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Pile group effect on the wave loading of a slender pile: A small-scale model study
Ocean Engineering 108 (2015) 449–461
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Pile group effect on the wave loading of a slender pile: A small-scale model study Lisham Bonakdar n, Hocine Oumeraci Leichtweiss-Institute for Hydraulic Engineering and Water Resources, Technische Universität Braunschweig, Beethovenstrasse. 51a, 38106 Braunschweig, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 22 December 2014 Accepted 14 August 2015
Closely-spaced piles are the most essential components for the structural safety of some types of coastal and offshore structures. Wave-induced loading of such piles is among the most uncertain issues in the design of pile group-supported marine structures. Therefore, the correct estimation of the wave loading of a pile within a pile-group in different arrangements is crucial for both safety and costs. Unlike the case of single isolated piles, less attention have been paid to pile groups where wave–pile group interaction is more obvious due to interference effects that affect the wave-induced flow around the piles. In this study, the pile group effect on the wave loading of a slender pile due to non-breaking and focused breaking waves is investigated by means of small scale experiments performed in the 2 m-wide wave flume of Leichtweiss-Institute for Hydraulic Engineering and Water Resources (LWI) in Braunschweig, Germany. Different pile arrangements including single, side by side, tandem, 2 2 and staggered with relative spacing of SG/D ¼0.5–5 were tested. Based on the results of this study an improved understanding of the processes associated with the interaction of waves and pile groups has been achieved. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Wave load Slender pile Pile group effect Pile group arrangement Relative spacing KC number
1. Introduction Pile-supported structures commonly found in a coastal or offshore environment are generally built by means of a group of piles in different arrangements. In the offshore environment, these structures are used for offshore oil and gas platforms (Fig. 1). In the coastal environment, they are widely used in marine transportation systems, for instance for the construction of sea bridges, piers and jetties (Fig. 2). According to the angle of the connecting line of the piles relative to the wave direction, pile groups are commonly categorised into three basic types (Fig. 3): (i) Tandem arrangement, where the angle of the centre connexion line of the cylinders relative to the wave direction is 01, (ii) side by side arrangement, where the incident wave direction is orthogonal to the connecting line of the piles located next to each other, and (iii) staggered arrangement in which the angle is between 01 and 901.
n
Corresponding author. Tel.: þ 49 5313913938. E-mail address: [email protected] (L. Bonakdar).
http://dx.doi.org/10.1016/j.oceaneng.2015.08.021 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
In such structures vertical piles are closely spaced, so that the wave load on a single slender pile is significantly affected by the neighbouring piles and can thus not be calculated by the commonly applied formulae for a single isolated pile which are generally based on the concept of Morison et al. (1950). Severe forces may cause considerable damage to load carrying members and may also affect the overall stability of the structure. A failure of a marine structure would not only cause significant financial losses, but might also result in widespread environmental damages, thus underlining the importance of the safe design of pile group-supported offshore structures. In some other cases, where the arrangement of the piles in a group results in a reduction of wave forces on one or more piles, neglecting the grouping effect might cause an overestimation of the resulting wave force and, consequently, an overdesign of the pile groupsupported structure. In the case of slender piles where both drag and inertia forces induced by highly complex turbulent flow are important, an analytical solution is hardly feasible. Given the high complexity of the interaction between waves and pile groups in different arrangements, laboratory experiments still represent the most reliable alternative. While a large number of research studies are rather related to wave loads on single piles, only very few small and large scale laboratory tests have been conducted to study the interference effects of neighbouring piles. In general, two methods
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have been mostly applied to analyse non-breaking wave loads on a cylindrical pile within pile groups: (i) “Force coefficient approach”: In this approach, the inertia and drag coefficients (CM and CD) are determined based on the knowledge of both flow velocity and acceleration by applying for instance the least square fit. The latter has been applied among others by Chakrabarti (1981, 1982), Smith and Haritos (1997) and Hildebrandt et al. (2008). Using the calculated drag and inertia coefficients, Chakrabarti (1981, 1982) computed maximum wave forces and found a relatively good agreement with measured forces. Smith and Haritos (1996, 1997) reported that drag and inertia coefficients are dependent on the KC number (KC¼umaxT/ D) and relative spacing SG/D where umax is the maximum horizontal wave-induced flow velocity, T is wave period, D is pile diameter and SG is the gap between the surfaces of two neighbouring piles in a pile group. Drag and inertia coefficients are generally plotted versus KC number for different relative spacings SG/D in these studies. However, the proposed CD and CM values are noticeably scattered; i.e. significantly different CD and CM values are obtained for a given KC number. (ii) “Wave force approach”: In this approach, the wave loading of a slender pile within a pile group is generally determined in relation to the wave load on the same pile without any
Fig. 1. Pile group-supported offshore platform (www.fmctechnologies.com).
neighbouring piles and as a function of the most relevant influencing wave and structural parameters. Based on this approach, Li et al. (1993) stated that the wave-induced force on a slender pile within a group of piles depends on the KC number and relative spacing SG/D. Mindao et al. (1987) introduced two parameters named interference coefficient Kg and shelter coefficient Kz for side by side arrangement and tandem arrangement, respectively. Both Kg and Kz coefficients are representative for the force ratio (FGroup/FSingle) where FGroup is the total wave force on a slender pile within further neighbouring piles and FSingle is the total wave force on a single isolated pile. They stated that SG/D is the most significant parameter and proposed the two following formulae for the calculation of interference coefficient Kg and shelter coefficient Kz for side by side arrangement and tandem arrangement, respectively: ð1Þ K g ¼ 1:265 0:225 Ln SG =D for side by side arrangement K z ¼ 0:836 þ 0:141 Ln SG =D for tandem arrangement
ð2Þ
In the proposed formulae (Eqs. 1 and 2), the wave conditions have no influence on interference coefficient Kg and shelter coefficient Kz. For a given pile group arrangement, both coefficients only depend on relative spacing parameter SG/D which was varied from 0.5 to 3 in the laboratory tests. Li et al. (1993) introduced significant pile group effect KG1/3 for piles in side by side arrangement exposed to irregular waves. He found out that the maximum KG1/3 occurs when KC number is between 15 and 20 for the case of pure waves. They also showed that, for a given pile group configuration, the combination of wave and current results in smaller pile group effect compared to wave action only. For shallow water conditions, Kudeih et al. (2010) made an experimental study on the wave-induced as well as wave and currentinduced forces on three piles in an array. They stated that wave load values on the instrumented pile group are significantly affected by the gap size. The interaction of waves and slender piles in different pile group arrangements was also studied by means of extensive largescale laboratory tests performed in the Large Wave Flume (GWK). A single isolated pile and 14 pile group configurations including side by side, tandem and staggered arrangements with gaps of up to three times the pile diameter (1 rSG/D r3) were tested. The results were analysed by Sparboom et al. (2006), Sparboom and Oumeraci (2006), Hildebrandt et al. (2008), Bonakdar and Oumeraci (2012, 2014) and Bonakdar (2014). Some of the general conclusions drawn from these analyses are: (i) Pile group effect increases by decreasing the gap between the piles in side by side arrangement.
Fig. 2. Pile group-supported pier structure (www.asrltd.com).
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(ii) The amplification of the wave load on the middle pile in side by side arrangement is more noticeable than the side pile due to the influence of two neighbouring piles from both sides. (iii) For the tested regular waves (5 o KC o38), the resulting wave load on the middle pile in side by side arrangement increases up to 60% in comparison with that on the single isolated pile. For this pile group arrangement, pile group effect becomes negligible for SG/D ¼3 and all piles behave like a single isolated pile in terms of the wave load. (iv) For tandem arrangement with SG/D ¼1, which is the smallest relative spacing tested in GWK, no significant sheltering effect was observed for the tested regular waves (5 oKC o38).
Besides the aforementioned general outcomes, some of the main limitations of the GWK tests which were identified may be summarised as follows: (i) Pile group configurations with smaller relative spacing of SG/ Do 1, where higher amplification and reduction of wave loads on piles are expected, were not tested. Wave
SG
θ =0°
Tandem
D
θ =90° Wave
D
Side by side
SG
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(ii) The tested wave conditions only cover a small range of relative water depth h/L located in the transition zone (h/L¼0.084–0.197). Therefore, the shallow and deep water conditions were not investigated. Moreover, considering the tested KC number values (5oKCo38), the fully dominant drag regime and the fully dominant inertia regime were not fully covered. (iii) Values of the KC and Re parameters change from one section of the pile to another as a result of the variation of the flow velocity with depth. Only the total wave-induced moment on the instrumented pile was measured meaning that the flow velocity was averaged over the water depth. (iv) Cantilever piles (truncated with lower end far from the bottom of the flume) were used in the GWK model set-up. This might result in unrealistic flow behaviour around the group of pile due to the flow separation at the lower end (a more detailed discussion is given in Bonakdar, 2014). The available experimental studies have contributed to enhance the knowledge about the interaction between waves and pile groups. Nevertheless, several weaknesses still remain which should be overcome to achieve a reliable prediction of wave-induced forces on slender piles within pile groups and, consequently, a safe design of pile-supported marine structures. By means of new small scale experiments, this study primarily attempts to achieve the following two main objectives: (1) Further improvement of the understanding of the processes associated with the wave–pile group interaction by focussing on a more systematic identification of the most relevant influencing wave and structural parameters as well as on the relative importance of their effects on the resulting wave load on a slender pile within pile groups in different arrangements and based of this improved knowledge, (2) Development of a new set of formulae for the prediction of wave loads on a slender pile within pile groups in different arrangements as a function of the most relevant influencing hydrodynamic and structural parameters. For this purpose, a hybrid artificial intelligence (AI)-based model for data analysis was successfully implemented (Bonakdar, 2014).
0°<θ<90° Wave
D SG
Fig. 3. The three basic pile group arrangements.
Staggered
While Bonakdar et al. (2015) addresses the latter objective, this paper which primarily focuses on the former objective, is outlined as follows: The model set-up and the deployed measuring/observation techniques are described in Section 2. In the next section, the tested wave conditions and pile structure configurations are discussed. The obtained data are analysed and the results are discussed in Section 4. Finally, the key results are summarised and concluding remarks are drawn in Section 5.
Fig. 4. Plan view of model set-up of LWI tests, exemplarily for a pile group with side by side arrangement.
