Wave load formulae for prediction of wave-induced forces on a slender pile within pile groups

Transcription

1 Wave load ormulae or prediction o wave-induced orces on a slender pile within pile groups Lisham Bonakdar 1 *, Hocine Oumeraci 1 and Amir Etemad-Shahidi 2 1 Leichtweiss-Institute or Hydraulic Engineering and Water Resources, Technische Universität Braunschweig, Beethovenstrasse. 51a, Braunschweig, ermany 2 riith School o Engineering, riith University old Coast Campus, QLD 4222, Australia * Corresponding Author, Phone: , l.bonakdar@tu-braunschweig.de Abstract Pile-supported structures commonly ound in both oshore (e.g. oshore oil and gas platorms) and coastal environments (e.g. sea bridges, piers and jetties) are generally built by means o a group o piles in dierent arrangements. The correct prediction o the wave loading o closelyspaced piles o these structures is vital or both saety and economical viewpoints. Unlike single isolated piles, where a large number o studies are available together with the well-known Morison equation which is still widely applied or the calculation o wave-induced orce, less research studies have been made on wave-pile group interactions. In act, no reliable wave load ormula is yet available or the prediction o wave-induced orces on a slender pile, or which the pile diameter (D) is generally less than about 0.15 times the wave length (L), within a pile group. In this study, new wave load ormulae or the prediction o wave-induced orce on a slender pile in pile groups with dierent arrangements are developed using a series o laboratory data obtained rom systematic model tests conducted in the 2 m-wide wave lume o Leichtweiss- Institute or Hydraulic Engineering and Water Resources (LWI) in Braunschweig, ermany. For the analysis o the laboratory data and the development o the new prediction ormulae, an artiicial intelligence (AI) - based computational tool, named hybrid M5MT-P model, is implemented. The new hybrid model and the new wave load ormulae allow us to systematically assess the pile group eect (K ) as a unction o the low regime (KC number) and the relative spacing (S /D) or each tested pile group arrangement. The accuracy o the new ormulae in predicting pile group eect K is conirmed by the statistical indicators using agreement index I a, correlation coeicient CC and scatter index SI.

2 Keywords Pile group eect, Pile group arrangement, Relative spacing, KC number, Hybrid M5MT-P model, Wave load ormulae

3 1. Introduction There might be a common assumption that two or more piles in a low should have a similar behaviour to that o a single isolated pile, but this assumption is correct only when they are adequately apart (Zdravkovich, 1977). For closely-spaced piles in groups exposed to waves, the intererence eects between piles may signiicantly change the low around the piles, and thus the wave load as compared to that on a single isolated pile. In such structures, wave load on a single slender pile is signiicantly aected by the neighbouring piles and can thus not be estimated by the commonly applied ormulae or a single isolated pile which are generally based on the concept o Morison et al. (1950). According to the angle o the connecting line o the piles centres relative to the wave direction, pile groups are commonly categorized into three basic arrangements. These three arrangements include (i) tandem, where the angle o the centre connection line o the cylinders relative to the wave direction is 0, (ii) side by side, where the incident wave direction is orthogonal to the connecting line o the piles located next to each other, and (iii) staggered in which the angle is between 0 and 90 (Fig. 1). In the case o slender piles where both drag and inertia orces induced by highly complex turbulent low are important, an analytical solution is hardly easible. iven the high complexity o the interaction between waves and pile groups in dierent arrangements, laboratory experiments still represent the most reliable alternative. A number o laboratory studies have been carried out to investigate the intererence eects o neighbouring piles. The methods commonly used in laboratory studies to determine wave loads on a pile group may be classiied in two main categories: wave orce coeicient approach and wave orce approach. In the ormer approach, the inertia and drag coeicients (C M and C D ) are determined based on the knowledge o both low velocity and acceleration by applying or instance the least square it. This approach was used or instance by Chakrabarti (1981, 1982), Haritos and Smith (1995), Smith and Haritos (1996, 1997). Using the calculated drag and inertia coeicients, Chakrabarti (1981, 1982) computed maximum wave orces and ound a relatively good agreement with measured orces. Smith and Haritos (1996, 1997) reported that drag and inertia coeicients are dependent on the Keulegan Carpenter (KC) number (KC=u max T/D) and relative spacing S /D where u max is the maximum horizontal wave-induced low velocity, T is wave period, D is pile diameter and S is the gap between the suraces o two neighbouring piles in a group o piles. Due to the lack o any reliable measured velocities, and based on appropriate wave theories, the maximum horizontal wave-induced low velocity u max was calculated by Chakrabarti (1981, 1982) below still water level at the elevation o the instrumented section o the pile on which the

4 local wave-induced orce was measured while u max was computed by Haritos and Smith (1995) as well as Smith and Haritos (1996, 1997) at the water surace elevation as the total wave orce on the piles was considered. Drag and inertia coeicients were usually plotted versus KC number or dierent relative spacing S /D in these studies. However, the proposed C D and C M values were noticeably scattered demonstrating that dierent C D and C M values can be obtained or a given KC number. In the latter approach, the ratio o wave orce on a pile within a group to that on a single isolated pile is determined. This method was applied by Apelt and Piorewicz (1986), Mindao et al. (1987) and Li et al. (1993). Li et al. (1993) stated that the wave-induced orce on a slender pile within a group o piles depends on the KC number and relative spacing S /D. Mindao et al. (1987) introduced two parameters named intererence coeicient K g and shelter coeicient K z or side by side arrangement and tandem arrangement, respectively. Both K g and K z coeicients are representative or the orce ratio (F roup /F ) where F roup is the wave orce on a slender pile within urther neighbouring piles and F is the wave orce on a single isolated pile. They stated that S /D is the most signiicant parameter and proposed the two ollowing ormulae or the estimation o intererence coeicient K g and shelter coeicient K z or side by side arrangement and tandem arrangement, respectively: ( ) K = ln S D or side by side arrangement (1) g ( ) K = ln S D or tandem arrangement (2) z In the proposed ormulae (Eqs.1 and 2), the wave conditions (i.e. wave height, period, steepness and etc.) have no inluence on intererence coeicient K g and shelter coeicient K z. For a given pile group arrangement, both coeicients only depend on relative spacing parameter S /D which was varied rom 0.5 to 3 in the laboratory tests. It was also stated by Mindao et al. (1987) that the intererence coeicient K g and shelter coeicient K z proposed in Eqs.1 and 2 are the average o those obtained or the side and middle piles in the pile group arrangements. Li et al. (1993) introduced signiicant pile group eect K 1/3 or piles in side by side arrangement exposed to irregular waves. He ound out that the maximum K 1/3 occurs when KC number is between 15 and 20 or the case o pure waves. They also showed that, or a given pile group coniguration, the combination o wave and current results in smaller grouping eect compared to wave action only. The interaction o waves and slender piles in dierent pile group arrangements was also studied by means o extensive large-scale laboratory tests perormed in the Large Wave Flume (WK).

