed by Cuomo et al. (2003) and Cuomo (2005), based on wavelet transform. Further analysis of the original data allowed a deeper understanding of the loading process and a new interpretation of the measured data.

– horizontal loads (both seaward and shoreward) on beams or other projecting elements. A sketch of wave-in-deck loads acting on a jetty is shown in Fig. 1. The nature, occurrence and magnitude of these wave loadings vary significantly for different structures and wave conditions. Horizontal elements such as deck slabs may be subject to large vertical forces upward or downward (especially under conditions that inundate the deck). Vertically faced elements like beams and fenders can experience significant forces both horizontally and vertically (if of significant thickness). 2. Previous work In the last fifty years, many prediction methods have been developed to evaluate wave-in-deck loads on jetty structures. A literature review of the most important among these works is described in the following. El-Ghamry (1965) and Wang (1970) first performed physical model tests to investigate wave loads on horizontal decks subject to breaking and non-breaking wave attacks. The authors found substantial similarity between the mechanisms of wave impact on horizontal platforms and vertical barriers. Uplift pressures are

1.3. Wave load definitions Hydraulic loads applied by waves to the deck or other projecting elements (beams, fenders) can be defined as “wave-indeck 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);

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

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Fig. 2. Experimental setup: overall view.

characterised by an initial peak pressure of considerable magnitude but of short duration, followed by a slowly-varying uplift pressure of less magnitude but of considerable duration, typically first positive, then negative. For regular progressive waves, Wang's method for prediction of uplift pressure P reads:
 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 4c k  H  qw  g 2kd P¼  tanh ð1Þ  1− 2 l 2 L H where ρw is the water density, g the acceleration due to gravity, H is the wave height at the structure, cl the deck clearance above the swl, d the water depth and L the incoming wave length. Broughton and Horn (1987) proposed a simple approach for the prediction of wave loads based on the assumption that the force impulse is equal to the change in momentum when the wave crest hits the leading edge of the deck. This leads to the following expressions for the vertical (FV) and horizontal (FH) force per unit length: FV ¼

k  q  c  u y  bw  l 2 w

FH ¼ qw  c  ux  bw  h

ð2Þ ð3Þ

where h and l are respectively the height and length of wave crest in contact with the deck, bw the cellar deck breadth, c the wave celerity (according to Stoke's 5th order theory), and uy and ux respectively, the vertical and horizontal velocity of the water particles. Broughton and Horn performed physical model tests in a wave basin (scale 1:50) on three different platform configurations. Observed force time histories were characterised by positive (uplift) forces, followed by downward forces of the same magnitude order of the uplift. Due to the low sampling rate (20 Hz) used during the experiments, no information was retained on impulsive short-duration peaks. Shih and Anastasiou (1992) and Toumazis et al. (1989) analysed wave-induced forces and pressures on horizontal platform decks at small and very small scales. Forces were measured by means of strain gauges. Pressures were logged at a sample rate of 500 Hz. A high speed video camera (up to 1000 frames/s) was used to capture snapshots of the interaction between waves and the suspended structure. Impact pressure peaks on horizontal plates were found to increase with decreasing clearance and increasing wave height.

The following empirical relations for peak pressures PV were suggested: PV ¼ ð1:8  7:6Þ  qw  g  H

ð4Þ

PV ¼ ð4:0  8:0Þ  qw  g  Hs

ð5Þ

respectively for regular and random waves (Hs = significant wave height). The authors confirmed that air entrainment at impact affects impulsive pressures, generating scatter in data. Froude scaling does not account for difference in air compressibility at model and prototype scales, potentially leading to distortion of measurement of impact maxima and rise times. Recorded slowlyvarying positive pressures (Pqs+) were found to be always lower than the hydrostatic head. The following expression was suggested for both regular and random waves: Pqsþ ¼ 0:65  qw  g  ðg−cl Þ

ð6Þ

with η = wave crest elevation. “Ventilated shocks” (Lundgren, 1969) were observed on the vertical plate, and the following relation was suggested for the evaluation of horizontal impact pressures PH,max: PH; max ¼ a  qw  c2

ð7Þ

where α is an empirical coefficient. For waves impacting before the breaking condition was reached, the authors indicate that α is always less than 1.3, for more violent impacts a value of 4.1 has to be used. For deep water spilling breakers impinging on a vertical 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 and 10, whilst field measurements on seawalls by Blackmore and Hewson (1984) suggest α to vary between 0.5 and 4. Suchithra and Koola (1995) performed model tests to measure vertical forces on a horizontal slab. They tested different configurations (with and without stiffeners) to investigate the influence of down-stand beams on wave-induced loading. The impact vertical force on the slam was expressed in terms of the slamming coefficient Cs, as follows: 1 FS ¼  Cs  qw  A  u2y 2

ð8Þ

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

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

incoming wave and on the clearance of the element. The parameter Cns = Cs · d / L (average value = 1.7) was found to be independent from the wave frequency, and to increase with decreasing values of clearance. The presence of longitudinal beams was found to slightly increase applied forces, whilst transversal beams, causing air pockets to be entrapped between the 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 horizontal beams on offshore platforms. Moving from the original work by Morison et al. (1950), they developed a semi-analytical model for the evaluation of wave-in-deck time-history loads on both vertical and horizontal members. According to Kaplan, as the wave travels along the platform, it transfers its energy to the superstructure and the variation in time of wave-in-deck loads results from the combination of an inertia force (resulting from the variation of the momentum in terms of structural acceleration and added mass), and a drag force. Bolt (1999) reviewed the state-ofthe-art for wave-in-deck time-history load calculations and found the model by Kaplan to provide the most sophisticated representation of wave-in-deck loads. The model also accounts for relative location of element along the structure, providing the most 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):

where l (horizontal wetted length) and l ˙ are determined from the relative degree of wetting of the flat deck underside on which loading occurs, bl is deck length, Cd is the drag coefficient, h (vertical wetted length) and h˙ are determined from the relative degree of wetting of the vertical face of the beam where loading occurs, bt is the thickness of the deck. Further developments of Kaplan's model have been discussed, among the others, by Isaacson and Bhat (1994), Isaacson et al. (1994) and Cuomo (2005). An alternative semiempirical method, also accounting for the dynamic amplification of slamming due to dynamic response of structural elements 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 flume at HR Wallingford, UK. The flume was 40 m long, 1.5 m wide, with a maximum water depth of 1.2 m at the paddle. Waves were generated by a piston-type paddle at one end of the flume