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and moment on the instrumented pile. These transducers were designed and constructed at LWI based on the measurement principle of using strain gauges bonded on a stainless steel panel. The total force transducers initially developed for the TAPFOR project are capable of measuring positive and negative wave forces up to 60 N exerted in the direction of the wave propagation (more detailed information is given by Oumeraci et al., 2009 and Strusińska-Correia et al., 2014. In addition, local wave forces on short pile segments were also measured by the so-called ring transducers. For this purpose, the S-shaped force sensor KD40S with the length of 4 cm and capacity of measuring positive and negative inline force up to 2 N was used. These three ring transducers were located at the elevations 0.14, 0.30 and 0.46 m below still water level (SWL). In other words, for each test, the local force on the small (4 cm) sections which is hereafter called line force was simultaneously measured at three different relative elevations of the water column (z/h¼ 0.78, 0.53 and 0.28). An Acoustic Doppler Velocimeter (ADV) was installed far from the pile group to measure the undisturbed horizontal wave-induced flow velocity at the elevations of the ring transducers. This will allow us to avoid averaging the flow velocity over the depth. The deployment of three ring transducers at different elevations below SWL also allows us to
2. Model set-up and measuring techniques The plan view and cross section of the model set-up are exemplarily drawn for the case of side by side arrangement in Figs. 4 and 5, respectively. As seen, in addition to the pile group, an isolated single pile was also placed far from the pile group as a reference pile. The distance of 8D between the single isolated pile and the wave flume's wall was found to be sufficient to avoid any influence of the wall on the resulting wave load on the single pile. In order to avoid possible unrealistic flow behaviour at the end of piles, and unlike the piles tested in GWK, the constructed 1-m long piles were stretched to the bottom of the wave flume with a gap of only a 2-cm between piles and bottom (Fig. 5). This very narrow gap is needed for the measurement of total force and moment on the slender piles by means of the deployed force and moment transducers. As the wave-induced flow velocity is negligibly small near the bottom of the flume, the 2-cm gap hardly affects the resulting horizontal wave force. Water surface elevations (wave heights) at the piles were measured by wave gauges. As seen in Fig. 5, force and moment transducers were placed on the top of the 5 cm diameter piles to measure the total wave force
+1.25 m
Force transducer
D=0.05 m Ring transducers
ADV Wall
+0.64 m
1.0 m
Observation window
Moment transducer
Pile group
+0.50 m
Single pile Ring transducers
+0.34 m
+0.18 m
Wave gauge +0.02 m
≥ 8D
≥ 4D
+0.00 m
2D
4D
8D
Wave gauge 3 piles in side by side arrangement
Single isolated pile ADV
Fig. 5. Model set-up of LWI tests, exemplarily for a pile group with side by side arrangement; (a) cross-section; (b) snapshot: test with regular non-breaking wave; (c) snapshot: test with focused breaking wave.
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Force transducer
Moment transducer 8.5 cm 2.5 cm
2.5 cm
3 cm
Gap (1 mm) Ring transducer (25 mm) Gap (1 mm) Body
2.5 cm
3 cm
Ring Transducer (10 N) to measure line impact load above SWL at different elevations
2.5 cm 4 cm
Body
28 cm
Ring Transducer (2 N) to measure line force below SWL at different elevations
100 cm Gap (1 mm) Ring transducer (40 mm)
Force Transducer to measure total wave force
4 cm
Gap (1 mm) Body
12 cm 4 cm
Moment Transducer to measure inline moment
12 cm
4 cm
Pile diameter = 5 cm
14 cm
Fig. 6. Details of the main instrumented pile tested in the LWI model.
examine whether and how the group effect varies over the water depth. The positions of wave gauge and ADV and their distance from each other as well as from the piles are indicated in Figs. 4 and 5. The main instrumented pile, which is shown as the middle pile of the side by side arrangement in Fig. 5 has three more ring transducers above SWL. These 2.5-cm ring transducers for inline forces up to 10 N were designed to measure local impact forces induced by focused waves at different elevations over the entire impact area. The analysis of these measurements is, however, not part of this paper (see Bonakdar, 2014). The details of the main instrumented 5-cm diameter pile are illustrated in Fig. 6.
3. Pile configurations and wave conditions 3.1. Pile configurations In addition to the three basic pile group arrangements (side by side, tandem and staggered), a new pile group arrangement named hereafter “2 2 arrangement” was also tested which may be considered as a combination of the three basic arrangements. Fig. 8 shows all 25 configurations tested in the LWI wave flume. As seen, each pile arrangement was tested for a wide range of relative spacing SG/D varying from 0.5 where piles are very closely spaced to 5.0 where the piles are so widely spaced that no pile group effect is expected. The regular non-breaking wave conditions tested for each pile group configuration are summarised in Table 1. 3.2. Wave conditions
Fig. 7. Wave and structure conditions of the laboratory tests performed in GWK and LWI (modified from Oumeraci (2008)).
In the GWK tests, the wave heights and periods, together with the water depth, covered only a limited range of wave conditions which are all located in the transition zone, i.e. both deep water and shallow
water conditions were not considered. In the smaller scale model experiments (LWI), the test programme was therefore expanded to cover a broader range of hydrodynamic conditions. For the pile
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Pile Arrangement
Relative Spacing (SG /D)
Number of tests
-------------
24
0.5, 0.75, 1, 1.5, 3, 4 and 5
136
0.5, 0.75, 1, 1.5, 2, 3 and 5
146
0.6, 0.75, 1, 1.5, 3 and 5
120
0.5, 0.75, 1 and 2
83
D Single
SG
Tandem
SG
SG
Side by side
SG
Staggered
SG
SG 2x2
SG
Instrumented pile
Neighboring pile
Fig. 8. Pile group configurations performed in the LWI wave flume tests. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.) Table 1 Test programme for regular non-breaking waves. Test number
Water depth (m)
Wave height (m)
Wave period (s)
Wave steepness (H/L)
Relative pile diameter (D/H)
Relative water depth (h/L)
KC number at z/ h ¼ 0.78*
Re number at z/ h ¼0.78*
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64
0.044 0.062 0.080 0.087 0.088 0.105 0.133 0.149 0.153 0.154 0.209 0.213 0.275 0.255 0.275 0.217 0.217 0.228 0.215 0.29 0.345 0.305 0.328 0.233
0.8 1.0 1.2 2.5 3.5 5.0 1.2 2.0 2.5 3.0 4.7 6.0 4.7 5.5 6.0 1.5 2.0 2.5 3.0 3.5 4.7 5.5 6.0 1.5
0.044 0.04 0.036 0.014 0.01 0.008 0.061 0.031 0.0248 0.0205 0.0176 0.014 0.023 0.018 0.018 0.068 0.045 0.037 0.028 0.033 0.029 0.022 0.022 0.073
1.12 0.80 0.62 0.57 0.56 0.47 0.37 0.34 0.33 0.32 0.24 0.23 0.18 0.20 0.18 0.23 0.23 0.22 0.23 0.17 0.14 0.16 0.15 0.21
0.64 0.41 0.29 0.10 0.072 0.05 0.292 0.134 0.104 0.085 0.054 0.042 0.054 0.046 0.042 0.2 0.135 0.104 0.085 0.073 0.054 0.046 0.042 0.2
1.1 2.22 3.52 8.63 13.4 28.16 5.3 11.66 15.62 18.32 47.93 65 58.87 66 77.6 11 16.77 23.16 25.36 42.84 69.14 77.8 88.46 11.65
3450 5571 7336 8633 9572 14,082 11,046 14,576 15,620 15,272 25,498 27,085 31,313 30,037 32,332 18,342 20,967 23,166 21,136 30,606 36,779 35,367 36,859 19,424
n
Re and KC-values were calculated based on the maximum horizontal wave-induced flow velocity (umax) measured at the relative water elevation of z/h ¼0.78.
configurations shown in Fig. 8, 24 regular non-breaking wave tests with different wave heights and periods were performed (Table 1). As seen in Table 1, the wave steepness varies from 0.008 to 0.073 which was the maximum possible wave steepness without incipient breaking. Relative water depth h/L varies from 0.042 to 0.64 meaning that the full range of wave conditions from deep to shallow water is covered. The KC number changes from 1.1 where inertia regime dominates to 88 where drag regime dominates. Reynolds number varies from Re¼0.34 104 to Re¼3.68 104 indicating that the LWI model tests are located in the subcritical zone. Fig. 7 shows how much
the tested hydrodynamic conditions were expanded in the LWI tests as compared to the GWK tests. The green box shows the area covered in the GWK tests and the pink box shows the conditions tested in the LWI wave flume. As seen, both drag and inertia regimes as well as the transition regime were covered in the LWI experiments, including deep, transition and shallow water conditions. A number of tests with focused breaking waves were also performed in the LWI wave flume for side by side and tandem arrangements with relative spacing of 0.5rSG/Dr3. Focused breaking waves were generated by the superposition of Gaussian wave packets described in
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Table 2 Definition of the loading cases and description of the characteristic appearance of the breaking waves (modified from Wienke (2001)). Loading case (LC)
Description
1 Wave breaking far in front of the cylinder
Breaker tongue impinges the pile at SWL when it is falling downward into the water (broken wave at the pile)
2 Wave breaking in front of the cylinder
Breaker tongue inclined about 451 (breaking wave at the pile)
3 Wave breaking just in front of the cylinder
Breaker tongue impinges the cylinder at the height of the wave peak (breaking wave at the pile)
4 Wave breaking at the cylinder
Breaker tongue formed at the pile (partial breaking wave at the pile)
5 Wave breaking behind the cylinder
Wave breaks immediately behind the pile (non-breaking wave at the pile)
Bergmann (1985). Each of the wave packets was generated with a wave height of 0.231 m and a wave period of 2.35 s. Based on the video analysis, a classification of the tests into five loading cases was performed as proposed by Wienke (2001) and Wienke and Oumeraci (2005). These five loading cases (LC1–LC5) were obtained only by shifting the concentration (focal) point, while the characteristic wave parameters were kept constant. The loading cases varied from broken waves to a wave breaking just behind the instrumented pile as shown in Table 2. LC1 represents a broken wave hitting the pile at the still water level (SWL) exactly when it is falling downward into the water. Due to the entrapped air, the wave has a foamy front. The distance between the breaking point and cylinder front had its maximum for LC1 and was reduced for other load cases. In LC2, the wave hits the pile at a higher elevation before falling down into the water. In LC3, the breaker tongue hits the pile at the height of the wave peak. In LC4, the wave becomes unstable exactly at the pile (partially breaking wave). In LC5, the wave breaks immediately behind the pile representing a non-breaking wave at the pile (see Bonakdar, 2014 for more details).
4. Data analysis, results and discussions 4.1. Pile group effect for regular non-breaking waves A detailed analysis on the effect of non-dimensional wave parameters has been performed by Bonakdar (2014) including KC number, Reynolds number Re, relative water depth h/L and wave
steepness H/L on pile group effect KG. The latter represents the relative wave force ratio (KG ¼fGroup/fSingle) where fGroup is the line force (local force) on a slender pile within further neighbouring piles in different arrangements and fSingle is line force on an isolated single pile. Among all these non-dimensional wave parameters, KC number was identified as the most suitable parameter to describe the effect of wave conditions on pile group interaction (Bonakdar, 2014). In addition, KC number is a function of both wave period and flow velocity which make it an appropriate parameter to describe wave-induced flow conditions. KC number has widely been used in many previous studies on wave loads on single piles and on piles groups (e.g. Chakrabarti 1979, 1981, 1982; Chaplin et al., 1995; Li et al. 1993; Sarpkaya and Isaacson, 1981; Sumer and Fredsøe, 2006; Yuan and Huang, 2010). From the structural point of view, as can be concluded from Fig. 8 and also from the results of the previous studies on wave loads on pile groups (e.g. Chakrabarti, 1979, 1981, 1982; Li et al., 1993; Mindao et al., 1987), pile group arrangement and relative spacing parameter SG/D are the most significant parameters affecting the resulting wave load on a slender pile within other neighbouring piles. Overall, it can be stated that: f Group SG ð3Þ KG ¼ ¼ f KC; ; Pile group arrangement f Single D Considering these most relevant influencing parameters, pile group effect KG is discussed for different pile group arrangements including side by side, tandem, 2 2 and staggered arrangements.
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2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6
Fig. 9. Relationship between pile group effect KG and KC number for side by side arrangement exposed to regular non-breaking waves (KC is calculated based on umax measured at relative water elevation of z/h ¼0.78).