5 A single isolated pile and 14 pile group conigurations including side by side, tandem and staggered arrangements with gaps o up to three times the pile diameter (1 S /D 3) 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 o the general conclusions drawn rom these analyses are: (i) (ii) (iii) (iv) Pile group eect increases by decreasing the gap between the piles in side by side arrangement, The ampliication o the wave load on the middle pile in side by side arrangement is more noticeable than the side pile due to the inluence o two neighbouring piles rom both sides, For the tested regular waves (5<KC<38), 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 eect becomes negligible or S /D=3 and all piles behave like a single isolated pile in terms o the wave load, For tandem arrangement with S /D=1, which is the smallest relative spacing tested in WK, no signiicant sheltering eect was observed or the tested regular waves (5<KC<38). In addition to the aorementioned general outcomes, some o the main limitations o the WK tests which were identiied may be summarized as ollows: (i) (ii) (iii) Pile group conigurations with smaller relative spacing o S /D<1, where higher ampliication and reduction o wave loads on piles are expected, were not tested, The tested wave conditions only cover a small range o relative water depth h/l located in the transition zone (h/l= ). Thereore, the shallow and depth water conditions were not investigated. Considering the tested KC number values (5<KC<38), moreover, the dominant drag regime and the dominant inertia regime were not ully covered, Values o the KC and Re parameters change rom one section o the pile to another as a result o the variation o the low velocity with depth. Only the total wave-induced moment on the instrumented pile was measured meaning that the low velocity was averaged over the water depth,

6 (iv) Cantilever piles (truncated with lower end ar rom the bottom o the lume) were used in the WK model set-up. This might result in unrealistic low behaviour around the group o pile due to the low separation at the lower end (a more detailed discussion is given in Bonakdar, 2014). Overall, the ollowing knowledge gaps and limitations o the previous studies were identiied: (i) the lack o deeper understanding o the processes associated with wave-pile group interaction, (ii) the lack o reliable wave load ormulae or the prediction o wave-induced orces on a slender pile within other neighbouring piles in dierent arrangements and (iii) the limitations o the WK tests. Hence, urther research is needed towards improving the understanding o the involved processes, including a more precise and systematic identiication o the most relevant inluencing hydrodynamic and structural parameters. Based on this improved understanding, simple generic ormulae or the prediction o wave loads on the piles have to be derived. Thereore, new laboratory tests were designed and perormed in 2 m-wide wave lume o Leichtweiss-Institute or Hydraulic Engineering and Water Resources (LWI) in Braunschweig, ermany. The main objectives o this experimental research study were (i) to speciy more precisely and systematically the most relevant inluencing wave and structural parameters on the wave loading o a pile in a pile group with dierent arrangements and, consequently, to improve the understanding o the processes associated with the wave-pile group interaction, and (ii) to generate a comprehensive data set which covers a range o wave and structural conditions broad enough to achieve a substantially improved insight into the hydrodynamic processes involved and, consequently, to develop wave load ormulae or the prediction o wave loads on a slender pile within a group o piles in dierent arrangements. The ormer objective was addressed by Bonakdar and Oumeraci (2015b) while the latter objective, namely the development o wave load ormulae or prediction o wave loads on a slender pile within pile groups in dierent arrangements, is the ocus o this study. For the systematic analysis o the generated data resulting in the development o new generic wave load ormulae, an artiicial intelligence (AI) - based computational tool, a combination o M5 tree (M5MT) and genetic programming (P) named hybrid M5MT-P model, is implemented. This paper is outlined as ollows: The laboratory data is briely described in Section 2. Next, the hybrid M5MT-P model is described. In Section 4, the implementation o the hybrid M5MT-P model or the analysis o the laboratory data as well as or the development o prediction ormulae are provided and the obtained results are discussed. Finally, the summary o the key results and concluding remarks are drawn in Section 5.

7 2. Laboratory experiments (LWI tests) A large number o small scale laboratory tests were carried out in the LWI wave lume, called hereater LWI tests. The details o the model set-up, measuring technics and test programme are provided in Bonakdar (2014) and Bonakdar and Oumeraci (2015b). Thereore, only a very brie description o the LWI tests is provided in this section. Dierent pile arrangements including single, side by side, tandem, 2 2 and staggered arrangements were perormed and relative spacing S /D was varied rom 0.5 to 5.0 as depicted in Fig. 2. Regular non-breaking waves with 24 dierent combinations o wave heights and periods were tested pile to cover a broad range o hydrodynamic conditions. Wave steepness varies rom to which was the maximum possible wave steepness without having incipient breaking. Relative water depth h/l varies rom to 0.64 meaning that deep, transition and shallow water conditions were considered. The KC number changes rom 1.1 where the inertia regime dominates to 88 where the drag regime dominates. Reynolds number varies rom Re= to Re= indicating that the LWI model is located in the subcritical zone. 3. Data analysis methodology The most common method or empirical model development is the regression analysis. In the process o traditional regression analysis (e.g. simple linear, polynomial, and etc.), the unctional relationship between output and input parameters (variables) is pre-deined, and the goal is only to determine a set o empirical coeicients o the input parameters. For highly complex and unknown systems, however, a predeined unctional structure may not result in an accurate model. Artiicial intelligence (AI)-based methods, also called data mining or sot computing methods, are also utilized to identiy hidden relationships that exist in datasets by implementing dierent optimization algorithms (Kazeminezhad et al., 2010). AI-based methods can be implemented or predicting processes in complex systems, reconstructing highly nonlinear unctions, classiying data and developing rule-based models (Solomatine and Osteld, 2008). The general steps or solving a problem using an AI-based method are (i) investigating the physical problem to be modelled, (ii) generating required data by perorming experimental or numerical tests or collecting data rom the available recourses, (iii) selecting, building and optimizing the AI-based method or the speciic problem, and (iv) validating the developed model using a new set o data called test data (Solomatine and Osteld, 2008).

8 The most requently used data mining techniques are artiicial neural networks (ANNs), uzzy systems, support vector machine (SVM), M5 model tree (M5MT), genetic algorithm (A) and genetic programming (P). A detailed review o the aorementioned AI-based methods was perormed to identiy the most easible approach or the development o wave load ormulae. For this purpose, two main criteria were considered in developing the prospective prediction ormulae and choosing the most suitable approach. These two main criteria were accuracy and simplicity o the prediction ormulae to be developed by applying the prospective AI-based model. It was concluded that a combination o M5MT and P named hybrid M5MT-P model is the best solution among dierent possibilities. A more detailed discussion is provided by Bonakdar (2014) M5 model tree (M5MT) The M5 model tree was introduced by Quinlan (1992) and represents one o the most recent computational tools or data analysis which can be applied or prediction purposes. The concept o the model tree approach is based on dividing complex problems into smaller sub-problems and solving each sub-problem (Bhattacharya et al., 2007). The concept o M5MT is described in Fig. 3 by a simple example with two input parameters (X 1 and X 2 ). As shown in window A o Fig. 3, M5MT is similar to an inverse tree with a root node at the top and a number o leaves at the bottom. The implementation o M5MT generally includes three steps called building, pruning and smoothing. Building: Firstly, the M5MT algorithm constructs a tree by splitting the instance space (data points). Window A o Fig. 3 shows the constructed tree while window B illustrates the classiied data. The splitting condition is used to minimize the intra-subset variability in the values down rom the root through the branch to the node. The variability is measured by the standard deviation o the values that reach that node rom the root through the branch, the expected reduction in error being calculated as a result o testing each attribute at that node. In this way, the attribute (input parameter) that maximizes the expected error reduction is chosen. The splitting process is perormed only i either the output values o all the instances that reach the node, called lea, vary slightly or a ew instances remain. The standard deviation reduction (SDR) is calculated as (Quinlan 1992): Ti SDR = sd( T ) sd( Ti ) T i (3) where T is the set o examples that reach the node, T i are the sets that result rom splitting the node according to the chosen attribute and sd is the standard deviation (Wang and Witten 1997).