: k bw  l 2 k   2 1=2  g̈ þ qw  4  bw  l  l 8 1 þ blw  2 1 þ 12  blw  qw     2 3=2  g þ 2  bw  l  Cd  g  jgj þ qw 1 þ blw

FV ¼ qw 

 g  ðg−cl Þ  bw  bl
FH ¼

ð9Þ

 : 2 qw 4 2 :  bt  Cd  ux  jux j þ  qw  h  h  ux  bw  q  h ux þ 2 k w k

ð10Þ

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

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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 covered with absorbing matting and rocks to minimize wave reflection. Behind the test structure, the absorbing beach extended in height ± 0.15 m around swl, lying on a steeper 1:1.5 beach, see Fig. 2. A 1:25 Froude scale model of a jetty structure was bolted to the floor of the wave flume and instrumented to provide direct measurements of wave loading. The configurations of the test structure and measurement elements were determined by an analysis of existing jetties in UK, Oman, Kuwait, India and Caribbean. At 1:25 scale, model wave heights were above those used in studies by Allsop et al. (1995) and Howarth et al. (1996) to identify scale effects on wave pressures, where measurements of impulsive loads were found to be slightly conservative. During the study design, it was also noted that the chosen scale also simulates conditions at scales around 1:50–1:75 for “offshore platforms”, although the study had not been intended to address such structures per se.

Fig. 6. Matrix of test conditions. Circles represent wave conditions (two different water depths d ); solid lines are breaking limit and stroke limit; dashed and dotted lines represent three values of wave steepness sm, respectively two storm 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. The suspended structure was made of wood and was supported by six steel piles (5 cm diameter) mounted on a tubular base frame bolted 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 by metal elements (aluminium and steel) in two different positions along the jetty (first and third spans, hereinafter respectively named “external” and “internal”). Each of these elements were connected to a force transducer, housed in the wooden frame of down-standing beams, see Fig. 3. Each force transducer was able to record forces in two normal directions (vertical and horizontal). Two pressure transducers in the seaward beam of the platform at two opposite sides of the longitudinal axis of the jetty measured horizontal pressures. Force and pressure

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

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ditions for exposed jetty locations using JONSWAP spectra (γ = 3.3). Chosen wave conditions were based on three wave steepnesses (sm = 0.065, 0.040, 0.010), which represented two storm and one swell condition. The latter also represented waves 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 s to 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 platforms may be considered in three phases in Fig. 7. At the instance of contact between the wave crest and the element, the slam or impulsive force may be large in magnitude and short in duration. 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).

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.

transducers were logged at 200 Hz. The incoming wave field was monitored by three wave gauges along the wave flume. One gauge was located well away from the structure in order to represent the generated wave field. The remaining two gauges measured waves before and after the model structure, to describe the wave field at the structure and to provide a description of the wave energy dissipation through the interaction with the jetty. Three different configurations are shown in Fig. 5. The original configuration (no panels, NP) had the supporting beams facing downward, and no side plates to limit 3-dimensional effects. A second configuration (panels, P) used large side panels to limit 3-dimensional effects due to lateral inundation of the deck. The last configuration (flat deck, FD) inverted the deck and beams to investigate wave loads on the (now) flat underside. 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 or lowering the deck assembly over the piles by means of spacers. Random sea states were defined by scaling typical wave con-

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

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The first contact between the water and the element causes an abrupt transfer of wave momentum from the water to the structure, generating the impact force. Such high intensity forces acting on limited areas, even over a short time, may cause severe local damage, local yielding and fatigue failure. Impulsive loads vary substantially in both magnitude and duration even under nominally identical conditions, confirming previous observations from research on wave impacts. A comprehensive review is given in Cuomo (2005). As the wave propagates along the underside of a deck, jets of water may shoot out sideways as the contact area moves along the deck (unless otherwise restrained). These lateral jets generally disappear as the free surface rises above soffit level. A difference between water levels under the deck and that alongside the structure gives rise to the pulsating or quasi-static positive force. The magnitude of this force is consistently lower than any initial impact, but its duration is of order 0.25·Tm. Finally, the wave surface falls below soffit level and moves inward 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 substantially increased when the wave inundates the deck, adding the weight of green water above the deck, sometimes leading to the downward (negative) force reaching the same order of intensity as the quasi-static uplift (positive) force. Horizontal loads on beam elements often exhibit different characteristics from vertical loads. The magnitude of the first

impact load on an external beam (i.e. vertical element at the edge of the jetty) is generally lower than the corresponding vertical impact. Example time histories are mainly characterised by quasi-static (pulsating) components. For waves underneath any platform formed by beams and deck elements, interactions with the protruding elements are complex, and wave crests and air may be trapped between beam and deck. This may result in high horizontal impulsive loads on the seaward face of internal elements and noticeable horizontal forces acting seaward on the shoreward face of the vertical elements. Example histories of horizontal forces and pressures on vertical elements are shown in Fig. 8. Momentum transfer to the face of the external beam is more gradual, giving quasi-static or pulsating forces (Fig. 8a) confirmed by time histories recorded by pressure transducers A and B (Fig. 8b). High frequency oscillations in Fig. 8a are due to the dynamic response of the instrument in the front beam. Wave interaction with internal elements is more complicated, in some circumstances resulting in high intensity impact loads (see Fig. 8c recorded as the wave slams against the seaward face of the internal beam). 4.2. Observation and parameterisation of time history of wave-in-deck loads Force histories recorded as the wave travels along the test structure are plotted in Fig. 9. From top to bottom, the wave initially hits the seaward beam (horizontal, 9a and vertical, 9b)

Fig. 10. Vertical time-history load on deck element: recorded time history (left — model units) and wavelet transform in the time-period domain (right).