4.1.1. Side by side arrangement In the side by side arrangement, where the incident wave direction is orthogonal to the connecting line of the centres of the piles located next to each other, seven pile group configurations with relative spacing of SG/ D¼0.5, 0.75, 1.0, 1.5, 2.0, 3.0 and 5.0 were tested, including a total of 146 regular non-breaking wave tests. The results illustrating the relationship between pile group effect KG and KC number for side by side arrangement with relative spacing SG/D¼ 0.5–5 are plotted in Fig. 9. The first noticeable implication to be drawn from this figure is that pile group effect KG increases with decreasing relative spacing SG/D as often reported in the literature (e.g. Chakrabarti, 1979, 1981, 1982; Li et al., 1993; Mindao et al., 1987; Sparboom and Oumeraci, 2006). The second important implication from Fig. 9 is that for smaller KC-values (KCo30), two basically different behaviours of the relationship between KG and KC number are distinguished for closely-spaced piles (SG/Dr1.5) and for largely-spaced piles (SG/D41.5): (i) For largely-spaced piles (SG/D 41.5), KG values are more or less the same for the whole range of KC numbers, meaning that the pile group interaction does not anymore depend on the wave conditions. This is also the case for SG/Dr1.5 and larger KCvalues (KC 430). For SG/DZ3 data points are grouped around KG ¼1 for all KC values meaning that already at this value there is no interaction between piles and each pile behaves like a single isolated pile. Mindao et al. (1987) showed that there is no significant pile group effect for the configuration with SG/ D ¼3. Bonakdar and Oumeraci (2012) and Sparboom et al. (2006) found no pile group interaction for SG/D ¼3 for the regular non-breaking waves tested in GWK. (ii) For closely-spaced piles (SG/Dr1.5) where more pile group interaction is expected, the relationship between KG and KC number is similar for all tested relative spacing SG/D¼ 0.5, 0.75, 1.0 and 1.5. For each of these configurations, KG is almost constant for KCo 6. This is the range below which inertia dominates and drag can be completely neglected. It can, in other words, be stated that for inertia dominant conditions (KC o6), pile group interaction is only dependent on relative spacing SG/D and, by decreasing SG/D, KG increases up to 1.3 for the smallest tested spacing (SG/D¼0.5). As the pile is quite slender, KC o6 corresponds to waves with very small wave heights and periods. By increasing the KC number, pile group effect parameter KG sharply increases and reaches its maximum for KC ¼13. For all closely-spaced piles (SG/Dr1.5), the KG values obtained for KC¼ 10–20, which correspond to the range of the transition regime
for drag and inertia, are indeed significantly higher than for KC o10 and KC 420 (Fig. 9). The maximum KG ¼2.4 is reached for the configuration with the smallest tested spacing (SG/D ¼0.5). The pile group effect KG decreases with a lower rate up to KC values of about 30 35. For KC 430–35, further increase of KC does not noticeably affect pile group effect parameter KG and it depends only on relative spacing SG/D. This is the case in which drag dominates and inertia is negligible. For a better understanding of this behaviour, four curves are manually drawn to illustrate the relationship between pile group effect KG and KC number (KC46) for SG/Dr1.5 named as closelyspaced piles (Fig. 9). It can be concluded that the highest amplification of wave load on the middle pile in side by side arrangement occurs when KC number is about 13 where both inertia and drag are important. For the pure inertia regime (KCo6), where the water depth is relatively high and for the pure drag regime (KC430–35), wave pile group effect KG only depends on relative spacing SG/D. A similar behaviour was reported by Li et al. (1993) who investigated the pile group effect of two piles in side by side arrangement subject to irregular non-breaking waves. They introduced significant pile group effect KG1/3 plotted against peak period related KC number KCp in Fig. 10. KCp ¼umaxTp/D was calculated based on peak wave period Tp and maximum wave-induced flow velocity umax at the water surface while in the regular wave tests of this study, KC was calculated based on umax measured at relative water elevation of z/ h¼0.78. Although a direct quantitative comparison is hardly possible between the results of Li et al. (1993) with those obtained in this study (due to differences in the wave and structural conditions as well as the different approaches used for the KC number), both results are qualitatively well-comparable. As shown in Fig. 10, significant pile group effect parameter KG1/3 is around 1.1 for small KC number (e.g. KCp ¼5) and increases by increasing KC number until it reaches its maximum around KCp ¼16 for SC/D¼ (SG/D)þ1¼1.5 (SC and SG are the spacing between the axes of two adjacent piles and the gap between the surfaces of two neighbouring piles, respectively) which represents the smallest relative spacing tested by Li et al. (1993). By further increasing the KC number, the pile group effect decreases which is in agreement with the result of this study (Fig. 9). A further implication from Fig. 10 is that by increasing SC/D from 1.5 to 2, the position of the highest KG1/ 3 changes slightly from a KCp value of about 16 to 18. This might be due to the regression analysis applied to the data which resulted in the fitting curves drawn in Fig. 10. As the data points are not shown by Li et al. (1993), it is not possible to see whether the highest KG1/3 was obtained at different KC values for different SC/D or not.
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Significant pile group effect (KG1/3 )
4.1.2. Tandem arrangement In the case of tandem arrangement, for which the angle of the connecting line of the centres of the piles relative to the wave direction is 01, seven configurations with relative spacing SG/ D ¼0.5, 0.75, 1.0, 1.5, 3.0, 4.0 and 5.0 were performed. In total, 136 regular non-breaking waves were performed for this arrangement. The effect of KC number and relative spacing SG/D on the wave force on the sheltered pile is discussed and pile group effect KG is plotted versus KC number in Fig. 11 for SG/D ¼0.5–5.0. The magnitude of the wave-induced force on the downstream pile does not solely depend on relative spacing SG/D, but also on the KC number. Like for the side by side arrangement, indeed, the interference effect is a multivariate function of both wave and structural parameters. It is apparent from Fig. 11 that pile group effect KG decreases as the spacing between the instrumented piles located in the middle and the pile directly facing the incident waves decreases. In other words, the sheltering effect decreases as the relative spacing keeps increasing. The increase of the sheltering effect is more significant for the closely-spaced piles with SG/Dr1.5. The manually drawn curves provide a picture of the relationship between pile group effect KG and KC number for different relative spacing SG/D. In addition to SG/D, the incident hydrodynamic conditions (KC number) significantly affect the magnitude of the wave force on the shielded pile. For smaller KC numbers (KC o6), where the resulting wave load on the pile is primarily dominated by inertia, pile group effect KG ¼ 0.83–1.0 for different pile configurations. By increasing the KC number, however, the sheltering effect increases for all cases with SG/D r3, but it is more pronounced for the
Significant KC number (KCP) Fig. 10. Pile group effect KG1/3 for middle pile in side by side arrangement for purewave condition (Li et al., 1993).
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configurations with SG/Dr 1.5. A very high sheltering effect was obtained for very large KC values (KC 440), where the wave load is primarily dominated by drag, and the maximum sheltering effect was consequently obtained for the largest KC value of these tests (KC ¼88). In this case (KC ¼88), KG reaches 0.42 for the configuration with the smallest tested relative spacing (SG/D¼ 0.5). For SG/ D4 3, KG values are more or less the same for the whole range of KC numbers and data points are grouped around KG ¼1.0 for all KC values indicating that there is no interaction between piles and each pile behaves like a single isolated pile. 4.1.3. 2 2 Arrangement The so called 2 2 arrangement is commonly found in both offshore and coastal structures and might be considered as a combination of the two basic arrangements “side by side” and “tandem”. For this arrangement, four pile group configurations with relative spacing SG/D¼0.5, 0.75, 1.0 and 2.0 were tested. In total, 83 regular non-breaking waves were performed for this arrangement. The instrumented pile is one of the two downstream piles shown in Fig. 8 with the red circle. The relationship between pile group effect KG and KC number for the 2 2 arrangement is demonstrated in Fig. 12. The most interesting indication from Fig. 12 is that the pile group interaction of the 2 2 arrangement is clearly a combination of both pile group interaction observed in side by side and tandem arrangement. This means that the wave load amplification observed in side by side (Fig. 9) as well as the sheltering effect observed in tandem arrangement (Fig. 11) can also be seen in the 2 2 arrangement (Fig. 12). For 10oKCo30, the amplification effect is more noticeable than the sheltering effect and, consequently, the KG values are higher than 1.0. Interestingly, this range of KC number is the same where higher wave load amplification occurs in side by side arrangement (see Fig. 9). The amplification of the wave force on the measured pile in the 2 2 arrangement is, however, less than that obtained for the middle pile in side by side arrangement. This is due (i) to the presence of the upstream pile by which the instrumented (red) pile is shielded and (ii) to the absence of one of the neighbouring piles at one side of the downstream instrumented pile. As seen in Fig. 12, the instrumented pile in the 2 2 arrangement is, in fact, affected by only one pile while two neighbouring piles influence the resulting wave load on the middle pile in the side by side arrangement. Therefore, the lower wave load amplification for the downstream pile in the 2 2 arrangement as compared to that obtained for the middle pile in side by side arrangement is physically justified.
1.2
1
0.8
0.6
0.4
0.2
0
Fig. 11. Relationship between pile group effect KG and KC number for tandem arrangement exposed to regular non-breaking waves (KC is calculated based on umax measured at relative water elevation of z/h ¼ 0.78).
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1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Fig. 12. Relationship between pile group effect KG and KC number for the 2 2 arrangement exposed to regular non-breaking waves (KC is calculated based on umax measured at relative water elevation of z/h ¼0.78). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
Fig. 13. Relationship between pile group effect KG and KC number for staggered arrangement exposed to regular non-breaking waves (KC is calculated based on umax measured at relative water elevation of z/h ¼0.78).
By further increasing the KC number, the influence of the sheltering effect becomes more important while the amplification effect decreases. For large KC values (KC 430), where the resulting wave load on the pile is fully dominated by drag, pile group effect KG decreases. This is the range of KC number where higher sheltering effect was found compared to other KC values (KC o30) as shown in Fig. 11. The sheltering effect caused by the upstream pile in the 2 2 arrangement is, however, less than that obtained for tandem arrangement. This is because the instrumented pile in the 2 2 arrangement is also affected by the neighbouring side pile. For very small KC numbers (KC o6), where the resulting waveinduced force on the pile is primarily dominated by inertia, pile group effect KG is around 1.0 for all tested configurations with different SG/D values. The qualitative comparison of KG values for KCo 6 shown in Figs. 9, 11 and 12 actually indicates that KG values of the 2 2 arrangement are between those of side by side and tandem arrangement. Overall, it can be stated that, instrumented piles in the 2 2 arrangement have a mixed behaviour including that in side by side arrangement and that in tandem arrangement.