9 Ater the initial tree has been grown, the linear regression models are generated, using the data associated with that lea. Window C o Fig. 3 shows the possible linear regression models or the given example. Pruning: In the second step, all sub-trees are considered or pruning. Pruning occurs i the estimated error or the linear model at the root o a sub-tree is smaller or equal to the expected error or the sub-tree. In this way, the sub-trees which cannot improve the accuracy o the model are pruned. Ater pruning, there is a possibility that the pruned tree might have discontinuities between nearby leaves. Smoothing: Thereore, to compensate discontinuities among adjacent linear models in the leaves o the tree a regularization process is perormed, which is called smoothing process. In this process, the estimated value o the lea model is iltered along the path back to the root. At each node, that value is combined with the value predicted by the linear model (LM) or that node as ollows: np + kq P = n+ k (4) where P is the prediction passed up to the next higher node, p is the prediction passed to this node rom the below, q is the value predicted by the model at this node, n is the number o training instances that reach the node below, and k is a constant (Wang and Witten, 1997). M5MT has a unique algorithm meaning that or a given data set o input and output variables, the model provides a unique solution or any number o simulations. Recently, M5MT has been successully employed or water level discharge relationship (Bhattacharya and Solomatine, 2005), sediment transport (Bhattacharya et al., 2007), stability o rubble-mound breakwaters (Etemad-Shahidi and Bonakdar, 2009 and Etemad-Shahidi and Bali, 2011), prediction o wave run-up on rubble-mound breakwaters (Bonakdar and Etemad-Shahidi, 2011), prediction waveinduced scour around pile groups (Etemad-Shahidi and haemi, 2011), prediction o scour depth under submarine pipeline (Etemad-Shahidi et al., 2011), sand wave overtopping at rubble-mound structures (Jaari and Etemad-Shahidi, 2012). The main advantages o the model trees are that they are easily applied and yield simple and transparent ormulae. In addition, unlike other existing sot computing methods such as artiicial neural networks (ANNs), the M5 model tree is quite transparent and does not need optimization o network geometry and internal parameters. While the traditional regression method its a single unction to the whole data set, M5 model tree splits the data points into homogeneous subsets (leaves) and it a linear unction or each lea. Sorting the whole data point into

10 homogeneous sub-sets can result in a more accurate model which cannot be achieved by a common regression method. The major limitation o this method is that it can provide only a linear relationship between input and output parameters at each lea, while the relationships between the output and input parameters are not necessarily linear enetic programming (P) enetic Programming (P) is an evolutionary symbolic regression method where, unlike traditional regression methods, the unctional structure between output and input parameters is not pre-deined and is a result o the search process. P was irstly introduced by Koza (1992) as a powerul tool or solving complex problems and is similar to more widely known genetic algorithms (A). However, unlike A, where a set o numbers is the solution o the given problem, a ormula also called a computer programme is generated as the solution. P creates an initial population o unctional orms rom user-speciied building blocks stored as unction and terminal sets. These building blocks can consist o a range o operators, including addition, subtraction, multiplication, division, etc. Using a tree-based representation, the genotype is arranged such that the top and middle o the tree is created rom members o the unction set, and the leaves consist o members o the terminal set. Once the initial population has been created, the so called reproduction, mutation and crossover are used to generate ospring. The best ospring (equation) resulting rom this process is the solution o the problem. These steps are individually described as ollows: First, an initial population o individuals (equations or programs) o a certain size is created by randomly picking up a set o unctions, which consists o basic mathematical operators (e.g. addition, subtraction, multiplication, division, log, etc.) and constants, and the so called terminals which consists o independent variables (input parameters) and constants. The constants can either be physical constants (e.g. Earth s gravitational acceleration, density o water) or randomly generated constants. The unction set and the terminal set construct the main body o P. Hence, their appropriate identiication plays a pivotal role in developing a robust P model. The determination o the mathematical operators considered in the unction set depends upon the degree o complexity o the problem to be modelled and should be careully chosen, otherwise they might result in an extremely complex ormula which carries no physical insight. For example, ( x ) exp 1 3 4x2 can be considered as an equation (programme). The tree structure o this programme is shown in Fig. 4. A population o random trees representing the programmes is initially constructed and genetic operations are perormed on these trees to generate programmes (equations) with the help o the terminal and the unction sets. For the given

11 example shown in Fig. 4, {exp,, /, *} and {3, 4, x 1, x 2 } belong to unction and terminal sets, respectively, where x 1 and x 2 are input variables o the given example. Except or the root o the tree which must be a unction, a uniorm random probability distribution is usually considered or the unction and terminal selection (Koza, 1992). Second, the itness o each equation (programme or individual) in a population is evaluated based on the chosen criterion. For instance, the root mean squared error (RMSE) can be used as the criterion. In this case, a smaller value o RMSE shows a better itness. Third, at each generation, new sets o models are evolved by applying the genetic operators: crossover, mutation and reproduction (Koza, 1992). These new models are named ospring, and they orm the basis or the next generation. (i) Crossover: Crossover is applied on an individual (equation or programme) by simply switching one o its nodes with another node rom another individual in the population. With a tree-based representation, replacing a node means replacing the whole branch. Thereore, two new programmes (equations) are generated as shown in Fig. 5. The resulting individuals (equations) are inserted into the new population. (ii) Mutation: Mutation aects an individual in the population. In mutation, a sub-tree is replaced by another one, randomly, (Fig. 6) and the new sub-tree is inserted into the new population. (iii)reproduction: Based on the itness criterion, the best programme is selected and copied into the new population. While the role o the crossover operator is to generate new models, which did not exist in the old population, the mutation operator guards the search against premature convergence by constantly introducing new genetic material into the population (Elshorbagy et al., 2010). Forth, i the number o generations is equal to a certain value selected or the model, the programme is terminated. The equation (individual or programme) with the best itness is represented as the best solution (ormula) or the given problem. A enetic Programming lowchart is illustrated in Fig. 7 where P is population size. P has lately been applied or the prediction o wave height (Nitsure et al., 2012), low discharge in compound channels (Azamathulla and Zahiri, 2012), scour below submerged pipeline (Azamathulla et al., 2011) and scour around piles (uven et al., 2009). The main strengths o P may be summarized as ollows:

12 (iv) Like M5MT, P also provides the result o modelling in a orm o computer programme (equation) which can be interpreted by scientists. This mathematical representation provides a great beneit in empirical modelling o unknown phenomena where a theoretical model does not exist. This has resulted in the successul application o P to a wide range o practical problems o various degree o complexity over the last two decades. (v) Unlike the traditional data analysis methods, in which users have to speciy the overall unctional orm o the model in advance, P evolves both unctional orm o the model and its numerical (empirical) coeicients. (vi) P can develop non-linear relationships between the output and input parameters using dierent mathematical unctions while M5MT can only generate linear relationship between output and input parameters. Hence, P can provide a better approximation o the complex natural processes and more insight into the unctional relationship between the input variables The drawbacks o P might be summarized as ollows: (i) Like regression analysis, P also its a single overall unction to the entire data set while M5MT splits the data set into homogeneous sub-sets (classes) and it a linear unction or each sub-set which might result in a more understandable and accurate model. (ii) P is not as ast as M5MT and, depending on the complexity o the problem, the initial population and determined unctions, the modelling procedure might take hours or days Hybrid M5MT-P model Considering the strengths and limitations o the M5MT and P and making use o their respective strengths, a one-way coupling o M5MT and P was considered or the development o inal wave load ormulae. By this way, all data sets obtained rom laboratory experiments are classiied in dierent classes based on the criteria o M5 model tree algorithm. Once the classiication o the data is completed, P is applied to the classiied data. The P method is used as a non-linear approach to predict wave loads on a slender pile within urther neighbouring piles in dierent arrangements as a unction o the most relevant inluencing wave and structural parameters. The overview o the procedure o data analysis and the development o prediction ormulae or the wave load using the hybrid M5MT-P model, including the classiication, prediction and validation processes, are drawn in Fig. 8.