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– trapping wave crests underneath a soffit between downstanding beams may result in local amplification of applied loads; – phase differences between positive and negative loads may increase as the wave travels along the structure; – rise time of wave loads (tr, see Fig. 7) may be comparable to the characteristic periods of oscillation of structural elements, therefore the resulting loading process depends strongly on the dynamic response of the structure. Observing force histories during the experiments, large variations in both magnitude and shape of the signal were noticed, even for similar test conditions. Despite this variability, an idealised time history has been developed to represent the general shape drawn by the force signal during the loading. The suggested idealised time history, superimposed on a measured one for the external deck element in Fig. 7, consists of a shortduration triangular pulse (linear increase from zero to its peak value), followed by the quasi-static component. The proposed model is characterised by:

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

and external deck (vertical, 9c), then moves shoreward to the internal beam (horizontal, 9d and vertical, 9e) and deck (vertical, 9f). From the signals in Fig. 9, it is possible to derive the following general observations: – intensities of loads do not necessarily reduce as the wave travels under the structure; – internal elements are subjected to wave impacts as are the external elements;

– Fmax = maximum value reached by the signal within each event — considered as representative of a typical impact force, see Section 4.4; – tr = rise time of the force signal; – Fqs+ = maximum value of the quasi-static component of the signal within each event; – Fqs− = minimum value of the quasi-static component of the signal within each event. Within this project, wave loads (pulsating and impulsive) have been parameterised at 1/250 level (average of the top 1/250 values). This choice was made to reduce dependence on highly variable extreme loads; and to maintain consistency with the general use of 1/250 values for wave loads on walls, see Goda (2000), Allsop (2000) and Oumeraci et al. (2001). In assessing the exceedance level, the number of measured loads was taken as a proportion of the number of incoming zero-crossing waves, themselves calculated from the test duration and mean wave

Fig. 12. Finite element model of the test structure (right) and deck element (left), shape deformed according to the first mode of oscillation (vertical direction).

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undertaken on short-duration wave loads on some classes of maritime structures, clear definitions of short-duration loads as well as of quasi-static loads are still missing. As the loading process depends on both incoming wave field and the dynamic response of the structure, the dynamic characteristics of the structure should be taken into account when distinguishing load types. In this paper, “impacts” or “short-duration” loads are defined as those that act on the structure for durations shorter or comparable with the resonance period of the structure. Conversely, “quasi-static” (also called slowly-varying or pulsating) loads are those that act on the structure for longer. In figures: tr Fig. 13. Filtering out long-duration components from signal recorded by measurement devices.

period, Tm. For tests of 1000 waves, the 1/250 value was therefore evaluated by simply averaging the top 4 loads. 4.3. Impacts and quasi-static loads Distinguishing between impulsive and quasi-static wave loads is not straightforward. Although much research has been

V2  T0 impacts N2  T0 quasistatic loads

where tr is the load rise time and T0 is the resonant period for the mode corresponding to the applied load. In analysing measurements in the model, dynamic characteristics of the instrument and of the jetty model must be taken into account. In particular, defining any value of an impulsive wave load to be used later in design (either as a statically equivalent load for feasibility studies; or as time-history loads for dynamic analysis of more complex structures) requires filtering out corruptions from the dynamic response of the model setup (see Cuomo et al., 2003 and Cuomo, 2005).

Fig. 14. Comparison of vertical quasi-static forces on deck elements with existing prediction methods.

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4.4. Wavelet analysis of time-history loads Non-stationary signals are frequently encountered in wind, ocean and earthquake engineering. Force signals from these experiments show occasional high peaks marking an otherwise relatively undisturbed signal. A standard Fourier analysis will not represent correctly this signal because sine or cosine functions in the Fourier series are periodic. Spectral analysis methods are not therefore able to describe transient features of these short-duration phenomena. The need to preserve time dependence and to describe evolving spectral characteristics of non-stationary processes requires tools which allow localization of energy content in both time and frequency domains. Wavelet transformations retain transient signal characteristics beyond the capabilities of Fourier methods. Time and frequency analysis by wavelet transforms provides insight into transient signals through time–frequency maps of the time variant spectra missed by traditional approaches. 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 deck element is shown in Fig. 10 together with its wavelet transform in the time/period domain. A peak can be recognised about 0.25 s after the beginning of the event, with energy in almost all the resolved scales/periods. Some energy can be recognised in

667

periods between 0.03 s and 0.07 s on the left hand side of the amplitude contour graph. The oscillating signal after the impact in Fig. 11 (top panel) suggests that the measurement element was responding dynamically to the wave loading. De-noising of the recorded signal is easily accomplished with the inverse wavelet transform by eliminating or reducing coefficients for components that are related to low energy processes or noise. Editing components affected by the dynamic response of instrumentation is possible, but more difficult, as it requires identifying dynamic characteristics of the test elements. This was assessed here by modelling the dynamic response of the instrument using finite element models (Fig. 12). The period of oscillation of the model structure (left hand side of Fig. 12) was found to be far enough from the characteristic periods of the loading (the slowest mode in the vertical direction has period equal to 0.005 s) and therefore not to significantly affect the measurements. The model of the deck element is shown on the right hand side of Fig. 12, superimposed on the deformed shape corresponding 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 generalised as: M ðt Þ  xð̈ t Þ þ C ðt Þ  x ðt Þ þ K ðt Þ  xðt Þ ¼ F ðt Þ

ð11Þ

where M, C and K represent respectively the mass, the damping and the stiffness of the system (in our case the

Fig. 15. Comparison of quasi-static forces with prediction by Kaplan's model.