4.1.4. Staggered arrangement In the case of staggered arrangement, where the angle of the centre connexion line of the cylinders relative to the wave direction is 451, six configurations with relative spacing SG/D¼0.6, 0.75, 1.0, 1.5, 3.0 and 5.0 were tested. In total, 120 regular waves were performed for this arrangement. The pile group effect KG is plotted in Fig. 13 against KC number for staggered arrangement. No specific trend can be seen in this figure and a larger scatter of data around KG ¼1.0 line is noticeable. Almost all of the data points for different wave and structural conditions vary between KG ¼ 0.9 and KG ¼1.1, meaning that pile group effect KG depends neither on relative spacing SG/D nor on KC number. In order to study the effect of angle of incident waves on pile group interaction, further investigations on pile group arrangements with different angles (0–901) of the centre connexion line of the cylinders relative to the wave direction are needed. 4.1.5. Pile group effect at different elevations below still water level Pile group effect at different elevations below still water level (SWL) was studied to examine whether pile group effect KG changes over the water column. For this aim, local wave forces were measured on the
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small (4 cm) sections of the instrumented piles located at the different elevations below SWL using the force sensors named ring transducers (see Section 2). For the case of tandem arrangement with the relative spacing of SG/D¼0.75, as an example, Fig. 14 is given to illustrate the relationship between pile group effect KG and KC number at three relative elevations below SWL (z/h¼0.78, 0.53 and 0.28). As seen, the results obtained at z/h¼0.78, 0.53 and 0.28 are very similar meaning that, for the tested conditions, the pile group effect (KG) is not dependent on the elevation (z) at which the local wave force is measured and, therefore, does not change over the water depth (h). 4.2. Pile group effect due to focused breaking waves As the focus was rather put on regular non-breaking waves, only 35 tests with focused waves were conducted for side by side and tandem arrangements with relative spacing of 0.5rSG/Dr3. The results related to pile group effect KG for the focused waves with loading cases LC1–LC5 are shown in Figs. 15 and 16 for side by side and tandem arrangements, respectively. Unlike the case of regular non-breaking waves where pile group effect KG is described by the line wave force ratio (fGroup/fSingle), KG is defined as the total wave force ratio (FGroup/ FSingle) for the case of focused waves. The reason of using total wave force ratio (KG ¼ FGroup/FSingle) for the focused waves is that the loading case, which shows how the focused wave hits the instrumented pile, plays a very significant role on the resulting focused wave-induced force on the pile. In fact, the magnitude of the wave load on the piles and, consequently, pile group effect are mainly dependent on the loading case as the characteristic wave parameters were kept constant for the generation of the focused waves. 4.2.1. Side by side arrangement The first point to be concluded from Fig. 15 is that for side by side arrangement, pile group effect KG is strongly affected by relative spacing SG/D when the latter varies from 1 to 0.5. Another point is that the highest pile group effect KG was found for load case LC5 where the wave breaks immediately after the pile, meaning that the most critical condition which results in the highest amplification of the resulting wave load on a pile in side by side arrangement occurs for non-breaking waves. As an example, for the configuration with relative spacing of SG/D¼0.5, where pile group interaction is most pronounced, KG values for focused breaking waves with different loading cases from broken to partially breaking are less than KG values observed for non-breaking waves. This might be due to higher energy losses in breaking waves. As seen in Fig. 15, pile
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group effect KG decreases by going from partially breaking wave (LC4) to broken wave (LC1). Another possible reason might be the time needed for the load development which, as reported by Hildebrandt and Schlurmann (2012) and Hildebrandt (2013), is higher for non-breaking wave (LC5) than for the load cases with breaking and broken waves. Therefore, load case LC5 has the highest potential for the interference effects of adjacent piles. 4.2.2. Tandem arrangement For the downstream pile in a tandem arrangement, pile group effect KG is plotted versus KC number in Fig. 16 for the tested relative spacing values (0.5rSG/Dr3). It can be generally stated that for the generated focused wave, KG values of broken and breaking waves (LC1–4) are smaller than that of LC5 where wave breaks immediately after the instrumented pile. This finding is physically sound because nearbreaking waves are already instable when they hit the upstream pile directly facing waves. The interaction of near-breaking wave and the upstream pile decreases wave energy and the resulting breaking wave force on the downstream pile is less than that on the upstream pile resulting from the aforementioned interaction. As seen in Fig. 16, the highest sheltering effect occurs for load case LC3 with KG ¼0.56 which was obtained for the configuration with SG/ D¼0.5. For load cases LC2 to LC5, KG decreases by decreasing relative spacing SG/D as expected. However, for load case LC1 representing a broken wave hitting the pile at the SWL when it is falling downward into the water, a larger KG value obtained for SG/D¼ 0.5 in comparison with other configurations with larger SG/D. This unexpected result might be due to the highly stochastic nature of the broken wave as a mixture of water and entrapped air which makes it very difficult to be reproduced in the wave flume. A detailed discussion on the repeatability of the focused waves with different load cases in the LWI wave flume is provided by Bonakdar (2014). In particular, it was shown that the resulting load induced by a broken wave (LC1) greatly differs from one test to another, thus implying that LC1 is the most difficult load case to be reproduced properly in the wave flume.
5. Summary, concluding remarks and outlook 5.1. Summary and concluding remarks Pile groups exposed to regular non-breaking waves as well as to focused breaking waves were investigated, but the focus was put on the former.
1.2
1
0.8
0.6
0.4
0.2
0
Fig. 14. Pile group effect KG at three different elevations below SWL for tandem arrangement with SG/D ¼ 0.75.
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2.2 2
Pile group effect (KG)
1.8 1.6 1.4 1.2 1 0.8 0.6
0.4 0.2 0
0
1
2
3
4
5
Loading Case Fig. 15. Pile group effect KG for focused waves with different loading cases on the middle pile in side by side arrangement with different SG/D.
1 0.9
Pile group effect (KG)
0.8 0.7 0.6 0.5
0.4 0.3 0.2 0.1
0 0
1
2
3
4
5
Loading case Fig. 16. Pile group effect KG for focused waves with different loading cases on the downstream pile in tandem arrangement with different SG/D.
Therefore, a total of 509 small scale laboratory tests were performed in the LWI wave flume to investigate non-breaking wave load on a slender pile within a group of piles in different arrangements. For this aim, a test programme was considered covering a broad range of wave conditions (deep to shallow water) and pile group configurations. The KC number was varied from 1.1 where inertia regime dominates to 88 where drag regime dominates. Different pile arrangements including single, side by side, tandem, 2 2 and staggered were performed and relative spacing SG/D was varied from 0.5 to 5.0. The obtained key results may be summarised as follows: (i) Pile group arrangement, relative spacing SG/D and KC number were found as the most significant parameters affecting wave-induced loads on a slender pile within a pile group. (ii) For side by side arrangement, when KC is between 6 and 35, for which both inertia and drag are important, pile group effect KG is a multivariate function of both hydrodynamic (KC) and structural (SG/D) parameters. In this case, very high KG values (up to 2.4 at SG/D¼0.5) are obtained. For the pure inertia regime (KCo6) where the water depth (h/L¼0.29–0.64) is relatively large and for the pure drag regime (KC430–35) wave pile group
(iii)
(iv) (v)
(vi)
effect KG is independent of KC and only a function of relative spacing SG/D. For tandem arrangement, the highest sheltering effect is obtained for very large KC number (KC ¼88), where the wave load is primarily dominated by drag. In this case, KG reaches 0.42 for the configuration with the smallest tested spacing (SG/D¼0.5). Sheltering effect disappears for SG/D 43 and KG values are more or less equal to 1 for the whole tested range of KC number. For the 2 2 arrangement, the downstream piles behave like in both side by side and tandem arrangement. For staggered arrangement, no specific relationship could be found between pile group effect KG and KC number. In fact, KG varies from 0.9 to 1.1 for almost all performed tests. Therefore, the influence of wave direction on the resulting wave load on a pile in a pile group needs further investigations by testing pile group arrangements with different angles (0–901) of the centre connexion line of the cylinders relative to the wave direction. The relationship between pile group effect KG and KC number at three relative elevations (z/h ¼0.78, 0.53 and 0.28) was investigated and no noticeable effect of elevation z/h, was
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found, so that pile group effect KG can be considered as constant over the entire water depth h. Focused breaking wave tests were conducted for side by side and tandem arrangements. Considering loading cases LC1–LC5 and the pile group arrangements “side by side” and “tandem”, 35 tests with focused wave were performed. For side by side arrangement, the highest amplification of wave load on the instrumented pile was obtained for LC5 which corresponds to a non-breaking wave load as the wave breaks immediately after the instrumented pile. By varying the wave load from non-breaking to partially breaking, breaking and broken waves, pile group effect KG decreases. This might be due to the higher energy losses associated with the wave breaking processes. For tandem arrangement, the KG values of broken, breaking and partially breaking waves (LC1–LC4) are smaller than that of LC5 where the wave breaks just after the measuring pile. Based on the results of this study, a substantial improvement of the understanding of pile group effect on the wave loading of a slender pile has been achieved which might lead to a safer design of pile-supported marine structures exposed to breaking and nonbreaking waves. Based on the improved understanding which has been obtained from the laboratory investigations, a new set of formulae for the prediction of wave-induced force on slender piles within pile groups is derived by Bonakdar et al. (2015) using a hybrid computational data analysis tool consisting of M5 model tree (M5MT) and genetic programming (GP). 5.2. Outlook Based on the knowledge gained from this study, the following recommendations for further research may be drawn: (i) The influence of wave direction on the resulting wave loads on a pile within a pile group needs to be further studied by testing pile group arrangements with different angles between the line connecting the pile centres and the wave direction (0–901). (ii) Further laboratory tests might also be conducted to study the interaction of bores (e.g. broken waves, tsunami bores) and pile groups. (iii) A numerical model needs to be developed and systematically validated against the existing laboratory experiments. The validated numerical model will build an excellent tool for further understanding of the physical processes involved in the wave-pile group interaction. In fact, this will provide more flexible and more comprehensive output options (e.g. waveinduced flow patterns within and around pile groups, including the wave loading of each pile within the pile group). (iv) Wave run-up on a slender pile within pile groups with different arrangements, on which much less attention has been paid compared to the case of single isolated pile, might be investigated to examine the effect of further adjacent piles and further relative spacing SG/D on the wave run-up along the instrumented pile(s).
Acknowledgements The financial support of the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) for this study through the WaPiGS project (Ou 1/13-1) is acknowledged. The support of the German Academic Exchange Service (DAAD, Deutscher Akademischer Austauschdienst) is also acknowledged.