13 The one-way hybrid M5MT-P model is more comprehensively described in Fig. 9 by providing a simple example o a problem with two input variables o X 1 and X 2 and an output variable Y. Step 1: Application o M5MT As shown in window A o Fig. 9, M5MT is applied to all data. This application results in three outputs illustrated in window B. The irst output is an inverse tree where the relevant inluencing parameters together with their splitting values are located at the nodes o the inverse tree. The nodes at the end o the inverse tree are called leaves. These leaves represent, in act, the inal sub-sets o the developed tree. Based on the chosen variables and splitting values at the upper nodes o the tree, all data are grouped into leaves through the branches. The second output is the classiication o the data into the generated leaves (classes). Three groups o data sets are obtained (sub-sets 1-3). The third output is a set o linear models (LMs) which are generated using the data associated with each lea. The main limitation o the M5MT model consists in its inability to generate non-linear models as the relation between the output and input parameters are not necessarily linear. In order to overcome this crucial limitation, the P model is used in the next step. Step 2: Application o P The P model is applied to the data associated with each lea, individually. This process is demonstrated in window C. For the given example, 3 P models are needed or the three subsets 1-3 obtained rom M5MT. As mentioned beore an optimisation o the P model is required to provide the desired solution describing the relationship between output and input parameters based on physical behaviour o the system. Thereore, P model should be optimised by changing the settings o the model including mathematical unctions, population size and generations and etc. P provides a solution or each simulation. The procedure o P simulation is drawn in both Fig. 7 and window D o Fig. 9. For the example, 3 possible equations or 3 optimised P models can be seen in window E o Fig. 9. These so called M5MT-P-based equations are, in act, the solutions obtained rom three optimised P models applied to three sub-sets classiied by M5MT. 4. Development o wave load ormulae using hybrid M5MT-P model 4.1. Data classiication using M5MT

14 As described in the previous section, in the irst place, M5MT needs to be applied to all data and classiies them into homogeneous sub-sets so called leaves. Bonakdar (2014) perormed a comprehensive analysis on the eect o non-dimensional wave parameters including KC number, Reynolds number Re, relative water depth h/l and wave steepness H/L on pile group eect K. The latter represents the relative wave orce ratio (K = roup / ) where roup is the maximum line orce on a slender pile within a pile group in dierent arrangements and is the maximum line orce on an isolated single pile. Among all these parameters, KC number was identiied as the most suitable parameter to describe the eect o wave conditions on pile group interaction. It was stated that pile group eect K related to KC number is more appropriate than that related to other non-dimensional wave parameters. In addition, KC number is a unction o both wave period and low velocity which make it an appropriate parameter or describing waveinduced low conditions. Thereore, KC number was avoured as a parameter describing the low regime or the development o wave load ormulae. Bonakdar and Oumeraci (2014) compared pile group eect K values measured in LWI tests with Froude scaled ones obtained in the 6.5 larger scale model tests (WK), where Re number was ranged rom to , and ound pile group eect K to be similar or a given structural condition (e.g. same pile group arrangement relative spacing S /D) and the same KC number in both LWI and WK tests. Thereore, the non-dimensional pile group eect K values obtained in this study can be applied or other cases with higher Re values. From the structural point o view, as can be concluded rom the results o 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 S /D are the most signiicant parameters aecting the resulting wave load on a slender pile within other neighbouring piles. Overall, it can be stated that: roup S K = = KC,, Pile group arrangement D (5) Dierent pile group arrangements including side by side, tandem, staggered and 2 2 were individually analysed by M5MT. Pile group eect K was set as the output while KC number and relative spacing S /D were set as the inputs o the model representing the most relevant inluencing hydrodynamic and structural parameter, respectively. For all pile group arrangements, KC number varies rom 1.1 to 88 meaning that it covers all wave-induced low regimes. Relative spacing parameter S /D, however, diers or dierent pile group arrangements. Table 1 provides a comprehensive overview o the hydrodynamic and structural conditions o the data used or the development o the M5MT-P model.

15 4.1.1 Side by side arrangement In side by side arrangement, seven dierent pile group conigurations with relative spacing o S /D=0.5, 0.75, 1.0, 1.5, 2.0, 3.0 and 5.0 were tested. In total, 146 regular non-breaking waves were perormed or the side by side arrangement. Fig. 10 illustrates the relationship between pile group eect K and KC number or side by side arrangement as well as the developed tree showing splitting parameters and the corresponding splitting values at nodes and leaves (subsets), where data points are inally classiied. The classiied sub-sets are also demonstrated by manually drawn dash lines. As seen, M5MT classiied all data into 5 dierent sub-sets based on dierent combinations o KC number and relative spacing S /D. The irst splitting parameter located at the root o the inverse tree is relative spacing S /D and its splitting value is 1.5. As discussed by Bhattacharya et al. (2007), the splitting value does not necessarily have any physical interpretation and is obtained by minimizing the prediction error. However, this value distinguishes the so-called closely-spaced piles (S /D 1.5) where a greater pile group interaction is expected rom largely-spaced piles (S /D>1.5) where less interaction o piles occurs due to larger gaps between piles. For the conigurations with S /D 1.5, where the piles are closely spaced next to each other in an array, KC number becomes important. As seen on the let hand side o the tree shown in Fig. 10, data points with S /D 1.5 were grouped by KC number with the splitting value o 13. As shown by the dash-line, this is almost the value at which maximum ampliication o wave load on the closely-spaced piles in side by side arrangement occurs. Data with S /D 1.5 and KC>13 was classiied only in one group (sub-set 3 in Fig. 10). For data with S /D 1.5 and KC<13, another categorization is made by KC number as shown on the down-let hand side o the tree. The values that came down rom the root through the branch to this node, were classiied into two other group at KC=6. As can be concluded rom the depicted igure, pile group eect K is almost constant when KC is smaller than 6 (inertia dominated regime, sub-set 1) and shows dierent behaviour or the cases with KC>6 (sub-set 2 in Fig. 10). For the conigurations with S /D>1.5, as can be concluded rom the developed M5MT model, data points were sorted into two groups according to relative spacing parameter S /D with a splitting value o 2. This means that the M5MT model ound the data with S /D>1.5 more homogeneous than others with S /D 1.5. This is also apparent rom the plotted data. As seen in Fig. 10, K values are more or less the same or the whole range o KC values, meaning that pile group interaction is not dependent on the hydrodynamic conditions or S /D>1.5. The developed tree and relationship between K and KC number or dierent S /D shown in Fig. 10 indicate that M5MT is able to identiy the relationship between input and output parameters and to