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experimental setup), including added mass and hydrodynamic damping. Reading the displacement from the transducer (x) will lead to the measurement of:

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

xðt Þ ¼ fF ðt Þ−½M ðt Þ  xð̈ t Þ þ C ðt Þ  x ðt Þg=K ðt Þ:

t−s 1 ws;b ðt;s;sÞ ¼ pffiffi  w : s s

ð12Þ

Hanssen and Tørum (1999) filtered out corruptions due to dynamic response of measurement instruments by numerically evaluating the terms in squared brackets in Eq. (12), that is, solving Duhamel's integral in time for a single degree of freedom system. For our type of structure, this procedure is not straightforward, since the level of inundation of the structural elements changes with time, together with the added mass and the hydrodynamic damping terms in Eqs. (11) and (12). For this reason, an alternative procedure for filtering out corruption from dynamic response of measurement instruments has been developed based on the wavelet transform of recorded signals. The procedure is described in the following. Let the wavelet transform of the signal F(t) be given by Emery and Thomson (2001): Z l WTðs;sÞ ¼ F ðt Þ  ws;b ðt;s;sÞ  ds: ð13Þ −l

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

ð14Þ

In the analysis, we adopt the Morlet wavelet, defined as:     2  1 1 t−s 2k ws;b ðt;s;sÞ ¼ pffiffi  exp −  exp i  ðt−sÞ : s 2 s s

ð15Þ

From Eq. (15), it is possible to appreciate the similarity between 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,min b Tn b Tn,max] affected by 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 instrument being analysed (generally valid for instruments within their principal range), it is possible to obtain a “filtered” signal by summing contributions from every resolved frequency, but

Fig. 16. Comparison of vertical impact forces on deck elements with existing prediction methods.

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removing those affected by the resonance. The following signals can thus be obtained from the original records: – the reconstructed signal F tot(t) or “response”, that is the inverse transform using all resolved (equivalent) periods Ti; F ðt Þ ¼

1 C

Z

l

−l

Z

l

WTðt;s;bÞ  ws;b ðt;s;sÞ  ds  ds s2 −l

ð16Þ

Rl where C −1 ¼ −l GðxÞ=x  dx and G(ω) is the Fourier transform of ψ. – the filtered signal F f(t), that is the sum of components from all resolved (equivalent) periods Ti but those affected by the dynamics of measurement instrument; – the smoothed signal F s(t), that is the sum of components having (equivalent) periods larger than Tsmooth (this can be easily seen as a low-pass filter). The choice of Tsmooth depends on both the dynamic characteristics of the measurement instrument and the time-history loads; an initial estimate for Tsmooth is 2·T0. The suitability of Morlet wavelet as a base for the transformation of signals recorded during physical model tests is shown in Fig. 11, with the original time-series (top) superimposed to its inverse transform derived using Eq. (16) and extending the integral operator to cover respectively the whole range of resolved (equivalent) periods. The filtered signal Ff(t), obtained by integrating Eq. (16) over all resolved periods but those in the range [Tn,min b Tn b Tn,max] is also shown in the bottom panel of Fig. 11. In this case, filtering out the corrupted component does not affect significantly the impact load magnitude or duration. For the sake of completeness, Fig. 11 also shows the part of the signal affected by the dynamics of the measurement instrument F c (t) = F tot (t) − F f (t) (central panel), confirming that the dynamic response of the measurement element has corrupted the recorded signal. It is worth noticing that since the maximum duration of wave-in-deck loads is comparable with the incident wave period, the whole loading process develop within a relatively limited range of (short) periods, and thus the cone of influence of the wavelet transform (Torrence and Compo, 1998) is extremely narrow and almost no information is lost. To minimize corruption due to dynamic response of measurement instruments on the extraction of meaningful parameters defined in Section 4.2 from recorded signals, the following procedure has been adopted:

669

measurement instruments as Fmax might be larger then the real maximum force (max[F(t)]) acting on the measurement instrument during the loading. Indeed, F tot(tmax) in general differs from Ff(tmax), the difference between the two being given by the dynamic response of the measurement instrument and the following relation is valid: min{Ftot (tmax); Ff (tmax)} ≤ max[F(t)] ≤ max {Ftot (tmax); Ff (tmax)}. Nevertheless, the choice of the aforementioned method for the extraction of Fmax has been made based on the following facts: 1. Ff might underestimate the maximum impact force, as the energetic contribution to the real force acting on the frequencies affected by dynamic of the resonance instruments is indeed neglected in the filtering process; 2. for pulse-shapes similar to those observed in time-history loads recorded during physical model tests, the maximum amplification due to dynamic response of measurement instruments is less than 1.5 (Cuomo, 2005). In such cases, the maximum displacement occurs at time t N tmax (Fig. 11) and thus extracting Fmax at time t = tmax significantly compensates for such potential over-estimation. In our experience, the use of the aforementioned technique reduces corruption due to dynamics of measurement instruments more effectively than using Eq. (12) with potentially erroneous estimation of mass and damping coefficients. Furthermore, when a difference exists between the real load and the