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References Bergmann, J., 1985. Gaußsche Wellenpakete – Ein Verfahren zur Analyse des Seegangsverhaltens meerestechnischer Konstruktionen Ph.D. thesis. TU Berlin, Germany. Bonakdar, L., 2014. Pile Group Effect on the Wave Loading of a Slender Pile. Technische Universität Braunschweig, Germany, ISBN: 978-3-86948-383-2, Ph. D. thesis. Bonakdar, L., Oumeraci, H., 2012. Interaction of waves and pile group-supported offshore structures: a large scale model study. In: Proceedings of the 22nd International Offshore and Polar Engineering Conference (ISOPE), Rhodes, Greece, pp. 812–818. Bonakdar, L., Oumeraci, H., 2014. Small and large scale experimental investigations of wave loads on a slender pile within closely spaced neighbouring piles. In: Proceeding of the 33rd International Conference on Ocean, Offshore and Arctic Engineering (OMAE), San Francisco, USA. Bonakdar, L., Oumeraci, H., Etemad-Shahidi, A., 2015. Wave load formulae for prediction of wave-induced forces on a slender pile within pile groups. Coast. Eng. 102, 49–68. Chakrabarti, S.K., 1979. Wave Forces on Vertical Array of Tubes, Proceeding of Civil Engineering in the Oceans. ASCE, San Francisco, USA, pp. 241–259. Chakrabarti, S.K., 1981. Hydrodynamic coefficients for a vertical tube in an array. Appl. Ocean Res. 3, 2–12. Chakrabarti, S.K., 1982. Inline and transverse forces on a tube array in tandem with waves. Appl. Ocean Res. 4, 25–32. Chaplin, J.R., Subbiah, K., Irani, M., 1995. loading on a vertical cylinder in multidirectional waves. J. Offshore Mech. Arct. Eng. 117, 151–158. Hildebrandt, A., Sparboom, U., Oumeraci, H., 2008. Wave forces on groups of slender cylinders in comparison to an isolated cylinder due to non-Breaking waves. In: Proceeding of the 31st International Conference on Coastal Engineering (ICCE), ASCE, Hamburg, Germany. Hildebrandt, A., Schlurmann, T., 2012. Breaking wave kinematics, local pressures, and forces on a tripod support structure. In: Proceeding of the 33rd International Conference on Coastal Engineering (ICCE), ASCE, Santander, Spain. Hildebrandt, A., 2013. Hydrodynamics of Breaking Waves on offshore Wind Turbine Structures. Leibniz Universität Hannover, Franzius-Institut für Wasserbau, Ästuar- und Küsteningenieurwesen, Germany, Dissertation. Kudeih, M., Nistor, I., Cornett, A., 2010. An experimental study of wave and currentinduced forces on a compact linear array of vertical cylinders in shallow water. In: Proceeding of the 32nd International Conference on Coastal Engineering (ICCE), ASCE, Shanghai, China. Li, Y.C.h., Wang, F.L., Wang, H.R., 1993. Wave-current forces on vertical piles in sideby-side arrangement. In: Proceedings of the 3rd International Offshore and Polar Engineering Conference (ISOPE), Singapore. Mindao, G., Lihua, H., Shaoshu, S., 1987. Experimental study for the wave forces on pile groups due to regular waves. In: Proceedings of the 2nd International Conference on Coastal and Port Engineering in Developing Countries (COPEDEC), China Ocean Press, Beijing, pp. 1956–1965. Morison, J.R., O’Brien, M.P., Johnson, J.W., Schaaf, S.A., 1950. The force exerted by surface waves on piles. Pet. Trans. AIME 189, 149–154. Oumeraci, H., 2008. Vertiefervorlesung Küsteningenieurwesen I. TU Braunschweig, Germany, in German. Oumeraci, H., Reimann, S., Husrin, S., Strusińska-Correia, A., 2009. Development of Measuring Devices – Tsunami Attenuation Performance of Costal Forest TAPFOR. Internal report, TU Braunschweig, Germany. Sarpkaya, T., Isaacson, M., 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold Company, New York. Smith, D., Haritos, N., 1996. The effect of in-line spacing of two cylinder groups on the Morison force coefficients. 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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Pile group effect on the wave loading of a slender pile: A small-scale model study Lisham Bonakdar n, Hocine Oumeraci Leichtweiss-Institute for Hydraulic Engineering and Water Resources, Technische Universität Braunschweig, Beethovenstrasse. 51a, 38106 Braunschweig, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 22 December 2014 Accepted 14 August 2015
Closely-spaced piles are the most essential components for the structural safety of some types of coastal and offshore structures. Wave-induced loading of such piles is among the most uncertain issues in the design of pile group-supported marine structures. Therefore, the correct estimation of the wave loading of a pile within a pile-group in different arrangements is crucial for both safety and costs. Unlike the case of single isolated piles, less attention have been paid to pile groups where wave–pile group interaction is more obvious due to interference effects that affect the wave-induced flow around the piles. In this study, the pile group effect on the wave loading of a slender pile due to non-breaking and focused breaking waves is investigated by means of small scale experiments performed in the 2 m-wide wave flume of Leichtweiss-Institute for Hydraulic Engineering and Water Resources (LWI) in Braunschweig, Germany. Different pile arrangements including single, side by side, tandem, 2 2 and staggered with relative spacing of SG/D ¼0.5–5 were tested. Based on the results of this study an improved understanding of the processes associated with the interaction of waves and pile groups has been achieved. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Wave load Slender pile Pile group effect Pile group arrangement Relative spacing KC number
1. Introduction Pile-supported structures commonly found in a coastal or offshore environment are generally built by means of a group of piles in different arrangements. In the offshore environment, these structures are used for offshore oil and gas platforms (Fig. 1). In the coastal environment, they are widely used in marine transportation systems, for instance for the construction of sea bridges, piers and jetties (Fig. 2). According to the angle of the connecting line of the piles relative to the wave direction, pile groups are commonly categorised into three basic types (Fig. 3): (i) Tandem arrangement, where the angle of the centre connexion line of the cylinders relative to the wave direction is 01, (ii) side by side arrangement, where the incident wave direction is orthogonal to the connecting line of the piles located next to each other, and (iii) staggered arrangement in which the angle is between 01 and 901.
n
Corresponding author. Tel.: þ 49 5313913938. E-mail address: [email protected] (L. Bonakdar).
http://dx.doi.org/10.1016/j.oceaneng.2015.08.021 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
In such structures vertical piles are closely spaced, so that the wave load on a single slender pile is significantly affected by the neighbouring piles and can thus not be calculated by the commonly applied formulae for a single isolated pile which are generally based on the concept of Morison et al. (1950). Severe forces may cause considerable damage to load carrying members and may also affect the overall stability of the structure. A failure of a marine structure would not only cause significant financial losses, but might also result in widespread environmental damages, thus underlining the importance of the safe design of pile group-supported offshore structures. In some other cases, where the arrangement of the piles in a group results in a reduction of wave forces on one or more piles, neglecting the grouping effect might cause an overestimation of the resulting wave force and, consequently, an overdesign of the pile groupsupported structure. In the case of slender piles where both drag and inertia forces induced by highly complex turbulent flow are important, an analytical solution is hardly feasible. Given the high complexity of the interaction between waves and pile groups in different arrangements, laboratory experiments still represent the most reliable alternative. While a large number of research studies are rather related to wave loads on single piles, only very few small and large scale laboratory tests have been conducted to study the interference effects of neighbouring piles. In general, two methods
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have been mostly applied to analyse non-breaking wave loads on a cylindrical pile within pile groups: (i) “Force coefficient approach”: In this approach, the inertia and drag coefficients (CM and CD) are determined based on the knowledge of both flow velocity and acceleration by applying for instance the least square fit. The latter has been applied among others by Chakrabarti (1981, 1982), Smith and Haritos (1997) and Hildebrandt et al. (2008). Using the calculated drag and inertia coefficients, Chakrabarti (1981, 1982) computed maximum wave forces and found a relatively good agreement with measured forces. Smith and Haritos (1996, 1997) reported that drag and inertia coefficients are dependent on the KC number (KC¼umaxT/ D) and relative spacing SG/D where umax is the maximum horizontal wave-induced flow velocity, T is wave period, D is pile diameter and SG is the gap between the surfaces of two neighbouring piles in a pile group. Drag and inertia coefficients are generally plotted versus KC number for different relative spacings SG/D in these studies. However, the proposed CD and CM values are noticeably scattered; i.e. significantly different CD and CM values are obtained for a given KC number. (ii) “Wave force approach”: In this approach, the wave loading of a slender pile within a pile group is generally determined in relation to the wave load on the same pile without any
Fig. 1. Pile group-supported offshore platform (www.fmctechnologies.com).
neighbouring piles and as a function of the most relevant influencing wave and structural parameters. Based on this approach, Li et al. (1993) stated that the wave-induced force on a slender pile within a group of piles depends on the KC number and relative spacing SG/D. Mindao et al. (1987) introduced two parameters named interference coefficient Kg and shelter coefficient Kz for side by side arrangement and tandem arrangement, respectively. Both Kg and Kz coefficients are representative for the force ratio (FGroup/FSingle) where FGroup is the total wave force on a slender pile within further neighbouring piles and FSingle is the total wave force on a single isolated pile. They stated that SG/D is the most significant parameter and proposed the two following formulae for the calculation of interference coefficient Kg and shelter coefficient Kz for side by side arrangement and tandem arrangement, respectively: ð1Þ K g ¼ 1:265 0:225 Ln SG =D for side by side arrangement K z ¼ 0:836 þ 0:141 Ln SG =D for tandem arrangement
ð2Þ
In the proposed formulae (Eqs. 1 and 2), the wave conditions have no influence on interference coefficient Kg and shelter coefficient Kz. For a given pile group arrangement, both coefficients only depend on relative spacing parameter SG/D which was varied from 0.5 to 3 in the laboratory tests. Li et al. (1993) introduced significant pile group effect KG1/3 for piles in side by side arrangement exposed to irregular waves. He found out that the maximum KG1/3 occurs when KC number is between 15 and 20 for the case of pure waves. They also showed that, for a given pile group configuration, the combination of wave and current results in smaller pile group effect compared to wave action only. For shallow water conditions, Kudeih et al. (2010) made an experimental study on the wave-induced as well as wave and currentinduced forces on three piles in an array. They stated that wave load values on the instrumented pile group are significantly affected by the gap size. The interaction of waves and slender piles in different pile group arrangements was also studied by means of extensive largescale laboratory tests performed in the Large Wave Flume (GWK). A single isolated pile and 14 pile group configurations including side by side, tandem and staggered arrangements with gaps of up to three times the pile diameter (1 rSG/D r3) were tested. The results were analysed by Sparboom et al. (2006), Sparboom and Oumeraci (2006), Hildebrandt et al. (2008), Bonakdar and Oumeraci (2012, 2014) and Bonakdar (2014). Some of the general conclusions drawn from these analyses are: (i) Pile group effect increases by decreasing the gap between the piles in side by side arrangement.
Fig. 2. Pile group-supported pier structure (www.asrltd.com).
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(ii) The amplification of the wave load on the middle pile in side by side arrangement is more noticeable than the side pile due to the influence of two neighbouring piles from both sides. (iii) For the tested regular waves (5 o KC o38), the resulting wave load on the middle pile in side by side arrangement increases up to 60% in comparison with that on the single isolated pile. For this pile group arrangement, pile group effect becomes negligible for SG/D ¼3 and all piles behave like a single isolated pile in terms of the wave load. (iv) For tandem arrangement with SG/D ¼1, which is the smallest relative spacing tested in GWK, no significant sheltering effect was observed for the tested regular waves (5 oKC o38).
Besides the aforementioned general outcomes, some of the main limitations of the GWK tests which were identified may be summarised as follows: (i) Pile group configurations with smaller relative spacing of SG/ Do 1, where higher amplification and reduction of wave loads on piles are expected, were not tested. Wave
SG
θ =0°
Tandem
D
θ =90° Wave
D
Side by side
SG
451
(ii) The tested wave conditions only cover a small range of relative water depth h/L located in the transition zone (h/L¼0.084–0.197). Therefore, the shallow and deep water conditions were not investigated. Moreover, considering the tested KC number values (5oKCo38), the fully dominant drag regime and the fully dominant inertia regime were not fully covered. (iii) Values of the KC and Re parameters change from one section of the pile to another as a result of the variation of the flow velocity with depth. Only the total wave-induced moment on the instrumented pile was measured meaning that the flow velocity was averaged over the water depth. (iv) Cantilever piles (truncated with lower end far from the bottom of the flume) were used in the GWK model set-up. This might result in unrealistic flow behaviour around the group of pile due to the flow separation at the lower end (a more detailed discussion is given in Bonakdar, 2014). The available experimental studies have contributed to enhance the knowledge about the interaction between waves and pile groups. Nevertheless, several weaknesses still remain which should be overcome to achieve a reliable prediction of wave-induced forces on slender piles within pile groups and, consequently, a safe design of pile-supported marine structures. By means of new small scale experiments, this study primarily attempts to achieve the following two main objectives: (1) Further improvement of the understanding of the processes associated with the wave–pile group interaction by focussing on a more systematic identification of the most relevant influencing wave and structural parameters as well as on the relative importance of their effects on the resulting wave load on a slender pile within pile groups in different arrangements and based of this improved knowledge, (2) Development of a new set of formulae for the prediction of wave loads on a slender pile within pile groups in different arrangements as a function of the most relevant influencing hydrodynamic and structural parameters. For this purpose, a hybrid artificial intelligence (AI)-based model for data analysis was successfully implemented (Bonakdar, 2014).