16 classiy the data based on the physical behaviour o the system. The data associated with each lea will be considered or the development o prediction ormulae using P as M5MT can only generate a linear relationship between output and input parameters Tandem arrangement In the case o tandem arrangement, seven dierent pile group conigurations with relative spacing o S /D=0.5, 0.75, 1.0, 1.5, 3.0, 4.0 and 5.0 were tested. In total, 136 data were classiied by M5MT or this arrangement. As seen in Fig. 11, the developed tree is very simple and has only one root (node) and two leaves. All 136 data points were sorted into two groups based on S /D with a splitting value o 3. This is also demonstrated by the manually drawn dash-line splitting data with S /D 3 rom the rest o the data. It is apparent rom the developed tree that the data points were not categorized by KC number which was one o the inputs o the model meaning that M5MT discovered a similar relationship between K and KC number or dierent pile conigurations with S /D 3 (sub-set 1 in Fig. 11). For the conigurations with S /D>3, where piles are airly ar rom each other, data points were sorted into another group (sub-set 2 in Fig. 11). In this case K values are grouped around K =1 or all KC values indicating that there is no interaction between piles and each pile behaves like a single isolated pile (Fig. 11) arrangement For the 2 2 arrangement, our pile group conigurations with relative spacing o S /D=0.5, 0.75, 1.0 and 2.0 were tested. In total, 83 regular waves were perormed or this arrangement. Fig. 12 illustrates the relationship between pile group eect K and KC number or the 2 2 arrangement as well as the developed tree showing splitting parameters and the corresponding splitting values at nodes and leaves (sub-sets). The most interesting indication rom Fig. 12 is that the pile group interaction o 2 2 arrangement is clearly a combination o both pile group interactions observed in side by side and tandem arrangement. This means that both wave load ampliication seen in side by side (Fig. 10) and sheltering eect observed in tandem arrangement (Fig. 11) can be seen in the so called 2 2 arrangement. As seen in Fig. 12, M5MT model classiied all data into 4 dierent sub-sets based on both KC number and relative spacing S /D. Like or the case o side by side arrangement, the irst splitting parameter shown at the root o the inverse tree is relative spacing S /D and its splitting value is 1.5. Next, or cases with S /D>1.5 and S /D 1.5, KC number appears as the splitting parameter. In both nodes, the corresponding splitting value is 6. This means that in both closelyspaced piles (S /D 1.5) and S /D>1.5 M5MT model ound dierent physical behaviours in the

17 data or KC<6 (sub-sets 1 and 3 in Fig. 12) where the resulting wave load on the pile is primarily dominated by inertia and KC>6 (sub-sets 2 and 4 in Fig. 12) where both inertia and drag orces are important Staggered arrangement Six dierent pile group conigurations with relative spacing o S /D=0.6, 0.75, 1.0, 1.5, 3.0 and 5.0 were used or staggered arrangement where the angle between incident waves and the axis o pile groups is 45. In total, 120 regular waves cases were used or the staggered arrangement. Fig. 13 demonstrates the relationship between pile group eect K and KC number or staggered arrangement with dierent S /D. As seen, no speciic relationship can be seen between pile group eect K and KC number or the tested pile conigurations with dierent S /D values and almost all o the data points or dierent wave and structural conditions vary between 0.9 and 1.1. Applied M5MT model did not classiy data points into dierent sub-sets and only represented a model with only one lea. In act, K =1 was ound as the best itting line to data points or the downstream pile in staggered arrangement. This result is not, however, unexpected as the K values obtained or staggered arrangement (45 ) are between those gained or side by side (90 ) and tandem (0 ) arrangements. In order to study the eect o the direction o incident waves on pile group interaction, urther investigations by testing pile group arrangements with dierent angles (0-90 ) o the centre connection line o the cylinders relative to the wave direction are needed Overall M5MT model An overall model can be proposed including all generated models as constituents. This overall M5MT model is drawn in Fig. 14. As seen, the complete model, which consists o all models individually developed or each pile group arrangements, has 12 sub-sets (leaves) named rom A to L. The irst splitting parameter located at the root o the inverse tree is relative spacing S /D and the splitting value is 1.5. The pile group conigurations with S /D 1.5 were named closelyspaced piles where a greater pile group interaction is expected. The second splitting criterion o M5MT model is the pile group arrangement at the second node o the inverse tree based on which an appropriate arrangement is chosen among the our tested arrangements including side by side, tandem, 2 2 and staggered arrangements. From this node, depending on the type o pile group arrangement, urther splitting parameters including KC number and relative spacing S /D might become important and play a role in sorting data. By this way, urther categorizations o a test might be made depending on its speciic wave and structural conditions and it reaches the inal node called lea through the branches.

18 4.2. Development o wave load ormulae using P Ater the classiication o data into sub-sets perormed by means o the M5MT model, the prediction is made by applying the P model to the data associated with each lea. For each pile group arrangement, like or M5MT, KC number and relative spacing S /D are used as the inputs o P models while pile group eect K is considered as the output parameter. These parameters are, indeed, the terminal set o the P model. For the unction set, however, dierent mathematical operators need to be tested in order to optimise the P model and, consequently, to obtain the best solution (ormula). Two main criteria were considered or selecting the best solution (ormula) among a large number o possible solutions that can be developed by P. These two main criteria were accuracy and simplicity o the possible solution Side by side arrangement As shown in Fig. 10, all 146 data were classiied by M5MT into 5 dierent sub-sets based on both KC number and relative spacing S /D. A number o mathematical operators were tested or the optimisation o the model and the development o appropriate ormulae. The optimised M5MT-P model and ormulae are shown in Fig. 15. The itting curves o developed M5MT- P-based wave load ormulae are also drawn in Fig. 16 or dierent hydrodynamic conditions and pile conigurations. As seen in Fig. 15, ive wave load ormulae (Eqs. SS1-SS5) were obtained or the prediction o non-breaking wave loads on a slender pile in side by side arrangement. Two o these ive ormulae show constant values or K. Pile group eect K is equal to 1 or conigurations with S /D>2 meaning that there is no ampliication o resulting wave loads on piles in a side by side arrangement and they can be treated as a single isolated pile. This inding is physically sound as no signiicant pile group interactions are expected, when they are placed airly ar rom each other (S /D>2). When 1<S /D 2, wave loads on the instrumented pile in the side by side arrangement is 10% larger those measured or a single isolated pile (K =1.1). It is also apparent rom Fig. 16 that the horizontal lines K =1.1 and K =1 match to the data points rom the LWI tests. For the so called closely-spaced piles (S /D 1.5), P developed three ormulae or the sub-sets obtained by the M5MT. For KC<6, where the resulting wave load on the pile is primarily dominated by inertia, the developed ormula (Eq. SS1 in Fig. 15), is solely dependent on the spacing between the piles. This result is in agreement with the data points plotted in Fig. 16. As seen in this igure, K is not dependent on KC number or KC<6. The negative exponent o S /D in Eq. SS1 shows that