– Fqs+ has been taken as the maximum value reached by the smoothed signal within each loading event; – Fqs− has been taken as the minimum value reached by the smoothed signal within each loading event; – Fmax has been taken as: max {F tot(tmax); F f(tmax)}, the maximum between the reconstructed (F tot) and the filtered (Ff) signals at time t = tmax (time at which the filtered signal reaches the maximum value within each loading event). Values of Fmax extracted by means of the aforementioned technique might still be affected by the dynamics of the

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

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value extracted by the analysis program, the latter is always (slightly) larger than the former. Applying a high-pass filter to the inverse wavelet transform also eliminates long-duration distortions like drift or non-linear displacement of the instruments, without affecting the description of the main processes (Fig. 13). Further details on this analysis, as well as on the dynamics of the experimental setup used in the physical model tests, can be found in Cuomo (2005). 5. Comparison with existing prediction methods Tirindelli (2004) and Cuomo (2005) reviewed prediction methods for wave-induced forces on beam and deck elements of exposed jetties and offshore platforms, including most recent works. In this section predictions by selected methods are

compared with wave-in-deck loads measured during physical model tests. Quasi-static vertical (upwards) loads (at exceedance level F1/250) measured during these new tests are compared with predictions by some of the reviewed methods of Section 2 in Fig. 14 for both external (○) and internal (⁎) deck elements. The scatter is large over the range of measurements for most methods used in the comparison. Quasi-static vertical loads on external and internal elements are compared with predictions by Kaplan in Fig. 15. According to what was suggested by Kaplan et al. (1995) wave kinematics in Eqs. (9) and (10) has been evaluated using linear theory but assuming the wave amplitude to be equal to ηmax; drag coefficient Cd in Eqs. (9) and (10) has been taken respectively equal to 2 and 1. The scatter is relatively large, with most predicted

Fig. 18. Relative importance of Hs on wave-in-deck loads. From left to right: horizontal (seaward) forces on beams, vertical (uplift) forces on beams and vertical (uplift) forces on decks. From top to bottom: quasi-static forces (NP), quasi-static forces (P), impacts (NP) and impacts (P). All data refer to experiments carried out with d = 0.75 m.

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values falling within the “unsafe” regions (upper left corner for positive loads – on the left hand side – and lower right corner for negative loads – on the right hand side). This is particularly true for horizontal loads on internal elements, where Kaplan suggests assuming only the drag component of the hydrodynamic force to act, resulting in a general under-estimation of total loads. Vertical (upwards) impulsive loads measured during these tests are compared with predictions by the existing methods in Fig. 16 for both external and internal deck elements. Scatter around predictions is large over the range of measurements for almost all methods in the comparison. Predictions of impact loads have improved significantly in recent times, at least partially due to improvements in measurement and data acquisition methods. Impacts on external and internal elements are compared with predictions by Kaplan in Fig. 17 for both vertical and horizontal loads. In general, slam forces on suspended elements are underestimated by Kaplan's model, but predictions of slam (both vertical and horizontal) were compared satisfactorily with measurements (at 1/250 level) on the seaward face of the external beam, where severe impacts were rarely recorded during the experiments. Differences between predictions and measurements are greater for internal elements, probably because the model assumes wave flows not to be affected by the presence of the structure. Kaplan's simple method cannot therefore include local amplification of pressures by trapping wave crests underneath the structure, or by 3-dimensional flows above.

671

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

corruption of the wave load measurements (described above), and from having to represent too many variations in the loading process. This paper therefore describes methods to refine and extend such predictions. Further non-dimensional analysis was carried out to reduce scatter around predictions, providing physically-based prediction equations less influenced by spurious correlations. Wave force results used in the following sections were extracted from signals filtered using wavelet transforms as in Section 4.

6. New prediction method 6.2. Parametric analysis 6.1. General Guidance for evaluating wave loads for decks and beams was derived within the “Exposed Jetties” research project, see McConnell et al. (2003, 2004) and Tirindelli et al. (2003a). New equations/coefficients were developed to provide designers with safe and user-friendly prediction methods. The “Exposed Jetties” data on wave loads did however suffer from some

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

Through the review of previous prediction methods for wave loading on platforms/decks, a series of geometric and hydrodynamic variables influencing forces F applied to a plate are identified in Eq. (17).  : : ð17Þ F ¼ f Hmax;Hs ;d;g max ;Tm ;L;sm ;c;u;u;bl ;bw ;bt ;cl ;h; h;l;l

:

where terms not previously defined are Hmax = maximum wave height, ηmax = maximum crest elevation, u = local water velocity vector, u˙ = local water acceleration vector. Among the variables in Eq. (17), the most informative appear to be Hs, ηmax, c, cl, Tm, and sm. Analysis of the new data confirmed the importance of these variables. Clear trends were found between vertical and horizontal loads (at 1/250 exceedance level) and Hs, ηmax, and cl (see Tirindelli et al., 2003b, Tirindelli, 2004 and Cuomo, 2005). Regardless of other wave and geometry parameters, the magnitude of forces depends strongly on wave height. The new data show clear increases of wave loads with Hs, see Fig. 18 (top panel: quasi static load, bottom panel: impacts). The variable most closely linked to Hs is ηmax. Although both wave height and maximum crest elevation represent the energy of the wave field approaching the jetty, ηmax is more strongly linked to the loading process as it takes into account the non-linear dependence of the wave profile from both wave period and water depth. This is particularly true for structures in

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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, the probability of occurrence of wave loading itself varies according to the probability distribution of ηmax (see Cuomo, 2005 and Bentiba et al., 2004).