0°<θ<90° Wave
D SG
Fig. 3. The three basic pile group arrangements.
Staggered
While Bonakdar et al. (2015) addresses the latter objective, this paper which primarily focuses on the former objective, is outlined as follows: The model set-up and the deployed measuring/observation techniques are described in Section 2. In the next section, the tested wave conditions and pile structure configurations are discussed. The obtained data are analysed and the results are discussed in Section 4. Finally, the key results are summarised and concluding remarks are drawn in Section 5.
Fig. 4. Plan view of model set-up of LWI tests, exemplarily for a pile group with side by side arrangement.
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and moment on the instrumented pile. These transducers were designed and constructed at LWI based on the measurement principle of using strain gauges bonded on a stainless steel panel. The total force transducers initially developed for the TAPFOR project are capable of measuring positive and negative wave forces up to 60 N exerted in the direction of the wave propagation (more detailed information is given by Oumeraci et al., 2009 and Strusińska-Correia et al., 2014. In addition, local wave forces on short pile segments were also measured by the so-called ring transducers. For this purpose, the S-shaped force sensor KD40S with the length of 4 cm and capacity of measuring positive and negative inline force up to 2 N was used. These three ring transducers were located at the elevations 0.14, 0.30 and 0.46 m below still water level (SWL). In other words, for each test, the local force on the small (4 cm) sections which is hereafter called line force was simultaneously measured at three different relative elevations of the water column (z/h¼ 0.78, 0.53 and 0.28). An Acoustic Doppler Velocimeter (ADV) was installed far from the pile group to measure the undisturbed horizontal wave-induced flow velocity at the elevations of the ring transducers. This will allow us to avoid averaging the flow velocity over the depth. The deployment of three ring transducers at different elevations below SWL also allows us to
2. Model set-up and measuring techniques The plan view and cross section of the model set-up are exemplarily drawn for the case of side by side arrangement in Figs. 4 and 5, respectively. As seen, in addition to the pile group, an isolated single pile was also placed far from the pile group as a reference pile. The distance of 8D between the single isolated pile and the wave flume's wall was found to be sufficient to avoid any influence of the wall on the resulting wave load on the single pile. In order to avoid possible unrealistic flow behaviour at the end of piles, and unlike the piles tested in GWK, the constructed 1-m long piles were stretched to the bottom of the wave flume with a gap of only a 2-cm between piles and bottom (Fig. 5). This very narrow gap is needed for the measurement of total force and moment on the slender piles by means of the deployed force and moment transducers. As the wave-induced flow velocity is negligibly small near the bottom of the flume, the 2-cm gap hardly affects the resulting horizontal wave force. Water surface elevations (wave heights) at the piles were measured by wave gauges. As seen in Fig. 5, force and moment transducers were placed on the top of the 5 cm diameter piles to measure the total wave force
+1.25 m
Force transducer
D=0.05 m Ring transducers
ADV Wall
+0.64 m
1.0 m
Observation window
Moment transducer
Pile group
+0.50 m
Single pile Ring transducers
+0.34 m
+0.18 m
Wave gauge +0.02 m
≥ 8D
≥ 4D
+0.00 m
2D
4D
8D
Wave gauge 3 piles in side by side arrangement
Single isolated pile ADV
Fig. 5. Model set-up of LWI tests, exemplarily for a pile group with side by side arrangement; (a) cross-section; (b) snapshot: test with regular non-breaking wave; (c) snapshot: test with focused breaking wave.
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Force transducer
Moment transducer 8.5 cm 2.5 cm
2.5 cm
3 cm
Gap (1 mm) Ring transducer (25 mm) Gap (1 mm) Body
2.5 cm
3 cm
Ring Transducer (10 N) to measure line impact load above SWL at different elevations
2.5 cm 4 cm
Body
28 cm
Ring Transducer (2 N) to measure line force below SWL at different elevations
100 cm Gap (1 mm) Ring transducer (40 mm)
Force Transducer to measure total wave force
4 cm
Gap (1 mm) Body
12 cm 4 cm
Moment Transducer to measure inline moment
12 cm
4 cm
Pile diameter = 5 cm
14 cm
Fig. 6. Details of the main instrumented pile tested in the LWI model.
examine whether and how the group effect varies over the water depth. The positions of wave gauge and ADV and their distance from each other as well as from the piles are indicated in Figs. 4 and 5. The main instrumented pile, which is shown as the middle pile of the side by side arrangement in Fig. 5 has three more ring transducers above SWL. These 2.5-cm ring transducers for inline forces up to 10 N were designed to measure local impact forces induced by focused waves at different elevations over the entire impact area. The analysis of these measurements is, however, not part of this paper (see Bonakdar, 2014). The details of the main instrumented 5-cm diameter pile are illustrated in Fig. 6.
3. Pile configurations and wave conditions 3.1. Pile configurations In addition to the three basic pile group arrangements (side by side, tandem and staggered), a new pile group arrangement named hereafter “2 2 arrangement” was also tested which may be considered as a combination of the three basic arrangements. Fig. 8 shows all 25 configurations tested in the LWI wave flume. As seen, each pile arrangement was tested for a wide range of relative spacing SG/D varying from 0.5 where piles are very closely spaced to 5.0 where the piles are so widely spaced that no pile group effect is expected. The regular non-breaking wave conditions tested for each pile group configuration are summarised in Table 1. 3.2. Wave conditions
Fig. 7. Wave and structure conditions of the laboratory tests performed in GWK and LWI (modified from Oumeraci (2008)).
In the GWK tests, the wave heights and periods, together with the water depth, covered only a limited range of wave conditions which are all located in the transition zone, i.e. both deep water and shallow
water conditions were not considered. In the smaller scale model experiments (LWI), the test programme was therefore expanded to cover a broader range of hydrodynamic conditions. For the pile
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Pile Arrangement
Relative Spacing (SG /D)
Number of tests
-------------
24
0.5, 0.75, 1, 1.5, 3, 4 and 5
136
0.5, 0.75, 1, 1.5, 2, 3 and 5
146
0.6, 0.75, 1, 1.5, 3 and 5
120
0.5, 0.75, 1 and 2
83
D Single
SG
Tandem
SG
SG
Side by side
SG
Staggered
SG
SG 2x2
SG
Instrumented pile
Neighboring pile
Fig. 8. Pile group configurations performed in the LWI wave flume tests. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.) Table 1 Test programme for regular non-breaking waves. Test number
Water depth (m)
Wave height (m)
Wave period (s)
Wave steepness (H/L)
Relative pile diameter (D/H)
Relative water depth (h/L)
KC number at z/ h ¼ 0.78*
Re number at z/ h ¼0.78*
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64 0.64
0.044 0.062 0.080 0.087 0.088 0.105 0.133 0.149 0.153 0.154 0.209 0.213 0.275 0.255 0.275 0.217 0.217 0.228 0.215 0.29 0.345 0.305 0.328 0.233
0.8 1.0 1.2 2.5 3.5 5.0 1.2 2.0 2.5 3.0 4.7 6.0 4.7 5.5 6.0 1.5 2.0 2.5 3.0 3.5 4.7 5.5 6.0 1.5
0.044 0.04 0.036 0.014 0.01 0.008 0.061 0.031 0.0248 0.0205 0.0176 0.014 0.023 0.018 0.018 0.068 0.045 0.037 0.028 0.033 0.029 0.022 0.022 0.073
1.12 0.80 0.62 0.57 0.56 0.47 0.37 0.34 0.33 0.32 0.24 0.23 0.18 0.20 0.18 0.23 0.23 0.22 0.23 0.17 0.14 0.16 0.15 0.21
0.64 0.41 0.29 0.10 0.072 0.05 0.292 0.134 0.104 0.085 0.054 0.042 0.054 0.046 0.042 0.2 0.135 0.104 0.085 0.073 0.054 0.046 0.042 0.2
1.1 2.22 3.52 8.63 13.4 28.16 5.3 11.66 15.62 18.32 47.93 65 58.87 66 77.6 11 16.77 23.16 25.36 42.84 69.14 77.8 88.46 11.65
3450 5571 7336 8633 9572 14,082 11,046 14,576 15,620 15,272 25,498 27,085 31,313 30,037 32,332 18,342 20,967 23,166 21,136 30,606 36,779 35,367 36,859 19,424
n
Re and KC-values were calculated based on the maximum horizontal wave-induced flow velocity (umax) measured at the relative water elevation of z/h ¼0.78.
configurations shown in Fig. 8, 24 regular non-breaking wave tests with different wave heights and periods were performed (Table 1). As seen in Table 1, the wave steepness varies from 0.008 to 0.073 which was the maximum possible wave steepness without incipient breaking. Relative water depth h/L varies from 0.042 to 0.64 meaning that the full range of wave conditions from deep to shallow water is covered. The KC number changes from 1.1 where inertia regime dominates to 88 where drag regime dominates. Reynolds number varies from Re¼0.34 104 to Re¼3.68 104 indicating that the LWI model tests are located in the subcritical zone. Fig. 7 shows how much
the tested hydrodynamic conditions were expanded in the LWI tests as compared to the GWK tests. The green box shows the area covered in the GWK tests and the pink box shows the conditions tested in the LWI wave flume. As seen, both drag and inertia regimes as well as the transition regime were covered in the LWI experiments, including deep, transition and shallow water conditions. A number of tests with focused breaking waves were also performed in the LWI wave flume for side by side and tandem arrangements with relative spacing of 0.5rSG/Dr3. Focused breaking waves were generated by the superposition of Gaussian wave packets described in
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Table 2 Definition of the loading cases and description of the characteristic appearance of the breaking waves (modified from Wienke (2001)). Loading case (LC)
Description
1 Wave breaking far in front of the cylinder
Breaker tongue impinges the pile at SWL when it is falling downward into the water (broken wave at the pile)
2 Wave breaking in front of the cylinder
Breaker tongue inclined about 451 (breaking wave at the pile)
3 Wave breaking just in front of the cylinder
Breaker tongue impinges the cylinder at the height of the wave peak (breaking wave at the pile)
4 Wave breaking at the cylinder
Breaker tongue formed at the pile (partial breaking wave at the pile)
5 Wave breaking behind the cylinder
Wave breaks immediately behind the pile (non-breaking wave at the pile)
Bergmann (1985). Each of the wave packets was generated with a wave height of 0.231 m and a wave period of 2.35 s. Based on the video analysis, a classification of the tests into five loading cases was performed as proposed by Wienke (2001) and Wienke and Oumeraci (2005). These five loading cases (LC1–LC5) were obtained only by shifting the concentration (focal) point, while the characteristic wave parameters were kept constant. The loading cases varied from broken waves to a wave breaking just behind the instrumented pile as shown in Table 2. LC1 represents a broken wave hitting the pile at the still water level (SWL) exactly when it is falling downward into the water. Due to the entrapped air, the wave has a foamy front. The distance between the breaking point and cylinder front had its maximum for LC1 and was reduced for other load cases. In LC2, the wave hits the pile at a higher elevation before falling down into the water. In LC3, the breaker tongue hits the pile at the height of the wave peak. In LC4, the wave becomes unstable exactly at the pile (partially breaking wave). In LC5, the wave breaks immediately behind the pile representing a non-breaking wave at the pile (see Bonakdar, 2014 for more details).