19 interaction eect decreases by increasing the spacing between the piles, which is physically sound. For KC>6, the generated ormulae (Eq. SS2 and SS3) are multivariate unctions o both relative spacing S /D and KC number. By increasing KC number rom approximately 7 to 13, where the highest pile group interaction occurs, grouping eect parameter K sharply increases. The positive exponent o KC shows that Eq. SS2 can correctly represent this behaviour. For KC>13, K values decrease or closely-spaced piles (S /D 1.5). P developed Eq. SS3 or this condition (lea) made by M5MT. In this case, the relationship between K and KC number is based on a combination o exponential and power operators. This relationship provides a smooth slope or the ormula (see itting curves in Fig. 16) by which K decreases until KC is about 35~40. From this point, as shown in Fig. 16, KC does not aect grouping eect anymore and K is solely dependent on relative spacing S /D. This is the case that the drag orce is completely dominant and inertia component is negligible. The negative power o S /D in Eq. SS2 and SS3 demonstrate that interaction eect decreases as relative spacing S /D keeps increasing. It should be noted that the proposed ormulae are valid or the tested conditions where KC number is between 1.1 and 88 and 0.5 S /D 5. However, all other side by side conigurations with S /D 3 can be treated like single isolated piles as there is no pile group interaction. It was also reported by Chakrabarti (1979) that pile group interaction or slender piles in side by side arrangement completely disappears at S /D=4. The perormance o the developed M5MT-P-based ormulae was quantitatively evaluated using statistical indicators such as agreement index I a, correlation coeicient CC, scatter index SI, and Bias deined as ollow: I a = 1 n i= 1 ( x y ) i ( x x + y y ) i i i 2 2 (6) CC = T (1/ n)[( x x) ( y y)] i (1 / n)( x x) (1 / n)( y y) i i 2 2 i (7) 2 1/ n ( yi xi) SI = (8) x Bias= y - x (9)

20 where x i and y i denote the predicted and the measured values, respectively and n is the number o measurements (data). x and y are the corresponding mean values o the predicted and measured parameters. For the 146 train data used or the development o the M5MT-P model, the scatter diagram o the measured and predicted K values as well as the statistical values are shown in Fig. 17. As seen, the scatter between measured and predicted K values is very small and the data points are concentrated around the optimal line. The statistical parameters also indicate that the developed M5MT-P model system can precisely predict non-breaking wave loads on a slender pile in side by side arrangement. As seen, the agreement index (I a ) is 0.99 and the scatter index (SI) is only 5.3% Tandem arrangement As shown in Fig. 11, M5MT classiied all 136 data into two groups (sub-sets) based on S /D with a splitting value o 3. Some P models with dierent mathematical unctions and settings were built or the classiied data and the optimised model was obtained. The M5MT-P model and the corresponding ormulae are shown in Fig. 18. For S /D 3, the developed ormula is a unction o both KC number and relative spacing S /D (Eq. T1 in Fig. 18). As can be concluded rom this equation, the maximum K value is close to 1 or either very small KC-values (KC 0) or or very large S /D values (S /D>3). Pile group eect K decreases when relative spacing S /D decreases meaning that sheltering eect becomes more signiicant by decreasing the gap between the piles in tandem arrangement. By increasing KC number, pile group eect K decreases and reaches its minimum value at the largest KC number indicating that the highest sheltering occurs or very large KC values where the resulting wave load is solely dominated by drag. For S /D>3, the data points were sorted into another group by M5MT model. For this case, K was ound to be 1 or all tested KC values (KC=1-88). This means that no sheltering eect was observed or the cases with large spacing between the piles as expected. For this condition, nonbreaking wave load on a protected pile in tandem arrangement can be calculated by Morison s ormula as the sheltered pile behaves like a single isolated pile. The curves o the developed M5MT-P-based wave load ormulae are plotted together with the data points rom the LWI lume tests or tandem arrangements in Fig. 19, demonstrating that the developed ormulae are able to predict properly non-breaking wave load on the sheltered piles in a tandem arrangement. Fig. 20 demonstrates the scatter between the measured and predicted K values or tandem arrangement or all 136 data with dierent tested pile conigurations and hydrodynamic conditions. The values o the our statistical indicators are also shown in Fig. 20. As seen, the

21 data points are concentrated on the 45 degree line which represents the ideal correlation. The statistical indicators with I a =0.975 and SI=4.5% demonstrate the high capability o the developed M5MT-P model to reproduce the experimental data arrangement For 2 2 arrangement, M5MT classiied all 83 data into 4 dierent sub-sets based on both relative spacing S /D and KC number parameters as shown in Fig. 12. The developed M5MT-P model and corresponding ormulae obtained rom optimised P models are shown in Fig. 21. As mentioned beore, the irst splitting parameter shown at the root o the inverse tree is relative spacing with a splitting value o S /D=1.5 which distinguishes the so called closely-spaced piles cases (S /D 1.5) rom the largely-spaced piles cases (S /D>1.5). For both closely-spaced and largely-spaced piles, the second splitting parameter is KC number with a splitting value o KC=6 which separates inertia dominant conditions (KC<6) rom cases where both drag and inertia or only drag are important (KC>6). For KC<6, it was ound that the instrumented pile in the so called 2 2 arrangement behaves like a single isolated pile and, thereore, K is equal to 1 (Eqs and in Fig. 21) or all tested conigurations with dierent S /D values (Fig. 22). The comparison o Fig. 16, Fig. 19 and Fig. 22 or the cases with KC<6 indicates that the wave load on the instrumented piles in 2 2 is somehow the average o the results measured or side by side and tandem arrangements. For KC>6, two ormulae were developed by P shown in Fig. 21. Eq in Fig. 21 obtained or closely-spaced piles (S /D 1.5) is a multivariate unction o both KC and S /D parameters, while Eq in Fig. 21 depends only on KC number or largely-spaced piles. However, in contrast to side by side and tandem arrangements, only one pile coniguration was perormed or S /D>1.5, namely S /D=2. In order to determine the threshold value o the spacing at which the grouping eect becomes insigniicant or the 2 2 arrangement, urther tests with larger spacing between the piles are needed. In this arrangement, thereore, the proposed ormulae are only valid or the tested conditions (0.5 S /D 2). The scatter diagram and statistical values indicate that the proposed model is accurate in predicting K values or the tested conditions (Fig. 23). For this arrangement, agreement index I a is and scatter index SI is 7.3%. For staggered arrangement, as shown in Fig. 13 and discussed in section 4.1.4, no speciic relationship was ound between pile group eect K and all KC numbers or all tested pile conigurations with dierent S /D. The range o K is rom 0.9 to 1.1 or almost all data. As stated beore, urther investigations on pile group arrangements with dierent angles between the

22 piles centre connection line and wave direction (0-90 ) are needed to study how the variation o wave direction aects resulting wave loads on piles in a group Overall M5MT-P model and ormulae The individual P models developed and optimised or each type o the pile group arrangements are brought together to build an overall M5MT-P model which is summarized in Fig. 24. This overall model includes (i) M5MT model classiying the entire data sets and (ii) P-based ormulae developed or the classiied data (Eqs ). K roup S = = 1.14 D 0.19 or side by side, S /D 1.5 & KC 6 (10) K roup S = = 0.87 D 0.51 ( KC) 0.26 or side by side, S /D 1.5 & 6<KC 13 (11) K 0.46 roup S = = 1.4 exp 52.7 D 2.22 ( ( KC) ) or side by side, S /D 1.5 & KC>13 (12) K roup = = 1 or 2 2 arrangement, S /D 1.5 & KC 6 (13) K 0.32 roup S KC = = exp D 56 or 2 2 arrangement, S /D 1.5 & KC>6 (14) K 0.8 roup S KC = = exp D 56 or tandem arrangement & S /D 3 (15) K K roup = = 1 or staggered arrangement (16) roup = = 1 or tandem arrangement & S /D>3 (17) K roup = = 1 or 2 2 arrangement, S /D>1.5 & KC 6 (18)