Unfortunately, although being more informative than Hs, values of ηmax are not always easily available. With this in mind, 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),

Fig. 22. Quasi-static vertical (upward) forces on deck elements solid lines obey Eq. (18) with coefficients a and b given in Table 1.

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the maximum wave height Hmax is calculated assuming the most appropriate local wave height distribution. The value of ηmax is then taken as the maximum crest elevation for a wave having H = Hmax, T = Tm and propagating in water of constant depth d. Results presented here have been obtained by describing the incoming wave profile according to Fenton's Fourier Transform, which is valid for regular waves in the range of parameter tested. Forces and pressures measured during these experiments show strong dependence on ηmax, evaluated as above, with wave loads increasing with increasing ηmax. The parameter that best synthesizes the geometrical information that must be taken into account for wave loading on exposed jetties is the clearance cl. The composite variable (ηmax − cl) provides an effective measure of water that inundates the deck or beam. The variable (ηmax − cl) can therefore be considered as the most useful variable for calculations of wave loadings on exposed jetties. 6.3. Dimensionless analysis Bearing in mind that the most important variables for providing prediction of wave loading on structures as jetties are Hs, ηmax and cl, and considering that a dimensionless approach is required for generalisation of results, Eq. (18) (forces) and Eq. (19) (pressures) have been fitted to the new dataset. ⁎ F1=250 ¼

⁎ P1=250

g −c  Fqs 1=250 l ¼ a  max þb qw  g  Hs  A d

g −c  Pqs 1=250 l ¼ ¼ a  max þb qw  g  Hs d

ð18Þ

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), A is the area of the element (orthogonal to the direction of application of the load), and a and b are empirical fitting coefficients.

673

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

These equations make forces or pressures dimensionless through the use of Hs, and identify linear trends between dimensionless forces and (ηmax − cl) / d including the main dependences in Section 4.1. Results of the experimental fitting to Eqs. (18) and (19) are reported in Section 7. 7. Results 7.1. Horizontal quasi-static loads on beams 7.1.1. Positive (shoreward) loads The main components of horizontal loads on external beams are quasi-static, and for this case coefficients a and b in Eq. (19) are derived for pressure data (for which the sampling rate at 200 Hz is high enough to cover the quasi-static component of the signal). Responses of the pressure transducers will not have been corrupted by their dynamic response as were the force measuring devices. Dimensionless pressures P⁎qs 1/250 =Pqs 1/250 /(ρw·g·Hs) on external beam are plotted against (ηmax −cl) /d in Fig. 19. The line given by

Fig. 23. Quasi-static vertical (downward) forces on deck elements, solid lines obey Eq. (18) with coefficients a and b given in Table 2.

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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 three configurations. The relatively high scatter is due to inherent spatial variability; the pressure transducers only give a local view rather than spatially averaged. For configurations without side panels, strong 3-dimensional effects increased spatial effects. Where wave

effects are constrained to be primarily 2-dimensional, the side panels channel waves towards the structure, limiting dispersion and generating the highest pressures of all configurations studied. The assumption that horizontal loading is constant across the width of the flume is well-supported. For all three configurations,

Table 1 Coefficients a and b for fit lines and values of R2 for Eqs. (18) and (19), positive loads; sε in model units: pressure [kPa] and force [N] Parameter

Direction

Element

Position

Configuration

a

b

R2

se

Pressure Pressure Pressure Force Force Force Force Force Force Force Force Force Force Force

Horizontal Horizontal Horizontal Horizontal Vertical Vertical Vertical Vertical Vertical Vertical Vertical Vertical Vertical Vertical

Beam Pa and Pb Beam Pa and Pb Beam Pa and Pb Beam Beam Beam Beam Beam Deck Deck Deck Deck Deck Deck

Ext Ext Ext Int Ext Ext Ext Int Ext Ext Ext Int Int Int

FD P NP NP FD P NP NP FD P NP FD P NP

1.19 1.19 1.19 0.56 1.74 0.71 1.10 1.36 2.31 1.23 1.57 0.83 0.58 1.57

0.43 0.43 0.43 0.75 0.14 0.57 0.46 0.46 0.05 0.51 0.52 0.13 0.19 0.73

0.90 0.87 0.96 0.90 0.96 0.97 0.96 0.89 0.95 0.96 0.84 0.69 0.67 0.95

0.34 0.22 0.17 6.84 1.68 1.24 1.61 2.27 6.78 7.28 7.64 9.80 6.57 11.21

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675

Table 2 Coefficients 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 Force Force Force Force Force Force Force Force Force Force Force Force Force

Horizontal Horizontal Horizontal Horizontal Vertical Vertical Vertical Vertical Vertical Vertical Vertical Vertical Vertical Vertical

Beam Beam Beam Beam Beam Beam Beam Beam Deck Deck Deck Deck Deck Deck

Ext Ext Ext Int Ext Ext Ext Int Ext Ext Ext Int Int Int

FD P NP NP FD P NP NP FD P NP FD P NP

− 0.77 − 0.56 − 0.84 0.00 − 1.89 0.00 − 0.04 − 0.23 − 1.95 0.00 − 0.66 − 0.52 − 0.08 − 1.35

0.00 − 0.04 0.04 − 0.22 − 0.12 − 0.49 − 0.48 − 0.29 0.03 − 0.51 − 0.36 − 0.05 − 0.06 − 0.29

0.87 0.91 0.75 0.47 0.86 0.69 0.69 0.87 0.94 0.72 0.61 0.89 0.94 0.89

2.15 0.86 1.08 1.35 1.90 1.74 2.25 0.76 7.80 7.57 8.74 2.24 0.81 4.90

the magnitude of horizontal dimensionless pressures on external beam is slightly higher (∼20%) than that of horizontal forces, see Fig. 20, suggesting that predicted pressures will generally give conservative results for design purposes. 7.1.2. Negative (seaward) loads As the wave travels through the structure, it may apply a reverse or seaward force on the shoreward face of the vertical elements, giving a net negative (seaward) force, whose magnitude may be comparable to the positive (shoreward) quasi-

static component if the element is still immersed. Pressure signals measured only on the external face of the element do not therefore give a reliable picture of the overall process, so values for negative loads are extracted only from force time histories. ⁎ Dimensionless negative horizontal forces Fqs−1/250 =Fqs− 1/250 / (ρw·g·Hs·A) on the external beam are plotted against (ηmax − cl) / d in Fig. 21. The linear fits provide good estimates of the forces for the three different configurations, confirming that horizontal loads on external elements are not significantly affected by the structural configuration.