4. Data analysis, results and discussions 4.1. Pile group effect for regular non-breaking waves A detailed analysis on the effect of non-dimensional wave parameters has been performed by Bonakdar (2014) including KC number, Reynolds number Re, relative water depth h/L and wave
steepness H/L on pile group effect KG. The latter represents the relative wave force ratio (KG ¼fGroup/fSingle) where fGroup is the line force (local force) on a slender pile within further neighbouring piles in different arrangements and fSingle is line force on an isolated single pile. Among all these non-dimensional wave parameters, KC number was identified as the most suitable parameter to describe the effect of wave conditions on pile group interaction (Bonakdar, 2014). In addition, KC number is a function of both wave period and flow velocity which make it an appropriate parameter to describe wave-induced flow conditions. KC number has widely been used in many previous studies on wave loads on single piles and on piles groups (e.g. Chakrabarti 1979, 1981, 1982; Chaplin et al., 1995; Li et al. 1993; Sarpkaya and Isaacson, 1981; Sumer and Fredsøe, 2006; Yuan and Huang, 2010). From the structural point of view, as can be concluded from Fig. 8 and also from the results of the previous studies on wave loads on pile groups (e.g. Chakrabarti, 1979, 1981, 1982; Li et al., 1993; Mindao et al., 1987), pile group arrangement and relative spacing parameter SG/D are the most significant parameters affecting the resulting wave load on a slender pile within other neighbouring piles. Overall, it can be stated that: f Group SG ð3Þ KG ¼ ¼ f KC; ; Pile group arrangement f Single D Considering these most relevant influencing parameters, pile group effect KG is discussed for different pile group arrangements including side by side, tandem, 2 2 and staggered arrangements.
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2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6
Fig. 9. Relationship between pile group effect KG and KC number for side by side arrangement exposed to regular non-breaking waves (KC is calculated based on umax measured at relative water elevation of z/h ¼0.78).
4.1.1. Side by side arrangement In the side by side arrangement, where the incident wave direction is orthogonal to the connecting line of the centres of the piles located next to each other, seven pile group configurations with relative spacing of SG/ D¼0.5, 0.75, 1.0, 1.5, 2.0, 3.0 and 5.0 were tested, including a total of 146 regular non-breaking wave tests. The results illustrating the relationship between pile group effect KG and KC number for side by side arrangement with relative spacing SG/D¼ 0.5–5 are plotted in Fig. 9. The first noticeable implication to be drawn from this figure is that pile group effect KG increases with decreasing relative spacing SG/D as often reported in the literature (e.g. Chakrabarti, 1979, 1981, 1982; Li et al., 1993; Mindao et al., 1987; Sparboom and Oumeraci, 2006). The second important implication from Fig. 9 is that for smaller KC-values (KCo30), two basically different behaviours of the relationship between KG and KC number are distinguished for closely-spaced piles (SG/Dr1.5) and for largely-spaced piles (SG/D41.5): (i) For largely-spaced piles (SG/D 41.5), KG values are more or less the same for the whole range of KC numbers, meaning that the pile group interaction does not anymore depend on the wave conditions. This is also the case for SG/Dr1.5 and larger KCvalues (KC 430). For SG/DZ3 data points are grouped around KG ¼1 for all KC values meaning that already at this value there is no interaction between piles and each pile behaves like a single isolated pile. Mindao et al. (1987) showed that there is no significant pile group effect for the configuration with SG/ D ¼3. Bonakdar and Oumeraci (2012) and Sparboom et al. (2006) found no pile group interaction for SG/D ¼3 for the regular non-breaking waves tested in GWK. (ii) For closely-spaced piles (SG/Dr1.5) where more pile group interaction is expected, the relationship between KG and KC number is similar for all tested relative spacing SG/D¼ 0.5, 0.75, 1.0 and 1.5. For each of these configurations, KG is almost constant for KCo 6. This is the range below which inertia dominates and drag can be completely neglected. It can, in other words, be stated that for inertia dominant conditions (KC o6), pile group interaction is only dependent on relative spacing SG/D and, by decreasing SG/D, KG increases up to 1.3 for the smallest tested spacing (SG/D¼0.5). As the pile is quite slender, KC o6 corresponds to waves with very small wave heights and periods. By increasing the KC number, pile group effect parameter KG sharply increases and reaches its maximum for KC ¼13. For all closely-spaced piles (SG/Dr1.5), the KG values obtained for KC¼ 10–20, which correspond to the range of the transition regime
for drag and inertia, are indeed significantly higher than for KC o10 and KC 420 (Fig. 9). The maximum KG ¼2.4 is reached for the configuration with the smallest tested spacing (SG/D ¼0.5). The pile group effect KG decreases with a lower rate up to KC values of about 30 35. For KC 430–35, further increase of KC does not noticeably affect pile group effect parameter KG and it depends only on relative spacing SG/D. This is the case in which drag dominates and inertia is negligible. For a better understanding of this behaviour, four curves are manually drawn to illustrate the relationship between pile group effect KG and KC number (KC46) for SG/Dr1.5 named as closelyspaced piles (Fig. 9). It can be concluded that the highest amplification of wave load on the middle pile in side by side arrangement occurs when KC number is about 13 where both inertia and drag are important. For the pure inertia regime (KCo6), where the water depth is relatively high and for the pure drag regime (KC430–35), wave pile group effect KG only depends on relative spacing SG/D. A similar behaviour was reported by Li et al. (1993) who investigated the pile group effect of two piles in side by side arrangement subject to irregular non-breaking waves. They introduced significant pile group effect KG1/3 plotted against peak period related KC number KCp in Fig. 10. KCp ¼umaxTp/D was calculated based on peak wave period Tp and maximum wave-induced flow velocity umax at the water surface while in the regular wave tests of this study, KC was calculated based on umax measured at relative water elevation of z/ h¼0.78. Although a direct quantitative comparison is hardly possible between the results of Li et al. (1993) with those obtained in this study (due to differences in the wave and structural conditions as well as the different approaches used for the KC number), both results are qualitatively well-comparable. As shown in Fig. 10, significant pile group effect parameter KG1/3 is around 1.1 for small KC number (e.g. KCp ¼5) and increases by increasing KC number until it reaches its maximum around KCp ¼16 for SC/D¼ (SG/D)þ1¼1.5 (SC and SG are the spacing between the axes of two adjacent piles and the gap between the surfaces of two neighbouring piles, respectively) which represents the smallest relative spacing tested by Li et al. (1993). By further increasing the KC number, the pile group effect decreases which is in agreement with the result of this study (Fig. 9). A further implication from Fig. 10 is that by increasing SC/D from 1.5 to 2, the position of the highest KG1/ 3 changes slightly from a KCp value of about 16 to 18. This might be due to the regression analysis applied to the data which resulted in the fitting curves drawn in Fig. 10. As the data points are not shown by Li et al. (1993), it is not possible to see whether the highest KG1/3 was obtained at different KC values for different SC/D or not.
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Significant pile group effect (KG1/3 )
4.1.2. Tandem arrangement In the case of tandem arrangement, for which the angle of the connecting line of the centres of the piles relative to the wave direction is 01, seven configurations with relative spacing SG/ D ¼0.5, 0.75, 1.0, 1.5, 3.0, 4.0 and 5.0 were performed. In total, 136 regular non-breaking waves were performed for this arrangement. The effect of KC number and relative spacing SG/D on the wave force on the sheltered pile is discussed and pile group effect KG is plotted versus KC number in Fig. 11 for SG/D ¼0.5–5.0. The magnitude of the wave-induced force on the downstream pile does not solely depend on relative spacing SG/D, but also on the KC number. Like for the side by side arrangement, indeed, the interference effect is a multivariate function of both wave and structural parameters. It is apparent from Fig. 11 that pile group effect KG decreases as the spacing between the instrumented piles located in the middle and the pile directly facing the incident waves decreases. In other words, the sheltering effect decreases as the relative spacing keeps increasing. The increase of the sheltering effect is more significant for the closely-spaced piles with SG/Dr1.5. The manually drawn curves provide a picture of the relationship between pile group effect KG and KC number for different relative spacing SG/D. In addition to SG/D, the incident hydrodynamic conditions (KC number) significantly affect the magnitude of the wave force on the shielded pile. For smaller KC numbers (KC o6), where the resulting wave load on the pile is primarily dominated by inertia, pile group effect KG ¼ 0.83–1.0 for different pile configurations. By increasing the KC number, however, the sheltering effect increases for all cases with SG/D r3, but it is more pronounced for the
Significant KC number (KCP) Fig. 10. Pile group effect KG1/3 for middle pile in side by side arrangement for purewave condition (Li et al., 1993).
457
configurations with SG/Dr 1.5. A very high sheltering effect was obtained for very large KC values (KC 440), where the wave load is primarily dominated by drag, and the maximum sheltering effect was consequently obtained for the largest KC value of these tests (KC ¼88). In this case (KC ¼88), KG reaches 0.42 for the configuration with the smallest tested relative spacing (SG/D¼ 0.5). For SG/ D4 3, KG values are more or less the same for the whole range of KC numbers and data points are grouped around KG ¼1.0 for all KC values indicating that there is no interaction between piles and each pile behaves like a single isolated pile. 4.1.3. 2 2 Arrangement The so called 2 2 arrangement is commonly found in both offshore and coastal structures and might be considered as a combination of the two basic arrangements “side by side” and “tandem”. For this arrangement, four pile group configurations with relative spacing SG/D¼0.5, 0.75, 1.0 and 2.0 were tested. In total, 83 regular non-breaking waves were performed for this arrangement. The instrumented pile is one of the two downstream piles shown in Fig. 8 with the red circle. The relationship between pile group effect KG and KC number for the 2 2 arrangement is demonstrated in Fig. 12. The most interesting indication from Fig. 12 is that the pile group interaction of the 2 2 arrangement is clearly a combination of both pile group interaction observed in side by side and tandem arrangement. This means that the wave load amplification observed in side by side (Fig. 9) as well as the sheltering effect observed in tandem arrangement (Fig. 11) can also be seen in the 2 2 arrangement (Fig. 12). For 10oKCo30, the amplification effect is more noticeable than the sheltering effect and, consequently, the KG values are higher than 1.0. Interestingly, this range of KC number is the same where higher wave load amplification occurs in side by side arrangement (see Fig. 9). The amplification of the wave force on the measured pile in the 2 2 arrangement is, however, less than that obtained for the middle pile in side by side arrangement. This is due (i) to the presence of the upstream pile by which the instrumented (red) pile is shielded and (ii) to the absence of one of the neighbouring piles at one side of the downstream instrumented pile. As seen in Fig. 12, the instrumented pile in the 2 2 arrangement is, in fact, affected by only one pile while two neighbouring piles influence the resulting wave load on the middle pile in the side by side arrangement. Therefore, the lower wave load amplification for the downstream pile in the 2 2 arrangement as compared to that obtained for the middle pile in side by side arrangement is physically justified.
1.2
1
0.8
0.6
0.4
0.2
0
Fig. 11. Relationship between pile group effect KG and KC number for tandem arrangement exposed to regular non-breaking waves (KC is calculated based on umax measured at relative water elevation of z/h ¼ 0.78).
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1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Fig. 12. Relationship between pile group effect KG and KC number for the 2 2 arrangement exposed to regular non-breaking waves (KC is calculated based on umax measured at relative water elevation of z/h ¼0.78). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
Fig. 13. Relationship between pile group effect KG and KC number for staggered arrangement exposed to regular non-breaking waves (KC is calculated based on umax measured at relative water elevation of z/h ¼0.78).