23 K roup KC = = exp 30 or 2 2 arrangement, S /D>1.5 & KC>6 (19) K K roup = = 1.1 or side by side arrangement & 1.5<S /D 2 (20) roup = = 1 or side by side arrangement & S /D>2 (21) The proposed M5MT-P model is quite simple, compact and easy to use. For the purpose o this study, the irst question to be answered is about the value o relative spacing S /D between the piles, as shown at the root o the inverse tree. The case will be identiied either as closelyspaced pile group or S /D 1.5 and largely spaced pile group or S /D>1.5. Next, the pile group arrangement should be determined, including, side by side, 2 2, tandem and staggered arrangements as drawn at the second node o the inverse tree. From this stage on and depending on the type o pile group arrangement, KC number, relative spacing S /D or both might need to be considered or urther classiications. Finally, M5MT leads to the appropriate lea (class) based on the related hydrodynamic and structural conditions. At this step, equation number at the lea determines which P-based ormula to be applied or the calculation o pile group eect K o the instrumented pile within a pile group or the speciic case considered. The scatter diagram o the measured and predicted K values is drawn in Fig. 25 or all 485 data points used or the development o the M5MT-P model. As seen, the predicted and measured K values are in a very good agreement and the scatter between them is very small as the data points are concentrated around the optimal line. The perormance o the complete M5MT-P model as well as that o each pile group arrangement were quantitatively evaluated using statistical indicators including agreement index I a, correlation coeicient CC, scatter index SI, and Bias and the results are given in Table 2. Though only K =1 was obtained or the staggered arrangement with dierent wave and structural conditions, values o the statistical parameters indicate that the developed M5MT-P model can precisely reproduce the experimental results or non-breaking wave loads on a slender pile in a group o piles. As shown in Table 2, the agreement index (I a ) and scatter index (SI) o the model or 485 tests are, and 5.8%, respectively Validation o the developed ormulae

24 A set o new data measured in LWI wave lume was applied or the validation o the developed hybrid M5MT-P model. Table 3 shows the hydrodynamic and structural conditions o the testing data. A more detail description o this data set is provided by Bonakdar (2014). As given in Table 3, relative spacing S /D o the test data varies rom 0.5 to 3 (0.5 S /D 3) meaning that both closely-spaced pile groups (S /D 1.5) and those with largely-spaced pile groups (S /D>1.5) were covered. KC number o tandem arrangement is between 6 and 86.3, while that o side by side arrangement varies rom 6 to For the 2 2 and staggered arrangements, no urther tests were conducted apart rom those used or the development o the model. Thereore, these two arrangements were not considered or the validation process. In total, 124 data were used or the validation o the developed M5MT-P-based model. The scatter diagram o the measured and predicted K values is drawn in Fig. 26 or testing data where the corresponding statistical values are also provided. It can be concluded rom this igure that the proposed M5-MT-P-based model is well validated as it can precisely predict K values or the new testing data which were not used or the development o the model. As seen, the model s agreement index (I a ) is or 124 test data points which is very close to what obtained or the 485 data used or the development o the model. Likewise agreement index, the scatter index o the testing data is 10.2% which is only 4.4% larger than what calculated or data applied in developing the model. 5. Summary and concluding remarks Based on the knowledge o the processes associated with the interaction o waves and pile groups, which was gained rom a series o systematic laboratory investigations (e.g. Bonakdar and Oumeraci 2012, 2014, 2015a, 2015b and Bonakdar 2014), new physically-based and more generic wave load ormulae were derived or the prediction o wave loads on a slender pile within a group o piles in dierent arrangements. For the analysis o the laboratory data and development o the wave load ormulae, an artiicial intelligence (AI) - based computational tool, a combination o M5 tree (M5MT) and genetic programming (P) named hybrid M5MT-P model, was developed. Pile group eect K, which was set as the output hybrid M5MT-P model, is a unction o (i) KC number and (ii) pile group arrangement and relative spacing S /D representing the most relevant inluencing hydrodynamic and structural parameter, respectively. The developed M5MT-P model and ormulae are summarized in Fig. 27. The concluding remark drawn rom the new M5MT-P model and ormulae may be summarized as ollows:

25 (i) The new AI-based model and ormulae are simple, compact, transparent and physicallybased and allow us to systematically assess pile group eect K depending on the low regime (KC) and the structural conditions (pile group arrangement, S /D). As can be concluded rom Fig. 27, the proposed ormulae needs to be applied or the estimation o the wave-induced orce on slender piles within urther neighbouring piles unless K =1 where each pile in the group can be treated like a single isolated pile. (ii) The high accuracy o the new ormulae in predicting pile group eect K is conirmed by the values o agreement index I a, correlation coeicient CC and scatter index SI o the overall M5MT-P model obtained rom the analysis o all 485 train data, which are 0.987, and 5.8%, respectively. (iii)new data sets were used or the validation o the developed M5MT-P model. Agreement index I a, correlation coeicient CC and scatter index SI o 0.975, and 10.2% were respectively obtained, indicating a high accuracy o the model and the prediction ormulae, even or a set o new data. (iv) The developed ormulae are valid or non-breaking waves with the range o hydrodynamic and structural conditions as summarized in Table 1. Acknowledgments The inancial support o the erman Research Foundation (DF, Deutsche Forschungsgemeinschat) or this study through the WaPiS project (Ou 1/13-1) is acknowledged. The support o the erman Academic Exchange Service (DAAD, Deutscher Akademischer Austauschdienst) is also acknowledged. The irst author is also grateul to riith University or providing him a visiting scholar opportunity.

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30 Figure Captions Fig. 1: The three basic pile group arrangements Fig. 2: Pile group conigurations perormed in the LWI wave lume tests or non-breaking waves Fig. 3: Example o M5 model tree (Developed tree, leaves, classiied data and linear equations) Fig. 4: Programme ( x ) exp 1 3 4x2 in the orm o a tree structure Fig. 5: enetic operator Crossover Fig. 6: enetic operator Mutation Fig. 7: enetic Programming lowchart (ater Koza, 1992) Fig. 8: Overview o one-way coupled M5MT-P modelling or data analysis and the development o new prediction wave load ormulae Fig. 9: Principle and organisation structure o a hybrid M5MT-P model (exemplarily or two input variables X 1 & X 2 and an output Y) Fig. 10: Developed M5MT model or side by side arrangement and relationship between pile group eect K and KC number or dierent S /D Fig. 11: Developed M5MT model or tandem arrangement and relationship between pile group eect K and KC number or dierent S /D Fig. 12: Developed M5MT model or 2 2 arrangement and relationship between pile group eect K and KC number or dierent S /D Fig. 13: Relationship between pile group eect K and KC number or staggered arrangement Fig. 14: Overall M5MT model or dierent pile group arrangements exposed to non-breaking waves Fig. 15: M5MT-P model and ormulae or side by side arrangement Fig. 16: M5MT-P-based ormulae or the prediction o pile group eect K or side by side arrangement Fig. 17: Comparison o predicted and measured K or side by side arrangement

31 Fig. 18: M5MT-P model and ormulae or tandem arrangement Fig. 19: M5MT-P-based ormulae or the prediction o pile group eect K or tandem arrangement Fig. 20: Comparison o predicted and measured K or tandem arrangement Fig. 21: M5MT-P model and ormulae or 2 2 arrangement Fig. 22: M5MT-P-based ormulae or the prediction o pile group eect K or 2 2 arrangement Fig. 23: Comparison o predicted and measured K or 2 2 arrangement Fig. 24: Overall M5MT-P model system or dierent pile group arrangements Fig. 25: Comparison o predicted and measured K or all 485 data used or the development o M5MT-P model (train data) Fig. 26: Validation o M5MT-P-based model using testing data Fig. 27: Overview o the new wave load ormulae obtained rom the overall M5MT-P model or the dierent pile group arrangements