Fig. 26. Peak forces versus quasi-static vertical (upward) forces on deck elements, solid lines obey Eq. (22) with coefficient a given in Table 3.

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Table 3 Coefficient 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 Force Force Force Force Force Force Force Force Force Force Force

Beam Beam Beam Beam Beam Deck Deck Deck Beam Deck Deck Deck

0.90 0.89 0.94 0.48 0.32 0.93 0.64 0.85 0.69 0.98 0.88 0.96

1.10 25.62 5.38 1.24 3.41 15.81 7.28 20.41 8.36 21.32 6.57 26.60

Horizontal Horizontal Vertical Vertical Vertical Vertical Vertical Vertical Vertical Vertical Vertical Vertical

Ext Int Ext Ext Ext Ext Ext Ext Int Int Int Int

All NP FD P NP FD P NP NP FD P NP

2.45 3.35 2.87 1.74 2.28 2.35 1.99 2.22 2.59 2.35 1.84 2.29

7.2. Vertical quasi-static loads on beams and decks 7.2.1. Positive (upward) loads The most complete data from the experiments are for vertical uplift forces on horizontal elements. Deck elements (and to a lesser degree beam elements) expose significant horizontal areas to wave action and may therefore be subject to important quasi-static uplift loads. Dimensionless uplift forces F⁎qs+1/250 = Fqs+1/250 / (ρw·g·Hs·A) on external beams and external and internal decks are plotted in Fig. 22 against (ηmax − cl) / d for each of the configurations tested. Data for the internal beam element are not presented here because some force transducer results may have been corrupted during the experiments. The linear response of forces with (ηmax − cl) /d is clear from all the plots, particularly for external elements (first two rows in Fig. 22), which are less influenced by the different configura-

tions, particularly for the most seaward facing element (external beam, first row of Fig. 22), where lines for the three different configurations are very similar. The configurations with side panels (P) seem to provide the highest loads for external elements. Without the side panels (NP), 3-dimensional wave interactions with the complex structure make forces on the internal deck less predictable (see third row of Fig. 22). Forces on internal elements are more sensitive to any 3-dimensional effects, see Section 7.2.2. Even for these elements, however, simple linear equations still give reasonable predictions for uplift loads. 7.2.2. Negative (downward) loads Downward forces may be by suction where sideways flows are restrained by continuous beams, but will be substantially increased when the deck is inundated by waves. Even for this ⁎ 1/250 are still cortype of loading, dimensionless forces Fqs− related with (ηmax − cl) / d, see Fig. 23. Downward forces on both external and internal deck elements are shown in Fig. 23a for the flat deck (FD). Inundation of the deck is limited and downward forces are mainly due to suction, reducing as the wave travels along the jetty from external to internal elements. The relative importance of inundation on internal deck elements is shown in Fig. 23b. Negative vertical loads are substantially higher, and longer-lasting, when 3-dimensional effects can act. Effects of inundation on downward forces are further illustrated in Fig. 24, showing example vertical force time histories recorded by external and internal deck elements under 3-dimensional attack (NP). Green water overtopping significantly increases downward loads; when lateral inundation of the deck is not prevented, downward loads on internal elements might be larger and longer-lasting than on external elements.

Fig. 27. Peak forces versus quasi-static forces on beams: horizontal (left) and vertical (right), solid lines obey Eq. (22) with coefficient a given in Table 3.

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

N P FD All

El-Ghamry (1965)

Wang (1970)

Broughton and Horn (1987)

Shih and Anastasiou (1992)

Suchithra and Koola (1995) (min)

(max)

0.120 0.144 0.192 0.084

0.130 0.149 0.153 0.082

0.080 0.095 0.584 0.155

0.104 0.087 0.252 0.083

0.135 0.157 0.592 0.169

0.344 0.364 1.362 0.393

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 Fqs

1=250

 ðqw  g  Hs  AÞ:

ð20Þ

Quasi-static loads on structural elements of exposed jetties as measured during the physical model tests are compared with predictions by Eq. (20) in Fig. 25. The comparison shows reasonable agreement, with values of a and b listed in Tables 1 and 2, together with the corresponding estimates of goodness of fit in terms of R2 (taken as the ratio of the sum of the squares of the regressions and the total sum of the squares, and evaluated using robust fitting) and the standard error of the estimate, that is: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP u Nt u ð yi − yî Þ2 t se ¼ i¼1 ðNt −2Þ

ð21Þ

where yi and ŷi are respectively measured and predicted loads and Nt is the number of tests used in each fit. It should be noted that values of sϵ are given in model units, that is pressure in kiloPascal [kPa] and force in Newton [N]. Compared with predictions by other methods in Figs. 14 and 15, the equations and coefficients derived here show significant improvement, considering that a single unique relationship has been used to predict somewhat variable loads on 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-static loads. Nevertheless, some important information about impulsive loads has also been derived from the revised data. Using definitions discussed earlier, impulsive loads on beams and decks can reach 3 times the value of corresponding quasi-static loads. Ratios between ⁎ 1/250 =Fmax 1/250 / ( ρw·g·Hs·A) (at 1/250 dimensionless impact Fmax ⁎ (at 1/250 level) are level) and quasi-static positive forces Fqs+ shown in Fig. 26 for both external and internal deck elements. Fitting lines in Fig. 26 have the following expression: ⁎ ⁎ Fmax 1=250 ¼ a  Fqsþ1=250

where a is an empirical coefficient, given in Table 3.