By further increasing the KC number, the influence of the sheltering effect becomes more important while the amplification effect decreases. For large KC values (KC 430), where the resulting wave load on the pile is fully dominated by drag, pile group effect KG decreases. This is the range of KC number where higher sheltering effect was found compared to other KC values (KC o30) as shown in Fig. 11. The sheltering effect caused by the upstream pile in the 2 2 arrangement is, however, less than that obtained for tandem arrangement. This is because the instrumented pile in the 2 2 arrangement is also affected by the neighbouring side pile. For very small KC numbers (KC o6), where the resulting waveinduced force on the pile is primarily dominated by inertia, pile group effect KG is around 1.0 for all tested configurations with different SG/D values. The qualitative comparison of KG values for KCo 6 shown in Figs. 9, 11 and 12 actually indicates that KG values of the 2 2 arrangement are between those of side by side and tandem arrangement. Overall, it can be stated that, instrumented piles in the 2 2 arrangement have a mixed behaviour including that in side by side arrangement and that in tandem arrangement.
4.1.4. Staggered arrangement In the case of staggered arrangement, where the angle of the centre connexion line of the cylinders relative to the wave direction is 451, six configurations with relative spacing SG/D¼0.6, 0.75, 1.0, 1.5, 3.0 and 5.0 were tested. In total, 120 regular waves were performed for this arrangement. The pile group effect KG is plotted in Fig. 13 against KC number for staggered arrangement. No specific trend can be seen in this figure and a larger scatter of data around KG ¼1.0 line is noticeable. Almost all of the data points for different wave and structural conditions vary between KG ¼ 0.9 and KG ¼1.1, meaning that pile group effect KG depends neither on relative spacing SG/D nor on KC number. In order to study the effect of angle of incident waves on pile group interaction, further investigations on pile group arrangements with different angles (0–901) of the centre connexion line of the cylinders relative to the wave direction are needed. 4.1.5. Pile group effect at different elevations below still water level Pile group effect at different elevations below still water level (SWL) was studied to examine whether pile group effect KG changes over the water column. For this aim, local wave forces were measured on the
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small (4 cm) sections of the instrumented piles located at the different elevations below SWL using the force sensors named ring transducers (see Section 2). For the case of tandem arrangement with the relative spacing of SG/D¼0.75, as an example, Fig. 14 is given to illustrate the relationship between pile group effect KG and KC number at three relative elevations below SWL (z/h¼0.78, 0.53 and 0.28). As seen, the results obtained at z/h¼0.78, 0.53 and 0.28 are very similar meaning that, for the tested conditions, the pile group effect (KG) is not dependent on the elevation (z) at which the local wave force is measured and, therefore, does not change over the water depth (h). 4.2. Pile group effect due to focused breaking waves As the focus was rather put on regular non-breaking waves, only 35 tests with focused waves were conducted for side by side and tandem arrangements with relative spacing of 0.5rSG/Dr3. The results related to pile group effect KG for the focused waves with loading cases LC1–LC5 are shown in Figs. 15 and 16 for side by side and tandem arrangements, respectively. Unlike the case of regular non-breaking waves where pile group effect KG is described by the line wave force ratio (fGroup/fSingle), KG is defined as the total wave force ratio (FGroup/ FSingle) for the case of focused waves. The reason of using total wave force ratio (KG ¼ FGroup/FSingle) for the focused waves is that the loading case, which shows how the focused wave hits the instrumented pile, plays a very significant role on the resulting focused wave-induced force on the pile. In fact, the magnitude of the wave load on the piles and, consequently, pile group effect are mainly dependent on the loading case as the characteristic wave parameters were kept constant for the generation of the focused waves. 4.2.1. Side by side arrangement The first point to be concluded from Fig. 15 is that for side by side arrangement, pile group effect KG is strongly affected by relative spacing SG/D when the latter varies from 1 to 0.5. Another point is that the highest pile group effect KG was found for load case LC5 where the wave breaks immediately after the pile, meaning that the most critical condition which results in the highest amplification of the resulting wave load on a pile in side by side arrangement occurs for non-breaking waves. As an example, for the configuration with relative spacing of SG/D¼0.5, where pile group interaction is most pronounced, KG values for focused breaking waves with different loading cases from broken to partially breaking are less than KG values observed for non-breaking waves. This might be due to higher energy losses in breaking waves. As seen in Fig. 15, pile
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group effect KG decreases by going from partially breaking wave (LC4) to broken wave (LC1). Another possible reason might be the time needed for the load development which, as reported by Hildebrandt and Schlurmann (2012) and Hildebrandt (2013), is higher for non-breaking wave (LC5) than for the load cases with breaking and broken waves. Therefore, load case LC5 has the highest potential for the interference effects of adjacent piles. 4.2.2. Tandem arrangement For the downstream pile in a tandem arrangement, pile group effect KG is plotted versus KC number in Fig. 16 for the tested relative spacing values (0.5rSG/Dr3). It can be generally stated that for the generated focused wave, KG values of broken and breaking waves (LC1–4) are smaller than that of LC5 where wave breaks immediately after the instrumented pile. This finding is physically sound because nearbreaking waves are already instable when they hit the upstream pile directly facing waves. The interaction of near-breaking wave and the upstream pile decreases wave energy and the resulting breaking wave force on the downstream pile is less than that on the upstream pile resulting from the aforementioned interaction. As seen in Fig. 16, the highest sheltering effect occurs for load case LC3 with KG ¼0.56 which was obtained for the configuration with SG/ D¼0.5. For load cases LC2 to LC5, KG decreases by decreasing relative spacing SG/D as expected. However, for load case LC1 representing a broken wave hitting the pile at the SWL when it is falling downward into the water, a larger KG value obtained for SG/D¼ 0.5 in comparison with other configurations with larger SG/D. This unexpected result might be due to the highly stochastic nature of the broken wave as a mixture of water and entrapped air which makes it very difficult to be reproduced in the wave flume. A detailed discussion on the repeatability of the focused waves with different load cases in the LWI wave flume is provided by Bonakdar (2014). In particular, it was shown that the resulting load induced by a broken wave (LC1) greatly differs from one test to another, thus implying that LC1 is the most difficult load case to be reproduced properly in the wave flume.
5. Summary, concluding remarks and outlook 5.1. Summary and concluding remarks Pile groups exposed to regular non-breaking waves as well as to focused breaking waves were investigated, but the focus was put on the former.
1.2
1
0.8
0.6
0.4
0.2
0
Fig. 14. Pile group effect KG at three different elevations below SWL for tandem arrangement with SG/D ¼ 0.75.
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2.2 2
Pile group effect (KG)
1.8 1.6 1.4 1.2 1 0.8 0.6
0.4 0.2 0
0
1
2
3
4
5
Loading Case Fig. 15. Pile group effect KG for focused waves with different loading cases on the middle pile in side by side arrangement with different SG/D.
1 0.9
Pile group effect (KG)
0.8 0.7 0.6 0.5
0.4 0.3 0.2 0.1
0 0
1
2
3
4
5
Loading case Fig. 16. Pile group effect KG for focused waves with different loading cases on the downstream pile in tandem arrangement with different SG/D.
Therefore, a total of 509 small scale laboratory tests were performed in the LWI wave flume to investigate non-breaking wave load on a slender pile within a group of piles in different arrangements. For this aim, a test programme was considered covering a broad range of wave conditions (deep to shallow water) and pile group configurations. The KC number was varied from 1.1 where inertia regime dominates to 88 where drag regime dominates. Different pile arrangements including single, side by side, tandem, 2 2 and staggered were performed and relative spacing SG/D was varied from 0.5 to 5.0. The obtained key results may be summarised as follows: (i) Pile group arrangement, relative spacing SG/D and KC number were found as the most significant parameters affecting wave-induced loads on a slender pile within a pile group. (ii) For side by side arrangement, when KC is between 6 and 35, for which both inertia and drag are important, pile group effect KG is a multivariate function of both hydrodynamic (KC) and structural (SG/D) parameters. In this case, very high KG values (up to 2.4 at SG/D¼0.5) are obtained. For the pure inertia regime (KCo6) where the water depth (h/L¼0.29–0.64) is relatively large and for the pure drag regime (KC430–35) wave pile group
(iii)
(iv) (v)
(vi)
effect KG is independent of KC and only a function of relative spacing SG/D. For tandem arrangement, the highest sheltering effect is obtained for very large KC number (KC ¼88), where the wave load is primarily dominated by drag. In this case, KG reaches 0.42 for the configuration with the smallest tested spacing (SG/D¼0.5). Sheltering effect disappears for SG/D 43 and KG values are more or less equal to 1 for the whole tested range of KC number. For the 2 2 arrangement, the downstream piles behave like in both side by side and tandem arrangement. For staggered arrangement, no specific relationship could be found between pile group effect KG and KC number. In fact, KG varies from 0.9 to 1.1 for almost all performed tests. Therefore, the influence of wave direction on the resulting wave load on a pile in a pile group needs further investigations by testing pile group arrangements with different angles (0–901) of the centre connexion line of the cylinders relative to the wave direction. The relationship between pile group effect KG and KC number at three relative elevations (z/h ¼0.78, 0.53 and 0.28) was investigated and no noticeable effect of elevation z/h, was
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found, so that pile group effect KG can be considered as constant over the entire water depth h. Focused breaking wave tests were conducted for side by side and tandem arrangements. Considering loading cases LC1–LC5 and the pile group arrangements “side by side” and “tandem”, 35 tests with focused wave were performed. For side by side arrangement, the highest amplification of wave load on the instrumented pile was obtained for LC5 which corresponds to a non-breaking wave load as the wave breaks immediately after the instrumented pile. By varying the wave load from non-breaking to partially breaking, breaking and broken waves, pile group effect KG decreases. This might be due to the higher energy losses associated with the wave breaking processes. For tandem arrangement, the KG values of broken, breaking and partially breaking waves (LC1–LC4) are smaller than that of LC5 where the wave breaks just after the measuring pile. Based on the results of this study, a substantial improvement of the understanding of pile group effect on the wave loading of a slender pile has been achieved which might lead to a safer design of pile-supported marine structures exposed to breaking and nonbreaking waves. Based on the improved understanding which has been obtained from the laboratory investigations, a new set of formulae for the prediction of wave-induced force on slender piles within pile groups is derived by Bonakdar et al. (2015) using a hybrid computational data analysis tool consisting of M5 model tree (M5MT) and genetic programming (GP). 5.2. Outlook Based on the knowledge gained from this study, the following recommendations for further research may be drawn: (i) The influence of wave direction on the resulting wave loads on a pile within a pile group needs to be further studied by testing pile group arrangements with different angles between the line connecting the pile centres and the wave direction (0–901). (ii) Further laboratory tests might also be conducted to study the interaction of bores (e.g. broken waves, tsunami bores) and pile groups. (iii) A numerical model needs to be developed and systematically validated against the existing laboratory experiments. The validated numerical model will build an excellent tool for further understanding of the physical processes involved in the wave-pile group interaction. In fact, this will provide more flexible and more comprehensive output options (e.g. waveinduced flow patterns within and around pile groups, including the wave loading of each pile within the pile group). (iv) Wave run-up on a slender pile within pile groups with different arrangements, on which much less attention has been paid compared to the case of single isolated pile, might be investigated to examine the effect of further adjacent piles and further relative spacing SG/D on the wave run-up along the instrumented pile(s).
Acknowledgements The financial support of the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) for this study through the WaPiGS project (Ou 1/13-1) is acknowledged. The support of the German Academic Exchange Service (DAAD, Deutscher Akademischer Austauschdienst) is also acknowledged.
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