32 Table captions Table 1: Range o the data used or development o M5MT-P model and corresponding ormulae (train data) Table 2: Perormance o the developed M5MT-P model or (i) all pile group arrangements, (ii) side by side, (iii) tandem and (iv) 2 2 arrangements (train data) Table 3: Range o the data used or the validation o M5MT-P-based wave load ormulae (testing data)

33 Wave S D θ =0 Tandem θ =90 Wave D Side by side S Wave D 0 <θ<90 Staggered S Fig. 1: The three basic pile group arrangements

34 Pile Arrangement Relative Spacing (S /D) Number o tests D Tandem S S 0.5, 0.75, 1, 1.5, 3, 4 and Side by side S 0.5, 0.75, 1, 1.5, 2, 3 and S Staggered S 0.6, 0.75, 1, 1.5, 3 and S 2 2 S 0.5, 0.75, 1 and 2 83 Instrumented pile Neighbouring pile Fig. 2: Pile group conigurations perormed in the LWI wave lume tests or non-breaking waves

35 A) Inverse Tree: 3 Root (Fi rs t node ) X2 3 Second node B) Classiied data: into homogeneous subsets based on the criteria o the minimization o the variation in the output values C) LM1 Lea 1 (L1) 5 Lea 2 (L2) LM2 Linear Models: X1 5 LM3 Lea 3 (L3) X Subset 2 Subset 3 Subset X1 The possible M5MT-based equations: I X2 < 3 Then : LM1: Y = a (X1) + b (X2) + c I X2 > 3 & X1 < 5 Then : LM2: Y = d (X1) + e (X2) + Y I X2 < 3 & X1 > 5 Then : LM3: Y = g (X1) + h (X2) + i a, b, c, d, e,, g, h, and i = Constant X1 & X2 = input variables Y = Output variable LM = Linear model Fig. 3: Example o M5 model tree (Developed tree, leaves, classiied data and linear equations)

36 = subtraction * = multiplication / = division exp = exponential X1 = input variable X2 = input variable exp _ * / X2 4 X1 3 Fig. 4: Programme ( x ) exp 1 3 4x2 in the orm o a tree structure

37 + = addition = subtraction * = multiplication / = division exp = exponential X1 = input variable X2 = input variable X3 = input variable exp / _ 2 * + * 3 X3 X2 5 X1 7 Crossover / * exp _ * X3 X1 7 X2 5 Fig. 5: enetic operator Crossover

38 + = addition = subtraction * = multiplication / = division exp = exponential ^ = power X1 = input variable X2 = input variable X3 = input variable exp * / _ 3 X3 Mutation exp + / _ 3 X3 X1 7 ^ 9 X2 3 Fig. 6: enetic operator Mutation

39 eneration = 0 Create initial random population Termination criterion satisied? Yes Best solution No Evaluate itness o each individual in population End Individuals = 0 eneration = eneration + 1 Yes Individuals = P? No Reproduction Select one individual based on itness Select genetic operation probabilistically Crossover Select two individuals based on itness Mutation Select one individual based on itness Perorm reproduction Perorm crossover Perorm mutation Copy into new population Insert two ospring into new population Insert mutant into new population Individuals = Individuals + 1 Individuals = Individuals + 2 Individuals = Individuals + 1 Fig. 7: enetic Programming lowchart (modiied rom Koza, 1992)

40 Step 1 Classiication Process Laboratory data obtained rom small scale (LWI) model tests Applying M5 Model Tree (M5MT) Non-dimensional wave and structural parameters One-way coupling Classiied data into homogeneous subsets based on the principle o the minimization o the variation in the output values (the criteria o M5MT) Step 2 Prediction Process (Development o Wave Load Formulae) Applying enetic Programming (P) Classiied data based on the criteria o M5MT Wave load ormulae as a unction o the most signiicant wave and structural parameters Step 3 New data set (testing data) Applying Validation Process Developed M5MT-P-based wave load ormulae Validated wave load ormulae Fig. 8: Overview o one-way coupled M5MT-P modelling or data analysis and the development o new prediction wave load ormulae

41 B) Step 1: Application o M5MT Root 1. Decision Tree: (Fi rs t node ) 7 2. Classiied data: X2 6 into homogeneous subsets based on Second the criteria o the 4 Subset 2 Subset 3 node X2 minimization o the 3 variation in the 2 LM1 X1 Subset 1 output values 1 Lea 1 (L1) Lea 3 (L3) X1 Lea 2 (L2) LM2 LM3 One-way coupling Y 3. Linear Models The possible M5MT-based equations: Main drawback I X2 < 3 Then : LM1: Y = a (X1) + b (X2) + c o M5MT C) enetic Programming (P) applies to each subset data classiied by M5MT Example o possible P s basic settings: Function set: +,-,*, exp, power Population size: 200 Number o generations: 100 Probability o P mutation 0.1 Probability o P crossover 0.85 Probability o P reproduction 0.05 *sum o mutation, crossover and reproduction must be equal to 1. D) Step 2: Application o P eneration = X1 Y P process or each subset data (P1, P2, P3) X2 Create initial random population P2 Subset 2 Subset 3 Subset 1 P3 P1 I X2 > 3 & X1 < 5 Then : LM2: Y = d (X1) + e (X2) + I X2 < 3 & X1 > 5 Then : LM3: Y = g (X1) + h (X2) + i a, b, c, d, e,, g, h, and i = Constant Termination criterion satisied? Yes Best solution or each P A) Outcome M5 Model Tree (M5MT) applies to all data E) X1 & X2 = input variables Y = Output variable LM = Linear model Outcome Final P-based equation: or each subset data classiied by M5MT eneration = eneration + 1 Yes No Evaluate itness o each individual in population Individuals = 0 Individuals = P? End X X1 3 possible P equations, using the selected mathematical unctions and variables, or 3 dierent data sets classiied by M5MT: P equation 1: Y= a(x1) b +c(expx2 d ) P equation 2: Y= e + X1 g X2 h P equation 3: Select one individual based on itness Perorm reproduction Copy into new population Reproduction No Select genetic operation probabilistically Crossover Select two individuals based on itness Perorm crossover Insert two ospring into new population Mutation Select one individual based on itness Perorm mutation Insert mutant into new population Y X1 & X2 = input variables Y = Output variable Y= i + j X1 X2 k (expx1) Individuals = Individuals + 1 Individuals = Individuals + 2 Individuals = Individuals + 1 Fig. 9: Principle and organisation structure o a hybrid M5MT-P model (exemplarily or two input variables X 1 & X 2 and an output Y)

42 S S 2.6 Side by side 2.4 Instrumented pile Neighbouring pile S /D=0.5 S /D=0.75 S /D=1 Pile group eect (k ) Sub-set 1 (lea 1) Sub-set 2 (lea 2) Sub-set 3 (lea 3) S /D=1.5 S /D=2 S /D=3 S /D=5 1.2 Sub-set 4 (lea 4) Sub-set 5 (lea 5) KC number Fig. 10: Developed M5MT model or side by side arrangement and relationship between pile group eect K and KC number or dierent S /D