ð22Þ

Kaplan et al. (1995)

Present formulae

0.131 0.137 0.116 0.094

0.050 0.058 0.074 0.034

Similar trends are observed in Fig. 27 for horizontal (a) and vertical (b) loads on the external beams (FD). Horizontal impacts are on average 2.5 times the corresponding quasi-static loads, whereas for vertical impacts the ratio is higher (2.9). The highest impact ratio (a = 3.4 times quasi-static) are recorded on the internal elements, where the complex geometry of the structure may trap and amplify wave effects. It must be stressed that impulsive loads measured during physical model tests have relatively short rise times (tr b 0.1·Tm) that might fall within the range of the natural periods of vibration of prototype structures. This might result in amplification and/or reduction of actual loading (Oumeraci and Kortenhaus, 1994). Indeed, when significant impulsive loads are expected to act on the suspended deck structure, the evaluation of the impact load to be used in design analysis must account for the dynamics of the prototype structure. Guidance for the evaluation of the effective load to be adopted in early feasibility studies can be found in Cuomo (2005). 7.5. Comparison with existing methods Comparison of the new method with models reviewed previously confirms the improvement in prediction. Values of relative error E defined in Eq. (23) for a series of existing methods and for the new method are shown in Table 4. 1 E¼ Nt

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi Nt
X yi − yî 2 : yi 1

ð23Þ

As most present methods were developed to predict only vertical wave loads on horizontal slabs, values in Table 4 only refer to quasi-static uplift forces on external deck. 8. Summary and conclusions An extensive review of previous work has identified gaps and inconsistencies in prediction methods for wave-in-deck loads. The need for guidance for wave loads on suspended deck structures in exposed locations originally motivated research within the “Exposed Jetties” project, including a series of flume tests to measure wave-induced loads on deck and beam elements, and that work has been extended here. Time histories of vertical and horizontal forces have been reanalysed using wavelet transform, providing important insight into physical mechanisms of the loading process. An improved

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prediction method has been derived to include the key variables that influence the loading process, but retaining simplicity in the prediction formulae. Results from the experimental fitting of the new data have been fully described above. Each prediction line is parameterised by Eqs. (18) and (19) and slope and intercept coefficients a and b in Tables 1 and 2 with corresponding regression fit R2 and standard error se. Similarly, the values of coefficient a in Eq. (22) are listed in Table 3, together with their regression fit R2 and standard error se. By selecting appropriate values of the empirical coefficients a and b in Tables 1–3, the new methodology succeeds in accounting for different structural configurations of the jetty and for relative location of structural element along the jetty deck, fitting coefficients to each sub-set. Comparison with experimental data shows good agreement between measured and predicted forces. When compared to previous models, variability around predicted values is significantly reduced, see Table 4. Acknowledgements This analysis has been supported by Universities of Roma TRE and Bologna, and HR Wallingford. The authors gratefully acknowledge contributions from the DTI PII Project on Exposed Jetties (39/5/130 cc2035), Prof. Leopoldo Franco, University of Roma TRE, Prof. Alberto Lamberti, University of Bologna; Terry Hedges, University of Liverpool, Kirsty McConnell and Ian Cruickshank, HR Wallingford; and visiting researchers at HRW Amjad-Mohammed Saleem and Oliver de Rooij. The authors wish to thank the very valuable comments to an early version of this manuscript provided by Andreas Kortenhaus and Alf Tørum. References Allsop, N.W.H., 2000. Wave forces on vertical and composite walls. Chapter 4 In: Herbich, J. (Ed.), Handbook of Coastal Engineering. McGraw-Hill, New York, pp. 4.1–4.47. Allsop, N.W.H., Vann, A.M., Howarth, M., Jones, R.J., Davis, J.P., 1995. Measurements of wave impacts at full scale: results of fieldwork on concrete armour units. Conf. on Coastal Structures and Breakwaters ‘‘95, ICE. Thomas Telford, London. ISBN: 0 7277 2509 2, pp. 287–302. Bea, R.G., Iversen, R., Xu, T., 2001. Wave-in-deck forces on offshore platforms. J. Offshore Mech. Arct. Eng. 123, 10–21. Bentiba, R., Cuomo, G., Allsop, N.W.H., Bunn, N., 2004. Probability of occurrence of wave loading on jetty deck elements. Proc. 29th ICCE. World Scientific, Lisbon. ISBN: 981-256-298-2, pp. 4113–4125. Blackmore, P.A., Hewson, P.J., 1984. Experiments on full-scale wave impact pressures. Coast. Eng. 8, 331–346. Bolt, H.M., 1999. Wave-in-deck load calculation methods. BOMEL, Airgap Workshop, Imperial College. Health and Safety Executive, London. 16 pp. Broughton, P., Horn, E., 1987. Ekofisk Platform 2/4C: re-analysis due to subsidence. Proc. Inst. Civ. Eng. 1 (82), 949–979 (Oct.). Chan, E.S., Melville, W.K., 1988. Deep-water plunging wave pressures on a vertical plane wall. Proc. R. Soc. Lond., A 417, 95–131. Cuomo, G., 2005. Dynamics of wave-induced loads and their effects on coastal structures. Final Dissertation PhD in Science of Civil Engineering, University of Roma TRE, Italy.

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